Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks.In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.

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We consider the community search problem defined upon a large graph G: given a query vertex q in G, to find as output all the densely connected subgraphs of G, each of which contains the query v. As an online, query-dependent variant of the well-known community detection problem, community search enables personalized community discovery that has found widely varying applications in real-world, largescale graphs. In this paper, we study the community search problem in the truss-based model aimed at discovering all dense and cohesive k-truss communities to which the query vertex q belongs. We introduce a novel equivalence relation, k-truss equivalence, to model the intrinsic density and cohesiveness of edges in k-truss communities. Consequently, all the edges of G can be partitioned to a series of k-truss equivalence classes that constitute a space-efficient, trusspreserving index structure, EquiTruss. Community search can be henceforth addressed directly upon EquiTruss without repeated, time-demanding accesses to the original graph, G, which proves to be theoretically optimal. In addition, EquiTruss can be efficiently updated in a dynamic fashion when G evolves with edge insertion and deletion. Experimental studies in real-world, large-scale graphs validate the efficiency and effectiveness of EquiTruss, which has achieved at least an order of magnitude speedup in community search over the state-of-the-art method, TCP-Index.

Online social network analysis has attracted great attention with a vast number of users sharing information and availability of APIs that help to crawl online social network data. In this paper, we study the research studies that are helpful for user characterization as online users may not always reveal their true identity or attributes. We especially focused on user attribute determination such as gender, age, etc.; user behavior analysis such as motives for deception; mental models that are indicators of user behavior; user categorization such as bots vs. humans; and entity matching on different social networks. We believe our summary of analysis of user characterization will provide important insights to researchers and better services to online users

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