Showing papers on "Connectivity published in 2009"

On a novel connectivity index

Abstract: We present a novel connectivity index for (molecular) graphs, called sum-connectivity index and give several basic properties for this index, especially lower and upper bounds in terms of graph (structural) invariants. It appears that this and the original Randic connectivity index that we call product-connectivity index are highly intercorrelated molecular descriptors, the value of the correlation coefficient being 0.991 for trees representing lower alkanes. We determine the unique tree with fixed numbers of vertices and pendant vertices with the minimum value of the sum-connectivity index, and trees with the minimum, second minimum and third minimum, and the maximum, second maximum and third maximum values of this index. Additionally, we discuss the properties of this novel connectivity index for a class of trees representing acyclic hydrocarbons.

Supervised pattern classification based on optimum-path forest

Abstract: We present a supervised classification method which represents each class by one or more optimum-path trees rooted at some key samples, called prototypes. The training samples are nodes of a complete graph, whose arcs are weighted by the distances between the feature vectors of their nodes. Prototypes are identified in all classes and the minimization of a connectivity function by dynamic programming assigns to each training sample a minimum-cost path from its most strongly connected prototype. This competition among prototypes partitions the graph into an optimum-path forest rooted at them. The class of the samples in an optimum-path tree is assumed to be the same of its root. A test sample is classified similarly, by identifying which tree would contain it, if the sample were part of the training set. By choice of the graph model and connectivity function, one can devise other optimum-path forest classifiers. We present one of them, which is fast, simple, multiclass, parameter independent, does not make any assumption about the shapes of the classes, and can handle some degree of overlapping between classes. We also propose a general algorithm to learn from errors on an evaluation set without increasing the training set, and show the advantages of our method with respect to SVM, ANN-MLP, and k-NN classifiers in several experiments with datasets of various types. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 120–131, 2009. A preliminary version of the paper was presented at the 12th International Workshop on Combinatorial Image Analysis (Papa et al.,2008a).

Conditions for Synchronizability in Arrays of Coupled Linear Systems

Abstract: Synchronization in arrays of coupled continuous-time linear systems is studied. Sufficiency of certain conditions for the existence of a synchronizing feedback law are analyzed. It is shown that, for neutrally stable systems that are detectable from their outputs, a linear feedback law exists under which any number of coupled systems synchronize provided that the (directed, weighted) graph describing the interconnection is fixed and connected. An algorithm generating one such feedback law as well as the trajectory that the solutions converge to are presented. It is also shown that, for critically unstable systems, detectability is not sufficient, whereas full-state coupling is, for the existence of a linear feedback law that is synchronizing for all coupling configurations described by a connected graph.

Coordinated Path-Following in the Presence of Communication Losses and Time Delays

Abstract: This paper addresses the problem of steering a group of vehicles along given spatial paths while holding a desired time-varying geometrical formation pattern. The solution to this problem, henceforth referred to as the coordinated path-following (CPF) problem, unfolds in two basic steps. First, a path-following (PF) control law is designed to drive each vehicle to its assigned path, with a nominal speed profile that may be path dependent. This is done by making each vehicle approach a virtual target that moves along the path according to a conveniently defined dynamic law. In the second step, the speeds of the virtual targets (also called coordination states) are adjusted about their nominal values so as to synchronize their positions and achieve, indirectly, vehicle coordination. In the problem formulation, it is explicitly considered that each vehicle transmits its coordination state to a subset of the other vehicles only, as determined by the communications topology adopted. It is shown that the system that is obtained by putting together the PF and coordination subsystems can be naturally viewed as either the feedback or the cascade connection of the latter two. Using this fact and recent results from nonlinear systems and graph theory, conditions are derived under which the PF and the coordination errors are driven to a neighborhood of zero in the presence of communication losses and time delays. Two different situations are considered. The first captures the case where the communication graph is alternately connected and disconnected (brief connectivity losses). The second reflects an operational scenario where the union of the communication graphs over uniform intervals of time remains connected (uniformly connected in mean). To better root the paper in a nontrivial design example, a CPF algorithm is derived for multiple underactuated autonomous underwater vehicles (AUVs). Simulation results are presented and discussed.

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Abstract: Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics, and threshold phenomena. Recent work on heuristics and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and $st$-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side—which includes but is not limited to all problems with polynomial-time algorithms for satisfiability—is in P for the $st$-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, diameter and complexity of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

Hardness and Algorithms for Rainbow Connection

Abstract: An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In the first result of this paper we prove that computing $rc(G)$ is NP-Hard solving an open problem from \cite{Ca-Yu}. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon >0$, a connected graph with minimum degree at least $\epsilon n$ has {\em bounded} rainbow connection, where the bound depends only on $\epsilon$, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

Maintaining Limited-Range Connectivity Among Second-Order Agents

Abstract: In this paper we consider ad-hoc networks of robotic agents with double integrator dynamics. For such networks, the connectivity maintenance problems are as follows: (i) Do there exist control inputs for each agent to maintain network connectivity, and (ii) given desired controls for each agent, can we compute the closest connectivity-maintaining controls in a distributed fashion? The proposed solution is based on three contributions. First, we define and characterize admissible sets for double integrators to remain inside disks. Second, we establish an existence theorem for the connectivity maintenance problem by introducing a novel state-dependent graph, called the double-integrator disk graph. Specifically, we show that one can always maintain connectivity by maintaining a spanning tree of this new graph, but one will not always maintain connectivity of a particular agent pair that happens to be connected at one instant of time. Finally, we design a distributed “flow-control” algorithm for distributed computation of connectivity-maintaining controls.

Hardness and Algorithms for Rainbow Connectivity

Abstract: An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connectivity} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing $rc(G)$ is NP-Hard. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon >0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connectivity, where the bound depends only on $\epsilon$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

Rainbow Connection in Graphs with Minimum Degree Three

Abstract: An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that $rc(G) < \frac{3n}{4}$ for graphs with minimum degree three, which was conjectured by Caro et al. [Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster, On rainbow connection, The Electronic Journal of Combinatorics 15 (2008), #57.]

Moving formation convergence of a group of mobile robots via decentralised information feedback

Abstract: In this article, a decentralised information feedback mechanism is introduced to a group of mobile robots such that the robots asymptotically converge to a given moving formation. It is assumed that the robots can exchange only position information according to a pre-specified communication graph. Each node represents a robot. Two robots are neighbours of each other if there is an edge between the two nodes. A feedback controller is performed for each robot by only using its own velocity information and the position information from its neighbours. It is proven that if the graph is connected, then the convergence to the moving formation of the closed-loop system is guaranteed. Several numerical simulations are presented to illustrate the results.

Graph Generation with Prescribed Feature Constraints.

Abstract: In this paper, we study the problem of how to generate synthetic graphs matching various properties of a real social network with two applications, privacy preserving social network publishing and significance testing of network analysis results. We present a simple switching based graph generation approach to generate graphs preserving features of a real graph. We then investigate potential disclosures of sensitive links due to the preserved features. Our algorithms on graph generation with feature range and feature distribution constraints are based on the Metropolis-Hastings sampling. This is of importance for significance testing of network analysis results.

Trading off space for passes in graph streaming problems

Abstract: Data stream processing has recently received increasing attention as a computational paradigm for dealing with massive data sets. Surprisingly, no algorithm with both sublinear space and passes is known for natural graph problems in classical read-only streaming. Motivated by technological factors of modern storage systems, some authors have recently started to investigate the computational power of less restrictive models where writing streams is allowed. In this article, we show that the use of intermediate temporary streams is powerful enough to provide effective space-passes tradeoffs for natural graph problems. In particular, for any space restriction of s bits, we show that single-source shortest paths in directed graphs with small positive integer edge weights can be solved in O((n log3/2n)/√s) passes. The result can be generalized to deal with multiple sources within the same bounds. This is the first known streaming algorithm for shortest paths in directed graphs. For undirected connectivity, we devise an O((n log n)/s) passes algorithm. Both problems require Ω(n/s) passes under the restrictions we consider. We also show that the model where intermediate temporary streams are allowed can be strictly more powerful than classical streaming for some problems, while maintaining all of its hardness for others.

Persistent neural states: Stationary localized activity patterns in nonlinear continuous n-population, q-dimensional neural networks

Abstract: Neural continuum networks are an important aspect of the modeling of macroscopic parts of the cortex. Two classes of such networks are considered: voltage and activity based. In both cases, our networks contain an arbitrary number, n, of interacting neuron populations. Spatial nonsymmetric connectivity functions represent cortico-cortical, local connections, and external inputs represent nonlocal connections. Sigmoidal nonlinearities model the relationship between (average) membrane potential and activity. Departing from most of the previous work in this area, we do not assume the nonlinearity to be singular, that is, represented by the discontinuous Heaviside function. Another important difference from previous work is that we relax the assumption that the domain of definition where we study these networks is infinite, that is, equal to or . We explicitly consider the biologically more relevant case of a bounded subset Ω of , a better model of a piece of cortex. The time behavior of these networks is described by systems of integro-differential equations. Using methods of functional analysis, we study the existence and uniqueness of a stationary (i.e., time-independent) solution of these equations in the case of a stationary input. These solutions can be seen as 'persistent'; they are also sometimes called bumps. We show that under very mild assumptions on the connectivity functions and because we do not use the Heaviside function for the nonlinearities, such solutions always exist. We also give sufficient conditions on the connectivity functions for the solution to be absolutely stable, that is, independent of the initial state of the network. We then study the sensitivity of the solutions to variations of such parameters as the connectivity functions, the sigmoids, the external inputs, and, last but not least, the shape of the domain of existence Ω of the neural continuum networks. These theoretical results are illustrated and corroborated by a large number of numerical experiments in most of the cases 2 ≤ n ≤ 3, 2 ≤ q ≤ 3.

Comparison of network criticality, algebraic connectivity, and other graph metrics

Abstract: The study of robustness and connectivity properties are important in the analysis of complex networks. This paper reports on an effort to compare different network topologies according to their algebraic connectivity, network criticality, average node degree, and average node betweenness. We consider different network types and study the behavior of these various metrics as scale is increased. Based on extensive simulations, we suggest some guidelines for the design and simplification of networks. The main finding is that, algebraic connectivity, network criticality, average degree, and average node betweenness capture different properties of a graph. Depending on the nature of the problem at hand, one needs to select which one is appropriate to use as the main metric for network analysis.

A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2

Abstract: We present a 1.8-approximation algorithm for the following NP-hard problem: Given a connected graph G = (V, E) and an edge set E on V disjoint to E, find a minimum-size subset of edges F ⊆ E such that (V, E ∪ F) is 2-edge-connected. Our result improves and significantly simplifies the approximation algorithm with ratio 1.875 + e of Nagamochi.

Fast Recovery from Dual Link Failures in IP Networks

Abstract: This paper develops a novel mechanism for recover- ing from dual link failures in IP networks. The highlight of the developed routing approach is that a node re-routes a packet around the failed link without the knowledge of the second link failure. The proposed technique requires three protection addresses for every node, in addition to the normal address. Associated with every protection address of a node is a protection graph. Each link connected to the node is removed in at least one of protection graphs and every protection graph is guaranteed to be two-edge connected. The network recovers from the first failure by tunneling the packet to the next-hop node using one of the protection addresses of the next-hop node; and the packet is routed over the protection graph corresponding to that protection address. We prove that it is sufficient to provide up to three protection addresses per node to tolerate any arbitrary two link failures in a three-edge connected graph. We evaluate the effectiveness of the proposed technique over several network topologies. network recovers from the first failure using IP-in-IP tunneling (RFC2003) using one of the "protection addresses" of the next node in the path. Packets destined to the protection address of a node are routed over a protection graph where the failed link is not present. Every protection graph is guaranteed to be two-edge connected by construction, hence is guaranteed to tolerate another link failure. We develop an elegant technique to compute the protection graphs at a node such that each link connected to the node is removed in at least one of protection graphs, and every protection graph is two-edge connected. The highlight of our approach is that we prove that every node requires at most three protection graphs. The rest of the paper is organized as follows: Section II surveys the techniques developed for fast recovery from single link failures. Section III describes the network model. Sec- tion IV describes our approach for dual link failure recovery, proves the requirement of up to three protection addresses per node, and discusses two different approaches to route using colored trees in the protection (auxiliary) graphs. We evaluate the effectiveness of the proposed approach on several networks and present our results in Section V. Our conclusions are presented in Section VI.

Discovering Network Topology in the Presence of Byzantine Faults

Abstract: We pose and study the problem of Byzantine-robust topology discovery in an arbitrary asynchronous network. The problem is an abstraction of fault-tolerant routing. We formally state the weak and strong versions of the problem. The weak version requires that either each node discovers the topology of the network or at least one node detects the presence of a faulty node. The strong version requires that each node discovers the topology regardless of faults. We focus on non-cryptographic solutions to these problems. We explore their bounds. We prove that the weak topology discovery problem is solvable only if the connectivity of the network exceeds the number of faults in the system. Similarly, we show that the strong version of the problem is solvable only if the network connectivity is more than twice the number of faults. We present solutions to both versions of the problem. The presented algorithms match the established graph connectivity bounds. The algorithms do not require the individual nodes to know either the diameter or the size of the network. The message complexity of both programs is low polynomial with respect to the network size. We describe how our solutions can be extended to add the property of termination, handle topology changes, and perform neighborhood discovery.

Bounds on Harary index

Abstract: In this paper, we obtain the lower and upper bounds on the Harary index of a connected graph (molecular graph), and, in particular, of a triangle- and quadrangle-free graphs in terms of the number of vertices, the number of edges and the diameter. We give the Nordhaus–Gaddum-type result for Harary index using the diameters of the graph and its complement. Moreover, we compare Harary index and reciprocal complementary Wiener number for graphs.

On the Hull Number of Triangle-Free Graphs

Abstract: A set of vertices $C$ in a graph is convex if it contains all vertices which lie on shortest paths between vertices in $C$. The convex hull of a set of vertices $S$ is the smallest convex set containing $S$. The hull number $h(G)$ of a graph $G$ is the smallest cardinality of a set of vertices whose convex hull is the vertex set of $G$. For a connected triangle-free graph $G$ of order $n$ and diameter $d$ at least 4, we prove that $h(G)\leq(n-d+3)/3$ if $G$ has minimum degree at least 3 and that $h(G)\leq2(n-d+5)/7$, if $G$ is cubic. Furthermore for a connected graph $G$ of order $n$, girth $g$ at least 5, minimum degree at least 2, and diameter $d$, we prove $h(G)\leq2+(n-d-1)/\left\lceil\frac{g-1}{2}\right\rceil$. All bounds are best possible.

Approximating the smallest k-edge connected spanning subgraph by LP-rounding

Abstract: The smallest k-ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k-edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3-k for undirected graphs with k > 1 odd, 1 + 2-k for undirected graphs with k even, and 1 + 2-k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2-k. On the hardness side we show that for some absolute constant c > 0, for any integer k ≥ 2 (k ≥ 1), a polynomial-time algorithm approximating the smallest k-ECSS on undirected (directed) multigraphs to within ratio 1 + c-k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Map construction and exploration by mobile agents scattered in a dangerous network

Abstract: We consider the map construction problem in a simple, connected graph by a set of mobile computation entities or agents that start from scattered locations throughout the graph. The problem is further complicated by dangerous elements, nodes and links, in the graph that eliminate agents traversing or arriving at them. The agents working in the graph communicate using a limited amount of storage at each node and work asynchronously. We present a deterministic algorithm that solves the exploration and map construction problems. The end result is also a rooted spanning tree and the election of a leader. The total cost of the algorithm is O(n s m) total number of moves, where m is the number of links in the network and n s is the number of safe nodes, improving the existing O(m2) bound.