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.

Abstract-Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier [1], Lin et al.[2], etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.

Along with the growth of cloud computing and mobile devices, the importance of client device identity concern over cloud environment is emerging. To provide a lightweight yet reliable method for device identification, an application layer approach based on clock skew fingerprint is proposed. The developed experimental platform adapts AJAX technology to collect the timestamps of client devices in the cloud server during connection time, then calculate the clock skews of client devices. Few methods based on linear regression and piecewise minimum algorithm are developed to optimize the precision and shorten timestamp collection process. A jump point detection scheme is also proposed to resolve the offset drifting problem, which is usually caused by switching network or temporary disconnection. Finally, two experiments are conducted to study the effectiveness of clock skew fingerprint, and the results illustrate that the false positive rate and the false negative rate, in the worst case, are both no more than 8% when the tolerance threshold is set appropriately.

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0000−0002−9082−7401] , Yu-Yao Lin 2[0000−0001−9761−5156] , Jie Gao 3 [0000−0001−5083−6082] , and Xianfeng Gu 3[0000−0001−8226−5851]Abstract. In this paper, we consider the problem of approximately aligning/matching two graphs. Given two graphs G1 = (V1, E1) and G2 = (V2, E2), the objective is to map nodes u, v ∈ G1 to nodes u , v ∈ G2 such that when u, v have an edge in G1, very likely their corresponding nodes u , v in G2 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 G1 to a node in G2 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) 4 . 4 The source code of computing Ricci curvature and Ricci flow metric are available: https://github.com/saibalmars/GraphRicciCurvature