Entropy of Some Models of Sparse Random Graphs With Vertex-Names

TL;DR

This paper calculates the entropy growth rate for various sparse random graph models with uniquely named vertices, highlighting its relevance for data compression and complexity analysis.

Contribution

It provides explicit entropy calculations for several models of sparse graphs with vertex names, establishing a foundation for future theoretical work in graph data compression.

Findings

Entropy grows as cN log N for many models

Explicit calculation of the rate constant c for different models

Highlights the setting as useful for data compression research

Abstract

Consider the setting of sparse graphs on N vertices, where the vertices have distinct "names", which are strings of length O(log N) from a fixed finite alphabet. For many natural probability models, the entropy grows as cN log N for some model-dependent rate constant c. The mathematical content of this paper is the (often easy) calculation of c for a variety of models, in particular for various standard random graph models adapted to this setting. Our broader purpose is to publicize this particular setting as a natural setting for future theoretical study of data compression for graphs, and (more speculatively) for discussion of unorganized versus organized complexity.