Showing papers on "Connectivity published in 2008"

On Consensus Algorithms for Double-Integrator Dynamics

Abstract: This note considers consensus algorithms for double-integrator dynamics. We propose and analyze consensus algorithms for double-integrator dynamics in four cases: 1) with a bounded control input, 2) without relative velocity measurements, 3) with a group reference velocity available to each team member, and 4) with a bounded control input when a group reference state is available to only a subset of the team. We show that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected. We further show that consensus is reached asymptotically for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound. We also show that consensus is reached asymptotically for the fourth case if and only if the group reference state flows directly or indirectly to all of the vehicles in the team.

Towards identity anonymization on graphs

Abstract: The proliferation of network data in various application domains has raised privacy concerns for the individuals involved. Recent studies show that simply removing the identities of the nodes before publishing the graph/social network data does not guarantee privacy. The structure of the graph itself, and in its basic form the degree of the nodes, can be revealing the identities of individuals. To address this issue, we study a specific graph-anonymization problem. We call a graph k-degree anonymous if for every node v, there exist at least k-1 other nodes in the graph with the same degree as v. This definition of anonymity prevents the re-identification of individuals by adversaries with a priori knowledge of the degree of certain nodes. We formally define the graph-anonymization problem that, given a graph G, asks for the k-degree anonymous graph that stems from G with the minimum number of graph-modification operations. We devise simple and efficient algorithms for solving this problem. Our algorithms are based on principles related to the realizability of degree sequences. We apply our methods to a large spectrum of synthetic and real datasets and demonstrate their efficiency and practical utility.

Undirected connectivity in log-space

Abstract: We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log4/3(⋅) obtained by Armoni, Ta-Shma, Wigderson and Zhou (JACM 2000). As undirected st-connectivity is complete for the class of problems solvable by symmetric, nondeterministic, log-space computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic log-space computations). Independent of our work (and using different techniques), Trifonov (STOC 2005) has presented an O(log n log log n)-space, deterministic algorithm for undirected st-connectivity.Our algorithm also implies a way to construct in log-space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph. Specifically, we give log-space constructible universal-traversal sequences for graphs with restricted labeling and log-space constructible universal-exploration sequences for general graphs.

Graph cut based image segmentation with connectivity priors

Abstract: Graph cut is a popular technique for interactive image segmentation. However, it has certain shortcomings. In particular, graph cut has problems with segmenting thin elongated objects due to the ldquoshrinking biasrdquo. To overcome this problem, we propose to impose an additional connectivity prior, which is a very natural assumption about objects. We formulate several versions of the connectivity constraint and show that the corresponding optimization problems are all NP-hard. For some of these versions we propose two optimization algorithms: (i) a practical heuristic technique which we call DijkstraGC, and (ii) a slow method based on problem decomposition which provides a lower bound on the problem. We use the second technique to verify that for some practical examples DijkstraGC is able to find the global minimum.

Distributed Connectivity Control of Mobile Networks

Abstract: Control of mobile networks raises fundamental and novel problems in controlling the structure of the resulting dynamic graphs. In particular, in applications involving mobile sensor networks and multiagent systems, a great new challenge is the development of distributed motion algorithms that guarantee connectivity of the overall network. Motivated by the inherently discrete nature of graphs as combinatorial objects, we address this challenge using a key control decomposition. First, connectivity control of the network structure is performed in the discrete space of graphs and relies on local estimates of the network topology used, along with algebraic graph theory, to verify link deletions with respect to connectivity. Tie breaking, when multiple such link deletions can violate connectivity, is achieved by means of gossip algorithms and distributed market-based control. Second, motion control is performed in the continuous configuration space, where nearest-neighbor potential fields are used to maintain existing links in the network. Integration of the earlier controllers results in a distributed, multiagent, hybrid system, for which we show that the resulting motion always ensures connectivity of the network, while it reconfigures toward certain secondary objectives. Our approach can also account for communication time delays as well as collision avoidance and is illustrated in nontrivial computer simulations.

On Rainbow Connection

Abstract: An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number 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 this paper we prove several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee $rc(G)=2$. Among our results we prove that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc(G)

Decentralized estimation and control of graph connectivity in mobile sensor networks

Abstract: The ability of a robot team to reconfigure itself is useful in many applications: for metamorphic robots to change shape, for swarm motion towards a goal, for biological systems to avoid predators, or for mobile buoys to clean up oil spills. In many situations, auxiliary constraints, such as connectivity between team members and limits on the maximum hop-count, must be satisfied during reconfiguration. In this paper, we show that both the estimation and control of the graph connectivity can be accomplished in a decentralized manner. We describe a decentralized estimation procedure that allows each agent to track the algebraic connectivity of a time-varying graph. Based on this estimator, we further propose a decentralized gradient controller for each agent to maintain global connectivity during motion.

On the robustness of complex networks by using the algebraic connectivity

Abstract: The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, plays a special role for the robustness of networks since it measures the extent to which it is difficult to cut the network into independent components In this paper we study the behavior of the algebraic connectivity in a well-known complex network model, the Erdos-Renyi random graph We estimate analytically the mean and the variance of the algebraic connectivity by approximating it with the minimum nodal degree The resulting estimate improves a known expression for the asymptotic behavior of the algebraic connectivity [18] Simulations emphasize the accuracy of the analytical estimation, also for small graph sizes Furthermore, we study the algebraic connectivity in relation to the graph's robustness to node and link failures, ie the number of nodes and links that have to be removed in order to disconnect a graph These two measures are called the node and the link connectivity Extensive simulations show that the node and the link connectivity converge to a distribution identical to that of the minimal nodal degree, already at small graph sizes This makes the minimal nodal degree a valuable estimate of the number of nodes or links whose deletion results into disconnected random graph Moreover, the algebraic connectivity increases with the increasing node and link connectivity, justifies the correctness of our definition that the algebraic connectivity is a measure of the robustness in complex networks

Algorithmic Aspects of Graph Connectivity

Abstract: Algorithmic Aspects of Graph Connectivity is the first comprehensive book on this central notion in graph and network theory, emphasizing its algorithmic aspects. Because of its wide applications in the fields of communication, transportation, and production, graph connectivity has made tremendous algorithmic progress under the influence of the theory of complexity and algorithms in modern computer science. The book contains various definitions of connectivity, including edge-connectivity and vertex-connectivity, and their ramifications, as well as related topics such as flows and cuts. The authors comprehensively discuss new concepts and algorithms that allow for quicker and more efficient computing, such as maximum adjacency ordering of vertices. Covering both basic definitions and advanced topics, this book can be used as a textbook in graduate courses in mathematical sciences, such as discrete mathematics, combinatorics, and operations research, and as a reference book for specialists in discrete mathematics and its applications.

Auto-Contractive Maps: an artificial adaptive system for data mining. An application to Alzheimer disease.

Abstract: This article presents a new paradigm of Artificial Neural Networks (ANNs): the Auto-Contractive Maps (Auto- CM). The Auto-CM differ from the traditional ANNs under many viewpoints: the Auto-CM start their learning task without a random initialization of their weights, they meet their convergence criterion when all their output nodes become null, their weights matrix develops a data driven warping of the original Euclidean space, they show suitable topological properties, etc. Further two new algorithms, theoretically linked to Auto-CM are presented: the first one is useful to evaluate the complexity and the topological information of any kind of connected graph: the H Function is the index to measure the global hubness of the graph generated by the Auto-CM weights matrix. The second one is named Maximally Regular Graph (MRG) and it is an development of the traditionally Minimum Spanning Tree (MST). Finally, Auto-CM and MRG, with the support of the H Function, are applied to a real complex dataset about Alzheimer disease: this data come from the very known Nuns Study, where variables measuring the abilities of normal and Alzheimer subject during their lifespan and variables measuring the number of the plaques and of the tangles in their brain after their death. The example of the Alzheimer data base is extremely useful to figure out how this new approach can help to re design bottom-up the overall structure of factors related to a complex disease like this.

Distributed motion constraints for algebraic connectivity of robotic networks

Abstract: This paper studies connectivity maintenance of robotic networks that communicate at discrete times and move in continuous space. We propose a distributed algorithm that allows the robots to decide whether a desired collective motion breaks connectivity. Our algorithm works under imperfect information caused by delays in communication and the robots? mobility. We analyze the correctness of our algorithm by formulating it as a game against a hypothetical adversary who chooses system states consistent with observed information. The technical approach combines tools from algebraic graph theory, linear algebra, nonsmooth analysis, and systems and control.

Deterministic Rendezvous in Trees with Little Memory

Abstract: We study the size of memory of mobile agents that permits to solve deterministically the rendezvous problem, i.e., the task of meeting at some node, for two identical agents moving from node to node along the edges of an unknown anonymous connected graph. The rendezvous problem is unsolvable in the class of arbitrary connected graphs, as witnessed by the example of the cycle. Hence we restrict attention to rendezvous in trees, where rendezvous is feasible if and only if the initial positions of the agents are not symmetric. We prove that the minimum memory size guaranteeing rendezvous in all trees of size at most nis i¾?(logn) bits. The upper bound is provided by an algorithm for abstract state machines accomplishing rendezvous in all trees, and using O(logn) bits of memory in trees of size at most n. The lower bound is a consequence of the need to distinguish between up to ni¾? 1 links incident to a node. Thus, in the second part of the paper, we focus on the potential existence of pairs of finiteagents (i.e., finite automata) capable of accomplishing rendezvous in all bounded degreetrees. We show that, as opposed to what has been proved for the graph exploration problem, there are no finite agents capable of accomplishing rendezvous in all bounded degree trees.

Algorithms for identity anonymization on graphs

Abstract: The proliferation of network data in various application domains has raised privacy concerns for the individuals involved. Recent studies show that simply removing the identities of the nodes before publishing the graph/social network data does not guarantee privacy. The structure of the graph itself, and in is basic form the degree of the nodes, can be revealing the identities of individuals. To address this issue, a specific graph-anonymization framework is proposed. A graph is called k-degree anonymous if for every node v, there exist at least k−1 other nodes in the graph with the same degree as v. This definition of anonymity prevents the re-identification of individuals by adversaries with a priori knowledge of the degree of certain nodes. Given a graph G, the proposed graph-anonymization problem asks for the k-degree anonymous graph that stems from G with the minimum number of graph-modification operations. Simple and efficient algorithms are devised for solving this problem, wherein these algorithms are based on principles related to the realizability of degree sequences.

Network Failure Detection and Graph Connectivity

Abstract: We consider a model for monitoring the connectivity of a network subject to node or edge failures. In particular, we are concerned with detecting $(\epsilon,k)$-failures: events in which an adversary deletes up to $k$ network elements (nodes or edges), after which there are two sets of nodes $A$ and $B$, each at least an $\epsilon$ fraction of the network, that are disconnected from one another. We say that a set $D$ of nodes is an $(\epsilon,k)$-detection set if, for any $(\epsilon,k)$-failure of the network, some two nodes in $D$ are no longer able to communicate; in this way, $D$ “witnesses” any such failure. Recent results show that for any graph $G$, there is an $(\epsilon,k)$-detection set of size bounded by a polynomial in $k$ and $\epsilon$, independent of the size of $G$. In this paper, we expose some relationships between bounds on detection sets and the edge-connectivity $\lambda$ and node-connectivity $\kappa$ of the underlying graph. Specifically, we show that detection set bounds can be made considerably stronger when parameterized by these connectivity values. We show that for an adversary that can delete $k \lambda$ edges, there is always a detection set of size $O(\frac{k}{\epsilon}\log\frac{1}{\epsilon})$ which can be found by random sampling. Moreover, an $(\epsilon,\lambda)$-detection set of minimum size (which is at most $\frac{1}{\epsilon}$) can be computed in polynomial time. A crucial point is that these bounds are independent not just of the size of $G$ but also of the value of $\lambda$. Extending these bounds to node failures is much more challenging. The most technically difficult result of this paper is that a random sample of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ nodes is a detection set for adversaries that can delete a number of nodes up to $\kappa$, the node-connectivity. For the case of edge-failures we use VC-dimension techniques and the cactus representation of all minimum edge-cuts of a graph; for node failures, we develop a novel approach for working with the much more complex set of all minimum node-cuts of a graph.

Constructing an inference graph for a network

Abstract: Constructing an inference graph relates to the creation of a graph that reflects dependencies within a network. In an example embodiment, a method includes determining dependencies among components of a network and constructing an inference graph for the network responsive to the dependencies. The components of the network include services and hardware components, and the inference graph reflects cross-layer components including the services and the hardware components. In another example embodiment, a system includes a service dependency analyzer and an inference graph constructor. The service dependency analyzer is to determine dependencies among components of a network, the components including services and hardware components. The inference graph constructor is to construct an inference graph for the network responsive to the dependencies, the inference graph reflecting cross-layer components including the services and the hardware components.

A bird's eye view of the cut method and a survey of its applications in chemical graph theory

Abstract: A general description of the cut method is presented and an overview of its applications in chemical graph theory is given. Applications include the Wiener index, the Szeged index, the hyper-Wiener index, the PI index, the weighted Wiener index, Wiener-type indices, and classes of chemical graphs such as trees, benzenoid graphs and phenylenes. A computation of the Wiener index of an arbitrary connected graph using its canonical metric representation is described. Algorithmic issues are also briefly mentioned as well as are the recently introduced CI index and related polynomials.

A multi-objective fuzzy graph approach for modular formulation considering end-of-life issues

Abstract: Increased awareness of the negative environmental impact caused by electronic waste has driven electronics manufacturers to re-engineer their product design process and include product end-of-life considerations into their design criteria. Design for the Environment (DfE), as a possible solution, lacks an implementation framework. To address this problem, a fuzzy graph based modular product design methodology is developed to implement DfE strategies in product modular formulation considering multiple product life cycle objectives guided by DfE. A fuzzy connected graph approach is used to present the product structure, whereby fuzzy relationship values are determined by applying Analytic Hierarchy Process (AHP) to life cycle environmental objectives along with other functional and production concerns. Based on the fuzzy connected graph, an optimal modular formulation is searched using the graph-based clustering algorithm to identify the best module configuration. An example is provided to illustrate the me...

Weighted Voronoi region algorithms for political districting

Abstract: Political districting on a given territory can be modelled as bi-objective partitioning of a graph into connected components. The nodes of the graph represent territorial units and are weighted by populations; edges represent pairs of geographically contiguous units and are weighted by road distances between the two units. When a majority voting rule is adopted, two reasonable objectives are population equality and compactness. The ensuing combinatorial optimization problem is extremely hard to solve exactly, even when only the single objective of population equality is considered. Therefore, it makes sense to use heuristics. We propose a new class of them, based on discrete weighted Voronoi regions, for obtaining compact and balanced districts, and discuss some formal properties of these algorithms. These algorithms feature an iterative updating of the distances in order to balance district populations as much as possible. Their performance has been tested on randomly generated rectangular grids, as well as on real-life benchmarks; for the latter instances the resulting district maps are compared with the institutional ones adopted in the Italian political elections from 1994 to 2001.

Words Maps and Spectra of Random Graph Lifts

Abstract: We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by s_i. We investigate the random variable X_w^n that counts the fixed points in this permutation when the s_i are selected uniformly at random. A major ingredient of our work is a new categorization of words which considerably extends the dichotomy of primitive vs. imprimitive words. We establish some results and make a few conjectures about the relation between the expectation E(X_w^n) and this new categorization. This analysis contributes deeply to our study of the spectra of random lifts of graphs. Let G be a connected graph, and let the infinite tree T be its universal cover space. If L and R are the spectral radii of G and T respectively, then, as shown by J. Friedman, for almost every n-lift H of G, all "new" eigenvalues of H are < O(L^(1/2)R^(1/2)). We improve this upper bound to O(L^(1/3)R^(2/3)), and our aforementioned conjectures suggest a possible approach to proving an upper bound of O(R). This is a generalization of the problem of bounding the second eigenvalue in a random 2d-regular graph. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica, which determines, for every fixed w, the limit distribution (as n \to \infty) of X_w^n. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=u^d for some word u.

Embedded in the Shadow of the Separator

Abstract: Eigenvectors to the second smallest eigenvalue of the Laplace matrix of a graph, also known as Fiedler vectors, are the basic ingredient in spectral graph partitioning heuristics. Maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity. By exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the $n$ nodes of the graph in $n$-space so that their barycenter is the origin, the distance between adjacent nodes is bounded by one, and the nodes are spread as much as possible (the sum of the squared norms is maximized). For connected graphs we prove that, for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator. Furthermore, we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one.

Cyclic 7-edge-cuts in fullerene graphs

Abstract: A fullerene graph is a planar cubic graph whose all faces are pentagonal and hexagonal. The structure of cyclic edge-cuts of fullerene graphs of sizes at most 6 is known. In the paper we study cyclic 7-edge connectivity of fullerene graphs, distinguishing between degenerate and non-degenerate cyclic edge-cuts, regarding the arrangement of the 12 pentagons. We prove that if there exists a non-degenerate cyclic 7-edge-cut in a fullerene graph, then the graph is a nanotube unless it is one of the two exceptions presented. We determined that there are 57 configurations of degenerate cyclic 7-edge-cuts, and we listed all of them.

Reconfigurations in Graphs and Grids

Abstract: Let $G$ be a connected graph, and let $V$ and $V'$ be two $n$-element subsets of its vertex set $V(G)$. Imagine that we place a chip at each element of $V$ and we want to move them into the positions of $V'$ ($V$ and $V'$ may have common elements). A move is defined as shifting a chip from $v_1$ to $v_2$ ($v_1,v_2 \in V(G)$) on a path formed by edges of $G$ so that no intermediate vertices are occupied. We give upper and lower bounds on the number of moves that are necessary and analyze the computational complexity of this problem under various assumptions: labeled versus unlabeled chips, arbitrary graphs versus the case when the graph is the rectangular (infinite) planar grid, etc. We prove hardness and inapproximability results for several variants of the problem. We also give a linear time algorithm which performs an optimal (minimum) number of moves for the unlabeled version in a tree, and a constant-ratio approximation algorithm for the unlabeled version in a graph. The graph algorithm uses the tree algorithm as a subroutine.