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\section{Introduction}

Networks of nonlinear dynamical systems appear in many fields, especially in the natural sciences
where the nonlinearity is often a key feature in generating the observed behavior \cite{Golubitsky-Stewart_06,Jiang-Lai_19}.
The vast majority of analysis of such networks is done in a finite dimensional state space setting using coupled
systems of ordinary differential equations. In \cite{Gray-Ebrahimi-Fard_SCL21}, however, the authors describe an alternative
approach which uses only input-output models at each node of the network in the form of a locally convergent Chen-Fliess series \cite{Fliess_81,Fliess_83}.
These weighted infinite sums of iterated integrals provide a convenient algebraic framework for describing the
network's behavior without relying on any particular choice of coordinates as in the state space setting.
Series coefficients for each node can be estimated via system identification techniques \cite{Gray-etal_Auto20}.
Computational tools were developed in \cite{Gray-Ebrahimi-Fard_SCL21} to determine,
for example, how an input injected at one node affects the output observed at another node.
Nevertheless, there are still a number of
open questions regarding the basic properties of such networks. The focus here will be on so called {\em additive} networks, where the outputs
of the nodes are simply added together and injected into other nodes, including self-loops. Other classes of aggregation functions
such the multiplication of node outputs will not be addressed here.

This paper has two goals. The first goal to address the open problem stated in \cite{Gray-Ebrahimi-Fard_SCL21} regarding whether an additive network of
locally convergent Chen-Fliess series always yields mappings between nodes which have locally convergent Chen-Fliess series representations.
This hypothesis will be proved to be true and is independent of the network's topology.
The approach taken is to identify for a given network an associated {\em maximal network} whose
growth bounds on the coefficients of the generating series between nodes upper bound all the growth bounds of the original network and are much
easier to determine using conventional methods as presented in \cite{Sussmann_83}. The particular growth bound derived turns out to be exactly
equivalent to one discovered for a class of unity feedback systems described in \cite{Thitsa-Gray_12}.
The second goal
is to provide sufficient conditions under which the input-output map between a pair of nodes has well defined relative degree as defined by
its generating series \cite{Gray-etal_AUTO14,Gray-Venkatesh_SCL19}. A simple counterexample will be given first to show that
this property can fail to hold in certain situations.
The proofs of the sufficient conditions rely on identifying certain properties first described in \cite{Gray-Venkatesh_SCL19}
in relation to a subgraph connecting a given input node and output node.
It is also shown, however, that this relative degree property is {\em generic} in a certain sense.
Namely, if the generating series for every node has relative degree and the connection strengths between the nodes are random, then every node pair has
a generating series with well defined relative degree with probability one.
An obvious application for
this result is in the context of feedback linearization for networks \cite{Menara-etal_CDC20}, however, that application will not
be pursued here.

The paper is organized as follows. To keep the presentation as self-contained as possible, the required preliminaries are briefly
summarized in Section~\ref{sec:preliminaries}. The question regarding the convergence of Chen-Fliess series for
mappings between nodes is addressed in Section~\ref{sec:local-convergence}.
The subsequent section treats the property of relative degree. The paper's conclusions are summarized in the final section.

\section{Preliminaries}
\label{sec:preliminaries}

An {\em alphabet} $X=\{ x_0,x_1,$ $\ldots,x_m\}$ is any nonempty and finite set
of noncommuting symbols referred to as {\em
letters}. A {\em word} $\eta=x_{i_1}\cdots x_{i_k}$ is a finite sequence of letters from $X$.
The number of letters in a word $\eta$, written as $\abs{\eta}$, is called its {\em length}.
The empty word, $\emptyset$, is taken to have length zero.
The collection of all words having length $k$ is denoted by
$X^k$. Define $X^\ast=\bigcup_{k\geq 0} X^k$,
which is a monoid under the concatenation (Cauchy) product.
Any mapping $c:X^\ast\rightarrow
\re^\ell$ is called a {\em formal power series}.
Often $c$ is
written as the formal sum $c=\sum_{\eta\in X^\ast}\langle c,\eta\rangle\eta$,
where the {\em coefficient} $\langle c,\eta\rangle\in\re^\ell$ is the image of
$\eta\in X^\ast$ under $c$.
The {\em support} of $c$, ${\rm supp}(c)$, is the set of all words having nonzero coefficients.
A series $c$ is {\em proper} when $\emptyset\not\in{\rm supp}(c)$.
The set of all noncommutative formal power series over the alphabet $X$ is
denoted by $\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$. The subset of series with finite support, i.e., polynomials,
is represented by $\mbox{$\re^{\ell}\langle X \rangle$}$.
For any $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$, the scalar product is
$\langle c,d\rangle:=\sum_{\eta\in X^\ast} \langle c,\eta\rangle\langle d,\eta\rangle$, provided the sum is finite.
The set $\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is an associative $\re$-algebra under the concatenation product and an associative and commutative $\re$-algebra under
the {\em shuffle product}, that is, the bilinear product uniquely specified by the shuffle product of two words $x_i\eta,x_j\xi\in X^\ast$:
\begin{displaymath}
	(x_i\eta){\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}(x_j\xi)=x_i(\eta{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}(x_j\xi))+x_j((x_i\eta){\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} \xi),
\end{displaymath}
where $x_i,x_j\in X$ and with $\eta{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}\emptyset=\emptyset{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}\eta=\eta$ \cite{Fliess_81}.

\subsection{Chen-Fliess series}

Given any $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ one can associate a causal
$m$-input, $\ell$-output operator, $F_c$, in the following manner.
Let $\mathfrak{p}\ge 1$ and $t_0 < t_1$ be given. For a Lebesgue measurable
function $u: [t_0,t_1] \rightarrow\re^m$, define
$\norm{u}_{\mathfrak{p}}=\max\{\norm{u_i}_{\mathfrak{p}}: \ 1\le
i\le m\}$, where $\norm{u_i}_{\mathfrak{p}}$ is the usual
$L_{\mathfrak{p}}$-norm for a measurable real-valued function,
$u_i$, defined on $[t_0,t_1]$.  Let $L^m_{\mathfrak{p}}[t_0,t_1]$
denote the set of all measurable functions defined on $[t_0,t_1]$
having a finite $\norm{\cdot}_{\mathfrak{p}}$ norm and
$B_{\mathfrak{p}}^m(R)[t_0,t_1]:=\{u\in
L_{\mathfrak{p}}^m[t_0,t_1]:\norm{u}_{\mathfrak{p}}\leq R\}$.
Assume $C[t_0,t_1]$
is the subset of continuous functions in $L_{1}^m[t_0,t_1]$. Define
inductively for each word $\eta=x_i\bar{\eta}\in X^{\ast}$ the map $E_\eta:
L_1^m[t_0, t_1]\rightarrow C[t_0, t_1]$ by setting
$E_\emptyset[u]=1$ and letting
\[E_{x_i\bar{\eta}}[u](t,t_0) =
\int_{t_0}^tu_{i}(\tau)E_{\bar{\eta}}[u](\tau,t_0)\,d\tau, \] where
$x_i\in X$, $\bar{\eta}\in X^{\ast}$, and $u_0=1$. The
{\em Chen--Fliess series} corresponding to $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is
\begin{displaymath}
y(t)=F_c[u](t) =
\sum_{\eta\in X^{\ast}} \langle c,\eta\rangle \,E_\eta[u](t,t_0)
\end{displaymath}
\hspace*{-0.06in}\cite{Fliess_81}.
If there exist real numbers $K,M>0$ such that
\begin{displaymath}
\abs{\langle c,\eta\rangle}\le K M^{|\eta|}|\eta|!,\;\; \forall\eta\in X^{\ast},
\end{displaymath}
then $F_c$ constitutes a well defined mapping from
$B_{\mathfrak p}^m(R)[t_0,$ $t_0+T]$ into $B_{\mathfrak
q}^{\ell}(S)[t_0, \, t_0+T]$ for sufficiently small $R,T >0$ and some $S>0$,
where the numbers $\mathfrak{p},\mathfrak{q}\in[1,\infty]$ are
conjugate exponents, i.e., $1/\mathfrak{p}+1/\mathfrak{q}=1$  \cite{Gray-Wang_02}.
(Here, $\abs{z}:=\max_i \abs{z_i}$ when $z\in\re^\ell$.) The set of all such
{\em locally convergent} series is denoted by $\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$, and $F_c$ is
referred to as a {\em Fliess operator}.

\subsection{System interconnections}

Given Fliess operators $F_c$ and $F_d$, where $c,d\in\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$,
the parallel and product connections satisfy $F_c+F_d=F_{c+d}$ and $F_cF_d=F_{c{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} d}$,
respectively \cite{Fliess_81}.
When Fliess operators $F_c$ and $F_d$ with $c\in\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$ and
$d\in\mbox{$\re^{m}_{LC}\langle\langle X \rangle\rangle$}$ are interconnected in a cascade fashion, the composite
system $F_c\circ F_d$ has the
Fliess operator representation $F_{c\circ d}$, where
the {\em composition product} of $c$ and $d$
is given by
\begin{displaymath}
c\circ d=\sum_{\eta\in X^\ast} \langle c,\eta\rangle\,\psi_d(\eta)(\mathbf{1})
\end{displaymath}%
\hspace*{-0.07in}\cite{Ferfera_80}. Here $\mbf{1}$ denotes the monomial $1\emptyset$, and
$\psi_d$ is the continuous (in the ultrametric sense) algebra homomorphism
from $\mbox{$\re\langle\langle X \rangle\rangle$}$ to the vector space endomorphisms on $\mbox{$\re\langle\langle X \rangle\rangle$}$, ${\rm End}(\allseries)$, uniquely specified by
$\psi_d(x_i\eta)=\psi_d(x_i)\circ \psi_d(\eta)$ with
$
\psi_d(x_i)(e)=x_0(d_i{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} e),
$
$i=0,1,\ldots,m$
for any $e\in\mbox{$\re\langle\langle X \rangle\rangle$}$,
and where $d_i$ is the $i$-th component series of $d$
($d_0:=\mbf{1}$). By definition,
$\psi_d(\emptyset)$ is the identity map on $\mbox{$\re\langle\langle X \rangle\rangle$}$.

\subsection{Relative degree of a generating series}

Let $X=\{x_0,x_1\}$.
Following \cite{Gray-etal_AUTO14}, a series $c\in\mbox{$\re\langle\langle X \rangle\rangle$}$ has relative degree $r$ if and only if it
has the decomposition
\begin{displaymath}
c=c_N+Kx_0^{r-1}x_1+x_0^{r-1}e
\end{displaymath}
for some $K\neq 0$ and proper $e\in\mbox{$\re\langle\langle X \rangle\rangle$}$ with $x_1\not\in{\rm supp}(e)$.
This definition of relative degree is consistent with the classical definition
whenever $y=F_c[u]$
is realizable \cite{Gray-etal_AUTO14,Gray-Ebrahimi-Fard_SIAM17}.
The following results
will be of central
importance in the work that follows.

\begin{theorem} \cite{Gray-Venkatesh_SCL19} \label{th:r-parallel-sum-connection}
If $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$ have distinct relative degrees $r_c$ and $r_d$, respectively, then $c+d$ has relative
degree $\min(r_c,r_d)$. On the other hand, if $r_c = r_d =: r$, then $c+d$ has relative degree $r$ if and
only if  $\langle c,x_{0}^{r-1}x_{1}\rangle + \langle d,x_{0}^{r-1}x_{1}\rangle \neq 0$.
\end{theorem}

\begin{corollary} \label{co:relative-degree-multi-sum-distinct}
If $c_1,c_2,\ldots,c_m$ have relative degree $r_1,r_2,\ldots,r_m$, respectively,
with $r_i\neq r_j$ when $i\neq j$,
then the relative degree of $c_1+c_2+\cdots+c_m$ is $\min_i(r_i)$.
\end{corollary}

\begin{corollary} \label{co:relative-degree-multi-sum-not-distinct}
Suppose $c_1,c_2,\ldots,c_m$ have relative degree $r_1,r_2,\ldots,r_m$, respectively.
Let $s_j$ denote the multiplicity of relative degree $r_j$.
If for each $s_j>1$ the series $c_{k_1},c_{k_2},\ldots,c_{k_{s_j}}$ having relative degree $r_j$
satisfy
\begin{displaymath}
\langle c_{k_1},x_0^{r_j-1}x_1\rangle+\langle c_{k_2},x_0^{r_j-1}x_1\rangle+\dots+\langle c_{k_{s_j}},x_0^{r_j-1}x_1\rangle\neq 0,
\end{displaymath}
then the relative degree of $c_1+c_2+\cdots+c_m$ is $\min_i(r_i)$.
\end{corollary}

\begin{theorem} \cite{Gray-Venkatesh_SCL19} \label{th:relative-degree-casecade}
If $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$ have relative degrees $r_c$ and $r_d$, respectively,
then $r_{c\circ d}$ has relative degree $r_c+r_d$.
\end{theorem}

\subsection{Formal realizations and representations}

It is shown in \cite{Kawski-Sussmann_97} that a given Chen-Fliess series $y=F_c[u]$ can be written in terms
of a state $z$ evolving on a formal Lie group $\mathcal{G}(X)$ with Lie algebra $\widehat{\mathcal L}(X)$ and output
map $y=\langle c,z\rangle$. This notion of a {\em universal control system} was generalized in
\cite{Gray-Ebrahimi-Fard_SCL21} as follows to describe networks of Chen-Fliess series.

\begin{definition}
Let $V_i$ be a vector field on $\mathcal{G}^n(X):=\mathcal{G}(X)\times\mathcal{G}(X)\times\cdots\times \mathcal{G}(X)$, $i=0,1,\ldots,m$ with
\begin{align*}
V_i&:{\mathcal G}^n(X)\rightarrow T_z{\mathcal G}^n(X) \\
&z=(z_1,\ldots,z_n)\mapsto V_i(z)=(V_{i1}(z)z_1,\ldots,V_{in}(z)z_n),
\end{align*}
where $V_{ij}(z(t))\in \widehat{\mathcal L}(X)$.
The $j$-th component of the corresponding state equation on ${\mathcal G}^n(X)$ is
\begin{displaymath}
\dot{z}_j=\sum_{i=0}^m V_{ij}(z)z_j u_{ij} , \;\; z_j(0)=z_{j0}.
\end{displaymath}
Given $\hat{c}_{k}\in\mbox{$\re_{LC}^{\otimes n}\langle\langle X \rangle\rangle$}$, $k=1,2,\ldots,\ell$, the $k$-th output equation is defined to be
\begin{displaymath}
y_{k}=\hat{c}_{k}(z).
\end{displaymath}
Collectively, $(V,z_0,\hat{c})$ is a {\em formal realization} on ${\mathcal G}^n(X)$ of the formal input-output map
$u\mapsto y$.
\end{definition}

Analogous to the standard finite dimensional theory \cite{Isidori_95,Nijmeijer-vanderSchaft_90}, a series $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is said to have a {\em formal representation} when
there exists a formal realization with the property that every coefficient of $c$ can be written in terms of iterated Lie derivatives of the vectors
fields acting on the output map and evaluated at $z_0$, i.e., $\langle c,x_{i_1}\cdots x_{i_k}\rangle=L_{V_{i_k}}\cdots L_{V_{i_1}}\hat{c}(z_0)$.

\section{Additive Networks of Chen-Fliess Series: Local Convergence}
\label{sec:local-convergence}

In this section it is shown that every network of additively interconnected locally convergent Fliess operators has the
property that the input-output maps between any two nodes can be represented by a locally convergent Fliess operator.
The first definition describes the specific class of networks under consideration.

\begin{definition}
A set of $m$ single-input, single-output Chen-Fliess series mapping $u_i$ to $y_i$ with generating series
$c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$, where $X_i=\{x_0,x_i\}$
is said to be an {\em additively interconnected network} ${\mathcal N}_m$
with weighting matrix $W\in\re^{m\times m}$
if $u_i=v_i+\sum_{j=1}^mW_{ij}y_j$, $i=1,2,\ldots,m$.
\end{definition}

A network ${\mathcal N}_m$ can therefore be viewed as a directed graph connecting $m$ nodes,
where the $i$-th node corresponds to a Chen-Fliess series with generating series $c_i$, $i=1,2,\ldots,m$.
Henceforth, it will be assumed that the connection weights are normalized so that $W_{ij}\in[0,1]$, $i,j=1,2,\ldots,m$.
The following theorem follows directly from Theorem 5.1 in \cite{Gray-Ebrahimi-Fard_SCL21}.

\begin{theorem} \label{th:additive-interconnections}
The input-output map $v_i\mapsto y_j$ in any additively interconnected network ${\mathcal N}_m$
has generating series $d_{ji}\in\re\langle\langle X_i\rangle\rangle$ which can be computed from a formal
representation in terms of the vector fields
\begin{align*}
V_0(z)&=
\left[
\begin{tabular}{p{0.5cm}}
\hspace*{-0.1cm}$x_0z_1$ \\
\hspace*{-0.1cm}$x_0z_2$ \\
\hspace*{0.2cm}\vdots \\
\hspace*{-0.1cm}$x_0z_m$
\end{tabular}
\right]+{\rm diag}(x_1z_1,\ldots,x_m z_m)W
\left[
\begin{tabular}{p{1cm}}
\hspace*{-0.02cm}$\langle c_1,z_1\rangle$ \\
\hspace*{-0.02cm}$\langle c_2,z_2\rangle$ \\
\hspace*{0.4cm}\vdots \\
\hspace*{-0.13cm}$\langle c_m,z_m\rangle$
\end{tabular}
\right] \\
V_i(z)&=x_iz_i\mbf{e}_i
\end{align*}
acting on
$\hat{c}_j=\mbf{1}\otimes\cdots\otimes\mbf{1}\otimes c_j\otimes\mbf{1}\cdots\otimes\mbf{1}\in\mbox{$\re_{LC}^{\otimes m}\langle\langle X \rangle\rangle$}$ ($c_j$ appears in the $j$-th position)
and evaluated at $z_{j0}=\mbf{1}$, $i,j=1,2\ldots,m$.
\end{theorem}

The next theorem states the main convergence result concerning additive networks.

\begin{theorem}
If ${\mathcal N}_m$ is an additively interconnected network where the generating series for each node $c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$,
then the generating series for every input-output map $d_{ji}\in\re_{LC}\langle\langle X_i\rangle\rangle$. More specifically, if $K_i,M_i$ denote the growth
constants for $c_i$, then for all $i,j=1,2,\ldots,m$
\begin{displaymath}
\abs{\langle d_{ji},\eta\rangle}<KM^{\abs{\eta}}\abs{\eta}!,\;\;\forall \eta\in X^\ast
\end{displaymath}
for some $K>0$ and any $M>M_{\rm inf}$, where
\begin{equation} \label{eq:M-inf-additive-network}
M_{\rm inf}=\frac{\bar{M}}{1-m\bar{K} \ln\left(1+\frac{1}{m\bar{K}}\right)}
\end{equation}
with $\bar{K}=\max_{i} K_i$ and $\bar{M}=\max_{i} M_i$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
It is first shown that each generating series $d_{ji}$ is locally convergent. Consider the case where
every node series $c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$ is a {\em maximal series}
$\bar{c}_i:=\sum_{\eta\in X^\ast}K_iM_i^{\abs{\eta}}\abs{\eta}!\, \eta$.
That is, every coefficient of $\bar{c}_i$ is growing at its maximal rate. While $y_i=F_{c_i}[u_i]$ may not have a finite
dimensional state space realization,
it is easily shown that a maximal series has the realization
\begin{displaymath}
\dot{z}_i=\frac{M_i}{K_i}z_i^2(1+u_i),\;\;z_i(0)=K_i,\;\;y_i=z_i
\end{displaymath}
\hspace*{-0.07in}\cite[Lemma 3]{Thitsa-Gray_12}.
Therefore, the corresponding network can be realized by
\begin{equation}
\label{eq:maximal-series-state-space}
\dot{z}_i=\frac{M_i}{K_i}z_i^2\left(1+\sum_{j=1}^m W_{ij}z_j+v_i\right)\!\!,\;\;z_i(0)=K_i,\;\;y_i=z_i,
\end{equation}
$i=1,2,\ldots,m$. As this realization of the input-output map $v\mapsto y$ is polynomial, it is clearly real analytic.
Therefore, every generating series for $v_i\mapsto y_j$, say $\bar{d}_{ji}$, must be locally convergent \cite[Lemma 4.2]{Sussmann_83}.
The claim now is that $d_{ji}$ must also be locally convergent since $\abs{\langle d_{ji},\eta\rangle}\leq \langle \bar{d}_{ji},\eta\rangle$
for all $\eta\in X^\ast$. This inequality is most easily deduced from the formal realization of $v_i\mapsto y_j$ given in Theorem~\ref{th:additive-interconnections},
where the Lie derivatives used to compute the coefficients of $d_{ji}$ will all be upper bounded in magnitude by the Lie derivatives computed using maximal series.

Next, a suitable geometric growth constant for the network ${\mathcal N}_m$ is determined.
First observe that the growth constants $\bar{K}$ and $\bar{M}$ constitute a worst case maximum growth rate for every
node in the network.
In light of the formal representation of any $d_{ji}$ in Theorem~\ref{th:additive-interconnections}, the growth
rate of $d_{ji}$ is upper bounded by the growth rate of the natural response $\langle \bar{d}_{ji},x_0^k\rangle=L_{V_0}^k\hat{c}_j(\mbf{1})$, $k\geq 0$,
where $W_{ij}=1$ for all $i,j$, and every non-trivial component of $\hat{c}_j$ is the maximal series
$\bar{c}=\sum_{\eta\in X^\ast}\bar{K} \bar{M}^{\abs{\eta}}\abs{\eta}!\, \eta$.
(See \cite[Lemma 7]{Thitsa-Gray_12} for an alterative approach when $m=1,2$.)
From the symmetry of such a {\em maximal network}, $z_i=z_j$ for all $i,j$. Applying these conditions to \rref{eq:maximal-series-state-space},
the natural response at each node is
given by the solution of the Abel differential equation
\begin{equation} \label{eq:maximal-network-Abel-equation}
\dot{z}=\frac{\bar{M}}{\bar{K}}(z^2+mz^3),\;\;z(0)=\bar{K}.
\end{equation}
It can be directly verified that this equation has the solution
\begin{displaymath}
z(t)=\frac{-\frac{1}{m}}{1+\mathcal{W}\left[-\left(1+\frac{1}{m\bar{K}}\right)\exp\left(\frac{\bar{M}}{m\bar{K}}t-\left(1+\frac{1}{m\bar{K}}\right)\right)\right]},
\end{displaymath}
where $\mathcal{W}$ denotes the Lambert $W$-function, that is, the inverse of the function
$f(x)=x\exp(x)$ corresponding to the principal branch of this multi-valued function \cite{Corless-etal_96}.
As $\mathcal{W}$ is known to be holomorphic on the complex plane,
$z(t)$ will therefore be analytic at $t=0$. The corresponding Taylor series has a radius of convergence determined by the singularity nearest to the origin,
in this case
\begin{displaymath}
t^\ast=\frac{1}{\bar{M}}\left(1-m\bar{K}\ln\left(1+\frac{1}{m\bar{K}}\right)\right).
\end{displaymath}
Applying a well known theorem from complex analysis (see \cite[Theorem~2.4.3]{Wilf_94}) gives
the infimum of all geometric growth constants for the maximal network, namely $M_{\rm inf}=1/t^\ast$.
(Note that the function $\lambda(x)=1-x\ln(1+{1/x})$ is a decreasing function, which further justifies using the maximum $K_i$ in
the network as the worst case.) Since for any $M>M_{\rm inf}$ there is a $K>0$ to upper bound the fastest coefficient growth
in the maximal network,
the generating series for every node in the original network must also be upper bounded by this growth rate.
\hfill\bull \vspace*{0.05in}

It is worth noting that \rref{eq:M-inf-additive-network} is in fact identical to the growth constant identified for
unity feedback systems with $m$ inputs as described in \cite[Corollary~2]{Thitsa-Gray_12}. While the network topologies
are clearly distinct, this point of tangency is derived from the fact that unity feedback systems and
additive maximal networks both have natural responses satisfying \rref{eq:maximal-network-Abel-equation}.

\begin{table}[tb]
\caption{Integer sequences generated by maximal additive network with unity growth constants}
\label{tbl:network-series}
\vspace*{-0.1in}
\begin{center}
\begin{tabular}{c|l|c|c}
	\toprule
	$m$ & \hspace*{0.8in}$a_n$ & $M_{\rm inf}$ & $\hat{M}_n$ \\
	\midrule
	1 & 1, 2, 10, 82, 938, 13778, 247210, $\ldots$            & 3.2589     & 3.22634 \\
	2 & 1, 3, 24, 318, 5892, 140304, $\ldots$         & 5.2891     & 5.23618 \\
	3 & 1, 4, 44, 804, 20556, 675588, $\ldots$        & 7.3017     & 7.22873 \\
	4 & 1, 5, 70, 1630, 53120, 2225480, $\ldots$      & 9.3088     & 9.21567 \\
	5 & 1, 6, 102, 2886, 114294, 5819190, $\ldots$    & 11.3132    & 11.2001 \\
	6 & 1, 7, 140, 4662, 217308, 13022688,$\ldots$   & 13.3163    &  13.1831 \\
	\bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.55]{three_node_simulation_worst_case}
\end{center}
\vspace*{-0.2in}
\caption{Natural response of three node maximal network in Example~\ref{ex:three-node-maximal-network-worst-case}.}
\label{fig:three-node-maximal-network-worst-case}
\end{figure}

\begin{example} \label{ex:three-node-maximal-network-worst-case}
{\rm
Consider a maximal additive network where $K_i=M_i=1$, $i=1,2,\ldots,m$. The Taylor series
of the natural response has integer coefficients $a_n$, $n\geq 1$ as shown in Table~\ref{tbl:network-series}.
The coefficients when $m=1$ correspond to the OEIS integer sequence
A112487 \cite{OEIS}. The table also shows the growth rate $M_{\rm \inf}$ computed from \rref{eq:M-inf-additive-network}
and an estimate of the growth constant $M$ computed
from $\hat{M}_n=na_{n}/a_{n-1}$ when $n=50$.
The corresponding three node network was simulated in MatLab for the zero input case. The node responses,
which are identical, are
shown in Figure~\ref{fig:three-node-maximal-network-worst-case}.
Since the coefficients of every generating series are positive,
it is known that the natural response of every node will have
a finite escape time at $t=t^\ast$ (see \cite[Theorem 11]{Thitsa-Gray_12}).
In this case, $t^\ast=1/M_{\rm inf}=0.1379$, which is what was observed
in the simulation.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.55]{three_node_simulation_generic}
\end{center}
\vspace*{-0.1in}
\caption{Natural response of three node network in Example~\ref{ex:three-node-maximal-network-generic}.}
\label{fig:three-node-maximal-network-generic}
\end{figure}

\begin{example} \label{ex:three-node-maximal-network-generic}
{\rm
Consider a three node additive network involving maximal series with $K_i=i$, $M_i=5-i$ and
\begin{displaymath}
W=\left[\begin{array}{cccc}
1 & 0.5 & 1  \\
1 & 1 & 0 \\
0.25 & 1 & 1
\end{array}\right].
\end{displaymath}
Thus, $\bar{K}=3$, $\bar{M}=4$, and $M_{\rm inf}=77.2867$.
The node natural responses are
shown in Figure~\ref{fig:three-node-maximal-network-generic}.
As this network is not maximal,
$t^\ast=1/M_{\rm inf}=0.01294$ provides only a lower bound on the escape times
of each node.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\section{Additive Networks of Chen-Fliess Series: Relative Degree}
\label{sec:relative-degree}

In this section the following question is addressed: When does the generating series of the
mapping $v_i\mapsto y_j$ in an additively interconnected
network ${\mathcal N}_m$ have a well defined relative degree?
The treatment starts with the easiest case first as described next.
It is assumed throughout that ${\mathcal N}_m$ is
comprised of systems with generating series $c_i$ which have relative degree $r_i$ for
$i=1,2,\ldots,m$.

\begin{definition}
The $i$-th node in a network ${\cal N}_m$ is said to be {\em fully connected} if $W_{ij}\neq 0$ for all $j\neq i$.
A network ${\cal N}_m$ is said to be {\em fully connected} if every node is fully connected.
\end{definition}

Note that self-loops, i.e., when $W_{ii}\neq 0$, are not important in the present context as proportional output feedback
is easily shown to preserve relative degree \cite{Gray-Venkatesh_SCL19}.

\begin{theorem} \label{th:relative-degree-fully-connected}
If
the $i$-th node in ${\mathcal N}_m$ is fully connected, then the generating series $d_{ji}$ for mapping
$v_i\mapsto y_j$ has relative degree $r_{ji}=r_j+r_i$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
Observe that the full output at node $j$ is
\begin{align*}
y_j&=F_{c_j}\left[v_j+\sum_{k=1}^m W_{jk} y_k \right] \\
&=F_{c_j}\left[v_j+\sum_{k,l=1}^m W_{jk} F_{d_{kl}}[v_l]\right].
\end{align*}
For any $i\neq j$, that part of $y_j$ in response to $v_i$ acting alone (i.e., $v_l=0$ for $l\neq i$) is
given by
\begin{align*}
y_{j}&=F_{c_j}\Bigg[W_{ji}F_{c_i}[v_i]+\sum_{k=1\atop k\neq i}^m W_{jk} F_{d_{ki}}[v_i]+ \\
&\hspace{0.2in} \sum_{k,l=1\atop l\neq i}^m W_{jk} F_{d_{kl}}[0]\Bigg].
\end{align*}
Note that for all $k\neq i$, ${\rm supp}(d_{ki})\subseteq x_0^{r}X^\ast$, where $r\geq r_i+1$, since
$v_i$ passes through $F_{c_i}$ in every path leading to the $j$-th node. In which case, the argument of $F_{c_j}$ above
has a generating series with relative degree $r_i$. The conclusion then follows immediately from Theorem~\ref{th:relative-degree-casecade}.
\hfill\bull \vspace*{0.05in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{diamond_network}
\end{center}
\vspace*{-0.2in}
\caption{Four node network in Example~\ref{ex:four-node-counterexample}.}
\label{fig:four-node-counterexample}
\end{figure}

\begin{example} \label{ex:four-node-counterexample}
{\rm
Consider the network shown in Figure~\ref{fig:four-node-counterexample}.
The corresponding weighting matrix is
\begin{displaymath}
W=\left[\begin{array}{cccc}
0 & 0 & 0 & 0 \\
W_{21} & 0 & 0 & 0 \\
W_{31} & 0 & 0 & 0 \\
W_{41} & W_{42} & W_{43} & 0
\end{array}\right].
\end{displaymath}
The network is clearly {\em not} fully connected,
but node $4$ is fully connected assuming $W_{4j}\neq 0$, $j=1,2,3$.
Therefore, applying the theorem above gives, for example, that
$r_{41}=r_4+r_1$.

Suppose now that $W_{41}=0$ so that the theorem no longer applies. Further assume that $r_2=r_3=r$.
Observe that
\begin{displaymath}
u_4=W_{42}F_{c_2}[W_{21}F_{c_1}[u_1]]+W_{43}F_{c_3}[W_{31}F_{c_1}[v_1]],
\end{displaymath}
and thus,
\begin{displaymath}
d_{41}=c_4\circ [W_{42}(c_2\circ (W_{21}c_1))+W_{43}(c_3\circ (W_{31}c_1))].
\end{displaymath}
Both $c_2\circ (W_{21}c_1)$ and $c_3\circ (W_{31}c_1)$ have relative degree $r+r_1$,
but $d_{41}$ can fail to have relative degree.
As a simple example, suppose $c_1=c_2=c_4=x_1$ and $c_3=-x_1$ so that
$d_{41}=(W_{42}W_{21}-W_{43}W_{31}) x_0^2x_1$.
If $W$ is such that $W_{42}W_{21}=W_{43}W_{31}$, then $d_{41}=0$ does not have relative degree.
On the other hand, if the symmetry condition $r_2=r_3$ is broken, then it follows that $d_{41}$ has
relative degree $r_{41}=r_4+\min (r_2,r_3)+r_1$.
}
\hfill\openbull \end{example} \vspace*{0.1in}

The final case in the example above suggests a sufficient condition for the general case. Namely,
in the absence of these degenerate situations where a node is presented with an input whose underlying
generating series does not have relative degree, the relative degree for $d_{ji}$ will be well defined and determined
by a path from node $i$ to node $j$ whose {\em accumulated} relative degrees is minimal. To make this
claim more precise, the following language adapted from signal flow graph theory will be useful.

Let ${\mathcal N}_m$ be a given additive network.
An {\em edge} is a directed line segment connecting two nodes.
A {\em path} is a continuous set of edges connecting two nodes in ${\mathcal N}_m$ and traversed in the direction indicated.
A {\em forward path} is a path in which no node is encountered more than once.
A {\em loop} is a path that originates and ends on the same node in which no node is encountered more than once.
Finally, the {\em subgraph} $G_{ji}$ from node $i$ to node $j$ is the simple graph (i.e., all loops are omitted)
consisting of all forward paths connecting node $i$ and node $j$.

The following theorems provide a sufficient condition under which the relative degree is well defined for a given
input-output map $v_i\mapsto y_j$ in an additive network. Given a subgraph $G_{ji}$, the {\em accumulated relative degree} of node $i$ is
$r_i^+=r_i$. If node $k\neq i$ in $G_{ji}$ has $N$ incoming edges from nodes $i_1,i_2,\ldots, i_N$ with accumulated relative degrees
$r^+_{i_1},r^+_{i_2},\ldots,r^+_{i_N}$, respectively, then the {\em accumulated relative degree} at node $k$ is
\begin{displaymath}
r^+_k=r_k+\min\{r^+_{i_1},r^+_{i_2},\ldots,r^+_{i_N}\}.
\end{displaymath}
Note this definition does not imply that any mappings defined by the network have relative degree, it simply
computes the {\em potential} relative degree of such a mapping should it be well defined.

\begin{theorem} \label{th:relative-degree-additive-network-distinct-condition}
Let $i$ and $j$ be fixed nodes in ${\mathcal N}_m$.
If at every node $l{\not\in}\{i,j\}$  the accumulated relative degrees
of the nodes from every incoming edge are distinct, then
the generating series $d_{ji}$ for $v_i\mapsto y_j$ in ${\mathcal N}_m$ has well defined relative degree
equivalent to $r_{ji}=r_j^+$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
As feedback loops do not affect the relative degree of any forward path, it is sufficient
to consider only the subgraph $G_{ji}$.
The claim then follows directly from Corollary~\ref{co:relative-degree-multi-sum-distinct},
Theorem~\ref{th:relative-degree-casecade}, and the definition of accumulated relative degree.

\hfill\bull \vspace*{0.05in}

The distinctness condition in the above theorem can be relaxed by
utilizing instead the condition in Corollary~\ref{co:relative-degree-multi-sum-not-distinct}.

\begin{theorem} \label{th:relative-degree-additive-network-repeated-condition}
Let $i$ and $j$ be fixed nodes in ${\mathcal N}_m$.
If at every node $l{\not\in}\{i,j\}$ the accumulated relative degrees
of the nodes from every incoming edge satisfy the condition in Corollary~\ref{co:relative-degree-multi-sum-not-distinct},
then the generating series $d_{ji}$ for $v_i\mapsto y_j$ in ${\mathcal N}_m$ has well defined relative degree
equivalent to $r_{ji}=r_j^+$.
\end{theorem}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{double_diamond_network}
\end{center}
\vspace*{-0.1in}
\caption{Network in Example~\ref{ex:double-diamond-example}.}
\label{fig:double-diamond-example}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{double_diamond_network_subgraph}
\end{center}
\vspace*{-0.1in}
\caption{Subgraph of forward paths for $v_1\mapsto y_7$ in Example~\ref{ex:double-diamond-example}. The relative
degree of each generating series $c_i$ is the circled number. The accumulated relative degree at each node is the number in the triangle.}
\label{fig:double-diamond-example-subgraph}
\end{figure}

\begin{example} \label{ex:double-diamond-example}
{\rm
Consider the network shown in Figure~\ref{fig:double-diamond-example}, where each weight $W_{ij}\in \{0,1\}$ (i.e., $0\sim\mbox{not connected}$,
$1\sim\mbox{connected}$), and
the generating series for the nodes are:
\begin{align*}
c_1 &= K_1 x_1 + 2 x_0  x_1 \\
c_2 &= x_0+K_2 x_0^2  x_2 \\
c_3 &= K_3 x_0  x_3 + 3 x_0^2  x_3^2 \\
c_4 &= 1+K_4 x_0  x_4 - x_0^2  x_4x_0 \\
c_5 &= 4x_0+K_5 x_0^2  x_5 -2x_0^4x_5 \\
c_6 &= K_6 x_6 -x_6^2\\
c_7 &= x_0+2+K_7 x_7+4x_0x_7
\end{align*}
with $K_i\neq 0$ in every case.
The subgraph of forward paths is shown in Figure~\ref{fig:double-diamond-example-subgraph}.
The relative degree of the generating series at each node is the circled number shown next to each node.
The accumulated relative degree at each node is the number in the triangle
The goal is to determine the relative degree of the mapping $v_1\mapsto y_7$, provided it is well defined.
Observe that only nodes $4$, $5$ and $7$, have more than one incoming edge. In each case, the accumulated relative degrees
are distinct, namely, $3,4$; $4,5$; and $6,7$, respectively. Therefore, Theorem~\ref{th:relative-degree-additive-network-distinct-condition}
applies,
and $r_{71}=7$.
To independently verify this claim,
the generating series $d_{71}$ was computed using
the full network via Theorem~\ref{th:additive-interconnections} with the aid of Mathematica and found to be
\begin{displaymath}
d_{71}=d_{71,N}+K_1K_3K_4K_6K_7x_0^6x_1+x_0^6e,
\end{displaymath}
where
\begin{align*}
d_{71,N} = & {\,} x_0+
(4 K_7 + K_6 K_7)x_0^2  +(16 + 4 K_6 - K_7)x_0^3+ \\
& {\,} (-4 + K_5 K_7 + 2 K_4 K_6 K_7) x_0^4 + (4 K_5 + 8 K_4 K_6- \\
& {\,} 8 K_4 K_7 + K_5 K_7 + 2 K_4 K_6 K_7) x_0^5 + \cdots\\
e = & {\,} (4 K_1 K_3 K_4 K_6 - 6 K_1 K_3 K_4 K_7 + K_1 K_2 K_5 K_7+ \\
& {\,}K_1 K_2 K_4 K_6 K_7 + 2 K_3 K_4 K_6 K_7) x_0 x_1 + \cdots
\end{align*}
The relative degree of $d_{71}$ is 7 as expected.
}
\hfill\openbull \end{example} \vspace*{0.1in}

An additive network ${\mathcal N}_m$ is said to have {\em complete relative degree} if every mapping
$v_i\mapsto y_j$, $i,j=1,2,\ldots,m$ has relative degree. From Theorem~\ref{th:relative-degree-fully-connected} it
is immediate that fully connected networks have this property. Another class of networks sharing this property is given
in the following theorem. It states that in some sense the property of a network having complete relative degree is {\em generic}.

\begin{theorem}
Consider an additive network ${\mathcal N}_m$ where the weighting matrix has entries $W_{ij}\in\{0,1\}$. If the
unity weights are replaced with continuous random variables, then every sample network has complete relative degree.
\end{theorem}

\noindent{\em Proof:}$\;\;$
At any given node, the incoming nodes may or may not have distinct accumulated relative degree. In the case where they do,
then Theorem~\ref{th:relative-degree-additive-network-distinct-condition} applies, otherwise, Theorem~\ref{th:relative-degree-additive-network-repeated-condition}
applies provided the condition for multiplicities greater than one can be met. Specifically, at node $k$ with incoming edges from nodes
$j_1,j_2,\ldots, j_N$ with accumulated relative degrees
$r^+_{j_1},r^+_{j_2},\ldots,r^+_{j_N}$, it is required that if $r^+_j$ is repeated $s_j>1$ times then
\begin{align*}
&W_{ij(1)}\langle d_{j(1)i},x_0^{r^+_j-1}x_i\rangle+
W_{ij(2)} \langle d_{j(2)i},x_0^{r^+_j-1}x_i\rangle+
\cdots \\
&\hspace*{0.2in}+W_{ij(s_j)}\langle d_{j(s_j)i},x_0^{r^+_j-1}x_i\rangle\neq 0,
\end{align*}
where $j(l)\in\{j_1,j_2,\ldots, j_N\}$, and the $W_{ij_l}$ are random variables with any continuous distribution(s).
But this condition is always true with probability one, and hence, the theorem is proved.
\hfill\bull \vspace*{0.05in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.6]{random_network}
\end{center}
\vspace*{-0.1in}
\caption{Estimate of density function for $\abs{\langle d_{41},x_0^2x_1\rangle}$ in Example~\ref{exer:random-network}.}
\label{fig:random-network-density-function}
\end{figure}

\begin{example} \label{exer:random-network}
{\rm
Reconsider Example~\ref{ex:four-node-counterexample}, where $W_{41} = 0$
and now $W_{21}$, $W_{31}$, $W_{42}$ and $W_{43}$ are i.i.d.
random variables with a uniform distribution on $[0,1]$.
An estimate of the density function for the random variable $\abs{\langle d_{41},x_0^2x_1\rangle}$ is shown in
Figure~\ref{fig:random-network-density-function}. In every case of the 1000
random networks generated, $d_{41}$ had relative degree $r=3$ as expected.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\section{Conclusions}

Two basic properties were established for an additive network of input-output systems where each
node of the network is modeled by a convergent Chen-Fliess series.
First it was shown that every input-output map between a pair of nodes has
a locally convergence Chen-Fliess series representation. An explicit and in some cases achievable growth bound on the
coefficients was computed using the notion of a maximal network. Second, sufficient
conditions were given under which the input-output map between a pair
of nodes has a well defined relative degree as defined by
its generating series. This analysis led to the conclusion
that this relative degree property is generic when the connection
strengths between nodes are randomized.

\section{Introduction}

Networks of nonlinear dynamical systems appear in many fields, especially in the natural sciences
where the nonlinearity is often a key feature in generating the observed behavior \cite{Golubitsky-Stewart_06,Jiang-Lai_19}.
The vast majority of analysis of such networks is done in a finite dimensional state space setting using coupled
systems of ordinary differential equations. In \cite{Gray-Ebrahimi-Fard_SCL21}, however, the authors describe an alternative
approach which uses only input-output models at each node of the network in the form of a locally convergent Chen-Fliess series \cite{Fliess_81,Fliess_83}.
These weighted infinite sums of iterated integrals provide a convenient algebraic framework for describing the
network's behavior without relying on any particular choice of coordinates as in the state space setting.
Series coefficients for each node can be estimated via system identification techniques \cite{Gray-etal_Auto20}.
Computational tools were developed in \cite{Gray-Ebrahimi-Fard_SCL21} to determine,
for example, how an input injected at one node affects the output observed at another node.
Nevertheless, there are still a number of
open questions regarding the basic properties of such networks. The focus here will be on so called {\em additive} networks, where the outputs
of the nodes are simply added together and injected into other nodes, including self-loops. Other classes of aggregation functions
such the multiplication of node outputs will not be addressed here.

This paper has two goals. The first goal to address the open problem stated in \cite{Gray-Ebrahimi-Fard_SCL21} regarding whether an additive network of
locally convergent Chen-Fliess series always yields mappings between nodes which have locally convergent Chen-Fliess series representations.
This hypothesis will be proved to be true and is independent of the network's topology.
The approach taken is to identify for a given network an associated {\em maximal network} whose
growth bounds on the coefficients of the generating series between nodes upper bound all the growth bounds of the original network and are much
easier to determine using conventional methods as presented in \cite{Sussmann_83}. The particular growth bound derived turns out to be exactly
equivalent to one discovered for a class of unity feedback systems described in \cite{Thitsa-Gray_12}.
The second goal
is to provide sufficient conditions under which the input-output map between a pair of nodes has well defined relative degree as defined by
its generating series \cite{Gray-etal_AUTO14,Gray-Venkatesh_SCL19}. A simple counterexample will be given first to show that
this property can fail to hold in certain situations.
The proofs of the sufficient conditions rely on identifying certain properties first described in \cite{Gray-Venkatesh_SCL19}
in relation to a subgraph connecting a given input node and output node.
It is also shown, however, that this relative degree property is {\em generic} in a certain sense.
Namely, if the generating series for every node has relative degree and the connection strengths between the nodes are random, then every node pair has
a generating series with well defined relative degree with probability one.
An obvious application for
this result is in the context of feedback linearization for networks \cite{Menara-etal_CDC20}, however, that application will not
be pursued here.

The paper is organized as follows. To keep the presentation as self-contained as possible, the required preliminaries are briefly
summarized in Section~\ref{sec:preliminaries}. The question regarding the convergence of Chen-Fliess series for
mappings between nodes is addressed in Section~\ref{sec:local-convergence}.
The subsequent section treats the property of relative degree. The paper's conclusions are summarized in the final section.

\section{Preliminaries}
\label{sec:preliminaries}

An {\em alphabet} $X=\{ x_0,x_1,$ $\ldots,x_m\}$ is any nonempty and finite set
of noncommuting symbols referred to as {\em
letters}. A {\em word} $\eta=x_{i_1}\cdots x_{i_k}$ is a finite sequence of letters from $X$.
The number of letters in a word $\eta$, written as $\abs{\eta}$, is called its {\em length}.
The empty word, $\emptyset$, is taken to have length zero.
The collection of all words having length $k$ is denoted by
$X^k$. Define $X^\ast=\bigcup_{k\geq 0} X^k$,
which is a monoid under the concatenation (Cauchy) product.
Any mapping $c:X^\ast\rightarrow
\re^\ell$ is called a {\em formal power series}.
Often $c$ is
written as the formal sum $c=\sum_{\eta\in X^\ast}\langle c,\eta\rangle\eta$,
where the {\em coefficient} $\langle c,\eta\rangle\in\re^\ell$ is the image of
$\eta\in X^\ast$ under $c$.
The {\em support} of $c$, ${\rm supp}(c)$, is the set of all words having nonzero coefficients.
A series $c$ is {\em proper} when $\emptyset\not\in{\rm supp}(c)$.
The set of all noncommutative formal power series over the alphabet $X$ is
denoted by $\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$. The subset of series with finite support, i.e., polynomials,
is represented by $\mbox{$\re^{\ell}\langle X \rangle$}$.
For any $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$, the scalar product is
$\langle c,d\rangle:=\sum_{\eta\in X^\ast} \langle c,\eta\rangle\langle d,\eta\rangle$, provided the sum is finite.
The set $\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is an associative $\re$-algebra under the concatenation product and an associative and commutative $\re$-algebra under
the {\em shuffle product}, that is, the bilinear product uniquely specified by the shuffle product of two words $x_i\eta,x_j\xi\in X^\ast$:
\begin{displaymath}
	(x_i\eta){\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}(x_j\xi)=x_i(\eta{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}(x_j\xi))+x_j((x_i\eta){\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} \xi),
\end{displaymath}
where $x_i,x_j\in X$ and with $\eta{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}\emptyset=\emptyset{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,}\eta=\eta$ \cite{Fliess_81}.

\subsection{Chen-Fliess series}

Given any $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ one can associate a causal
$m$-input, $\ell$-output operator, $F_c$, in the following manner.
Let $\mathfrak{p}\ge 1$ and $t_0 < t_1$ be given. For a Lebesgue measurable
function $u: [t_0,t_1] \rightarrow\re^m$, define
$\norm{u}_{\mathfrak{p}}=\max\{\norm{u_i}_{\mathfrak{p}}: \ 1\le
i\le m\}$, where $\norm{u_i}_{\mathfrak{p}}$ is the usual
$L_{\mathfrak{p}}$-norm for a measurable real-valued function,
$u_i$, defined on $[t_0,t_1]$.  Let $L^m_{\mathfrak{p}}[t_0,t_1]$
denote the set of all measurable functions defined on $[t_0,t_1]$
having a finite $\norm{\cdot}_{\mathfrak{p}}$ norm and
$B_{\mathfrak{p}}^m(R)[t_0,t_1]:=\{u\in
L_{\mathfrak{p}}^m[t_0,t_1]:\norm{u}_{\mathfrak{p}}\leq R\}$.
Assume $C[t_0,t_1]$
is the subset of continuous functions in $L_{1}^m[t_0,t_1]$. Define
inductively for each word $\eta=x_i\bar{\eta}\in X^{\ast}$ the map $E_\eta:
L_1^m[t_0, t_1]\rightarrow C[t_0, t_1]$ by setting
$E_\emptyset[u]=1$ and letting
\[E_{x_i\bar{\eta}}[u](t,t_0) =
\int_{t_0}^tu_{i}(\tau)E_{\bar{\eta}}[u](\tau,t_0)\,d\tau, \] where
$x_i\in X$, $\bar{\eta}\in X^{\ast}$, and $u_0=1$. The
{\em Chen--Fliess series} corresponding to $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is
\begin{displaymath}
y(t)=F_c[u](t) =
\sum_{\eta\in X^{\ast}} \langle c,\eta\rangle \,E_\eta[u](t,t_0)
\end{displaymath}
\hspace*{-0.06in}\cite{Fliess_81}.
If there exist real numbers $K,M>0$ such that
\begin{displaymath}
\abs{\langle c,\eta\rangle}\le K M^{|\eta|}|\eta|!,\;\; \forall\eta\in X^{\ast},
\end{displaymath}
then $F_c$ constitutes a well defined mapping from
$B_{\mathfrak p}^m(R)[t_0,$ $t_0+T]$ into $B_{\mathfrak
q}^{\ell}(S)[t_0, \, t_0+T]$ for sufficiently small $R,T >0$ and some $S>0$,
where the numbers $\mathfrak{p},\mathfrak{q}\in[1,\infty]$ are
conjugate exponents, i.e., $1/\mathfrak{p}+1/\mathfrak{q}=1$  \cite{Gray-Wang_02}.
(Here, $\abs{z}:=\max_i \abs{z_i}$ when $z\in\re^\ell$.) The set of all such
{\em locally convergent} series is denoted by $\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$, and $F_c$ is
referred to as a {\em Fliess operator}.

\subsection{System interconnections}

Given Fliess operators $F_c$ and $F_d$, where $c,d\in\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$,
the parallel and product connections satisfy $F_c+F_d=F_{c+d}$ and $F_cF_d=F_{c{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} d}$,
respectively \cite{Fliess_81}.
When Fliess operators $F_c$ and $F_d$ with $c\in\mbox{$\re^{\ell}_{LC}\langle\langle X \rangle\rangle$}$ and
$d\in\mbox{$\re^{m}_{LC}\langle\langle X \rangle\rangle$}$ are interconnected in a cascade fashion, the composite
system $F_c\circ F_d$ has the
Fliess operator representation $F_{c\circ d}$, where
the {\em composition product} of $c$ and $d$
is given by
\begin{displaymath}
c\circ d=\sum_{\eta\in X^\ast} \langle c,\eta\rangle\,\psi_d(\eta)(\mathbf{1})
\end{displaymath}%
\hspace*{-0.07in}\cite{Ferfera_80}. Here $\mbf{1}$ denotes the monomial $1\emptyset$, and
$\psi_d$ is the continuous (in the ultrametric sense) algebra homomorphism
from $\mbox{$\re\langle\langle X \rangle\rangle$}$ to the vector space endomorphisms on $\mbox{$\re\langle\langle X \rangle\rangle$}$, ${\rm End}(\allseries)$, uniquely specified by
$\psi_d(x_i\eta)=\psi_d(x_i)\circ \psi_d(\eta)$ with
$
\psi_d(x_i)(e)=x_0(d_i{\, {\mathchoice{\shuff{5pt}{3.5pt}}{\shuff{5pt}{3.5pt}}{\shuff{3pt}{2.6pt}}{\shuff{3pt}{2.6pt}}} \,} e),
$
$i=0,1,\ldots,m$
for any $e\in\mbox{$\re\langle\langle X \rangle\rangle$}$,
and where $d_i$ is the $i$-th component series of $d$
($d_0:=\mbf{1}$). By definition,
$\psi_d(\emptyset)$ is the identity map on $\mbox{$\re\langle\langle X \rangle\rangle$}$.

\subsection{Relative degree of a generating series}

Let $X=\{x_0,x_1\}$.
Following \cite{Gray-etal_AUTO14}, a series $c\in\mbox{$\re\langle\langle X \rangle\rangle$}$ has relative degree $r$ if and only if it
has the decomposition
\begin{displaymath}
c=c_N+Kx_0^{r-1}x_1+x_0^{r-1}e
\end{displaymath}
for some $K\neq 0$ and proper $e\in\mbox{$\re\langle\langle X \rangle\rangle$}$ with $x_1\not\in{\rm supp}(e)$.
This definition of relative degree is consistent with the classical definition
whenever $y=F_c[u]$
is realizable \cite{Gray-etal_AUTO14,Gray-Ebrahimi-Fard_SIAM17}.
The following results
will be of central
importance in the work that follows.

\begin{theorem} \cite{Gray-Venkatesh_SCL19} \label{th:r-parallel-sum-connection}
If $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$ have distinct relative degrees $r_c$ and $r_d$, respectively, then $c+d$ has relative
degree $\min(r_c,r_d)$. On the other hand, if $r_c = r_d =: r$, then $c+d$ has relative degree $r$ if and
only if  $\langle c,x_{0}^{r-1}x_{1}\rangle + \langle d,x_{0}^{r-1}x_{1}\rangle \neq 0$.
\end{theorem}

\begin{corollary} \label{co:relative-degree-multi-sum-distinct}
If $c_1,c_2,\ldots,c_m$ have relative degree $r_1,r_2,\ldots,r_m$, respectively,
with $r_i\neq r_j$ when $i\neq j$,
then the relative degree of $c_1+c_2+\cdots+c_m$ is $\min_i(r_i)$.
\end{corollary}

\begin{corollary} \label{co:relative-degree-multi-sum-not-distinct}
Suppose $c_1,c_2,\ldots,c_m$ have relative degree $r_1,r_2,\ldots,r_m$, respectively.
Let $s_j$ denote the multiplicity of relative degree $r_j$.
If for each $s_j>1$ the series $c_{k_1},c_{k_2},\ldots,c_{k_{s_j}}$ having relative degree $r_j$
satisfy
\begin{displaymath}
\langle c_{k_1},x_0^{r_j-1}x_1\rangle+\langle c_{k_2},x_0^{r_j-1}x_1\rangle+\dots+\langle c_{k_{s_j}},x_0^{r_j-1}x_1\rangle\neq 0,
\end{displaymath}
then the relative degree of $c_1+c_2+\cdots+c_m$ is $\min_i(r_i)$.
\end{corollary}

\begin{theorem} \cite{Gray-Venkatesh_SCL19} \label{th:relative-degree-casecade}
If $c,d\in\mbox{$\re\langle\langle X \rangle\rangle$}$ have relative degrees $r_c$ and $r_d$, respectively,
then $r_{c\circ d}$ has relative degree $r_c+r_d$.
\end{theorem}

\subsection{Formal realizations and representations}

It is shown in \cite{Kawski-Sussmann_97} that a given Chen-Fliess series $y=F_c[u]$ can be written in terms
of a state $z$ evolving on a formal Lie group $\mathcal{G}(X)$ with Lie algebra $\widehat{\mathcal L}(X)$ and output
map $y=\langle c,z\rangle$. This notion of a {\em universal control system} was generalized in
\cite{Gray-Ebrahimi-Fard_SCL21} as follows to describe networks of Chen-Fliess series.

\begin{definition}
Let $V_i$ be a vector field on $\mathcal{G}^n(X):=\mathcal{G}(X)\times\mathcal{G}(X)\times\cdots\times \mathcal{G}(X)$, $i=0,1,\ldots,m$ with
\begin{align*}
V_i&:{\mathcal G}^n(X)\rightarrow T_z{\mathcal G}^n(X) \\
&z=(z_1,\ldots,z_n)\mapsto V_i(z)=(V_{i1}(z)z_1,\ldots,V_{in}(z)z_n),
\end{align*}
where $V_{ij}(z(t))\in \widehat{\mathcal L}(X)$.
The $j$-th component of the corresponding state equation on ${\mathcal G}^n(X)$ is
\begin{displaymath}
\dot{z}_j=\sum_{i=0}^m V_{ij}(z)z_j u_{ij} , \;\; z_j(0)=z_{j0}.
\end{displaymath}
Given $\hat{c}_{k}\in\mbox{$\re_{LC}^{\otimes n}\langle\langle X \rangle\rangle$}$, $k=1,2,\ldots,\ell$, the $k$-th output equation is defined to be
\begin{displaymath}
y_{k}=\hat{c}_{k}(z).
\end{displaymath}
Collectively, $(V,z_0,\hat{c})$ is a {\em formal realization} on ${\mathcal G}^n(X)$ of the formal input-output map
$u\mapsto y$.
\end{definition}

Analogous to the standard finite dimensional theory \cite{Isidori_95,Nijmeijer-vanderSchaft_90}, a series $c\in\mbox{$\re^{\ell} \langle\langle X \rangle\rangle$}$ is said to have a {\em formal representation} when
there exists a formal realization with the property that every coefficient of $c$ can be written in terms of iterated Lie derivatives of the vectors
fields acting on the output map and evaluated at $z_0$, i.e., $\langle c,x_{i_1}\cdots x_{i_k}\rangle=L_{V_{i_k}}\cdots L_{V_{i_1}}\hat{c}(z_0)$.

\section{Additive Networks of Chen-Fliess Series: Local Convergence}
\label{sec:local-convergence}

In this section it is shown that every network of additively interconnected locally convergent Fliess operators has the
property that the input-output maps between any two nodes can be represented by a locally convergent Fliess operator.
The first definition describes the specific class of networks under consideration.

\begin{definition}
A set of $m$ single-input, single-output Chen-Fliess series mapping $u_i$ to $y_i$ with generating series
$c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$, where $X_i=\{x_0,x_i\}$
is said to be an {\em additively interconnected network} ${\mathcal N}_m$
with weighting matrix $W\in\re^{m\times m}$
if $u_i=v_i+\sum_{j=1}^mW_{ij}y_j$, $i=1,2,\ldots,m$.
\end{definition}

A network ${\mathcal N}_m$ can therefore be viewed as a directed graph connecting $m$ nodes,
where the $i$-th node corresponds to a Chen-Fliess series with generating series $c_i$, $i=1,2,\ldots,m$.
Henceforth, it will be assumed that the connection weights are normalized so that $W_{ij}\in[0,1]$, $i,j=1,2,\ldots,m$.
The following theorem follows directly from Theorem 5.1 in \cite{Gray-Ebrahimi-Fard_SCL21}.

\begin{theorem} \label{th:additive-interconnections}
The input-output map $v_i\mapsto y_j$ in any additively interconnected network ${\mathcal N}_m$
has generating series $d_{ji}\in\re\langle\langle X_i\rangle\rangle$ which can be computed from a formal
representation in terms of the vector fields
\begin{align*}
V_0(z)&=
\left[
\begin{tabular}{p{0.5cm}}
\hspace*{-0.1cm}$x_0z_1$ \\
\hspace*{-0.1cm}$x_0z_2$ \\
\hspace*{0.2cm}\vdots \\
\hspace*{-0.1cm}$x_0z_m$
\end{tabular}
\right]+{\rm diag}(x_1z_1,\ldots,x_m z_m)W
\left[
\begin{tabular}{p{1cm}}
\hspace*{-0.02cm}$\langle c_1,z_1\rangle$ \\
\hspace*{-0.02cm}$\langle c_2,z_2\rangle$ \\
\hspace*{0.4cm}\vdots \\
\hspace*{-0.13cm}$\langle c_m,z_m\rangle$
\end{tabular}
\right] \\
V_i(z)&=x_iz_i\mbf{e}_i
\end{align*}
acting on
$\hat{c}_j=\mbf{1}\otimes\cdots\otimes\mbf{1}\otimes c_j\otimes\mbf{1}\cdots\otimes\mbf{1}\in\mbox{$\re_{LC}^{\otimes m}\langle\langle X \rangle\rangle$}$ ($c_j$ appears in the $j$-th position)
and evaluated at $z_{j0}=\mbf{1}$, $i,j=1,2\ldots,m$.
\end{theorem}

The next theorem states the main convergence result concerning additive networks.

\begin{theorem}
If ${\mathcal N}_m$ is an additively interconnected network where the generating series for each node $c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$,
then the generating series for every input-output map $d_{ji}\in\re_{LC}\langle\langle X_i\rangle\rangle$. More specifically, if $K_i,M_i$ denote the growth
constants for $c_i$, then for all $i,j=1,2,\ldots,m$
\begin{displaymath}
\abs{\langle d_{ji},\eta\rangle}<KM^{\abs{\eta}}\abs{\eta}!,\;\;\forall \eta\in X^\ast
\end{displaymath}
for some $K>0$ and any $M>M_{\rm inf}$, where
\begin{equation} \label{eq:M-inf-additive-network}
M_{\rm inf}=\frac{\bar{M}}{1-m\bar{K} \ln\left(1+\frac{1}{m\bar{K}}\right)}
\end{equation}
with $\bar{K}=\max_{i} K_i$ and $\bar{M}=\max_{i} M_i$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
It is first shown that each generating series $d_{ji}$ is locally convergent. Consider the case where
every node series $c_i\in\re_{LC}\langle\langle X_i\rangle\rangle$ is a {\em maximal series}
$\bar{c}_i:=\sum_{\eta\in X^\ast}K_iM_i^{\abs{\eta}}\abs{\eta}!\, \eta$.
That is, every coefficient of $\bar{c}_i$ is growing at its maximal rate. While $y_i=F_{c_i}[u_i]$ may not have a finite
dimensional state space realization,
it is easily shown that a maximal series has the realization
\begin{displaymath}
\dot{z}_i=\frac{M_i}{K_i}z_i^2(1+u_i),\;\;z_i(0)=K_i,\;\;y_i=z_i
\end{displaymath}
\hspace*{-0.07in}\cite[Lemma 3]{Thitsa-Gray_12}.
Therefore, the corresponding network can be realized by
\begin{equation}
\label{eq:maximal-series-state-space}
\dot{z}_i=\frac{M_i}{K_i}z_i^2\left(1+\sum_{j=1}^m W_{ij}z_j+v_i\right)\!\!,\;\;z_i(0)=K_i,\;\;y_i=z_i,
\end{equation}
$i=1,2,\ldots,m$. As this realization of the input-output map $v\mapsto y$ is polynomial, it is clearly real analytic.
Therefore, every generating series for $v_i\mapsto y_j$, say $\bar{d}_{ji}$, must be locally convergent \cite[Lemma 4.2]{Sussmann_83}.
The claim now is that $d_{ji}$ must also be locally convergent since $\abs{\langle d_{ji},\eta\rangle}\leq \langle \bar{d}_{ji},\eta\rangle$
for all $\eta\in X^\ast$. This inequality is most easily deduced from the formal realization of $v_i\mapsto y_j$ given in Theorem~\ref{th:additive-interconnections},
where the Lie derivatives used to compute the coefficients of $d_{ji}$ will all be upper bounded in magnitude by the Lie derivatives computed using maximal series.

Next, a suitable geometric growth constant for the network ${\mathcal N}_m$ is determined.
First observe that the growth constants $\bar{K}$ and $\bar{M}$ constitute a worst case maximum growth rate for every
node in the network.
In light of the formal representation of any $d_{ji}$ in Theorem~\ref{th:additive-interconnections}, the growth
rate of $d_{ji}$ is upper bounded by the growth rate of the natural response $\langle \bar{d}_{ji},x_0^k\rangle=L_{V_0}^k\hat{c}_j(\mbf{1})$, $k\geq 0$,
where $W_{ij}=1$ for all $i,j$, and every non-trivial component of $\hat{c}_j$ is the maximal series
$\bar{c}=\sum_{\eta\in X^\ast}\bar{K} \bar{M}^{\abs{\eta}}\abs{\eta}!\, \eta$.
(See \cite[Lemma 7]{Thitsa-Gray_12} for an alterative approach when $m=1,2$.)
From the symmetry of such a {\em maximal network}, $z_i=z_j$ for all $i,j$. Applying these conditions to \rref{eq:maximal-series-state-space},
the natural response at each node is
given by the solution of the Abel differential equation
\begin{equation} \label{eq:maximal-network-Abel-equation}
\dot{z}=\frac{\bar{M}}{\bar{K}}(z^2+mz^3),\;\;z(0)=\bar{K}.
\end{equation}
It can be directly verified that this equation has the solution
\begin{displaymath}
z(t)=\frac{-\frac{1}{m}}{1+\mathcal{W}\left[-\left(1+\frac{1}{m\bar{K}}\right)\exp\left(\frac{\bar{M}}{m\bar{K}}t-\left(1+\frac{1}{m\bar{K}}\right)\right)\right]},
\end{displaymath}
where $\mathcal{W}$ denotes the Lambert $W$-function, that is, the inverse of the function
$f(x)=x\exp(x)$ corresponding to the principal branch of this multi-valued function \cite{Corless-etal_96}.
As $\mathcal{W}$ is known to be holomorphic on the complex plane,
$z(t)$ will therefore be analytic at $t=0$. The corresponding Taylor series has a radius of convergence determined by the singularity nearest to the origin,
in this case
\begin{displaymath}
t^\ast=\frac{1}{\bar{M}}\left(1-m\bar{K}\ln\left(1+\frac{1}{m\bar{K}}\right)\right).
\end{displaymath}
Applying a well known theorem from complex analysis (see \cite[Theorem~2.4.3]{Wilf_94}) gives
the infimum of all geometric growth constants for the maximal network, namely $M_{\rm inf}=1/t^\ast$.
(Note that the function $\lambda(x)=1-x\ln(1+{1/x})$ is a decreasing function, which further justifies using the maximum $K_i$ in
the network as the worst case.) Since for any $M>M_{\rm inf}$ there is a $K>0$ to upper bound the fastest coefficient growth
in the maximal network,
the generating series for every node in the original network must also be upper bounded by this growth rate.
\hfill\bull \vspace*{0.05in}

It is worth noting that \rref{eq:M-inf-additive-network} is in fact identical to the growth constant identified for
unity feedback systems with $m$ inputs as described in \cite[Corollary~2]{Thitsa-Gray_12}. While the network topologies
are clearly distinct, this point of tangency is derived from the fact that unity feedback systems and
additive maximal networks both have natural responses satisfying \rref{eq:maximal-network-Abel-equation}.

\begin{table}[tb]
\caption{Integer sequences generated by maximal additive network with unity growth constants}
\label{tbl:network-series}
\vspace*{-0.1in}
\begin{center}
\begin{tabular}{c|l|c|c}
	\toprule
	$m$ & \hspace*{0.8in}$a_n$ & $M_{\rm inf}$ & $\hat{M}_n$ \\
	\midrule
	1 & 1, 2, 10, 82, 938, 13778, 247210, $\ldots$            & 3.2589     & 3.22634 \\
	2 & 1, 3, 24, 318, 5892, 140304, $\ldots$         & 5.2891     & 5.23618 \\
	3 & 1, 4, 44, 804, 20556, 675588, $\ldots$        & 7.3017     & 7.22873 \\
	4 & 1, 5, 70, 1630, 53120, 2225480, $\ldots$      & 9.3088     & 9.21567 \\
	5 & 1, 6, 102, 2886, 114294, 5819190, $\ldots$    & 11.3132    & 11.2001 \\
	6 & 1, 7, 140, 4662, 217308, 13022688,$\ldots$   & 13.3163    &  13.1831 \\
	\bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.55]{three_node_simulation_worst_case}
\end{center}
\vspace*{-0.2in}
\caption{Natural response of three node maximal network in Example~\ref{ex:three-node-maximal-network-worst-case}.}
\label{fig:three-node-maximal-network-worst-case}
\end{figure}

\begin{example} \label{ex:three-node-maximal-network-worst-case}
{\rm
Consider a maximal additive network where $K_i=M_i=1$, $i=1,2,\ldots,m$. The Taylor series
of the natural response has integer coefficients $a_n$, $n\geq 1$ as shown in Table~\ref{tbl:network-series}.
The coefficients when $m=1$ correspond to the OEIS integer sequence
A112487 \cite{OEIS}. The table also shows the growth rate $M_{\rm \inf}$ computed from \rref{eq:M-inf-additive-network}
and an estimate of the growth constant $M$ computed
from $\hat{M}_n=na_{n}/a_{n-1}$ when $n=50$.
The corresponding three node network was simulated in MatLab for the zero input case. The node responses,
which are identical, are
shown in Figure~\ref{fig:three-node-maximal-network-worst-case}.
Since the coefficients of every generating series are positive,
it is known that the natural response of every node will have
a finite escape time at $t=t^\ast$ (see \cite[Theorem 11]{Thitsa-Gray_12}).
In this case, $t^\ast=1/M_{\rm inf}=0.1379$, which is what was observed
in the simulation.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.55]{three_node_simulation_generic}
\end{center}
\vspace*{-0.1in}
\caption{Natural response of three node network in Example~\ref{ex:three-node-maximal-network-generic}.}
\label{fig:three-node-maximal-network-generic}
\end{figure}

\begin{example} \label{ex:three-node-maximal-network-generic}
{\rm
Consider a three node additive network involving maximal series with $K_i=i$, $M_i=5-i$ and
\begin{displaymath}
W=\left[\begin{array}{cccc}
1 & 0.5 & 1  \\
1 & 1 & 0 \\
0.25 & 1 & 1
\end{array}\right].
\end{displaymath}
Thus, $\bar{K}=3$, $\bar{M}=4$, and $M_{\rm inf}=77.2867$.
The node natural responses are
shown in Figure~\ref{fig:three-node-maximal-network-generic}.
As this network is not maximal,
$t^\ast=1/M_{\rm inf}=0.01294$ provides only a lower bound on the escape times
of each node.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\section{Additive Networks of Chen-Fliess Series: Relative Degree}
\label{sec:relative-degree}

In this section the following question is addressed: When does the generating series of the
mapping $v_i\mapsto y_j$ in an additively interconnected
network ${\mathcal N}_m$ have a well defined relative degree?
The treatment starts with the easiest case first as described next.
It is assumed throughout that ${\mathcal N}_m$ is
comprised of systems with generating series $c_i$ which have relative degree $r_i$ for
$i=1,2,\ldots,m$.

\begin{definition}
The $i$-th node in a network ${\cal N}_m$ is said to be {\em fully connected} if $W_{ij}\neq 0$ for all $j\neq i$.
A network ${\cal N}_m$ is said to be {\em fully connected} if every node is fully connected.
\end{definition}

Note that self-loops, i.e., when $W_{ii}\neq 0$, are not important in the present context as proportional output feedback
is easily shown to preserve relative degree \cite{Gray-Venkatesh_SCL19}.

\begin{theorem} \label{th:relative-degree-fully-connected}
If
the $i$-th node in ${\mathcal N}_m$ is fully connected, then the generating series $d_{ji}$ for mapping
$v_i\mapsto y_j$ has relative degree $r_{ji}=r_j+r_i$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
Observe that the full output at node $j$ is
\begin{align*}
y_j&=F_{c_j}\left[v_j+\sum_{k=1}^m W_{jk} y_k \right] \\
&=F_{c_j}\left[v_j+\sum_{k,l=1}^m W_{jk} F_{d_{kl}}[v_l]\right].
\end{align*}
For any $i\neq j$, that part of $y_j$ in response to $v_i$ acting alone (i.e., $v_l=0$ for $l\neq i$) is
given by
\begin{align*}
y_{j}&=F_{c_j}\Bigg[W_{ji}F_{c_i}[v_i]+\sum_{k=1\atop k\neq i}^m W_{jk} F_{d_{ki}}[v_i]+ \\
&\hspace{0.2in} \sum_{k,l=1\atop l\neq i}^m W_{jk} F_{d_{kl}}[0]\Bigg].
\end{align*}
Note that for all $k\neq i$, ${\rm supp}(d_{ki})\subseteq x_0^{r}X^\ast$, where $r\geq r_i+1$, since
$v_i$ passes through $F_{c_i}$ in every path leading to the $j$-th node. In which case, the argument of $F_{c_j}$ above
has a generating series with relative degree $r_i$. The conclusion then follows immediately from Theorem~\ref{th:relative-degree-casecade}.
\hfill\bull \vspace*{0.05in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{diamond_network}
\end{center}
\vspace*{-0.2in}
\caption{Four node network in Example~\ref{ex:four-node-counterexample}.}
\label{fig:four-node-counterexample}
\end{figure}

\begin{example} \label{ex:four-node-counterexample}
{\rm
Consider the network shown in Figure~\ref{fig:four-node-counterexample}.
The corresponding weighting matrix is
\begin{displaymath}
W=\left[\begin{array}{cccc}
0 & 0 & 0 & 0 \\
W_{21} & 0 & 0 & 0 \\
W_{31} & 0 & 0 & 0 \\
W_{41} & W_{42} & W_{43} & 0
\end{array}\right].
\end{displaymath}
The network is clearly {\em not} fully connected,
but node $4$ is fully connected assuming $W_{4j}\neq 0$, $j=1,2,3$.
Therefore, applying the theorem above gives, for example, that
$r_{41}=r_4+r_1$.

Suppose now that $W_{41}=0$ so that the theorem no longer applies. Further assume that $r_2=r_3=r$.
Observe that
\begin{displaymath}
u_4=W_{42}F_{c_2}[W_{21}F_{c_1}[u_1]]+W_{43}F_{c_3}[W_{31}F_{c_1}[v_1]],
\end{displaymath}
and thus,
\begin{displaymath}
d_{41}=c_4\circ [W_{42}(c_2\circ (W_{21}c_1))+W_{43}(c_3\circ (W_{31}c_1))].
\end{displaymath}
Both $c_2\circ (W_{21}c_1)$ and $c_3\circ (W_{31}c_1)$ have relative degree $r+r_1$,
but $d_{41}$ can fail to have relative degree.
As a simple example, suppose $c_1=c_2=c_4=x_1$ and $c_3=-x_1$ so that
$d_{41}=(W_{42}W_{21}-W_{43}W_{31}) x_0^2x_1$.
If $W$ is such that $W_{42}W_{21}=W_{43}W_{31}$, then $d_{41}=0$ does not have relative degree.
On the other hand, if the symmetry condition $r_2=r_3$ is broken, then it follows that $d_{41}$ has
relative degree $r_{41}=r_4+\min (r_2,r_3)+r_1$.
}
\hfill\openbull \end{example} \vspace*{0.1in}

The final case in the example above suggests a sufficient condition for the general case. Namely,
in the absence of these degenerate situations where a node is presented with an input whose underlying
generating series does not have relative degree, the relative degree for $d_{ji}$ will be well defined and determined
by a path from node $i$ to node $j$ whose {\em accumulated} relative degrees is minimal. To make this
claim more precise, the following language adapted from signal flow graph theory will be useful.

Let ${\mathcal N}_m$ be a given additive network.
An {\em edge} is a directed line segment connecting two nodes.
A {\em path} is a continuous set of edges connecting two nodes in ${\mathcal N}_m$ and traversed in the direction indicated.
A {\em forward path} is a path in which no node is encountered more than once.
A {\em loop} is a path that originates and ends on the same node in which no node is encountered more than once.
Finally, the {\em subgraph} $G_{ji}$ from node $i$ to node $j$ is the simple graph (i.e., all loops are omitted)
consisting of all forward paths connecting node $i$ and node $j$.

The following theorems provide a sufficient condition under which the relative degree is well defined for a given
input-output map $v_i\mapsto y_j$ in an additive network. Given a subgraph $G_{ji}$, the {\em accumulated relative degree} of node $i$ is
$r_i^+=r_i$. If node $k\neq i$ in $G_{ji}$ has $N$ incoming edges from nodes $i_1,i_2,\ldots, i_N$ with accumulated relative degrees
$r^+_{i_1},r^+_{i_2},\ldots,r^+_{i_N}$, respectively, then the {\em accumulated relative degree} at node $k$ is
\begin{displaymath}
r^+_k=r_k+\min\{r^+_{i_1},r^+_{i_2},\ldots,r^+_{i_N}\}.
\end{displaymath}
Note this definition does not imply that any mappings defined by the network have relative degree, it simply
computes the {\em potential} relative degree of such a mapping should it be well defined.

\begin{theorem} \label{th:relative-degree-additive-network-distinct-condition}
Let $i$ and $j$ be fixed nodes in ${\mathcal N}_m$.
If at every node $l{\not\in}\{i,j\}$  the accumulated relative degrees
of the nodes from every incoming edge are distinct, then
the generating series $d_{ji}$ for $v_i\mapsto y_j$ in ${\mathcal N}_m$ has well defined relative degree
equivalent to $r_{ji}=r_j^+$.
\end{theorem}

\noindent{\em Proof:}$\;\;$
As feedback loops do not affect the relative degree of any forward path, it is sufficient
to consider only the subgraph $G_{ji}$.
The claim then follows directly from Corollary~\ref{co:relative-degree-multi-sum-distinct},
Theorem~\ref{th:relative-degree-casecade}, and the definition of accumulated relative degree.

\hfill\bull \vspace*{0.05in}

The distinctness condition in the above theorem can be relaxed by
utilizing instead the condition in Corollary~\ref{co:relative-degree-multi-sum-not-distinct}.

\begin{theorem} \label{th:relative-degree-additive-network-repeated-condition}
Let $i$ and $j$ be fixed nodes in ${\mathcal N}_m$.
If at every node $l{\not\in}\{i,j\}$ the accumulated relative degrees
of the nodes from every incoming edge satisfy the condition in Corollary~\ref{co:relative-degree-multi-sum-not-distinct},
then the generating series $d_{ji}$ for $v_i\mapsto y_j$ in ${\mathcal N}_m$ has well defined relative degree
equivalent to $r_{ji}=r_j^+$.
\end{theorem}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{double_diamond_network}
\end{center}
\vspace*{-0.1in}
\caption{Network in Example~\ref{ex:double-diamond-example}.}
\label{fig:double-diamond-example}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.7]{double_diamond_network_subgraph}
\end{center}
\vspace*{-0.1in}
\caption{Subgraph of forward paths for $v_1\mapsto y_7$ in Example~\ref{ex:double-diamond-example}. The relative
degree of each generating series $c_i$ is the circled number. The accumulated relative degree at each node is the number in the triangle.}
\label{fig:double-diamond-example-subgraph}
\end{figure}

\begin{example} \label{ex:double-diamond-example}
{\rm
Consider the network shown in Figure~\ref{fig:double-diamond-example}, where each weight $W_{ij}\in \{0,1\}$ (i.e., $0\sim\mbox{not connected}$,
$1\sim\mbox{connected}$), and
the generating series for the nodes are:
\begin{align*}
c_1 &= K_1 x_1 + 2 x_0  x_1 \\
c_2 &= x_0+K_2 x_0^2  x_2 \\
c_3 &= K_3 x_0  x_3 + 3 x_0^2  x_3^2 \\
c_4 &= 1+K_4 x_0  x_4 - x_0^2  x_4x_0 \\
c_5 &= 4x_0+K_5 x_0^2  x_5 -2x_0^4x_5 \\
c_6 &= K_6 x_6 -x_6^2\\
c_7 &= x_0+2+K_7 x_7+4x_0x_7
\end{align*}
with $K_i\neq 0$ in every case.
The subgraph of forward paths is shown in Figure~\ref{fig:double-diamond-example-subgraph}.
The relative degree of the generating series at each node is the circled number shown next to each node.
The accumulated relative degree at each node is the number in the triangle
The goal is to determine the relative degree of the mapping $v_1\mapsto y_7$, provided it is well defined.
Observe that only nodes $4$, $5$ and $7$, have more than one incoming edge. In each case, the accumulated relative degrees
are distinct, namely, $3,4$; $4,5$; and $6,7$, respectively. Therefore, Theorem~\ref{th:relative-degree-additive-network-distinct-condition}
applies,
and $r_{71}=7$.
To independently verify this claim,
the generating series $d_{71}$ was computed using
the full network via Theorem~\ref{th:additive-interconnections} with the aid of Mathematica and found to be
\begin{displaymath}
d_{71}=d_{71,N}+K_1K_3K_4K_6K_7x_0^6x_1+x_0^6e,
\end{displaymath}
where
\begin{align*}
d_{71,N} = & {\,} x_0+
(4 K_7 + K_6 K_7)x_0^2  +(16 + 4 K_6 - K_7)x_0^3+ \\
& {\,} (-4 + K_5 K_7 + 2 K_4 K_6 K_7) x_0^4 + (4 K_5 + 8 K_4 K_6- \\
& {\,} 8 K_4 K_7 + K_5 K_7 + 2 K_4 K_6 K_7) x_0^5 + \cdots\\
e = & {\,} (4 K_1 K_3 K_4 K_6 - 6 K_1 K_3 K_4 K_7 + K_1 K_2 K_5 K_7+ \\
& {\,}K_1 K_2 K_4 K_6 K_7 + 2 K_3 K_4 K_6 K_7) x_0 x_1 + \cdots
\end{align*}
The relative degree of $d_{71}$ is 7 as expected.
}
\hfill\openbull \end{example} \vspace*{0.1in}

An additive network ${\mathcal N}_m$ is said to have {\em complete relative degree} if every mapping
$v_i\mapsto y_j$, $i,j=1,2,\ldots,m$ has relative degree. From Theorem~\ref{th:relative-degree-fully-connected} it
is immediate that fully connected networks have this property. Another class of networks sharing this property is given
in the following theorem. It states that in some sense the property of a network having complete relative degree is {\em generic}.

\begin{theorem}
Consider an additive network ${\mathcal N}_m$ where the weighting matrix has entries $W_{ij}\in\{0,1\}$. If the
unity weights are replaced with continuous random variables, then every sample network has complete relative degree.
\end{theorem}

\noindent{\em Proof:}$\;\;$
At any given node, the incoming nodes may or may not have distinct accumulated relative degree. In the case where they do,
then Theorem~\ref{th:relative-degree-additive-network-distinct-condition} applies, otherwise, Theorem~\ref{th:relative-degree-additive-network-repeated-condition}
applies provided the condition for multiplicities greater than one can be met. Specifically, at node $k$ with incoming edges from nodes
$j_1,j_2,\ldots, j_N$ with accumulated relative degrees
$r^+_{j_1},r^+_{j_2},\ldots,r^+_{j_N}$, it is required that if $r^+_j$ is repeated $s_j>1$ times then
\begin{align*}
&W_{ij(1)}\langle d_{j(1)i},x_0^{r^+_j-1}x_i\rangle+
W_{ij(2)} \langle d_{j(2)i},x_0^{r^+_j-1}x_i\rangle+
\cdots \\
&\hspace*{0.2in}+W_{ij(s_j)}\langle d_{j(s_j)i},x_0^{r^+_j-1}x_i\rangle\neq 0,
\end{align*}
where $j(l)\in\{j_1,j_2,\ldots, j_N\}$, and the $W_{ij_l}$ are random variables with any continuous distribution(s).
But this condition is always true with probability one, and hence, the theorem is proved.
\hfill\bull \vspace*{0.05in}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.6]{random_network}
\end{center}
\vspace*{-0.1in}
\caption{Estimate of density function for $\abs{\langle d_{41},x_0^2x_1\rangle}$ in Example~\ref{exer:random-network}.}
\label{fig:random-network-density-function}
\end{figure}

\begin{example} \label{exer:random-network}
{\rm
Reconsider Example~\ref{ex:four-node-counterexample}, where $W_{41} = 0$
and now $W_{21}$, $W_{31}$, $W_{42}$ and $W_{43}$ are i.i.d.
random variables with a uniform distribution on $[0,1]$.
An estimate of the density function for the random variable $\abs{\langle d_{41},x_0^2x_1\rangle}$ is shown in
Figure~\ref{fig:random-network-density-function}. In every case of the 1000
random networks generated, $d_{41}$ had relative degree $r=3$ as expected.
}
\hfill\openbull \end{example} \vspace*{0.1in}

\section{Conclusions}

Two basic properties were established for an additive network of input-output systems where each
node of the network is modeled by a convergent Chen-Fliess series.
First it was shown that every input-output map between a pair of nodes has
a locally convergence Chen-Fliess series representation. An explicit and in some cases achievable growth bound on the
coefficients was computed using the notion of a maximal network. Second, sufficient
conditions were given under which the input-output map between a pair
of nodes has a well defined relative degree as defined by
its generating series. This analysis led to the conclusion
that this relative degree property is generic when the connection
strengths between nodes are randomized.

redpajama_set_name: RedPajamaArXiv


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text:
Jazz Newz
Deseret News - Jazz could be more shorthanded as Lauri Markkanen added to injury report
Thread starter Sarah Todd
Utah Jazz forward Lauri Markkanen (23) drives past Phoenix Suns forward Torrey Craig during the first half of an NBA basketball game, Saturday, Nov. 26, 2022, in Phoenix.
Rick Scuteri, Associated Press
The Utah Jazz have already been operating without their full compliment of players but they could be even more shorthanded on Monday when the Chicago Bulls visit Vivint Arena.
Lauri Markkanen, the Jazz's leading scorer, is listed as questionable for Monday night's game with a right knee contusion.
The injury occurred at the 7:16 mark of the second quarter on Saturday against the Phoenix Suns, when Markkanen drove toward the basket for a layup and collided with Suns center Deandre Ayton.
After the play, Markkanen was limping and favoring the right leg but never left the game. The Jazz eventually fell to the Suns, 113-112.
Here's the play where Markkanen and Ayton hit knees. Markkanen was limping for a bit after this but stayed in the game. He's listed as questionable (right knee contusion) for Monday's Jazz game vs. Bulls pic.twitter.com/E4t14NYUU3
— Sarah Todd (@NBASarah) November 28, 2022
The Jazz are currently on a four-game losing streak, a skid that began after Mike Conley was sidelined with a left knee injury. The Jazz beat the Portland Trail Blazers the night that Conley suffered the injury, but since then the team has lost to the Los Angeles Clippers, Detroit Pistons, Golden State Warriors and Suns.
Conley has already been ruled out for Monday's game. Additionally, Rudy Gay has been out since Nov. 15 and is set to be reevaluated later this week for a finger sprain in his left hand, but he'll also be out on Monday night.
Further down the roster Leandro Bolmaro, who spent some time with the SLC Stars, is questionable as he continues in the NBA concussion protocol and two-way player Johnny Juzang suffered a wrist sprain while playing with the Stars and is out.
The Jazz have been navigating playing without Conley by experimenting with Collin Sexton, Talen Horton-Tucker and Nickeil Alexander-Walker in different situations. Though all three of the young players have shown flashes of growth, Conley's experience and expertise have been felt, especially in clutch situations.
Pistons head coach Dwane Casey pointed to Conley as one of the reasons that the Jazz have been so successful this season.
"I didn't believe the hype of a (complete) rebuild," Casey said last week. "Because you have Michael Conley, who is an All-Star, who still can play, who is a quarterback and one of the best point guards in our league."
Though Conley's scoring production is at one of the lower points in his career, he is averaging a career-high 7.9 assists per game this season with a career-tying low for turnovers per game at 1.5.
The Jazz, who have played 14 of their first 22 games on the road will start their first major home stand of the season on Monday and although being at home for eight of the next nine games will be a huge relief for the Jazz, the schedule will not be easy, especially if Markkanen misses any time.
The Finnish 7-footer is having a break out season, averaging career-highs in points, assists, and overall field-goal percentage and has earned considerable All-Star buzz through the early days of the 2022-23 season.
Full Story from the Deseret News...

redpajama_set_name: RedPajamaCommonCrawl


An evening in Bad Kissingen's Bayerische Spielbank casino has always been considered an atmospheric ending to a day at the spa. Even today, its sophisticated furnishings, with chandeliers, velvet curtains, and floor-to-ceiling windows are reminiscent of the glorious days of old. It is located in the north wing of the Luitpoldbad, which was reopened in 1968 and contains a café and restaurant.


Today marked the official opening of the academic year at St Andrew's Greek Orthodox Theological College in Sydney. A Doxology Service was celebrated in the College Chapel of St John the Evangelist and Theologian by the Dean, His Eminence Archbishop STYLIANOS of Australia, among faculty members, staff, alumni, students and guests.
"We are blessed to be able to gather and offer thanks to God for the beginning of the thirty-third year of our College's mission here in Australia. We have almost sixty students with us this year, from various parts of Australia and a few from overseas, most of whom are studying part time and online. Some of them are here with us today for our residential program and I thank them for making the effort to be present. We have a small group of full time students too, who live within the campus residence, and this is a very encouraging sign for the daily life of our College.
Theology today, unfortunately, as we all know, seems to be somehow neglected or overlooked. It is not estimated as it should be. And yet, the need for theology was never so great, so deep, and so profound. Because the more critical the times become, the more theologically interesting they become. Danger is always a factor which can bring those who would like to serve the goals of the Church to come closer and to learn to speak a more convincing language about our Christian faith, because only out of need does the human being become sincere and properly pious.
I do not need to stress the importance of the mission which our small College has in this country. I repeat this every time when we commence a new academic year. And yet, we know that we are not capable enough to respond to so many needs which are to be found in our parishes and in society, but we persevere nonetheless, and we keep in mind the words of the Apostle Paul who instructed us to stand fast in one spirit, and to strive together with one mind for the faith of the gospel, and not in any way to be terrified by our opponents.
His Eminence then presented a copy of the critical edition of the Greek New Testament to new seminarian students: Dionysios Bozikis (Syd), Terry Gerovasilis (Syd), Nicholas Haridemos (Canberra), Timothy Klintsaris (Syd), Stavros Kosmatos (Bris), Nickolas Krikelis (Melb), Melhem Makdessi (Syd), Adam Neou (Syd), Aristotle Papailiadis (Melb), and John Varipatis (Syd).
The customary group photograph in the courtyard of the College was followed by a lunch generously prepared by ladies from the Central Philoptohos Committee of the Archdiocese in Sydney.


text:
Some NHL Play-In teams now have incentive to lose for Alexis Lafreniere
Ever since the "mystery team" won the NHL Draft Lottery this past Friday night, a bunch of teams have fans debating if losing the qualifying round would be better. That's because all eights teams that get knocked out will have a 12.5% chance of getting generational talent Alexis Lafreniere with the No. 1 pick.
It is an odd conversation to have, but we are talking about a decent shot at landing a franchise player that could have the offensive potential of a Sidney Crosby. Where do you stand? Let's discuss.
Who is NHL Prospect Alexis Lafreniere?
Alexis Lafreniere is a LW coming out of the QMJHL, oddly enough for the same team Sidney Crosby played for, the Rimouski Oceanic. The kid has hands, vision, skating agility, speed, high hockeyIQ, and a great shot on top of all that. He is by far and away the number one choice in this draft.
"He's already one of the best players in the world, in my opinion, so he'll step right in," NHL Central Scouting senior manager David Gregory said. "Unless something really dramatic happens, he's going to go to a team that needs immediate help."
Well the dramatic did happen and if a team like the Toronto Maple Leafs did get knocked out could add Lafreniere who scored 112 points in 52 games to their stacked lineup. I mean could you imagine Lafreniere playing with John Tavares or Auston Matthews out of the gate? That is why 16 "play-in" teams are having this debate.
Run at the Cup or a chance at Lafreniere
These are no ordinary playoffs. All the teams in the 24 team Return to Play format will have not played a single game in over three months, so throw those records out the door. All teams are starting fresh and more importantly healthy. If there was ever a postseason where one could say it's anybody's tournament, this is the one.
In a recent online poll run by David Pagnotta of The Fourth Period Magazine, he asked the question to fans if you'd rather your team lose and win the draft lottery for Alexis Lafreniere. With over 5000 responses, 72% of fans voted overwhelmingly "YES".
Would you rather see your team lose in the qualifying round and win the draft lottery?
— David Pagnotta (@TheFourthPeriod) June 27, 2020
Maybe I am just too old school, but I'd rather try and win it all. Instead it seems like many would take their 12.5% lottery ticket for a player who would still need to prove it on the big stage. Even if that player is being talked about as the next Sidney Crosby, I'm going for the big silver Cup everytime.


The Crossing's project, The Gulf (between you and me), commissioned Santa Fe composer Chris Jonas, experimental, Philadelphia-based composer Gene Coleman, and internally recognized British composer Gabriel Jackson to write new choral works. These compositions were based on a cycle of poems by French-American writer Pierre Joris, which address the increasing gap between humans and the natural environment. Jonas' work was paired with a video by artist and animator Dan Cole, and Philadelphia composer Coleman collaborated with Japanese sound artist Toshimaru Nakamura on a work that incorporated natural sounds heard in the Gulf of Mexico. Ancillary activities included a reading by Joris and performances by Jonas' quartet Sun Spits Cherries and Coleman's ensemble N_JP. The project culminated in performances and a recording of the three works, and was the centerpiece of The Crossing's 2013 Month of Moderns annual concert series.


text:
<?php
/**
 * LinkedinOAuthService class file.
 *
 * Register application: https://www.linkedin.com/secure/developer
 * Note: Intagration URL should be filled with a valid callback url.
 *
 * @author Maxim Zemskov <nodge@yandex.ru>
 * @link http://github.com/Nodge/yii-eauth/
 * @license http://www.opensource.org/licenses/bsd-license.php
 */

require_once dirname(dirname(__FILE__)) . '/EOAuthService.php';

/**
 * LinkedIn provider class.
 *
 * @package application.extensions.eauth.services
 */
class LinkedinOAuthService extends EOAuthService {

	protected $name = 'linkedin';
	protected $title = 'LinkedIn';
	protected $type = 'OAuth';
	protected $jsArguments = array('popup' => array('width' => 900, 'height' => 550));

	protected $key = '';
	protected $secret = '';
	protected $scope = 'r_basicprofile'; // 'r_fullprofile r_emailaddress';
	protected $providerOptions = array(
		'request' => 'https://api.linkedin.com/uas/oauth/requestToken',
		'authorize' => 'https://www.linkedin.com/uas/oauth/authenticate', // https://www.linkedin.com/uas/oauth/authorize
		'access' => 'https://api.linkedin.com/uas/oauth/accessToken',
	);

	protected function fetchAttributes() {
		$info = $this->makeSignedRequest('http://api.linkedin.com/v1/people/~:(id,first-name,last-name,public-profile-url)', array(), false); // json format not working :(
		$info = $this->parseInfo($info);

        $this->attributes['info'] = $info;
		$this->attributes['id'] = $info['id'];
		$this->attributes['name'] = $info['first-name'] . ' ' . $info['last-name'];
		$this->attributes['url'] = $info['public-profile-url'];
	}

	/**
	 *
	 * @param string $xml
	 * @return array
	 */
	protected function parseInfo($xml) {
		/* @var $simplexml SimpleXMLElement */
		$simplexml = simplexml_load_string($xml);
		return $this->xmlToArray($simplexml);
	}

	/**
	 *
	 * @param SimpleXMLElement $element
	 * @return array
	 */
	protected function xmlToArray($element) {
		$array = (array)$element;
		foreach ($array as $key => $value) {
			if (is_object($value)) {
				$array[$key] = $this->xmlToArray($value);
			}
		}
		return $array;
	}
}

redpajama_set_name: RedPajamaGithub


Telltale Games has released the first screenshots of "In Harm's Way",'the next episode of The Walking Dead: Season Two.
The Walking Dead: Season Two has been excellent so far. If you're still thinking about picking up this Season, be sure to check out our reviews of Episode 1 and Episode 2.
Here's a gallery of the first official screenshots from the episode.
Tags: Telltale Games, The Walking Dead. Bookmark the permalink.


Enjoying its own street frontage this spacious villa is sure to impress. Offers comfortable gas heated lounge with air conditioning, sep kitchen meals area with as new cupboards & tiled floors. Also includes dishwasher & elec cooking. Renovated bathroom with sep shower and 2 dble bedrooms with BI robes complete the package. Other extras include sep laundry, private garden with storage & dble carport. Decorated in neutral tones this unit is also fitted with floating timber floors, vertical blinds & security shutters. Comfortable home or investment.