A tutorial on MDL hypothesis testing for graph analysis

When analysing graph structure, it can be difficult to determine whether patterns found are due to chance, or due to structural aspects of the process that generated the data. Hypothesis tests are often used to support such analyses. These allow us to make statistical inferences about which null models are responsible for the data, and they can be used as a heuristic in searching for meaningful patterns. The minimum description length (MDL) principle [6, 4] allows us to build such hypothesis tests, based on efficient descriptions of the data. Broadly: we translate the regularity we are interested in into a code for the data, and if this code describes the data more efficiently than a code corresponding to the null model, by a sufficient margin, we may reject the null model. This is a frequentist approach to MDL, based on hypothesis testing. Bayesian approaches to MDL for model selection rather than model rejection are more common, but for the purposes of pattern analysis, a hypothesis testing approach provides a more natural fit with existing literature. 1 We provide a brief illustration of this principle based on the running example of analysing the size of the largest clique in a graph. We illustrate how a code can be constructed to efficiently represent graphs with large cliques, and how the description length of the data under this code can be compared to the description length under a code corresponding to a null model to show that the null model is highly unlikely to have generated the data.