Giant Component in Random Multipartite Graphs with Given Degree Sequences

TL;DR

This paper characterizes the emergence of a giant component in random multipartite graphs with specified degree sequences, using spectral and branching process techniques to determine when such a component exists and its size.

Contribution

It provides a new criterion based on Perron-Frobenius norm for the existence of a giant component in multipartite graphs with given degree sequences.

Findings

Giant component exists if Perron-Frobenius norm exceeds one

Explicit size of the giant component when it exists

Extension of Molloy-Reed exploration process to multipartite graphs

Abstract

We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with $p$ parts generated according to a given degree sequence $n_i^{\mathbf{d}}(n)$ which denotes the number of vertices in part $i$ of the multipartite graph with degree given by the vector $\mathbf{d}$. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed combined with techniques from the theory of multidimensional Galton-Watson processes to establish this result.