The connected component of the partial duplication graph

TL;DR

This paper analyzes the degree distribution of the connected component in a partial duplication graph model, revealing a power law tail in the subcritical case when the duplication parameter is below a critical threshold.

Contribution

It provides a detailed description of the degree distribution in the subcritical phase, using Markov chain analysis, and confirms the power law tail predicted by previous theories.

Findings

Degree distribution converges to a limit described by a Markov chain stationary distribution.

When p<e^{-1}, the degree distribution exhibits an approximate power law tail.

The power law index aligns with predictions by Ispolatov, Krapivsky, and Yuryev.

Abstract

We consider the connected component of the partial duplication model for a random graph, a model which was introduced by Bhan, Galas and Dewey as a model for gene expression networks. The most rigorous results are due to Hermann and Pfaffelhuber, who show a phase transition between a subcritical case where in the limit almost all vertices are isolated and a supercritical case where the proportion of the vertices which are connected is bounded away from zero. We study the connected component in the subcritical case, and show that, when the duplication parameter $p<e^{-1}$, the degree distribution of the connected component has a limit, which we can describe in terms of the stationary distribution of a certain Markov chain and which follows an approximately power law tail, with the power law index predicted by Ispolatov, Krapivsky and Yuryev. Our methods involve analysing the quasi-stationary distribution of a certain continuous time Markov chain associated with the evolution of the graph.