Cycles and zeta functions

The number of sequences of n cells that can occur repeatedly, corresponding to cycles in the network, is given in terms of the adjacency matrix m by Tr[MatrixPower[m,n]]. These numbers can also be obtained as the coefficients of xⁿ in the series expansion of x ∂_x Log[ζ[m, x]], with the so-called zeta function, which is always a rational function of x, given by

ζ[m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x]

and corresponds to the product over all cycles of 1/(1 - xⁿ).