Connected Subgraph Research Articles

Mining Evolving Patterns of Connection Subgraphs over Evolving Graphs

探讨演变图(即随时间变化的图)的挖掘,重点研究在演变图中挖掘连接子图的演变模式集合.提出一种连 接子图的相似度函数及其快速计算算法.基于该相似度函数,提出一种发现演变模式集合的多项式时间复杂度的动 态规划算法.模拟数据集上的实验结果表明,该算法具有较低的误差率和较高的效率.真实数据集上的实验结果表 明,挖掘结果在真实应用中具有实际意义.;This paper investigates into the problem of mining evolving graphs, i.e. graphs changing over time. It focuses on mining evolving pattern set of connection subgraphs between given vertices in an evolving graph. A similarity function of connection subgraphs and the algorithm to compute it have been presented. Based on this similarity function, a dynamic programming algorithm with polynomial time complexity is proposed to find evolving pattern set. The experimental results on synthetic datasets show that the proposed algorithm has low error rate and high efficiency. The mining results on real datasets illustrate that the mining results have practical significance in real applications.

An ILP formulation and genetic algorithm for the Maximum Degree-Bounded Connected Subgraph problem

A general instance of a Degree-Constrained Subgraph problem may be found in an edge-weighted or vertex-weighted graph G whereas the objective is to find an optimal weighted subgraph, subject to certain degree constraints on the vertices of the subgraph. This class of combinatorial problems has been extensively studied due to its numerous applications in network design. If the input graph is bipartite, these problems are equivalent to classical transportation and assignment problems in operational research. This paper is an illustration of a research of the N P -hard Maximum Degree-Bounded Connected Subgraph problem (MDBCS). It is a classical N P -complete problem. Moreover this paper offers a first integer linear programming formulation of the (MDBCS), and a formal proof that it is correct. A genetic algorithm for obtaining the optimal solution of (MDBCS) has also been provided. The proposed solution comprises a genetic algorithm (GA) that uses binary representation, fine-grained tournament selection, one-point crossover, simple mutation with frozen genes and caching technique.

Subgraph queries by context-free grammars.

We describe a method for querying vertex- and edge-labeled graphs using context-free grammars to specify the class of interesting paths. We introduce a novel problem, finding the connection subgraph induced by the set of matching paths between given two vertices or two sets of vertices. Such a subgraph provides a concise summary of the relationship between the vertices. We also present novel algorithms for parsing subgraphs directly without enumerating all the individual paths. We evaluate experimentally the presented parsing algorithms on a set of real graphs derived from publicly available biomedical databases and on randomly generated graphs. The results indicate that parsing the connection subgraph directly is much more effective than parsing individual paths separately. Furthermore, we show that using a bidirectional parsing algorithm, in most cases, allows for searching twice as long paths as using a unidirectional search strategy.

Subgraph Queries by Context-free Grammars

SummaryWe describe a method for querying vertex- and edge-labeled graphs using context-free grammars to specify the class of interesting paths. We introduce a novel problem, finding the connection subgraph induced by the set of matching paths between given two vertices or two sets of vertices. Such a subgraph provides a concise summary of the relationship between the vertices. We also present novel algorithms for parsing subgraphs directly without enumerating all the individual paths. We evaluate experimentally the presented parsing algorithms on a set of real graphs derived from publicly available biomedical databases and on randomly generated graphs. The results indicate that parsing the connection subgraph directly is much more effective than parsing individual paths separately. Furthermore, we show that using a bidirectional parsing algorithm, in most cases, allows for searching twice as long paths as using a unidirectional search strategy.

Measuring and extracting proximity graphs in networks

Measuring distance or some other form of proximity between objects is a standard data mining tool. Connection subgraphs were recently proposed as a way to demonstrate proximity between nodes in networks. We propose a new way of measuring and extracting proximity in networks called “cycle-free effective conductance” (CFEC). Importantly, the measured proximity is accompanied with a proximity subgraph which allows assessing and understanding measured values. Our proximity calculation can handle more than two endpoints, directed edges, is statistically well behaved, and produces an effectiveness score for the computed subgraphs. We provide an efficient algorithm to measure and extract proximity. Also, we report experimental results and show examples for four large network datasets: a telecommunications calling graph, the IMDB actors graph, an academic coauthorship network, and a movie recommendation system.

Random walk with restart: fast solutions and applications

How closely related are two nodes in a graph? How to compute this score quickly, on huge, disk-resident, real graphs? Random walk with restart (RWR) provides a good relevance score between two nodes in a weighted graph, and it has been successfully used in numerous settings, like automatic captioning of images, generalizations to the “connection subgraphs”, personalized PageRank, and many more. However, the straightforward implementations of RWR do not scale for large graphs, requiring either quadratic space and cubic pre-computation time, or slow response time on queries. We propose fast solutions to this problem. The heart of our approach is to exploit two important properties shared by many real graphs: (a) linear correlations and (b) block-wise, community-like structure. We exploit the linearity by using low-rank matrix approximation, and the community structure by graph partitioning, followed by the Sherman–Morrison lemma for matrix inversion. Experimental results on the Corel image and the DBLP dabasets demonstrate that our proposed methods achieve significant savings over the straightforward implementations: they can save several orders of magnitude in pre-computation and storage cost, and they achieve up to 150 × speed up with 90%+ quality preservation.

2-Connected spanning subgraphs with low maximum degree in locally planar graphs

In this paper, we prove that there exists a function a : N 0 × R + → N such that for each ε > 0 , if G is a 4-connected graph embedded on a surface of Euler genus k such that the face-width of G is at least a ( k , ε ) , then G has a 2-connected spanning subgraph with maximum degree at most 3 in which the number of vertices of degree 3 is at most ε | V ( G ) | . This improves results due to Kawarabayashi, Nakamoto and Ota [K. Kawarabayashi, A. Nakamoto, K. Ota, Subgraphs of graphs on surfaces with high representativity, J. Combin. Theory Ser. B 89 (2003) 207–229], and Böhme, Mohar and Thomassen [T. Böhme, B. Mohar, C. Thomassen, Long cycles in graphs on a fixed surface, J. Combin. Theory Ser. B 85 (2002) 338–347].

Discovering informative connection subgraphs in multi-relational graphs

Discovering patterns in graphs has long been an area of interest. In most approaches to such pattern discovery either quantitative anomalies, frequency of substructure or maximum flow is used to measure the interestingness of a pattern. In this paper we introduce heuristics that guide a subgraph discovery algorithm away from banal paths towards more "informative" ones. Given an RDF graph a user might pose a question of the form: "What are the most relevant ways in which entity X is related to entity Y?" the response to which is a subgraph connecting X to Y. We use our heuristics to discover informative subgraphs within RDF graphs. Our heuristics are based on weighting mechanisms derived from edge semantics suggested by the RDF schema. We present an analysis of the quality of the subgraphs generated with respect to path ranking metrics. We then conclude presenting intuitions about which of our weighting schemes and heuristics produce higher quality subgraphs.

On the existence of [formula omitted] edge-disjoint 2-connected spanning subgraphs

We prove that every 6 k -connected graph contains k edge-disjoint rigid (and hence 2-connected) spanning subgraphs. By using this result we can settle special cases of two conjectures, due to Kriesell and Thomassen, respectively: we show that every 12 -connected graph G has a spanning tree T for which G - E ( T ) is 2 -connected, and that every 18 -connected graph has a 2 -connected orientation.

On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality

In this paper we investigate the problem of finding a 2-connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, for both the edge and the vertex connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [ 1 , 1 + ε ] , for an arbitrary small ε > 0 . This result implies the first explicit lower bound on the approximability of the general version (i.e., for arbitrary graphs) of the problem. On the other hand, if the input graph satisfies the sharpened β -triangle inequality, then a 2 3 + 1 3 · β 1 - β -approximation algorithm is designed. This ratio tends to 1 with β tending to 1 2 , and it improves the previous known bound of 3 2 , holding for graphs satisfying the triangle inequality, as soon as β < 5 7 . Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened β -triangle inequality with β < 2 3 , then this ratio can be improved to β 1 - β .

2-connected graphs with small 2-connected dominating sets

Let G be a 2-connected graph. A subset D of V( G) is a 2- connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ 2( G) denote the minimum size of a 2-connected dominating set of G. Let δ( G) be the minimum degree of G. For an n-vertex graph G, we prove that γ 2(G)⩽n ln δ(G) δ(G) (1+o δ(1)) where o δ (1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3–8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202–211, Lovász, Discrete Math. 13 (1975) 383–390 and Payan, Cahièrs Centre Etudes Rech. Opér. 17 (1975) 307–317).

Approximating minimum size {1,2}-connected networks

The problem of finding the minimum size 2-connected subgraph is a classical problem in network design. It is known to be NP-hard even on cubic planar graphs and MAX SNP-hard. We study the generalization of this problem, where requirements of 1 or 2 edge or vertex disjoint paths are specified between every pair of vertices, and the aim is to find a minimum size subgraph satisfying these requirements. For both problems we give 3 2 -approximation algorithms. This improves on the straightforward 2-approximation algorithms for these problems, and generalizes earlier results for 2-connectivity. We also give analyses of the classical local optimization heuristics for these two network design problems.

Steiner k-Edge Connected Subgraph Polyhedra

In this paper we consider the Steiner k-edge survivable network problem. We discuss the polytope associated with the solutions to that problem. We show that when the graph is series-parallel and k is even, the polytope is completely described by the trivial constraints and the so called Steiner-cut constraints. This generalizes recent work of Baiou and Mahjoub, SIAM J. Discrete Mathematics, vol. 10, pp. 505–514, 1997 for the case k = 2. As a consequence, we obtain in this case a linear description of the polyhedron associated with the problem when multiple copies of an edge are allowed.

Recognizing the P4-structure of block graphs

A 4-uniform hypergraph represents the P 4-structure of a graph G, if its hyperedges are the vertex sets of the induced paths P 4 in G. We shall give in this paper a simple algorithm that recognizes the P 4-structure of a block graph in polynomial time. Here, block graphs are connected graphs in which all maximal 2-connected subgraphs are cliques. Our algorithm is based on a similar approach as that for trees by the authors and S. Olariu using weighted 2-section graphs of hypergraphs.

On 2-Connected Spanning Subgraphs with Low Maximum Degree

Given a graph G , let a k -trestle of G be a 2-connected spanning subgraph of G of maximum degree at most k . Also, let χ ( G ) be the Euler characteristic of G . This paper shows that every 3-connected graph G has a (10−2 χ ( G ))-trestle. If χ ( G )⩽−5, this is improved to 8−2 χ ( G ), and for χ ( G )⩽−10, this is further improved to 6−2 χ ( G ). This result is shown to be best possible for almost all values of χ ( G ) by the demonstration of 3-connected graphs embedded on each surface of Euler characteristic χ ⩽0 which have no (5−2 χ )-trestle. Also, it is shown that a 4-connected graph embeddable on a surface with non-negative Euler characteristic has a 3-trestle, approaching a conjecture of Nash-Williams.

Strong orientations without even directed circuits

We characterize the graphs for which all 2-connected non-bipartite subgraphs have a strongly connected orientation in which each directed circuit has an odd number of edges. We also give a polynomial-time algorithm to find such an orientation in these graphs. Moreover, we give an algorithm that given any orientation of such a graph, determines if it has an even directed circuit. The proofs of these results are based on a constructive characterization of these graphs.

Steiner 2-Edge Connected Subgraph Polytopes on Series-Parallel Graphs

Given a graph G=(V,E) with weights on its edges and a set of specified nodes $S\subseteq V$, the Steiner 2-edge survivable network problem is to find a minimum weight subgraph of G such that between every two nodes of S there are at least two edge-disjoint paths. This problem has applications to the design of reliable communication and transportation networks. In this paper, we give a complete linear description of the polytope associated with the solutions to this problem when the underlying graph is series-parallel. We also discuss related polyhedra.

On finding two-connected subgraphs in planar graphs

We consider a basic problem in the design of reliable networks, namely, that of finding a minimum-weight 2-connected subgraph spanning a given set K of vertices in a planar graph. We show a relationship between this problem and that of finding shortest trails and cycles enclosing K, and derive a polynomial-time algorithm in the case when the vertices of K all lie on the same face.

The dominant of the 2-connected-Steiner-subgraph polytope for W4-free graphs

This paper presents a linear-inequality description of the dominant of the polytope of the 2-connected Steiner subgraphs of a given W 4-free graph. For the special case of 2-connected spanning subgraphs, a description of the polytope is given. The latter contains the Traveling-Salesman polytope for W 4-free graphs as a face.

2-Connected Spanning Subgraphs of Planar 3-Connected Graphs

We prove that every planar 3-connected graph has a 2-connected spanning subgraph of maximum valence 15. We give an example of a planar 3-connected graph with no spanning 2-connected subgraph of maximum valence five.