I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes.

Each weight between neurons is 1/N, meaning that if node n has fired the probability that any connected node k will fire in the next time step is 1/N (there is no temporal integration of inputs to any neuron).

My question is as follows. If node n fires at t=0 what is the probability that a specific node m will fire at t=d ( d time steps later)?

I know that if the weighted adjacency matrix W was a stochastic matrix then the probability would be ${W}_{mn}^{d}$. However the matrix is not stochastic (since the rows do not sum to 1). Furthermore this calculation fails after direct experimentation.

This problem is related to the question: What is the probability of any path of length n between the two nodes in a random graph where the existence of any edge has probability 0.7/N?

It is most useful if I could do this in terms of W. I think the answer may be related to Schur decomposition, as this is alluded to in "Networks: An Introduction" by M.E.J. Newman.

Thank you for your help.


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