On cliques in graphs
TL;DR: In this article, the maximum number of cliques possible in a graph with n nodes is determined and bounds are obtained for the number of different sizes of clique possible in such a graph.
Abstract: A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.
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TL;DR: Two backtracking algorithms are presented, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique, and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed.
Abstract: Description bttroductian. A maximal complete subgraph (clique) is a complete subgraph that is not contained in any other complete subgraph. A recent paper [1] describes a number of techniques to find maximal complete subgraphs of a given undirected graph. In this paper, we present two backtracking algorithms, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique. The first version is a straightforward implementation of the basic algorithm. It is mainly presented to illustrate the method used. This version generates cliques in alphabetic (lexicographic) order. The second version is derived from the first and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed. This version tends to produce the larger cliques first and to generate sequentially cliques having a large common intersection. The detailed algorithm for version 2 is presented here. Description o f the algorithm--Version 1. Three sets play an important role in the algorithm. (1) The set compsub is the set to be extended by a new point or shrunk by one point on traveling along a branch of the backtracking tree. The points that are eligible to extend compsub, i.e. that are connected to all points in compsub, are collected recursively in the remaining two sets. (2) The set candidates is the set of all points that will in due time serve as an extension to the present configuration of compsub. (3) The set not is the set of all points that have at an earlier stage already served as an extension of the present configuration of compsub and are now explicitly excluded. The reason for maintaining this set trot will soon be made clear. The core of the algorithm consists of a recursively defined extension operator that will be applied to the three sets Just described. It has the duty to generate all extensions of the given configuration of compsub that it can make with the given set of candidates and that do not contain any of the points in not. To put it differently: all extensions of compsub containing any point in not have already been generated. The basic mechanism now consists of the following five steps:
2,405 citations
Book•
12 Mar 2012TL;DR: Comprehensive and coherent, this hands-on text develops everything from basic reasoning to advanced techniques within the framework of graphical models, and develops analytical and problem-solving skills that equip them for the real world.
Abstract: Machine learning methods extract value from vast data sets quickly and with modest resources They are established tools in a wide range of industrial applications, including search engines, DNA sequencing, stock market analysis, and robot locomotion, and their use is spreading rapidly People who know the methods have their choice of rewarding jobs This hands-on text opens these opportunities to computer science students with modest mathematical backgrounds It is designed for final-year undergraduates and master's students with limited background in linear algebra and calculus Comprehensive and coherent, it develops everything from basic reasoning to advanced techniques within the framework of graphical models Students learn more than a menu of techniques, they develop analytical and problem-solving skills that equip them for the real world Numerous examples and exercises, both computer based and theoretical, are included in every chapter Resources for students and instructors, including a MATLAB toolbox, are available online
1,474 citations
Cites background from "On cliques in graphs"
...This shows that a graph can have an exponentially large number of maximal cliques[211]....
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TL;DR: A framework is proposed for evaluating algorithms' ability to detect overlapping nodes, which helps to assess overdetection and underdetection, and for low overlapping density networks, SLPA, OSLOM, Game, and COPRA offer better performance than the other tested algorithms.
Abstract: This article reviews the state-of-the-art in overlapping community detection algorithms, quality measures, and benchmarks. A thorough comparison of different algorithms (a total of fourteen) is provided. In addition to community-level evaluation, we propose a framework for evaluating algorithms' ability to detect overlapping nodes, which helps to assess overdetection and underdetection. After considering community-level detection performance measured by normalized mutual information, the Omega index, and node-level detection performance measured by F-score, we reached the following conclusions. For low overlapping density networks, SLPA, OSLOM, Game, and COPRA offer better performance than the other tested algorithms. For networks with high overlapping density and high overlapping diversity, both SLPA and Game provide relatively stable performance. However, test results also suggest that the detection in such networks is still not yet fully resolved. A common feature observed by various algorithms in real-world networks is the relatively small fraction of overlapping nodes (typically less than 30p), each of which belongs to only 2 or 3 communities.
1,166 citations
TL;DR: A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed.
Abstract: The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].
1,065 citations
Cites background from "On cliques in graphs"
...As they pointed out, this was the best one could hope for since the Moon and Moser graphs [238] have 3n=3 maximal cliques....
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TL;DR: It is shown that this family of graphs can be uniquely represented by a tree where the leaves of the tree correspond to the vertices of the graph.
872 citations
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