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I am trying to mathematically model the following idea that describes the dynamical evolution of a quantity over a graph.

Let us imagine we have a directed graph, with $n$ nodes and $m$ edges. Each directed edge from node $i$ to node $j$ has weight $w_{ij}$ and a reciprocal edge from node $j$ to node $i$ exists with inverse weight: $w_{ji} = 1/w_{ij}$.

I am assigning each node a scalar variable $x_i$ and I would like to model the temporal evolution of the scalar variables $x_i$ over all nodes, as if nodes communicate the quantity $x_i$ multiplied by the edge weights.

At every step, the variable $x_i$ should increase by the sum of the products of its neighbors' $x_j$ multiplied by the incident weights $w_{ji}$ and decrease by the sum of the products of its outgoing weights multiplied by $x_i$.

In other words:

$$ x_i(t+1) = x_i(t) + \sum_{j\neq i}w_{ji}x_j - \sum_{j\neq i}w_{ij} x_{i} $$

or, given the condition that $w_{ij}=1/w_{ji}$ we could also write

$$ x_i(t+1) = x_i(t) + \sum_{j\neq i}w_{ji}x_j - \sum_{j\neq i}\frac{1}{w_{ji}} x_{i} $$

In the limit $\Delta t \to 0$ I expect to describe this system by an ODE whose asymptotical behaviour I would like to study, but I am struggling to find a good description of this system in terms of matrices and linear algebra.

I believe this can be modeled in terms of master equation in statistical mechanics, but the presence of the inverse term makes things a bit more complicated.

How is it possible to write down this system of equations in terms of an ODE? Has this system already been studied? Could this system be related to chemical reaction networks?

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  • $\begingroup$ The Fokker-Planck-Equation may be of interest to you. A good source is the chapter on classical nonequilibrium in this book. $\endgroup$
    – denklo
    May 21, 2019 at 9:26
  • $\begingroup$ I think that at steady state, the detailed balance will hold, but I am having problems to express this problem in terms of matrices. $\endgroup$
    – linello
    May 21, 2019 at 9:28
  • $\begingroup$ The main problem is that the weights are not the classical rates, but basically the ratio between $x_i$ and $x_j$. For example a weight $w_{ij}=2$ means that one unit of a quantity in node $i$ becomes 2 units of another quantity in node $j$. $\endgroup$
    – linello
    May 22, 2019 at 9:10
  • $\begingroup$ I believe the simplest solution here it to model it as $\frac{dx}{dt} = -L x$ where $L=D^{out}-W^T$, with $W$ the weighted adjacency matrix and $D^{out}$ the diagonal matrix of the outgoing node degrees. $\endgroup$
    – linello
    Jun 3, 2019 at 15:42

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