Predicting Network Edge Count and Fragmentation under Vertex Percolation Processes

Network and graph-based percolation theory are applicable to a broad range of disciplines throughout the natural, life, and social sciences. We present a graph-based vertex (site) percolation study empirically quantifying – as a function of vertex occupation fraction – the number of edges (bonds) formed, the extent of fragmentation of the network, and the scaling of percolation thresholds with the mean vertex degree of the graph. The edge count is shown to be quadratically dependent upon the vertex occupation fraction with no unknown fitting parameters, thus applying universally - that is, to all networks - with minimal error. It may be used to predict, for example, the number of nearest neighbor bonds remaining in a lattice (e.g. translational degrees of freedom in a spatial lattice) or the number of friendships remaining as a social network is deconstructed (e.g. due to account closure on social media or mortality). For well-behaved networks with a reasonably low variance in the vertex degree distribution, the latter relations may be used to predict fragmentation: both the percolation threshold and, subsequently, the number of distinct connected components in the network at a given occupation fraction.