Asymptotic structure and singularities in constrained directed graphs

TL;DR

This paper investigates the asymptotic behavior and phase transitions of large directed graphs with constrained edge and subgraph densities, revealing uniform or bipodal structures and their transitions.

Contribution

It introduces a new modeling approach for directed graphs with fixed subgraph densities, avoiding limitations of traditional ERGMs, and characterizes their asymptotic structures and phase transitions.

Findings

Large graphs exhibit uniform or bipodal structures.

Phase transitions occur when fixing one density and controlling the other.

Only bipodal structures are found when fixing both densities, with no phase transition.

Abstract

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large graphs have either uniform or bipodal structure. When edge density (resp. $p$-star density) is fixed and $p$-star density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and $p$-star density are fixed, we find only bipodal structures and no phase transition.