text:
\section*{Nomenclature}
\begin{longtable}[l]{ll}
$\alpha$ & 	constant related to chain stiffness and volume exclusion\tabularnewline
$a'$  & scaling factor for average shortest path\tabularnewline
$\boldsymbol{A}$  & adjacency matrix\tabularnewline
$b_{k}$  & disconnected components of the network\tabularnewline
$c_{e},\;c_{i},\;c_{m}$  & reaction propensity coefficients \tabularnewline
$c_{k},\;c(A)$  & local and global clustering coefficients\tabularnewline
$d_{i}$  & degree of node $i$\tabularnewline
$d_{i,\text{free}}$  & free degree of node $i$\tabularnewline
$d_{i,\text{max}}$  & maximum degree of node $i$\tabularnewline
$E_{\text{max}}$  & maximum number of edges in the network\tabularnewline
$f_{i}$  & concentration of fatty acids with $i$ unsaturations\tabularnewline
$g_{i}$  & hindrance effect \tabularnewline
$g_{01}(\chi)$  & gel indicator function\tabularnewline
$H(\chi)$  & Heaviside function\tabularnewline
$k_{p},\;k_{c},\;k_{m}$  & reaction rate constants \tabularnewline
$l$  & average length of a chemical bond \tabularnewline
$n$  & number of nodes in the network \tabularnewline
$n_{A}$  & Avogadro number\tabularnewline
$N$  & number of samples in MC\tabularnewline
$O(n)$  & big O notation for asymptotic behaviour of a function\tabularnewline
$p_{i,j}$  & shortest path between nodes $i,j$ \tabularnewline
$p$  & average shortest path \tabularnewline
$P(A)_{i,j}$  & probability of the next reaction between monomers $i,j$ at given topology $A$\tabularnewline
$P^{*}(A)_{i,j}$  & proposal probability for the MCMC scheme\tabularnewline
$\mathcal{P}(k)$  & Poisson distribution\tabularnewline
$Q$  & stochastic matrix for computing steric hindrance\tabularnewline
$R$  & effective reaction distance \tabularnewline
$\rho$  & molar density\tabularnewline
$s_{1}(q),\;s_{2}(q)$  & Markov chain for MCMC sampling\tabularnewline
$S_{1}(n),\;S_{2}(n)$  & size of largest, and second largest components for an $n-$monomer system\tabularnewline
$t$  & reaction time\tabularnewline
$\tau$  & waiting time between reactions\tabularnewline
$u_{i}$  & uniformly distributed random variable\tabularnewline
$U[0,1]$  & uniform distribution\tabularnewline
$V$  & volume\tabularnewline
$W_{n}$  & concentration of TAGs of functionality n\tabularnewline
$\Phi(\boldsymbol{A})_{i,j}$  & cyclisation rate for nodes $i,j$\tabularnewline
$\chi$  & conversion of edges\tabularnewline
$\chi_{g}$  & Conversion at gel transition\tabularnewline
$z$  & current number of edges in the network\tabularnewline
\end{longtable}

\section{Introduction}

Polymer science profits from the rapid development in the computing
field in a very straightforward way: the complexity of macromolecular
architecture or geometry is, to a growing extent, captured by simulation
of the most basic physical processes. Instead of following heuristics
leading to the classical mathematical equations. We anticipate that
properties of continuum can be obtained as emerging from repeated
simulations of the many-body physics at the nano-level, a strategy
that seems to be more properly placed in stochastic simulations than
in deterministic mathematical modelling. That said, one must realise
that such simulations imply an enormous spread of length and time
scales: even the simple case of a linear polymer exhibits geometrical
structure from the scale of a chemical bond (1A) to the scale of the
gyration radius (100A), and collective length scales in dense materials
often are even much larger. Moreover, our drive for modelling is not
limited to linear polymers but also is directed at tree-like branched
structures, cross-linked polymers, and even infinite molecular networks,
\latin{i.e.} \emph{gels}. In this paper we propose a strategy that draws a
consensus between what can be computed on the one hand and pursuing
fundamental roots of the matter at hand on the other.

The process of assembly or polymerisation of molecular networks usually
goes through a few temporal stages: disconnected monomers, linear
polymer, branched polymer, and gel (an infinite network) \cite{dusek2012,dusek2013, kryven2014a}. Furthermore, the gel itself should
not be perceived as a single, final state of polymer topology. After
the gel point the connectivity patterns of the network continue to
evolve\cite{akagi2009,zhou2012,armitage2000}. In fact, in many cases
a major part of the whole process of conversion may occur in the gel
regime\cite{kryven2014b}.

Side by side with new developments in Polymer Reaction Engineering,
a novel field of \emph{network science} has been introduced recently.
In a much broader sense, the main object of studies in network science
is networks regardless their background, \latin{i.e.} not necessarily molecular
networks. In chemistry, not surprisingly as chemical bonding leads
to connectivity problems, one of the main toolboxes in network science,
graph theory is already in use, for instance, in the form of the Wiener
index \cite{rouvray2002}. However, apart from this line of research,
the applicability of graph theory to large or infinite molecular networks, has
received little attention. The current paper presents a cross dissemination
between polymer science and network science as disciplines, with special
care devoted to polymer-related issues. The problem we deal with in the present paper has three main features.\\
\emph{a)}
The functionality of the monomer is limited and satisfies a predefined
frequency distribution. The monomer functionality is crucial for the
gel formation and properties of the material\cite{schwenke2011,lange2011,lang2003}. A bi-functional monomer just forms linear chains. Monomers with higher functionality may lead to gels or infinite networks. Star
macro-monomers may carry many functional groups, thus being a perfect
base for complex molecular networks.\\
 \emph{b)} The average reaction rates may decrease by orders of magnitude
due to diffusion limitation as a consequence of the decrease of the
free volume caused by the formation of the polymer network, \latin{e.g.} as in Ref. \cite{goodner2002}. The Trommsdorff or gel effects in radical polymerisation
is an example of this. Moreover, the mobility of the larger molecules
is affected more strongly than that of small molecules. This may affect
the rates of reaction, depending on the type of molecules involved
in those reactions. The issue of 'chain length dependent termination'
is a frequently studied example of this phenomenon \cite{kattner2015}.
Furthermore, due to steric hindrance effect some monomers might be
less available for the reaction being shielded by the surrounding
topology of the network. The available polymerisation models accounting
for shielding\cite{eichinger2005} have a mean-field nature and consequently
do not allow for correct interpretation of possible inhomogeneities
or irregularities present in network topology. \\
\emph{c)} The topological
distance between reacting units has to be accounted for. The distance
between reacting units is not an issue in systems where these units
make part of separate larger assemblies moving so fast that their
average distance - or concentration - is the dominating factor. In large
 networks, however, the units are relatively fixed and morel likely to react if the distance is small.
In atomistic simulations position and reactivity are addressed
by the 'capture sphere growth approach'\cite{eichinger2005}.
In percolation theory monomer position and reactivity is described using a grid in 3D space.
However, even without performing simulations in actual three-dimensional space, it is possible to incorporate
geometrical information inferred from the network topology \cite{kryven2013c},
from analytical results on random walks\cite{dusek2014,wang1952,dusek1978},
or from population balance equations \cite{kryven2014g,kryven2014d}.

Here, we present a new approach based on Random Graph Modelling, which
must be considered as the first attempt ever in non-atomistic modelling,
to account for all the three features described above. It is based
on the kinetic Monte Carlo concept\cite{wang2009a,wang2009b,lazzari2014}, where we explicitly deal with position-reactivity by using the graph-theoretical
concept of the shortest path between monomer pairs. This is one significant
step further than recent kinetic MC modelling studies, where the impact
of size of individual (polymer) molecules on reactivity is accounted
for \cite{mastan2015}. In this paper we will show that this is an ambitious
but still feasible approach, as it is not computationally very demanding.
Given the fact that kinetic MC typically requires $10^9$
to $10^{11}$ units to achieve statistically relevant
results, one might expect that introducing the three features described
above would lead to excessively expensive calculations, but we will
show that this is not the case. It is indeed the higher functionality
of the monomers that creates a computationally very favourable
condition concerning the required size of the domain simulated, typically
$10^4$ units.

This paper is structured as follows. To introduce the idea of application
Random Graph Models to polymerisation problems, we explain the key
concepts, such as: adjacency matrix, relation between topological
and spatial distances, relation between steric hindrance and node
centrality. Then, this concepts are used to build a Gillespie's stochastic
simulation algorithm that describes evolution of network topology
from monomers up to the full conversion of bonds. We also explain
how the stochastic simulations can be accelerated applying the Markovian
Chain Monte Carlo (MCMC) principle. Finally, the theoretical results
are fortified with an extensive study on a realistic system: polymerisation
of triacylglycerides, which is more commonly known as the 'drying'
of oil paint. Simulation results include widely used in the reaction
engineering community descriptors as, for example, size distribution,
gel point conversion, distance between crosslinks. Furthermore, novel
descriptors for molecular networks that are especially suitable for
post gel region, are introduced. Among these descriptors: modularity
of network, clustering coefficient, average path, and cluster-size distribution.

\section{The model system}

\paragraph{Star macromonomers}

\begin{figure}
\center \includegraphics[width=0.7\textwidth]{star_monomer} \caption{\emph{a)} A schematic representation for the multifunctional macromonomer
as a star polymer molecule. Shaded nodes carry free functional groups
and thus provide means for further polymerisation into a complex network.
\emph{b)} Coarse representation for a monomer as a 'blob' retaining
the number of functional groups. }
\label{fig:star}
\end{figure}

Star polymers consist of a few arms joined together on one end \cite{schwenke2011}.
The arms may carry multiple functional groups, (Figure~\ref{fig:star}),
making star polymers a very attractive basis for construction of larger
macromolecular networks. In the current paper we develop a model for
polymerisation of macromonomers that are essentially star polymers
of various functionality. Since in this study the focus is on the
connectivity patterns of such monomers incorporated into a network,
we will use terms monomer and node interchangeably, depending on the
chemical or topological nature of the issue at hand. The monomers
will be distinguished according to their functionalities $d_{i,max}.$
A frequency distribution of functionalities is assumed to be known.
\begin{figure}
\center \includegraphics[width=0.6\textwidth]{A} \caption{ An example of a network model with 6 nodes, the corresponding adjacency
matrix $\boldsymbol{A},$ and maximum degree vector $d_{max}$. There
are six nodes in the system: nodes 4 and 5 have a loop; nodes 3,4,5
have all functionalities used, $d_{i,free}=0;$ nodes 1,3,6 have free
functionalities available. The adjacency matrix $\boldsymbol{A}$
is block diagonalised to emphasise the fact that there are two connected
components in the system. The maximum number of edges is $E_{max}=16,$
and the current number of edges used is 12, or as a conversion fraction,
$\chi=\frac{3}{4}$. }
\label{fig:A}
\end{figure}

\paragraph{Graph representation of a molecular network}

In chemical context, graphs have been used to represent connectivity
of (carbon) atoms in an organic molecule with a priori defined structure\cite{janezic2015,rouvray2002}.
Here, we represent the whole molecular system as a single undirected
graph, regardless of the system composition: as disconnected monomers,
ensemble of polymers, or a single connected network. Nodes numbered
with index $i=1,\dots,n$ represent monomers, and edges represent
chemical bonds between such monomers. Each node can have at most $d_{i,max}$
edges. The adjacency matrix of the molecular network is defined as
a square matrix,
\begin{equation}
\boldsymbol{A}_{i,j}=\begin{cases}
1, & \text{node }i\text{ has bond with node }j,\\
0, & \text{no bond between }i\text{ and }j
\end{cases}
\end{equation}
The degree of node $i$ may be lower then its maximum degree, so we
distinguish between the maximum degree $d_{max}$ (monomer functionality
defined beforehand), the actual degree,
\[
d_{i}(\boldsymbol{A})=\!\sum _{1\leq i\leq j\leq n}\!\!\boldsymbol{A}_{i,j}+\boldsymbol{A}_{i,i},
\]
and the free degree,
\[
d_{i,free}=d_{i,max}-d_{i}(\boldsymbol{A}).
\]
The adjacency matrix $\boldsymbol{A}$ together with the maximum degree
$d_{i,max}$ constitute the model representation for a state of a
particular molecular system. Figure~\ref{fig:A} gives a simple example
of a 6-monomer system and the corresponding matrix representation.
In view of limitations posed on node's degree, $d_{i,max},$ the
maximum number of edges in the whole network is also limited, $E_{max}\leq0.5\sum_{i}d_{i,max}.$
The last equation becomes equality if $d_{i,max},\;i=1,\dots,n$ is
a \emph{graphic sequence}\cite{eggleton1979}, namely: $\sum_{i}d_{i,max}$
is an even number, and for any $r\leq n-1,$ the following inequality
holds,
\begin{equation}
\sum _{i=1}^{n}d_{i,max}<r(r-1)+\sum _{i=r+1}^{n}\min(r,d_{i,max}).
\end{equation}
The graphic sequence is an important concept for finite simulation
systems. Since the degrees of nodes are completely random (up to a
specific frequency distribution), it is useful to constrain the degree
sequence of a finite simulation system to be graphic and thus eliminating
small-size system effects that are not in an infinite network. This
idea may be compared to imposing periodic boundary conditions in a
simulation box for partial differential problems in order to avoid
the effect of boundary layers.

Knowing a maximum number of edges allows defining a natural way of
quantifying the polymerisation progress that evolves from $0$ to
$E_{max}$ edges,
\begin{equation}
z=\sum _{i=1}^{n}d_{i}(\boldsymbol{A}),\;z=0,\dots,E_{max},
\end{equation}
or as a relative conversion $\chi=\frac{z}{E_{max}}\in[0,1].$ Before
proceeding to defining the principle of choosing a pair of nodes receiving
a new edge on each polymerisation step, we need to introduce two
concepts first: the idea of distance between nodes, and a measure
of the node's centrality, which will allow us to address steric hindrance
effects.

\paragraph{Topological distance}

The probability of a reaction between two monomers incorporated in
the network is smaller if they are positioned far apart in space.
Our model does not explicitly include spatial configurations, but
certain information on the distance between any two nodes is encoded in
the connectivity pattern defined by the adjacency matrix $\boldsymbol{A}$.
The shortest path from node $i$ to node $j$ is the shortest sequence
of intermediate nodes that link $i,j$ together. The length of the
shortest path is the number of edges in this sequence. If nodes $i,j$
are adjacent then the shortest path is 1; if they do not belong to
the same connected component then the path is defined to have length
0. While the shortest path $p_{i,j}$ does not explicitly tell us
how far the nodes are in space, it sets an upper bound on the distance:
the two nodes cannot be further apart then the shortest path times
length of the edge, $l$. Later on, we will exploit this fact to develop
a notion for network density. Let $p_{i,j}$ denote the shortest path
between nodes $i$ and $j$. One can always find such a path by applying
a path-finding algorithm directly on the adjacency matrix $A$, for
instance the Dijkstra algorithm\cite{jungnickel2008}. In a finite
network, the average shortest path,
\[
p=\frac{1}{n\cdot(n-1)}\cdot\sum_{i\ne j}p_{i,j}
\]
is proportional to the radius of gyration and hence there is a proportionality
between the volume occupied by the monomer network and the volume
of the gyration sphere: $\rho n=V\propto p^{3}$. Assuming a homogenous
density of the network $\rho$, augmenting the number of nodes from
$n$ up to $a\cdot n$ will cause a scale-up of the average shortest
path to $a'\cdot p$. Th exact value of $a'$ depends on spatial configuration
of the monomers but the lower bound is given by
\begin{equation}
a'\geq\frac{\left(\frac{4}{3\pi}a\rho n\right)^{1/3}}{\left(\frac{4}{3\pi}\rho n\right)^{1/3}}=a^{1/3}\label{eq:scaling}
\end{equation}
Thus, any polymerisation model has to respect the scaling ratio $\frac{a'}{a^{1/3}}\geq1$,
otherwise the produced topologies will not have any feasible spatial
configurations. Even though this condition is necessary but not sufficient,
it allows us to restrict ranges for input parameters for the model,
post factum.
\begin{figure}
\center \includegraphics[width=1\textwidth]{Distance_Shielding}
\caption{ Network topology as seen from the perspective of a single node (marked
as 'EGO'). Numbers indicate the shortest path distance to EGO. a)
The reaction probability of two monomers incorporated in the network
depends on the probability of the path configuration that brings these
nodes together (b-c). Both the immediately surrounding and the more
remote nodes contribute to the steric hindrance effect. }
\label{fig:EGO}
\end{figure}

In order to formulate the reaction rate between two nodes incorporated
in the network ($p_{i,j}>0$), we will follow a similar reasoning.
This time we additionally account for spatial configurations, assuming
that the shortest paths can be mimicked by self-avoiding random walks.
Two monomers need to be at minimal reaction distance $R$ to allow
reaction. For a sub-chain approximated by a random walk\cite{lang2005},
the probability of two ends being at distance $R$ is proportional
to
\begin{equation}
\Phi^{*}(A)_{i,j}\left(\frac{3}{2\pi l^{2}\,p_{i,j}}\right)^{3/2}e^{-\frac{3R^{2}}{2l^{2}p_{i,j}}}.\label{eq:32}
\end{equation}
Now, let us estimate the probability that a random walk is self-avoiding
given that its end-to-end distance is $R$. We assume that all monomers
are distributed uniformly in the the gyration volume $V=ap^{3/2}.$
Here $a$ is a constant related to the stiffness of a polymer chain.
Since the positions of all monomers $x_{k}=0,\dots,x_{p}$ are distinct,
the probability that position $x_{k}$ does not coincide with all
previous positions is $1-k\,ap^{-3/2}.$ Hence, the total probability
that the whole chain is self avoiding is given by
\[
\prod _{k=1}^{p}(1-k\,a\,p^{-3/2})\approx e^{-a\,p^{3/2}\sum _{k=1}^{p}k}=e^{-0.5\,a\,p^{-3/2}\,(1+p)p}\approx e^{-\alpha p^{-1/2}}.
\]
Note that a similar argument has been proposed by Flory\cite{flory1949}.
Flory, furthermore, derived the value of $\alpha$ from measurable physical
constants related to solvent properties, the stiffness of a chain,
and monomer volume. Combining the last relation with \eqref{eq:32}
and taking $R=l$ gives the probability of a self-avoiding cyclic
chain configuration,
\begin{equation}
\Phi(\boldsymbol{A})_{i,j}=Cp_{i,j}^{-3/2}e^{-\frac{3}{2}p_{i,j}^{-1}-\alpha p_{i,j}^{1/2}}.\label{eq:Phi}
\end{equation}
The constant $C$ is chosen to satisfy $\Phi(2)=1$ allowing to relate
the intramolecular reaction rate constant to the propagation rate
constant, as will be shown later on. Another constant appearing in
\eqref{eq:32}, $\alpha,$ defines the rate of decline of the reaction
probability, asymptotically at large distances. Note that $\Phi(\boldsymbol{A})_{i,j}=0$
means that these nodes do not belong to the same connected component
and that their spatial distance cannot be inferred from the topology.
In this case the reactivity will be defined by a different mechanism
based on the chemical rate constant and the degree of steric hindrance.

\paragraph{Steric hindrance as a centrality measure}

Here, we will construct a centrality measure, defined by numbers $g_{i}\in[0,1]$
that will reflect how much a node is obscured by other nodes in the
network, and thus experience a steric hindrance effect. Matrix function
$g(\boldsymbol{A})$ is set up according to three principles: \\
 1) the nodes of higher degree are more obscured: if $d_{i}>d_{j}$
then $g_{i}<g_{j}$;\\
 2)if the node is not obscured then $g_{i}=1$; $g_{i}=0$ means full
obscuration;\\
 3) from two nodes of the same degree, the node adjacent to a more
obscured one is more obscured.\\
 In order to satisfy the first principle it is sufficient to choose
$g_{i}$ proportional to $1/d_{i},$ which also has a logical interpretation
from the excluded volume point of view (Figure~\ref{fig:EGO}b). The
second principle might be achieved by an appropriate rescaling. It
is only the third principle that requires to incorporate information
on the specific topology $A$ (Figure~\ref{fig:EGO}c). Altogether,
the principles can be reformulated in the form of an implicit equation
for the reciprocal $g_{i}^{*}=\frac{1}{g_{i}}$,
\begin{equation}
g_{i}^{*}=\lambda_{i}d_{i}\sum _{j\sim i}g_{j}^{*},\label{eq:gi}
\end{equation}
where the summation is performed over all nodes $j$ adjacent to node
$i$, and $\lambda_{i}$ is a normalisation factor required to ensure
existence of the solution. The implicit equation \eqref{eq:gi} rewritten
in a matrix form yields an eigenvalue problem,
\begin{equation}
\begin{aligned}g^{*}= & \;\,Qg^{*},\\
Q_{i,j}= & \;d_{i}\sum _{j}\frac{A'_{i,j}}{\sum _{k}A'_{k,j}d_{j}}
\end{aligned}
\label{eq:eig}
\end{equation}

Here matrix $\boldsymbol{A}'$ coincides with $\boldsymbol{A}$ except
for the diagonal where $(\boldsymbol{A}')_{i,i}=1.$ Matrix $Q$ has
the same size and number of zero elements as the adjacency matrix
A. With no loss in generality we may consider $d_{i}>0$ for all nodes
(otherwise $d_{i}=0$ would yield a trivial solution $g^{*}=g_{i}=1$
due to the second principle). The eigenvalue problem as formulated
in \eqref{eq:eig} is ready to use in the case of a fully connected
network (\latin{i.e.} gel). In the case of a few network components that are
not connected to each other, $\boldsymbol{A}$ has block-diagonal form.
Solving the eigenvalue problem in each block of matrix $\boldsymbol{A}$
would lead to a correct ranking within each separate component, but
it would not allow to relate the extent of steric hindrance of two
different components. To overcome this problem, we normalise $g$
in each component of the network (each block of block-diagonalised
$\boldsymbol{A}$):
\[
g_{i}=\frac{\min\limits _{i\in b_{k}}(g_{i}^{*})}{g_{i}^{*}},\;\;i\in b_{k},
\]
where $k=1,\dots,m$ counts connected components $b_{k}$ of the network
$\boldsymbol{A}$. An example of a network topology with $g_{i}$
computed for each node is given in Figure~\ref{fig:Centrality}.

The fact that the matrix $Q$ in the eigenvalue problem \eqref{eq:eig}
is normalised, allows to alternatively view $Q$ as a Markov (left-stochastic)
matrix defining a random walk on the network\cite{newman2010}. Recall
that $Q_{i,j}>0$ if and only if $A_{i,j}=1,$ hence Q defines a weighted
graph with the same edges as in $A$. From this perspective, we may
view the solution $g_{i}^{*}$ as a frequency node-visiting by the random walk on the weighted graph $Q$. This means that instead
of solving the expensive eigenvalue problem \eqref{eq:eig}, it is
sufficient to consider a fixed point iteration $g^{*}=Qg^{*}$ starting
with an arbitrary vector with all non-zero components .
\begin{figure}
\center \includegraphics[width=0.7\textwidth]{shield} \caption{ A sample of a network topology with values of steric hindrance factors,
$g_{i}(A),$ indicated for each node. The example shows that $g_{i}(A)$
is defined by both the degree of a node and its position in the topology. }
\label{fig:Centrality}
\end{figure}

\section{Stochastic simulation}

We consider the process of network formation (or polymerisation) as
a stochastic process $\boldsymbol{A}(z),\;z=0,\dots,E_{max}$ starting
from an empty adjacency matrix $\boldsymbol{A}(0)$ and taking it to a matrix with
one edge $\boldsymbol{A}(1)$, and so on up to, finally, $\boldsymbol{A}(E_{max}).$
This evolution proceeds by means of sampling steps: matrix $\boldsymbol{A}(z+1)$
is obtained from matrix $\boldsymbol{A}(z)$ by placing an additional
edge. Generally speaking, there are $E_{max}-z$ alternatives to decide
where to put a new edge. This choice is performed according to a conditional
probability accounting for all possibilities for the new edge to appear,
obeying the reaction propensity coefficients $c_{e},c_{i},c_{m}$
and given the topology $\boldsymbol{A}$,
\begin{equation}
P(\boldsymbol{A})_{i,j}=\lambda_{z}\cdot\left\{ \begin{aligned} & \,c_{e}\,d_{i,free}d_{j,free}\Big(g_{i}(\boldsymbol{A})g_{j}(\boldsymbol{A})\Big)^{\beta} & i\neq j, & \;\Phi(\boldsymbol{A})_{i,j}=0;\\
 & \,c_{i}\,d_{i,free}d_{j,free}\Phi(\boldsymbol{A})_{i,j}, & i\neq j, & \;\Phi(\boldsymbol{A})_{i,j}>0;\\
 & \,c_{m}\,\binom{d_{i,free}}{2}, & i=j.
\end{aligned}
\right.\label{eq:P}
\end{equation}
Here the normalisation coefficient $\lambda_{z}$ is selected to satisfy
\begin{equation}
\sum _{i,j}P(\boldsymbol{A}(z))_{i,j}=1.\label{eq:lambda}
\end{equation}
The first line of \eqref{eq:P} refers to a new edge between nodes
forming distinct components of the network (in the case of an intermolecular
reaction, no path exists between $i,j:$ $\Phi(\boldsymbol{A})_{i,j}=0$).
The probability $P(\boldsymbol{A})$ here is defined via the node's
degree $d_{i}$ and its steric hindrance factor $g_{i}$. The intermolecular
propensity constant $c_{e}$ is related to dimerisation rate, $c_{e}=\frac{2k_{p}}{Vn_{A}},$
where $V=\frac{n}{n_{A}\rho}$ denotes the reaction volume, and $k_{p}$
is the usual rate constant for the reaction,
\[
P_{1}+P_{1}\xrightarrow{k_{p}}P_{2},
\]
where $P_{1}$ denotes monomer, and $P_{2}$ dimer. The second line
of \eqref{eq:P} refers to a new edge between nodes from the same
connected component (in case of an intramolecular reaction, a path
between $i,j$ exists: $\Phi(A)_{i,j}>0$). The probability $P(\boldsymbol{A})$
incorporates the free degree of the nodes $d_{i}$ and the probability
that $i,j$ are positioned close in space, $\Phi(\boldsymbol{A}).$
The intramolecular propensity constant $c_{i}$ is not dependent on
volume this time, but it is directly equal to cyclisation rate constant
$k_{c}$, according to the reaction equation,
\[
P_{3}\xrightarrow{k_{c}}P_{3}^{c},
\]
where $P_{3}$ denotes a trimer, and $P_{3}^{c}$ a trimer with a
cycle. Finally, the third line of \eqref{eq:P} refers to appearance
of a loop (in graph theoretical sense), which represents a reaction
firing inside a monomer reducing its functionality by one. The sampling
probability \eqref{eq:P} is proportional to the combinatorial number
of possibilities to select a pair of bonds out of the available free
functionalities, leading to the binomial coefficient $\binom{d_{i,free}}{2}.$
The propensity constant $c_{m}$ is directly equal to reaction rate
constant $k_{m}$ in
\[
M_{i}\xrightarrow{k_{m}}M_{i-1},
\]
where $M_{i}$ denotes an $i$-functional monomer.

\paragraph{Gillespie's algorithm}

The basis of Gillespie's algorithm is formed by the concept of the
homogeneous Poisson process, that counts reaction events occuring at rate $\lambda$. In this process, the number
of events occurring in a time interval $(t,t+\tau]$ follows a Poisson
distribution with an associated parameter $\lambda\tau$,
\[
\mathcal{P}(k)=\frac{e^{-\lambda\tau}(\lambda\tau)^{k}}{k!}.
\]
Thus, the waiting times between events are distributed according to
\begin{equation}
p(\tau)=\mathcal{P}(0)=e^{-\lambda\tau}.\label{eq:time}
\end{equation}
Gillespie proposed a stochastic simulation algorithm (SSA), where
an exponential distribution \eqref{eq:time} is used to estimate the time
intervals between reaction firings \cite{gillespie2007}. In the SSA
the event rate is not constant but depends on the reaction propensity
$\lambda_{z},$ (in the current case introduced in \eqref{eq:lambda}).
The actual time between the start and stage $z$ is a sum of $z$
waiting times:
\begin{equation}
t(z)=-\sum _{i=1}^{z}\frac{\ln(u_{i,0})}{\lambda_{i}},\;u_{i,0}\sim U[0,1]\label{eq:wait.times}
\end{equation}
where $u\sim U[0,1]$ refers to a uniformly distributed random variable.
Therefore, the reaction time $t(z)$ is a random variable itself,
that is determined by $z$ samples from a uniform distribution.

On every stage of the reaction $z,$ a pair of nodes $i,j$ is selected
to receive an edge according to the probability distribution $P(\boldsymbol{A})_{i,j},$
given in \eqref{eq:P}. Each of these selections is uniquely defined
by two uniformly distributed random variables, $u_{z,1},u_{z,2}\sim U[0,1].$
This implies that the adjacency matrix at stage $z$ is uniquely defined
by a specific sampling history. More precisely, the adjacency matrix
$A$ on stage $z$ can be put into a bijective correspondence to a
$z\times2$ matrix that defines samples on the preceding stages:
\begin{equation}
\boldsymbol{A}(z)=F(\boldsymbol{U}),\boldsymbol{U}=\left(\begin{array}{c}
u_{0,1},\;\;\;\;u_{0,2}\\
u_{1,1},\;\;\;\;u_{1,2}\\
\dots\\
u_{z-1,1},u_{z-1,2}
\end{array}\right).\label{eq:U}
\end{equation}
This reflects the usual idea behind Monte Carlo simulations that all
networks $F(U)$ for all sample matrices $\boldsymbol{U}$ are considered
as equiprobable. Consequently, any statistical property (\latin{e.g.} average
degree, component size distribution, density) of the network $\phi(\boldsymbol{A})$
may be determined by an integration over the set of sampling variables
$\Omega=[0,1]^{2t}$. If independent sampling matrices $\boldsymbol{U}$
were drawn $N$ times, the integral is approximated by a sum, the
Monte Carlo estimator:
\begin{equation}
\int\limits _{\Omega}\phi(F(\boldsymbol{U}))du\approx\frac{1}{N}\sum _{i=1}^{N}\phi(F(\boldsymbol{u}_{i})).\label{eq:est}
\end{equation}
Here, $\boldsymbol{U}_{i},\;i=1,\dots,N$ are randomly generated instances
of matrix \eqref{eq:U}. The error of estimator \eqref{eq:est} decreases
asymptotically as $N^{-\frac{1}{2}}$ when $N\rightarrow\infty.$

\paragraph{Markov Chain Monte Carlo, simulation of a large system}

The stochastic simulation that has been described above, proceeds
by drawing samples distributed according to the two-dimensional probability
distribution $P(\boldsymbol{A})_{i,j}$ given in \eqref{eq:P}. This
can be done by drawing two successive samples from one-dimensional
distributions. The first sample, $s_{1},$ is drawn from the marginal
distribution $\sum _{i=1}^{n}P(\boldsymbol{A})_{i,j}$; the
second, $s_{2}$ - from the conditional probability distribution $P(\boldsymbol{A})_{s_{1},j}\Big(\sum _{j=1}^{n}P(\boldsymbol{A})_{s_{1},j}\Big)^{-1}.$
This procedure requires recomputing all entries of the matrix $P(\boldsymbol{A})_{i,j}$
, which is computationally the most expensive step of the whole algorithm.
Observe, that even though the adjacency matrix $\boldsymbol{A}$ is
very sparse, matrix $P(\boldsymbol{A})_{i,j}$ with its upper bound
on density $E_{max}/n^{2},$ is a full one.

The computational cost of the algorithm may be significantly reduced
utilising a Markov Chain Monte Carlo (MCMC) algorithm, which is also
commonly used in statistical mechanics and other disciplines. The
MCMC algorithm replaces the draws by a specific Markov process having
$P(\boldsymbol{A})_{i,j}$ as the stationary distribution \cite{brooks2011}.
More specifically, for a fixed matrix $\boldsymbol{A}(z)$, we design
a sequence of pairs $\big(s_{1}(q),s_{2}(q)\big)$ that is initiated
by two random numbers from $1\,\dots,n.$ The transition from pair
$q$ to $q+1$ is realised according to the following principle:
\begin{equation}
(s_{1}(q+1),s_{2}(q+1))=\begin{cases}
\big(s_{1}^{*}(q),s_{2}^{*}(q)\big), & \text{if}\;u(q)<\min\Big(\frac{P_{s_{1}^{*}(q),s_{2}^{*}(q)}}{P_{s_{1}(q),s_{2}(q)}}\frac{P_{s_{1}(q),s_{2}(q)}^{*}}{P_{s_{1}^{*}(q),s_{2}^{*}(q)}^{*}},1\Big)\\
\big(s_{1}(q),s_{2}(q)\big), & \text{otherwise}
\end{cases}\label{eq:MCMC}
\end{equation}
where $u(q)\sim U[0,1]$, and $(s_{1}^{*}(q),s_{2}^{*}(q))$ is drawn
from the proposal distribution $P^{*}(\boldsymbol{A})$. The proposal
distribution coincides with the original $P^{*}(\boldsymbol{A})_{i,j}=P(\boldsymbol{A})_{i,j}$
when $i=j,$ and assumes a simpler form for $i\neq j,$
\begin{equation}
P^{*}(\boldsymbol{A})_{i,j}=\lambda^{*}\cdot\left\{ \begin{aligned} & \,c_{e}\,d_{i,free}d_{j,free}\Big(g_{i}(\boldsymbol{A})g_{j}(\boldsymbol{A})\Big)^{\beta} & \text{ intermolecular };\\
 & \,c_{i}\,d_{i,free}d_{j,free}, & \text{intramolecular},i\neq j;\\
 & \,c_{m}\,\binom{d_{i,free}}{2}, & i=j.
\end{aligned}
\right.\label{eq:Pstar}
\end{equation}
Although the MCMC scheme \eqref{eq:MCMC} provides only a means of
approximate sampling, it possesses one very important feature: it
is not necessary to compute all the elements of matrix $P_{i,j}$
but only certain selected values. This feature allows to considerably save on computational
resources for large adjacency matrices.

The downside of MCMC is that one has to ensure that the Markov chain
\eqref{eq:MCMC} operates in a stationary regime before terminating
it and accepting the final values as a random sample that mimics samples
from $P_{i,j}$.

\section{Discussion/application to linseed oil network}

In this section we discuss the kind of information one may extract
from the random molecular networks generated by the stochastic simulations
using the Random Graph Model as introduced above. The results are
formulated as graph-theoretical properties that may be calculated
with the MC estimator \eqref{eq:est}. These data are useful in improving
our understanding of the system, but also might be used for physical
interpretation of the topological information and its influence on
the observable and measurable properties of the final material.

\begin{figure}[h]
\center \includegraphics[width=0.9\textwidth]{TAG} \caption{ Conceptual model for representation of the linseed oil network. Try-acyl
glyceride is treated as a single node. Depending on composition of
the nodes, they may have maximum functionality ranging between 0 and
9. }
\label{fig:TAG}
\end{figure}

\begin{figure}
\center \includegraphics[width=0.65\textwidth]{degree} \caption{ Evolution of degree distribution during polymerisation. Fraction
of edges, $\chi\in[0,1],$ depicts the progress of the stochastic
process. The final distribution at $\chi=1$ coincides with distribution
of TAG-unit functionalities in linseed oil.}
\label{fig:degrees}
\end{figure}

\begin{figure}
\center \includegraphics[width=0.65\textwidth]{times} \caption{ (\emph{Top:}) time span between reaction events as a function of
the conversion. (\emph{Bottom:}) the conversion of edges versus total reaction
time. }
\label{fig:times}
\end{figure}

In order to demonstrate the RGM approach on a practical case we consider
linseed oil polymerisation, which is more commonly known as the 'drying'
of oil paint. The binding medium linseed oil consists of triacylglycerides
(TAG) and the unsaturated groups on the three fatty acid (C19) chains
cause the TAG-units to polymerise into a network. This curing process
is a complex process involving radicals, oxygen, peroxides, etc.,
which is only partly understood, and only little quantitative kinetic
information is available \cite{iedema2014Mel}. Here, we simplify this
complex process to the coupling of the TAG-unit star monomers according
to an estimated 'chemical' coupling rates, $k_{c}$. and $k_{p}$,
which then is affected by decreasing functionality, distance within
the network and steric hindrance as is now possible with the RGM model.
We expect that this model with strongly simplified kinetics will nevertheless
contribute to a basic understanding of the linseed oil network, as
influenced by its curing behaviour, but also by degradation, for instance
by hydrolysis of the ester bonds. The conceptual model of TAG-unit
network is illustrated in Figure~\ref{fig:TAG}. TAGs contain double
bonds that are responsible for the formation of the cross-links. Since
there can not be more then 3 unsaturations per fatty acid in the TAG
node, the maximum number of connections node $i$ can form is limited
to 9, $0\leq d_{i,max}\leq9$. More precisely $d_{i,max}$ are distributed
according to frequency distribution $W_{n}=\sum _{i+j+k=n}f_{i}f_{j}f_{k},\;n=0,...,9,$
where $f_{i}$ denotes concentration of fatty acids with $i$ unsaturations.
The frequency that we have assumed in this study is based on the natural
abundance of oleic, linoleic and linolenic acid in linseed oil, having,
respectively, one, two and three unsaturations (see Figure 5). From
the point of view of a single node the polymerisation process looks
simple: the degree of each node, $d_{i}$, will transit from 0 to,
eventually $d_{i,max}$. Thus the evolution of the frequency distribution
of node degrees,
\[
f_{i,degree}(\boldsymbol{A})=\frac{1}{n}\sum _{j=1}^{n}\delta(d_{j}(A)-i).
\]
has a simple form: in the beginning of the polymerisation process
the distribution describes disconnected monomers, $f_{i,degree}(\boldsymbol{A}(0))=0,i>0,$
and $f_{0,degree}(\boldsymbol{A}(0))=1.$ At the full conversion the distribution
of node degrees approaches the distribution of functionalities of
pure monomers (Figure~\ref{fig:degrees}).

The global properties of the network have less trivial dynamics. In order to extract various network properties we simulated ensembles of $10^4$ monomers and averaged the results over 100 of such ensembles. These
global properties in turn affect reactivity rates and the overall
kinetics of the system. Even the timing of reaction events, given
in \eqref{eq:wait.times}, strongly depends on the ratio of rate constants
$\nicefrac{k_{c}}{k_{p}}.$ For time intervals between the reaction
events and the overall conversion/reaction time plot see panels in
Figure~\ref{fig:times}. Alternatively, instead of increasing $\nicefrac{k_{c}}{k_{p}},$
one may consider to decrease concentration of monomers, which by
enhancing cyclisation has an identical effect on the system.

\paragraph{Gel transition}

\begin{figure}
\center \includegraphics[width=0.7\textwidth]{gel_path2000} \caption{ Gel point conversion as a function of reactivity ratio as obtained
from the RGM simulation, confidence interval for $p=0.95$. The dashed lines indicate the point when the average shortest path length is maximum. }
\label{fig:gelpoint}
\end{figure}

Polymerisation of multifunctional monomers is known to pass through
the gel transition: a transition that drastically affects the physical
properties of the material. The gel point is experimentally well observable
due to considerable differences in the material properties between
the gel and sol phases, such as solubility and viscosity \cite{bower2002}.
Hence, it is a common practice to define the gel transition point as
a point where the observable properties change. Here, we will define
the gel transition entirely on topological properties of the corresponding
graph. Let's consider a sequence of adjacency matrices $A(n)$ constructed
for identical input parameters and given number of nodes, $n$. We
associate the gel part of the network with the giant component (in
the sense of random graph theory \cite{spencer2010}). Thus, the gel
point is the time instant in the reaction progress at which the giant
component emerges. More specifically, the largest component in the
network is called gel if its size $S_{1}(n)=O(n)$ when $n$ approaches
infinity. Furthermore, if gel is present, the size of the second largest
component is $S_{2}(n)=O(\log(n))$. In the pre-gel regime both $S_{1}(n)$ and $S_{2}(n)$
are equal to $O(\log(n)).$ Thus distinguishing between pre- and post-gel
regimes is equivalent to distinguishing between two asymptotical modes
of $S_{2}(n)/S_{1}(n)$ when $n\rightarrow\infty.$ We use this asymptotic
estimates to determine whether the system operates in the gel regime
for a given progress of the reaction $z$ (or conversion $\chi$),
\[
g_{01}(\chi)=\begin{cases}
0, & \text{if }\;S_{2}(n)/S_{1}(n)\rightarrow const\\
1, & \text{if }\;S_{2}(n)/S_{1}(n)\rightarrow0
\end{cases}
\]
where $S_{1}(n)$ is the size of the largest component for a system
with $n$ monomers and $z$ edges, and $S_{2}(n)$ - size of second
second-largest component. Clearly, $g_{01}(\chi)$ itself is a random
variable and its mean has to be estimated by the MC estimator. When
the number of samples in the MC estimator $N$ becomes large, the
gel indicator function $g_{01}(\chi)$ approaches the Heaviside function
$H(\chi-\chi_{g}),$ where $\chi_{g}$ is a 'breaking point' separating
the pre-gel and gel regimes. In practice, for a fixed $N$ , we identify
$\chi_{g}$ by solving a least-squares problem that minimises the
residual $\int\limits _{\chi}|H(\chi-\chi_{g})-g_{01}(\chi)|d\chi.$

The gel point conversion $\chi_{g}$ is plotted versus $\nicefrac{k_{c}}{k_{p}}$
in Figure~\ref{fig:gelpoint}. A clear discontinuity can be seen:
the fraction equals to 0 up to the gel transition. Higher levels of
$\nicefrac{k_{c}}{k_{p}}$ postpone the gel transitions up to the
point when gelation does not occur at all. This is a logical result,
since a higher $k_{c}$ implies a stronger intramolecular cross-linking,
which obviously competes the gel formation.

\paragraph{Molecular sizes}

As shown in Figure~\ref{fig:times} the reaction rates are not constant,
but slow down in a non-linear fashion. One of the causes for this
effect is the emergence of large, connected components that restrict
their internal monomers in reactivity. Molecular sizes in terms of
number of monomers are equal to the number of nodes in connected components
of the network that are not connected to each other. It is straightforward
to identify the connected components of a graph in linear time using,
for instance, breadth-first search\cite{jungnickel2008}. Figure~\ref{fig:sizes}
shows how the distribution of sizes of these components (molecular
size distribution) evolves in time.
\begin{figure}
\center \includegraphics[width=0.8\textwidth]{sizes1} \includegraphics[width=0.8\textwidth]{sizes30}
\includegraphics[width=0.8\textwidth]{sizes60}

\caption{Evolution of size distributions (excluding gel) for three values of
$\nicefrac{k_{c}}{k_{p}}$. }
\label{fig:sizes}
\end{figure}

Eventually, a connected component of the same order of magnitude as
the whole network emerges (the giant component). The point in time/conversion
when this event takes place marks the usual gel transition. The size
of the giant component increases rapidly as it becomes connected to
the rest of the components. Reactions between large components (including
the giant component) dominate over reactions between small-sized components.
This is why the size distributions in Figure~\ref{fig:sizes} shifts
back towards the origin after the gel transition. Thus, one observes
that the gel point detected from the topology alone, as explained
above, indeed marks the turning point of the size distribution. For
high values of $\nicefrac{k_{c}}{k_{p}}$ the gel transition is postponed
up to the full conversion (Figure~\ref{fig:sizes}b) or does not happen
at all (Figure~\ref{fig:sizes}c). According to the general convention
the data presented in Figure~\ref{fig:sizes} excludes the giant
component itself as it only refer to the sol part. In a complementary
fashion, the fraction of nodes incorporated in the giant component
is given in Figure~\ref{fig:gel.fraction}.
\begin{figure}
\center \includegraphics[width=0.7\textwidth]{gel_fraction} \caption{ Fraction of monomers involved in giant component (gel) as a function
of conversion for various values of $\nicefrac{k_{c}}{k_{p}}$. }
\label{fig:gel.fraction}
\end{figure}

\paragraph{Structure of the network}

The characterisation of the polymer network in terms of numbers and
sizes of connected components is a widely used in the polymer community.
However, in cases where the topology is not a priori known (\latin{e.g.} linear
or regularly branched polymers), we have to invent new characteristic
measures to describe how individual monomers are arranged inside the
single connected component that the network eventually becomes. Although
being randomly interconnected, the nodes may occasionally line up
into certain \emph{motifs} of special interest. The search for motifs
that are frequently occurring in the network is a new structural characterisation
method that is easy to quantify. For instance, the degree distribution
is nothing else than the frequency of nodes having zero, one, two,
etc., adjacent nodes.

Besides structural information, the size distribution of linear fragments
plays an important role in the elasticity of polymeric networks \cite{gunduz2008,ngai1994}.
The number-average length of linear fragments strongly depends on
the conversion; it increases at early stages of the reaction and eventually
decreases to almost to 1. High values of $\nicefrac{k_{c}}{k_{p}}$
are associated with shorter linear fragments at intermediate stages
of conversion, but ultimately, with longer linear fragments at later
stages (Figure~\ref{fig:linear}).
\begin{figure}
\center \includegraphics[width=0.7\textwidth]{linear} \caption{ Average length of linear fragments as a function of conversion for
various values of $\nicefrac{k_{c}}{k_{p}}.$ The illustration depicts
a sample network with four linear fragments of lengths: 1, 2, 4, 4. }
\label{fig:linear}
\end{figure}

Another motif-related property is the local clustering coefficient.
For node $k$, it is equal to fraction of pairs connected to $k$
that are also connected to each other(\latin{i.e.} forms a triadic clousure),
\[
c_{k}(\boldsymbol{A})=\frac{\sum _{i<j}\boldsymbol{A}_{i,j}\boldsymbol{A}_{i,k}\boldsymbol{A}_{j,k}}{d_{k}(d_{k}-1)}.
\]
This coefficient is a number between 0 and 1, see the example of Figure
12. In order to transit from local clustering per node to average
clustering of the whole network we apply averaging $\boldsymbol{A},$
$c(\boldsymbol{A})=\frac{1}{n}\sum c_{k}(\boldsymbol{A})$.
\begin{figure}
\center \includegraphics[width=0.7\textwidth]{clustcoeff} \caption{Clustering coefficient as a function of conversion for various values
of $\nicefrac{k_{c}}{k_{p}}$. Three examples of a 4-node network
are given. The clustering coefficient of the node marked 'EGO' is
equal to 0, 1/3, or 1 depending on the connections between EGO's neighbouring
nodes. }
\label{fig:clustcoeff}
\end{figure}

The average clustering coefficient gives a good idea of small scale
patterns that may arise in the network. At the beginning of the polymerisation,
the average clustering increases monotonically in time. For systems with
low values of $\nicefrac{k_{c}}{k_{p}}$ the average clustering becomes
significant only in the gel regime. It is interesting to note that
the average clustering decreases at conversions close to 1. This effect
is caused by steric hindrance in the ultimately overcrowded network
that eventually suppresses clustering.

\begin{figure}
\center \includegraphics[width=0.98\textwidth]{path_length} \caption{ Average path length as a function of conversion. Left panel depicts
results in the case of no intramonomeric reactions, right panel -
high intramonomeric reaction rate ($\nicefrac{k_{i}}{k_{p}}=10$). }
\label{fig:path.length}
\end{figure}

\paragraph{Average path length}

The average path length of a network is a concept that defines the
average number of steps along the shortest paths for all possible
pairs of network nodes,
\[
L(\boldsymbol{A})=\frac{1}{n(n-1)}\sum_{i\ne j}p_{i,j},
\]
Here, $p_{i,j}$ denotes the distance between nodes $i$ and $j$
in network $\boldsymbol{A}$, if the nodes $i,j$ are not connected,
$p_{i,j}=0.$ One may think of average path length as a measure of
'network density'. Various theories connect the shortest path to observable
data related to light scattering, radius of gyration, etc. \cite{rouvray2002}.
Figure~\ref{fig:path.length} shows how the normalised average path
length evolves during reaction progress for various values of $\nicefrac{k_{c}}{k_{p}}.$
Note that even though the average path length is normalised by dividing
by the total number of nodes $n$, it still depends on $n$ and special
care has to be taken, when comparing the average path lengths of two
systems of different size. The two panels presented in Figure~\ref{fig:path.length}
represent cases with and without intramonomeric reaction. A pattern
that is typical for many polymer properties, can be seen: the average
path increases in the pre-gel regime and decreases thereafter. However,
closer investigation reveals that the maximum of the average path
does not exactly coincide with the gel transition but is shifted.
This is demonstrated by Figure~\ref{fig:gelpoint}, where the conversion
at maximum average path length and at the gel point is plotted for various
values of $\nicefrac{k_{c}}{k_{p}}.$ The intramonomeric reaction leads to a more sparse network
by consuming functional groups, and thus, reducing functionality of the monomers. Since the intramonomeric reaction
is not hindered by the resulting topology, the influence of this reaction
is most visible after the gel transition that is reflected by a suppressed
decrease of the average path length (see Figures~\ref{fig:path.length},\ref{fig:gelpoint}).

\begin{figure}
\center \includegraphics[width=0.8\textwidth]{modularity} \caption{ Modularity as a function of conversion for various values of $\nicefrac{k_{c}}{k_{p}}.$
The two samples depict networks with low/high modularities. }
\label{fig:modularity.pdf}
\end{figure}

\begin{figure}
\center \includegraphics[width=0.8\textwidth]{cluster_distribution}
\caption{ Cluster size distribution for various levels of conversion }
\label{fig:cluster.distribution}
\end{figure}

\paragraph{Structural non-homogeneity}

The evolving structure of networks may be more or less homogeneous,
for instance microgels may (temporarily) evolve as a form of heterogeneity\cite{seiffert2015,asai2013}.
It turns out that structural non-homogeneity is an inherited property
of the polymerisation models that account for distance between monomers.
Certain values of input parameters lead to topologies that, although
being connected, can be partitioned into distinct clusters, comparable
to microgels. These clusters are loosely connected between each other
but are densely interconnected within themselves. This effect can
be achieved when cyclisation dominates propagation (high values of
$\nicefrac{k_{c}}{k_{p}}$). For example, in diluted systems the chance
that a monomer will connect to its direct neighbour (or another node
from the same component) is higher than the chance of encountering
a disconnected component. One way to measure the degree of inhomogeneity
(clustering) present in the network is the measure of \emph{modularity,}
which may be calculated using a special optimisation algorithm involving
the adjacency matrix \cite{newman2006}. The evolution of modularity
as presented in Figure~\ref{fig:modularity.pdf} exhibits the increase-decrease
pattern with a maximum at intermediate levels of conversion. High
values of $\nicefrac{k_{c}}{k_{p}}$ postpone the maximum towards
the full conversion, up to the point where the modularity increases over
the whole time span of the process. It can be observed that a certain
degree of structural inhomogeneity is prevalent even for very low
values of $\nicefrac{k_{c}}{k_{p}}$ at intermediate conversions.
However, at the final conversion a structure with distinct clusters is
preserved only for high values of $\nicefrac{k_{c}}{k_{p}}.$

As similar to the component size distribution, we may consider a cluster
size distribution that evolves over time with reaction progress (Figure~\ref{fig:cluster.distribution}).
Even though the value of $\nicefrac{k_{c}}{k_{p}}$ defines how well
separated the clusters are, it has little effect on cluster sizes.
However, cluster sizes are strongly affected by excluded volume
constant $\alpha,$ that appears in \eqref{eq:Phi}.

\begin{figure}
\center \includegraphics[width=0.8\textwidth]{scaling} \caption{Scaling fraction vs $k_{c}/k_{p}$ as obtained for three values of
chain stiffness constant, $\alpha$. The bands represent 0.9 coinfidence
interval around mean value based on Monte Carlo sampled data. }
\label{fig:scaling}
\end{figure}

\paragraph{Scaling}

The average path length is related to the gyration radius of individual
molecules and to the density of the gel in space. The scaling of the
average path length for gel networks has to satisfy the inequality
\eqref{eq:scaling}. Testing this inequality for various values of
input parameters helps to narrow down the range of valid values for
these parameters. Figure~\ref{fig:scaling} shows the normalised
scaling factor $\frac{a'}{a^{1/3}}$ of the final topology ($\chi=1$)
as a function of $\nicefrac{k_{c}}{k_{p}}.$ Low cyclisation to propagation
ratios lead to topologies that possess a scaling factor under critical
level, which implies that they are too dense to become embedded in
a three-dimensional space.

\paragraph{Timeline of network topology}

Even though the final degree distribution of the monomers is fixed,
various input parameters yield a vast range of different network topologies.
We will now comment on the complex process assembling these topologies
by highlighting a few important stages. The input parameters influence
the positions of these stages in time/conversion but the order remains
unaltered. In all cases the polymerisation obviously starts with all
nodes disconnected, $d_{free}=d_{max}$ (see Figures \ref{fig:timelineA}-I
and \ref{fig:timelineB}-II). We now separately consider scenarios
for weak ($k_{c}/k_{p}$ is small) and strong ($k_{c}/k_{p}$ is high)
cyclisatios. When cyclisation is weak, shortly after the start of
the polymerisation, many small disconnected components with tree-topology
are formed (Figure~\ref{fig:timelineA}-II). The sizes of the components
increase but remain $o(n)$; the average length of the linear fragments
continues to increase until it reaches the global maximum (Figure~\ref{fig:timelineA}-III).
The giant component of size $O(n)$ emerges, while cycles start to
appear in all components (Figure~\ref{fig:timelineA}-IV). These
cycles remain being short and consequently do not considerably affect
the average shortest path length. As the network is becoming more
dense, long cycles start to appear occasionally. These long cycles seem
to force the average shortest path length to start declining (Figure~\ref{fig:timelineA}-V).
As the density of the network continues to increase, modularity reaches
its maximal value: clear structural non-homogeneity is present. (Figure~\ref{fig:timelineA}-VI).
At the full conversion the network consist of a single connected component.
The last reactions, even at very low rates, connect the previously
formed clusters and thus reduce the non-homogeneity. The final network
is quite dense and homogeneous (Figure~\ref{fig:timelineA}-VII).

Whereas in the case of strong cyclisation, small molecules with
many cycles form shortly after the start of the reaction (Figure~\ref{fig:timelineB}-II).
The disconnected components grow in size for a much longer span of
conversion before gel transition is achieved. Most of the new edges
appear inside the connected components, enhancing the clustering coefficient
(Figure~\ref{fig:timelineB}-III). The giant component emerges at
a relatively high conversion and it has a non-homogenous structure.
(Figure~\ref{fig:timelineB}-IV). Finally, at the full conversion the
major part of the network is occupied by the giant component, although
other components of small size, $o(n)$, remain present too. The structure
is highly non-homogenous. The full conversion is also the point at which
the modularity and the average shortest path length reach maximum
values.
\begin{figure}
\center \includegraphics[width=1.1\textwidth]{timelineA} \caption{Timeline of network assembly, $k_{c}/k_{p}=10$ (weak cyclisation)}
\label{fig:timelineA}
\end{figure}

\begin{figure}
\center \includegraphics[width=1.1\textwidth]{timelineB} \caption{Timeline of network assembly, $k_{c}/k_{p}=300$ (strong cyclisation)}
\label{fig:timelineB}
\end{figure}

\paragraph{Role of excluded volume principle}

An important and distinct feature of the RGM approach is that reaction
rates depend on distances between monomers that are not known directly,
but are implicitly inferred from the topological configuration of
the network. In order to define the connection between topology and
physical distance we employ the 'random walk' concept. The question
arises, which of the random-walk models should be chosen: the commonly used random coil model, $\Phi^{*}$ as given
in \eqref{eq:32}, or the self-avoiding random walk (excluded volume
principle $\Phi,$ Equation \ref{eq:Phi})? A further question is:
how would accuracy be improved in the final results by adopting the
more realistic model? It turns out that the choice between the two
models is not just a matter of accuracy, since using
the random coil model would lead to completely erroneous results for
large networks. The reason behind this phenomenon is the asymptotic
behaviour of the models on increasingly large distance.

Let's consider a connected infinite network of monomers being homogeneously
distributed in space. Two spheres of radii $p,p+\delta,$ are centred
at a selected monomer. The number of monomers positioned in the volume
between the two spheres is proportional to $\delta p^{2}$ when $\delta\rightarrow0$.
Thus, for any small $\delta>0,$ the asymptotical behaviour of the
reaction probability is given by,
\[
\begin{aligned}\label{eq:ref}\lim\limits _{p\rightarrow\infty}\delta p^{2}\Phi^{*}(A) & =\infty,\\
\lim\limits _{p\rightarrow\infty}\delta p^{2}\Phi(A) & =0.
\end{aligned}
\]
In other words, in a model equipped with $\Phi^{*}(A)$ (unlike $\Phi(A)$)
the reaction probability increases for increasing distance between
the reacting monomers. This eliminates the random coil model
$\Phi^{*}$ as a proper candidate to be employed in simulations of
infinite network.

\section{Conclusions}

We designed a new stochastic approach for modelling molecular networks
that lies at the intersection of two disciplines: polymer reaction
engineering and network science. The kinetics is accounted for by
a Markov Chain accelerated Gillespie Monte Carlo routine. The main
output is the network topology as presented by an adjacency matrix.
Utilising adjacency matrices allows analyses with the whole ensemble
of tools coming from graph theory: average shortest path, clustering,
modularity, giant component transition, etc. These results are then
used to estimate how the spatial position of monomers incorporated
in the network affects their reactivity. The model turns out to reproduce
polymer (network) characteristics like gel point transition, molecular
size distributions, length of linear fragments. Additionally, the
model gives insight into nuances of topological structure, for instance:
the measure of non-homogeneity or cluster (micro-gel) distribution.

We have successfully applied the RGM algorithm to linseed oil polymerisation,
more commonly known as the drying of oil paint. This process produces
a TAG network that involves monomers of various functionality up 9,
forming the building blocks the network. We showed that even though
the degree distribution is fixed, as it is dictated by the natural
abundance of fatty acids of different unsaturation in linseed oil,
a wide range of different topologies emerge. Strong cyclisation (caused
for instance by diluting the system) leads to highly non-homogeneous
networks that are also less robust. In contrast, weak cyclisation
causes more robust and homogenous structures.

If intra-monomeric reactions, that reduce functionality
of monomers, take place even at a very slow rate they will considerably contribute
to the final structure of the network in the post-gel regime, when rates of all other competing reactions are slowed down.

The resulting network topology is presented as an adjacency matrix,
which allows employing graph theory oriented mechanical models
to study elastic, transport, or rheological properties of the materials.

The main outcome of this work is an algorithm generating molecular
topologies. However, alongside we have introduced a new vocabulary
inspiring a new way of discussing and comparing random molecular topologies,
even regardless of the the methodology adopted to construct them.

\section*{Acknowledgement}

This study has been funded by the PAinT (Paint Alterations in Time)
project as part of the NWO (Dutch Science Foundation) Science4Arts
Program. The authors (Jorien, Ivan, Joen, Piet) thank dr.
Katrien Keune (Rijksmuseum/UvA), dr. Annelies van Loon (UvA), dr.
Maartje Witlox-Stols (UvA), and Marta Mouro, MSc (UvA) for the inspiring
discussions that have led to the problem formulation.
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


The BizPal Company, LLC announces its founder, Becky Bruso will be presenting a session at the upcoming Nichols College Empowering Women in Business Conference in Dudley, MA.
Becky will be conducting her session, entitled: The Art of Networking: Conquering Your Inner Scaredy-Cat.
The Empowering Women in Business conference provides women with the dynamic opportunity to advance their career, grow their business, and help design their future by pursuing professional development within a welcoming environment of career-focused women leaders. Women can expect to gain a newfound inspiration, a sense of achievement, and an increased business network within the local community.
Click here to find out more about this session and the conference.


A mass-wasting event can occur any time a slope becomes unstable. Sometimes, as in the case of creep or solifluction, the slope is unstable all of the time and the process is continuous. In other instances, triggering events can occur causing sudden instability.
Modification of a slope, either by humans or by natural causes, can result in a slope angle that is no longer tenable. A sudden shock, such as an earthquake, may trigger slope instability. Minor shocks like heavy trucks rambling down the road, trees blowing in the wind, or human-made explosions, can also serve this purpose over time. A mass-wasting event can then restore the slope to its natural angle of repose.
Example: In the 1870s there was a large demand for slate to make blackboards throughout Europe. To meet this demand, miners near Elm, Switzerland, began digging a slate quarry at the base of a steep cliff. Slate is a metamorphic rock with an excellent planar foliation that breaks smoothly along the foliation planes. By 1876 a "v"-shaped fissure formed above the cliff, about 360 m above the quarry. By 1881, the quarry had been excavated to where it was 180 m long and 60 m into the hill below the cliff. The "v"-shaped fissure had opened to 30 m wide.
Falling rocks were frequent in the quarry and there were almost continuous loud noises heard coming from within the overhang above the quarry. Realizing that the slope had become unstable, the miners abandoned the quarry, thinking that the rock mass would probably collapse and fall.
On September 11, 1881, the mass of rock above the quarry suddenly did fall, sliding rapidly down the slope. But it did not stop when it hit the quarry floor. Instead, the 10 million m3 of rock broke into pieces and rebounded into the air. Residents in Untertal, on the opposite side of the valley, saw the rebounded wall of shattered rock coming uphill at the them. The rock continued up the walls of the valley and buried the residents of Untertal alive. The avalanche then turned and ran an additional 2,230 m as a dry avalanche traveling at 180 km/hr and burying the entire village of Elm. The event killed 148 people.


text:
X5 Retail Group reports net retail sales increase
By Planet Retail2012-01-25T07:00:00+00:00
X5 Retail Group has reported a 32% increase consolidated net retail sales of RUB452.5bn (£9.3bn) for 2011. Like-for-like sales rose 6%, with a 1% drop in customer traffic and a 7% increase in average basket. However, like-for-like sales growth at Pyaterochka more than halved to 5% compared with last year while Perekrestok posted a 10% like-for-like rise. The grocery retailer added a total of 577 stores.
Analysis: Is Lidl starting to close the gap on Aldi in the discounter battle?
Over the last 12 months, momentum has shifted away from Aldi towards Lidl in the battle of the discounters. Retail Week looks at what Lidl has done to close the gap on its rival.
The chief executives of the UK's largest grocers have warned the government of "significant disruption to food supplies" to Northern Ireland after March 31 unless there is a closer agreement with the EU.
More International
Analysis: Is UK retail turning its back on internationalism?
With both Tesco and Superdry pulling out of international ventures on the same day, Retail Week asks whether UK retail is scaling back its global ambitions.
Coronavirus to wipe £1.7trn off global retail sales
The coronavirus crisis will wipe $2.1trn (£1.7trn) off of global retail sales in 2020, new data has predicted.
Analysis: Chinese etailers' awe-inspiring coronavirus reaction
The coronavirus outbreak has sparked fears and headlines worldwide since emerging in the Chinese city of Wuhan at the start of the year.

redpajama_set_name: RedPajamaCommonCrawl


What is our purpose, what defines our course and what does life demand from us? What is the reason of conflict? Do freedom and free will actually exist? What is the power dictating the forms and ways of human relationships? Can we really reach the core of truth and love? Dimitris Vithos' speech on the TEDxPatras 2018 stage will be an attempt to deal with all these questions. Using simple and practical examples showcasing the value of acceptance, Dimitris will explain why acceptance of ourselves and of "others" is the key element to a path leading us from acting as individuals towards becoming part of the big "Whole". Getting there can ultimately allow us to access true understanding of the whys and hows of our very existence, of human relationships and of that power determining the whole universe… Come and be part of the TEDxPatras audience to witness a different speech, approaching "Values" through the prism of philosophy.
Dimitris Vythos has been acting as Creative Director and Copywriter at a Sentimental Branding level, with expertise and international experience in implemented communication and advertising. He is a postgraduate specializing in postmodernism, stereotypes and Social Media and Public Advocacy. He is actively engaged in traditional archery, Ninjutsu and Chi-Gong. A "Seeker of Truth", as he defines himself, Mr Vythos has participated in various mystical and philosophical communities, while he considers himself an empiricist agnostic. His family origins come from Zakynthos, but he was born in Australia, where he has lived for many years. Currently, he works and lives in Cyprus.


Brad and Sheena had been traveling in their 1984 Volkswagen bus for eight months, all the way from Arizona, USA, to Columbia in South America, when their transmission failed, leaving them stranded in a remote area somewhere in Columbia. To make things worse, the Colombian government had passed a law prohibiting the importation of used car parts, making it virtually impossible to get a new transmission. Ironically, it turned out that this was one of the better things that happened to them on their trip. Not the transmission failure but what followed as a result of it.
Brad and Sheena met Hernando and Constanza, a farmer couple that was able to offer them a little cabin in the middle of the Colombian mountains. "Our patio overlooked a meadow of tall green grass with weeping willow trees, grazing dairy cows, and little frolicking baby cows," wrote Brad in a blog post."Each day, our hosts would show up at our door bearing housewarming gifts. Some days they would have a pitcher of fresh squeezed juice from one of their fruit trees, or a platter of fresh fruit. One day Hernando dropped off a bag of fresh coffee that he had just roasted using beans from a neighboring farm...Somehow, the importance of choosing an option for how to fix our transmission seemed a distant second to living the good life...Being stranded in Susacón was really a blessing and we didn't much care to leave."
It seems obvious to ask why you would ever travel the world in a car that's almost 30 years old and destined to break down at some point. But when you consider what happened to Brad and Sheena in Columbia, it seems to make sense: It's all about the journey and what you experience on the way. It's an adventure, risky and uncertain of what's going to happen. The many different vehicle failures they experienced on their way forced them to stop at places they would've otherwise probably just passed by.
Let's rewind eight months and find out how it all started. Brad and Sheena decided to quit their jobs at the end of 2011 to travel the world in their 1984 Volkswagen Vanagon. "The idea was borne of coincidence, an unsettling feeling of complacency, and a festering itch for adventure," said Brad. How do you finance a trip like that? Both of them actually saved money for two and a half years, following a strict savings plan. "We identified the expensive aspects of our life, and created an attack plan to kill (or severely maim) each one. We were ultimately able to reduce our spending by more than half. This allowed us to put all of my (Brad's) paychecks into savings while we lived off of Sheena's," explained Brad in a blog post that outlines the entire plan.
Using a 30-year old VW bus is not the only way Brad and Sheena tried to challenge themselves. Brad told us in an email that they actually drove all through Central America without a map (not even a paper one - just the overview maps in their Lonely Planet). "We figured it would be a good way to get in touch with locals, by stopping to ask directions. Well, it did get us out of the car talking to locals, but there was one thing we didn't count on: most locals don't have cars, and have no idea how to get to the next place. There's also another problem: they're too proud to tell you that they don't know, so they just make stuff up," wrote Brad in an email. After a couple of months in a state of continual lost they had enough and bought a Garmin nüvi in Panama. "The rest is history; I won't turn the key until the GPS is up and running now," Brad continued.
Now how did Brad and Sheena solve the problem with their transmission in Columbia? Hop over to their blog to check out the full story. We'll make sure to follow Brad and Sheena as they travel through Asia over the next couple of months, so stay tuned for more posts right here on the Garmin blog. If you enjoy what they write on their blog, you can also help them fund their book project on Kickstarter.


text:
Bickley: Special bond links Cardinals' Patrick Peterson, Tyrann Mathieu
Dan Bickley
azcentral sports
Patrick Peterson and Tyrann Mathieu are more than longtime teammates. They are like brothers. They are bonded in triumph and tough times.
They share something special, and they knew it from the first moment they met in Baton Rouge, La., when Peterson hosted Mathieu's official campus visit to LSU.
"I'll never forget what he told me," Peterson said. "He said, 'Hey, man. Are you Patrick?' We had just met. And then he says, 'Man, I'm going to be better than you.' "
Peterson laughs hard at the memory. Mathieu smiles sheepishly, attempting to crawl inside his skin.
"I don't know what made me say that," Mathieu said. "He was the Number 1 recruit coming out of high school. I wanted to be just like him."
The gap has never been closer, and same goes with the friendship. Peterson is now 25, a polished Pro Bowler with a big contract and a new daughter. He's a shut-down cornerback in the mold of Deion Sanders, in the midst of a tremendous season. He has corrected health issues that marginalized his transcendent talent, and the results are obvious.
You don't see quarterbacks testing him or fellow competitors taunting him anymore.
"He's that premier cornerback that everybody hopes they have," Mathieu said. "He's been showing that since high school, showing he's the best player on the football field. He makes my job easier because I'm not chasing the best receiver."
Peterson also has a stronger locker room presence in 2015, assuming a more forceful brand of leadership. He even cancelled Victory Monday for all defensive players following a win against the 49ers, and nobody said a word.
"He's definitely more vocal this year," Mathieu said.
Arizona Cardinals vs. Philadelphia Eagles scouting report, prediction
Meanwhile, Mathieu is fulfilling his own prophecy of a "savage season." He is a tenacious force on the field, liberated from that balky knee brace he was forced to wear last season. He has successfully flushed his addiction problem, becoming the gold standard of discipline, drive and accountability. Once, Peterson had to vouch for Mathieu, putting his professional reputation on the line for a player who had been kicked out of college. Mathieu is proof that second chances can work in the NFL.
"Ty is a great person," Peterson said. "A lot of people have this guy misunderstood. That mistake he made at LSU, in the media's perspective, made him what he was. But that wasn't Tyrann. I knew he was a great kid at heart. I knew he always meant well. It's just that some of the decisions he's made have haunted him.
"By me standing on the table for him, I knew he wasn't going to make that mistake anymore because he didn't want to miss football anymore. I knew he wanted to be back in the spotlight again, and have that 'Honey Badger' type of year he had in 2010, going out there, taking the ball away, just being so focused on football. It was just about getting his head on straight."
Challenges are nothing new for Mathieu. Growing up in New Orleans, he was picked on because of his diminutive size and for the lighter tone of his skin. His biological father is in prison, while his biological mother recently pleaded guilty to aiding and abetting a case of Medicare fraud in New Orleans.
At his very lowest, Mathieu lived with Peterson's parents in Florida, where he reconfigured his life.
Cardinals' offensive line can't get no satisfaction
"My biological mother, who is a great woman, we still have a relationship," Mathieu said. "It just isn't a motherly-son relationship. I'm adopted by my uncle and aunt, who I look to as my parents and people who really did everything they could to put me on the right path. I'm really grateful to them. I don't think I'd be here without them, Patrick and a lot of other people."
Yet it's also evident that both players are tired of the mentor-protégé narrative. They are less than two years apart in age, and more like contemporaries. And truth is, Mathieu has done most of the heavy lifting, reclaiming his beloved status in the sport.
"It's his size, his attitude, the Badger-hawk (haircut), the way he attacks the game at all times," Peterson said. "He plays with such a high energy at all times. No matter what point in time it is during a ballgame, you know what you're going to get from Number 32. And I believe that's what the world likes.
"He really resembles the Honey Badger, flying around the field, taking the ball when he wants, getting a thousand tackles, and he's always on TV. So what do you not like about him?"
Peterson even liked how Mathieu approached him in 2010, when Peterson was the big man on campus, often asked to host the really important recruits. That's when Mathieu threw down a competitive gauntlet, and almost six years later, life has come full circle.
Bickley: Sellout sports fans are hurting Valley's image
Mathieu has been mentioned as an NFL Defensive Player of the Year candidate. Except some think Peterson is the Cardinals' defensive MVP. So who is better than whom?
Someone's going to have make a very tough call.
"We knew one of us was going to be up for (the award)," Peterson said. "Back in February, we said it's going to take both of us to get where we needed to be. We set the goal for one of us to have the opportunity to hoist that defensive MVP trophy. He's doing a great job. I'm doing a great job … but I'm definitely pulling for Ty to win."
Said Mathieu: "It doesn't matter who wins. We're both going to share anyway."
Like brothers.
Reach Bickley at dan.bickley@arizonarepublic.com or 602-444-8253. Follow him at twitter.com/danbickley. Listen to "Bickley and Marotta," weekdays from noon-2 p.m. on Arizona Sports 98.7 FM.


M and I had date night last night. We went to hear a friend of mine who is a blues singer. I was one of the two friends who showed up to support her. It wasn't like they didn't know she was performing. I had personally called and sent text messages to all of our mutual friends to remind them. Some said they were coming and didn't..others didn't respond. I could tell she was disappointed with the low turnout. As I looked around the club, it made me think about friendship and how as humans, we take our relationships with others for granted.
My friendships are precious to me. I surround myself with very few people that I can actually call friends. I have many acquaintances but in my opinion, the word friend is so overused. I learned a long time ago that not everyone can wear the friendship label because like any relationship it requires work and not everyone is prepared to do the work of friendship. It means remembering birthdays or important dates, losing sleep occasionally while a friend talks through their issues and it means pitching in when needed.
I sometimes think that technology is a curse. With emails, text messages and the like, most people in society no longer take the time to handwrite thank you notes, or visit with friends the way it was done 30 or 40 years ago. We're all just too busy. However, life is short and we all need to remember to appreciate those around us that stand tall and strong with us through any storm.
I've had friends who have traveled around the world with me to witness the traditional Kerala ceremony between M and I. That's some love. I've had friends who have stayed up all night with me while I cried over a broken relationship. That's even more love. LOL.
M and I differ on the friendship thing. He's a loner and from what I've been told by him and his family, he always has been for the most part. He can take people or leave them and will quickly call me an agony aunt when I'm nurturing a friend through hard times. It's important for me to have friendships and if I hadn't seen it over the last few years I wouldn't believe that M could live without having friends. My dear sweet nerdy M simply reads, plays with the dogs, does Sudoku or watches television in his spare time. My ideology is that if he's happy with his friendless life, it's fine with me. My job is to be friend, wife, nurse and psychologist whenever he needs it.
How do you nurture your friendships? Do you feel the art of friendship is changing? Until the next blog..


Canvas bags are yet another type of bags that are environment-friendly in nature. These are biodegradable and can be reused and recycled over and over again. Canvas bags are great alternatives to those poisonous plastic bags. Therefore, canvas bags are effective choice to bring about a positive change in the environment.
Canvas bags are tough and do not rip easily like plastic bags which are generally used at avariety of stores. These bags consist of handles which make easier for its users to carry them and travel long distances. Sometimes, canvas bags are used as an alternative to shopping carrier baskets at some stores. By using these bags, consumers can conveniently estimate the amount of products that they could carry and many times they are amazed at the capability of these canvas bags to be able to provide ample space. Another strategy that consumers could follow is to keep canvas shopping bags always with them for all those unplanned and impulse buying. The more convenient such habits would be, the more likable are these habits to be continued.
Canvas tote bags are also great for outings. These can be used to carry on- perishable food and other items which are necessary for a day out. These do not easily get worn out and can withhold substantial amount. As the canvas bags do not rip off easily, they can be reused over many times by quick washing methods.
Canvas bags are also agreat replacement for gift bags. As we know canvas bags are reusable, the bag itself serves as a gift which further can be used for shopping, gardening, craft items or for any other household purposes. It also serves as a green alternative to all those attractive gift wrapping papers.
Giving away food to friends are relatives to distant places cam also be another great utilization of canvas bags. People generally use plastic or paper bags to transport food items from one place to another. These paper bags or plastic bags can wear out easily and thus become useless. Moreover, these paper and plastic bags also contribute in polluting the environment.Canvas tote bags are, therefore, effective alternative for transporting products or food items. Users of such bags are more assured having proper handles on these cloth bags. There is less possibility of products falling out of creating a mishap. Once unpacked and cleaned, the bags are ready to use again for other purposes. Thus, plastic bags or paper bags do not get litter and create pollution.
Therefore, opting for canvas bags over of plastic bags is one of the wisest decisions that one could make. Being cost effective and affordable, these bags are widely popular in many countries. Also being multi-purpose in its function, these bags are must haves for each one of the citizens of the country. Not only they help consumers to serve various purposes, but they are also an effective measure to keep the environment clean and safe. Thus, cloth bags like this are rapidly becoming the green choice for all those environment conscious people.


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News, jobs & events in Pittsburgh for our community of tech professionals
Sep. 20, 2021 3:06 pm
Chicken vs. egg: 5 VCs on the dos and don'ts of raising money in Pittsburgh
"As you think about the Pittsburgh region, what's a higher priority? More startup capital or more investment-ready startups?" Partners from local firms and beyond weigh in on the future of venture capital in the Steel City.
By Sophie Burkholder / staff
About Abridge AI Find Out More
Venture capitalists discuss the future of local investing at the inaugural RustBuilt Pittsburgh conference.
(Photo by Josh Lucas via Twitter)
Pittsburghers in the know think about the area's venture capital availability like a chicken and egg problem.
Does the city need to have more local investors to support founders at early and medium growth stages in their businesses, or do Pittsburgh's entrepreneurs need to do a better job at building startups that are capable of attracting big funding rounds?
As part of the inaugural RustBuilt Pittsburgh Conference, five regional venture capitalists with investment experience in Pittsburgh got together last week to talk about the limitations and opportunities in finding funding for young businesses here. The venture capitalists, all partners at early-stage funds ranging from $20 million to $50 million, fielded questions from panel moderator Josh Lucas, founder of Work Hard Pittsburgh.
"So as you think about the Pittsburgh region, what's a higher priority? More startup capital or more investment-ready startups?" asked Lucas.
"Is that a trick question?" joked Zach Malone, a partner at Magarac Venture Partners.
Ilana Diamond. (Photo via LinkedIn)
It's an issue that founders and investors alike often grapple with, including Ilana Diamond, a managing partner at 412 Venture Fund, who also has experience and growing and exiting companies.
"What's lacking is somebody to tell founders how they can get involved in funding other founders," she said, adding that she herself initially struggled to learn about pathways into venture capital so that she could use the benefits of her company exits to support others.
But sometimes, as StartFast Managing Director Nasir Ali argued, there needs to be a critical mass of people more frequently investing small amounts in entrepreneurs' ideas for there to be a better culture of venture capital investments in general.
"One of the things that is absolutely critical is that people should be willing to write small checks," he said. "And that's at the individual level. If you're not ready to do that, you really can't build a community, because that's what gives someone the confidence to say, 'OK, I can start down this path.'"
While Ali sees a chain reaction that small check donations can have on readying and encouraging entrepreneurs to pursue bigger investments, other venture capitalists pointed out that dollars alone rarely prepare founders enough to be the kinds of executives firms expect when they invest in a startup. In fact, all of the panelists shared red flags they often spot in new founders that keep them from taking a deal further.
Kelauni Jasmyn. (Courtesy photo)
Kelauni Jasmyn, the founder and CEO of Black Tech Nation as well as a general partner in the organization's new venture capital arm Black Tech Nation Ventures, shared that she and other investors she's spoken with see weaknesses in entrepreneurs who are too focused on the small details in business plans.
Instead of justifying each line item on a financial spreadsheet that will most likely be wrong anyway, "know every single thing about your market like the back of your hand," Jasmyn said. "Because you want to convince VCs that there is a space for your company, and then be able to quickly and efficiently say how you're going to solve [your problem] and be the biggest player in that space."
To be able to have that market awareness, though, Mountain State Capital Managing Director Matt Harbaugh said the best thing entrepreneurs can do is "get out of Pittsburgh." Though the city's tech ecosystem is growing, the industry is far-reaching, and founders need to have an understanding of what's happening in their market beyond local competition, investors and other key partners.
"A lot of times we do see entrepreneurs coming to us telling us about how they're going to change some industry, and we feel like we know their industry better than they do," Harbaugh said. "That's not a good place to be."
While Pittsburgh's venture capital raises in 2021 have been less than stellar so far, the city attracted half a billion dollars in 2020 investments and $1.34 billion in 2019, with the latter largely due to the autonomous vehicle sector. The panelists weren't surprised at all by Pittsburgh's recent successes.
Jasmyn, who got her start in Pittsburgh tech as a 2016 graduate of coding bootcamp Academy Pittsburgh, said more people are starting to notice "what I think I was seeing five years ago." She's increasingly noticed the way the local tech ecosystem has grown to include so many different sectors in the industry — that "it's just a matter of all [of that] coming together to make the nucleus and then an atomic bomb," she said.
"And once that happens, the entire country's going to be looking at Pittsburgh, and we're already gonna be here."
Sophie Burkholder is a 2021-2022 corps member for Report for America, an initiative of The Groundtruth Project that pairs young journalists with local newsrooms. This position is supported by the Heinz Endowments.
Companies: RustBuilt, 412 Venture Fund, Black Tech Nation
People: Ilana Diamond, Kelauni Jasmyn
Growth | pittsburgh
The rise of RareBreed Ventures: How Mac Conwell used Twitter and hard-won savvy to grow a $10M fund
Power Moves: 3 Pittsburgh tech firms name big leadership changes and 2022 growth plans
Professional Development | pittsburgh
Where tech meets education: At TRETC, Pittsburgh focused on the future of their overlap
Historically, disruption equals opportunity. What does that mean for 2022?


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Graduate Catalog 2018-19 > School of Sciences, Mathematics, and Education
Forensic and Legal Psychology
Dean: Dr. Catherine M. Wehlburg
The core mission of the School of Sciences, Mathematics, and Education is to enable students to serve as agents of positive change for individuals and in the global community.
At the graduate level, students may pursue degrees in counseling, education, and forensic and legal psychology.
Minimum Grade Requirements: For degree-seeking students, unless otherwise indicated, a minimum grade of C is needed to receive credit for a graduate course in the school. See individual programs for further requirements.
Nondegree Admission: Graduate nondegree admission, in programs which permit it, is limited to a total of nine (9) credits and two consecutive semesters in the School of Sciences, Mathematics, and Education. Students who are denied admission to a degree program in the School of Sciences, Mathematics, and Education will need to be approved by the department before being accepted as a nondegree student. See individual program descriptions for further restrictions or requirements.


text:
Al-Falah Transformation Plan focuses on five main thrusts
Home Entertainment A Look At Ayra Starr's Headies Nominated '19 & Dangerous' Album |...
A Look At Ayra Starr's Headies Nominated '19 & Dangerous' Album | ALBUM REVIEW
Mavin teenage star, Ayra Starr, has dropped her new album titled '19 and Dangerous' and we have opinions about it.
Mavin teenage star Oyinkansola Sarah Aderigbe, popularly known as Ayra Starr, has dropped her new album titled '19 and Dangerous' and it is such a solid project.
At 19, Ayra Starr was signed by Nigeria's heavyweight label Mavin Records, led by Don Jazzy, in the early part of 2021. Ayra wasted no time in throwing herself into conversations with her self-titled EP 'Ayra Starr', which dropped the second day she was signed. The EP housed four songs and it was a decent attempt for a newbie. That EP showed a promising talent, but this new album has solidified the fact that Ayra Starr isn't here to play.
Making a project entails direction, it entails a theme, it entails sampling a lot of things before reaching a conclusion, it entails recording over 50 songs and putting out 10, among other things. It is work, especially if you get it right. Kudos to the Mavin team (Tega Oghenejobo, the COO of Mavin, should get his flowers as well) for highly being involved in the 'project-inclined' movement.
Before now, the Nigerian music industry paid attention to singles over projects but that era is gradually fading off as artiste are now expected to bring more to the table than just singles. Putting out a project most times guarantees the versatility and depth of an artist, even if the project is themed around a single idea.
At the top of that 'project-inclined' movement is Mavin records who are setting pace for others. They announced Rema with an EP, Ruger (who can be said to be an associate artiste) did the same, and it is amazing to see that their female star, Ayra Starr, has an EP and album out in the space of eight months! A newbie! That is insane!
Tega Oghenejobo and Ayra Starr
Delving into the album, it was exciting to see that Ayra Starr was given the role of a 'Lionel Messi', freely dictating the game – a playmaker. As a teenager, the project largely revolves around the issues of a typical teenage – heartbreaks, freedom, youthful exorbitance, etc.
You are introduced to the album with a song called 'Cast', a.k.a 'Gen Z Anthem'. Interestingly, most Nigerian female pop singers mostly revolve the theme of their songs around love, heartbreak, and 'being a bad bitch'. Ayra, however, reflects on the confusion plastered on the faces of most teenagers (most people actually) – not knowing who we are, trying to live our lives to please people. Ayra, however, isn't riding on that boat. She emphasizes on staying true to oneself and enjoying it to the fullest. She says "if I cast then I cast", which is equivalent to not giving a middle finger to what people think. After all, we all came to this world alone.
Another amazing thing about the intro is the sequence of the beat. It starts with a solemn key while an unknown voice speaks to your very soul about how much we make life problematic for ourselves. The beat progresses to the soothing voice of Ayra, before it drops and the kick kicks in to properly put you in the groove. A dope intro!
The second song, 'Fashion Killer', produced by Mavin-associate producer London, switches up totally to the typically afrobeats gbedu. Ayra is a classic bad bitch in this song. She declares how she is a fashion killer and even has thick thighs to accompany it. What else do you want? Really, what else?
This song has the same feeling as the previously released song off the album titled 'Bloody Samaritan', which was also produced by London. Both songs start with solemn keys ('Fashion Killer' –piano, while 'Bloody Samaritan' starts with a violin) and all of a sudden switch to a groovy tempo that would have you nodding; there's no escaping it.'
'Lonely' brings the heartbreak part of the story as the bad-girl Ayra, is pleading to have the company of a certain boy. Ayra established the fact that she can sing-sing on this record. Although it was a high-tempo song, she dominated it with her vocals and melodies. Also, she has this style (I'm not sure if there is a name for it) in which she sounds like she wants to break the pitch when singing, but she won't. I enjoy it!
London makes a comeback in the fourth song, 'Snitch', a soft trap song that had Ayra automatically switch back to her bad bitch vibe. Ayra's American influence came to play on the song, as she duly delivered on the hip-hop infused jam. She features American singer Fousheé, who came through with a soothing verse (she sounded 'Amaarae-ish' by the way). Big respect to the guest, but Ella Mia, Jorja Smith, or H.E.R would have been the perfect feature for the song. It might not necessarily be the people's favourite, but I love the song. It is not the regular tune you get from a Nigerian act.
'Toxic' follows in almost the same production direction – soft trap, diluted with afro-fusion elements, the dangerous Ayra Starr eats it up again. By this time of listening to the album, you are rest assured that she brought her A-game to the project.
Moving on, the tempo of the project is completely dropped and, at the same time, elevated to a celestial feel with the song 'In between'. The three-minute song is a Celtic tune that puts you on psychedelic trips. The production, vocal delivery, and lyrics aid that cause. It immediately puts you in a reflective realm where you want to pay attention to everything about the song; you are indeed 'In Between'.
You are introduced to the song with a calm and sexy French voice with smooth accent, going forward, the Yoruba chorus Ayra deployed in the song is commendable. Like angels singing into your ears, Ayra's voice leads while various voices back it up. Noticeable is the bass backup voice following her closely, singing: "Mi mo mi mo o wa ma lo/o ma wa, oma wa, oma bo…" – A spiritual record. It would be very interesting to get a visual for this song, directed by a cinematographer who can sync people properly into the song with a visual interpretation. A classic!
'Beggie Beggie' sees the vulnerable Arya back in her lover girl mode, to clearly state that even bad bitches do get emotional. The acoustic-dominated 'Beggie Beggie' is an afro-pop tune that revolves around begging to be loved. It features talented Nigerian singer and songwriter, Ckay, who delivers a smooth verse. For a minute, I paused, finding it hard to soak in the fact that Nigerian music has evolved so much, so much that the mainstream is gradually not being tied to the usual fast-tempo songs anymore. Just do you. Both artists did and in turn delivered a dope tune.
'Karma' and the earlier released 'Bloody Samaritan' follows. Both songs still see Ayra dominantly deliver a splendid performance.
The tenth song, 'Bridgertn', is another stand-out song for me on the album. It is what you call a 'happy song', like Pharrell's 'happy', Black Eyed Pea's 'I Gotta Feeling', Childish Gambino's 'Feels Like Summer', Robin Thicke's 'Blurred Lines' and so on. When the beat drops, you are assured that you are about to go on an ecstatic ride all through the song. The song themes around self-value, seeing Ayra flexing her riches, her class, her swag, her sauce, her personality! She is a queen and wouldn't hesitate to show it. It is my favourite song from the project!
Ayra concludes the project with 'Amin', a Yoruba word for 'Amen' – a religious positive declaration. Ayra gives hope in this song, assuring you that all you want will come, though she asks if you are really ready for them. The chorus delivered in the Yoruba language radiates positivity. She delivered it in an ancient style -the traditional folk way.
Ayra Starr has shown that she is a force to reckon with and, at this pace, she is definitely going to lead the Nigerian Gen Z music space. '19 and Dangerous' is a rich and solid album that would definitely stand the test of time. Oh! by the way, the album became Apple music's number one album in the country the same day it dropped. So yes! Ayra Starr is a dangerous 19-year-old popstar and you should be scared!
Listen to Ayra Starr '19 & Dangerous' Album Below
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The post A Look At Ayra Starr's Headies Nominated '19 & Dangerous' Album | ALBUM REVIEW appeared first on NotjustOk.
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