Testing for directed information graphs

TL;DR

This paper develops a hypothesis testing method to identify directed graph structures among network nodes modeled as Markov processes, using directed information to infer causal links with high detection accuracy.

Contribution

It introduces a statistically rigorous test for directed information graphs, establishing asymptotic optimality and analyzing convergence based on true graph links.

Findings

Detection probability can be made arbitrarily close to one.

False alarm probability remains negligible.

The test is asymptotically optimal with sufficient samples.

Abstract

In this paper, we study a hypothesis test to determine the underlying directed graph structure of nodes in a network, where the nodes represent random processes and the direction of the links indicate a causal relationship between said processes. Specifically, a k-th order Markov structure is considered for them, and the chosen metric to determine a connection between nodes is the directed information. The hypothesis test is based on the empirically calculated transition probabilities which are used to estimate the directed information. For a single edge, it is proven that the detection probability can be chosen arbitrarily close to one, while the false alarm probability remains negligible. When the test is performed on the whole graph, we derive bounds for the false alarm and detection probabilities, which show that the test is asymptotically optimal by properly setting the threshold test and using a large number of samples. Furthermore, we study how the convergence of the measures relies on the existence of links in the true graph.