Entropy of Soft Random Geometric Graphs

Random Geometric Graphs (RGGs) are random network structures that model real-world structures with a spatial dependence or latent space embedding. They are formed by placing points randomly and independently in a space, and connecting the points with a probability that is dependent on their mutual distance. The Shannon entropy of a graph ensemble measures how difficult it is to predict the structure of a random graph sampled from the ensemble, and therefore gives a measure of how well a network model captures the properties of real network structures. Broadly, the aim of this project is to quantify the entropy of RGG ensembles, and how it changes when we change the underlying geometry, and the rule for connecting points. Specifically, we attempt to investigate the following. We aim to determine the changes in entropy as we increase the dimension of the underlying space, and whether the presence of periodic boundary conditions influences this change. Also, in lower dimensional spaces, can we quantify the effect of individual boundary conditions or non-convexities in the domain on the entropy? What is the behaviour of the entropy when we change the range at which points become likely to be connected, especially when the range is very small, and very large. Finally, how can we apply the techniques we have developed throughout the project to answer other questions about RGGs, or quantify the entropy of other related models?