Entropy of Spatial Network Ensembles

TL;DR

This paper introduces a graph entropy framework for analyzing the complexity of spatial network ensembles, modeling them as soft random geometric graphs with probabilistic links based on node distances.

Contribution

It derives bounds and formulas for the entropy of spatial networks, including the maximum entropy connection function, and applies the framework to wireless and flight networks.

Findings

Both studied networks are nearly maximally entropic.

The framework generalizes classical random geometric graph models.

Analytical bounds for network entropy are provided.

Abstract

We analyze complexity in spatial network ensembles through the lens of graph entropy. Mathematically, we model a spatial network as a soft random geometric graph, i.e., a graph with two sources of randomness, namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function (the "pair connection function") of their mutual distance. We consider the general case where randomness arises in node positions as well as pairwise connections (i.e., for a given pair distance, the corresponding edge state is a random variable). Classical random geometric graph and exponential graph models can be recovered in certain limits. We derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft (probabilistic) pair connection functions. Under this formalism, we derive the connection function that yields maximum entropy under general constraints. Finally, we apply our analytical framework to study two practical examples: ad hoc wireless networks and the US flight network. Through the study of these examples, we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles.