\begin{document}
\subsection{Description of functional transitions}
\label{subsection:parameterization}

For Markov chains describing interacting processes, i.e., automata, we are interested in  functional transitions, for which the entries of $\Q_{\nu}^{(\mu)}$ are functions.
Every time such an entry is needed one would have to evaluate the function, which requires knowledge of  the states of all automata.

To analyze this problem more precisely, we concentrate on Markov chains satisfying the following assumptions:
\begin{enumerate}
	\item \label{ass:(i)}
	The continuous-time Markov chain can be represented as a Stochastic Automata Network $\{\A_1, \dots, \A_d\}$ with $d$ automata.
	Each automaton $\A_{\mu}$ has a state space $\Space_{\mu}$ with $\vert \Space_{\mu} \vert = n_{\mu}$, and the state space of the network is given by $S = \bigtimes_{\mu = 1}^d \Space_{\mu}$.
	\item \label{ass:(ii)}
	There are only local and functional transitions.
	\item \label{ass:(iii)}
	For each automaton $\A_{\mu}$ there exists an ordering of the states $i_1 < i_2 < \cdots < i_{n_{\mu}}$ such that there is no transition from state $i$ to state $j$ for all $i > j$. We denote the set of possible transitions in $\Space_{\mu}$ by $T_{\mu} = \{ \left(i, j\right) ~ \vert ~ i < j \in \Space_{\mu} \}$.
	\item \label{ass:(iv)}
	The transition rate from state $x$ to $y$ in $\Space$, where $x$ and $y$ differ only in automaton $\A_{\nu}$ and satisfy $x[\nu] < y[\nu]$, is separable and can be represented with parameters $ \PM_{ \left(x[\nu], y[\nu] \right), x[\mu]} \geq 0$ by
	\begin{align*}
		Q[y, x] = 	\prod\limits_{\mu = 1}^d \PM_{ \left(x[\nu], y[\nu] \right), x[\mu]} .
	\end{align*}
	\item \label{ass:(v)}
	The time $t$ is an exponentially distributed random variable with mean $1$, i.e., $t \sim \operatorname{Exp}[1]$. 
\end{enumerate}
In the following we discuss these assumptions in more detail.
Due to assumption~\ref{ass:(ii)} and~\ref{ass:(iii)}, all possible transitions are covered in assumption~\ref{ass:(iv)}. 
All other transition rates are zero.
Each diagonal entry of the generator is given by minus its off-diagonal column sum, since each column sum is zero.
Hence, the whole infinitesimal generator is already defined by the parameters $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\mu}} \geq 0$ for all states $i_{\mu} \in \Space_{\mu}$ and $\left(i_{\nu},  j_{\nu}\right) \in T_{\nu}$, and we denote the corresponding generator as $\QPM$.
The parameter $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\mu}}$ with $\nu \neq \mu$ can be interpreted as a direct multiplicative effect of the state $i_{\mu} \in \Space_{\mu}$ on the local transition from $i_{\nu}$ to $j_{\nu}$ in $\Space_{\nu}$, while $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\nu}}$ is a  baseline rate of transition from $i_{\nu}$ to $j_{\nu}$ in $\Space_{\nu}$.
If a state $i_{\mu} \in \Space_{\mu}$ has no direct effect on the transition from $i_{\nu}$ to $j_{\nu}$ in $\Space_{\nu}$, then $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\mu}} = 1$.
As usually assumed, an automaton does not affect many others directly, and therefore many multiplicative effects, i.e., parameters $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\mu}}$, are equal to one.
This corresponds to a sparsity of the interactions between the automata.
Assumption~\ref{ass:(iv)} allows us to separate the functional transition rates and to transfer the parameters $\PM_{\left(i_{\nu}, j_{\nu}\right), i_{\mu}}$ to the matrices corresponding to automaton $\A_{\mu}$ within each Kronecker product in~\cref{eq:local_transition_part}.
Thus, we represent all functional transitions within the Kronecker product structure without any functional entries:
\begin{align}
\label{eq:Q_CP}
	\QPM &= \sum\limits_{\nu = 1}^d ~ \sum\limits_{\left(i,j\right) \in T_{\nu}} ~ \bigotimes\limits_{\mu = 1} ^d \QCore
	\\
	\text{ with } 	 \qquad
	\QCore &= 
		\begin{cases}
			\diag\left(\PM_{\left(i,j\right), \eta} ~\vert~ \eta \in \Space_{\mu} \right) & \text{if } \mu \neq \nu,
			\\[0.2cm]
			\PM_{\left(i,j\right), i} \left(\Delta_{j,i}^{\nu} - \Delta_{i,i}^{\nu}\right) & \text{if } \mu = \nu
		\end{cases}
	\nonumber
\end{align}
where $\Delta_{j,i}^{\nu} \in \R^{\Space_{\nu} \times \Space_{\nu}}$ has only one non-zero entry $\Delta_{j,i}^{\nu}[j,i] = 1$.
By assumption~\ref{ass:(iii)}, the generator $\QPM$ is similar to a lower triangular matrix where the similarity transformation is obtained by permuting states following the order of each automaton. 
Hence, the spectrum consists of all diagonal entries
\begin{align*}
	\sigma\left(\QPM - \Id\right) = \Bigg\{-1 - \sum\limits_{\nu = 1}^d ~ \sum\limits_{\left(x[\nu], j\right) \in T_{\nu} } ~ \prod\limits_{\mu = 1}^d 	\PM_{\left(x[\nu], j\right), x[\mu]} ~  \Bigg\vert ~x \in S\Bigg\} \subseteq \R_{\leq - 1}
\end{align*}
and the improper integral in~\cref{eq:marginal_distribution} exists.
	
\end{document}
{"config": "arxiv", "file": "2006.08135/tex/parameterization.tex"}