A Topological Sorting Criterion for Random Causal Directed Acyclic Graphs

TL;DR

This paper introduces a topological sorting criterion based on the monotonic increase of reachable nodes in random DAGs, aiding causal order recovery and impacting causal discovery evaluations.

Contribution

It demonstrates that the set of reachable nodes increases monotonically along the causal order in certain random DAGs and proposes using this for causal order recovery.

Findings

Relatives increase monotonically along the causal order in random DAGs.

This pattern can be exploited for causal order recovery.

Many simulations show this pattern as an effective proxy for causal order.

Abstract

Random directed acyclic graphs (DAGs) based on imposing an order on Erd\H{o}s-R\'enyi and scale free random graphs are widely used for evaluating causal discovery algorithms. We show that in such DAGs, the set of nodes reachable via open paths, termed relatives, increases monotonically along the causal order. We assess the prevalence of this pattern numerically, and demonstrate that it can be exploited for causal order recovery via sorting by the estimated number of relatives. We note that many simulations in the literature feature settings where this yields an excellent proxy for the causal order, and show that a strict increase of relatives along the causal order leads to a singular Markov equivalence class. We propose sampling time-series DAGs as a possible alternative and discuss implications for causal discovery algorithms and their evaluation on synthetic data.