Natural Orbital Networks

Given a finite set T of maps on a finite ring R, we look at the finite simple graph G = (V,E) with vertex set V = R and edge set E = {(a, b) | ∃t ∈ T, b = t(a), and b 6= a}. An example is when R = Zn and T consists of a finite set of quadratic maps Ti(x) = x 2 + ai. Graphs defined like that have a surprisingly rich structure. This holds especially in an algebraic setup when T is generated by polynomials on Zn. The characteristic path length μ and the mean clustering coefficient ν are interlinked by global-local quantity λ = −μ/ log(ν) which often appears to have a limit for n→∞ like for two quadratic maps on a finite field Zp. We see that for one quadratic map x+a, the probability to have connectedness goes to zero and for two quadratic maps, the probability goes to 1, for three different quadratic maps x + a, x + b, x + c on Zp, we always appear to get a connected graph for all primes. In Memory of Oscar Lanford III.