Preferential Attachment Graphs with Planted Communities

TL;DR

This paper introduces a modified preferential attachment model that includes planted communities, showing convergence properties and community-specific heavy-tailed degree distributions, with implications for understanding complex network structures.

Contribution

It presents a novel preferential attachment model with community labels and proves convergence of degree distributions and community-specific properties.

Findings

Fraction of half-edges converges for certain affinity matrices.

Degree distribution within communities is heavy-tailed with community-dependent decay.

Model extends classical preferential attachment to include community structure.

Abstract

A variation of the preferential attachment random graph model of Barab\'asi and Albert is defined that incorporates planted communities. The graph is built progressively, with new vertices attaching to the existing ones one-by-one. At every step, the incoming vertex is randomly assigned a label, which represents a community it belongs to. This vertex then chooses certain vertices as its neighbors, with the choice of each vertex being proportional to the degree of the vertex multiplied by an affinity depending on the labels of the new vertex and a potential neighbor. It is shown that the fraction of half-edges attached to vertices with a given label converges almost surely for some classes of affinity matrices. In addition, the empirical degree distribution for the set of vertices with a given label converges to a heavy tailed distribution, such that the tail decay parameter can be different for different communities. Our proof method may be of independent interest, both for the classical Barab\'asi -Albert model and for other possible extensions.