Isolated vertices in two duplication-divergence models with edge deletion

TL;DR

This paper introduces two new duplication-divergence models with edge deletion to better reflect real gene and protein networks, analyzing the conditions under which isolated vertices are prevalent or rare.

Contribution

The paper presents novel models incorporating random edge deletions and provides bounds on isolated vertices, advancing understanding of network evolution.

Findings

Bounds on the proportion of isolated vertices in large networks

Identification of parameter regimes with high or low isolated vertices

Simulations showing convergence of isolated vertex proportion to non-trivial limits

Abstract

Duplication-divergence models are a popular model for the evolution of gene and protein interaction networks. However, existing duplication-divergence models often neglect realistic features such as loss of interactions. Thus, in this paper we present two novel models that incorporate random edge deletions into the duplication-divergence framework. As in protein-protein interaction networks, with proteins as vertices and interactions as edges, by design isolated vertices tend to be rare, our main focus is on the number of isolated vertices; our main result gives lower and upper bounds for the proportion of isolated vertices, when the network size is large. Using these bounds we identify the parameter regimes for which almost all vertices are typically isolated; and also show that there are parameter regimes in which the proportion of isolated vertices can be bounded away from 0 and 1 with high probability. In addition, we find regimes in which the proportion of isolated vertices tends to be small. The proof relies on a standard martingale argument, which in turn requires a careful analysis of the first two moments of the expected degree distribution. The theoretical findings are illustrated by simulations, indicating that as the network size tends to infinity, the proportion of isolated vertices can converge to a limit that is neither 0 or 1.