Showing papers on "Connectivity published in 2011"

Graph-Theoretic Connectivity Control of Mobile Robot Networks This paper develops an analysis for groups of vehicles connected by a communication network; control laws are formulated to accomplish tasks requiring rendezvous, and swarm in group formations.

Abstract: In this paper, we provide a theoretical framework for controlling graph connectivity in mobile robot networks. We discuss proximity-based communication models composed of disk-based or uniformly-fading-signal-strength communica- tion links. A graph-theoretic definition of connectivity is pro- vided, as well as an equivalent definition based on algebraic graph theory, which employs the adjacency and Laplacian matrices of the graph and their spectral properties. Based on these results, we discuss centralized and distributed algorithms to maintain, increase, and control connectivity in mobile robot networks. The various approaches discussed in this paper range from convex optimization and subgradient-descent algo- rithms, for the maximization of the algebraic connectivity of the network, to potential fields and hybrid systems that main- tain communication links or control the network topology in a least restrictive manner. Common to these approaches is the use of mobility to control the topology of the underlying com- munication network. We discuss applications of connectivity control to multirobot rendezvous, flocking and formation con- trol, where so far, network connectivity has been considered an assumption.

Graph-theoretic connectivity control of mobile robot networks

Abstract: In this paper, we provide a theoretical framework for controlling graph connectivity in mobile robot networks. We discuss proximity-based communication models composed of disk-based or uniformly-fading-signal-strength communication links. A graph-theoretic definition of connectivity is provided, as well as an equivalent definition based on algebraic graph theory, which employs the adjacency and Laplacian matrices of the graph and their spectral properties. Based on these results, we discuss centralized and distributed algorithms to maintain, increase, and control connectivity in mobile robot networks. The various approaches discussed in this paper range from convex optimization and subgradient-descent algorithms, for the maximization of the algebraic connectivity of the network, to potential fields and hybrid systems that maintain communication links or control the network topology in a least restrictive manner. Common to these approaches is the use of mobility to control the topology of the underlying communication network. We discuss applications of connectivity control to multirobot rendezvous, flocking and formation control, where so far, network connectivity has been considered an assumption.

Neighborhood based fast graph search in large networks

Abstract: Complex social and information network search becomes important with a variety of applications. In the core of these applications, lies a common and critical problem: Given a labeled network and a query graph, how to efficiently search the query graph in the target network. The presence of noise and the incomplete knowledge about the structure and content of the target network make it unrealistic to find an exact match. Rather, it is more appealing to find the top-k approximate matches.In this paper, we propose a neighborhood-based similarity measure that could avoid costly graph isomorphism and edit distance computation. Under this new measure, we prove that subgraph similarity search is NP hard, while graph similarity match is polynomial. By studying the principles behind this measure, we found an information propagation model that is able to convert a large network into a set of multidimensional vectors, where sophisticated indexing and similarity search algorithms are available. The proposed method, called Ness (Neighborhood Based Similarity Search), is appropriate for graphs with low automorphism and high noise, which are common in many social and information networks. Ness is not only efficient, but also robust against structural noise and information loss. Empirical results show that it can quickly and accurately find high-quality matches in large networks, with negligible cost.

Divided Edge Bundling for Directional Network Data

Abstract: The node-link diagram is an intuitive and venerable way to depict a graph. To reduce clutter and improve the readability of node-link views, Holten & van Wijk's force-directed edge bundling employs a physical simulation to spatially group graph edges. While both useful and aesthetic, this technique has shortcomings: it bundles spatially proximal edges regardless of direction, weight, or graph connectivity. As a result, high-level directional edge patterns are obscured. We present divided edge bundling to tackle these shortcomings. By modifying the forces in the physical simulation, directional lanes appear as an emergent property of edge direction. By considering graph topology, we only bundle edges related by graph structure. Finally, we aggregate edge weights in bundles to enable more accurate visualization of total bundle weights. We compare visualizations created using our technique to standard force-directed edge bundling, matrix diagrams, and clustered graphs; we find that divided edge bundling leads to visualizations that are easier to interpret and reveal both familiar and previously obscured patterns.

Graph connectivity in sparse subspace clustering

Abstract: Sparse Subspace Clustering (SSC) is one of the recent approaches to subspace segmentation In SSC a graph is constructed whose nodes are the data points and whose edges are inferred from the L1-sparse representation of each point by the others It has been proved that if the points lie on a mixture of independent subspaces, the graphical structure of each subspace is disconnected from the others However, the problem of connectivity within each subspace is still unanswered This is important since the subspace segmentation in SSC is based on finding the connected components of the graph Our analysis is built upon the connection between the sparse representation through L1-norm minimization and the geometry of convex poly-topes proposed by the compressed sensing community After introduction of some assumptions to make the problem well-defined, it is proved that the connectivity within each subspace holds for 2- and 3-dimensional subspaces The claim of connectivity for general d-dimensional case, even for generic configurations, is proved false by giving a counterexample in dimensions greater than 3

Compressive sensing over graphs

Abstract: In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient l 1 minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph.

Optimal Synchronization for Networks of Noisy Double Integrators

Abstract: In this technical note, we present a novel synchronization protocol to synchronize a network of controlled discrete-time double integrators which are nonidentical, with unknown model parameters and subject to additive measurement and process noise. This framework is motivated by the typical problem of synchronizing a network of clocks whose speeds are nonidentical and are subject to variations. This synchronization protocol is formally studied in its synchronous implementation. In particular, we provide a completely distributed strategy that guarantees convergence for any undirected connected communication graph and we also propose an optimal design strategy when the underlaying communication graph is known. Moreover, this protocol can be readily used to study the effect of noise and external disturbances on the steady-state performance. Finally, some simulations including also randomized implementation of the proposed algorithm are presented.

Process Flexibility Revisited: The Graph Expander and Its Applications

Abstract: We examine how to design a flexible process structure for a production system to match supply with demand more effectively. We argue that good flexible process structures are essentially highly connected graphs, and we use the concept of graph expansion (a measure of graph connectivity) to achieve various insights into this design problem. Whereas existing literature on process flexibility has focused on the expected performance of process structure, we analyze in this paper the worst-case performance of the flexible structure design problem under a more general setting, which encompasses a large class of objective functions. Chou et al. [Chou, M. C., G. Chua, C. P. Teo, H. Zheng. 2010. Design for process flexibility: Efficiency of the long chain and sparse structure. Oper. Res.58(1) 43--58] showed the existence of a sparse process structure that performs nearly as well as the fully flexible system on average, but the approach using random sampling yields few insights into the nature of the process structure. We show that the ψ-expander structure, a variant of the graph expander structure (a highly connected but sparse graph) often used in communication networks, is within e-optimality of the fully flexible system for all demand scenarios. Furthermore, the same expander structure works uniformly well for all objective functions in our class. Based on this insight, we derive design guidelines for general nonsymmetrical systems and develop a simple and easy-to-implement heuristic to design flexible process structures. Numerical results show that this simple heuristic performs well for a variety of numerical examples previously studied in the literature and compares favourably even with the best solutions obtained via extensive simulation and known demand distribution.

Efficient and Complete Centralized Multi-Robot Path Planning.

Abstract: Multi-robot path planning is abstracted as the problem of computing a set of non-colliding paths on a graph for multiple robots. A naive search of the composite search space, although complete, has exponential complexity and becomes computationally prohibitive for problems with just a few robots. This work proposes an efficient and complete algorithm for solving a general class of multi-robot path planning problems, specifically those where there are at most n-2 robots in a connected graph of n vertices. The algorithm employs two primitives: a "push" operation where a robot moves toward its goal until no further progress can be made, and a "swap" operation that allows two robots to swap positions without altering the configuration of any other robot. Simulated experiments compare the proposed approach with several other centralized and decoupled planners, and show that the proposed technique has highly competitive computation time and easily scales to problems involving 100s of robots, solving them in under 5 seconds.

Modeling social networks with node attributes using the multiplicative attribute graph model

Abstract: Networks arising from social, technological and natural domains exhibit rich connectivity patterns and nodes in such networks are often labeled with attributes or features. We address the question of modeling the structure of networks where nodes have attribute information. We present a Multiplicative Attribute Graph (MAG) model that considers nodes with categorical attributes and models the probability of an edge as the product of individual attribute link formation affinities. We develop a scalable variational expectation maximization parameter estimation method. Experiments show that MAG model reliably captures network connectivity as well as provides insights into how different attributes shape the network structure.

Efficient and complete centralized multi-robot path planning

Abstract: Multi-robot path planning is abstracted as the problem of computing a set of non-colliding paths on a graph for multiple robots. A naive search of the composite search space, although complete, has exponential complexity and becomes computationally prohibitive for problems with just a few robots. This paper proposes an efficient and complete algorithm for solving a general class of multi-robot path planning problems, specifically those where there are at most n-2 robots in a connected graph of n vertices. This paper provides a full proof of completeness. The algorithm employs two primitives: “push”, where a robot moves toward its goal until no progress can be made, and “swap”, that allows two robots to swap positions without altering the position of any other robot. Additionally, this paper provides a smoothing procedure for improving solution quality. Simulated experiments compare the proposed approach with several other centralized and decoupled planners, and show that the proposed technique improves computation time and solution quality, while scaling to problems with 100s of robots, solving them in under 5 seconds.

Hyperbolicity and Chordality of a Graph

Abstract: Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ by at most $2\delta.$ The graph $G$ is $k$-chordal provided it has no induced cycle of length greater than $k.$ Brinkmann, Koolen and Moulton find that every $3$-chordal graph is $1$-hyperbolic and that graph is not $\frac{1}{2}$-hyperbolic if and only if it contains one of two special graphs as an isometric subgraph. For every $k\geq 4,$ we show that a $k$-chordal graph must be $\frac{\lfloor\frac{k}{2}\rfloor}{2}$-hyperbolic and there does exist a $k$-chordal graph which is not $\frac{\lfloor \frac{k-2}{2}\rfloor}{2}$-hyperbolic. Moreover, we prove that a $5$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does not contain any of a list of five special graphs as an isometric subgraph.

Succinct Greedy Geometric Routing Using Hyperbolic Geometry

Abstract: We describe a method for performing greedy geometric routing for any n-vertex simple connected graph G in the hyperbolic plane, so that a message M between any pair of vertices may be routed by having each vertex that receives M pass it to a neighbor that is closer to M's destination. Our algorithm produces succinct embeddings, where vertex positions are represented using O(\log n) bits and distance comparisons may be performed efficiently using these representations. These properties are useful, for example, for routing in sensor networks, where storage and bandwidth are limited.

Tree exploration with logarithmic memory

Abstract: We consider the task of network exploration by a mobile agent (robot) with small memory. The agent has to traverse all nodes and edges of a network (represented as an undirected connected graph), and return to the starting node. Nodes of the network are unlabeled and edge ports are locally labeled at each node. The agent has no a priori knowledge of the topology of the network or of its size, and cannot mark nodes in any way. Under such weak assumptions, cycles in the network may prevent feasibility of exploration, hence we restrict attention to trees. We present an algorithm to accomplish tree exploration (with return) using O(log n)-bit memory for all n-node trees. This strengthens the result from Diks et al. [2004], where O(log 2 n)-bit memory was used for tree exploration, and matches the lower bound on memory size proved there. We also extend our O(log n)-bit memory traversal mechanism to a weaker model in which ports at each node are ordered in circular manner, however, the explicit values of port numbers are not available.

The complexity of determining the rainbow vertex-connection of graphs

Abstract: A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing $rvc(G)$ is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether $rvc(G)=2$. We also prove that the following problem is NP-Complete: given a vertex-colored graph $G$, check whether the given coloring makes $G$ rainbow vertex-connected.

A Graph-based Method for Entity Linking

Abstract: In this paper, we formalize the task of finding a knowledge base entry that a given named entity mention refers to, namely entity linking, by identifying the most “important” node among the graph nodes representing the candidate entries. With the aim of ranking these entities by their “importance”, we introduce three degree-based measures of graph connectivity. Experimental results on the TACKBP benchmark data sets show that our graph-based method performs comparably with the state-of-the-art methods. We also show that using the name phrase feature outperforms the commonly used bagof-word feature for entity linking.

Network sampling and classification: An investigation of network model representations

Abstract: Methods for generating a random sample of networks with desired properties are important tools for the analysis of social, biological, and information networks. Algorithm-based approaches to sampling networks have received a great deal of attention in recent literature. Most of these algorithms are based on simple intuitions that associate the full features of connectivity patterns with specific values of only one or two network metrics. Substantive conclusions are crucially dependent on this association holding true. However, the extent to which this simple intuition holds true is not yet known. In this paper, we examine the association between the connectivity patterns that a network sampling algorithm aims to generate and the connectivity patterns of the generated networks, measured by an existing set of popular network metrics. We find that different network sampling algorithms can yield networks with similar connectivity patterns. We also find that the alternative algorithms for the same connectivity pattern can yield networks with different connectivity patterns. We argue that conclusions based on simulated network studies must focus on the full features of the connectivity patterns of a network instead of on the limited set of networkmetrics for a specific network type. This fact has important implications for network data analysis: for instance, implications related to the way significance is currently assessed.

Resistance Boundaries of Infinite Networks

Abstract: A resistance network is a connected graph (G, c). The conductance function \(c_{xy}\) weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form e produces a Hilbert space structure he on the space of functions of finite energy.

Local Natural Connectivity in Complex Networks

Abstract: In network theory, a complex network represents a system whose evolving structure and dynamic behavior contribute to its robustness. The natural connectivity is recently proposed as a spectral measure to characterize the robustness of complex networks. We decompose the natural connectivity of a network as local natural connectivity of its connected components and quantify their contributions to the network robustness. In addition, we compare the natural connectivity of a network with that of an induced subgraph of it based on interlacing theorems. As an application, we derive an inequality for eigenvalues of Erdos-Renyi random graphs.

Business network access protocol for the business network

Abstract: The present disclosure describes methods, systems, and computer program products for providing access to business network data. One method includes identifying a logical graph from business network linked graph data to be transformed into a resource graph, the logical graph including at least two nodes and at least one edge connecting a pair of nodes and defining a connection between the nodes. Each node is converted into a resource. A resource graph associated with the logical graph can be generated, where generation comprises, for each identified node, associating at least one attribute associated with the identified node as a resource attribute of the corresponding resource, adding at least one node connected to the identified node via an edge in the logical graph as a resource attribute of the corresponding resource, and dissolving at least one connection between the identified node and at least one other entity in the logical graph.

The radio antipodal and radio numbers of the hypercube

Abstract: A radio k-labeling of a connected graph G is an assignment f of non negative integers to the vertices of G such that |f(x) − f(y)| \ge k + 1 − d(x, y), for any two vertices x and y, where d(x, y) is the distance between x and y in G. The radio antipodal number is the minimum span of a radio (diam(G) − 1)-labeling of G and the radio number is the minimum span of a radio (diam(G))-labeling of G. In this paper, the radio antipodal number and the radio number of the hypercube are determined by using a generalization of binary Gray codes.

Responsible eigenvalue concept for the stability of a class of single-delay consensus dynamics with fixed topology

Abstract: This study investigates the stability of a class of single-delay interconnected dynamics where dynamics share only delayed information with each other while seeking consensus in a large scale network with fixed topology. The coupled dynamics, which are infinite dimensional because of the presence of delays, can remain stable for at most a certain amount of delays called the delay margin. This margin is intricately determined by the nature of the dynamics, the network connectivity and coupling strengths among the dynamics. Here we present a systematic approach to correlate the finite dimensional network graph properties to the delay margin associated with an infinite dimensional eigenvalue problem. In particular, the developed mathematical approach leads to the responsible eigenvalue concept, which becomes the one and only one eigenvalue that directly determines the delay margin of the entire network. Case studies are provided to demonstrate the effectiveness of the approach as well as the connections between delay margin and the eigenvalues of the corresponding graph Laplacian.

Corona: a stabilizing deterministic message-passing skip list

Abstract: We present Corona, a deterministic self-stabilizing algorithm for skip list construction in structured overlay networks. Corona operates in the low-atomicity message-passing asynchronous system model. Corona requires constant process memory space for its operation and, therefore, scales well. We prove the general necessary conditions limiting the initial states from which a self-stabilizing structured overlay network in message-passing system can be constructed. The conditions require that initial state information has to form a weakly connected graph and it should only contain identifiers that are present in the system. We formally describe Corona and rigorously prove that it stabilizes from an arbitrary initial state subject to the necessary conditions. We extend Corona to construct a skip graph.