Efficient Truss Maintenance in Evolving Networks

Truss was proposed to study social network data represented by graphs. A k-truss of a graph is a cohesive subgraph, in which each edge is contained in at least k-2 triangles within the subgraph. While truss has been demonstrated as superior to model the close relationship in social networks and efficient algorithms for finding trusses have been extensively studied, very little attention has been paid to truss maintenance. However, most social networks are evolving networks. It may be infeasible to recompute trusses from scratch from time to time in order to find the up-to-date $k$-trusses in the evolving networks. In this paper, we discuss how to maintain trusses in a graph with dynamic updates. We first discuss a set of properties on maintaining trusses, then propose algorithms on maintaining trusses on edge deletions and insertions, finally, we discuss truss index maintenance. We test the proposed techniques on real datasets. The experiment results show the promise of our work

Egomunities, Exploring Socially Cohesive Person-based Communities

In the last few years, there has been a great interest in detecting overlapping communities in complex networks, which is understood as dense groups of nodes featuring a low outbound density. To date, most methods used to compute such communities stem from the field of disjoint community detection by either extending the concept of modularity to an overlapping context or by attempting to decompose the whole set of nodes into several possibly overlapping subsets. In this report we take an orthogonal approach by introducing a metric, the cohesion, rooted in sociological considerations. The cohesion quantifies the community-ness of one given set of nodes, based on the notions of triangles - triplets of connected nodes - and weak ties, instead of the classical view using only edge density. A set of nodes has a high cohesion if it features a high density of triangles and intersects few triangles with the rest of the network. As such, we introduce a numerical characterization of communities: sets of nodes featuring a high cohesion. We then present a new approach to the problem of overlapping communities by introducing the concept of ego-munities, which are subjective communities centered around a given node, specifically inside its neighborhood. We build upon the cohesion to construct a heuristic algorithm which outputs a node's ego-munities by attempting to maximize their cohesion. We illustrate the pertinence of our method with a detailed description of one person's ego-munities among Facebook friends. We finally conclude by describing promising applications of ego-munities such as information inference and interest recommendations, and present a possible extension to cohesion in the case of weighted networks

Intrinsically Dynamic Network Communities

Community finding algorithms for networks have recently been extended to dynamic data. Most of these recent methods aim at exhibiting community partitions from successive graph snapshots and thereafter connecting or smoothing these partitions using clever time-dependent features and sampling techniques. These approaches are nonetheless achieving longitudinal rather than dynamic community detection. We assume that communities are fundamentally defined by the repetition of interactions among a set of nodes over time. According to this definition, analyzing the data by considering successive snapshots induces a significant loss of information: we suggest that it blurs essentially dynamic phenomena - such as communities based on repeated inter-temporal interactions, nodes switching from a community to another across time, or the possibility that a community survives while its members are being integrally replaced over a longer time period. We propose a formalism which aims at tackling this issue in the context of time-directed datasets (such as citation networks), and present several illustrations on both empirical and synthetic dynamic networks. We eventually introduce intrinsically dynamic metrics to qualify temporal community structure and emphasize their possible role as an estimator of the quality of the community detection - taking into account the fact that various empirical contexts may call for distinct `community' definitions and detection criteria.Comment: 27 pages, 11 figure

Analyzing the Facebook Friendship Graph

Online Social Networks (OSN) during last years acquired a\ud huge and increasing popularity as one of the most important emerging Web phenomena, deeply modifying the behavior of users and contributing to build a solid substrate of connections and relationships among people using the Web. In this preliminary work paper, our purpose is to analyze Facebook, considering a signi�cant sample of data re\ud ecting relationships among subscribed users. Our goal is to extract, from this platform, relevant information about the distribution of these relations and exploit tools and algorithms provided by the Social Network Analysis (SNA) to discover and, possibly, understand underlying similarities\ud between the developing of OSN and real-life social networks

Analysis of category co-occurrence in Wikipedia networks

Wikipedia has seen a huge expansion of content since its inception. Pages within this online encyclopedia are organised by assigning them to one or more categories, where Wikipedia maintains a manually constructed taxonomy graph that encodes the semantic relationship between these categories. An alternative, called the category co-occurrence graph, can be produced automatically by linking together categories that have pages in common. Properties of the latter graph and its relationship to the former is the concern of this thesis. The analytic framework, called t-component, is introduced to formalise the graphs and discover category clusters connecting relevant categories together. The m-core, a cohesive subgroup concept as a clustering model, is used to construct a subgraph depending on the number of shared pages between the categories exceeding a given threshold t. The significant of the clustering result of the m-core is validated using a permutation test. This is compared to the k-core, another clustering model. TheWikipedia category co-occurrence graphs are scale-free with a few category hubs and the majority of clusters are size 2. All observed properties for the distribution of the largest clusters of the category graphs obey power-laws with decay exponent averages around 1. As the threshold t of the number of shared pages is increased, eventually a critical threshold is reached when the largest cluster shrinks significantly in size. This phenomena is only exhibited for the m-core but not the k-core. Lastly, the clustering in the category graph is shown to be consistent with the distance between categories in the taxonomy graph

Finding the Hierarchy of Dense Subgraphs using Nucleus Decompositions

Finding dense substructures in a graph is a fundamental graph mining operation, with applications in bioinformatics, social networks, and visualization to name a few. Yet most standard formulations of this problem (like clique, quasiclique, k-densest subgraph) are NP-hard. Furthermore, the goal is rarely to find the "true optimum", but to identify many (if not all) dense substructures, understand their distribution in the graph, and ideally determine relationships among them. Current dense subgraph finding algorithms usually optimize some objective, and only find a few such subgraphs without providing any structural relations. We define the nucleus decomposition of a graph, which represents the graph as a forest of nuclei. Each nucleus is a subgraph where smaller cliques are present in many larger cliques. The forest of nuclei is a hierarchy by containment, where the edge density increases as we proceed towards leaf nuclei. Sibling nuclei can have limited intersections, which enables discovering overlapping dense subgraphs. With the right parameters, the nucleus decomposition generalizes the classic notions of k-cores and k-truss decompositions. We give provably efficient algorithms for nucleus decompositions, and empirically evaluate their behavior in a variety of real graphs. The tree of nuclei consistently gives a global, hierarchical snapshot of dense substructures, and outputs dense subgraphs of higher quality than other state-of-the-art solutions. Our algorithm can process graphs with tens of millions of edges in less than an hour