An appraisal of the performance of the MMSC subgraph generation algorithm on a Cyber System 170/720

An algorithm for generating maximal minimally strongly connected (MMSC) subgraphs or cliques in undirected graphs has been suggested by Daset al. in a recent paper. The proposed algorithm is based on a refinement of the technique of successive splitting described by Paull and Unger in the determination of maximal compatibles of states in the context of minimization of incomplete sequential machines. The present paper discusses the performance of the aforementioned subgraph generation algorithm in an implementation on Cyber System 170/720 using FORTRAN IV, in particular reference to cluster analysis.     

Magnitude ordering of degree complements of certain node pairs in an undirected graph and an algorithm to find a class of maximal subgraphs

An undirected or a symmetric graph consists of a set of nodes and a set of nonoriented edges connecting between pairs of nodes. In widely differing disciplines of science and engineering, symmetric graphs find important uses. In many of these application areas, an often encountered problem is that of finding all the maximal complete subgraphs of a symmetric graph. In this paper, borrowing the concept of strong connectedness in a nonsymmetric graph, the idea of minimally strongly connected (MSC) and maximal minimally strongly connected (MMSC) subgraphs in a symmetric graph is introduced. The MMSC subgraphs play a kind of role identical to that played by maximal complete subgraphs in symmetric graphs. Many important properties of MMSC subgraphs are discussed in the paper, and an explicit, computer-oriented algorithm is developed for finding all the MMSC subgraphs, given an undirected graph.     

Maximal prime subgraph decomposition of Bayesian networks

The authors present a method for decomposition of Bayesian networks into their maximal prime subgraphs. The correctness of the method is proven and results relating the maximal prime subgraph decomposition (MPD) to the maximal complete subgraphs of the moral graph of the original Bayesian network are presented. The maximal prime subgraphs of a Bayesian network can be organized as a tree which can be used as the computational structure for LAZY propagation. We also identify a number of tasks performed on Bayesian networks that can benefit from MPD. These tasks are: divide and conquer triangulation, hybrid propagation algorithms combining exact and approximative inference techniques, and incremental construction of junction trees. We compare the proposed algorithm with standard algorithms for decomposition of undirected graphs into their maximal prime subgraphs. The discussion shows that the proposed algorithm is simpler, more easy to comprehend, and it has the same complexity as the standard algorithms.     

最大频繁子图挖掘算法研究

随着图的广泛应用,图的规模不断扩大,因此提高频繁子图挖掘效率势在必行。本文针对频繁子图挖掘所产生的庞大的结果集,提出了一个最大频繁子图挖掘算法MFME,从而极大地减少了结果集的数量。MFME使用了映射的思想将图集中的边映射到边表中并在此表上进行子图挖掘,有效地提高了算法的效率。实验结果表明,MFME的效率较经典算法SPIN有明显提高。     

Enumerating all connected maximal common subgraphs in two graphs

We represent a new method for finding all connected maximal common subgraphs in two graphs which is based on the transformation of the problem into the clique problem. We have developed new algorithms for enumerating all cliques that represent connected maximal common subgraphs. These algorithms are based on variants of the Bron–Kerbosch algorithm. In this paper we explain the transformation of the maximal common subgraph problem into the clique problem. We give a short summary of the variants of the Bron–Kerbosch algorithm in order to explain the modification of that algorithm such that the detected cliques represent connected maximal common subgraphs. After introducing and proving several variants of the modified algorithm we discuss the runtimes for all variants by means of random graphs. The results show the drastical reduction of the runtimes for the new algorithms.     

Efficiently mining <Emphasis Type="Italic">δ</Emphasis>-tolerance closed frequent subgraphs

The output of frequent pattern mining is a huge number of frequent patterns, which are very redundant, causing a serious problem in understandability. We focus on mining frequent subgraphs for which well-considered approaches to reduce the redundancy are limited because of the complex nature of graphs. Two known, standard solutions are closed and maximal frequent subgraphs, but closed frequent subgraphs are still redundant and maximal frequent subgraphs are too specific. A more promising solution is δ-tolerance closed frequent subgraphs, which decrease monotonically in δ, being equal to maximal frequent subgraphs and closed frequent subgraphs for δ=0 and 1, respectively. However, the current algorithm for mining δ-tolerance closed frequent subgraphs is a naive, two-step approach in which frequent subgraphs are all enumerated and then sifted according to δ-tolerance closedness. We propose an efficient algorithm based on the idea of “reverse-search” by which the completeness of enumeration is guaranteed and for which new pruning conditions are incorporated. We empirically demonstrate that our approach significantly reduced the amount of real computation time of two compared algorithms for mining δ-tolerance closed frequent subgraphs, being pronounced more for practical settings.     

A note on the derivation of maximal common subgraphs of two directed or undirected graphs

In this note the problem is considered of finding maximal common subgraphs of two given graphs. A technique is described by which this problem can be stated as a problem of deriving maximal compatibility classes. A known «maximal compatibility classes» algorithm can then be used to derive maximal common subgraphs. The same technique is shown to apply to the classical subgraph isomorphism problem.     

Strong connectivity in symmetric graphs and generation of maximal minimally strongly connected subgraphs

In nonsymmetric graphs strong connectivity is an important concept. In this paper, extending the concept of strong connectivity of nonsymmetric graphs to the case of symmetric graphs, the idea of minimally strongly connected (MSC) and maximal minimally strongly connected (MMSC) subgraphs in a symmetric graph is introduced, and theoretical results are developed that pertain to certain useful properties of these subgraphs. A computer-oriented algorithm is also proposed for finding the MMSC subgraphs from a given symmetric graph, that seems efficient and simple, and tends to reduce computation in generating the subgraphs.     

Maximal Biclique Subgraphs and Closed Pattern Pairs of the Adjacency Matrix: A One-to-One Correspondence and Mining Algorithms

Maximal biclique (also known as complete bipartite) subgraphs can model many applications in Web mining, business, and bioinformatics. Enumerating maximal biclique subgraphs from a graph is a computationally challenging problem, as the size of the output can become exponentially large with respect to the vertex number when the graph grows. In this paper, we efficiently enumerate them through the use of closed patterns of the adjacency matrix of the graph. For an undirected graph G without self-loops, we prove that 1) the number of closed patterns in the adjacency matrix of G is even, 2) the number of the closed patterns is precisely double the number of maximal biclique subgraphs of G, and 3) for every maximal biclique subgraph, there always exists a unique pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, the problem of enumerating maximal bicliques can be solved by using efficient algorithms for mining closed patterns, which are algorithms extensively studied in the data mining field. However, this direct use of existing algorithms causes a duplicated enumeration. To achieve high efficiency, we propose an O(mn) time delay algorithm for a nonduplicated enumeration, in particular, for enumerating those maximal bicliques with a large size, where m and n. are the number of edges and vertices of the graph, respectively. We evaluate the high efficiency of our algorithm by comparing it to state- of-the-art algorithms on three categories of graphs: randomly generated graphs, benchmarks, and a real-life protein interaction network. In this paper, we also prove that if self-loops are allowed in a graph, then the number of closed patterns in the adjacency matrix is not necessarily even, but the maximal bicliques are exactly the same as those of the graph after removing all the self-loops.     

Clique detection for nondirected graphs: Two new algorithms

Making use of special tree search algorithms the present paper describes two new methods for determining all maximal complete subgraphs (cliques) of a finite nondirected graph. In both methods the blockwise generation of all cliques induces characteristic properties, which guarantee an efficient calculation of special clique subsets, especially the set of all cliques of maximal length. Moreover, by their structure both algorithms allow to calculate the complete clique set by parallel processing. The algorithms have been tested for many series of characteristic graphs and compared with the algorithm of Bron-Kerbosch (Algorithm 457 of CACM) the most efficient algorithm which is known to the authors.     

基于FSG的最大频繁子图挖掘算法*

图挖掘已成为数据挖掘领域研究的热点,然而挖掘全部频繁子图很困难且得到的频繁子图过多,影响结果的理解和应用。可通过挖掘最大频繁子图来解决挖掘结果数量巨大的问题,最大频繁子图挖掘得到的结果数量很少且不丢失信息,节省了空间和以后的分析工作。基于算法FSG提出了最大频繁子图挖掘算法FSG-MaxGraph;结合节点的度、标记及邻接列表来计算规范编码,提出两个定理来减少子图同构判断的次数,并应用改进后的决策树来计算支持度。实验证明,新算法解决了挖掘结果太多理解困难的问题,且提高了挖掘效率。     

利用MapReduce平台实现高效并行的频繁子图挖掘

频繁子图挖掘是数据挖掘领域的一个重要问题,并且有着广泛的应用。在Hadoop平台上实现了一种基于MapReduce的高效频繁子图挖掘算法Cloud-GFSG（cloud-global frequent subgraph）。该算法基于Apriori思想,在扩展边生成新的子图时,使用已经挖掘出的k-1阶的频繁子图生成k阶的频繁子图。同时,检查是否存在待扩展生成的子图,设定生成的频繁子图表示规则,保证了频繁子图信息的唯一性。较同类算法相比,该算法在挖掘频繁子图时更具通用性,并且在扩展边时避免产生大量的复制图,从而使得算法的正确性得以保证,且运行效率显著提高。     

Graphical model based continuous estimation of distribution algorithm

In this paper, a new estimation of distribution algorithm is introduced. The goal is to propose a method that avoids complex approximations of learning a probabilistic graphical model and considers multivariate dependencies between continuous random variables. A parallel model of some subgraphs with a smaller number of variables is learned as the probabilistic graphical model. In each generation, the joint probability distribution of the selected solutions is estimated using a Gaussian Mixture model. Then, learning the graphical model of dependencies among random variables and sampling are done separately for each Gaussian component. In the learning step, using the selected solutions of each Gaussian mixture component, the structure of a Markov network is learned. This network is decomposed to maximal cliques and a clique graph. Then, complete Bayesian network structures are learned for these subgraphs using an optimization algorithm. The proposed optimization problem is a 0–1 constrained quadratic programming which finds the best permutation of variables. Then, sampling is done from each Bayesian network of each Gaussian component. The introduced method is compared with the other network-based estimation of distribution algorithms for optimization of continuous numerical functions.     

基于关键词的RDF数据图查询模型研究

随着语义网数据的海量涌现,人们更加关注RDF图的数据查询效率,通过关键词匹配直接查询RDF数据图成为一个研究热点。针对关键词查询中普遍存在的结果冗余与偏离等问题,提出了一种基于关键词的RDF数据图查询模型。该模型首先采用提出的基于迭代的图查询算法(ISGR)对所查询关键词进行子图匹配,得到唯一且最大的结果子图集合;然后根据关键词图与结果子图之间的结构信息,利用统计语言模型,给出了一种结果子图排序方法(SimLM)。对比实验表明,提出的查询模型及排序方法在一致性和相关性方面的性能优于传统模型。     

加权最大频繁子图挖掘算法的研究

如何从大量的图中挖掘出令人感兴趣的子图模式已经成为数据挖掘领域研究的热点之一。传统的频繁子图挖掘方法对满足最小支持度阈值的子图同等对待,但在真实数据库中不同的子图往往具有不同的重要程度。为解决上述问题,提出了一种深度优先的挖掘加权最大频繁子图的新算法。首先给出了一种新的用于计算图的邻接矩阵规范编码的结点排序策略,大大降低了求图规范编码的复杂度,并可以加速子图规范编码匹配的速度。其次,给出了加权最大频繁子图的定义,不仅可以找出较为重要的最大频繁子图,而且可以使挖掘结果同样具有反单调性,从而可加速剪枝。实验结果表明,提出的算法不仅可以有效地减少挖掘结果的数量,而且具有较高的效率。     

基于不确定性PPI网络的最大稠密子图挖掘*

研究表明使用PPI数据进行蛋白质功能预测是很有意义的。然而,从生物学实验得到的PPI数据一般是含有噪声的、不完全的和不精确的,这使得将PPI网络作为不确定图来处理变得更加合理。提出了一种基于深度优先搜索策略和点扩展的挖掘算法,它可以有效地从不确定的PPI网络中挖掘最大稠密子图。该算法使用了几种高效的剪枝技术来提高挖掘的时间效率。在酵母菌PPI数据上的实验结果表明该算法在精度和效率上都有很好的表现。     

Dynamic detection of subgraphs in computer networks

Most distributed algorithms for computer networks are designed to work with arbitrary graph structures. Most networks, however, can usually be decomposed into subgraphs with a specific structure. Detecting and exploiting these subgraphs can considerably reduce the storage and communication cost of the algorithm. In this paper we propose a distributed algorithm for detecting and exploiting tree subgraphs. In a network with fixed topology, the algorithm is optimal in terms of communication complexity. The algorithm also dynamically adapts to changes in network topology caused by link failure and recovery. The dynamic operation of the algorithm is incremental as only nodes that may be affected by the change reinitiate the algorithm. Another important property of our algorithm is that it requires no node identities or sequence numbers. We examine how this idea can be extended to other subgraph structures.     

Dynamic detection of subgraphs in computer networks

Most distributed algorithms for computer networks are designed to work with arbitrary graph structures. Most networks, however, can usually be decomposed into subgraphs with a specific structure. Detecting and exploiting these subgraphs can considerably reduce the storage and communication cost of the algorithm. In this paper we propose a distributed algorithm for detecting and exploiting tree subgraphs. In a network with fixed topology, the algorithm is optimal in terms of communication complexity. The algorithm also dynamically adapts to changes in network topology caused by link failure and recovery. The dynamic operation of the algorithm is incremental as only nodes that may be affected by the change reinitiate the algorithm. Another important property of our algorithm is that it requires no node identities or sequence numbers. We examine how this idea can be extended to other subgraph structures.     

高连通子图的贪心挖掘算法

利用海量的生物网络数据发现功能模块越来越受到人们的重视,从蛋白质建模的网络图中挖掘高连通子图是其中一个很重要的问题,然而由于数据规模巨大,现有的算法在时间效率上无法胜任实际的应用需求。通过深入研究高连通图的性质定理,设计了一个高连通子图的贪心挖掘算法（HCSGM）算法,在时间复杂度上比HCS算法提高了一个数量级。实验结果表明,HCSGM算法在仿真数据上的挖掘结果优于HCS算法,并且能够从大规模网络图中快速地进行高连通子图挖掘,从而高效地从蛋白质相互作用数据库中挖掘出功能模块。