Network Alignment by Discrete Ollivier-Ricci Flow
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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 ∈ G_1 to nodes u',v'∈ 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. 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)[The source code of computing Ricci curvature and Ricci flow metric are available: https://github.com/saibalmars/GraphRicciCurvature].
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