Graphlet based network analysis

The majority of the existing works on network analysis, study properties that are related to the global topology of a network. Examples of such properties include diameter, power-law exponent, and spectra of graph Laplacians. Such works enhance our understanding of real-life networks, or enable us to generate synthetic graphs with real-life graph properties. However, many of the existing problems on networks require the study of local topological structures of a network. Graphlets which are induced small subgraphs capture the local topological structure of a network effectively. They are becoming increasingly popular for characterizing large networks in recent years. Graphlet based network analysis can vary based on the types of topological structures considered and the kinds of analysis tasks. For example, one of the most popular and early graphlet analyses is based on triples (triangles or paths of length two). Graphlet analysis based on cycles and cliques are also explored in several recent works. Another more comprehensive class of graphlet analysis methods works with graphlets of specific sizes—graphlets with three, four or five nodes ({3, 4, 5}-Graphlets) are particularly popular. For all the above analysis tasks, excessive computational cost is a major challenge, which becomes severe for analyzing large networks with millions of vertices. To overcome this challenge, effective methodologies are in urgent need. Furthermore, the existence of efficient methods for graphlet analysis will encourage more works broadening the scope of graphlet analysis. For graphlet counting, we propose edge iteration based methods (ExactTC and ExactGC) for efficiently computing triple and graphlet counts. The proposed methods compute local graphlet statistics in the neighborhood of each edge in the network and then aggregate the local statistics to give the global characterization (transitivity, graphlet frequency distribution (GFD), etc) of the network. Scalability of the proposed methods is further improved by iterating over a sampled set of edges and estimating the triangle count (ApproxTC) and graphlet count (Graft) by approximate rescaling of the aggregated statistics. The independence of local feature vector construction corresponding to each edge makes the methods embarrassingly parallelizable. We show this by giving a parallel edge iteration method ParApproxTC for triangle counting. For graphlet sampling, we propose Markov Chain Monte Carlo (MCMC) sampling based methods for triple and graphlet analysis. Proposed triple analysis methods, Vertex-MCMC and Triple-MCMC, estimate triangle count and network transitivity. Vertex-MCMC samples triples in two steps. First, the method selects a node (using the MCMC method) with probability proportional to the number of triples of which the node is a center. Then Vertex-MCMC samples uniformly from the triples centered by the selected node. The method Triple-MCMC samples triples by performing a MCMC walk in a triple sample space. Triple sample space consists of all the possible triples in a network. MCMC method performs triple sampling by walking form one triple to one of its neighboring triples in the triple space. We design the triple space in such a way that two triples are neighbors only if they share exactly two nodes. The proposed triple sampling algorithms Vertex-MCMC and Triple-MCMC are able to sample triples from any arbitrary distribution, as long as the weight of each triple is locally computable. The proposed methods are able to sample triples without the knowledge of the complete network structure. Information regarding only the local neighborhood structure of currently observed node or triple are enough to walk to the next node or triple. This gives the proposed methods a significant advantage: the capability to sample triples from networks that have restricted access, on which a direct sampling based method is simply not applicable. The proposed methods are also suitable for dynamic and large networks. Similar to the concept of Triple-MCMC, we propose Guise for sampling graphlets of sizes three, four and five ({3, 4, 5}-Graphlets). Guise samples graphlets, by performing a MCMC walk on a graphlet sample space, containing all the graphlets of sizes three, four and five in the network. Despite the proven utility of graphlets in static network analysis, works harnessing the ability of graphlets for dynamic network analysis are yet to come. Dynamic networks contain additional time information for their edges. With time, the topological structure of a dynamic network changes—edges can appear, disappear and reappear over time. In this direction, predicting the link state of a network at a future time, given a collection of link states at earlier times, is an important task with many real-life applications. In the existing literature, this task is known as link prediction in dynamic networks. Performing this task is more difficult than its counterpart in static networks because an effective feature representation of node-pair instances for the case of a dynamic network is hard to obtain. We design a novel graphlet transition based feature embedding for node-pair instances of a dynamic network. Our proposed method GraTFEL, uses automatic feature learning methodologies on such graphlet transition based features to give a low-dimensional feature embedding of unlabeled node-pair instances. The feature learning task is modeled as an optimal coding task where the objective is to minimize the reconstruction error. GraTFEL solves this optimization task by using a gradient descent method. We validate the effectiveness of the learned optimal feature embedding by utilizing it for link prediction in real-life dynamic networks. Specifically, we show that GraTFEL, which uses the extracted feature embedding of graphlet transition events, outperforms existing methods that use well-known link prediction features