The eccentricity of a vertex is defined as the length of the longest shortest path to any other vertex. While eccentricity is an important measure of vertex centrality, directly computing exact eccentricity for all vertices on large-scale graphs is prohibitively costly. Takes and Kosters proposed an iterative algorithm that uses multiple runs of single-source shortest path (SSSP) to compute lower and upper bounds on eccentricity at every vertex. Their technique converges to exact eccentricity by performing SSSP from only a small percentage of vertices, when sources are efficiently selected. However, their source selection strategies do not always yield rapid convergence. We propose a pincer movement source selection algorithm that efficiently selects source vertices based on analysis of the lower and upper bounds produced by SSSP. We also leverage k-BFS, which runs breadth-first search (BFS) from multiple sources concurrently on HavoqGT, a high-performance vertex-centric message-passing graph processing framework, to achieve an additional significant performance improvement on distributed-memory systems. We demonstrate that our novel source vertex selection strategy has better performance on various real-world graph datasets compared with the previous strategy. In addition, we compute exact eccentricity for graphs with more than 1000X more edges (112B undirected edges) than graphs in the previous literature.

Given a large time-evolving graph, how can we model and characterize the temporal behaviors of individual nodes (and network states)? How can we model the behavioral transition patterns of nodes? We propose a temporal behavior model that captures the "roles" of nodes in the graph and how they evolve over time. The proposed dynamic behavioral mixed-membership model (DBMM) is scalable, fully automatic (no user-defined parameters), non-parametric/data-driven (no specific functional form or parameterization), interpretable (identifies explainable patterns), and flexible (applicable to dynamic and streaming networks). Moreover, the interpretable behavioral roles are generalizable and computationally efficient. We applied our model for (a) identifying patterns and trends of nodes and network states based on the temporal behavior, (b) predicting future structural changes, and (c) detecting unusual temporal behavior transitions. The experiments demonstrate the scalability, flexibility, and effectiveness of our model for identifying interesting patterns, detecting unusual structural transitions, and predicting the future structural changes of the network and individual nodes.

Given a network, intuitively two nodes belong to the same role if they have similar structural behavior. Roles should be automatically determined from the data, and could be, for example, "clique-members," "periphery-nodes," etc. Roles enable numerous novel and useful network-mining tasks, such as sense-making, searching for similar nodes, and node classification. This paper addresses the question: Given a graph, how can we automatically discover roles for nodes? We propose RolX (Role eXtraction), a scalable (linear in the number of edges), unsupervised learning approach for automatically extracting structural roles from general network data. We demonstrate the effectiveness of RolX on several network-mining tasks: from exploratory data analysis to network transfer learning. Moreover, we compare network role discovery with network community discovery. We highlight fundamental differences between the two (e.g., roles generalize across disconnected networks, communities do not); and show that the two approaches are complimentary in nature.

Given a graph, how can we extract good features for the nodes? For example, given two large graphs from the same domain, how can we use information in one to do classification in the other (i.e., perform across-network classification or transfer learning on graphs)? Also, if one of the graphs is anonymized, how can we use information in one to de-anonymize the other? The key step in all such graph mining tasks is to find effective node features. We propose ReFeX (Recursive Feature eXtraction), a novel algorithm, that recursively combines local (node-based) features with neighborhood (egonet-based) features; and outputs regional features -- capturing "behavioral" information. We demonstrate how these powerful regional features can be used in within-network and across-network classification and de-anonymization tasks -- without relying on homophily, or the availability of class labels. The contributions of our work are as follows: (a) ReFeX is scalable and (b) it is effective, capturing regional ("behavioral") information in large graphs. We report experiments on real graphs from various domains with over 1M edges, where ReFeX outperforms its competitors on typical graph mining tasks like network classification and de-anonymization.

Advances in data collection and storage capacity have made it increasingly possible to collect highly volatile graph data for analysis. Existing graph analysis techniques are not appropriate for such data, especially in cases where streaming or near-real-time results are required. An example that has drawn significant research interest is the cyber-security domain, where internet communication traces are collected and real-time discovery of events, behaviors, patterns, and anomalies is desired. We propose MetricForensics, a scalable framework for analysis of volatile graphs. MetricForensics combines a multi-level "drill down" approach, a collection of user-selected graph metrics, and a collection of analysis techniques. At each successive level, more sophisticated metrics are computed and the graph is viewed at finer temporal resolutions. In this way, MetricForensics scales to highly volatile graphs by only allocating resources for computationally expensive analysis when an interesting event is discovered at a coarser resolution first. We test MetricForensics on three real-world graphs: an enterprise IP trace, a trace of legitimate and malicious network traffic from a research institution, and the MIT Reality Mining proximity sensor data. Our largest graph has 3M vertices and 32M edges, spanning 4.5 days. The results demonstrate the scalability and capability of MetricForensics in analyzing volatile graphs; and highlight four novel phenomena in such graphs: elbows, broken correlations, prolonged spikes, and lightweight stars.

We introduce a novel Bayesian framework for hybrid community discovery in graphs. Our framework, HCDF (short for Hybrid Community Discovery Framework), can effectively incorporate hints from a number of other community detection algorithms and produce results that outperform the constituent parts. We describe two HCDF-based approaches which are: (1) effective, in terms of link prediction performance and robustness to small perturbations in network structure; (2) consistent, in terms of effectiveness across various application domains; (3) scalable to very large graphs; and (4) nonparametric. Our extensive evaluation on a collection of diverse and large real-world graphs, with millions of links, show that our HCDF-based approaches (a) achieve up to 0.22 improvement in link prediction performance as measured by area under ROC curve (AUC), (b) never have an AUC that drops below 0.91 in the worst case, and (c) find communities that are robust to small perturbations of the network structure as defined by Variation of Information (an entropy-based distance metric).