Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution

Abstract

In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius. Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity. Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling. These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.

Effect of different monomer precursors with identical functionality on the properties of the polymer network

“…Although formation of small cycles do not significantly affects the degree distribution, they delay the onset of gelation, that is the moment in time when an extensive connected component emerges. Note that the mean-field theories predict the gel point exclusively on the basis of the degree distribution, 18,20,21,35,36 which unlike self loops and small cycles, was shown to be insensitive to the monomer type. The onset of gelation is Figure 3: Two-dimensional representation of the network and cell-complex developing during free-radical polymerisation of DDDA.…”