Relating Vertex and Global Graph Entropy in Randomly Generated Graphs

Abstract

Combinatoric measures of entropy capture the complexity of a graph but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of vertex level measures that do not suffer from this pathological computational complexity, but that can be shown to be effective at quantifying graph complexity. In this paper, we consider whether these local measures are fundamentally equivalent to global entropy measures. Specifically, we investigate the existence of a correlation between vertex level and global measures of entropy for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate strong correlation for this subset of graphs and outline how this may arise theoretically.

Quantum Mesh Dynamics (QMD)

“…In particular, it is possible, at least approximately, to reduce the definition of graph entropy to a sum over the constituent node's entropy, that becomes exact when the graph is highly regular ([6, 34, 35]. This closest form of vertex entropy to the behavior of structural entropy [34], is defined in terms of the degree k of each vertex, and the total number of edges in the graph |E|. As the graph is extremely large, this number of edges can be assumed to be a constant for spacetime, which for later analysis we can safely ignore, as we are only concerned with entropy changes as matter quanta coalesce.…”

Balancing Capacity and Epidemic Spread in the Global Airline Network

“…In addition to investigating the removal of hubs of large node degree, we also explore the use of vertex entropy, an informational measure of node importance [12,13,14,15], as a schema for selecting route reduction by closing airports. Vertex Entropy is closely correlated with centrality measures, and does not in general focus on the hubs in a network, but instead the 'pinch points' in connectivity.…”

Balancing capacity and epidemic spread in the global airline network

“…With this addition we can assess the dismantling approaches with respect to the reduction in total carrying capacity of the network, which we can then balance against a simple model of epidemic spread on that network. Together with a metric of graph entropy that measures the information content of the graph on a node by node basis, and known as Vertex Entropy (VE) (Tee et al 2017(Tee et al , 2018, we compare a number of dismantling schema on capacity and epidemic spread. In the cited studies of network dismantling the leading metric, as measured by the 'Robustness' metric is the Betweenness Centrality (BC) (Wandelt et al 2018), which identifies nodes that lie on the most shortest paths between node pairs in the network.…”