Dynamics of Random Graphs with Bounded Degrees

TL;DR

This paper models the formation of regular random graphs through a dynamic process, analyzing the emergence of giant components and fluctuations in the formation time.

Contribution

It provides a formal solution to the evolution equations of the graph formation process and calculates percolation thresholds for regular random graphs.

Findings

Percolation thresholds for d>=3 are derived.

The time until the giant component spans the system is estimated.

Formation times exhibit non self-averaging fluctuations.

Abstract

We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this linking step until a regular random graph, where all nodes have degree d, forms. We view this process as a multivariate aggregation process, and formally solve the evolution equations using the Hamilton-Jacoby formalism. We calculate the nontrivial percolation thresholds for the emergence of the giant component when d>=3. Also, we estimate the number of steps until the giant component spans the entire system and the total number of steps until the regular random graph forms. These quantities are non self-averaging, namely, they fluctuate from realization to realization even in the thermodynamic limit.