# The Mathematics

#### Definition

A finite set of maps T_{1}, ... T

_{d}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 T

_{k}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 T
_{k}are polynomials. For example, if T_{1}(x) = x^{2}+ a and T_{2}(x) = x^{2}+ b. - If V is a group and T
_{i}x = a_{i}x, then G is the Cayley graph of the finite presentation of V. - if V is the group Z
_{n}T_{1}x=x^{2}and T_{2}x = x^{3}then the graph G is connected if and only if n is a Pierpont prime, a prime of the form 2^{t}3^{u}+1.

#### Deterministic Small world

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