The Topological Behavior of Preferential Attachment Graphs

TL;DR

This paper explores the algebraic-topological properties of scale-free networks modeled as preferential attachment graphs, revealing phase transitions and asymptotic behaviors of their clique complexes.

Contribution

It provides the first detailed analysis of the Betti numbers and homotopy connectivity of preferential attachment graph complexes, verifying conjectures and identifying phase transitions.

Findings

Betti numbers grow asymptotically almost surely with network size

Homotopical phase transitions occur at critical thresholds

Numerical evidence suggests Betti numbers follow power-law distributions

Abstract

We investigate the higher-order connectivity of scale-free networks using algebraic topology. We model scale-free networks as preferential attachment graphs, and we study the algebraic-topological properties of their clique complexes. We focus on the Betti numbers and the homotopy-connectedness of these complexes. We determine the asymptotic almost sure orders of magnitude of the Betti numbers. We also establish the occurence of homotopical phase transitions for the infinite complexes, and we determine the critical thresholds at which the homotopy-connectivity changes. This partially verifies Weinberger's conjecture on the homotopy type of the infinite complexes. We conjecture that the mean-normalized Betti numbers converge to power-law distributions, and we present numerical evidence. Our results also highlight the subtlety of the scaling limit of topology, which arises from the tension between topological operations and analytical limiting process. We discuss such tension at the end of the Introduction.