Connectivity of inhomogeneous random K-out graphs

TL;DR

This paper studies the connectivity of inhomogeneous random K-out graphs where nodes belong to different classes with varying connection strategies, revealing conditions under which the graph is connected with high probability.

Contribution

It extends the understanding of connectivity in inhomogeneous K-out graphs, especially when the smallest class selects only one node, identifying the impact of the maximum class degree.

Findings

Graph is connected whp if the maximum class degree tends to infinity.

Bounded maximum class degree results in a positive probability of disconnection.

Simulation confirms theoretical results for finite networks.

Abstract

We propose inhomogeneous random K-out graphs $\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$, where each of the $n$ nodes is assigned to one of $r$ classes independently with a probability distribution $\pmb{\mu} = \{\mu_1, \ldots, \mu_r\}$. In particular, each node is classified as class-$i$ with probability $\mu_i>0$, independently. Each class-$i$ node selects $K_{i,n}$ distinct nodes uniformly at random from among all other nodes. A pair of nodes are adjacent in $\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$ if at least one selects the other. Without loss of generality, we assume that $K_{1,n} \leq K_{2,n} \leq \ldots \leq K_{r,n}$. Earlier results on homogeneous random K-out graphs $\mathbb{H}(n; K_n)$, where all nodes select the same number $K$ of other nodes, reveal that $\mathbb{H}(n; K_n)$ is connected with high probability (whp) if $K_n \geq 2$ which implies that $\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$ is connected whp if $K_{1,n} \geq 2$. In this paper, we investigate the connectivity of inhomogeneous random K-out graphs $\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$ for the special case when $K_{1,n}=1$, i.e., when each class-$1$ node selects only one other node. We show that $\mathbb{H}\left(n;\pmb{\mu},\pmb{K}_n\right)$ is connected whp if $K_{r,n}$ is chosen such that $\lim_{n \to \infty} K_{r,n} = \infty$. However, any bounded choice of the sequence $K_{r,n}$ gives a positive probability of $\mathbb{H}\left(n;\pmb{\mu},\pmb{K}_n\right)$ being not connected. Simulation results are provided to validate our results in the finite node regime.