Isotemporal classes of diasters, beachballs, and daisies

TL;DR

This paper introduces a simple formula for counting isotemporal classes of diasters, a specific graph structure, based on their automorphisms and temporal path preservation, advancing understanding of temporal network equivalences.

Contribution

The paper provides a novel, straightforward formula for counting isotemporal classes of diasters, extending it to other pseudographs using a new theorem on automorphisms and edge label transpositions.

Findings

Formula for N(D(a,b)) when a ≠ b: ab + a + b + 1

Formula for N(D(a,a)): (a^2 + 3a + 2)/2

Extension of the formula to five additional pseudograph types

Abstract

If the vertices composing a network interact at distinct time points, the temporal ordering of these interactions and the network's graph structure are sufficient to convey the routes by which information can flow in the network. Two networks with real-valued edge labels are temporally isomorphic if there exists a graph isomorphism f:N->M that preserves temporal paths - paths in which sequential edge labels are strictly increasing. An equivalence class of temporally ismorphic networks is known as an isotemporal class. Methods to determine the number of isotemporal classes of a particular graph structure N(G) are non-obvious, and refractory to traditional techniques such as P\'olya enumeration (P\'olya, 1937). Here, I present a simple formula for the number of isotemporal classes of diasters, graphs composed of a vertex of degree a+1 connected to a vertex of degree b+1, with all other vertices of degree 1 (denoted D(a,b)). In particular, N(D(a,b)))=ab+a+b+1 if a is not equal to b, and N(D(a,a)))=(1/2) (a^2+3a+2) otherwise. This formula is then extended to five additional types of pseudograph by application of a theorem that states N(G) is preserved between two graph types if edge adjacencies and automorphisms are preserved, and provided that any two networks are members of the same isotemporal class if and only if they are isomorphic by transpositions of sequential edge labels on non-adjacent edges.