# Direction Matters: On Influence-Preserving Graph Summarization and Max-cut Principle for Directed Graphs

## TL;DR

This paper introduces a novel graph summarization method for directed graphs that preserves edge directionality and minimizes reconstruction error, enabling more effective analysis of large-scale directed networks.

## Contribution

The paper proposes a new model based on Max-Cut principles and non-negative constraints for directed graph summarization, with a multiplicative update algorithm and theoretical guarantees.

## Key findings

- The method accurately preserves directed edge information.
- It demonstrates robustness across various experiments.
- The approach effectively captures group-level features.

## Abstract

Summarizing large-scaled directed graphs into small-scale representations is a useful but less studied problem setting. Conventional clustering approaches, which based on "Min-Cut"-style criteria, compress both the vertices and edges of the graph into the communities, that lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features which is useful for efficient interventions of population behavior. In this paper, we present a model, based on minimizing reconstruction error with non-negative constraints, which relates to a "Max-Cut" criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and on the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.