Many complex networks in the real world have community structures -- groups of well-connected nodes with important functional roles. It has been well recognized that the identification of communities bears numerous practical applications. While existing approaches mainly apply statistical or graph theoretical combinatorial methods for community detection, in this paper, we present a novel geometric approach which enables us to borrow powerful classical geometric methods and properties. By considering networks as geometric objects and communities in a network as a geometric decomposition, we apply curvature and discrete Ricci flow, which have been used to decompose smooth manifolds with astonishing successes in mathematics, to break down communities in networks. We tested our method on networks with ground-truth community structures, and experimentally confirmed the effectiveness of this geometric approach.
Unlike discrete Ricci curvature, discrete Ricci flow has not been studied as much. Chow and Luo introduced the first discrete Ricci flow on surfaces @cite_65 . In @cite_32 , Weber al suggested applying Forman-Ricci flow for anomaly detection in the complex network. In @cite_38 , Ni al used the Ollivier-Ricci curvature flow to compute the Ricci flow metric as edge weights for the problem of network alignment (noisy graph matching).
@cite_38 @cite_65 @cite_32
2891096760 1970564715 2780222030 2792707137
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. 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). We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound. We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices. The discrete version of the Ollivier-Ricci (OR) curvature, applicable to networks, has recently found utility in diverse fields. OR curvature requires solving an optimal mass transport problem for each edge, which can be computationally expensive for large and or dense networks. We propose two alternative proxies of curvature to OR that are motivated by the Jaccard index and are demonstrably less computationally intensive. Jaccard curvature (JC) is a simple shift and scaling of the Jaccard index that captures the overlap of edge node neighborhoods. Generalized Jaccard curvature (gJC) captures the shortest path distances in a mass exchange problem. We study the goodness of approximation between the proposed curvatures and an alternative metric, Forman-Ricci curvature, with OR curvature for several network models and real networks. Our results suggest that the gJC exhibits a reasonably good fit to the OR curvature for a wide range of networks, while the JC is shown to be a good proxy only for certain scenarios.