# Weak equivalence of local independence graphs

## TL;DR

This paper investigates the complexity of determining Markov equivalence in local independence graphs of multivariate stochastic processes, introduces weaker equivalence relations, and develops feasible algorithms for their analysis.

## Contribution

It proves coNP-completeness of Markov equivalence decision and introduces a hierarchy of weak equivalence relations with practical algorithms and concise representations.

## Key findings

- Deciding Markov equivalence of DMGs is coNP-complete.
- Introduces weaker equivalence relations with feasible algorithms.
- Provides hierarchical structure linking different equivalence levels.

## Abstract

Classical graphical modeling of multivariate random vectors uses graphs to encode conditional independence. In graphical modeling of multivariate stochastic processes, graphs may encode so-called local independence analogously. If some coordinate processes of the multivariate stochastic process are unobserved, the local independence graph of the observed coordinate processes is a directed mixed graph (DMG). Two DMGs may encode the same local independences in which case we say that they are Markov equivalent. Markov equivalence is a central notion in graphical modeling. We show that deciding Markov equivalence of DMGs is coNP-complete, even under a sparsity assumption. As a remedy, we introduce a collection of equivalence relations on DMGs that are all less granular than Markov equivalence and we say that they are weak equivalence relations. This leads to feasible algorithms for naturally occurring computational problems related to weak equivalence of DMGs. The equivalence classes of a weak equivalence relation have attractive properties. In particular, each equivalence class has a greatest element which leads to a concise representation of an equivalence class. Moreover, these equivalence relations define a hierarchy of granularity in the graphical modeling which leads to simple and interpretable connections between equivalence relations corresponding to different levels of granularity.