Dynamics of random graphs with bounded degrees

Abstract

We investigate the dynamic formation of regular random graphs. In our model, we pick a pairof nodes at random and connect them with a link if both of their degrees are smaller thand. Starting with a set of isolated nodes, we repeat this linkingstep until a regular random graph, where all nodes have degreed, forms. We view this process as a multivariate aggregation process, and formally solvethe evolution equations using the Hamilton–Jacobi formalism. We calculate thenontrivial percolation thresholds for the emergence of the giant component whend ≥ 3. Also, we estimate the number of steps that have occurred before the giant componentspans the entire system and the total number of steps that have occurred before theregular random graph forms. These quantities are non-self-averaging, namely,they fluctuate from realization to realization even in the thermodynamic limit.