Connectivity of a Family of Bilateral Agreement Random Graphs

TL;DR

This paper proves La and Kabkab's conjecture that connectivity in bilateral agreement random graphs occurs when the parameter exceeds a threshold of 1, and introduces an asymptotic for the average degree of these graphs.

Contribution

The paper provides a proof for the conjectured connectivity threshold in bilateral agreement random graphs and derives an asymptotic expression for their average degree.

Findings

Connectivity occurs at t>1 with high probability.

Graph is disconnected for t<1.

Derived asymptotic for average degree.

Abstract

Bilateral agreement based random undirected graphs were introduced and analyzed by La and Kabkab in 2015. The construction of the graph with $n$ vertices in this model uses a (random) preference order on other $n-1$ vertices and each vertex only prefers the top $k$ other vertices using its own preference order; in general, $k$ can be a function of $n$. An edge is constructed in the ensuing graph if and only if both vertices of a potential edge prefer each other. This random graph is a generalization of the random $k^{th}$-nearest neighbor graphs of Cooper and Frieze that only consider unilateral preferences of the vertices. Moharrami \emph{et al.} studied the emergence of a giant component and its size in this new random graph family in the limit of $n$ going to infinity when $k$ is finite. Connectivity properties of this random graph family have not yet been formally analyzed. In their original paper, La and Kabkab conjectured that for $k(t)=t \log n$, with high probability connectivity happens at $t>1$ and the graph is disconnected for $t<1$. We provide a proof for this conjecture. We will also introduce an asymptotic for the average degree of this graph.