Network Alignment by Discrete Ollivier-Ricci Flow

TL;DR

This paper introduces a novel graph alignment method using discrete Ricci flow metrics, which enhances robustness and accuracy in matching complex networks like social, internet, and biological graphs.

Contribution

The work proposes a new network alignment approach based on Ricci flow metrics, improving robustness to network modifications and outperforming existing methods on real-world data.

Findings

Outperforms existing methods on complex network datasets

Robust to node and edge insertions/deletions

Effective in aligning biological and social networks

Abstract

In this paper, we consider the problem of approximately aligning/matching two graphs. Given two graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, the objective is to map nodes $u, v \in G_1$ to nodes $u',v'\in G_2$ such that when $u, v$ have an edge in $G_1$, very likely their corresponding nodes $u', v'$ in $G_2$ are connected as well. This problem with subgraph isomorphism as a special case has extra challenges when we consider matching complex networks exhibiting the small world phenomena. In this work, we propose to use `Ricci flow metric', to define the distance between two nodes in a network. This is then used to define similarity of a pair of nodes in two networks respectively, which is the crucial step of network alignment. %computed by discrete graph curvatures and graph Ricci flows. Specifically, the Ricci curvature of an edge describes intuitively how well the local neighborhood is connected. The graph Ricci flow uniformizes discrete Ricci curvature and induces a Ricci flow metric that is insensitive to node/edge insertions and deletions. With the new metric, we can map a node in $G_1$ to a node in $G_2$ whose distance vector to only a few preselected landmarks is the most similar. The robustness of the graph metric makes it outperform other methods when tested on various complex graph models and real world network data sets (Emails, Internet, and protein interaction networks)\footnote{The source code of computing Ricci curvature and Ricci flow metric are available: https://github.com/saibalmars/GraphRicciCurvature}.