4,056 research outputs found
An algorithm to discover the k-clique cover in networks
In social network analysis, a k-clique is a relaxed clique, i.e., a k-clique is a quasi-complete sub-graph. A k-clique in a graph is a sub-graph where the distance between any two vertices is no greater than k. The
visualization of a small number of vertices can be easily performed in a graph.
However, when the number of vertices and edges increases the visualization
becomes incomprehensible. In this paper, we propose a new graph mining approach based on k-cliques. The concept of relaxed clique is extended to the whole graph, to achieve a general view, by covering the network with k-cliques.
The sequence of k-clique covers is presented, combining small world concepts
with community structure components. Computational results and examples are
presented
Pure simplicial complexes and well-covered graphs
A graph is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex is called
pure if all of its facets have the same cardinality. Let be the
class of graphs with some disjoint maximal cliques covering all vertices. In
this paper, we prove that for any simplicial complex or any graph, there is a
corresponding graph in class with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class is well-covered or not. To do
this characterization, we use an algebraic interpretation according to
zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1009.524
Edge Clique Cover of Claw-free Graphs
The smallest number of cliques, covering all edges of a graph , is
called the (edge) clique cover number of and is denoted by . It
is an easy observation that for every line graph with vertices,
. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26;
MR1761707] extended this observation to all quasi-line graphs and questioned if
the same assertion holds for all claw-free graphs. In this paper, using the
celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour,
we give an affirmative answer to this question for all claw-free graphs with
independence number at least three. In particular, we prove that if is a
connected claw-free graph on vertices with , then and equality holds if and only if is either the graph of
icosahedron, or the complement of a graph on vertices called twister or
the power of the cycle , for .Comment: 74 pages, 4 figure
Motif Clustering and Overlapping Clustering for Social Network Analysis
Motivated by applications in social network community analysis, we introduce
a new clustering paradigm termed motif clustering. Unlike classical clustering,
motif clustering aims to minimize the number of clustering errors associated
with both edges and certain higher order graph structures (motifs) that
represent "atomic units" of social organizations. Our contributions are
two-fold: We first introduce motif correlation clustering, in which the goal is
to agnostically partition the vertices of a weighted complete graph so that
certain predetermined "important" social subgraphs mostly lie within the same
cluster, while "less relevant" social subgraphs are allowed to lie across
clusters. We then proceed to introduce the notion of motif covers, in which the
goal is to cover the vertices of motifs via the smallest number of (near)
cliques in the graph. Motif cover algorithms provide a natural solution for
overlapping clustering and they also play an important role in latent feature
inference of networks. For both motif correlation clustering and its extension
introduced via the covering problem, we provide hardness results, algorithmic
solutions and community detection results for two well-studied social networks
Partitioning networks into cliques: a randomized heuristic approach
In the context of community detection in social networks, the term community can be grounded in the strict way that simply everybody should know each other within the community. We consider the corresponding community detection problem. We search for a partitioning of a network into the minimum number of non-overlapping cliques, such that the cliques cover all vertices. This problem is called the clique covering problem (CCP) and is one of the classical NP-hard problems. For CCP, we propose a randomized heuristic approach. To construct a high quality solution to CCP, we present an iterated greedy (IG) algorithm. IG can also be combined with a heuristic used to determine how far the algorithm is from the optimum in the worst case. Randomized local search (RLS) for maximum independent set was proposed to find such a bound. The experimental results of IG and the bounds obtained by RLS indicate that IG is a very suitable technique for solving CCP in real-world graphs. In addition, we summarize our basic rigorous results, which were developed for analysis of IG and understanding of its behavior on several relevant graph classes
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