New node anomaly detection algorithm based on nonnegative matrix factorization for directed citation networks

Abstract

Outlier detection is a crucial task for network data analysis, which identifies abnormal entities that deviate from the rest of the dataset. Ranking in outlierness is often used for identifying abnormal nodes in directed citation networks containing citation relationship among nodes. A challenging issue in outlier ranking is how to leverage the rich graph data of complex citation networks. In this paper, we propose a cluster-based outlier score function to identify outliers in citation networks based on nonnegative matrix factorization (NMF). We first represent the citation data as a directed graph, and cluster the directed graph into logical groupings of nodes using NMF. Based on the clustering results, we obtain the outlier score and ranking for each node using the proposed outlier scoring function. The proposed method leverages the direct and indirect citation links between nodes to measure the graph-based outlierness. We validate the proposed outlier ranking method using small artificial dataset and the real-world U.S. patent data.

Appendix: Proof of proposition 1

Suppose that a graph G is an acyclic directed graph with n nodes, then the length of the longest path in the graph is at most $n-1$. Let $\tau$ be the longest path in the graph G. For $l>\tau$ and for all $1\le i$, $j\le n$, $[\mathbf A ]_{ij}^{(l)}=0$ since there is no path with length l from node i to node j. Therefore $\mathbf A ^l=\mathbf 0$, for all $l>\tau$. Let $\mathbf C =\sum _{l=1}^\infty \beta ^l \mathbf A ^l$, then $\mathbf C =\sum _{l=1}^\tau \beta ^l \mathbf A ^l$. With $0<\beta <1$, C can be rewritten as