On the Strength of Connectivity of Inhomogeneous Random K-out Graphs

TL;DR

This paper analyzes the connectivity properties of inhomogeneous random K-out graphs, establishing precise thresholds for k-connectivity and showing that simple conditions suffice for near-complete connectivity in large networks.

Contribution

It provides a sharp zero-one law for k-connectivity in inhomogeneous K-out graphs, extending known results and clarifying the scaling needed for different connectivity levels.

Findings

Sharp zero-one law for k-connectivity established

K_n must scale as (1/(1-μ))(log n + (k-2) log log n + ω(1))

K_n ≥ 2 ensures a large connected component whp

Abstract

Random graphs are an important tool for modelling and analyzing the underlying properties of complex real-world networks. In this paper, we study a class of random graphs known as the inhomogeneous random K-out graphs which were recently introduced to analyze heterogeneous sensor networks secured by the pairwise scheme. In this model, first, each of the $n$ nodes is classified as type-1 (respectively, type-2) with probability $0<\mu<1$ (respectively, $1-\mu)$ independently from each other. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively, $K_n$ arcs towards $K_n$ distinct nodes) selected uniformly at random, and then the orientation of the arcs is ignored. From the literature on homogeneous K-out graphs wherein all nodes select $K_n$ neighbors (i.e., $\mu=0$), it is known that when $K_n \geq2$, the graph is $K_n$-connected asymptotically almost surely (a.a.s.) as $n$ gets large. In the inhomogeneous case (i.e., $\mu>0$), it was recently established that achieving even 1-connectivity a.a.s. requires $K_n=\omega(1)$. Here, we provide a comprehensive set of results to complement these existing results. First, we establish a sharp zero-one law for $k$-connectivity, showing that for the network to be $k$-connected a.a.s., we need to set $K_n = \frac{1}{1-\mu}(\log n +(k-2)\log\log n + \omega(1))$ for all $k=2, 3, \ldots$. Despite such large scaling of $K_n$ being required for $k$-connectivity, we show that the trivial condition of $K_n \geq 2$ for all $n$ is sufficient to ensure that inhomogeneous K-out graph has a connected component of size $n-O(1)$ whp.