Showing papers on "Connectivity published in 2017"

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 the estimation of graph Laplacian matrices from some observed data under given structural constraints (eg, graph connectivity and sparsity level) From a probabilistic perspective, the problems of interest correspond to maximum a posteriori parameter estimation of Gaussian–Markov random field 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

Distributed Time-Varying Quadratic Optimization for Multiple Agents Under Undirected Graphs

Abstract: This paper considers a class of distributed quadratic optimization problem under an undirected and connected graph. Different from most of the existing distributed optimization works that consider the optimal solutions to be constants, the optimal solution and the objective functions at the optimal solution are both assumed to be time varying. For the case where there is no constraint on the decision variables, gradient-based searching methods are proposed to track the unknown optimal solution. The tracking errors are proven to be asymptotically stable. For the case where there exists a local compact convex constraint set for each agent, projected gradient-based methods are proposed for both neighboring coupled and generally coupled objective functions, and the tracking errors are proven to be uniformly ultimately bounded with arbitrarily small bound.

On extra connectivity and extra edge-connectivity of balanced hypercubes

Abstract: The balanced hypercube , as a new variation of the hypercube, possesses many attractive properties such that the hypercube dose not have. Given a connected graph G and a non-negative integer g, the g-extra connectivity resp. g-extra edge-connectivity of G, denoted by resp. , is the minimal cardinality of a set of vertices resp. edges of G, if exists, whose deletion disconnects G and each remaining component contains more than g vertices. In this paper, we show that the 2-extra connectivity of is 4n−4 and 2-extra edge-connectivity of is 6n−4 for . Also, we determine 3-extra connectivity of for .

A novel multi-agent formation control law with collision avoidance

Abstract: In this paper a stable formation control law that simultaneously ensures collision avoidance has been proposed. It is assumed that the communication graph is undirected and connected. The proposed formation control law is a combination of the consensus term and the collision avoidance term U+0028 CAT U+0029. The first order consensus term is derived for the proposed model, while ensuring the Lyapunov stability. The consensus term creates and maintains the desired formation shape, while the CAT avoids the collision. During the collision avoidance, the potential function based CAT makes the agents repel from each other. This unrestricted repelling magnitude cannot ensure the graph connectivity at the time of collision avoidance. Hence we have proposed a formation control law, which ensures this connectivity even during the collision avoidance. This is achieved by the proposed novel adaptive potential function. The potential function adapts itself, with the online tuning of the critical variable associated with it. The tuning has been done based on the lower bound of the critical variable, which is derived from the proposed connectivity property. The efficacy of the proposed scheme has been validated using simulations done based on formations of six and thirty-two agents respectively.

The Complexity of Optimal Design of Temporally Connected Graphs

Abstract: We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u ź v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels.

Are We Connected? Optimal Determination of Source–Destination Connectivity in Random Networks

Abstract: This paper investigates the problem of optimally determining source-destination connectivity in random networks. Viewing the network as a random graph, we start by investigating the Erdos–Renyi (ER) graph, as well as a structured graph where, interesting, the problem appears to be open. The problem examined is that of determining whether a given pair of nodes, a source $S$ , and a destination $D$ are connected by a path. Assuming that at each step one edge can be tested to see if it exists or not, we determine an optimal policy that minimizes the total expected number of steps. The optimal policy has several interesting features. In order to establish the connectivity of $S$ and $D$ , a policy needs to check all edges on some path to see if they all exist, but to establish the disconnectivity it has to check all edges on some cut to see if none of them exists. The optimal policy has the following form. At each step, it examines the condensation multigraph formed by contracting each known connected component to a single node, and then checks an edge that is simultaneously on a shortest $S$ – $D$ path as well as in a minimum $S$ – $D$ cut. Among such edges, it chooses that which lead to the most opportunities for connection. Interestingly, for an ER graph with $n$ nodes, where there is an edge between two nodes with probability $p$ , the optimal strategy does not depend on $p$ or $n$ , even though the entire graph itself undergoes a sharp transition from disconnectivity to connectivity around $p = \ln n/n$ . The policy is efficiently implementable, requiring no more than $30\log _{2} n$ operations to determine which edge to test next. The result also extends to some more general graphs and, meanwhile, provide useful insights into the connectivity determination in random networks.

On Connectivity and Robustness in Random Intersection Graphs

Abstract: Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for $k$ -connectivity and $k$ -robustness, as well as the asymptotically exact probability of $k$ -connectivity, for any positive integer $k$ . The $k$ -connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while $k$ -robustness measures the effectiveness of local-information-based consensus algorithms (which do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, we present asymptotically exact probabilities of $k$ -connectivity and asymptotic zero-one laws for $k$ -robustness.

Complex dynamics of memristive circuits: Analytical results and universal slow relaxation.

Abstract: Networks with memristive elements (resistors with memory) are being explored for a variety of applications ranging from unconventional computing to models of the brain. However, analytical results that highlight the role of the graph connectivity on the memory dynamics are still few, thus limiting our understanding of these important dynamical systems. In this paper, we derive an exact matrix equation of motion that takes into account all the network constraints of a purely memristive circuit, and we employ it to derive analytical results regarding its relaxation properties. We are able to describe the memory evolution in terms of orthogonal projection operators onto the subspace of fundamental loop space of the underlying circuit. This orthogonal projection explicitly reveals the coupling between the spatial and temporal sectors of the memristive circuits and compactly describes the circuit topology. For the case of disordered graphs, we are able to explain the emergence of a power-law relaxation as a superposition of exponential relaxation times with a broad range of scales using random matrices. This power law is also universal, namely independent of the topology of the underlying graph but dependent only on the density of loops. In the case of circuits subject to alternating voltage instead, we are able to obtain an approximate solution of the dynamics, which is tested against a specific network topology. These results suggest a much richer dynamics of memristive networks than previously considered.

Erdźs---Gallai-type results for colorful monochromatic connectivity of a graph

Abstract: A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored with one same color. An edge-coloring of G is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in G. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors used in an MC-coloring of G. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we study two kinds of Erdźs---Gallai-type problems for mc(G), and completely solve them.

On maximum Wiener index of trees and graphs with given radius

Abstract: Let G be a connected graph of order n. The long-standing open and close problems in distance graph theory are: what is the Wiener index W(G) or average distance \(\mu (G)\) among all graphs of order n with diameter d (radius r)? There are very few number of articles where were worked on the relationship between radius or diameter and Wiener index. In this paper, we give an upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and characterize the extremal graphs. Moreover, from this result we give an upper bound on \(\mu (G)\) in terms of order and independence number of graph G. Also we present another upper bound on Wiener index of graphs in terms of number of vertices n, radius r and maximum degree \(\Delta \), and characterize the extremal graphs.

Bounds on the burning numbers of spiders and path-forests

Abstract: Graph burning is one model for the spread of memes and contagion in social networks. The corresponding graph parameter is the burning number of a graph $G$, written $b(G)$, which measures the speed of the social contagion. While it is conjectured that the burning number of a connected graph of order $n$ is at most $\lceil \sqrt{n} \rceil$, this remains open in general and in many graph families. We prove the conjectured bound for spider graphs, which are trees with exactly one vertex of degree at least 3. To prove our result for spiders, we develop new bounds on the burning number for path-forests, which in turn leads to a $\frac 3 2$-approximation algorithm for computing the burning number of path-forests.

Graffinity: Visualizing Connectivity in Large Graphs

Abstract: Multivariate graphs are prolific across many fields, including transportation and neuroscience. A key task in graph analysis is the exploration of connectivity, to, for example, analyze how signals flow through neurons, or to explore how well different cities are connected by flights. While standard node-link diagrams are helpful in judging connectivity, they do not scale to large networks. Adjacency matrices also do not scale to large networks and are only suitable to judge connectivity of adjacent nodes. A key approach to realize scalable graph visualization are queries: instead of displaying the whole network, only a relevant subset is shown. Query-based techniques for analyzing connectivity in graphs, however, can also easily suffer from cluttering if the query result is big enough. To remedy this, we introduce techniques that provide an overview of the connectivity and reveal details on demand. We have two main contributions: 1 two novel visualization techniques that work in concert for summarizing graph connectivity; and 2 Graffinity, an open-source implementation of these visualizations supplemented by detail views to enable a complete analysis workflow. Graffinity was designed in a close collaboration with neuroscientists and is optimized for connectomics data analysis, yet the technique is applicable across domains. We validate the connectivity overview and our open-source tool with illustrative examples using flight and connectomics data.

Exponential Convergence of a Distributed Algorithm for Solving Linear Algebraic Equations

Abstract: In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form $Ax = b$ assuming that the equation has at least one solution. The equation is presumed to be solved by $m$ agents assuming that each agent knows a subset of the rows of the matrix $[A \; b]$, the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph $N(t)$ whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on $N(t)$ were derived under which the algorithm can cause all agents' estimates to converge exponentially fast to the same solution to $Ax = b$. These conditions were also shown to be necessary for exponential convergence, provided the data about $[A \; b]$ available to the agents is "non-redundant". The aim of this paper is to relax this "non-redundant" assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

Distance Domination in Graphs

Abstract: For an integer k ≥ 1, a (distance) k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G) ∖ S is at distance at most k from some vertex of S. The k-domination number, γk(G), of G is the minimum cardinality of a k-dominating set of G. In this chapter, we survey selected results on the k-domination number of a graph.

Critical window for connectivity in the Configuration Model

Abstract: We identify the asymptotic probability of a configuration model CM n (d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

Distributed Bounds on the Algebraic Connectivity of Graphs With Application to Agent Networks

Abstract: This paper establishes several bounds on the algebraic connectivity and spectral radius of graphs. Before deriving these bounds, a directed graph with a leader node is first investigated, for which some bounds on the spectral radius and the smallest real part of all the eigenvalues of $ {M=L+D}$ are obtained using the properties of ${M}$ -matrix and non-negative matrix under a mild assumption, where $L$ is the Laplacian matrix of the graph and $ {D={\mathrm{ diag}}\{d_{1},d_{2},\ldots ,d_{N}\}}$ with $ {d_{i}>0}$ if node $ {i}$ can access the information of the leader node and 0 otherwise. Subsequently, by virtue of the results on directed graphs, the bounds on the algebraic connectivity and spectral radius of an undirected connected graph are provided. Besides establishing these bounds, another important feature is that all these bounds are distributed in the sense of only knowing the information of edge weights’ bounds and the number of nodes in a graph, without using any information of inherent structures of the graph. Therefore, these bounds can be in some sense applied to agent networks for reducing the conservatism where control gains in control protocols depend on the eigenvalues of matrices $ {M}$ or $ {L}$ , which are global information. Also some examples are provided for corroborating the feasibility of the theoretical results.

Optimal subgraph structures in scale-free configuration models

Abstract: Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model.

Partial domination - the isolation number of a graph

Abstract: We prove the following result: If G be a connected graph on n ≥ 6 vertices, then there exists a set of vertices D with │D│≤ n/3 and such that V(G) n N[D] is an independent set, where N[D] is the closed neighborhood of D. Furthermore, the bound is sharp. This seems to be the first result in the direction of partial domination with constrained structure on the graph induced by the non-dominated vertices, which we further elaborate in this paper.

An Adaptive Parallel Algorithm for Computing Connected Components

Abstract: We present an efficient distributed memory parallel algorithm for computing connected components in undirected graphs based on Shiloach-Vishkin’s PRAM approach. We discuss multiple optimization techniques that reduce communication volume as well as load-balance the algorithm. We also note that the efficiency of the parallel graph connectivity algorithm depends on the underlying graph topology. Particularly for short diameter graph components, we observe that parallel Breadth First Search (BFS) method offers better performance. However, running parallel BFS is not efficient for computing large diameter components or large number of small components. To address this challenge, we employ a heuristic that allows the algorithm to quickly predict the type of the network by computing the degree distribution and follow the optimal hybrid route. Using large graphs with diverse topologies from domains including metagenomics, web crawl, social graph and road networks, we show that our hybrid implementation is efficient and scalable for each of the graph types. Our approach achieves a runtime of 215 seconds using 32 K cores of Cray XC30 for a metagenomic graph with over 50 billion edges. When compared against the previous state-of-the-art method, we see performance improvements up to 24 $\times$ .

Steiner Distance in Graphs--A Survey

Abstract: For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. In this paper, we summarize the known results on the Steiner distance parameters, including Steiner distance, Steiner diameter, Steiner center, Steiner median, Steiner interval, Steiner distance hereditary graph, Steiner distance stable graph, average Steiner distance, and Steiner Wiener index. It also contains some conjectures and open problems for further studies.

On 2-Step and Hop Dominating Sets in Graphs

Abstract: Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a 2-step dominating set of G if every vertex is 2-step dominated by some vertex of S. A subset S of vertices of G is a hop dominating set if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $$\gamma _{h}(G)$$ , of G is the minimum cardinality of a hop dominating set of G. It is known that for a connected graph G, $$\gamma _{h}(G) = |V(G)|$$ if and only if G is a complete graph. We characterize the connected graphs G for which $$\gamma _{h}(G) = |V(G)|-1$$ , which answers a question posed by Ayyaswamy and Natarajan [An. Stt. Univ. Ovidius Constanta 23(2):187–199, 2015]. We present probabilistic upper bounds for the hop domination number. We also prove that almost all graphs $$G=G(n,p(n))$$ have a hop dominating set of cardinality at most the total domination number if $$p(n)\ll 1/n$$ , and almost all graphs $$G=G(n,p(n))$$ have a hop dominating set of cardinality at most $$1+np(1+o(1))$$ , if p is constant. We show that the decision problems for the 2-step dominating set and hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.