Natural orbital networks

TL;DR

This paper explores the structure of graphs generated by finite sets of polynomial maps on finite rings, revealing phase transitions in connectivity and clustering properties as the number of maps increases.

Contribution

It introduces a novel algebraic graph model based on polynomial maps on finite rings and analyzes their structural properties and connectivity behavior.

Findings

Graphs with one quadratic map tend to be disconnected as size grows.

Adding a second quadratic map increases the likelihood of connectivity.

For three quadratic maps, the graphs are almost always connected across all primes.

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.