Showing papers on "Connectivity published in 2013"

Ligra: a lightweight graph processing framework for shared memory

Abstract: There has been significant recent interest in parallel frameworks for processing graphs due to their applicability in studying social networks, the Web graph, networks in biology, and unstructured meshes in scientific simulation. Due to the desire to process large graphs, these systems have emphasized the ability to run on distributed memory machines. Today, however, a single multicore server can support more than a terabyte of memory, which can fit graphs with tens or even hundreds of billions of edges. Furthermore, for graph algorithms, shared-memory multicores are generally significantly more efficient on a per core, per dollar, and per joule basis than distributed memory systems, and shared-memory algorithms tend to be simpler than their distributed counterparts.In this paper, we present a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write. The framework has two very simple routines, one for mapping over edges and one for mapping over vertices. Our routines can be applied to any subset of the vertices, which makes the framework useful for many graph traversal algorithms that operate on subsets of the vertices. Based on recent ideas used in a very fast algorithm for breadth-first search (BFS), our routines automatically adapt to the density of vertex sets. We implement several algorithms in this framework, including BFS, graph radii estimation, graph connectivity, betweenness centrality, PageRank and single-source shortest paths. Our algorithms expressed using this framework are very simple and concise, and perform almost as well as highly optimized code. Furthermore, they get good speedups on a 40-core machine and are significantly more efficient than previously reported results using graph frameworks on machines with many more cores.

Distance Spectra of Graphs: A Survey

Abstract: In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.

Dynamic graph connectivity in polylogarithmic worst case time

Abstract: The dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes which is undergoing a sequence of edge insertions and deletions, answer queries of the form q(a, b): "Is there a path between nodes a and b?" While data structures for this problem with polylogarithmic amortized time per operation have been known since the mid-1990's, these data structures have Θ(n) worst case time. In fact, no previously known solution has worst case time per operation which is o(√n).We present a solution with worst case times O(log4n) per edge insertion, O(log5n) per edge deletion, and O(log n/log log n) per query. The answer to each query is correct if the answer is "yes" and is correct with high probability if the answer is "no". The data structure is based on a simple novel idea which can be used to quickly identify an edge in a cutset.Our technique can be used to simplify and significantly speed up the preprocessing time for the emergency planning problem while matching previous bounds for an update, and to approximate the sizes of cutsets of dynamic graphs in time O(min{|S|, |V\S|}) for an oblivious adversary.

Finding small separators in linear time via treewidth reduction

Abstract: We present a method for reducing the treewidth of a graph while preserving all of its minimal s-t separators up to a certain fixed size k. This technique allows us to solve s-tCut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices.Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H, C, K)- and (H, C,≤K)-coloring problems as well, which are cardinality constrained variants of the classical H-coloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.

Graph measures and network robustness

Abstract: Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness.

Push and rotate: cooperative multi-agent path planning

Abstract: In cooperative multi-agent path planning, agents must move between start and destination locations and avoid collisions with each other. Many recent algorithms require some sort of restriction in order to be complete, except for the Push and Swap algorithm [Luna and Bekris, 2011], which claims only to require two unoccupied locations in a connected graph. Our analysis shows, however, that for certain types of instances Push and Swap may fail to find a solution.We present the Push and Rotate algorithm, an adaptation of the Push and Swap algorithm, and prove that by fixing the latter's shortcomings, we obtain an algorithm that is complete for the class of instances with two unoccupied locations in a connected graph. In addition, we provide experimental results that show our algorithm to perform competitively on a set of benchmark problems from the video game industry.

The k-metric dimension of a graph

Abstract: As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i) e d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.

Representing software defined networks using a programmable graph model

Abstract: System, method, and computer program product to represent a network using a programmable graph model, by generating a directed graph to represent a topology of the network, wherein each of a plurality of network elements in the network are represented, in the directed graph, by one of the plurality of nodes, identifying, through the directed graph, a subset of network elements, of the plurality of network elements, upon which to apply a requested operation, and applying the requested operation to the subset of network elements in a distributed manner through the directed graph.

To stay or not to stay: modeling engagement dynamics in social graphs

Abstract: Given a large social graph, how can we model the engagement properties of nodes? Can we quantify engagement both at node level as well as at graph level? Typically, engagement refers to the degree that an individual participates (or is encouraged to participate) in a community and is closely related to the important property of nodes' departure dynamics, i.e., the tendency of individuals to leave the community. In this paper, we build upon recent work in the field of game theory, where the behavior of individuals (nodes) is modeled by a technology adoption game. That is, the decision of a node to remain engaged in the graph is affected by the decision of its neighbors, and the "best practice" for each individual is captured by its core number - as arises from the k-core decomposition. After modeling and defining the engagement dynamics at node and graph level, we examine whether they depend on structural and topological features of the graph. We perform experiments on a multitude of real graphs, observing interesting connections with other graph characteristics, as well as a clear deviation from the corresponding behavior of random graphs. Furthermore, similar to the well known results about the robustness of real graphs under random and targeted node removals, we discuss the implications of our findings on a special case of robustness - regarding random and targeted node departures based on their engagement level.

Immune networks: multi-tasking capabilities near saturation

Abstract: Pattern-diluted associative networks were introduced recently as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T- and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with $N_T$ T-lymphocytes can manage a number $N_B!=!\order(N_T^\delta)$ of B-lymphocytes simultaneously, with $\delta!

Static Network Reliability Estimation via Generalized Splitting

Abstract: We propose a novel simulation-based method that exploits a generalized splitting GS algorithm to estimate the reliability of a graph or network, defined here as the probability that a given set of nodes are connected, when each link of the graph fails with a given small probability. For large graphs, in general, computing the exact reliability is an intractable problem and estimating it by standard Monte Carlo methods poses serious difficulties, because the unreliability one minus the reliability is often a rare-event probability. We show that the proposed GS algorithm can accurately estimate extremely small unreliabilities and we exhibit large examples where it performs much better than existing approaches. It is also flexible enough to dispense with the frequently made assumption of independent edge failures.

Connectedness index of uncertain graph

Abstract: In practical applications of graph theory, non-deterministic factors are frequently encountered. This paper employs uncertainty theory to deal with non-deterministic factors in problems of graph connectivity. The concepts of uncertain graph and connectedness index of uncertain graph are proposed in this paper. It presents two algorithms to calculate connectedness index of an uncertain graph.

Faster deterministic fully-dynamic graph connectivity

Abstract: We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in O(log2n/log log n) amortized time and connectivity queries in O(log n/log log n) worst-case time, where n is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J. ACM 2001) and Thorup (STOC 2000) which both have O(log2n) amortized update time and O(log n/log log n) worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer machine with standard AC0 instructions.

Neighbor Sum Distinguishing Index

Abstract: We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndiΣ(G). In the paper we conjecture that for any connected graph G ? C 5 of order n ? 3 we have ndiΣ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndiΣ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ? 2 and ndiΣ(G) ≤ 8 if G is cubic.

Immune networks: multitasking capabilities near saturation

Abstract: Pattern-diluted associative networks were recently introduced as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T- and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with NT T-lymphocytes can manage a number NB = O(N δ T ) of B-lymphocytes simultaneously, with δ< 1. Here we study this model in the extensive load regime NB = αNT , with a high degree of pattern dilution, in agreement with immunological findings. We use graph theory and statistical mechanical analysis based on replica methods to show that in the finite-connectivity regime, where each T-lymphocyte interacts with a finite number of B-lymphocytes as NT →∞ , the T-lymphocytes can coordinate effective immune responses to an extensive number of distinct antigen invasions in parallel. As α increases, the system eventually undergoes a second order transition to a phase with clonal cross-talk interference, where the system’s performance degrades gracefully. Mathematically, the model is equivalent to a spin system on a finitely connected graph with many short loops, so one would expect the available analytical methods, which all assume locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results are supported by numerical simulations.

The ABC Index Conundrum

Abstract: The atom-bond connectivity (ABC) index of a graph G is defined as the sum over all pairs of adjacent vertices u;v, of the terms √ (d(u) + d(v) 2)=(d(u)d(v)), where d(v) denotes the degree of the vertex v of the graph G. Whereas the finding of the graphs with the greatest ABC-value is an easy task, the characterization of the graphs with smallest ABC-value, in spite of numerous attempts, is still an open problem. What only is known is that the connected graph with minimal ABC index must be a tree, and some structural features of such trees have been determined. Several conjectures on the structure of the minimal-ABC trees, were disproved by counterexamples. In this review we present the state of art of the search for minimal-ABC trees, and provide a complete bibliography on ABC index.

Some Properties of the Distance Laplacian Eigenvalues of a Graph

Abstract: The distance Laplacian of a connected graph G is defined by \(\mathcal{L} = Diag(Tr) - \mathcal{D}\), where \(\mathcal{D}\) is the distance matrix of G, and Diag(Tr) is the diagonal matrix whose main entries are the vertex transmissions in G. The spectrum of \(\mathcal{L}\) is called the distance Laplacian spectrum of G. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.

Proper Interval Vertex Deletion

Abstract: The NP-complete problem Proper Interval Vertex Deletion is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 6410, pp. 232---243, 2010) showed that this problem can be solved in $\mathcal {O}((14k +14)^{k+1} kn^{6})$ time. We improve this result by presenting an $\mathcal {O}(6^{k} kn^{6})$ time algorithm for Proper Interval Vertex Deletion. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw,net,tent,C4,C5,C6}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves Proper Interval Vertex Deletion on {claw,net,tent,C4,C5,C6}-free graphs in $\mathcal {O}(n+m)$ time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of Proper Interval Vertex Deletion.

On Laplacian Energy

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥· · · ≥ μn−1 >μ n =0 . The Laplacian energy of the graph G is defined as

Extra edge connectivity of hypercube-like networks

Abstract: Extra edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks. Given a graph G and a non-negative integer g , the g - extra edge connectivity of G is the minimum cardinality of an edge subset in G , if it exists, whose deletion disconnects G and causes every remaining component to have more than g vertices. This paper shows that the 3-extra edge connectivity of an n -dimensional hypercube-like network is for .

On the strong metric dimension of Cartesian and direct products of graphs

Abstract: Let $G$ be a connected graph. A vertex $w$ {\em strongly resolves} a pair $u, v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a {\em strong resolving set} for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. The smallest cardinality of a strong resolving set for $G$ is called the {\em strong metric dimension} of $G$. It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian product graphs and direct product graphs.

Relationships between Harmonic Index and other Topological Indices

Abstract: The harmonic index of a connected graph G, denoted by H(G), is defined as H(G )= uv∈E(G) 2 du+dv , where dv is the degree of a vertex v in G. In this note, we established some relationships between harmonic index and several other topological indices. Mathematics Subject Classification: 05C35; 05C12; 05C90

Toward finding hidden communities based on user profile

Abstract: We consider the community detection problem from a partially observable network structure where some edges are not observable. Previous community detection methods are often based solely on the observed connectivity relation and the above situation is not explicitly considered. Even when the connectivity relation is partially observable, if some profile data about the vertices in the network is available, it can be exploited as auxiliary or additional information. We propose to utilize a graph structure (called a profile graph) which is constructed via the profile data, and propose a simple model to utilize both the observed connectivity relation and the profile graph. Furthermore, instead of a hierarchical approach, based on the modularity matrix of the network structure, we propose an embedding approach which utilizes the regularization via the profile graph. Various experiments are conducted over two social network datasets and comparison with several state-of-the-art methods is reported. The results are encouraging and indicate that it is promising to pursue this line of research.

Meyniel's conjecture on the cop number: a survey

Abstract: Meyniel's conjecture is one of the deepest open problems on the cop number of a graph. It states that for a connected graph $G$ of order $n,$ $c(G) = O(\sqrt{n}).$ While largely ignored for over 20 years, the conjecture is receiving increasing attention. We survey the origins of and recent developments towards the solution of the conjecture. We present some new results on Meyniel extremal families containing graphs of order $n$ satisfying $c(G) \ge d\sqrt{n},$ where $d$ is a constant.

On the Connectivity of Sensor Networks Under Random Pairwise Key Predistribution

Abstract: We investigate the connectivity of wireless sensor networks under the random pairwise key predistribution scheme of Chan Under the assumption of full visibility, this reduces to studying the connectivity in the so-called random K-out graph H (n;K); here, n is the number of nodes and K <; n is an integer parameter affecting the number of keys stored at each node. We show that if K ≥ 2 (respectively, K=1), the probability that H (n;K) is a connected graph approaches 1 (respectively, 0) as n goes to infinity. For the one-law this is done by establishing an explicitly computable lower bound on the probability of connectivity. Using this bound, we see that with high probability, network connectivity can already be guaranteed (with K ≥ 2) by a relatively small number of sensors. This corrects earlier predictions made on the basis of a heuristic transfer of connectivity results available for Erdos-Renyi graphs.