The Mathematics

Definition

A finite set of maps T1, ... Td from a finite set V onto itself defines a finite simple graph G=(V,E), where E={ (x,y) | x ≠ y, exists k such that Tk x=y }.

Why look at them?

• These dynamical graphs show interesting structures and bring arithmetic structures to live.
• Their statistical properties of path length, global cluster and vertex degree are interesting.
• Some examples lead to deterministic small world examples with small diameter and large cluster.
• The graphs display rich-club phenomena, where high degree nodes are more interconnected.
• They feature garden of eden states, unreachable configurations like transient trees.
• They can feature attractors like cycle sub graphs but also more complex structures.
• By definition, these graphs are factors of Cayley graphs on the permutation group of V.
• While arithmetic graphs are the focus they are universal: any finite simple graph is obtained.

Examples:

• V is a ring and Tk are polynomials. For example, if T1(x) = x2 + a and T2(x) = x2 + b.
• If V is a group and Ti x = ai x, then G is the Cayley graph of the finite presentation of V.
• if V is the group Zn T1x=x2 and T2 x = x3 then the graph G is connected if and only if n is a Pierpont prime, a prime of the form 2t 3u+1.

Deterministic Small world

Deterministic rewiring:
 n=20, k=2 n=20, k=4 n=200, k=4