Subgraph Detection Using Eigenvector L1 Norms

When working with network datasets, the theoretical framework of detection theory for Euclidean vector spaces no longer applies. Nevertheless, it is desirable to determine the detectability of small, anomalous graphs embedded into background networks with known statistical properties. Casting the problem of subgraph detection in a signal processing context, this article provides a framework and empirical results that elucidate a "detection theory" for graph-valued data. Its focus is the detection of anomalies in unweighted, undirected graphs through L1 properties of the eigenvectors of the graph's so-called modularity matrix. This metric is observed to have relatively low variance for certain categories of randomly-generated graphs, and to reveal the presence of an anomalous subgraph with reasonable reliability when the anomaly is not well-correlated with stronger portions of the background graph. An analysis of subgraphs in real network datasets confirms the efficacy of this approach.

[1]  Jure Leskovec,et al.  Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters , 2008, Internet Math..

[2]  Christos Faloutsos,et al.  Graphs over time: densification laws, shrinking diameters and possible explanations , 2005, KDD '05.

[3]  Azadeh Iranmehr,et al.  Trust Management for Semantic Web , 2009, 2009 Second International Conference on Computer and Electrical Engineering.

[4]  Lawrence B. Holder,et al.  Anomaly detection in data represented as graphs , 2007, Intell. Data Anal..

[5]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[6]  David J. Marchette,et al.  Scan Statistics on Enron Graphs , 2005, Comput. Math. Organ. Theory.

[7]  Padhraic Smyth,et al.  A Spectral Clustering Approach To Finding Communities in Graph , 2005, SDM.

[8]  Patrick J. Wolfe,et al.  Toward signal processing theory for graphs and non-Euclidean data , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[9]  Weixiong Zhang,et al.  An Efficient Spectral Algorithm for Network Community Discovery and Its Applications to Biological and Social Networks , 2007, Seventh IEEE International Conference on Data Mining (ICDM 2007).

[10]  Diane J. Cook,et al.  Graph-based anomaly detection , 2003, KDD '03.

[11]  Jimeng Sun,et al.  Neighborhood formation and anomaly detection in bipartite graphs , 2005, Fifth IEEE International Conference on Data Mining (ICDM'05).

[12]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[13]  Kenji Yamanishi,et al.  Network anomaly detection based on Eigen equation compression , 2009, KDD.

[14]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[15]  Tom Mifflin.  Detection theory on random graphs , 2009, 2009 12th International Conference on Information Fusion.

[16]  Christos Faloutsos,et al.  R-MAT: A Recursive Model for Graph Mining , 2004, SDM.

[17]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Hisashi Kashima,et al.  Eigenspace-based anomaly detection in computer systems , 2004, KDD.

[19]  Chris H. Q. Ding,et al.  Nonnegative Matrix Factorization for Combinatorial Optimization: Spectral Clustering, Graph Matching, and Clique Finding , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[20]  Jimeng Sun,et al.  Less is More: Compact Matrix Decomposition for Large Sparse Graphs , 2007, SDM.

[21]  Christos Faloutsos,et al.  Graph mining: Laws, generators, and algorithms , 2006, CSUR.