Network Alignment by Discrete Ollivier-Ricci Flow (1809.00320v2)

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}.

• The paper presents a novel Ricci flow-based technique that leverages intrinsic curvature properties to robustly align graphs.
• It defines a distance metric via weighted edges and landmark-based node mapping, ensuring efficiency and resilience against network changes.
• Experiments on synthetic and real-world datasets demonstrate superior alignment accuracy compared to traditional methods.

Network Alignment by Discrete Ollivier-Ricci Flow

The paper discusses a novel methodology for approximately aligning or matching two graphs, a task that finds utility in numerous applications such as database schema matching, protein interaction alignment, and social network analysis. The primary challenge lies in ensuring that the connected nodes in one graph find a corresponding pair in another graph such that the connectivity is preserved to the extent possible. While traditional approaches for network alignment often rely on heuristic algorithms or direct computation of structural similarities, this work proposes a fundamentally different approach by leveraging the intrinsic properties of Ricci curvature and curvature flow.

Summary of Methodology

The core innovation lies in utilizing the Ricci flow metric to define the distance between nodes in a graph. The Ricci curvature of an edge abstractly measures how well local neighborhoods of the nodes are connected, essentially providing a mechanism to understand the local geometric structure of the graph. By employing a discrete version of the Ricci curvature and Ricci flow, the proposed method allows the calculation of a robust metric that remains insensitive to typical perturbations in the network structure, such as node or edge insertions and deletions.

The methodology involves:

1. Graph Weighting: The edges in a graph are weighted according to the Ricci curvature, which is determined by considering local neighborhood transport properties. This weighting effectively normalizes the diverse edge curvatures, contributing to a more stable measure of node distances.
2. Distance Metric: A Ricci flow metric is defined as the shortest path metric where edge weights are adjusted to make curvatures uniform. This metric offers notable robustness against structural changes, allowing for more reliable alignment of graphs with differing topologies.
3. Node Mapping via Landmarks: The algorithm utilizes preselected landmarks (a small subset of nodes with known correspondences across graphs) to inform the node similarity scoring. A landmark-based coordinate system for each node is established, enabling a computationally efficient matching process through established algorithms like the Hungarian min-cost matching algorithm.

Evaluation and Results

The paper reports on exhaustive experiments conducted across synthetic (random regular, Erdős–Rényi, preferential attachment, and Kleinberg's small-world models) and real-world databases (such as internet AS graphs and protein interaction networks). The results consistently showcase the superiority of the Ricci flow metric in maintaining alignment accuracy under graph perturbations compared with traditional methods like IsoRank, NSD, spectral embedding, and spring embedding.

1. Robustness: The proposed Ricci flow metric outperforms other methods significantly, maintaining matching accuracy even when subjected to random node/edge deletions.
2. Efficiency: In cases of real-world graph alignment, the method reconciles both high accuracy and computational efficiency, making it a versatile tool for dynamic network analysis.
3. Comparison with Other Metrics: The Ricci flow metric shows minimal sensitivity to edge modifications -- a critical advantage in complex, evolving networks, unlike the spectral and spring embeddings affected by such changes due to their inherent structural dependency.

Implications and Future Directions

The approach delineated in this paper not only elucidates the potential of leveraging intrinsic graph properties borne out of differential geometry but also sets a promising direction towards fortified methodologies in graph analytics. Potential implications extend beyond mere graph alignment to areas necessitating high resilience against network changes, such as anomaly detection and feature prediction. Further research is expected to refine the theoretical underpinnings of discrete Ricci flow on graphs and explore its cross-disciplinary applications in computational biology, social network analysis, and beyond.

In conclusion, incorporating Ricci flow metrics provides a mathematically principled and computationally effective solution for graph matching challenges, unveiling new frontiers in understanding and aligning complex networks.