Large Communities in a scale-free network

TL;DR

This paper proves that in a preferential attachment model with an edge-step, the resulting large-scale network contains a very large complete subgraph, demonstrating the emergence of significant community structures.

Contribution

It establishes the existence of large complete subgraphs in the GLP preferential attachment model with an edge-step, quantifying their size as a power of the network.

Findings

Existence of a complete subgraph of size t^α w.h.p.

Size of the subgraph scales as a power of the network size.

Results hold for a range of parameters p and ε.

Abstract

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we prove that the random graph $G_{t}$ produced by the so-called GLP model at time $t$ contains a complete subgraph of order $t^\alpha$, where $\alpha = (1-\varepsilon)\frac{1-p}{2-p}$, $\varepsilon$ is any number such that $0<\varepsilon<1$, and $0<p<1$ is a parameter of the model.