State-Driven Dynamic Graphon Model

TL;DR

This paper introduces a novel state-driven dynamic graphon model that overcomes the limitations of traditional graphon dynamics by constructing time-evolving, permutation-invariant probability measures on the universal graph space, enabling analysis and preservation of graph limit properties.

Contribution

It proposes a new state-driven approach to model dynamic graphons, establishing a bijection with the graphon space and generalizing the model using Aldous-Hoover representation.

Findings

Model preserves graph limit definitions

Bijection between measures and graphons established

Analysis of dynamic graphon behavior demonstrated

Abstract

This paper shows the equivalence class definition of graphons hinders a direct development of dynamics on the graphon space, and hence proposes a state-driven approach to obtain dynamic graphons. The state-driven dynamic graphon model constructs a time-index sequence of the permutation-invariant probability measures on the universal graph space by assigning i.i.d. state random processes to $\mathbbm{N}$ and edge random variables to each of the unordered integer pairs. The model is justified from three perspectives: graph limit definition preservation, genericity, and analysis availability. It preserves the graph limit definition of graphon by applying a bijection between the permutation-invariant probability measures on the universal graph space and the graphon space to obtain the dynamic graphon, where the existence of the bijection is proved. Also, a generalized version of the model is proved to cover the graphon space by an application of the celebrated Aldous-Hoover representation, where generalization is achieved by adding randomness to the edge-generating functions. Finally, analysis of the behavior of the dynamic graphon is shown to be available by making assumptions on the state random processes and the edge random variables.