A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

In this thesis, we propose a system for finding friend groups in Blogosphere. This system includes two parts: The first part can traverse the Blogosphere so as to obtain the friend network; and the second part is used to find friend groups from the friend network. Our practical performance was tested on Wretch, which is the largest Blogosphere in Taiwan. In today's blog service environment, the establishments of friend relationships are usually unidirectional, i.e., a blogger can add any other as his friend without confirmation. Traditional methods such as clique/club or 2-clique/club are not suitable because the bidirectional link is built incompletely in the social network under such circumstances. To solve this problem, we propose the 1.5-club based on transitive extension. We further make a comparison among the results of finding groups by 1-club, 1.5-club, 2-club and k-clique, and analyze the historical data of social networks from over almost one year. The experimental results show that our proposed method is effective and promising.

No description available.

This dissertation establishes mathematical foundations for the properties exhibited by generalizations of cliques, as well as algorithms to find such objects in a network. Cliques are a model of an ideal group with roots in social network analysis. They have since found applications as a part of grouping mechanisms in computer vision, coding theory, experimental design, genomics, economics, and telecommunications among other fields. Because only groups with ideal properties form a clique, they are often too restrictive for identifying groups in many real-world networks. This motivated the introduction of clique relaxations that preserve some of the various defining properties of cliques in relaxed form. There are six clique relaxations that are the focus of this dissertation: s-clique, s-club, s-plex, k-core, quasi-clique, and k-connected subgraphs. Since cliques have found applications in so many fields, research into these clique relaxations has the potential to steer the course of much future research. The focus of this dissertation is on bringing organization and rigorous methodology to the formation and application of clique relaxations. We provide the first taxonomy focused on how the various clique relaxations relate on key structural properties demonstrated by groups. We also give a framework for how clique relaxations can be formed. This equips researchers with the ability to choose the appropriate clique relaxation for an application based on its structural properties, or, if an appropriate clique relaxation does not exist, form a new one. In addition to identifying the structural properties of the various clique relaxations, we identify properties and prove propositions that are important computationally. These assist in creating algorithms to find a clique relaxation quickly as it is immersed in a network. We give the first ever analysis of the computational complexity of finding the maximum quasi-clique in a graph. Such analysis identifies for researchers the appropriate set of computational tools to solve the maximum quasiclique problem. We further create a polynomial time algorithm for identifying large 2-cliques within unit disk graphs, a special class of graphs often arising in communication networks. We prove the algorithm to have a guaranteed 1=2-approximation ratio and finish with computational results.

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

Content Based Image Retrieval (CBIR) systems represent images in the database by color, texture, and shape information. In this thesis, we concentrate on tex- ture features and introduce a new generic texture descriptor, namely, Statistical Analysis of Structural Information (SASI). Moreover, in order to increase the re- trieval rates of a CBIR system, we propose a new method that can also adapt an image retrieval system into a con&macr / gurable one without changing the underlying feature extraction mechanism and the similarity function. SASI is based on statistics of clique autocorrelation coe&plusmn / cients, calculated over structuring windows. SASI de&macr / nes a set of clique windows to extract and measure various structural properties of texture by using a spatial multi- resolution method. Experimental results, performed on various image databases, indicate that SASI is more successful then the Gabor Filter descriptors in cap- turing small granularities and discontinuities such as sharp corners and abrupt changes. Due to the &deg / exibility in designing the clique windows, SASI reaches higher average retrieval rates compared to Gabor Filter descriptors. However, the price of this performance is increased computational complexity. Since, retrieving of similar images of a given query image is a subjective task, it is desirable that retrieval mechanism should be con&macr / gurable by the user. In the proposed method, basically, original feature space of a content-based retrieval system is nonlinearly transformed into a new space, where the distance between the feature vectors is adjusted by learning. The transformation is realized by Arti&macr / cial Neural Network architecture. A cost function is de&macr / ned for learning and optimized by simulated annealing method. Experiments are done on the texture image retrieval system, which use SASI and Gabor Filter features. The results indicate that con&macr / gured image retrieval system is signi&macr / cantly better than the original system.

Zugl.: Siegen, Univ., Diss., 2007

Zugl.: Halle, Univ., Diss., 2009

abstract: Every graph can be colored with one more color than its maximum degree. A well-known theorem of Brooks gives the precise conditions under which a graph can be colored with maximum degree colors. It is natural to ask for the required conditions on a graph to color with one less color than the maximum degree; in 1977 Borodin and Kostochka conjectured a solution for graphs with maximum degree at least 9: as long as the graph doesn't contain a maximum-degree-sized clique, it can be colored with one fewer than the maximum degree colors. This study attacks the conjecture on multiple fronts. The first technique is an extension of a vertex shuffling procedure of Catlin and is used to prove the conjecture for graphs with edgeless high vertex subgraphs. This general approach also bears more theoretical fruit. The second technique is an extension of a method Kostochka used to reduce the Borodin-Kostochka conjecture to the maximum degree 9 case. Results on the existence of independent transversals are used to find an independent set intersecting every maximum clique in a graph. The third technique uses list coloring results to exclude induced subgraphs in a counterexample to the conjecture. The classification of such excludable graphs that decompose as the join of two graphs is the backbone of many of the results presented here. The fourth technique uses the structure theorem for quasi-line graphs of Chudnovsky and Seymour in concert with the third technique to prove the Borodin-Kostochka conjecture for claw-free graphs. The fifth technique adds edges to proper induced subgraphs of a minimum counterexample to gain control over the colorings produced by minimality. The sixth technique adapts a recoloring technique originally developed for strong coloring by Haxell and by Aharoni, Berger and Ziv to general coloring. Using this recoloring technique, the Borodin-Kostochka conjectured is proved for graphs where every vertex is in a large clique. The final technique is naive probabilistic coloring as employed by Reed in the proof of the Borodin-Kostochka conjecture for large maximum degree. The technique is adapted to prove the Borodin-Kostochka conjecture for list coloring for large maximum degree. / Dissertation/Thesis / Ph.D. Mathematics 2013

Indiana University-Purdue University Indianapolis (IUPUI) / Network analysis is an important research task which explains the relationships among various entities in a given domain. Most of the existing approaches of network analysis compute global properties of a network, such as transitivity, diameter, and all-pair shortest paths. They also study various non-random properties of a network, such as graph densifi cation with shrinking diameter, small diameter, and scale-freeness. Such approaches enable us to understand real-life networks with global properties. However, the discovery of the local topological building blocks within a network is an important task, and examples include clique enumeration, graphlet counting, and motif counting. In this paper, my focus is to fi nd an efficient solution of k-clique enumeration problem. A clique is a small, connected, and complete induced subgraph over a large network. However, enumerating cliques using sequential technologies is very time-consuming. Another promising direction that is being adopted is a solution that runs on distributed clusters of machines using the Hadoop mapreduce framework. However, the solution suffers from a general limitation of the framework, as Hadoop's mapreduce performs substantial amounts of reading and writing to disk. Thus, the running times of Hadoop-based approaches suffer enormously. To avoid these problems, we propose an e cient, scalable, and distributed solution, kc-spark , for enumerating cliques in real-life networks using the Apache Spark in-memory cluster computing framework. Experiment results show that kc-spark can enumerate k-cliques from very large real-life networks, whereas a single commodity machine cannot produce the same desired result in a feasible amount of time. We also compared kc-spark with Hadoop mapreduce solutions and found the algorithm to be 80-100 percent faster in terms of running times. On the other hand, we compared with the triangle enumeration with Hadoop mapreduce and results shown that kc-spark is 8-10 times faster than mapreduce implementation with the same cluster setup. Furthermore, the overall performance of kc-spark is improved by using Spark's inbuilt caching and broadcast transformations.