text:
\section{Introduction}

In this paper we apply complex network analysis to discrete, disordered,
deterministic dynamical systems. The set of all trajectories of such a system
can be described as a directed network. Each dynamical state is represented by
a node and linked to its unique temporal successor by a directed link, giving
the state space network (SSN).  Thus the out-degree of a node is one.  The
irreversibility of the dynamics implies that a node can potentially have many
in-coming links.  The number of in-coming links at a node, or its in-degree,
can vary from node to node, depending on the dynamical system considered.  A
wide dispersity of degrees characterizes many complex
networks~\cite{NewmanSIAMReview}. Therefore, complex network analysis may
offer a useful alternative to traditional analyses of dynamical systems, such
as the characterization of spatiotemporal patterns~\cite{NKS,
  Peter1986IntJPhys, Badii, Bialek,Crutchfield}. The first results of such an
analysis on one dimensional cellular automata (CA) showed that heterogeneity
of the SSNs at both the local and global scales distinguishes ``complex''
dynamics from simple dynamics~\cite{Shreim2007CA}.  Here we exploit complex
network theory to characterize {\it disordered} dynamical systems by examining
SSNs for ensembles of random Boolean networks.

Random Boolean networks (RBNs) were introduced by Kauffman~\cite{Kauffman1969}
as models of gene regulation and have been extensively studied over the years
by physicists as examples of strongly disordered
systems~\cite{drosselRBNreview, aldanaRBN2003,
  derridaStauffer1986,flyvbjerg1988opn,BastollaParisi}.  The dynamics of each
of the $N$ Boolean elements in an RBN is given by a Boolean function of $K$
randomly chosen input elements. Different realizations of the input elements
and the Boolean functions for an RBN lead to an ensemble of RBNs for a given
$(N,K)$.

In the thermodynamic ($N\rightarrow \infty$) limit RBNs exhibit a phase
transition between chaotic and frozen phases passing through a critical phase
for $K=2$~\cite{DerridaPomeau}. In the frozen phase, $K<2$, the Hamming
distance between two perturbations of the same state quickly die out.  On the
other hand, in the chaotic phase, $K>2$, perturbations grow exponentially in
time.  Substantial analytical work has focused on the number of attractors as
well as their lengths~\cite{Drossel2005PRE, Drossel2005PRL,samuelsson2003RBN},
which have been found exactly for $K=1$ ~\cite{flyvbjerg1988esk}. Krawitz and
Shmulevich~\cite{ShmulevichEntropyRBN} computed the entropy of basin sizes and
showed that for critical RBNs it scales with the system size.  Otherwise, it
asymptotes to a constant.

In unrelated developments, a variety of statistical methods have been
established to characterize the structure of complex networks. Measures
include probability distributions for the node degree, clustering (the
tendency of nodes that share a common neighbor to also be linked to each other
directly), motifs (overrepresented subgraphs in the network),
etc.~\cite{NewmanSIAMReview, AlbertBarabasiRevModPhys, AlonMotifs2002}.  Many
real world networks such as regulatory networks~\cite{AlbertCellScience}, the
world-wide web~\cite{BarabasiWWW}, or the correlation structure of
earthquakes~\cite{MayaNetworksEarthquakes,MayaNetworksEarthquakes2}) differ
markedly from a random graph -- where the degree distribution is Poisson and
clustering is absent. They often display ``fat-tailed'' or even scale-free
degree distributions.

In this paper we show that strong deviations from random graph behavior also
occur in the SSNs of RBNs.  In particular, for sufficiently small $K$, the
probability distribution for the number of incoming links to nodes in the SSN,
$P(k)$, has a fat-tail. In contrast, a randomly rewired null model for the SSN
({\it i.e.}, a random map) has Poissonian in-degree distributions. Also for
small $K$, the maximum in-degree of any node in the SSN, $k_{max}$, displays
scaling behavior with respect to size of the SSN, ${\cal N}=2^N$.  While
$k_{max}$ is a measure of local heterogeneity, we use the path diversity,
${\cal D}$, to characterize global heterogeneity~\cite{Shreim2007CA}. We show
that ${\cal D}$ grows linearly with $N$ for $K=1$, while it grows faster than
linearly with $N$ for $2 \leq K \leq 5$. For $K\geq 6$, ${\cal D}$ scales with
SSN size ${\cal N}$.  In addition, for $K=2$ (where RBNs are critical) the
sample to sample fluctuations of ${\cal D}$ are the largest and might be
non-self-averaging~\cite{chamati2002fss, wiseman1998fss}.  We speculate that
SSN fluctuations at these three different scales (local, global, and sample to
sample) for $K=2$ RBNs are associated with criticality in the thermodynamic
limit ${\cal N} \rightarrow \infty$.

\subsection{Summary}

In Section~\ref{def} we discuss the procedure used to construct RBNs, and their
SSNs. The in-degree and the path diversity are also defined.  In
Section~\ref{indeg}, we present results for the behavior of the nodes'
in-degrees in an ensemble of SSN.  In Section~\ref{K1}, we discuss the SSNs
for $K=1$ RBNs, which have unique features not shared by other $K$s. In
Section~\ref{diversity}, we examine the behavior of the path diversity.
Discussion of our results and concluding remarks are found in
Section~\ref{discussion}.

\section{Definitions}
\label{def}

An RBN consists of $N$ Boolean variables $\sigma_i \in {0,1}$ with $ i=1 \cdots
,N$.  The dynamics of each element is determined by a Boolean function of $K$
randomly chosen input elements,
\begin{equation}
  \sigma_i(t+1)   = f_i(\sigma_{i_1}(t),\sigma_{i_2}(t),\ldots,\sigma_{i_K}(t))  \quad ,
  \label{function}
\end{equation}
where $\sigma_{i_j}(t)$ is the value of the $j^{th}$ input to $\sigma_i$ at
time $t$. The function $f_i$ is randomly chosen to be $1$ with probability $p$
and $0$ with probability $1-p$ for each set of values of its arguments.  We
consider only the case of unbiased RBNs with $p=1/2$, except when $K=1$, where
we study two different cases. For $K=1$ there are four possible boolean
functions for each element. Instead of choosing them uniformly, using only the
copy ($f(\sigma) = \sigma$) or the invert function ($f(\sigma) = {\rm
  NOT}(\sigma)$) for each $\sigma_i$ (with equal probability), leads to a
\emph{critical} $K=1$ RBN~\cite{drosselRBNreview}.

We use synchronous update for the dynamics, {\it i.e.} all the Boolean
elements in the network are updated in parallel at each time step. The
networks are set up by choosing $K$ different random inputs for each
$\sigma_i$.  While allowing self-connections, we do not allow multiple
connections. We do not impose any connectivity constraint on the RBNs.

An RBN with $N$ elements has ${\cal N} =2^N$ different dynamical states.
These states are nodes of a directed network.  A link from ${\bf A}$ to ${\bf
  B}$ indicates that ${\bf A}$ evolves to ${\bf B}$ in one time step, making
${\bf B}$ the image of ${\bf A}$, or ${\bf A}$ a pre-image of ${\bf B}$.  This
directed network forms the state space network (SSN).The SSN typically
consists of disconnected clusters, or basins of attraction.  Each basin
contains transient states, which are visited no more than once on any
dynamical trajectory, and attractor states that may be visited infinitely
often.  Garden of Eden states (GoE) are transient states that cannot be
reached from any other state, {\it i.e} they have no pre-images. Examples of
some state space clusters are shown in Fig.~\ref{networks} for $N=9$ and
different values of $K$.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/fig1}
\caption{\label{networks}(Color online).  One connected cluster of an SSN for
  RBNs with $N=9$ and different values of $K$ displayed using the program
  'Pajek'~\cite{Pajek}.  Nodes on the attractors are drawn in red; the other
  colors indicate distance from the attractor. For instance, for $K=1$, green
  nodes are distance one from the attractor, blue nodes are distance two and
  magenta nodes are distance three. Note that for $K=1$ all nodes are either
  Garden of Eden states or hubs with all the hubs having the the same
  in-degree.  RM stands for a random map.}
\end{center}
\end{figure}

Random maps are the limit of RBNs when $N,K \rightarrow
\infty$~\cite{DerridaFlyvbjerg}.  By construction, the state space of a random
map forms an Erd{\"o}s-Reyni random graph with a Poisson in-degree
distribution, $P(k)=e^{-1}/k!$ with mean $\langle k \rangle = 1$. We
constructed random map SSNs by picking the image of each node uniformly
randomly from the ${\cal N} =2^N$ possible nodes.

Identifying all attractors states and lumping them into a single node turns
the SSN into a rooted tree.  To study the global heterogeneity of this tree,
we use path diversity $\cal D$. It quantifies topological variations in the
paths connecting the GoE states to the root. It is similar to other global
measures of tree like structures such as tree diversity~\cite{Huberman} and
topological depth~\cite{Badii}. ${\cal D}$ measures the number of
non-equivalent choices encountered when following each reversed path from the
root to the GoE states.

Specifically, $\cal D$ is computed as follows: each GoE state is assigned a
diversity equal to one; a state with a single pre-image has the same diversity
as its unique pre-image; and the diversity of a node with more than one
incoming link is the sum of all \emph{distinct} diversities of its pre-images
plus one. For instance, if a node has in-degree 6, and three of its pre-images
have diversity 4, two have diversity 5, and the last has diversity 8, then
that node's diversity is $4+5+8+1=18$. Finally, the path diversity ${\cal D}$
of the entire SSN is the diversity of the root.

Table~\ref{TableOfNotation} provides a summary of the notation used. Note that
we refer to nodes of the RBNs as ``elements'' and otherwise reserve the phrase
``node'' only for SSNs.

\begin{table}[!h]
\begin{center}
\begin{tabular}{|l||l|}
\hline
RBN              & random Boolean network       \\
\hline
$N$              & number of elements of an RBN  \\
\hline
$K$              & number of inputs to each element of an RBN \\
\hline
SSN              & state space network of an RBN \\
\hline
$\cal N$         & number of nodes in an SSN (${\cal N}=2^N$) \\
\hline
$k$              & in-degree of a node in an SSN \\
\hline
$k_{max}$         & largest in-degree of a node in an SSN \\
\hline
$\cal D$         & path diversity \\
\hline
\end{tabular}
\caption{\label{TableOfNotation} Notations used in this paper.}
\end{center}
\end{table}

\section{Results}

\subsection{In-Degree}
\label{indeg}

An elementary description of local heterogeneity of a network is its degree
distribution.  We computed in-degree distributions, $P(k)$, of the SSNs. These
SSNs were obtained for RBNs with $1 \leq K \leq 10$ and $10 \leq N \leq 24$,
as well as random maps of system sizes $10 \leq N \leq 24$. For $K=1$ we
distinguish between RBNs constructed using all four boolean functions and the
critical $K=1$ RBN discussed earlier.  In Section ~\ref{K1} we discuss the
in-degrees for $K=1$ RBNs, where we obtain exact analytic results. We present
in Section ~\ref{K2} numerical results for $K>2$ RBNs.

\subsubsection{K=1 Networks}
\label{K1}

We start with the case of $K=1$ critical RBNs, where all functions
$f_i(\sigma_{i_1})$ are either copy or invert. If none of the boolean elements
is a leaf on the RBN, then the value of every element $\sigma_i$ at the next
time step is determined by the current value of of some element $\sigma_j$. If
the state $\bf{S}$ differs from $\bf{S}'$ in the value of $\sigma_i$, their
pre-images will differ in the value of $\sigma_j$.  Thus on a $K=1$ critical RBN
without leaves, the mappings are one-to-one, there are no GoE states and each
hub has in-degree one.

When the RBN contains $L$ leaves in addition to $Q$ non-leaf elements, the
state of the $L+Q$ elements can be written as a concatenation,
\begin{equation}
\bf{S} = \bf{L} \oplus \bf{Q} \label{concatenation}.
\end{equation}
If $\sigma_i$ is among the $L$ leaves, it cannot effect the next step value of
any element. The next value of $\sigma_i$ itself is determined by the current
value of a non-leaf element $\sigma_j$.  The image of $\bf{S}$ under the RBN
rules will then be of the form,
\begin{equation}
{\cal I}(\bf{S})={\cal F}(\bf{Q}) \oplus {\cal G}(\bf{Q}),
\end{equation}
where ${\cal F}$ gives the next-step state of the leaves, and is a function
only of the non-leaves. This means that at least $2^L$ states that differ only
in the values of the $L$ leaf elements will be mapped to the same image,
{\it i.e.} the in-degree of each non-GoE state (``hubs''), is $k_h \geq 2^L$.
To see that $k_h=2^L$, we note that
\begin{eqnarray}
\label{notsame}
{\cal I}({\bf S}')\neq{\cal I}({\bf S})  {~\rm if}  {~\bf Q}'\neq {\bf Q}.
\end{eqnarray}
Consider two states ${\bf Q}$ and ${\bf Q}'$ of the non-leaf elements that
differ in the state of the element $\sigma_a$.  If $\sigma_a$ is an input to
at least one non-leaf element, then ${\cal G}(\bf{Q}')\neq{\cal G}(\bf{Q})$.
If $\sigma_a$ is an input to at least one leaf element, then ${\cal
  F}(\bf{Q}')\neq{\cal F}(\bf{Q})$.  Since $\sigma_a$ is not a leaf, at least
one of these two cases has to be true, proving Eq.~(\ref{notsame}).  Thus the
cardinality of the set $\{{\cal I}({\bf S})\}$ is at least equal to the
cardinality of the set $\{{\bf Q}\}$, which is $2^Q$. If for each hub
$k_h>2^L$, the sum of hub in-degrees $\geq 2^Q 2^L = 2^{Q+L} = 2^N$. Since
there are only $2^N$ nodes in the SSN, this inequality has to be saturated,
giving us that $k_h = 2^L$.

The boolean function used for $K=1$ critical RBNs are either copy or invert.
For the general $K=1$ case, there are four possible functions, two of which
are copy and invert. The other two are constant functions, which map all
aruguements to one value (either zero or one). If an element $\sigma_i$ is
input only to elements with constant functions, it behaves effectively like a
leaf. We will refer to such elements along with the leafs, as effective
leaves.  Using this observation we can extend the above argument for $K=1$
critical RBNs to the general case of $K=1$ RBNs. Thus all hubs in the SSN of a
$K=1$ RBN have the same in-degree of the form $k_h=2^l$, where $l$ is the
number of effective leaves.  However, $k_h$ will vary over the different
realizations of the RBN.

Construction of an instance of a $K=1$ RBN can be considered as $N$ rolls of
an $N$-faced die.  Each face represents an element. The outcome $j$ of the
$\tau$-th roll is the input to the $\tau$-th element of the RBN. The function
$f_{\tau}$ in Eq.~(\ref{function}) that determines the updates of the
$\tau$-th element is chosen randomly over the function's arguments, $\sigma_j$
in the present case.  Since for $K=1$ there is only one input, $f_\tau$ will
be a constant function if it maps both values of the input (0,1) to the same
value, either $1$ (with probability $p^2$) or $0$ (with probability
$(1-p)^2$).  Therefore, the probability that the chosen function is not a
constant function is $q=1-p^2 - (1-p)^2$. We flip a coin, after each roll, to
determine if the function $f_{\tau}$ is a constant function or not.  If it is
not a constant function, we mark the displayed face of the die, and keep track
of the number of marked faces until $\tau=N$.  The marked faces exclude the
candidates for effective leaves. After $N$ die rolls and coin flips, the
number of marked faces will be $m = N-l$, where $l$ is the the number of
effective leaves in the constructed RBN. The distribution of $m$ evolves from
the $\tau$-th to the $\tau+1$-th roll according to,
\begin{equation}
M_{\tau+1,m}= q\left(1-\frac{m-1}{N}\right)M_{\tau,m-1}+\left(1-q(1-\frac{m}{N})\right)M_{\tau,m},
\label{dist-marked}
\end{equation}
under the boundary conditions that
\begin{eqnarray*}
	M_{\tau,m}=0, {~\rm if} ~\ m<0 {~\rm or} ~\  m>\tau,
\end{eqnarray*}
and the initial condition $M_{0,0}=1$.

 Thus $q=1$ in Eq.~(\ref{dist-marked}) generates the
distribution of leaves for $K=1$ critical RBNs. In this case, the solution to
Eq.~(\ref{dist-marked}) is
\begin{equation}
\label{q1hubexact}
	M_{N,m} = \frac{1}{N^N}{N\choose m}C_{N,m}m!,
\end{equation}
where $C_{N,m}$ are Stirling numbers of the second kind, and enumerate
partitions of an $N$-set into $m$ non-empty subsets \cite{abramowitz1965hmf}.
For $q\neq1$ analytical expressions are hard to obtain. However, solving
Eq.~(\ref{dist-marked}) numerically is straightforward.

We can use the distribution of $m$ to generate the distribution of ${\rm
  log}_2 k_h = N-m$.  In Figs. ~\ref{PDFlogkmaxK1crit} and ~\ref{PDFlogkmaxK1}
we present the distribution of logarithm of hub in-degrees ${\rm log}_2 k_h$
for the $K=1$ critical and $K=1$ non-critical RBNs respectively. The
hub-degrees were obtained for an ensemble of $2,000$ randomly chosen RBNs with
$10 \leq N \leq 24$. We also show our analytical results obtained from
Eq.~(\ref{dist-marked}) with $q=1$ for the $K=1$ critical RBNs and with
$q=1/2$ for the $K=1$ RBNs.

\begin{figure}[htbp]
\begin{center}
	\includegraphics*[width=8cm]{Figures/K1critkmaxwAna.eps}
        \caption{\label{PDFlogkmaxK1crit}(Color online). The PDF $P(\log_2 k)$
          of the log of the hub in-degree, $\log_2 k_h$ for $K=1$ critical RBNs.
          The three dashed lines with empty symbols were obtained using
          Eq.~(\ref{dist-marked}) with $q=1$.}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
	\includegraphics*[width=8cm]{Figures/K1kmaxwAna.eps}
        \caption{\label{PDFlogkmaxK1}(Color online). The PDF $P(\log_2 k)$
          of the log of the hub in-degree, $\log_2 k_h$ for $K=1$ RBNs. The
          three dashed lines with empty symbols were obtained using
          Eq.~(\ref{dist-marked}) with $q=1/2$.}
\end{center}
\end{figure}

We can use Eq.~(\ref{dist-marked}) to obtain the mean number of
effective leaves,
\begin{eqnarray*}
\langle m \rangle_{\tau+1}&\equiv& \sum_{m=0}^{\tau+1} m M_{\tau+1,m}\\
&=& \sum_{m=0}^{\tau+1} m \left[q\left(1-\frac{m-1}{N}\right)M_{\tau,m-1}\right]\\
&+&\sum_{m=0}^{\tau+1} \left[\left(1-q(1-\frac{m}{N})\right)M_{\tau,m}\right],
\end{eqnarray*}
which reduces to,
\begin{eqnarray*}
\langle m \rangle_{\tau+1} = q + \left(1-\frac{q}{N}\right)\langle m \rangle_{\tau}.
\end{eqnarray*}
This recursion gives the solution for the mean number of
effective leaves $N-\langle m \rangle_N$, which is also the mean of ${\rm
  log}_2 k_h$,
\begin{eqnarray}
\label{K1meanleaves}
\langle {\rm log}_2 k_h \rangle &=& N\left(1-\frac{q}{N}\right)^N\to N e^{-q}~\ {\rm as}~\ N\to\infty.
\end{eqnarray}

\subsubsection{Networks with $K\geq 2$}
\label{K2}
For $K>1$ RBNs the boolean functions are not the simple copy/invert and
constant functions as in the case of $K=1$. For example, consider the truth
table in Table ~\ref{funcegg}. $\sigma_1$ and $\sigma_2$ are inputs to
$\sigma_3$ in a $K=2$ RBN. $\sigma_2$ effects the value of $\sigma_3$ only
when $\sigma_1 = 1$.  Conversely, $\sigma_1$ effects the value of $\sigma_3$
only when $\sigma_2 = 1$. Such interactions between the inputs for $K>1$ RBNs
make it harder to extend the analytical aruguements that we applied to the
$K=1$ case.

\begin{table}[!h]
\begin{center}
\begin{tabular}{|l||l|}
\hline
$\sigma_1$   $\sigma_2$              & $\sigma_3$\\
\hline
0  ~\  0 & 0\\
\hline
0   ~\ 1 & 0\\
\hline
1   ~\ 0 & 0\\
\hline
1   ~\ 1 & 1\\
\hline
\end{tabular}
\caption{\label{funcegg} A function for $K=2$ RBNs}
\end{center}
\end{table}

Fig.~\ref{PDFInDegN18} presents $P(k)$ for SSNs with $N=18$ (${\cal N} =
2^{18}$).  It is broad for $K = 1$, becomes narrower with increasing $K$, and
converges for $K\to N$ to the in-degree distribution of the random map.  The
random map in-degree distribution itself converges to a Poissonian with
$\langle k \rangle = 1$ when $N \to \infty$.  Note that $k$ for $K=1$ is
either zero for the GoE states, or takes the same power of two value for the
hubs.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/PDFInDegN18.eps}
\caption{\label{PDFInDegN18}(Color online). The in-degree distribution
  function, $P(k)$, for various connectivities, $K$, of the RBN, on a log-log
  scale. The distribution is computed for RBNs of size $N=18$ by averaging the
  PDFs over $200$ different realizations for each $K$. We used $k+1$ instead
  of $k$ on the $x$-axis, so that we could also show values for $k=0$. The
  black solid line shows a Poissonian with $\langle k \rangle = 1$ for
  comparison. $K=1C$ refers to $K=1$ critical RBNs.}
\end{center}
\end{figure}

The in-degree distribution $P(k)$ does not exhibit easily described behavior
such as scaling.  However, the results of Ref.~\cite{Shreim2007CA} suggest
that a more useful quantity is the largest in-degree $k_{max}$.  Scaling
behavior in $k_{max}$ was shown to be a necessary but not sufficient signature
of complex dynamics.  Fig.~\ref{checklinearmeanlogkmax}a shows that
\begin{equation}
  \langle \log_2 k_{max} \rangle \sim \nu_K N {~\rm for } ~\ K \leq 6,
         \label{eq2}
\end{equation}
where the angular brackets indicate an average over different realizations of
the RBN.  The exponents $\nu_K$ appear to obey the relation:
\begin{equation}
\nu_K = -0.07 K + 0.68 {~\rm for } ~\ 1 \leq K \leq 6,
\end{equation}
where the error is $\pm 1$ in the last digit. For the system sizes studied, it is not
possible to determine the asymptotic limit for larger $K$. However, the
behavior for large $K$ approaches the random map result.

For the random map, $P(k)$ tends to a Poisson distribution when $N \to
\infty$. If $\cal N$ different values of $k$ are chosen from a distribution
$P(k)$, the expected maximum $k_{max}$ can be related to $\cal N$ by
\begin{equation}
\label{kmaxrmeq}
\sum_{k=k_{max}}^{\infty} P(k) = \frac{1}{\cal N}.
\end{equation}
For a Poissonian with mean one,
\begin{eqnarray*}
\sum_{k=k_{max}}^{\infty} \frac{e^{-1}}{k!} = \frac{1}{\cal N}.
\end{eqnarray*}
This gives the following bounds on $k_{max}$:
\begin{eqnarray*}
  \frac{e^{-1}}{k_{max}!} \leq \frac{1}{\cal N} \leq \frac{e^{-1}}{k_{max}!} \sum_{m=0}^{\infty} \frac{1}{(k_{max})^{m}}.
\end{eqnarray*}
The sum in the last term in the equation evaluates to $1 - 1/k_{max}$ which
goes to one for large $k_{max}$, allowing us to write
\begin{eqnarray*}
\frac{e^{-1}}{k_{max}!} \approx \frac{1}{\cal N}.
\end{eqnarray*}
Taking logarithms and using Stirling's approximation, the above equations
yields for large ${\cal N}$
\begin{equation}
  k_{max}\ln k_{max} \approx  \ln{\cal N} = N\ln 2.
\end{equation}
Finally we take logarithms again to get,
\begin{equation}
  \log_2 k_{max} \approx  \log_2 N.     \label{kmaxN}
\end{equation}

As shown in from Fig.~\ref{checklinearmeanlogkmax}a, $\langle \log_2 k_{max}
\rangle$ is very well described by Eq.(\ref{kmaxN}) for random maps and RBNs
with large $K$ in the limit $N\to\infty$.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/checklinearitymeanlogkmax.eps}
\begin{center}
(a)
\end{center}
\includegraphics*[width=8cm]{Figures/checklinearitylogmeankmax.eps}
\begin{center}
(b)
\end{center}
\caption{\label{checklinearmeanlogkmax}(Color online). (a) The ratio
  $\frac{\langle \log_2 k_{max} \rangle}{N}$ is plotted against $N$. (b) The
  ratio $\frac{\log_2 \langle k_{max} \rangle}{N}$ is plotted against $N$.
  Both figures show systematic deviations from scaling for $K>6$.  The solid
  lines in (a) and (b) are $ \frac{\log N}{N} $ and were plotted for
  comparison. The statistics, in this figure and
  Fig.~\ref{meanlogkmaxDivlogmeankmax} were obtained by sampling $2000$
  different realizations for each $N$ and $K$.}
\end{center}
\end{figure}

In order to see whether $k_{max}/N$ is self averaging, {\it i.e.} whether the
fluctuations of $k_{max}/N$ become negligible for large $N$, we also
plot in Fig.~\ref{checklinearmeanlogkmax} the ratio $N^{-1}\log_2 \langle
k_{max} \rangle$. A first look at Fig.~\ref{checklinearmeanlogkmax} might
suggest that both ways of averaging lead indeed to the same results, and
self-averaging is satisfied. That this is not the case is demonstrated in
Fig.~\ref{meanlogkmaxDivlogmeankmax}, where we show the ratio $\langle \log_2
k_{max} \rangle/\log_2\langle k_{max} \rangle$ versus $N$.  We see from this
figure that $\langle k_{max} \rangle$ scales with ${\cal N}$ as $\langle
k_{max} \rangle \sim {\cal N}^{\tau_K}$ for $K \leq6$, {\it i.e.}  $\log_2
\langle k_{max}\rangle \sim \tau_K N$, similarly to Eq.(\ref{eq2}). However,
$\tau_K \neq \nu_K$. For large $K$ it seems that that the fluctuations are
less important, $\langle \log_2 k_{max} \rangle/\log_2\langle k_{max} \rangle
\to 1$ for $K\to\infty$.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/meanlogkmaxDivlogmeankmaxVsNallKs.eps}
\caption{\label{meanlogkmaxDivlogmeankmax} (Color online). The ratio between
  the mean of the log and the log of the mean of the largest in-degree,
  $y\equiv \frac{\langle \log_{2} k_{max} \rangle}{\log_{2} \langle k_{max}
    \rangle}$, is plotted against the size of the RBN, $N$. If the two
  quantities were scaling with the same exponent, all curves should tend to
  $y=1$ for large $N$. The plot shows that $\langle \log_{2} k_{max} \rangle$
  and $\log_{2} \langle k_{max} \rangle$ are scaling with different
  exponents for $K \leq 6$.}
\end{center}
\end{figure}

The sample-to-sample fluctuations in $k_{max}$ are captured by its probability
distribution, shown in Fig.~\ref{RescaledPDFkmaxK246}.  We plot the
probability distribution of $\log k_{max}$ multiplied by its standard
deviation, $\sigma(\log_2 k_{max})$. The mean and standard deviation were
computed independently for each $N$ and $K$.  The plots are well-fitted by a
Gaussian for $2 \leq K \leq 6$, which suggests that $P(k_{max})$ is a
log-normal, for sufficiently large $N$. For $K>6$, the plots indicate
deviations from log-normal behavior.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/RescaledPDFlogkmaxK246.eps}
\begin{center}
(a)
\end{center}
\includegraphics*[width=8cm]{Figures/RescaledPDFlogkmaxK79RM.eps}
\begin{center}
(b)
\end{center}
\caption{\label{RescaledPDFkmaxK246}(Color online). The rescaled PDFs of the
  log of the largest in-degree $P(\log_2 k_{max})$ for (a)$K=$ 2, 4, and 6,
  and (b) $K=$ 7, 9, and the random map. The dashed lines are Gaussian
  distributions with mean zero and variance one. The plots indicate that
  $k_{max}$ is distributed according to a log normal distribution for $K=$ 2,
  4 and 6. Similar results hold for $K =$ 3 and 5.  Deviations from the
  Gaussian distribution are seen for $K>6$.  In (a) the distributions for
  $K=4$ and $K=6$ were offset by $7.5$ and $15$ units on the x-axis for
  clarity of presentation. In (b) the distributions for $K=9$ and the random
  map were offset by $10$ and $20$ units.}
\end{center}
\end{figure}

\subsection{Path Diversity}
\label{diversity}

Scaling of $k_{max}$ with system size for $K\leq6$, indicates asymptotic local
heterogeneity of the SSNs. As shown in Ref.~\cite{Shreim2007CA} local
heterogeneity is often insufficient to distinguish simple from complex
dynamics.  A complimentary global topological measure, the path diversity,
detects global heterogeneity in SSNs. In Ref.~\cite{Shreim2007CA} it was found
that complex cellular automata show scaling behavior in both local and global
measures. Here we investigate the path diversity of SSNs of RBNs.

Fig.~\ref{meanlogD} suggests that $\langle \log_2{\cal D} \rangle \sim \zeta_K
N$ for $K \geq 5$ and for the random map. However, the plots are not
conclusive as to whether or not $\langle \log_2{\cal D} \rangle$ becomes
linear in $N$ for $K<5$, for large system sizes, or with large finite size
corrections. Fig.~\ref{DerivmeanlogDVslogN} shows $\Delta_{\log_2N} \langle
\log_2{\cal D} \rangle$, the discrete derivative of $\langle \log_2{\cal D}
\rangle$ with respect to $\log_2N$. For $K=1$, the figure indicates that
$\langle \log_2{\cal D} \rangle \sim \log_2 N$. A faster than logarithmic
growth of $\langle \log_2{\cal D} \rangle$ is seen for $K=2$.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/meanlogDDivN.eps}
\caption{\label{meanlogD} (Color online). Log path diversity as a function of
  the system size. $\langle \log_2{\cal D} \rangle/N$, is plotted as a
  function of $N$, for various values of $K$ and the random map. For $K \geq 5$,
  $\langle \log_2{\cal D} \rangle$ asymptotes to $\zeta_K N$, while the plots
  have the opposite curvatures for $K\leq4$.}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/DerivmeanlogDVslogN.eps}
\caption{\label{DerivmeanlogDVslogN}(Color online). The discrete derivative of
  $\langle \log_2{\cal D} \rangle$ with respect to $\log_2N$,
  $\Delta_{\log_2N} \langle \log_2{\cal D} \rangle$, is plotted against
  $\log_2N$ for $K = 1,2$ and $K = 1$ critical. $\Delta_{\log_2N} \langle
  \log_2{\cal D} \rangle$ is defined as $\frac{ \langle \log_2{\cal D}_i
    \rangle - \langle \log_2{\cal D}_{i-1} \rangle}{\log_2N_i -
    log_2N_{i-1}}$.  A horizontal line of this plot indicates a relationship
  of the form $\langle \log_2{\cal D} \rangle \sim \log_2 N$. For the system
  sizes studied, the data suggests a logarithmic growth of $\langle
  \log_2{\cal D} \rangle$ with respect to $N$. For $K=2$, $\Delta_{\log_2N}
  \langle \log_2{\cal D} \rangle$ show growth trends which are indicative of a
  faster than logarithmic growth of $\langle \log_2{\cal D} \rangle$ with
  $N$. The data is inconclusive for $K=1$ critical.}
\end{center}
\end{figure}

An example of a simple tree that exhibits logarithmic behavior of path
diversity is the Cayley tree. On such an SSN each transient state has exactly
$z$ pre-images, except for the root which has $z+1$ pre-images. Given the
symmetry of the Cayley tree, all the nodes at the same distance from the root
will have the same path diversity. Furthermore, the path diversity is
incremented by one with each step towards the root. This makes the path
diversity equal to the depth of the tree which can be expressed in terms of
the number of nodes ${\cal N}$:
\begin{equation}
\label{diversityTree}
{\cal D} = \log_z\{(z -1){\cal N} + 1\}.
\end{equation}

For a maximally heterogeneous SSN each transient node will contribute to the
path diversity. The path diversity of an SSN is the diversity of the root.
Diversity of a node is the sum of all distinct diversities of its pre-images
plus one if it has more than one pre-image. Thus the diversity of a node is
bounded above by,
\begin{eqnarray*}
{\cal D}_i \leq 1+ \sum_{j \prec i} {\cal D}_j
\end{eqnarray*}
where $j \prec i$ indicates that $j$ is a pre-image of $i$. In the general
case, only distinct values of the diversities will contribute to the sum on
the right hand side. We can iteratively use this upper bound to calculate the
path diversity of an SSN. Denoting the root of the SSN by $\bullet$,
\begin{eqnarray*}
{\cal D}&\leq& 1 + \sum_{i_1 \prec \bullet}{\cal D}_{i_1}\\
&=&1 + \sum_{i_1 \prec \bullet} 1 + \sum_{i_2 \prec i_1}\sum_{i_1 \prec \bullet}{\cal D}_{i_2}\\
&=&1 + \sum_{i_1 \prec \bullet} 1 + \sum_{i_2 \prec i_1}\sum_{i_1 \prec \bullet} 1 +  \sum_{i_3 \prec i_2}\sum_{i_2 \prec i_1}\sum_{i_1 \prec \bullet}{\cal D}_{i_3}
\end{eqnarray*}
and so on. The subscript $h$ in $i_h$ represents the distance of the node
$i_h$ from the root. If $i_h$ is a GoE state, its diversity is defined to be
one. The first term on the right hand side of the last line counts the root,
the first sum counts the root's pre-images, and the two nested sums count the
pre-images of the root's pre-images. Iteratively continuing this calculation
we can see that the right hand side evaluates to ${\cal N}$, the size of the
SSN. Thus we have as the upper bound to the path-diversity,
\begin{equation}
{\cal D}\leq {\cal N}=2^N,
\end{equation}
or that,
\begin{equation}
{\rm log_2} {\cal D}\leq N.
\end{equation}

This bound is nearly reached for binary tress where single leaves branch off
at every branching point.

The logarithmic growth of $\langle \log_2{\cal D} \rangle$ for $K=1$ networks
suggests a different class of dynamical complexity than for $K\geq 2$
networks.  Meanwhile, random map-like-scaling of $\langle \log_2 k_{max}
\rangle$ for RBNs with sufficiently large $K$ suggests relatively simple
dynamics.  However, on the basis of those two measures, one cannot cleanly
distinguish the behavior of $K=2$ critical RBN from that of $K>2$ RBNs.  We
now show that sample-to-sample fluctuations in $\log_2{\cal D}$ enable such a
distinction.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/PDFlogDallKsN24.eps}
\caption{\label{PDFlogD} (Color online). The log of the PDF of the log of the
  path diversity, $P(\log_2{\cal D})$, for various values of $K$ and $N=24$.
  $P(\log_2{\cal D})$ is the broadest for $K=2$. $P(\log_2{\cal D})$ becomes
  narrow for large values of $K$.}
\end{center}
\end{figure}

We report the distribution of $\log_2 {\cal D}$, $P(\log_2{\cal D})$, in
Fig.~\ref{PDFlogD}.  For a fixed system size, $P(\log_2{\cal D})$ is broadest
for $K=2$ RBNs.  To quantify the width of $P(\log_2{\cal D})$, we study its
variance.  Fig.~\ref{varlogD} shows that the variance of $\log_2{\cal D}$,
$\sigma^2(\log_2{\cal D})$, grows fastest with $N$ for $K=2$.  Unlike $\langle
\log_2 k_{max} \rangle$ and $\langle \log_2{\cal D} \rangle$,
$\sigma^2(\log_2{\cal D})$ shows non-monotonic behavior as a function of $K$.
We interpret this as an indication that the criticality of $K=2$ RBNs can be
associated with large sample-to-sample fluctuations of its SSNs.

Generally, large sample-to-sample fluctuations in disordered media may lead to
the absence of self-averaging.  A canonical measure of self-averaging is the
ratio:
\begin{equation}
   {\cal R}({\cal D}) = \frac{\langle {\cal D}^2 \rangle - \langle {\cal D}
     \rangle^2}{\langle {\cal D}  \rangle^2}.
 \end{equation}
 A system is said to be self-averaging with respect to ${\cal D}$ if ${\cal
   R}({\cal D})$ goes to zero for large $N$~\cite{chamati2002fss,
   wiseman1998fss}; otherwise, it is said to lack self-averaging.  For the
 system sizes we are able to study, the evidence for or against self-averaging
 is not completely conclusive. Nevertheless, as indicated in the inset of
 Fig~\ref{varlogD}, $K=2$ RBNs show the largest values for ${\cal R}$ and are
 therefore the most likely to exhibit non-self-averaging behavior in the
 thermodynamic limit of large system size.

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{Figures/varlogDWithInset.eps}
\caption{\label{varlogD} (Color online). The variance of the log of the path
  diversity $\sigma^2(\log_2{\cal D})$ is plotted as function of the size of
  the RBN, $N$, for various values of $K$. $\sigma^2(\log_2{\cal D})$ shows
  non-monotonic behavior as a function of $K$, it grows the fastest for $K=2$.
  On the other hand, $\sigma^2(\log_2{\cal D})$ appears to tend to a constant
  or decreases with increasing $N$, for large values of $K$. For the random
  map, $\sigma^2(\log_2{\cal D})$ is a decreasing function of $N$. The inset
  shows a plot of ${\cal R}({\cal D}) = \frac{\sigma^2(\log_2{\cal
      D})}{\langle \log_2{\cal D} \rangle ^2}$ as function of $N$. The data
  for $K=2$ suggests that $\log_2{\cal D}$ might be non-self-averaging in the
  thermodynamic limit.}
\end{center}
\end{figure}

\section{Discussion and Conclusion}
\label{discussion}
We have studied the topology of state space networks (SSNs) for ensembles of
random Boolean networks.  Each dynamical state of the Boolean network
corresponds to a node in the SSN and is linked to its successor state.  We
characterize the heterogeneity of these SSNs at the local, node, scale by the
distribution of the in-degrees and the scaling of the largest in-degree.
Global heterogeneity over all paths in the SSN is characterized by the path
diversity. For elementary 1-d cellular automata, it was demonstrated in
Ref.~\cite{Shreim2007CA} that simultaneous scaling behavior in both $k_{max}$
and ${\cal D}$ indicates ``complex'' spatio-temporal dynamics (Wolfram class
IV and partly class III).  On the other hand, it was found in
\cite{Shreim2007CA} that one or both of these does not scale for CA in class I
or II -- which all have simple dynamics.

As is well-known, RBNs exhibit a phase transition between chaotic ($K> 2$) and
frozen ($K<2$) behavior.  $K=2$ RBNs are critical and therefore more complex
than RBNs with $K\neq 2$ which mainly show frozen ($K<2$) or chaotic ($K>2$)
behavior~\cite{OriginOrder}.  In fact, RBNs in the frozen phase (e.g. $K=1$)
resemble class II CA as they almost always rapidly go to one of many
attractors with a short period.  In addition, RBNs in the chaotic phase
resemble some class III CA in that they have long transients and attractors
with large periods.  We have investigated whether or not the different phases
of RBNs can be distinguished on the basis of a topological analysis of the
corresponding ensembles of SSNs, and the extent to which these phases overlap
in the behavior of their SSNs.

Many analytical results are known for $K=1$
RBNs~\cite{flyvbjerg1988esk,Drossel2005PRL} (and also for the random map,
\cite{DerridaFlyvbjerg}). In the present paper we derived analytical results
for the structure of the state space network for $K=1$ RBNs, and verified them
using numerical methods. We related the hub in-degree $k_h$ to $l$, the number
of effective leaves in the RBN. Simple arguments show that all the hubs have
the same in-degree $k_h = 2^l$. We also presented a stochastic process in
Eq.~(\ref{dist-marked}) that generates the distribution of $l$ for any $K=1$
RBN. For $K=1$ critical RBNs we could obtain an analytical expression for the
distribution of the hub in-degree $k_h$, and compared numerical solutions to
Eq.~(\ref{dist-marked}) with the simulations for the general $K=1$ RBNs.

Using Eq.~(\ref{dist-marked}) we derived that for $K=1$, ${\rm log}_2 k_h$
scales with the size of the RBNs as $\langle{\rm log}_2 k_h\rangle = e^{-q}N$,
where $q$ is the fraction of non-constant functions.  Numerically we find that
for $2 \leq K \leq 6$, $\log_2 k_{max}$ also exhibits scaling as a function of
the size of the RBN, $\langle \log_2 k_{max} \rangle \sim \nu_K N$.  The
scaling exponent $\nu_K$ is largest for $K=1$ and monotonically decreases for
larger values of $K$.  For $K=1$ the hub in-degree $k_h$ depends exponentially
on the number of effective leaves, $l$, which scales linearly with $N$. The
state of effective leaves is forgotten after the first time step during
evolution of the RBN. For $K\geq 2$, there are more than one inputs to each
element.  Whether the state of an element is forgotten or not depends on the
state of other elements in the RBN. Observed linear scaling of ${\rm log}_2
k_{max}$ with $N$ for $2\leq N \leq 6$ indicates an explanation similar to the
one we presented for $K=1$.  However, more analytical work is required to
fully understand the relationship between $k_{max}$ and the structure of the
RBN for $K>1$.  For larger $K$, we notice that the $K,N \to \infty$ limit of
the RBNs is the random map~\cite{DerridaFlyvbjerg}. We have shown analytically
that for the random map, $\log_2 k_{max} \sim \log_2 N$.

On the other hand, our numerical results indicate that $\langle \log_2{\cal D}
\rangle$ scales with the size of the SSN only for sufficiently large values of
$K$ ($K \geq 5$), $\langle \log_2{\cal D} \rangle \sim \zeta_K N$. $\zeta_K$
is largest for the random map and monotonically decreases with decreasing $K$
(which is the opposite of $\nu_K$).  The logarithmic scaling with $N$ of
$\langle \log_2{\cal D}\rangle$ for $K=1$ translates to a logarithmic scaling
of ${\cal D}$ with the size of the SSN, ${\cal N}$. This compares with the
path diversity of a Cayley tree in Eq.~(\ref{diversityTree}).  On an SSN with
Cayley tree structure, all nodes, except GoE nodes, have the same in-degree.
In fact we have shown the same for $K=1$ SSNs.  For $ 2 \leq K \leq 5$, it is
not clear from the data whether $\langle \log_2{\cal D} \rangle$ scales
linearly with $N$.  It is clear, however, that $\langle \log_2{\cal D}
\rangle$ grows faster-than-linear with $\log_2{N}$.

Together, the absence of scaling for $k_{max}$ and ${\cal D}$ correctly rule
out the random map (and most likely RBNs with large $K$) and $K=1$ RBNs, from
reporting high dynamical complexity.  However, neither of these measures
addresses sample-to-sample fluctuations, which are generally important in the
characterization of disordered systems.  We find that the probability
distribution function of $k_{max}$ converges to a log-normal distribution for
$2\leq K \leq 6$.  While this is also seen for $\log_2{\cal D}$, unlike the
monotonic dependence of the variance of $\log_2 k_{max}$, the variance of
$\log_2{\cal D}$ does not vary monotonically with $K$. In fact, the variance
grows the fastest with the size of the SSN for critical $K=2$ RBNs.  Numerical
results also suggest that $\log_2{\cal D}$ may possibly exhibit
non-self-averaging behavior for $K=2$.

Our results support the conclusion that heterogeneity in SSNs can be
associated with complex dynamics in disordered systems. $K=2$ RBNs are
distinguished from other RBNs by simultaneously exhibiting three kinds of
network heterogeneity.  The first is a local heterogeneity on the node level
indicated by the scaling of $\log_2 k_{max}$ with the size of the SSN.  The
second is a global heterogeneity on the trajectories level indicated by a
faster-than-linear growth of $\log_2{\cal D}$ with $\log_2N$. Finally, there
is a heterogeneity on the level of samples of RBNs, indicated by the fast
growth of the $\sigma^2(\log_2{\cal D})$ for RBNs with $K=2$.

\begin{acknowledgments}
  We would like to thank one of the referees for suggesting a way to prove
  that the in-degrees of a $K=1$ RBN only comes as a power of two. VS would
  like to thank Fanny Dufour for insightful discussions.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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The Lord requires that we return to Him in tithes and offerings a portion of the goods He has lent us. He accepts these offerings as an act of humble obedience on our part and a grateful acknowledgment of our indebtedness to Him for all the blessings we enjoy. Then let us offer willingly, saying with David: "All things come of Thee, and of Thine own have we given Thee." Withholding more than is meet tends to poverty. God will bear long with some, He will test and prove all; but His curse will surely follow the selfish, world-loving professor of truth. God knows the heart; every thought and every purpose is open to His eye. He says: "Them that honor Me I will honor, and they that despise Me shall be lightly esteemed." He knows whom to bless and who are deserving of His curse. He makes no mistakes, for angels are keeping a record of all our works and words.
Had some men of limited ideas been on the ground they would have opened their eyes in horror. Like Judas they would have asked: "To what purpose is this waste?" "Why not make everything in the cheapest manner?" But the sanctuary was not designed to honor man, but the God of heaven. He had given specific directions how everything was to be done. The people were to be taught that He was a being of greatness and majesty, and that He was to be worshiped with reverence and awe.


GVD Corporation (GVD) proposes to develop high performance, solventless and perfluorooctanoic acid (PFOA)-free, non-stick mold release coatings based on its novel polytetrafluoroethylene (PTFE) fluoropolymer technology. Most commercial mold release agents make use of organic solvents that, when liberated, during the drying process may have a significant air quality impact. The drying/curing process also may be energy intensive, requiring additional consumption of fossil fuels and the attendant emission of greenhouse gases. The environmental problems associated with these mold release agents and tighter regulations imposed on their use are generating urgency among manufacturers to identify green alternative release agents with the same level of performance.
GVD's green PTFE coatings offer such an alternative. PTFE offers an unparalleled combination of low surface energy (anti-stick), chemical resistance and thermal stability. GVD produces thin, conformal, environmentally benign PTFE coatings at low temperature via the novel initiated Chemical Vapor Deposition (iCVD) process. In this solvent-free process, GVD creates the pure PTFE fluoropolymer from reactive precursor vapors and deposits the PTFE as a coating at the same time. Using iCVD, GVD can optimize PTFE's surface energy, adhesion and durability during deposition. Reactive fluorocarbon vapors deposit uniformly across the mold surface, forming the lubricious PTFE release coating. Further, these PTFE coatings are applied at room-temperature and do not require curing or other post-processing (conventional wet-applied PTFE coatings require sintering at > 330°C). GVD's PTFE coatings already have shown excellent mold release performance in some niche applications. However, improved PTFE coating durability is needed when "stickier" polymers are being molded and demolded, such as those used in fabricating fiber-resin composites. GVD proposes to markedly improve the durability of its PTFE coatings, offering manufacturers an affordable, green and high-performance alternative to conventional solvent-based mold release agents.
In Phase I, GVD will conduct a series of experiments designed to improve the durability performance of GVD coatings. The most promising coating formulations will be downselected through a series of increasingly rigorous testing. Wet-applied mold release agents will be tested for comparison. By the end of Phase I, GVD will have generated PTFE-based coating formulations showing enhanced durability and release properties superior to that of conventional mold release products. In Phase II, promising recipes from Phase I will be subjected to further testing and process scale-up will be undertaken. Subsequently, mold release coatings will be commercialized through contract coating of mold elements followed by a license and equipment sale model once volumes increase.


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Three CleanFlake materials are now available on a thin rPET23 liner. The 'switchable' CleanFlake adhesive is designed to separate cleanly from PET bottles during the recycling process so that contamination of PET flakes is avoided – an important factor in ensuring that recycled PET can be recycled rather than downcycled. A fourth material – a high clarity ClearCut PP50 TOP CLEAR-S7000-rPET23 construction – is considerably thinner than today's market reference (PP60 with PET30), and offers high speed conversion and dispensing using the same thin rPET23 liner.
The rPET liner has been designed to convert in the same way as a conventional PET liner, with no noticeable differences in performance.


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Profligate spending on 'affordable'…
Profligate spending on 'affordable' housing in L.A.
A woman holds up a sign during a committee meeting at Los Angeles City Hall on the then-proposed developer linkage fee which would charge homeowners and developers $5 to $15 per square foot for new projects under a key affordable housing initiative proposed by Mayor Eric Garcetti. (Photo by David Crane, Los Angeles Daily News/SCNG)
By The Editorial Board | opinion@scng.com |
PUBLISHED: August 28, 2019 at 6:01 p.m. | UPDATED: August 28, 2019 at 6:01 p.m.
Good intentions, when exercised through government, have a way of costing a lot of money and delivering paltry results.
Such is the case with Los Angeles's effort to build permanent housing for people facing homelessness. USA Today recently reported on the high price tag of apartment units financed in part by a bond approved by L.A. voters.
The apartments, located in Koreatown, will have been completed at a cost of $690,692 for each unit. Two pending projects are expected to exceed $700,000 in costs per unit.
As is often the case, City Controller Ron Galperin served as a voice of reason out of Los Angeles city government. "This kind of cost is utterly unacceptable," he said to USA Today. "I believe we need a fundamental course correction."
Of course, it's not cost-free to do nothing about the city's chronically homeless either, when one considers the high medical and other costs of so many people living on the street versus living in housing of some sort.
But that does not justify profligacy in the name of tackling the problem. Nor does such profligacy particularly serve the interests of those who are ostensibly supposed to benefit from the help.
Practically speaking, if Los Angeles doesn't get a better handle on the costs of constructing housing for the homeless, it won't be able to achieve the construction of 10,000 permanent housing units promised in the campaign for the $1.2 billion bond approved by Los Angeles voters in 2016.
In a city with over 27,000 unsheltered homeless individuals, the more units that can be built, the better and the bigger the impact.
Galperin is right that a course correction is needed if the city of Los Angeles is going to try to address the large and growing homeless population. There is plenty of skepticism around about the ability of Los Angeles officials to do so. But there's clearly a lot of money at stake here. The $1.2 billion bond doesn't come for free; it comes, in fact, at substantial cost.
If Los Angeles cannot prove it can spend the money optimally, voters here and elsewhere will have reason to be skeptical of future calls for more funding.
The editorial board and opinion section staff are independent of the news-gathering side of our organization. Through our staff-written editorials, we take positions on important issues affecting our readership, from pension reform to protecting our region's unique natural resources to transportation. The editorials are unsigned because, while written by one or more members of our staff, they represent the point of view of our news organization's management. In order to take informed positions, we meet frequently with government, community and business leaders on important issues affecting our cities, region and state. During elections, we meet with candidates for office and the proponents and opponents of ballot initiatives and then make recommendations to voters.
opinion@scng.com


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The most reliable cars for towing
Looking for a tow car that won't let you down? Here are the most reliable cars for towing, according to warranty claims data
Working out the best cars for towing might seem straightforward given the plethora of information on the subject you can source online. It's easy to find the stats of maximum towing capacity, towball weight, engine pulling power, luggage capacity, but have you thought about the reliability of your towing car?
Thanks to data from Warrantywise, choosing a reliable option doesn't have to be such a risk as we've put together the most reliable cars for towing. These cars are not only trusty tow cars, they should also provide you with years of trouble-free motoring whatever you use them for.
Given that the data is based on actual warranty claims, you can rest assured that these cars should give you less hassle than some of their competitors. The benefit to these cars is that they should have the muscle to haul heavy loads about and be tough enough to do so without developing significant issues.
The following are our picks of the eight most reliable tow cars. Click the button to use the BuyaCar website to find your next car and don't pay more than you need to. Still working out your budget? Click the button above to get started with finance.
Most reliable cars for towing
1. Jeep Renegade
The Jeep Renegade achieved the maximum score in the Warrantywise study, with the fewest number of warranty claims and the lowest cost to fix a problem, should the worst happen. That makes it a great used tow car choice, and theoretically the best car on this list, but make sure it suits your lifestyle before diving in - there are plenty of other exciting offers to come.
Launched in 2015, the Renegade features the kind of chunky styling you'd expect from a Jeep. Not that the Renegade is all-American. On the contrary, the Renegade shares the platform it's built on with the Fiat 500X and is built at the same factory in Italy.
The 2.0-litre 'Multijet' diesel engine offers the best pulling power, with a maximum towing weight of 1,500kg. Four-wheel-drive is standard on the 2.0-litre diesel, but we'd opt for the 170hp version rather than the lesser-powered 140hp unit for its extra punch, especially if you're towing something large. Make no mistake, this is one of the most capable small SUVs you can buy.
JEEP RENEGADE BUYERS' GUIDE
Used Jeep Renegade deals
2. Kia Ceed
Used deals from £7,500
Of all the cars in the top eight most reliable cars for towing, the Kia Cee'd is the oldest in terms of average age. This means you can be sure that a Cee'd will provide years of problem-free service even after the market-leading seven-year warranty has expired.
You're going to want the 115hp 1.6-litre 'CRDi' diesel engine for towing duties, as it offers a maximum towing capacity of 1,500kg. You can expect some seriously low running costs, including free road tax on models registered on or before 31 March 2017.
Although the claimed fuel economy figures are likely to prove hard to achieve on real roads, you can expect to see between 50mpg and 55mpg - although that will undoubtedly drop when you have something large and heavy hooked up to the tow bar.
We'd recommend the '3' trim, as it comes with a seven-inch touchscreen sat-nav system, a reversing camera, dual-zone air-conditioning and 16-inch alloy wheels.
2012-2018 KIA CEED BUYERS' GUIDE
Used Kia Cee'd deals
3. Dacia Duster
Britain's cheapest SUV offers a terrific blend of space, equipment and price. In fact, we think this is the best budget tow car you can buy.
According to the warranty data, the Duster can be one of the cheapest tow cars to fix should the worst happen. Although the front-wheel drive Duster is the least expensive version to buy and run, we'd recommend getting a 4x4 version for towing duties. The 1.5-litre 'dCi' diesel engine provides plenty of pulling power and excellent fuel economy.
Stability control was an option on older models, so you may want to make sure the Duster you go for has this fitted before towing anything. Likewise, avoid the lower trim levels in favour of better-equipped versions like Prestige and Comfort models.
DACIA DUSTER BUYERS' GUIDE
Used Dacia Duster deals
4. Volkswagen Polo
The Volkswagen Polo might seem like an unlikely tow car, but the little supermini is capable of punching above its weight. It also performed well in the Warrantywise study, with the results based on the largest number of policyholders. Of the 2,500 policyholders, just 1.1% made a claim.
It's what we'd class as an 'ultralight' tow car. In other words, you're not going to use a Polo for towing anything other than a superlight caravan or garden rubbish to the recycling centre. However, that doesn't detract from its appeal as a small tow car.
One of the larger petrol engines, or better still a diesel, would be best suited to towing anything of substantial weight.
VOLKSWAGEN POLO BUYERS' GUIDE
Used Volkswagen Polo deals
5. Ford Mondeo
If you're after a brilliant tow car with a petrol engine, look no further than the Ford Mondeo 1.5-litre EcoBoost. Although it lacks the ultimate pulling power of a diesel engine, you'll enjoy the smoothness and relative quietness of the turbocharged petrol unit.
The Mondeo finished fifth in the Warrantywise study, despite having the highest average mileage of any car in the top eight. This suggests that the Mondeo will provide years of loyal service, even when the mileage is approaching 100,000.
An electrically retractable tow bar with trailer sway control was an £850 option when new, so you may want to track down a Mondeo with this fitted. Alternatively, you can buy a new Ford Mondeo with trailer sway control fitted as standard. Check out some of the prices on the BuyaCar website and you'll see that you could have a used Mondeo delivered to your door for the cost of a new Fiesta.
FORD MONDEO BUYERS' GUIDE
Used Ford Mondeo deals
The Volvo XC60 is a multi award-winning tow car, with a number of towing gongs to its name. Although the previous-generation XC60 lacks the sophisticated interior and modern safety kit of the newer model, it remains a formidable towing vehicle.
Automatic versions, badged Geartronic, offer a maximum towing weight of 2,000kg, but this drops to 1,600kg in XC60 models with a manual gearbox. Although the 2.0-litre D4 diesel engine is adequate, we'd recommend the 2.4-litre D5 diesel, not least for its substantial 215hp and 440Nm of torque in the automatic version.
The most impressive thing about the Volvo XC60 is how stable and assured it feels when towing a caravan or trailer. The optional reversing camera will come in handy when hitching up to a trailer. Thanks to a cavernous boot and a spacious cabin, the XC60 provides enough luggage space for a two-week holiday in the sun.
2008-2017 VOLVO XC60 BUYERS' GUIDE
Used Volvo XC60 deals
7. Audi A3
Just 1.2% of Warrantywise policyholders made a claim on their Audi A3, which is why it makes the list of the most reliable cars for towing. Although it might seem like an unlikely tow car, the combination of punchy diesel engines and quattro four-wheel drive makes the A3 a formidable towing vehicle.
A maximum towing weight of 1,800kg is available on an A3 with a 2.0-litre 'TDI' diesel engine with four-wheel drive, so it's ideal for towing a lightweight caravan or small trailer. Best of all, you can enjoy a cabin that's rich in quality and loaded with high-tech gadgets.
While the A3 Sportback is the most popular model, the A3 Saloon is worthy of your attention. It's more exclusive than the Sportback - the five-door hatchback - arguably nicer looking, and you can hide your valuables in the boot.
2012-2020 AUDI A3 BUYERS' GUIDE
Used Audi A3 deals
8. Nissan Juke
The final car on the list of reliable cars for towing is the Nissan Juke. This is one of Britain's most popular crossovers, but you might be surprised to find it on a list of tow cars. The Warrantywise data suggests it could be quite expensive if things go wrong, but the Juke scores highly for the fact that just 1.8% of policyholders had to make a claim.
The Nissan Juke 1.5 'dCi' diesel offers a maximum towing weight of 1,250kg, while a standard-fit reversing camera makes it easy to hitch up. This means that like with the Polo, you won't be towing the heaviest loads, but it still offers good scope for a car of this size.
If you're considering the Juke, chances are that you'll love the raised driving position and funky styling, which still turns heads a decade after its launch. Opt for a Tekna model and you'll benefit from a 360-degree camera, lane-departure warning and blind-spot monitoring - three things that could come in handy when you're towing a caravan or trailer.
2012-2019 NISSAN JUKE BUYERS' GUIDE
Used Nissan Juke deals
Most reliable cars 2021
Best medium size cars for towing
Best used cars for towing a caravan


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A Thing of Beauty _ My award-winning 1955 Mercedes-Benz 220 Coupe
Laura Peters, with Gary Anderson
Denis L. Tanney
I still remember my first glimpse of a Mercedes-Benz: a lovely SL convertible. I ran over to get a closer look. The sleek lines and small size particularly appealed to me; they matched my own ideas of grace and understated beauty. Upon my return home after seeing that SL, I announced to my mother that I was going to own a car just like it someday. After a few years of looking at other people's vehicles at car shows, I concluded that my holy grail was a W187 220 Coupe, a variant of the 1951-1955 220 Cabriolet, built for just two years, 1954 and 1955.
Laura Peters with Gary Anderson Images Denis L Tanney
For as long as I can remember, I've always had an appreciation for interesting automobiles. Growing up as the child of merchants in a small town in rural Georgia an hour east of Atlanta, I was always alert to the sound of a car or truck coming down the street. By the time I was about 10, I could correctly identify the brands of most of the cars I saw.
I still remember my first glimpse of a Mercedes-Benz: a lovely SL convertible. I ran over to get a closer look. The sleek lines and small size particularly appealed to me; they matched my own ideas of grace and understated beauty.
A life-long love affair
Upon my return home after seeing that SL, I announced to my mother that I was going to own a car just like it someday. By the late 1960s, as a student in the business school at the University of Georgia, I was finally able to buy a Mercedes-Benz of my own: I've owned one ever since. As soon as I became aware that there was a club for people like me who admired the marque, I joined the Peachtree Section of the Mercedes-Benz Club of America.
Being settled in as manager of the family's furniture and appliance store – with a growing understanding of the history of Mercedes-Benz and a deepening respect for good design – I started combing classified ads in The Star magazine looking for an earlier model. In 2001, a red-and-black coupe for sale in Eugene, Oregon, caught my eye; these were the colors of my alma mater. So I bought my first classic, a W111 220SE Ponton Coupe built at the end of 1959.
The coupe was in pretty good condition. With a little work I was able to bring it to a level that I was proud to present in local shows. I discovered that I enjoyed sharing my love for its design, details and history with other enthusiasts. The bug had well and truly bitten. I started thinking about what model I might like to acquire next.
My holy grail
I liked the cars of the 1950s and 1960s; I particularly liked the coupes' practicality and styling. After a few years of looking at other people's vehicles at car shows, I concluded that my holy grail was a W187 220 Coupe, a variant of the 1951-1955 220 Cabriolet, built for just two years, 1954 and 1955. It had the same stunning and dignified tall grille and sweeping fenders of the 220 sedans and cabriolets, but the more understated styling of the coupe's rear end seemed much better proportioned to me than the body styles of either of its older siblings.
There was only one catch: Only 85 of the coupes were built, and they were only sold in Europe. One might speculate that the decision to produce so few of these coupes was perhaps a response to customers with close ties to the factory who appreciated the lines of the separate-fendered luxury-level 300S coupes. These vehicles had been hand-built in very small numbers at very high prices, but production ended when the more up-to-date Ponton coupes were introduced.
Fantasy and rough reality
At this point, my dream seemed more like a fantasy that might entertain me for decades but never be realized. In June 2008, after years of telling everyone who might have a lead what I was looking for, I saw an ad posted by Leland Gohlike in Stillwater, Minnesota: He had a Stone Grey 220 coupe for sale, reportedly found in a barn, that he had purchased in 1991 through Mercedes-Benz specialist Alex Dearborn. Gohlike cautioned me that the coupe was very rough, but it was reasonably complete and the interior and exterior colors were original.
Tossing aside the rigorous due diligence I had learned so well in business school, I immediately decided to buy it. After all, everyone needs an unrealistic project on the side to balance all of the serious and sensible concerns of daily life.
Having gotten to know the experts at Bud's Benz during the years that I had owned my Ponton, I shipped my new project to them in Douglasville, Georgia. Right away, the team at Bud's cautioned me about the challenges and pitfalls of a full restoration, but I knew I had to do it. When only 85 of something have ever existed – and I owned a precious example – how could I not accept the duty of preserving it in original form?
Timeline with no end
The 220 project would extend over the next seven years. A big stumbling block was that many small parts were missing. Not only were they hard to find, experts would often disagree about which part was correct for a specific application. Andre Horvath, Henry Magno and Ricardo Jimenez were a great help with finding parts and separating truth from fiction.
Even when I found someone who said they had what we were looking for, we were often disappointed when what arrived was clearly wrong. Equally frustrating, a parts order would often simply be canceled weeks or months after delivery had originally been promised.
Arranging for the elaborate and impressive interior wood fittings to be correctly refinished was very trying. After all, fine furniture was my family business and this feature was a major appeal of the 220 for me. However, the talented craftsman who had done my W128 was no longer available. Reaching out to my network of specialists, I found similar commitment to high-quality work at Madera Concepts on the West Coast.
In spite of the delays, over the years my patience and persistence have been rewarded.
Today, everything in the interior, including the optional radio, is period and model correct. I'm very proud of finds like the full set of original luggage and original tool kit and factory-original box of spare parts in the 220's trunk.
A show queen emerges
Finally, about six years ago, the coupe was finished at last. Since then, I have had the pleasure of sharing the car with other enthusiasts at regional concours events like Château Élan, as well as club gatherings. At StarFest® in 2018, the 220 was awarded Best of Show and featured on the cover of The Star soon after.
The only drawback is that with my small stature I have difficulty reaching across the fenders to raise the center-hinged hood, so I have friends help with even simple maintenance. Stop-and-go traffic is also a challenge with a 4-speed manual transmission and stiff clutch.
But there is no more sublime pleasure than taking the car out for exercise early on a Sunday morning before people start to stir; a beautiful car at speed on the idyllic back roads of rural Georgia. I like the idea that – just like a piece of well-designed antique furniture – this rare and lovely automobile will live on after I'm gone because I invested the time and resources to bring it back to life.
Images and Captions
At once practical and elegant, the 1955-1956 W187 220 coupe has the dignified grille and flowing fenders of its older model siblings, the 220 sedans and cabriolets, but a more understated and better-proportioned rear end.
Lustrous leather and gleaming wood welcome the driver and passenger to this 1955 220 coupe's refined interior. In the trunk, factory tool roll, spare-parts box, and a set of optional fitted luggage are all period and model correct.
Proud owner Laura Peters has taken great satisfaction overseeing the painstaking years-long restoration of this rare 1955 220 coupe in its original colors and finishes, as verified by her Mercedes-Benz certificate of originality.
The Star® Magazine is a benefit of Membership in the Mercedes-Benz Club of America.
Become a Member to get exclusive access to the full Printed and Digital editions of The Star®


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Home » World » Middle East » YouTube Blocked in Egypt Over Anti-Islam Film
YouTube Blocked in Egypt Over Anti-Islam Film
in Middle East Leave a comment
YouTube has been blocked in Egypt. This will be carried out by the government as per a Court order. The Court order the ban as YouTube was hosting an anti-Islam film.
YouTube has been blocked for 30 days.
But, the order can be appealed and based on the current information, well, it might just not be enforced.
Judge Hassouna Tawfiq ordered YouTube blocked for carrying the film, which he described as "offensive to Islam and the Prophet (Muhammad)." He made the ruling in the Egyptian capital where the first protests against the film erupted last September before spreading to more than 20 countries, killing more than 50 people.
Egypt's new constitution includes a ban on insulting "religious messengers and prophets."
Lawyer Mohammed Hamid Salim, who filed the lawsuit in Cairo, alleged the film constitutes a threat to Egypt's security, adding that YouTube refused to remove the film despite its offensive content.
YouTube's parent company, Google, declined requests to remove the video from the website last year, but restricted access to it in certain countries, including Egypt, Libya and Indonesia, because it says the video broke laws in those countries. At the height of the protests in September, YouTube was ordered blocked in several countries, including Iran, Pakistan and Afghanistan, and Saudi Arabia's King Abdullah issued an order blocking all websites with access to the anti-Islam film in the conservative Kingdom.
Two more court cases have been filed in Egyptian courts regarding the film. The cases are against the government and Google. The case against Google suggests a complete ban on Google's search engine and wants the firm to pay a $2 billion fine.
YouTube nor Google has yet to publicly comment on the ruling. Well mostly they will remain silent until the rule is enforced.
As of now, YouTube is accessible in Egypt.
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Baba Dochia, the band with 12 lambskins
Andrei Bucureci Nov 06, 2017
Baba Dochia is a Cluj based electronic-rock four piece band with it's origins in the medieval city of Sighișoara. The band have released one critically praised conceptual studio album called XII Lambskins a bridge between electronic, post-rock, film score and eclectic sampling. About the band sound and its future plans, Leon Lotoțchi was kind to respond to our questions.
What can you tell us about the beginnings of Baba Dochia in Sighișoara? How was the atmosphere in your hometown, a place with a solid reputation for it's medieval sounds?
I think that, on an unconscious level, the exposure to medieval music and the influence of rock artists that used to play acoustic spontaneous sets all over the city during the famous medieval festival influenced us on some level to get into music. We were very young and not very aware of all the different musical subcultures, but because music has such a deep influence on human emotion it clearly touched us.
I remember hearing the first Prodigy album, Experience, the impact of its energy made me say "i wanna do that!". Soon after, I started to make music on a computer, showing my future band colleague Marius some of the possibilities a DAW can offer. It all started as a hobby but after listening Team Sleeps' first album, we decided to make music together, blending our two favourite music styles at the time, rock and electronica. Our first 'studio' was in my grandmother's rented apartment. All of my gear was already there and, so we brought Marius' equipment to, a computer, keyboard, guitar, a couch and some clothes. That very first session produced our debut ideas, one of which is the song now known as Starfall, still a strong point in our sets to this day.
You have your headquarters in Cluj now. How do you vibe with the new city? Do you still find inspiration in the city?
We love the energy Cluj is emanating, and the feel of potential is omnipresent, and I don't mean just in a musical sense. It's still not quite the same as Bucharest, but you can still draw inspiration by randomly going out and inhaling the opportunities for interesting interactions. The Cluj music scene is great, it really has that 'underground' vibe to it. I still remember that as a freshman student in the city, going at rehearsals of the ska-punk band called Dance Trauma, somewhere in an unkempt basement near the city center. Ignoring the awful conditions, the band jammed with, what seemed at the time, unstoppable energy. I think there was even a puddle of water in the room, but nobody cared, that wasn't important. The focus was upon the process of creation. That was the first time I witnessed in real time the creation of a new song by a band. When I heard it afterwards it felt awesome, knowing that I witnessed the spark between the band members.I think the song was Antichristophor Columb or something like that.
You have some interesting native voice samples in your music. Do you use them just to add colour or do you want to emphasize a certain atmosphere of your music?
Sampling helps us. Sometimes you get stuck at a certain point, and a fresh sample can provide that necessary boost to get things going. A sample might emphasize the mood of a specific spot in a song, or might just fill up an otherwise bland portion where you have run out of ideas.
Another thing we are trying out more and more now, is to record random sounds and incorporate them into the songs, maybe with some processing if it is necessary. For example in one of our newer tracks, Venus, around the first half of the song there is a whooshing sound that accompanies the kick drum, and this creates tension. Marius recorded himself breathing heavily, sort of HAAAAAA like and blend it with the rest of the song. I think it sounds really good, even though when I first heard the raw sample recording I made fun of him that he sounds creepy, hahaha. But we know each other for a couple of decades now and it's all made with good intentions and fun, so this kind of teasing helps us in the creative process.
For the 12 Lambskins album you worked with a music producer outside the band, Alexandru Șeidiu aka MGCH. How was this collaboration?
Cheers Alex and thank you! Yeah, he made our first album sound great, and of course that is always a big emotional hurdle to overcome, the hope that your first material sounds as good as you want it to be, and he helped us get over it. His reputation as a "musical chameleon" made him great for the role we needed, our songs ranged from post-rock to techno and from trip-hop to experimental (not forgetting the electronica-rock hybrid) and he seemed comfortable in each setting. He wasn't shy about pointing out where there was room for improvement, where he felt it was needed, and we salute him for that.
You worked so far only with lady voices, Andreea on TRAKT, Ioana Lefter from Fine It's Pink on Wind and I know that you also tried to collaborate with Irina Bucescu from Temple Invisible and Delia Panait. There are nice features that brings a lot of passion to your music.
I think that a female voice complements nicely the overall dark vibe of our songs. It gives a nice contrast and a moody lifeline, to put it into words. We're not throwing away the possibility of collaborating with male voices, but I guess that the circumstances up until now haven't favored this kind of collaboration.
The pretty lonely process of creating a new electronic song might be the fault of the instrumental direction of the album. We sit there, jam a bit, come up with an idea or two, and progress from there, more or less forgetting and/or not taking into account that you might need or use a voice at some point in the track.
We have now some tracks in progress with female singers, but we are still at the ideas exchange phase, so it might take a bit for those tracks to see the light of day.
XII Lambskins by Baba Dochia
I have a great consideration for your live show, music and visual both. You told me recently that you moved, for the live part at least, from Reason to Ableton. What made consider switching the DAW?
We used Propellerhead Reason until recently, as it was our first digital audio workstation. But we felt it wasn't the greatest option to reproduce electronic songs in a live setting, so we changed to perform using Ableton Live. Even though Reason is a great tool for creating sounds, we found it to be a bit cumbersome in creating a fluid live set and eliminate the "just press some buttons and the song will play itself" feel. We drew inspiration from Apparat and Trentemoller in this aspect, as they first produced the songs in a studio setting, creating as many tracks as they liked, and then discussing it to see what would work live. Do we really need this extra sound? Can this be reproduced on a guitar instead and can we control this more efficiently?
Nowadays most releases are using only the digital platforms. What is your input regarding this, taking into consideration that your first album was also released on a CD with a very nice booklet?
Even though online digital formats are probably the most widespread format nowadays, we didn't want to underestimate the importance of having a physical material in your hand. The CD came with the booklet so that the listener has something to immerse himself into while listening to the record. There was a lot of work done for it as well, and we thank Marius Rosu for that. There were also some discussions about releasing a vinyl format, but it didn't happen after all, mostly because of us, but we hope to fix this in the future.
I think that the path of our passion for music intertwines itself pretty much with the path of our explorations of synthesis that we described earlier. Also, all those rock concerts that we attended as youngsters helped a lot, as we opened ourselves up to a lot of influences from people that were passionate about music and surely more experienced than us.
There are for years now since the beginning of your musical journey with Baba Dochia. After touring Romania and playing in all kind of venues could you describe how are people are reacting to your sound?
The local scenes are very eager to try out new sounds and you can definitely see that in the energy of the venue. We played for both jam packed clubs or for locations where there were just a handful of people that just seemed to stare at us lifelessly throughout the whole concert. Sometimes the public opened up after the first riffs of the first track, sometimes only towards the end of the set, but one constant was the personal "thank you" messages received in person from people in attendance, and that makes it all more worth playing for.
What do you think about the future of Romanian music? Are we likely to have an international reputation and do we have an authentic sound that promoters outside the country are looking for?
The future is bright, the future's a Romanian. Forgetting the allusion to a certain commercial, the Romanian musical scene seems to expect the moment to make itself heard, and it surely has something to say!
There are a lot of original projects around, each with a distinct flavor to it. Regarding the international online reputation, I read an article not so long ago about the Romanian underground minimal scene: it served as a great appetizer about the things to come. That article, even though written with aplomb, barely scratched the surface of what could be offered musically from this country. This kind of question has a bigger socio-political connotation, and because of this I think it would be best to have it expanded in a future discussion with more time allocated for a subject such as this.
We can't finish this interview without asking about your Indian tour. How did the local community look like? Did you interact with Indian music?
It wasn't just a tour, it was also a spiritual journey as well, because we visited a lot of temples, haha. It was of course great, from the big things, like the warm reception of the organisers and the audience, to the small things, emotional moments of unpacking the flight cases and hoping everything is intact after flying a couple of thousands kilometers. The people we interacted with were very warm, humble and at the same time curious about us and our culture. We played at an international festival organized at the campus of the biggest University of Chennai, but Indian music took a large part of the schedule. The passion that the Indian people have for dancing and expressing themselves through movement was particularly fascinating. Did you see all those complicated traditional Indian dance moves? Yeah, those look WAY more awesome live, their flow is amazing.
#12 lambskins#baba dochia#electronica#leon lotoțchi#mgch
Gallery of Dream Diggers and Valerinne concert
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The Scottsdale Historical Society is giving residents their chance to visit historic Prescott, capital of Territorial Arizona, on a day-long bus tour led by knowledgeable museum guides.
The Thursday, March 10 event includes the Sharlot Hall and Fort Whipple Museums, with a special lunch at the legendary Hassayampa Hotel, for an all-inclusive fee of $95. Reservations close Feb. 10.