# PenPC: A Two-step Approach to Estimate the Skeletons of High Dimensional Directed Acyclic Graphs

arXiv: https://arxiv.org/abs/1405.1603 · HTML: https://mlsanity.com/paper/1405.1603/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2014.

## Built on by (most cited first, 3 of 3)

- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Learning linear non-Gaussian directed acyclic graph with diverging number of nodes](https://mlsanity.com/paper/2111.00740/) (2021)
- [Bayesian Network Structure Discovery Using Large Language Models](https://mlsanity.com/paper/2511.00574/) (2025)

## Abstract (arXiv)

Estimation of the skeleton of a directed acyclic graph (DAG) is of great importance for understanding the underlying DAG and causaleffects can be assessed from the skeleton when the DAG is notidentifiable. We propose a novel method named PenPC toestimate the skeleton of a high-dimensional DAG by a two-stepapproach. We first estimate the non-zero entries of a concentrationmatrix using penalized regression, and then fix the differencebetween the concentration matrix and the skeleton by evaluating aset of conditional independence hypotheses. For high dimensionalproblems where the number of vertices p is in polynomial orexponential scale of sample size n, we study the asymptoticproperty of PenPC on two types of graphs: traditionalrandom graphs where all the vertices have the same expected numberof neighbors, and scale-free graphs where a few vertices may have alarge number of neighbors. As illustrated by extensive simulationsand applications on gene expression data of cancer patients, PenPChas higher sensitivity and specificity than the standard-of-the-artmethod, the PC-stable algorithm.