Existence and Size of the Giant Component in Inhomogeneous Random K-out Graphs

TL;DR

This paper analyzes the existence and size of giant components in inhomogeneous random K-out graphs, showing conditions under which large connected sub-networks persist despite node failures or removals, with implications for distributed network design.

Contribution

It provides new theoretical results on the connectivity and robustness of inhomogeneous random K-out graphs with bounded degree, extending understanding of their giant component properties.

Findings

A giant component of size n - O(1) exists when K_n ≥ 2.

A giant component persists after removing O(1) nodes.

A giant component of size n(1 - o(1)) remains after removing o(n) nodes.

Abstract

Random K-out graphs are receiving attention as a model to construct sparse yet well-connected topologies in distributed systems including sensor networks, federated learning, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes differ in resources and requirements, inhomogeneous random K-out graphs, denoted by $H(n;\mu,K_n)$, were proposed recently. Motivated by practical settings where establishing links is costly and only a bounded choice of $K_n$ is feasible ($K_n = O(1)$), we study the size of the largest connected sub-network of $H(n;\mu,K_n)$, We first show that the trivial condition of $K_n \geq 2$ for all $n$ is sufficient to ensure that $H(n;\mu,K_n)$, contains a giant component of size $n-O(1)$ whp. Next, to model settings where nodes can fail or get compromised, we investigate the size of the largest connected sub-network in $H(n;\mu,K_n)$, when $d_n$ nodes are selected uniformly at random and removed from the network. We show that if $d_n=O(1)$, a giant component of size $n- \OO(1)$ persists for all $K_n \geq 2$ whp. Further, when $d_n=o(n)$ nodes are removed from $H(n;\mu,K_n)$, the remaining nodes contain a giant component of size $n(1-o(1))$ whp for all $K_n \geq 2$. We present numerical results to demonstrate the size of the largest connected component when the number of nodes is finite.