text:
\section{Introduction}
The interest on networks of dynamical systems is increasing in recent years, especially because of their capability of modeling and describing a large variety of phenomena and behaviors.
Particular attention has been oriented towards the emergence of complicated phenomena from interconnections of simple models (see, for example, \cite{CziBar99,LevRap00,FaxMur04,LiuYad08}).\\
Principal advantages provided by networked systems are three: a modular approach to design, the possibility of directly introducing redundancy, and the realization of distributed and parallel algorithms. All these advantages have led to increased utilization of realizationof  networked systems in the realization of many devices (see e.g. \cite{Gee05,KadBhu08}).\\
Interconnected systems are also successfully exploited to develop novel modeling approaches in many fields, such as
Economics (see e.g. \cite{NayRos07,ManSta00}),
Biology (see e.g. \cite{EisSpe98,RavSom02,DelNin08}),
Cognitive Sciences (see e.g. \cite{BroDin04}),
Ecology (see e.g. \cite{BunUrb00,UrbKei01})
and
Geology (see e.g. \cite{BaiMon06,MonBai05}),
especially when the investigated phenomena are characterized by spatial distributions where multivariate analysis is preferred \cite{InnMat08}.\\
While networks of dynamical systems are well studied and analyzed in physics \cite{BocLat06,GirNew02,NewGir04} and engineering \cite{ZhaLiu07,Olf07,SchRib08}, there are fewer results that address the problem of reconstructing an unknown dynamical network, since it poses formidable theoretical and practical challenges \cite{Kol09}.
However, unravelling the interconnectedness of a set of processes is of significant interest in many fields, and the necessity for general tools  is rapidly increasing (see \cite{Tim07}, \cite{BocIva07} and \cite{NapSau08} and the bibliography therein for recent results).
Existing results derive a network topology from sampled data (see e.g. \cite{ManSta00,Tim07,NapSau08,OzeUzu08}) or to determine the presence of substructures (see e.g. \cite{NewGir04,BocIva07}).
The Unweighted Pair Group Method with Arithmetic mean (UPGMA) \cite{MicSok57} is one of the first techniques proposed to reveal an unknown topology.
It has found widespread use  in the reconstruction of phylogenetic trees, and is widely employed in other areas such as communication systems and for resource allocation \cite{FreHar02}.
UPGMA identifies a tree topology relying on the observation of leaf nodes, theoretically guaranteeing a correct identification only on the strong assumption that an ultrametric is defined among the leaves.
Another well-known technique for the identification of a tree network is developed in \cite{ManSta00} for the analysis of a stock portfolio.
The authors identify a tree structure according to the following procedure: i) a metric based on the correlation index is defined among the nodes; ii) such a metric is employed to extract the Minimum Spanning Tree \cite{Die06} which forms the reconstructed topology.
Many improvements over \cite{ManSta00} have been devised especially using shrinking techniques to estimate the correlation matrix \cite{LedWol04,TumLil08} or refining its estimate via high frequency sampling \cite{TotKer09}. A reliability index for any link can also be defined using bootstrap techniques \cite{TumLil08}.
However, in \cite{InnMat08} a severe limit of this strategy is highlighted,
where it it is shown that, even though the actual network is a tree, the presence of dynamical connections or delays can lead to the identification of a wrong topology.
In \cite{MatInn09} a similar strategy, where the correlation metric is replaced by a metric based on the coherence function, is numerically shown to provide an exact reconstruction for tree topologies. Finally, in \cite{MatInn08} it is shown that a correct reconstruction can be guaranteed for any topology with no cycles.\\
An approach for the identification of more general topologies is developed in the area of Machine Learning for Bayesian Networks (BNs) \cite{GetFri02,FriKol03}.
The results presented in this paper present have strong connections with this area.
Indeed, the topology reconstruction of BNs is based on the fact that the conditional expectation operator to estimate one signal is determined only by the signals belonging to its Markov Blanket \cite{Pea88}. However, the most significant difference between BNs and the network models of this paper is that in BNs the network graph can not have loops.
The fact that the Wiener filter applied to the class of network models studied in this paper has properties similar to BNs even in presence of loops can not be considered trivial.
Moreover, we tackle the problem of the robustness of the reconstruction in terms of power spectral densities of the noise corrupting the signals.\\
In \cite{BocIva07} different techniques for quantifying and evaluating the modular structure of a network are compared and a new one is proposed trying to combine both the topological and dynamic information of the complex system. However, the network topology is only qualitatively estimated.
In \cite{Tim07} a method to identify a network of dynamical systems is described. However, primary assumptions of the technique are the possibility to manipulate the input of every single node and the possibility of conducting many experiments to detect the link connectivity.\\
In \cite{MarPel08a} an interesting and novel approach based on auto-regressive models and Granger-causality \cite{Gra69} is proposed for reconstructing a network of dynamical systems.
This technique relies on multivariate identification procedure to detect the presence of a link, but still no theoretical sufficient or necessary conditions are derived to check the correctness of the results.\\
More recently, in \cite{NapSau08} and \cite{CanWak08} interesting equivalences between the identification of a dynamical network and a $l_0$ sparsification problem are highlighted, suggesting the difficulty of the reconstruction procedure \cite{CanTao05}.\\
Summarizing, to the best knowledge of the authors, apart from the results in \cite{MatInn08}, which are limited to tree topologies, no general theoretical guarantees about the correct reconstruction of network links are provided if there is no possibility of directly manipulating the system.\\
In this paper the problem of reconstructing a network of dynamical systems where every node represents an observable scalar signal and the dynamics is linear and represented by the connecting links is addressed.
The problem, when analyzed from a systems theory point of view, provides a method for correctly identifying a topology that belongs to the pre-specified class of self-kin networks. Moreover, if the network does not belong to such a class, conditions about the optimality of the identified topology according to a certain criterion is estabilished.
From this perspective, sufficient conditions for the exact reconstruction of a large class of networks, which we name self-kin, are derived. Examples of self-kin networks are given by (but not limited to) trees, and ring topologies \cite{JovBam05}. In the case the network is not self-kin, the reconstructed topology is guaranteed to be the smallest self-kin network containing the actual one.
The theory developed is not bayesian and relies directly on Wiener filtering theory.
Conditions derived for the detection of links are based on sparsity properties of the (non-causal) Wiener filter modeling the network.
Indeed, conditions under which the Wiener filter smoothing a signal of the network is ``local'' are derived.
From a different perspective, another important contribution of the paper is given by providing conditions for a local and distributed implementation of the Wiener filter.

The paper is organized as follows.
In Section \ref{sec:intuition} examples are provided to provide the basic intuition behind central ideas;
in Section \ref{sec:preliminaries} definitions are provided based on standard notions of graph theory;
in Section \ref{sec:problem} the main problem is formulated;
in Section \ref{sec:non-causal} the main results are provided for non-causal Wiener filtering;
in Section \ref{sec:causal} the results are extended to causal Wiener filtering and Granger causality;
in Section \ref{sec:algorithm} the implementation of algorithms for the detection of network topologies are discussed for different scenarios;
in Section \ref{sec:corrupted} the robustness of the identification is addressed;
eventually, in Section \ref{sec:example} numerical simulations illustrating the effectiveness of the methodology are presented.

~\\~\\
\noindent{\bf Notation:}\\
The symbol $:=$ denotes a definition\\
$\|x\|$: $2$-norm of a vector $x$\\
$W^T$: the transpose of a matrix or vector $W$ \\
$W^*$: the conjugate transpose of a matrix or vector $W$ \\
$x_i$ or $\{x\}_i$: the $i$-th element of a vector $x$ \\
$W_{ji}$: the entry $(j,i)$ of a matrix $W$ \\
$W_{j*}$: j-th row of a matrix $W$\\
$W_{*i}$: i-th column of a matrix $W$\\
$x_V$: when $V=(v_1,...,v_n)$ is a $n$-tuple of natural numbers denotes the vector $(x_{v_1} ~...~x_{v_n})^T$\\
$|A|$: cardinality (number of elements) of a set $A$\\
$E[\cdot]$: mean operator; \\
$R_{XY}(\tau):= E[X(t)Y^T(t+\tau)]$: cross-covariance function of wide-sense stationary vector processes $X$ and $Y$;\\
$R_{X}(\tau):= R_{XX}(\tau)$: autocovariance; \\
$\mathcal{Z}(\cdot)$: Zeta-transform of a signal;\\
$\Phi_{XY}(z):=\mathcal{Z}(R_{XY}(\tau))$: cross-power spectral density;\\
$\Phi_{X}(z):=\Phi_{XX}(z)$: power spectral density;\\
with abuse of notation, $\Phi_{XY}(\omega)=\Phi_{XY}(e^{i\omega})$ for spectral densities and $W(\omega)=W(e^{i\omega})$ for transfer functions;\\
$b_{i}:$ $i$-th element of the canonical base of $\mathbb{R}^{n}$.\\

In this sections a representation of dynamical networks in terms of oriented graphs is presented.\\
In this representation, every node $N_j$ represents a scalar time-discrete wide-sense stationary stochastic process $x_j$, while every directed arc form a node $N_i$ to a node  $N_j$ represents a possibly non-causal transfer function $H_{ji}(z)\neq 0$.
The absence of such an arc implies that $H_{ji}(z)=0$.
Every node signal is also implicitly considered affected by an additive process noise $e_j$. Given the graphical representation, the dynamics of the network is described by
\begin{align}
	x_j = e_j + \sum_{i}H_{ji}(z)x_i.
\end{align}
For example, in Figure~\ref{fig:example} a network is represented, the dynamics of which corresponds to
\begin{align*}
	&x_1 = e_1\\
	&x_2 = e_2 + H_{25}(z)x_5\\
	&x_3 = e_3 + H_{31}(z)x_1+H_{32}(z)x_2\\
	&x_4 = e_4 + H_{43}(z)x_3\\
	&x_5 = e_5 + H_{51}(z)x_1+H_{54}(z)x_4.
\end{align*}
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=0.6\columnwidth]{example_dynamics}
	\caption{An example of the graphical representation of dynamical networks. Every node represents a stochastic signal and the edge from a node $N_i$ to a node $N_j$ represents the transfer function $H_{ji}(z)$. Every node signal is also implicitely affected by a process noise.\label{fig:example}}
\end{figure}

\section{Illustrative examples}\label{sec:intuition}
In this section special network configurations of dynamical systems are presented where the Wiener filter producing the estimate of a node signal $x_0$ from the other node signals $x_i$ has the characteristic of being sparse.
In most cases, the presence of a non-null entry in the Wiener filter corresponds to the presence of a direct link between the signal $x_i$ and the signal $x_0$.
In the following sections it will be shown that this result holds in general for the class of self-kin networks, a class of networks defined later.
Moreover if the network is not self-kin it will be shown that the presence of a non-null entry in the Wiener filter identifies the presence of a link in the ``smallest'' (in the sense of number of edges) self-kin network containing the original one.

\subsection{Wiener Filtering of a downstream signal}
Consider a network of four systems as represented in Figure~\ref{fig:cascade dw} where
\begin{align*}
	& x_{3}=e_{3}\\
	& x_{i-1}=e_{i-1}+H_{i-1,i}(z)x_{i}
\end{align*}
with $H_{i-1,i}(z)\neq 0$ three possibly non-causal SISO transfer functions for $i=1,2,3$ and with signals $e_i$ mutually uncorrelated for $i=0,1,2,3$.
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=0.8\columnwidth]{cascade_dw}
	\caption{A cascade network. The Wiener filter estimating $x_0$ from the other signals makes use only of the ``parent'' signal $x_1$ which is directly linked to $x_0$.\label{fig:cascade dw}}
\end{figure}
Consider the Wiener filter that provides the estimate $\hat x_0$ of $x_0$ based on the other signals, $x_1$, $x_2$ and $x_3$. It can be shown that
\begin{align*}
	\hat x_{0}&=\left( W_{01}(z)~~ W_{02}(z)~~ W_{03}(z) \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right)=\\
		&=\left(H_{01}(z)~~ 0~~ 0 \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right)
\end{align*}
is the best estimate of $x_0$ in the least squares sense. Indeed,
\begin{align}
	E[(x_0 - \hat x_0)x_i]=E[e_0 x_i]=0
\end{align}
for $i=1,2,3$. Thus, from the Hilbert projection theorem \cite{Lue69},
\begin{align}
	W_0(z) := \left( W_{01}(z)~~ W_{02}(z)~~ W_{03}(z) \right)
\end{align}
is the Wiener filter providing the best estimate for $x_0$ \cite{Cai87}.
In this case, the only non-null entry of $W_0$ correctly detects that there is a connection between the node $0$ and the node $1$.

\subsection{Wiener Filtering of an upstream signal}
Consider a network of four systems as represented in Figure~\ref{fig:cascade up} with
\begin{align}
	& x_{0}=e_{0}\\
	& x_{i+1}=H_{i+1,i}(z)x_{i}+e_{i+1}
\end{align}
where $H_{i+1,i}(z)$ are three possibly non-causal SISO transfer functions for $i=1,2,3$ and with signals $e_i$ mutually not correlated for $i=0,1,2,3$.
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=0.8\columnwidth]{cascade_up}
	\caption{A cascade network. The Wiener filter estimating $x_0$ from the other signals makes use only of the ``child'' signal $x_1$ which is directly linked to $x_0$. \label{fig:cascade up}}
\end{figure}
The Wiener filter that provides the estimate $\hat x_0$ of $x_0$ making use of the other signals $x_1$, $x_2$ and $x_3$ is such that
\begin{align*}
	\hat x_{0}&=\left( W_{01}(z)~~ W_{02}(z)~~ W_{03}(z) \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right)=\\
		&=\left( \frac{\Phi_{x_0 x_0}(z)H_{10}^*(z)}{|H_{10}(z)|^2\Phi_{x_0}(z)+\Phi_{e_1}}~~ 0~~ 0 \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right).
\end{align*}
Again in this case, there is one only non-null entry in the Wiener filter corresponding to the link between the node $0$ and the node $1$.

\subsection{Wiener Filtering of a loop}
In the two previous cases the network configurations did not involve loops and the resulting Wiener filter had the property of having non-null entries corresponding to the node signals immediately connected to the node of interest.
The identification of network topologies with loops is a challenging problem \cite{ManSta00, InnMat08}. Indeed, most techniques deal with networks with no cycles \cite{MatInn08}.
The presence of a loop leads to more complex relations between the node signals, especially in terms of the covariance function and (cross)-spectral densities.
Thus, it is interesting to note that the absence or presence of loops does not seem to affect the sparsity of the Wiener filter as shown in the following example \cite{MatInn08}.\\
Consider a network of four systems as represented in Figure~\ref{fig:loop}a where
\begin{align*}
	& x_{i}=H_{i,[i-1]_{mod~4}}(z)x_{[i-1]_{mod~4}}+e_{i}
\end{align*}
for $i=0,...,3$ and
\begin{align}
	& [n]_{mod~m}:=\min\{q| q=n+km\geq 0 \text{ and } k\in \mathbb{Z} \}.
\end{align}
\begin{figure}[hbt!]
	\centering
	\begin{tabular}{cc}
	\includegraphics[width=0.4\columnwidth]{loop} &
	\includegraphics[width=0.4\columnwidth]{nonkin4}\\
	(a) & (b)
	\end{tabular}
	\caption{A loop network (a): the Wiener filter estimating $x_0$ from the other signals makes use only of the ``child'' signal $x_1$ and the parent signal $x_3$ which are directly linked to $x_0$\label{fig:loop} A more general network (b): the Wiener filter estimating $x_0$ from the other signals makes use of the signal $x_2$ which is directly linked to it, but also of the signal $x_1$ which is not.\label{fig:nonkin4}
	}
\end{figure}
Assume also that the signals $e_{i}$ are mutually not correlated.
It is possible to show that the best least squares estimate of $x_0$, based on $x_1$, $x_2$ and $x_3$, is given by
\begin{align*}
	&\hat x_{0}=\left( W_{01}(z)~~ W_{02}(z)~~ W_{03}(z) \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right)=\\
		&=\left( \frac{\Phi_{e_0}H_{10}(z)^{*}}{\Phi_{e_0}|H_{10}|^2+\Phi_{e_1}}~~ 0~~ (1-\frac{\Phi_{e_0}|H_{10}|^2}{\Phi_{e_0}|H_{10}|^2+\Phi_{e_1}})H_{03}(z) \right) \cdot \\
		&\qquad\cdot\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right).
\end{align*}
It is to be noted that the signal $x_2$, which is not directly connected to $x_0$, is not used in the optimal estimate of $x_0$.
Thus, in all the examples presented so far the node signals actually employed by the Wiener Filter are the signals corresponding to nodes directly connected to the node signal being estimated.
This example, along with the previous ones, leads to the conjecture that the Wiener Filter can be used as a tool to identify which nodes are connected to a specific node of interest by checking wether the corresponding entries of the Wiener filter are null or not.
Unfortunately, the situation is more complex as shown in the following example.

\subsection{Wiener filter using a not directly connected signal}
Consider a network of four systems as represented in Figure~\ref{fig:nonkin4}b where
\begin{align}
	& x_{0}=e_{0}\\
	& x_{1}=e_{1}\\
	& x_{2}=H_{2,0}(z)x_{0}+H_{2,1}(z)x_{1}+e_{2}\\
	& x_{3}=H_{3,2}(z)x_{2}+e_{3}
\end{align}
and the noises $e_{j}$ are mutually not correlated.
In this example, the Wiener filter providing the estimate $\hat x_0$ of $x_{0}$ using the other signals of the network is given by
\begin{align*}
	\hat x_{0}=\left( W_{01}(z)~~ W_{02}(z)~~ W_{03}(z) \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right)=\\
		=\frac{\Phi_{e_0}(z)H_{20}^*(z)}{\Phi_{e_0}(z)|H_{20}(z)|^2+\Phi_{e_2}(z)}
		\left( -H_{21}(z)~~ 1~~ 0 \right)
		\left(\begin{array}{c}
			x_{1}\\ x_{2}\\ x_{3}
		\end{array}\right).
\end{align*}
In this case the Wiener filter estimating $x_0$ from the other signals makes use not only  of the signal $x_2$ which is directly linked to it, but also of the signal $x_1$ which is not.\\

After the analysis of these examples, it is natural to ask to what extent and under which assumptions it is possible to reconstruct the topology of a network of linear dynamical systems measuring the node signals and if the Wiener filter can be a useful tool for accomplishing this identification procedure.

\section{Preliminary definitions}\label{sec:preliminaries}
In this section, basic notions of graph theory, which are functional to the following developments, will be recalled.
For an extensive overview see \cite{Die06}.
First, the standard definition of undirected and oriented graphs is provided.\\
\begin{define}[Directed and Undirected Graphs]
	An undirected graph $G$ is a pair $(V,A)$ where $V$ is a set of vertices or nodes and $A$ is a set of edges or arcs, which are unordered subsets of two distinct elements of $V$.\\
	A directed (or oriented) graph $G$ is a pair $(V,A)$ where $V$ is a set of vertices or nodes and  $A$ is a set of edges or arcs, which are ordered pairs of elements of $V$.\\
\end{define}
In the following, if not specified, oriented graphs are considered.\\
\begin{define}[Topology of a graph]
	Given an oriented graph $G=(V,A)$, its topology is defined as the undirected graph $G'=(V,A')$ such that $\{N_i,N_j\}\in A'$ if and only if $(N_i,N_j)\in A$ or $(N_j,N_i)\in A$, and $top(G):=G'$.\\
\end{define}
The topology of an oriented graph $G$ is the undirected graph $G'$ obtained by removing the orientation from any edge.
An example of a directed graph and its topology is represented in Figure~\ref{fig:topexample}.
\begin{figure}[tb]
	\centering
	\begin{tabular}{cc}
		\includegraphics[width=0.45\columnwidth]{generic_net} &
		\includegraphics[width=0.45\columnwidth]{generic_top}\\
		(a) & (b)
	\end{tabular}
	\caption{A directed graph (a) and its topology (b)\label{fig:topexample}.}
\end{figure}
~\\
\begin{define}[Children and Parents]
	Given a graph $G=(V,A)$ and a node $N_j\in~V$, we define the children of $N_j$ as
	$\mathcal{C}_G(N_j):=\{N_i| (N_j,N_i)\in A\}$ and the parents of $N_j$ as
	$\mathcal{P}_G(N_j):=\{N_i| (N_i,N_j)\in A\}$.\\
	Extending the notation, children and the parents of a set of nodes are denoted as follows
	\begin{align*}
		& \mathcal{C}_G(\{N_{j_1},...,N_{j_m}  \}):=
			\cup_{k=1}^m \mathcal{C}_G(N_{j_k})\\
		& \mathcal{P}_G(\{N_{j_1},...,N_{j_m}  \}):=
			\cup_{k=1}^m \mathcal{P}_G(N_{j_k}).
	\end{align*}
\end{define}
\begin{define}[Kins]
	Given an oriented graph $G=(V,A)$ and a node $N_j\in V$, kins of $N_j$ are defined as
	\begin{align*}
		\mathcal{K}_G(N_j):=\{N_i| N_i\neq N_j \text{ and }
					N_i \in \mathcal{C}_G(N_j)~\cup\\
			\qquad\qquad \cup~\mathcal{P}_G(N_j)~
						\cup~\mathcal{P}_G(\mathcal{C}_G(N_j)) \}.
	\end{align*}
	Kins of a set of nodes are defined in the following way
	\begin{align*}
		& \mathcal{K}_G(\{N_{j_1},...,N_{j_m}  \}):=
			\cup_{k=1}^m \mathcal{K}_G(N_{j_k}).
	\end{align*}
\end{define}
\begin{define}[Proper Parents and Children]
	Given an oriented graph $G=(V,A)$ and a node $N_j$, $N_i$ is a proper parent (child) of $N_j$ if it is a parent (child) of $N_j$ and
	$N_i\notin \mathcal{P}_G(\mathcal{C}_G(N_j))$.
	$N_i$ is a proper kin if it is a kin and
	$N_i\notin \mathcal{P}_G(N_j)\cup \mathcal{C}_G(N_j)$.\\
\end{define}
\begin{figure}[tb]
	\centering
	\begin{tabular}{cc}
		\includegraphics[width=0.5\columnwidth]{nonselfkin} &
		\includegraphics[width=0.4\columnwidth]{nonselfkin_kinof}\\
		(a) & (b)
	\end{tabular}
	\caption{An oriented graph (a) and its kin topology (b)\label{fig:nonselfkin}.}
\end{figure}
Note that the kin relation is symmetric, in the sense that $N_i \in \mathcal{K_G}(N_j)$ if and only if $N_j \in \mathcal{K_G}(N_i)$.\\
\begin{define}[Kin-graph]
	Given an oriented graph $G=(V,A)$, its kin-graph is
	the undirected graph $\tilde G=(V,\tilde A)$ where
	\begin{align*}
		\tilde A := \{\{N_i, N_j\}| N_i \in \mathcal{K}_G(N_j)~ \text{for all } j \}.
	\end{align*}
	and it is denoted as $kin(G)=\tilde G$.\\
\end{define}
A directed graph and its kin-graph are represented in Figure~\ref{fig:nonselfkin}.
Note that the kin-graph of $G$ is an undirected graph. It could be defined as a directed graph, but, because of the symmetry of the kin relation, a directed graph contains exactly the same information. Moreover such a choice is motivated by the following definition
\begin{define}[Self-kin Graph]
	An oriented graph $G$ is self-kin if
	\begin{align*}
		top(G)=kin(G).
	\end{align*}
\end{define}

An example of self-kin network is given in Figure~\ref{fig:selfkinring}.
\begin{figure}[tb]
	\centering
	\includegraphics[width=0.75\columnwidth]{selfkinring}
	\caption{A ring network is always a self-kin network.\label{fig:selfkinring}}
\end{figure}
Many graphs are self-kin, such as oriented trees and rings \cite{Die06}.

\begin{define}
	Let $\mathcal{E}$ be a set containing time-discrete scalar, zero-mean, wide-sense, stationary random processes such that, for any $e_i,e_j\in\mathcal{E}$, the power spectral density $\Phi_{e_i e_j}(z)$ exists, is real rational with no poles on the unit circle and given by
	\begin{align}
		\Phi_{e_i e_j}(z)=\frac{A(z)}{B(z)},
	\end{align}
	where $A(z)$ and $B(z)$ are polynomials with real coefficients such that $B(z)\neq 0$ for any $z\in \mathbf{C}$, with $|z|=1$.
	Then, $\mathcal{E}$ is a set of rationally related random processes.\\
\end{define}

\begin{define}
	The set $~{}^0\mathcal{F}$ is defined as the set of real-rational SISO transfer function with no poles on the unit circle $\{z\in\mathbf{C}|~~|z|=1\}$.\\
\end{define}

\begin{define}
	Given a SISO transfer function $H(z)\in~{}^0\mathcal{F}$, represented as
	\begin{align}
		H(z)=\sum_{k=-\infty}^{\infty}h_k z^{-k},
	\end{align}
	we define the causal truncation operator as
	\begin{align}
		\{H(z)\}_{C}:=\sum_{k=0}^{\infty}h_k z^{-k}.
	\end{align}
\end{define}

\begin{lem}
	For every $H(z)\in~{}^0\mathcal{F}$, we have that $\{H(z)\}_{C}\in~{}^0\mathcal{F}$.
\end{lem}

\begin{define}
	The set $~{}^0\mathcal{F}^{+}$ is defined as the set of real-rational SISO transfer functions in $~{}^0\mathcal{F}$ such that
	\begin{align}
		\{H(z)\}_{C}=H(z).
	\end{align}
\end{define}

\begin{define}
	Let $\mathcal{E}$ be a set of rationally related random processes.
	The set ${}^{0}\mathcal{F}\mathcal{E}$ is defined as
	\begin{align*}
		{}^0\mathcal{F}\mathcal{E}:=
		\left\{x=\sum_{k=1}^{m} H_k(z)e_k~|~e_k\in\mathcal{E}, H_k(z)\in {}^0\mathcal{F}, m\in\mathbb{N} \right\}.
	\end{align*}
\end{define}
\begin{lem}
	The set ${}^0\mathcal{F}\mathcal{E}$ is a vector space with the field of real numbers. Let
\begin{align*}
	<x_1,x_2>:=R_{x_1 x_2}(0)=\int_{-\pi}^{\pi}\Phi_{x_1 x_2}(\omega),
\end{align*}
which defines an inner product on ${}^0\mathcal{F}\mathcal{E}$ with the assumption that two processes $x_1$ and $x_2$ are considered identical if $x_1(t)=x_2(t)$, almost always for any $t$.
\end{lem}
\begin{IEEEproof}
      See supplemental material.
\end{IEEEproof}

For any $x\in {}^0\mathcal{F}\mathcal{E}$, the norm induced by the inner product is defined as $\|x\|:=\sqrt{<x,x>}$.
\begin{define}
	For a finite number of elements $x_1,...,x_m\in{}^0\mathcal{F}\mathcal{E}$, tf-span is defined as
	\begin{align*}
		&\text{tf-span}\{x_1,...,x_m\}:=\\
		&:=\left\{x=\sum_{i=1}^m \alpha_{i}(z)x_i
		~\lvert~\alpha_{i}(z)\in{}^0\mathcal{F}
		\right\}.
	\end{align*}
\end{define}
\begin{lem}
	The tf-span operator defines a subspace of ${}^0\mathcal{F}\mathcal{E}$.\\
\end{lem}
\begin{IEEEproof}
      The proof is left to the reader.
\end{IEEEproof}

\begin{define}
	For a finite number of elements $x_1,...,x_m\in{}^0\mathcal{F}\mathcal{E}$, c-tf-span is defined as
	\begin{align*}
		&\text{c-tf-span}\{x_1,...,x_m\}:=\\
		&:=\left\{x=\sum_{i=1}^m \alpha_{i}(z)x_i
		~\lvert~\alpha_{i}(z)\in{}^0\mathcal{F}^{+}
		\right\}.
	\end{align*}
\end{define}
\begin{lem}
	The c-tf-span operator defines a subspace of ${}^0\mathcal{F}\mathcal{E}$.\\
\end{lem}
\begin{IEEEproof}
      The proof is left to the reader.
\end{IEEEproof}

The following definition provides a class of models for a network of dynamical systems.
\begin{define}
	Let $I=(1,...,n)^T$ and consider the triplet $\mathcal{G}=(G,\mathcal{H},e_I)$ where $G=(V,A)$ is a graph with $n$ vertexes $\{N_j\}_{j=1,...,n}$; $\mathcal{H}:A\rightarrow~ {}^0\mathcal{F}$ is a function associating a, possibly non-causal, transfer function $H_{ji}(z)\in {}^0\mathcal{F}$, with no poles on the unit circle, to any edge $(N_i, N_j)\in A$; and $\mathcal{E}:=e_I=(e_1,...,e_n)^T$ is an ordered $n$-tuple of rationally related random processes.
	$\mathcal{G}$ is a Linear Dynamic Graph (LDG) if the following conditions are satisfied
	\begin{itemize}
		\item	$(N_i,N_j)\in A$ implies $(N_j,N_i)\notin A$
		\item	$\Phi_{e_i e_j}(\omega)=0$ for $i\neq j$ and for all $\omega\in \mathbb{R}$
	\end{itemize}
	The output $x_j$ of the dynamic of a LDG is defined as
	\begin{align*}
		x_j=e_j+\sum_{i}H_{ji}(z)x_i.
	\end{align*}
	for any $j=1,...,n$.
	In a matrix form it is possible to write the network dynamics of a LDG as
	\begin{align}
		x_{I}=e_{I}+H(z)x_{I}
	\end{align}
	where the entry $(j,i)$ of the transfer matrix $H(z)$ is given by the transfer function $H_{ji}(z)$.
	The LDG is {\it well-posed} if the matrix $(\mathcal{I}-H(z))$ is invertible, and both $(\mathcal{I}-H(z))$ and $(\mathcal{I}-H(z)) ^{-1}$ have full normal rank and no poles on the unit circle.
	This implies that there is a transfer matrix $T(z)=[\mathcal{I}-H(z)]^{-1}$ such that $x_{I}=T(z)e_{I}$. Interpreting $T(z)$ as a possibly non-causal, linear transformation, it follows that $x_{i}\in {}^0\mathcal{F}\mathcal{E}$ for any $i=1,...,n$.\\
	The LDG is {\it causal} if each entry of $H(z)$ and each entry of $T(z)$ is in ${}^0\mathcal{F}^{+}$.
\end{define}
A LDG is a complex interconnection of linear transfer functions $H_{ji}(z)$ connected according to a graph $G$ and forced by stationary additive mutually uncorrelated noise.
The following definitions will be useful for determining sufficient conditions for detection of links in a network.
\begin{define}
	A LDG $\mathcal{G}=(G,\mathcal{H},e_I)$ is topologically detectable if
	$\Phi_{e_i}(\omega)>0$ for any $\omega\in[-\pi,\pi]$ and for any $i=1,...,n$.\\
\end{define}

\section{Problem Formulation}\label{sec:problem}
\begin{prob}
	Consider a well-posed LDG $\mathcal{G}=(G,\mathcal{H},\mathcal{E})$ where the graph $G$ is unknown.
	Given the Power (Cross-)~Spectral Densities of $\{x_j\}_{i=1,\ldots,n}$,
	reconstruct the unknown topology of the graph $G$ of the LDG $\mathcal{G}$.
\end{prob}

\section{Sparsity of the non-causal Wiener Filter}\label{sec:non-causal}

First, a lemma is provided that guarantees that any element in $\text{tf-span}\{x_i\}_{i=1,...,n}$ admits a unique representation if the cross-spectral density matrix of its generating processes has full normal rank.
\begin{lem}\label{lem:unique representation}
	Let $q$ and $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$, and
	suppose that $x\in\text{tf-span}\{x_i\}_{i=1,...,n}$ and that $\Phi_{x_{I}x_{I}}(\omega)>0$ almost for any $\omega\in~[-\pi, \pi]$. Then there exists a unique transfer matrix $H(z)$ such that $q=H(z)x_{I}$.
\end{lem}
\begin{IEEEproof}
	Note that if $H(z)$ is such that $H(z)x_I=0$, then
	\begin{align}
		\Phi_{qq}(\omega)=0=H(\omega)\Phi_{x_{I}x_{I}}(\omega)H^{*}(\omega).
	\end{align}
	Since $\Phi_{x_{I}x_{I}}(\omega)>0$ almost everywhere, we have $H(\omega)=0$ almost everywhere which implies that $H(z)=0$.
	Now, by contradiction assume that $q=H_{1}(z)x_{I}=H_{2}(z)x_{I}$, with $H_1(z)\neq H_2(z)$. Then $0=[H_2(\omega)-H_1(\omega)]\Phi_{x_I x_I}(\omega)[H_2(\omega)-H_1(\omega)]^*$ implying that $H_1(z)=H_2(z)$.
\end{IEEEproof}
A specific formulation of the non-causal Wiener filter is introduced for the defined spaces.
\begin{prop}\label{eq: my wiener}
	Let $x$ and $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$ and $X:=\text{tf-span}\{x_1,...,x_n\}$. Consider the problem
	\begin{align}\label{eq: cost general wiener}
		\inf_{q \in X} \|x-q\|^2.
	\end{align}
	If $\Phi_{x_I}(\omega)>0$, for $\omega\in[-\pi,\pi]$, the solution $\hat x\in X$ exists, is unique and has necessarily the form $\hat x=W(z)x_I$ where
	\begin{align*}
		W(z)=\Phi_{x x_{I}}(z)\Phi_{x_{I}}(z)^{-1}.
	\end{align*}
	Moreover, it holds that $\hat x$ is the only element in $X$ satisfying, for any $q\in X$,
	\begin{align}\label{eq:perp hilbert projection thm}
		<x-\hat x,q>=0.
	\end{align}
\end{prop}
\begin{IEEEproof}
	Observe that, since $q\in X$, the cost function satisfies
	\begin{align*}
		&\|x-W(z)x_{I}\|^2=\\
		&\quad=\int_{-\pi}^{\pi}\Phi_{xx}(\omega)+W(\omega)\Phi_{x_{I}x_{I}}(\omega)W^*(\omega)+\\
		&\qquad-\Phi_{x_{I}x}(\omega)W^{*}(\omega)-W(\omega)\Phi_{xx_{I}}(\omega).
	\end{align*}
	The integral is minimized by minimizing the integrand for all $\omega\in [-\pi,\pi]$.
	It is straightforward to find that the minimum is achieved for
	\begin{align*}
		W(\omega)=\Phi_{x x_{I}}(\omega)\Phi_{x_{I} x_{I}}(\omega)^{-1}.
	\end{align*}
	Defining the filter $W(z)=\Phi_{x x_{I}}(z)\Phi_{x_{I} x_{I}}(z)^{-1}$ a real-rational transfer matrix is obtained with no poles on the unit circle that has the specified frequency response. Thus $\hat x=W(z)x_I\in X$ minimizes the cost (\ref{eq: cost general wiener}).
	As a consequence of the Hilbert projection theorem (for pre-Hilbert spaces) Equation (\ref{eq:perp hilbert projection thm}) is satisfied for $\hat x$ if and only if it is the unique element of the subspace $X$ minimizing (\ref{eq: cost general wiener}) \cite{Lue69}.
	If $\Phi_{x_{I} x_{I}}(z)>0$, the uniqueness of $W(z)$ follows from Lemma \ref{lem:unique representation}.
\end{IEEEproof}
In the following definition a notion of conditional non-causal Wiener-uncorrelation is given.
\begin{define}
	Let $q$, $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$ and $X:=\text{tf-span}\{x_1,...,x_{n}\}$.
	For any $i\in\{1,...,n\}$, the process $x_i$ is non-causally Wiener-uncorrelated with $q$ given the processes $\{x_{k}\}_{k\neq i}$ if the $i$-th entry of the Wiener filter to estimate $q$ from $x_I$ is zero, that is
	\begin{align}
		\Phi_{q x_I}\Phi_{x_I x_I}^{-1}b_i=0.
	\end{align}
\end{define}
The following lemma provides an immediate relationship between non-causal Wiener-uncorrelation and the inverse of the cross-spectral density matrix.
This result presents strong similarities with the property of the inverse of the covariance matrix for jointly Gaussian random-variables. Indeed, it is well-known that the entry $(i,j)$ of inverse of the covariance matrix of $n$ random variables $x_1,...,x_n$ is zero if and only if $x_i$ and $x_j$ are conditionally independent given the othe variables.
\begin{lem}\label{lem:topological filter}
	Let $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$. Assume that $\Phi_{x_I}$ has full normal rank.
	The process $x_i$ is non-causally Wiener-uncorrelated with $x_j$ given the processes $\{x_k\}_{k\neq i,j}$, if and only if
	the entry $(i,j)$, or equivalently the entry $(j,i)$, of $\Phi_{x_I}^{-1}(z)$ is zero, that is, for $i\neq j$,
	\begin{align}
		b_{j}^{T}\Phi_{x_I}^{-1}b_{i}=b_{i}^{T}\Phi_{x_I}^{-*}b_{j}=0.
	\end{align}
\end{lem}
\begin{IEEEproof}
	Without any loss of generality, let $j=n$.
	Define the vector $x_{\overline{n}}=(x_1,...,x_{n-1})^T$ and determine the non-causal Wiener filter $W_{n\overline{n}}$ estimating $x_n$ from $x_{\overline{n}}$.
	\begin{align}
		x_{n}=\varepsilon_n +W_{n\overline{n}}(z)x_{\overline{n}}
	\end{align}
	where the error $\varepsilon_n$ has the property that $\Phi_{\varepsilon_n x_{\overline{n}}}(z)=0$. Define $r:=(x_{\overline{n}^T}, \varepsilon_n )$ and observe that
	\begin{align}
		r=\left(\begin{array}{cc}
			\mathcal{I} & 0\\ -W_{n\overline{n}}(z) & 1
		\end{array}\right)x; \qquad
		x_{I}=\left(\begin{array}{cc}
			\mathcal{I} & 0\\ W_{n\overline{n}}(z) & 1
		\end{array}\right)r.
	\end{align}
	Compute
	\begin{align}
		&\Phi_{x_I}^{-1}=
		\left(\begin{array}{cc}
			\mathcal{I} & W_{n\overline{n}}(z)^{*}\\ 0 & 1
		\end{array}\right)
		\left(\begin{array}{cc}
			\Phi_{x_{\overline{n}}}^{-1} & 0\\ 0 & \Phi_{\varepsilon_{n}}^{-1}
		\end{array}\right)
		\left(\begin{array}{cc}
			\mathcal{I} & 0\\ W_{n\overline{n}}(z) & 1
		\end{array}\right)=\\
		&=
		\left(\begin{array}{cc}
			\Phi_{x_{\overline{n}}}+
			\frac{W_{n\overline{n}}^*W_{n\overline{n}}}{\Phi_{\varepsilon_n}}
			& W_{n\overline{n}}^* \Phi_{\varepsilon_n}^{-1}\\
			\Phi_{\varepsilon_n}^{-1} W_{n\overline{n}} & \Phi_{\varepsilon_n}^{-1}
		\end{array}\right).
	\end{align}
	The assertion is proven by premultiplying by $b_n^T$ and post-multiplying by $b_{i}$
\end{IEEEproof}
The following theorem provides a sufficient condition to determine if two nodes in a LDG are kins.
\begin{theo}\label{thm:kinship non-null}\label{thm:sparse wiener}
	Consider a well-posed and topologically detectable LDG.
	Let $x_1,...,x_n\in {}^0\mathcal{F}\mathcal{E}$ be the signals associated with the $n$ nodes of its graph.
	Define $X_j=\text{tf-span}\{x_i\}_{i\neq j}$.
	Consider the problem of approximating the signal $x_j$ with an element $\hat x_j\in X_j$, as defined below
	\begin{align}\label{eq:min problem}
		\inf_{\hat x_j\in X_j}
		\left\|x_j-\hat x_j \right\|^2.
	\end{align}
	Then the optimal solution $\hat x_j$ exists, is unique and
	\begin{align}
		\hat x_j =\sum_{i\neq j} W_{ji}(z)x_i
	\end{align}
	where $W_{ji}(z)\neq 0$ implies $(N_i, N_j) \in kin(G)$.
\end{theo}
\begin{IEEEproof}
	The LDG dynamics is given by $x=(\mathcal{I}-H(z))^{-1}e$ implying that $\Phi_{xx}$ can also be represented as
	\begin{align}
		\Phi_{xx}^{-1}=(\mathcal{I}-H)^{*}\Phi_{e}^{-1}(\mathcal{I}-H).
	\end{align}
	Consider the $j$-th line of $\Phi_{xx}^{-1}$. We have
	\begin{align}
		b_{j}^{T} \Phi_{xx}^{-1}=(b_{j}^{T}-H_{*j}^*)\Phi_{e}^{-1}(\mathcal{I}-H).
	\end{align}
	The row vector $(b_{j}^{T}-H_{*j}^*)$ has zero entries corresponding any index $k$ such that $k\neq j$ that is not a parent of $j$.
	Since $\Phi_{e}$ is diagonal the $i$-th column of $\Phi_{e}^{-1}(\mathcal{I}-H)$ has zero entries for any $k\neq i$ and is not a parent of $i$. Given $i\neq j$, if $i$ is not a parent of $j$ and $i$ is not a child of $j$ and $i$ and $j$ have no common children (they are not coparents), it follows that the entry $(j,i)$ of $\Phi_{xx}^{-1}$ is not zero. Using Lemma (\ref{lem:topological filter}) the assertion is proven.
\end{IEEEproof}
The following result provides a sufficient condition for the reconstruction of a link in a LDG.
\begin{cor}
	Consider a LDG $\mathcal{G}=(G,\mathcal{H},e_I)$ with $G=(V,A)$
	Assume $V=\{N_i\}_{i=1,...,n}$ and let $X:=\{x_i\}_{i=1,...,n}$ be the set of processes $x_i$ corresponding to the nodes $N_i$.
	Let $W_{ji}(z)$ be entry of the non-causal Wiener filter estimating $x_j$ from $\{x_k\}_{k\neq j}$ corresponding to the process $x_i$.
	If $\mathcal{G}$ is self-kin, then $W_{ji}(z)\neq 0$ implies $(N_j, N_i) \in top(G)$.
\end{cor}
\begin{IEEEproof}
	Since $\mathcal{G}$ is self-kin,
	$\mathcal{P}_G(N_j)\cup \mathcal{C}_G(N_j) \cup \mathcal{P}_G(\mathcal{C}_G(N_j))=
		\mathcal{C}_G(N_j) \cup \mathcal{P}_G((N_j))$.
	Thus, from the previous theorem the assertion follows immediately.
\end{IEEEproof}

The following lemma is a key result to explicitly determine the expression of the Wiener filter for a LDG in the non-causal and in the causal scenarios.
\begin{lem}\label{lem:smart span}
	Consider a well-posed LDG $\mathcal{G}=(G,\mathcal{H},e_I)$.
	Assume $V=\{N_i\}_{i=1,...,n}$ and let $X:=\{x_i\}_{i=1,...,n}$ be the set of processes $x_i$ corresponding to the nodes $N_i$.
	Fix $j\in\{1,...,n\}$ and define the set
	\begin{align*}
		& C:=\{c| N_c\in \mathcal{C}_G(N_j)\}=\{c_1, ... c_{n_c}\}
	\end{align*}
	containing the indexes of the $n_c$ children of $N_j$.
	Then, for $i \neq j$,
	\begin{align*}
		x_i \in \text{tf-span}
		\left\{
			\left\{
				\bigcup_{k \in C } (e_k+H_{kj}(z)e_j)
			\right\}
			\cup
			\left\{
				\bigcup_{k \notin C \cup \{j\}} \{e_k\}
			\right\}
		\right\}.
	\end{align*}
	Furthermore, if $\mathcal{G}$ is causal,
	\begin{align*}
		x_i \in \text{c-tf-span}
		\left\{
			\left\{
				\bigcup_{k \in C } (e_k+H_{kj}(z)e_j)
			\right\}
			\cup
			\left\{
				\bigcup_{k \notin C \cup \{j\}} \{e_k\}
			\right\}
		\right\}.
	\end{align*}
\end{lem}
\begin{IEEEproof}
	Define
	\begin{align}
		&\varepsilon_{j}:=0 \nonumber\\
		&\varepsilon_{k}:=e_k+H_{kj}(z)e_j	&\text{if } k \in C\nonumber\nonumber\\
		&\varepsilon_{k}:=e_k		&\text{if }k \notin \{C \} \cup \{j\}\nonumber\\
		&\xi_{k}:=\sum H_{ki}(z)x_{i}	&\text{if } k=j&\nonumber\\
		&\xi_{k}:=x_k		&\text{if }k \neq j \label{eq:xi when not i}
	\end{align}
and, by inspection, observe that
\begin{align*}
	[\mathcal{I}-H(z)]\xi_{I}=\varepsilon_{I}.
\end{align*}
	Since $\mathcal{G}$ is well posed, $[\mathcal{I}-H(z)]$ is invertible implying that the signals $\{\xi_i\}_{i=1,...,n}$ are a linear transformation of the signals $\{\varepsilon_i\}_{i=1,...,n}$. For $i\neq j$, we have
\begin{align*}
	x_i = \xi_i \in \text{tf-span}\{\varepsilon_k\}_{k=1,...,n}=\text{tf-span}\{\varepsilon_k\}_{k\neq j}
\end{align*}
	where the first equality follows from (\ref{eq:xi when not i}) and  last equality follows from the fact that $\varepsilon_{j}=0$.
	The causality of $\mathcal{G}$ also implies that
	\begin{align*}
		x_i =\text{c-tf-span}\{\varepsilon_k\}_{k\neq j}.
	\end{align*}
	This proves the assertion.
\end{IEEEproof}
In the case of a well-posed and topologically detectable LDG, the following theorem
completes Theorem~\ref{thm:kinship non-null}
providing an explicit expression for the non-causal Wiener filter $W$ since the solution of the minimization problem (\ref{eq:min problem}) is known to be unique.
\begin{theo}\label{thm:kinship explicit}
	Consider a well posed and topologicaly detectable LDG $\mathcal{G}=(G,\mathcal{H},e_I)$ with $G=(V,A)$, $V=\{N_1,...,N_n\}$.
	Define $I=(1,...,n)^T$ and $I_j=(1,...,j-1,0,j+1,...,n)$.
	Consider an auxiliary process $e_0$ such that $\Phi_{e_0 e_I}=0$ and $\Phi_{e_0}=1$.
	Define,
	\begin{align}\label{eq: child filter}
		\hat C_{ji}(z)=
			\{\Phi_{e_j}H^{*}_{*j}(z)
			[\Phi_{e_j}(z)H_{*j}(z)H^{*}_{*j}(z)+\Phi_{e_{I_j}}(z)]^{-1}\}_i
	\end{align}
	\begin{align}\label{eq: parent filter}
		\hat P_{ji}(z)=
			\left(
				1-\hat C_{j*}(z)H_{*j}(z)
			\right)H_{ji}(z)
	\end{align}
	\begin{align}\label{eq: kin filter}
		\hat K_{ji}(z)=
			\left\{\begin{array}{ll}
				0 & \text{  if } i=j\\
				-\hat C_{j*}(z)H_{*i}(z) & \text{otherwise}.
			\end{array}\right.
	\end{align}
	These definitions do not depend on the choice on $e_0$.
	Consider the estimate $\hat x_j$ for $x_j$ provided by the non-causal Wiener filter using the other signals $\{x_i\}_{i\neq j}$
	\begin{align}
		\hat x_{j}=\sum_{i\neq j}W_{ji}x_i.
	\end{align}
	Then, we have that
	\begin{align}
		W_{ji}(z)= \hat C_{ji}(z)+ \hat P_{ji}(z)+\hat K_{ji}(z).
	\end{align}
\end{theo}
\begin{IEEEproof}
	Define the following set of indexes
	\begin{align*}
		& C:=\{c| N_c\in \mathcal{C}_G(N_j)\}=\{c_1, ... c_{n_c}\}\\
		& P_l:=\{p| N_p\in \mathcal{P}_G(N_l)\}/\{j\}=\{p_{l1}, ... p_{ln_{pl}}\}\\
		& K:=\{k| N_k\in \mathcal{K}_G(N_j)\}=\{k_{1}, ... k_{n_{k}}\}
	\end{align*}
	for any $l=1,...,n$.\\
	The set $C$ contains all the indexes of the $n_c$ children of $N_n$ while the set $K$ contains all the $n_k$ indexes of the kins of $N_j$. The set $P_l$ contains all the $n_{p_l}$ parents (but $N_j$) of a generic node $N_l$.\\
	For any $i\neq j$, define
	\begin{align}\label{eq:def yc}
		\varepsilon_i:=x_i-\sum_{k\neq j}H_{ij}(z)H_{jk}(z)x_k
			-\sum_{k\neq j}H_{ik}(z)x_k.
	\end{align}
	Note that, for any $i\neq j$,
	\begin{align}\label{eq:eps_i as in smart lemma}
		\varepsilon_{i}=e_{i}+H_{ij}(z)e_{j}.
	\end{align}
	Also note that
	\begin{align}\label{eq: ej function of kins}
		e_j:=x_j-\sum_{k}H_{jk}(z)x_k,
	\end{align}
	The solution to the minimization problem
	\begin{align}
		&\arg\min_{q\in \text{tf-span}\{\varepsilon_i\}_{i\neq j}}\|e_j-q\|^2=\\
		&\qquad =\arg\min_{\stackrel{q\in \text{tf-span}\{\varepsilon_i\}_{i\neq j,0}}{\hat C_{jj}(z)\in~{}^{0}\mathcal{F}} }\|e_j-q\|^2+\|C_{jj}(z)e_0\|^2=\\
		&\qquad \arg\min_{q\in \text{tf-span}\{\varepsilon_i\}_{i\neq j,0}}\|e_j-q\|
	\end{align}
	is given by
	\begin{align}
		\hat e_j = \sum_{\stackrel{i=1}{i\neq j}}^{n} \hat C_{ji}(z)\varepsilon_{i},
	\end{align}
	where the process $e_0$ has played the role of a placeholder forcing $\hat C_{jj}=0$, without affecting the values of $\hat C_{ji}(z)$, for $i\neq j$.
	Now, let us consider the problem
	\begin{align}
		\arg\min_{q\in \text{tf-span}\{x_i\}_{i\neq j}}\|x_j-q\|.
	\end{align}
	From (\ref{eq: ej function of kins}), its solution $\hat x_j$ satisfies
	\begin{align}
		\hat x_j 	&= \sum_{k}H_{jk}(z)x_k+
			\arg\min_{q\in \text{tf-span}\{x_i\}_{i\neq j}}\|e_j-q\|=\\
			&= \sum_{k}H_{jk}(z)x_k+
			\arg\min_{q\in \text{tf-span}\{\varepsilon_i\}_{i\neq j}}\|e_j-q\|
	\end{align}
	where the last equality has been obtained by using Lemma (\ref{lem:smart span}).
	Thus we have
	\begin{align}
		\hat x_j & = \sum_{i\neq j} W_{ji}x_i
			= \sum_{k}H_{jk}(z)x_k + \sum_{i\neq j} \hat C_{ji}\varepsilon_{i}
	\end{align}
	Substituting the espression of $\varepsilon_i$, $i\neq j$, as a function of $x_i$, $i \neq j$, the assertion is proven.
\end{IEEEproof}

Theorem \ref{thm:kinship explicit} gives the expression of the entry $W_{ji}$ of the topological filter as the sum of the three components $\hat C_{ji}$, $\hat P_{ji}$ and $\hat K_{ji}$.
The three components have a graphical interpretation:
$\hat C_{ji}(z)$ is a the contribution corresponding to the fact that $N_i$ is a child of $N_j$; $\hat P_{ji}(z)$ is the contribution present when $N_i$ is a parent of $N_j$; and $\hat K_{ji}(z)$ is a term present when there is a ``co-parent'' relation.
\begin{lem}\label{lem: three components}
	Given a well posed LDG, consider Equation (\ref{eq: child filter}), Equation (\ref{eq: parent filter}) and Equation (\ref{eq: kin filter}). Then
	\begin{itemize}
		\item	$\hat C_{ji}(z)\neq 0$ if and only if $N_i \in \mathcal{C}_G(N_j)$
		\item	$\hat P_{ji}(z)\neq 0$ if and only if $N_i \in \mathcal{P}_G(N_j)$
		\item	$\hat K_{ji}(z)\neq 0$ implies $N_i \in \mathcal{P}_G(\mathcal{C}_G(\mathcal{N}_j))\backslash N_j$.
	\end{itemize}
\end{lem}
\begin{IEEEproof}
	Evaluate
	\begin{align*}
		&\hat C_{j*}=\Phi_{e_j}H^{*}_{*j}(z)
			[\Phi_{e_j}(z)H_{*j}(z)H^{*}_{*j}(z)+\Phi_{e_{I_j}}(z)]^{-1}=\\
		&~=\Phi_{e_j}(z) H_{*j}^{*}(z)\Phi_{e_{I_j}}^{-1}(z)
			\left[
				\mathcal{I}+\Phi_{e_j}(z)H_{*j}(z)H_{*j}^*(z)\Phi_{e_{I_j}}(z)^{-1}
			\right]^{-1}=\\
		&~=\Phi_{e_j}(z) H_{*j}^{*}(z)\Phi_{e_{I_j}}^{-1}(z)
			\sum_{m=0}^{+\infty}
				\frac{(-1)^m}{m!}\|H_{*j}^*(z)\Phi_{e_{I_j}}^{-1}H_{*j}(z)\|^m\Phi_{e_j}(z)^m=\\
		&~=\Phi_{e_j}(z) H_{*j}^{*}(z)\Phi_{e_{I_j}}^{-1}(z)
		\frac{1}{1+\|H_{*j}^*(z)\Phi_{e_{I_j}}^{-1}(z)H_{*j}(z)\|\Phi_{e_j}(z)}.
	\end{align*}
	Thus $\hat C_{ji}\neq 0$ for any $N_i \in \mathcal{C}_G(N_j)$.
	Furthermore, it follows
	\begin{align}
		\|\hat C_{j*}(\omega) H_{*j}(\omega)\|<1,
	\end{align}
	implying, from (\ref{eq: parent filter}) , that
	\begin{align}
		\hat P_{ji}(\omega)\neq 0.
	\end{align}
	for any $N_i \in \mathcal{P}_G(N_j)$.
\end{IEEEproof}
From Lemma \ref{lem: three components} Theorem \ref{thm:sparse wiener} can be recovered, since $\hat C_{ji}(z)=\hat P_{ji}(z)=\hat K_{ji}(z)=0$ for all the nodes $N_i$ that are not kin of $N_j$. However the only conditions according to which $W_{ji}=W_{ij}=0$ even though $\{N_i,N_j\} \in kin(G)$, making the link not detectable, are found.
\begin{theo}\label{thm:kinship null entry}
	Consider a well posed LDG $\mathcal{G}=(G,\mathcal{H},\mathcal{E})$ with $G=(V,A)$, $V=\{N_1,...,N_n\}$. $\{N_i, N_j\}\in kin(G)$ and $W_{ji}=W_{ij}=0$ if and only if one of the following condition is satisfied
	\begin{itemize}
		\item	$\hat C_{ji}(z)=-\hat K_{ji}(z)$ and $\hat P_{ij}(z)=-\hat K_{ij}(z)$
		\item   $\hat P_{ji}(z)=-\hat K_{ji}(z)$ and $\hat C_{ij}(z)=-\hat K_{ij}(z)$
	\end{itemize}

\end{theo}
\begin{IEEEproof}
	Indeed, since in a LDG $\mathcal{C}_G(N_j) \cap \mathcal{P}_G(N_j)=\emptyset$, it follows that, for any couple $(i,j)$, $\hat C_{ji}(z)=0$ or $\hat P_{ji}(z)=0$.
	The conditions, then, follow in a straightforward manner.
\end{IEEEproof}
Theorem \ref{thm:kinship null entry} shows that the condition under which a link present in the kin topology leads to two null entries ($W_{ji}=W_{ij}=0$) in the topological filter can be considered a pathological case. Indeed two simultaneous cancellations between the kin term and the other term (parent or child) must happen.
Moreover, if a node is simple, such a cancellation is not possible.

\section{Sparsity of causal filtering operators}\label{sec:causal}
First, we need to introduce the following lemma.
\begin{lem}\label{lem:zero span from generators}
	Let $x$ and $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$. Assume that $\Phi_{xx_I}(\omega)=0$ for all $\omega\in [-\pi, \pi]$.
	Then $<x,y>=0$ for all $y\in \text{tf-span}\{x_i\}_{i=1,...,n}$.
\end{lem}
\begin{IEEEproof}
	As $y\in \text{tf-span}\{x_i\}_{i=1,...,n}$, it follows that there exist $\alpha_i(z)\in {}^0\mathcal{F}$ such that
	\begin{align*}
		y=\sum_{i=1}^{n}\alpha_i(z)x_i=\alpha_I(z)x_I,
	\end{align*}
	where $\alpha_{I}(z)=[\alpha_{1}(z), ... , \alpha_{n}(z)]$ is a row vector of real-rational transfer functions.
	Then it follows that
	\begin{align*}
		<x,y>=\int_{-\pi}^{\pi} \Phi_{x x_I}(\omega)\alpha_I(\omega)^{*}=0.
	\end{align*}
\end{IEEEproof}
Now, a specific formulation of the causal Wiener filter is introduced for the defined spaces.
\begin{prop}\label{eq: my causal wiener}
	Let $x$ and $x_1,...,x_n$ be processes in the space $^0\mathcal{F}\mathcal{E}$.
	Define $I:=(1,....,n)$ and $X:=\text{c-tf-span}\{x_1,...,x_n\}$. Consider the problem
	\begin{align}\label{eq: cost general wiener (causal)}
		\inf_{q \in X} \|x-q\|^2.
	\end{align}
	Let $S(z)$ be the spectral factorization of $\Phi_{x_I}(\omega)=S(\omega)S^{*}(\omega)$.
	If $\Phi_{x_I}(\omega)>0$, for $\omega\in[-\pi,\pi]$, the solution $\hat x^{(c)}\in X$ exists, is unique and has the form
	\begin{align*}
		\hat x^{(c)}=W^{(c)}(z)x_I
	\end{align*}
	where
	\begin{align*}
		W^{(c)}(z)=\{\Phi_{x x_{I}}(z)\Phi_{x_{I}}(z)^{-1}S(z)\}_{C}S^{-1}(z).
	\end{align*}
	Moreover $\hat x^{(c)}$ is the only element in $X$ such that, for any $q\in X$, it holds that
	\begin{align}\label{eq:perp hilbert projection thm (causal)}
		<x-\hat x^{(c)},q>=0.
	\end{align}
\end{prop}
\begin{IEEEproof}
	Observe that $W^{(c)}(z)$ is rational and causal. Let us prove that, for any causal $H(z)$, we have
	\begin{align}
		<x-\hat x^{(c)},H(z) x_{I}>=0
	\end{align}
	Define $r_{I}:=S^{-1}(z)x_{I}$, and observe that
	$S(z), S(z)^{-1}(z)\in~{}^0\mathcal{F}^{+}$,
	since $S(z)$ is the spectral factor of $\Phi_{x_I}(z)$ that is real-rational and has full rank on the unit circle. The signal $r_I$ is white since
	\begin{align}
		\Phi_{r_I}(z)=\mathcal{I}_{n}.
	\end{align}
	Define
	\begin{align}
		\hat x^{(c)}& =\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{C}r_I\\
				& =\{\Phi_{x x_I}(z)\Phi^{-1}_{x_I}(z)S(z)\}_{C}S^{-1}(z)x_I.
	\end{align}
	observing that $\hat x^{(c)}\in X$ since it is obtained by composing transformations in $~{}^0\mathcal{F}^{+}$.
	Also observe that $\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)$ is the non-causal Wiener filter estimating $x$ from $r_I$. Let the strictly anticausal component of such a filter part be
	\begin{align}
		\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{SA}:=
			\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)-\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{C}.
	\end{align}
	Now, for a $H(z)\in~{}^0\mathcal{F}$, compute
	\begin{align}
		& <x-\hat x, Hx_I>=<x-W^{(c)}Sr_I x, HSr_I>=\\
		& <x-\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)r_I+\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{AC}r_I, HSr_I>=\\
		& <\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{SA}r_I, HSr_I>=0.
	\end{align}
	where Lemma \ref{lem:zero span from generators} has been used to prove that
	$<x-\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)r_I,H(z)S(z) r_I>=0$ and the last equality follows from the causality of $H(z)S(z)$, the strictly anti-causality of $\{\Phi_{x r_I}(z)\Phi^{-1}_{r_I}(z)\}_{SA}$ and the whiteness of $r_{I}$.
	From the Hilbert projection theorem, $\hat x^{(c)}$ is the unique process minimizing the cost (\ref{eq: cost general wiener (causal)}).
\end{IEEEproof}

The following theorem proves the sparsity of the causal Wiener filter stating that the causal Wiener filter estimating $x_{j}$ from the signals $x_{i}$, $i\neq j$, has non-zero entries corresponding to the kin signals of $x_{j}$.
\begin{theo}\label{thm:sparse causal wiener}
	Consider a well-posed, causal and topologically detectable LDG.
	Let $x_1,...,x_n\in {}^0\mathcal{F}\mathcal{E}$ be the signals associated with the $n$ nodes of its graph.
	Define $X_j=\text{c-tf-span}\{x_i\}_{i\neq j}$.
	Consider the problem of approximating the signal $x_j$ with an element $\hat x_j\in X_j$, as defined below
	\begin{align}
		\min_{\hat x_j\in X_j}
		\left\|x_j-\hat x_j \right\|^2.
	\end{align}
	Then the optimal solution $\hat x_j$ exists, is unique and
	\begin{align}
		\hat x_j =\sum_{i\neq j} W_{ji}(z)x_i
	\end{align}
	where $W_{ji}(z)\neq 0$ implies $(N_i, N_j) \in kin(G)$.
\end{theo}
\begin{IEEEproof}
	For any $i\neq j$, define $\varepsilon_i$ as in (\ref{eq:def yc}) and observe that $\varepsilon_i$ can be represented as in (\ref{eq:eps_i as in smart lemma}).
	Also note that
	\begin{align}\label{eq: e_j causal}
		e_j:=x_j-\sum_{i}H_{ji}(z)x_i.
	\end{align}
	Consider $\hat e_j$ defined as
	\begin{align}
		\hat e_j
		:=\arg\min_{q\in \text{c-tf-span}\{\varepsilon_i\}_{i\neq j}}\|e_j-q\|
		=\sum_{i\neq j} C^{(c)}_{ji}(z)\varepsilon_{i}
	\end{align}
	where the transfer functions $C^{(c)}_{ji}(z)$ are given by the causal Wiener filter estimating $e_j$ from $\{\varepsilon_{i}\}_{i\neq j}$. Notice that, by (\ref{eq:eps_i as in smart lemma}), $C^{(c)}_{ji}(z)$ is equal to zero if $N_i$ is not a child of $N_j$.
	Now, let us consider the optimization problem
	\begin{align}
		\hat x_{j}:=\arg\min_{q\in \text{c-tf-span}\{x_i\}_{i\neq j}}\|x_j-q\|=\sum_{i\neq j}W_{ji}(z)x_i
	\end{align}
	where $W_{ji}(z)$ are the entries of the causal Wiener filter solving it.
	Its solution $\hat x_j$ satisfies also
	\begin{align}
		\hat x_j 	&= \sum_{i\neq j}H_{ji}(z)x_i+
			\arg\min_{q\in \text{c-tf-span}\{x_i\}_{i\neq j}}\|e_j-q\|=\\
			&= \sum_{i}H_{ji}(z)x_i+
			\arg\min_{q\in \text{c-tf-span}\{\varepsilon_i\}_{i\neq j}}\|e_j-q\|
	\end{align}
	where the first equality derives from (\ref{eq: e_j causal}) and the last one has been obtained by using Lemma \ref{lem:smart span}.
	Thus we have
	\begin{align}
		\hat x_j & = \sum_{i\neq j} W_{ji}x_i
			= \sum_{i}H_{ji}(z)x_i + \sum_{i\neq j} C_{ji}\varepsilon_{i}
	\end{align}
	Substituting the espression of $\varepsilon_i$, $i\neq j$, as a function of $x_i$, $i \neq j$, the assertion is proven.
\end{IEEEproof}

The following theorem proves the sparsity of the one step prediction operator (or granger-causal operator).
Granger-causality is a wide-spread technique in econometrics to test the causal dependence of time series.
If the stronger hypothesis of strictly causal transfer functions $H_{ji}(z)$ is met Granger-causality provides an exact reconstruction of parent-child links in a LDG.
\begin{theo}\label{thm:sparse granger}
	Consider a well-posed, strictly causal and topologically detectable LDG.
	Let $x_1,...,x_n\in {}^0\mathcal{F}\mathcal{E}$ be the signals associated with the $n$ nodes of its graph.
	Define $X_j=\text{c-tf-span}\{x_i\}$.
	Consider the problem of approximating the signal $z~x_j$ with an element $\hat x_j\in X_j$, as defined below
	\begin{align}
		\min_{\hat x_j\in X_j}
		\left\|z x_j-\hat x_j \right\|^2.
	\end{align}
	Then the optimal solution $\hat x_j$ exists, is unique and
	\begin{align}
		\hat x_j =\sum_{i=1}^{n} W_{ji}(z)x_i
	\end{align}
	where $W_{ji}(z)\neq 0$ implies $i=j$ or $N_i$ is a parent of $N_i$.
\end{theo}
\begin{IEEEproof}
	For any $i\neq j$, define $\varepsilon_i$ as in (\ref{eq:def yc}) and observe that $\varepsilon_i$ can be represented as in (\ref{eq:eps_i as in smart lemma}). Also define $\varepsilon_j:=e_j$.
	Note that
	\begin{align}\label{eq: e_j granger}
		e_j:=x_j-\sum_{i}H_{ji}(z)x_i.
	\end{align}
	Consider the minimization problem
	\begin{align}
		\hat e_{j}:=\arg\min_{q\in \text{c-tf-span}\{\varepsilon_i\}_{i}}\|ze_j-q\|= \sum_{i\neq j} C^{(g)}_{ji}(z)\varepsilon_{i}
	\end{align}
	where the transfer functions $C^{(g)}_{ji}(z)$ are elements of ${}^{0}\mathcal{F}$.
	We have that $C^{(g)}_{ji}(z)=0$ for any $i\neq j$. Indeed, it holds that
	\begin{align}
		&\arg\min_{q\in \text{c-tf-span}\{\varepsilon_i\}_{i=1}^{n}}\|ze_j-q\|=
		\arg\min_{q\in \text{c-tf-span}\{e_i\}_{i=1}^{n}}\|ze_j-q\|=\\
		&\qquad = \arg\min_{q\in \text{c-tf-span}\{e_j\}}\|ze_j-q\|
	\end{align}
	and, since $\Phi_{e_ie_j}(\omega)=0$ for $i\neq j$.
	Conversely, from (\ref{eq: my causal wiener}), we find$C^{(g)}_{jj}(z)=\{zS_{j}(z)\}_{C}z^{-1}S_j(z)$ where $S_j(z)$ is the spectral factor of $e_j$.
	Now, let us consider the problem
	\begin{align}
		\arg\min_{q\in \text{c-tf-span}\{x_i\}_{i\neq j}}\|zx_j-q\|.
	\end{align}
	Its solution $\hat x_j$ is
	\begin{align*}
		\hat x_j 	&= \sum_{k}zH_{jk}(z)x_k+
			\arg\min_{q\in \text{c-tf-span}\{x_i\}_{i}}\|ze_j-q\|=\\
			&= \sum_{k}zH_{jk}(z)x_k+C^{(g)}_{jj}(z)e_j\\
			&= C^{(g)}_{jj}(z)x_j+\sum_{k\neq j} [zH_{jk}(z)-C^{(g)}_{jj}(z)H_{jk}(z)]x_k.
	\end{align*}
	This proves the assertion.
\end{IEEEproof}

\section{A reconstruction algorithm}\label{sec:algorithm}
The previous section provides theoretical results allowing for the reconstruction of a topology via Wiener filtering. It needs to be stressed that even in the case of sparse graphs, the reconstruction of the kinship topology can be considered a practical solution. The reasons are two-fold.
In many situations it is possible to measure the outputs of many nodes, while it is important to identify a reduced number of possible interconnections among those nodes. For example, DNA-microarrays are devices that allow the measurement of gene expression of a cell of a cell.
Such data can be useful in understanding which genes interact together and realize a specific metabolic pathway and how they are related \cite{RavSom02}.
Indeed, a cell can express tens of thousands of genes while only a few tens must be involved in a gene regulatory network. The possibility of reducing the set of involved genes to test the presence of actual interactions with targeted experiments is of significant importance \cite{MarPel08b}.
Analogously, in Finance, quantifying the strongest interconnections among a set of market stocks can suggest good strategies to balance a given portfolio \cite{ManSta00}, thus it is important to have a quantitative tool to group together different stocks or, at least, to detect a limited set of possible dynamical connections.
Similar problems are also faced in neuroscience in order to understand neural interconnections \cite{BroDin04}.
A second reason why the presented analysis is of practical importance is that as a byproduct of the reconstruction an optimal model for the node dynamics which can be used for smoothing procedures (in the case non-causal Wiener filter are derived) or predictive ones (in the strictly causal case which is out of the scope of this article) can be obtained.

The following algorithm is the natural pseudocode implementation of the reconstruction technique that was developed in the previous section.
\subsection*{\tt Reconstruction algorithm}
\begin{itemize}
	\item[{\tt 0.}] Initialize the set of edges $A=\{ \}$
	\item[{\tt 1.}] For any signal $x_j$
	\item[{\tt 2.}]\quad solve the minimization problem (\ref{eq:min problem})
	\item[{\tt 3.}]\quad For any $W_{ji}(z) \ncong 0$
	\item[{\tt 4.}]\quad\quad add $\{N_i,N_j\}$ to $A$
	\item[{\tt 5.}]\quad end
	\item[{\tt 7.}] end
	\item[{\tt 8.}] return $A$
\end{itemize}

Step {\tt 2} can be computed efficiently for a large number of signals using techniques based on Gram-Schmidt orthogonalization as those described in \cite{GolRee98}.
When checking the condition at step {\tt 3}, the condition $W_{ji} \ncong 0$ can be implemented as $\|W_{ji}(z)\|>\sigma_{thr}(x_i,x_j)$ where $\sigma_{thr}(x_i,x_j)$ is a threshold that might depend on both the signals $x_i$ and $x_j$. Moreover, a proper norm must be defined on the space of transfer functions.\\

\section{Discussion on the robustness of the reconstruction}\label{sec:corrupted}
In the previous sections an ideal scenario was analyzed, where the node signals are  affected by the process noises $e_{j}$ which are actively driving the network.
This leads to the result that the Wiener filter is a useful tool for identifying the network topology providing the topology of the smallest self-kin network that contains the original system.
In most pratical cases the signals are not accessible but are also affected by measurement noise.
Unfortunately, the presence of measurement noise destroys the sparsity property of the Wiener filter.
However, it will be shown that when the measurement noise is known to be sufficiently small, the introduction of a threshold in the algorithm again guarantees the reconstruction of the topology of the minimal self-kin network.
In different terms, the Wiener filter property presented in order to reconstruct a topology offers a certain degree of robustness that will be quantified.\\
\begin{define}
	A Corrupted LDG (CLDG) is a pair $(\mathcal{G},\eta_I)$ where $\mathcal{G}=(G,\mathcal{H},e_{I})$ is a LDG with dynamics $X=\{x_1 ,..., x_{n}\}$ and $\eta_I=\{\eta_1,...,\eta_n\}$ is a set of noises with the property that they are mutually not correlated and not correlated with the signals $\{e_{j}\}\in \mathcal{E}$, either.
	The output of the CLDG as the set $Y$ of signals is defined as
	\begin{align*}
		y_j=x_j+\eta_j.
	\end{align*}
	Introducing an auxiliary signal $y_0$ which is not correlated with any $y_j$ $(j=1,...,n)$,
	the additively corrupted topological filter of $\mathcal{G}$ with respect to the measures $y_{j}$ is defined as
	\begin{align}
		\hat W(z)=\left(\begin{array}{c}
				\Phi_{y_1 y_{I_{1}}}(z)\Phi_{y_{I_1} y_{I_1}}^{-1}(z)\\
				\vdots \\
				\Phi_{y_n x_{I_{n}}}(z)\Phi_{y_{I_n} y_{I_n}}^{-1}(z)
			\end{array}\right).
	\end{align}
\end{define}
The following important lemma about the inverse of a matrix is recalled.
\begin{lem}\label{lem:inverse lemma}
	If $Q$ and $Q+\Delta$ are two invertible matrices the following equality holds
	\begin{align}
		(Q+\Delta)^{-1}-Q^{-1}=-Q^{-1}(Q^{-1}+\Delta^{-1})^{-1}Q^{-1}.
	\end{align}
\end{lem}
The following result establishes a bound on the difference between the topological filter $W(z)$ which can be obtained by measuring directly the dynamics of a LDG and the corrupted topological filter $\hat W(z)$ obtained by measuring its output.
\begin{theo}\label{thm:bound on corrupted}
	Consider a CLDG $(\mathcal{G},\eta)$. Let $W(z)$ be the topological filter of $\mathcal{G}$ and let $\hat W(z)$ be its corrupted topological filter.
	If
	\begin{align}
		\frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|\Phi_{\eta}(z)\|>0,
	\end{align}
	then, for any $z\in \mathbf{C}$,
	\begin{align}
		\|\hat W_{j}(z) - W_{j}(z) \|&\leq
		\frac{\|\Phi_{y_{j} y_{I_j}}(z) \|\|\Phi_{y_{I_j} y_{I_j}}(z)^{-1} \| \|\Phi_{\eta}(z) \|}
		{\left( \frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|\Phi_{\eta}(z)\|
		\right)}.
	\end{align}
\end{theo}
\begin{IEEEproof}
	First note that
	\begin{align}
		\Phi_{y_{j} y_{I_j}}(z)=\Phi_{x_{j} x_{I_j}}(z),
	\end{align}
	thus
	\begin{align}
		\hat W_{j}(z) - W_{j}(z)=\Phi_{y_{j} y_{I_j}}(z)
			\left[
			\Phi_{y_{I_j} y_{I_j}}^{-1}(z)-\Phi_{x_{I_j} x_{I_j}}^{-1}(z)
			\right]=\\
			\Phi_{y_{j} y_{I_j}}(z)
			\left[
			\Phi_{y_{I_j} y_{I_j}}^{-1}(z)-
				\left(
					\Phi_{y_{I_j} y_{I_j}}(z)-\Phi_{\eta}(z)
				\right)^{-1}
			\right].
	\end{align}
	By applying Lemma~\ref{lem:inverse lemma}, it follows that
	\begin{align}
		&Q^{-1}-(Q-\Delta)^{-1}=Q^{-1}(Q^{-1}-\Delta^{-1})^{-1}Q^{-1}=\\
		& =Q^{-1}(\Delta-Q)^{-1}\Delta,
	\end{align}
	which implies
	\begin{align}
		&\|Q^{-1}-(Q-\Delta)^{-1}\|\leq \|Q^{-1}\|\|\Delta\|\|(Q-\Delta)^{-1}\|=\\
		&=\frac{\|Q^{-1}\|\|\Delta\|}{\min sp(Q-\Delta)}
			\leq\frac{\|Q^{-1}\|\|\Delta\|}{\min sp(Q)-\max sp(\Delta)}=\\
		&=\frac{\|Q^{-1}\|\|\Delta\|}{\frac{1}{\|Q^{-1}\|}-\|\Delta\|}
	\end{align}

\end{IEEEproof}
An immediate way to apply Theorem~\ref{thm:bound on corrupted} is when a function $D(z)$ bounding $\|\Phi_{\eta}(z)\|$ is known as illustrated by the following corollary.
\begin{cor}
	Consider a CLDG $(\mathcal{G},\eta)$. Let $\hat W(z)$ be its corrupted topological filter.
	Assume that there exists a real valued function $D(z)$ such that
	\begin{align}
		\|\Phi_{\eta}(z)\|<D(z).
	\end{align}
	If, for some $z\in \mathbf{C}$,
	\begin{align}
		\frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|D(z)\|>0
	\end{align}
	and
	\begin{align}
		\|\hat W_{ji}(z)\|\geq
		\frac{\|\Phi_{y_{j} y_{I_j}}(z) \| \|\Phi_{\eta}(z) \|}
		{\left( \frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|\Phi_{\eta}(z)\|
		\right)},
	\end{align}
	then
	\begin{align}
		(N_j, N_i)\in kin(G).
	\end{align}
\end{cor}
\begin{IEEEproof}
	By contradiction, assume that $(N_j, N_i)\notin kin(G)$. Then $W_{ji}(z)=0$.
	By Theorem~\ref{thm:bound on corrupted} we have
	\begin{align}
		\|\hat W_{ji}(z)\|
			&\leq
		\frac{\|\Phi_{y_{j} y_{I_j}}(z) \| \|\Phi_{\eta}(z) \|}
		{\left( \frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|\Phi_{\eta}(z)\|
		\right)}\\
			& <
		\frac{\|\Phi_{y_{j} y_{I_j}}(z) \| \|\Phi_{\eta}(z) \|}
		{\left( \frac{1}{\|\Phi_{y_{I_j} y_{I_j}}^{-1}(z)\|}-\|D(z)\|
		\right)}
	\end{align}
	which is a contradiction.
\end{IEEEproof}

Note that the inequality of Theorem~\ref{thm:bound on corrupted} needs to hold for any $z$, so, in this sense, it constitutes a sharp criterion to detect the presence of a link in a CLDG.

\section{Numerical Examples}\label{sec:example}
In this section illustrative applications of the theoretical results are provided.
It is worth observing that results were developed for the general class of linear models.
Indeed, no assumptions were made on the order and causality property of considered transfer functions.

\subsection{Self-kin network}\label{sec:self-kin example}
Consider a ring network of $15$ nodes where the dynamics of the links is given by $5$-th order FIR filters and the noise power is the same on every node. The network topology is provided in Figure~\ref{fig:true_ring}
\begin{figure}[bth]
	\centering
	\begin{tabular}{cc}
		\includegraphics[width=0.45\columnwidth]{ring_true} &
		\includegraphics[width=0.45\columnwidth]{ring_rec} \\
		(a) & (b)
	\end{tabular}
	\caption{A ring network of $15$ nodes as the one considered in Example ~\ref{sec:self-kin example} (a).\label{fig:true_ring}
	The reconstructed topology of Example ~\ref{sec:self-kin example}. Every single links has been detected and, since the network is self-kin, the topology does not contain any spurious link (b). \label{fig:rec_ring}
	}
\end{figure}
The network was simulated for $1000$ steps and an implementation of the developed algorithm was applied to the data providing the topology of Figure~\ref{fig:rec_ring}(b).

\subsection{Generic network}\label{sec:generic example}
Consider a network of $24$ nodes for $1000$ steps as reported in Figure~\ref{fig:true_net}(a) and the reconstructed topology obtained is depicted in Figure~\ref{fig:rec_net}(b).
\begin{figure}[bth]
	\centering
	\begin{tabular}{cc}
		\includegraphics[width=0.45\columnwidth]{net24_true} &
		\includegraphics[width=0.45\columnwidth]{net24_rec}\\
		(a) & (b)
	\end{tabular}
	\caption{
		A network of $24$ nodes as the one considered in Example ~\ref{sec:generic example} (a) and the reconstructed topology (b).
		Every single links has been detected, but, since the network is not self-kin, as expected, the topology contains the additional links between the ``kins''.
		\label{fig:true_net}
		\label{fig:rec_net}
	}
\end{figure}

\section{Conclusions}
This work has illustrated a simple but effective procedure to identify
the general structure of a network of linear dynamical systems.
To the best knowledge of the authors, a general analytical formulation of the problem of identifying a network is not tackled in scientific literature yet.
The approach followed is based on Wiener Filtering in order to detect the  existing links of a network.
When the topology of the original graph is described by a self-kin network, the method developed guarantees an exact reconstruction. Self-kin networks provide a non-trivial class of networks since they allow the presence of loops, nodes with multiple inputs and lack of connectivity.
Moreover, the paper also provides results about general networks. It is shown that, for a general graph, the developed procedure reconstructs the topology of the smallest self-kin graph containing the original one. Thus, the method is optimal in this sense.
Numerical examples illustrate the correctness and also the reliability of the identification technique.

\section*{Acknowlgements}
Donatello Materassi wants to thank Lipeng Ning for the useful discussions they had on the topics developed in this paper.

redpajama_set_name: RedPajamaArXiv


Are you going to move the code base from http://svn.hive.pt to https://github.com/hivesolutions at some point?
It's not that easy, maybe in the future ;)... We have a lot of infra-structure around our svn: verifications on commit, automatic unit test running, continuous integration, etc.. Meanwhile we use git (at github) for our smaller projects that require a distributed version control system.
No need to change your existing setup; the only thing you would need to do is put a repo onto github and sync them using git-svn http://www.markus-gattol.name/ws/scm.html#svn_and_git Imo having Colony on github would give you more exposure/feedback, thus boost development, mainly through growing a community around it.
We've tried git-svn but we had some problems with the svn:external references and many other things. Anyway we'll get back to the git issue in some time that's for sure.
Thank you for your insight on this complex subject.


Chalk Menuboard is back, with a BANG!!
Cloud based digital menuboard application is one step forward to the solution. Cloud based digital menuboard design application, like DSMenu, enables restaurant owners to create menuboards using the "Chalk board themes" that they had being using for so long. Only difference is Chalks are replaced with some specialized fonts which gives Chalk type effect.
There are applications, like DSMenu, where a restaurant owner can create digital menuboard for their restaurant and publish those on a url using the App.
Menuboards can be seen from anywhere using the url.
Menuboards can be submitted on social networking website like facebook, twitter etc.
Menuboards can be mailed as a url link to potential customers.
5 ways in which Digital menuboards can boost restaurant business.
Ability to create a menuboard every day.
Helps in both indoor and outdoor promotion of menus.
Ability to create Fresh content with menuboard slideshow.
Ability to create promotions on-the-fly. Menu upselling method.
Ability to Edit menuboard, Anytime, Anywhere.
16th of October each year is celebrated as World Food Day which is also famously known as the Food Engineer day. World Food Day (WFD) was established by FAO's Member Countries at the Organization's 20th General Conference in November 1979. WFD first event was ideated by Dr. Pal Romany from Hungary. The day is habitually observed in more than 150 countries world-wide.
We can call digital menuboards as a "Silent Seller". Digital menuboards are just another "creative way" by which restaurant owners sells their menus. Promote products that are in high inventory - "Menu up selling". Digital menuboards helps in making quick offers and combo offers and sell menu items. Adding new items to the menu is just so easy with Digital Menuboards. Menu editing is just so FAST and SIMPLE using the digital menuboard application.
There are few basic differences between static menuboard and animated menubard. Animated menuboards can highlight a particular content on a menuboard more than static menuboards. Animated menuboards can create more lasting impression on audience?s mind than static menuboards. There are few disadvantages too for animated menuboard.
Digital Menuboards are known for its focus on customer retention but Digital Menuboards are fast becoming an important brand building tool for restaurant owners. Here are some tips to get the best digital menuboard application that will ensure both customer retention and Brand building initiatives: Company Logo, Corporate colors, Own Fonts, Picture upload, Know the chef, Custom Contents, Important Announcements and Social Networking.
A "Time Saving" solution for restaurant owners.
"Time is money". The true essence of these words reverberates in restaurant business. Studies shows that restaurant owners spends most of their time focusing on menus, customer service, quick order processing etc. As they focus more on these important business issues other smaller issues gets less attention and hence "opportunity lost" in the form of "Brand Building", "menu upselling" etc.
A "Revolutionary" way to design digital menuboards.
Cloud computing made its debut in food and restaurant industry in the form of Cloud based digital menuboard application which makes it a very "Revolutionary"idea. It has redefined the way digital menuboards are created for restaurants, considering that manual menuboards is meant for museum. Cloud based digital menuboard application based on enhanced SAAS business model is an emerging trend which more and more restaurants are adapting to. This article is to articulate the focus of these application. There are many utilities of such application but this article will focus on the follow 5 parameters: Quick Solution, Cost Effective, User friendliness, Platform Independence and Collaborative work.
What does a coffee shop needs?
Research found that 3 things sells the most in a coffee shop: Quick Offers so that a customer can quickly choose and order really FAST, Attractive graphics of the menus so that the customer feels like ordering and Coffee shop Ambiance.
Halloween Specials - Best Business With Spooky Dinners !
Labor Day Menu Board - Workers Day Out!
Mexican Cuisine - Divinity In Simplicity !
Trends In Bakery - An Icing On The Cake!
Father's Day Celebration with a Menu. And much more.
This Thanksgiving Cook-up a Menu with Keyboards in Digital Signage!
Addressing the 5 questions that the restaurant owners ask.
Not sure of using Digital Menuboards? - 12 facts to help in decision making.
Digital menu boards offer more than just prices... It "adds value"
A simple App for digital signage menuboard design.


7 Comments Published by Tiffany on Wednesday, September 21, 2005 at 11:58 AM.
Reagan tagged me yesterday! I feel so loved. Or molested. I can't decide which. Either way, here are my answers!
~Quit watching the OC. I have a problem.
~My name is Tiffany, and I've been appointed as your attorney. Can you tell my why you haven't been paying your child support?
Ha ha ha ha ha! Now it's your turn!
"Marie! It's Tim! Open up!"


Next time you're offered cheap cigarettes in the pub, do yourself a favour and think twice. Smokers looking for a bargain have been warned not to buy a bootleg brand of cigarettes from Russia, after it was found to contain asbestos.
The warning came from trading standard chiefs after a batch of Jin Ling cigarettes was reportedly offered around British pubs and clubs.
The cigarettes, which come in yellow packs with the words Jin Ling and USA emblazoned across the front, are thought to be twice as strong as an ordinary cigarettes.
Many of those tested have been revealed to contain toxins such as industrial chemicals and asbestos-lined Chinese drywall.
It is believed the cigarettes are created specifically for the black market, as the brand on offer is not available over the counter.
A fitter who worked at West Cumberland Hospital in Whitehaven has died from exposure to asbestos. Edward Nicholls was 76.
Postmortem examinations revealed prominent tumours in Mr Nicholl's right lung. During a hearing, Mr Nicholl's statement on his working history was read out. Mr Nicholls said he worked at the West Cumberland Hospital as a fitter's mate between 1963 and 1998. During this time, he undertook a variety of jobs, from fitting light bulbs to working in the boiler house. Part of his job was to repair pipes which had been lagged with asbestos. He said he had to crawl under ducts where there was asbestos and he was often exposed to asbestos dust.
The cause of his death was bronchopneumonia caused by mesothelioma.
A statement written by a man four days before he died from asbestos-related cancer was read out as evidence at an inquest into his own death.
Evidence from Clifford Baker, of Knowle, Bristol, about how he was exposed to asbestos in his working life was read out at the hearing at Flax Bourton Coroner's Court.
The retired carpenter and joiner, of Langton Road, had been exposed to asbestos dust for most of his working life, since he first became an apprentice at a carpentry and joinery firm in Clifton in 1961, the inquest heard.
Mr Baker, 64, would often have to sweep up asbestos dust, cut fire doors containing asbestos and fix asbestos ceiling panels on local properties.
After his apprenticeship and throughout his working life, he was on a number of occasions exposed to a lot of dust, which he believed to contain asbestos.
His wife Jane, a housewife, and his daughters Charlotte and Catherine were in court for the hearing on Friday.
Just read that one of favourite actors, Steve McQueen, was actually diagnosed with the asbestos disease mesothelioma, resulting in his early death. He worked in ship yards when he was younger and was exposed to asbestos whilst working there.
We had an asbestos removal Wiltshire job last week, dismantling an asbestos roof off a scout hut.
We had a group of scouts who kept asking us if they could help. Unfortunately this will contravene child labour laws and health and safety laws.
b) a bit crooked and is taking his chances with the law.
When Did the UK Stop Using Asbestos Insulating Board?
We had a call from someone a few days ago asking when did the UK stop using asbestos insulating board? Now I'm more concerned with getting rid of asbestos and keeping places safe than history dates, but my colleague Dan informed that insulation board was banned in the UK in 1985 along with other asbestos containing material.


LISA Armstrong could make it big in the US, after being axed from Britain's Got Talent.
Lisa Armstrong has had a tough year, after divorcing Ant McPartlin and losing her job as the head make-up artist on Britain's Got Talent.
However, things could be looking up for the stunning star after it was reported she was heading to LA to focus on her career Stateside.
According to the Daily Star Online, PR expert Tom Bourlet from Bayagan Group reckons she could make it big in the US, if she makes the right decisions.
He said: "Depending on how Lisa handles the situation and whether she makes the most of the short and fast opportunity, she can transfer her career towards the celebrity world.
"National support for her is at an all time high and she has been a mainstay in the media.
He added: "She will undoubtedly have opportunities to either expand herself from a MUA [makeup artist] angle, or alternatively to go down the reality TV angle, both of which have potential benefits from a career perspective but with very altering results.
The 42 year old, who divorced Ant McPartlin in October, was reported to be making plans to head to LA after she spent time there recuperating earlier this year.
And, according to reports, long-time friend Simon Cowell has "promised to put some work her way" while she is there.


text:
Defending the indefensible?
Editorial COLUMNIST
published : 4 Dec 2019 at 04:00
This week, both the government and the Defence Ministry went on the defensive. They have publicly countered the opposition's quest to scrutinise the military's "off-budget" spending and abolish conscription. Unfortunately, their defence has ended up as almost indefensible -- simply because their arguments lack any real substance.
The issue of the Defence Ministry's 18-billion-baht off-budget spending was raised last Friday by Future Forward Party (FFP) leader Thanathorn Juangroongruangkit upon his resignation from a House committee investigating the 2020 fiscal budget.
This off-budget funding is off limits to public disclosure and scrutiny by the Lower House. There is no information provided by the ministry about who is authorised to use this money and precisely how it will be spent, Mr Thanathorn pointed out.
The FFP leader also revealed that the Financial and Fiscal Discipline Act of 2018, passed by the former military regime's lawmakers, provides a loophole for the Defence Ministry to seek an exemption from the Finance Ministry to have this budget bypass compliance with laws and regulations applicable to all other government agencies. Defence Ministry regulations also allow this kind of budget to be subject solely to internal audit.
Mr Thanathorn's quest for transparency drew a response from Deputy Prime Minister Prawit Wongsuwon who called for the public to trust the military and its senior soldiers, saying the budget expenditure has been overseen and audited internally by all levels of supervisors in the armed forces.
The money is used for defence personnel welfare benefits and on public services, he said. Nevertheless, this explanation of the budget's use was too vague and general to give the public a clear idea about what the money is needed for.
Instead of asking for trust, the government and the ministry should disclose the details of this line of spending to parliament and the public.
Also weak in substance was Defence Ministry spokesman Lt Gen Kongcheep Tantravanich's rebuke of the FFP's anti-conscription campaign. He said it is too soon for the country to put an end to conscription and rely instead on volunteer recruits. This could "affect national security", he insisted, adding it could result in a shortage of military personnel as the number of volunteers who apply each year is far from sufficient for the ministry's needs.
But what is insufficient is his explanation. Lt Gen Kongcheep should have come up with figures and detailed operational plans for the use of draftees. He should also have elaborated on what the imminent threats to national security are and explained in detail why existing professional soldiers, military hardware and weapons, and volunteer recruits are insufficient to deal with the potential threats.
The government and the military should stop relying on the same old, empty rhetoric whenever they mount a defence against these criticisms. They should start to commit themselves to the same level of transparency and accountability imposed on other ministries and agencies.
If all ministries and government agencies were able to enjoy such large amounts of off-budget spending without being subject to external audit and public disclosure, corruption, which is already rampant in the public sector, would be even more widespread.
The same goes for conscription. If all government agencies refuse any organisational restructuring or reform, in time they will become irrelevant and inefficient.
Bangkok Post editorial column
These editorials represent Bangkok Post thoughts about current issues and situations.
Email : ploenpotea@bangkokpost.co.th
Corrections Dept goes on fake news warpath
The Department of Corrections is looking to file charges against the administrator of the "Anny Anny Khon Thai Hua Jai Pi Tak Put" (Anny Anny with A Heart to Protect Buddhism) Facebook page, for disseminating fake news.

redpajama_set_name: RedPajamaCommonCrawl


There it is, the statue youve been waiting for!
The Nick Saban statue was installed Thursday afternoon outside Bryant-Denny Stadium. By tonight, about 100 fans had stopped by snapping photos, said security guards Jama Brown and Marcus Williams.
The two were assigned to guard the statue from 6 p.m. until 6 a.m. Friday. Read more about their job in tomorrow's newspaper.
Hear Alabama student Jeremy Davis talk about his sketches that turned into the statue of Nick Saban set to be unveiled at Saturday's A-Day game. The video goes with this story that ran in Wednesday's paper.
Almost something to see here.
The long line of Nick Saban statue posts is about to end. Yes, it apparently exists and will be unleashed to the world before Saturday's A-Day game in Tuscaloosa.
I just got done speaking with UA student Jeremy Davis whose drawings and prototype sculpture inspired the final product that will stand in the "Walk of Champions" for eternity.
Saturday's unveiling will end a 16-month saga that took a few twists and turns before fading from the spotlight.
Read the paper tomorrow for the story behind the process of the statue's creation.
About to fly south, but there are a few Alabama football notes of interest.
First, UA formally announced the Nick Saban statue will be unveiled before the A-Day game April 16. Only a year late. Could be worse, I guess.
Secondly, senior linebacker Alex Watkins will miss the rest of spring practice with an unspecified knee injury.
Nothing to see here folks, still.
As you can see in this photo taken Tuesday, the statue is still either transparent or nonexistent.
We're currently checking to see if there's anything new on the Nick Saban statue mystery and will report any new developments as soon as they're available.
It's been a topic of much mystery but little talk. What's the deal with that statue of Nick Saban that was supposed to go up outside Bryant-Denny Stadium this fall?
One smart reporter (not me) asked Saban himself about the issue at this morning's news conference. He deferred to his wife.
No, that is just a light-hearted fan standing where Nick Saban's mysteriously absent statue was supposed to go back on A-Day. Now it's almost December and still nothing.
I'm on it tomorrow. I'll see what I can find out about this statue debacle and get the info to you ASAP.
Until then, have a nice Sunday.
Nice sitting area, not much to see.
This is the semi-regular no update update concerning the invisible statue of Nick Saban on the Walk of Champions.
Obviously, it isn't there and I haven't heard any time table for an unveiling. I'll check into that Monday and get back to you.
What about that Saban statue?
Nick Saban's invisible statue as seen Saturday afternoon.
I have no concrete update on the status of the still absent statue on the Walk of Champions.
It's just a picture of the still-empty pedestal that was supposed to be home to a likeness of Nick Saban after winning last season's national title.
I've left messages for Alabama officials who would have updates on the status of the monument, but none have been returned. I'll get back on that Monday since it seems to be an issue in the universe that is Alabama football.


Back in the 1990s I was just rediscovering what prog was all about and after spending much time perusing the old GEPR web site, I came across a band with a very lengthy discography that I had never heard of before. That band was called Camel. The descriptions made them out to be in the same league as our trusty favorites from that era: Genesis, Yes, ELP, Van der Graaf Generator, etc. As to why I had never heard of them is beyond me. Their music was never played on the radio in the 1980s. Their albums apparently went unnoticed to me as I perused used record stores all those years ago. Thanks to the Internet, another undiscovered gem has graced my ears.
After having found Camel on the Internet, finding them in CD stores was another story (maybe I just live on the wrong part of the planet?). But perseverance and a little luck brought me to Echoes, a 2-CD compilation released in 1993. Right from the first track I could tell I came across something special and right then I knew the descriptions of this band as being one of the best of the 1970s progressive rock era were absolutely true. As the disc continued to play, I became ever more enthralled with my discovery. Freefall, Lady Fantasy, Rhayader, Song Within a Song, and Lunar Sea (one of the greatest songs ever!). What an amazing band this was. Latimer and Bardens were quite a skillful keyboard/guitar duo. Their interplay was impeccable.
Then I started to hear the same signs that many of the prog greats went through. As the disc progressed and then onto the second disc, I literally saw the transformation of a symphonic prog band into a band coerced into making radio-friendly pop rock. In many ways, Camel's transformation was worst than Genesis or Yes. At least for them, I could stand their pop rock. Camel was a different animal altogether. For most of the second disc, it was truly a skipping affair in my CD player.
Even though the second disc of Echoes leaves much to be desired, I still think this is an absolutely wonderful introduction to Camel. After all, it lead me to start acquiring all the important Camel albums of the 1970s. Also the historical notes contained within the accompanying booklet are very well done. However, in the grand scheme of things this compilation isn't for everyone. It's primarily for the uninitiated and completionists. If you're familiar with Camel, I would instead recommend getting their albums from their 1973-1976 period, skip over their 1980s output and jump right into their 1990s-2000s period. For those of you who have never heard of Camel, Echoes is a good start. Three stars.


Every year, millions of manufacturing jobs are being lost in the United States. This not only means millions of families losing income, it means a loss in the country's ability to produce goods for its own use.
If the country is going to get back to the self-sufficient nation it once was not so long ago, then Americans need to step up and support American industry. It's as simple as that. Buying cheap things made in China or Taiwan does not give me pride of ownership. Does it you? We can't afford to lose the skills and technology that have made our country what it is.
Saves or create jobs for American citizens so they can support their families and the U.S. economy.
The money made by American companies stays in America, thereby supporting our economy at home.
Products made in American are held to safety standards not found in other countries so the products you buy are safer and usually more reliable.
Saves on energy. Products made in other countries must be imported on cargo ships. These ships travel thousands of miles using thousands of gallons of fuel.
Makes our country more self-sufficient. We need to maintain our manufacturing abilities and technology so we aren't dependent on foreign countries for everything.
Gives us pride in ownership.
These are just a handful of reasons to buy American-made products. Can you think of others?
Some people get confused as to what products really are made in America and how to find them. The internet is a great source for locating American-made products. And be sure to check the label of any product before you buy. It's not always possible to buy Made in the U.S.A., but each time you do you are helping your country.
Remember, when you have a choice, please buy American! Your purchase will keep money in the U.S. and support families and communities across the country.
There are several internet sites that do nothing but show USA made product categorized, to help you become an informed shopper. In addition, when purchasing products, spend a little extra time to look for a product which is American made. The internet has made it easier for us to search and learn about companies, and be informed buyers. Obviously, in today's world, it is tougher and tougher to buy 100% American made products. Many electronics, which we can't live without, are mostly not made in the USA.
Knowing that it is not possible to buy everything American made, the best we can do to help is to look and try to buy made in the USA product, whenever it is available.
Even those companies that have revolved around automation have found a new way to create American jobs. They are doing this by coming back to American soil, opening up new manufacturing plants, and creating new jobs within the manufacturing industry. Automation and robots can only do so much and with the rise in technology there is a new need for operators and people who know how to perform maintenance on these instruments.


Our ASE Certified Service Writer will ensure you personalized customer service, and accurate and knowledgeable information to help customize a repair plan to get you on the road as quickly and safely as possible. You can also deal directly with the owner for any of your car repair needs!
Our Master techs get it right the first time!
We are a Preferred Tire Rack Installer!
If you have damage to an alloy wheel , we can fix it! We can also take the bend out of any alloy wheel causing shaking or vibration— this will also extend the life of your tires and give you a smoother ride!
We offer 4-wheel alignments, according to dealer specifications, on all makes and models. We have the most advanced and state of the art wheel alignment equipment available. All alignments are done according to dealer specifications, or any custom specifications you provide.
Although we do not have tow service ourselves, we do partner with 2 tow companies to have your vehicle brought to our facility. Your tow charge will be automatically added onto your repair order, its that easy!
Galati's Foreign Car Service does not set rates and is not responsible for any issues occurring during the tow process.


New Zealand wide pick up and drop off.
Travel in style and pay budget campervan rates in our FULLY PIMPED Campervans!
If you are budget conscious have a sense of humour and style, Sweet As Campers are the perfect vehicles for you.
You will love our racy exteriors with fibreglass body kits and batwings. You'll be sure to turn heads on your travels! Interiors can include anything from disco balls, chandeliers, neon lights, purple velour and mirrored celings to ensure you travel fully pimped!
We offer you the freedom to pick up and drop off your campervan from most locations in New Zealand.


text:
// Copyright (c) .NET Foundation. All rights reserved.
// Licensed under the Apache License, Version 2.0. See License.txt in the project root for license information.

using System;
using System.Collections.Generic;
using Microsoft.Extensions.Configuration;
using Microsoft.Extensions.DependencyInjection;

namespace Microsoft.Extensions.Hosting
{
    /// <summary>
    /// A program initialization abstraction.
    /// </summary>
    public interface IHostBuilder
    {
        /// <summary>
        /// A central location for sharing state between components during the host building process.
        /// </summary>
        IDictionary<object, object> Properties { get; }

        /// <summary>
        /// Set up the configuration for the builder itself. This will be used to initialize the <see cref="IHostingEnvironment"/>
        /// for use later in the build process. This can be called multiple times and the results will be additive.
        /// </summary>
        /// <param name="configureDelegate">The delegate for configuring the <see cref="IConfigurationBuilder"/> that will be used
        /// to construct the <see cref="IConfiguration"/> for the host.</param>
        /// <returns>The same instance of the <see cref="IHostBuilder"/> for chaining.</returns>
        IHostBuilder ConfigureHostConfiguration(Action<IConfigurationBuilder> configureDelegate);

        /// <summary>
        /// Sets up the configuration for the remainder of the build process and application. This can be called multiple times and
        /// the results will be additive. The results will be available at <see cref="HostBuilderContext.Configuration"/> for
        /// subsequent operations, as well as in <see cref="IHost.Services"/>.
        /// </summary>
        /// <param name="configureDelegate">The delegate for configuring the <see cref="IConfigurationBuilder"/> that will be used
        /// to construct the <see cref="IConfiguration"/> for the application.</param>
        /// <returns>The same instance of the <see cref="IHostBuilder"/> for chaining.</returns>
        IHostBuilder ConfigureAppConfiguration(Action<HostBuilderContext, IConfigurationBuilder> configureDelegate);

        /// <summary>
        /// Adds services to the container. This can be called multiple times and the results will be additive.
        /// </summary>
        /// <param name="configureDelegate">The delegate for configuring the <see cref="IServiceCollection"/> that will be used
        /// to construct the <see cref="IServiceProvider"/>.</param>
        /// <returns>The same instance of the <see cref="IHostBuilder"/> for chaining.</returns>
        IHostBuilder ConfigureServices(Action<HostBuilderContext, IServiceCollection> configureDelegate);

        /// <summary>
        /// Overrides the factory used to create the service provider.
        /// </summary>
        /// <typeparam name="TContainerBuilder"></typeparam>
        /// <param name="factory"></param>
        /// <returns>The same instance of the <see cref="IHostBuilder"/> for chaining.</returns>
        IHostBuilder UseServiceProviderFactory<TContainerBuilder>(IServiceProviderFactory<TContainerBuilder> factory);

        /// <summary>
        /// Enables configuring the instantiated dependency container. This can be called multiple times and
        /// the results will be additive.
        /// </summary>
        /// <typeparam name="TContainerBuilder"></typeparam>
        /// <param name="configureDelegate"></param>
        /// <returns>The same instance of the <see cref="IHostBuilder"/> for chaining.</returns>
        IHostBuilder ConfigureContainer<TContainerBuilder>(Action<HostBuilderContext, TContainerBuilder> configureDelegate);

        /// <summary>
        /// Run the given actions to initialize the host. This can only be called once.
        /// </summary>
        /// <returns>An initialized <see cref="IHost"/></returns>
        IHost Build();
    }
}

redpajama_set_name: RedPajamaGithub


How can you buy a copy of The Milk Lady at New Park Farm?
The Milk Lady can be ordered and purchased from any good book shop.
P&P is FREE within the UK. To ship abroad there will be additional charges. Please contact us for further details.
We hope you enjoy your purchase. However, if for any reason you are unhappy with it, you can return it to the address below within seven working days for a full refund providing the purchase is in its original condition. We reserve the right to refuse returns if the product is received otherwise than in accordance with this requirement. The expense of the return postage and packing must be borne by the customer.


Lighter components and more efficient manufacturing technologies: These aims are in the focus of researchers of the Institut für Integrierte Produktion Hannover gGmbH (IPH), the Institute of Forming Technology and Machines (IFUM) and the Institute of Materials Science (IW).
IPH researchers develop a novel lightweight design concept for car body construction. They investigate tailored hybrid tubes. Here, steel and aluminium are joined by laser brazing and soldering processes and then shaped by hydroforming.
Researchers at IFUM and IW work on a new realistic stress test for clinched components, the first test which considers thermal as well as mechanical stress. Another research topic at IFUM is to design a tool which allows forming and joining at the same time: The aim is to weld components already during the deep drawing process. Moreover, the researchers search for an alternative to welding: Upset bulging is a method to join sheets and tubes mechanically.


With over 3,000 active members, BIMA is the only organization that brings together New England's online media players exclusively.
Our BIMA Board of Directors hail from the largest agencies, publishers and adtech companies in Boston. We leverage our network to help your brand make meaningful connections.
With all of Boston's agencies as part of our ecosystem, aligning your brand with BIMA will ensure that all eyes are on you – a sure way for business doors to open!
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