Natural orbital networks

Abstract

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) | exists t in T, b=t(a), b not equal to a}. An example is when R=Z_n and T consists of a finite set of quadratic maps T_i(x)=x^2+a_i. Graphs defined like that have a surprisingly rich structure. This holds especially in an algebraic set-up when T is generated by polynomials on Z_n. The characteristic path length L and the mean clustering coefficient C are interlinked by global-local quantity LC=-L/log(C) which often appears to have a limit for n to infinity like for two quadratic maps on a finite field Z_p. We see that for one quadratic map x^2+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^2+a,x^2+b,x^2+c on Z_p, we always appear to get a connected graph for all primes.