Showing papers on "Connectivity published in 2016"

Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies

Abstract: We study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs. A frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity. Smoothly varying signals defined on the nodes are of particular interest in various applications, and tend to be approximately bandlimited in the frequency basis. Sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples. Most approaches to this problem require a computation of the frequency basis (i.e., the eigenvectors of the variation operator), followed by a search procedure using the basis elements. This can be impractical, in terms of storage and time complexity, for real datasets involving very large graphs. We circumvent this issue in our formulation by introducing quantities called graph spectral proxies, defined using the powers of the variation operator, in order to approximate the spectral content of graph signals. This allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements. We show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited. Furthermore, the proposed approach is valid for any choice of the variation operator, thereby covering a wide range of graphs and applications. We demonstrate its effectiveness through various numerical experiments.

Consensus of Switched Multiagent Systems

Abstract: In this brief, we consider the consensus problem of a switched multiagent system composed of continuous-time (CT) and discrete-time (DT) subsystems. By combining the classical consensus protocols of CT and DT multiagent systems, we propose a linear consensus protocol for switched multiagent system. Based on the graph theory and the Lyapunov theory, we prove that the consensus of switched multiagent system is solvable under arbitrary switching with undirected connected graph, directed graph, and switching topologies, respectively. Simulation examples are also provided to demonstrate the effectiveness of the theoretical results.

Generalized Connectivity of Graphs

Abstract: Noteworthy results, proof techniques, open problems and conjectures in generalized (edge-) connectivity are discussed in this book. Both theoretical and practical analyses for generalized (edge-) connectivity of graphs are provided. Topics covered in this book include: generalized (edge-) connectivity of graph classes, algorithms, computational complexity, sharp bounds, Nordhaus-Gaddum-type results, maximum generalized local connectivity, extremal problems, random graphs, multigraphs, relations with the Steiner tree packing problem and generalizations of connectivity. This book enables graduate students to understand and master a segment of graph theory and combinatorial optimization. Researchers in graph theory, combinatorics, combinatorial optimization, probability, computer science, discrete algorithms, complexity analysis, network design, and the information transferring models will find this book useful in their studies.

Graph Learning from Data under Laplacian and Structural Constraints

Abstract: Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretations and (iii) associated algorithms. Specifically, graph learning problems are posed as estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori (MAP) parameter estimation of Gaussian-Markov random field (GMRF) models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.

MST in Log-Star Rounds of Congested Clique

Abstract: We present a randomized algorithm that computes a Minimum Spanning Tree (MST) in O(log* n) rounds, with high probability, in the Congested Clique model of distributed computing. In this model, the input is a graph on n nodes, initially each node knows only its incident edges, and per round each two nodes can exchange O(log n) bits. Our key technical novelty is an O(log* n) Graph Connectivity algorithm, the heart of which is a (recursive) forest growth method, based on a combination of two ideas: a sparsity-sensitive sketching aimed at sparse graphs and a random edge sampling aimed at dense graphs. Our result improves significantly over the O(log log log n) algorithm of Hegeman et al. [PODC 2015] and the O(log log n) algorithm of Lotker et al. [SPAA 2003; SICOMP 2005].

Multi-objective compositions for collision-free connectivity maintenance in teams of mobile robots

Abstract: Compositional barrier functions are proposed in this paper to systematically compose multiple objectives for teams of mobile robots. The objectives are first encoded as barrier functions, and then composed using AND and OR logical operators. The advantage of this approach is that compositional barrier functions can provably guarantee the simultaneous satisfaction of all composed objectives. The compositional barrier functions are applied to the example of ensuring collision avoidance and static/dynamical graph connectivity of teams of mobile robots. The resulting composite safety and connectivity barrier certificates are verified experimentally on a team of four mobile robots.

Shuffles and Circuits: (On Lower Bounds for Modern Parallel Computation)

Abstract: The goal of this paper is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in the motivating application domains, such as graph connectivity problems.We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the "fan-in" of each machine is limited to s bits, where s is smaller than the input size n, and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems.We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the "unbounded width" version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any non-trivial monotone graph property --- such as connectivity --- requires a super-constant number of rounds when every machine can accept only a sub-polynomial (in n) number of input bits s.Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds require proving that P ≠ NC1.

Handbook of Graph Theory, Combinatorial Optimization, and Algorithms

Abstract: Basic Concepts and Algorithms Basic Concepts in Graph Theory and Algorithms Subramanian Arumugam and Krishnaiyan "KT" Thulasiraman Basic Graph Algorithms Krishnaiyan "KT" Thulasiraman Depth-First Search and Applications Krishnaiyan "KT" Thulasiraman Flows in Networks Maximum Flow Problem F. Zeynep Sargut, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Minimum Cost Flow Problem Balachandran Vaidyanathan, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Multi-Commodity Flows Balachandran Vaidyanathan, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Algebraic Graph Theory Graphs and Vector Spaces Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Incidence, Cut, and Circuit Matrices of a Graph Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Adjacency Matrix and Signal Flow Graphs Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Adjacency Spectrum and the Laplacian Spectrum of a Graph R. Balakrishnan Resistance Networks, Random Walks, and Network Theorems Krishnaiyan "KT" Thulasiraman and Mamta Yadav Structural Graph Theory Connectivity Subramanian Arumugam and Karam Ebadi Connectivity Algorithms Krishnaiyan "KT" Thulasiraman Graph Connectivity Augmentation Andras Frank and Tibor Jordan Matchings Michael D. Plummer Matching Algorithms Krishnaiyan "KT" Thulasiraman Stable Marriage Problem Shuichi Miyazaki Domination in Graphs Subramanian Arumugam and M. Sundarakannan Graph Colorings Subramanian Arumugam and K. Raja Chandrasekar Planar Graphs Planarity and Duality Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Edge Addition Planarity Testing Algorithm John M. Boyer Planarity Testing Based on PC-Trees Wen-Lian Hsu Graph Drawing Md. Saidur Rahman and Takao Nishizeki Interconnection Networks Introduction to Interconnection Networks S.A. Choudum, Lavanya Sivakumar, and V. Sunitha Cayley Graphs S. Lakshmivarahan, Lavanya Sivakumar, and S.K. Dhall Graph Embedding and Interconnection Networks S.A. Choudum, Lavanya Sivakumar, and V. Sunitha Special Graphs Program Graphs Krishnaiyan "KT" Thulasiraman Perfect Graphs Chinh T. Hoang and R. Sritharan Tree-Structured Graphs Andreas Brandstadt and Feodor F. Dragan Partitioning Graph and Hypergraph Partitioning Sachin B. Patkar and H. Narayanan Matroids Matroids H. Narayanan and Sachin B. Patkar Hybrid Analysis and Combinatorial Optimization H. Narayanan Probabilistic Methods, Random Graph Models, and Randomized Algorithms Probabilistic Arguments in Combinatorics C.R. Subramanian Random Models and Analyses for Chemical Graphs Daniel Pascua, Tina M. Kouri, and Dinesh P. Mehta Randomized Graph Algorithms: Techniques and Analysis Surender Baswana and Sandeep Sen Coping with NP-Completeness General Techniques for Combinatorial Approximation Sartaj Sahni epsilon-Approximation Schemes for the Constrained Shortest Path Problem Krishnaiyan "KT" Thulasiraman Constrained Shortest Path Problem: Lagrangian Relaxation-Based Algorithmic Approaches Ying Xiao and Krishnaiyan "KT" Thulasiraman Algorithms for Finding Disjoint Paths with QoS Constraints Alex Sprintson and Ariel Orda Set-Cover Approximation Neal E. Young Approximation Schemes for Fractional Multicommodity Flow Problems George Karakostas Approximation Algorithms for Connectivity Problems Ramakrishna Thurimella Rectilinear Steiner Minimum Trees Tao Huang and Evangeline F.Y. Young Fixed-Parameter Algorithms and Complexity Venkatesh Raman and Saket Saurabh

EdgeCentric: Anomaly Detection in Edge-Attributed Networks

Abstract: Given a network with attributed edges, how can we identify anomalous behavior? Networks with edge attributes are ubiquitous, and capture rich information about interactions between nodes. In this paper, we aim to utilize exactly this information to discern suspicious from typical behavior in an unsupervised fashion, lending well to the traditional scarcity of ground-truth labels in practical anomaly detection scenarios. Our work has a number of notable contributions, including (a) formulation: while most other graph-based anomaly detection works use structural graph connectivity or node information, we focus on the new problem of leveraging edge information, (b) methodology: we introduce EdgeCentric, an intuitive and scalable compression-based approach for detecting edge-attributed graph anomalies, and (c) practicality: we show that EdgeCentric successfully spots numerous such anomalies in several large, edge-attributed real-world graphs, including the Flipkart e-commerce graph with over 3 million product reviews between 1.1 million users and 545 thousand products, where it achieved 0.87 precision over the top 100 results.

Power System Connectivity Monitoring Using a Graph Theory Network Flow Algorithm

Abstract: A method of applying network flow analyses during real time power system operation, to provide better network connectivity visualization, is developed and presented. Graph theory network flow analysis is capable of determining the maximum flow that can be transported between two nodes within a directed graph. These network flow algorithms are applied to a graphical representation of a power system topology to determine the minimum number of system branches needed to be lost in order to guarantee disconnecting the two nodes in the system that are selected. The number of system branches that are found serves as an approximate indicator of system vulnerabilities. The method used in these connectivity analyses makes use of well known graph theory network flow maximum flow algorithms, but also introduces a new algorithm for updating an old network flow solution for the loss of only a single system branch. The proposed new algorithm allows for significantly decreased solution time that is desired in a real-time environment. The value of using the proposed method is illustrated by using a detailed example of the 2008 island formation that occurred in the Entergy power system. The method was applied to a recreation of the 2008 event using a 20,000-bus model of the Entergy system to show both the proposed method's benefits as well as practicality of implementation.

Graph connectivity and binomial edge ideals

Abstract: We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.

On the distance signless Laplacian of a graph

Abstract: The distance signless Laplacian of a connected graph is defined by , where is the distance matrix of , and is the diagonal matrix whose main entries are the vertex transmissions in . The spectrum of is called the distance signless Laplacian spectrum of . In the present paper, we study some properties of the distance signless Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance signless Laplacian eigenvalues. We prove several bounds on eigenvalues and establish a relationship between being a distance signless Laplacian eigenvalue of and containing a bipartite component.

Graph Learning from Data under Structural and Laplacian Constraints

Abstract: Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretations and (iii) associated algorithms. Specifically, graph learning problems are posed as estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori (MAP) parameter estimation of Gaussian-Markov random field (GMRF) models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.

On Computing the Edge-Connectivity of an Uncertain Graph

Abstract: Uncertain graphs are employed to describe graph models with imprecise expert data. In uncertain graphs, due to the existence of uncertain edges, edge-connectivity is essentially an uncertain variable. Different from that in a deterministic graph, it is more suitable to investigate the possibility (uncertain measure) that an uncertain graph is $k$ -edge-connected, which is the main aim of this paper. We first deduce a theoretical formula to calculate the uncertain measure, and on this basis, we then propose an algorithm, which is derived from maximum flow algorithms, to numerically calculate the uncertain measure. The proposed algorithm is also proved to be a polynomial time algorithm, and its effectiveness and efficiency are illustrated by numerical examples.

Hybrid constructive heuristics for the critical node problem

Abstract: We consider the Critical Node Problem: given an undirected graph and an integer number K, at most K nodes have to be deleted from the graph in order to minimize a connectivity measure in the residual graph. We combine the basic steps used in common greedy algorithms with some flavour of local search, in order to obtain simple hybrid heuristic algorithms. The obtained algorithms are shown to be effective, delivering improved performances (solution quality and speed) with respect to known greedy algorithms and other more sophisticated state of the art methods.

Graph Connectivity in Noisy Sparse Subspace Clustering

Abstract: Subspace clustering is the problem of clustering data points into a union of lowdimensional linear/ane subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work [4, 19, 24, 20] provided strong theoretical guarantee for sparse subspace clustering [4], the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from dierent subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious “graph connectivity problem” [15]. In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple postprocessing procedure is capable of delivering consistent clustering under certain “general position” or “restricted eigenvalue” assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3.

Automated drawing of network plots in network meta-analysis.

Abstract: In systematic reviews based on network meta-analysis, the network structure should be visualized. Network plots often have been drawn by hand using generic graphical software. A typical way of drawing networks, also implemented in statistical software for network meta-analysis, is a circular representation, often with many crossing lines. We use methods from graph theory in order to generate network plots in an automated way. We give a number of requirements for graph drawing and present an algorithm that fits prespecified ideal distances between the nodes representing the treatments. The method was implemented in the function netgraph of the R package netmeta and applied to a number of networks from the literature. We show that graph representations with a small number of crossing lines are often preferable to circular representations.

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.

On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

Abstract: A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G . It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3 , every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

Computing Top-k Closeness Centrality Faster in Unweighted Graphs

Abstract: Given a connected graph G = (V,E), the closeness centrality of a vertex v is defined as (n-1 / \Sigma_{w \in V} d(v,w). This measure is widely used in the analysis of real-world complex networks, and the problem of selecting the k most central vertices has been deeply analysed in the last decade. However, this problem is computationally not easy, especially for large networks: in the first part of the paper, we prove that it is not solvable in time O(|E|^{2-epsilon) on directed graphs, for any constant epsilon > 0, under reasonable complexity assumptions. Furthermore, we propose a new algorithm for selecting the k most central nodes in a graph: we experimentally show that this algorithm improves significantly both the textbook algorithm, which is based on computing the distance between all pairs of vertices, and the state of the art. For example, we are able to compute the top k nodes in few dozens of seconds in real-world networks with millions of nodes and edges. Finally, as a case study, we compute the 10 most central actors in the IMDB collaboration network, where two actors are linked if they played together in a movie, and in the Wikipedia citation network, which contains a directed edge from a page p to a page q if p contains a link to q.

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences†

Abstract: Let G be a simple undirected connected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a proper Δ-coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k-coloring, inline image, a Δ-coloring of G can be obtained by a sequence of inline image recolorings using only the original k colors unless – G is a complete graph or a cycle with an odd number of vertices, or – inline image, G is Δ-regular and, for each vertex v in G, no two neighbors of v are colored alike. We use this result to study the reconfiguration graph inline image of the k-colorings of G. The vertex set of inline image is the set of all possible k-colorings of G and two colorings are adjacent if they differ on exactly one vertex. We prove that for inline image, inline image consists of isolated vertices and at most one further component that has diameter inline image. This result enables us to complete both a structural and an algorithmic characterization for reconfigurations of colorings of graphs of bounded maximum degree.

A New Characterization of $$P_k$$Pk-Free Graphs

Abstract: Let $$G$$G be a connected $$P_k$$Pk-free graph, $$k \ge 4$$k?4. We show that $$G$$G admits a connected dominating set that induces either a $$P_{k-2}$$Pk-2-free graph or a graph isomorphic to $$P_{k-2}$$Pk-2. In fact, every minimum connected dominating set of $$G$$G has this property. This yields a new characterization for $$P_k$$Pk-free graphs: a graph $$G$$G is $$P_k$$Pk-free if and only if each connected induced subgraph of $$G$$G has a connected dominating set that induces either a $$P_{k-2}$$Pk-2-free graph, or a graph isomorphic to $$C_k$$Ck. We present an efficient algorithm that, given a connected graph $$G$$G, computes a connected dominating set $$X$$X of $$G$$G with the following property: for the minimum $$k$$k such that $$G$$G is $$P_k$$Pk-free, the subgraph induced by $$X$$X is $$P_{k-2}$$Pk-2-free or isomorphic to $$P_{k-2}$$Pk-2. As an application our results, we prove that Hypergraph 2-Colorability can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graphs is $$P_7$$P7-free.

Deep Haar scattering networks

Abstract: An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet transforms on the graph, with Haar wavelets having connected supports. It defines a representation which is invariant to local displacements of signal values on the graph. When the graph connectivity is unknown, unsupervised Haar learning can provide a consistent estimation of connected wavelet supports. Classification results are given on image data bases, defined on regular grids or graphs, with a connectivity which may be known or unknown.