Showing papers on "Connectivity published in 2006"

High-dimensional graphs and variable selection with the Lasso

Abstract: The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.

Logarithmic Lower Bounds in the Cell-Probe Model

Abstract: We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized $\Omega(\lg n)$ lower bound per operation for several data structural problems on $n$ elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of $\Omega(\lg n\,/\lg\lg n)$ for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first $\Omega(\log_B n)$ lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update.

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

Abstract: We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue $\lambda_2$ of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize $\lambda_2$ subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, , the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding” high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

Deterministic Rendezvous in Graphs

Abstract: Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

The k-Neighbors Approach to Interference Bounded and Symmetric Topology Control in Ad Hoc Networks

Abstract: Topology control, wherein nodes adjust their transmission ranges to conserve energy and reduce interference, is an important feature in wireless ad hoc networks. Contrary to most of the literature on topology control which focuses on reducing energy consumption, in this paper we tackle the topology control problem with the goal of limiting interference as much as possible, while keeping the communication graph connected with high probability. Our approach is based on the principle of maintaining the number of physical neighbors of every node equal to or slightly below a specific value k. As we will discuss in this paper, having a nontrivially bounded physical node degree allows a network topology with bounded interference to be generated. The proposed approach enforces symmetry on the resulting communication graph, thereby easing the operation of higher layer protocols. To evaluate the performance of our approach, we estimate the value of k that guarantees connectivity of the communication graph with high probability both theoretically and through simulation. We then define k-NEIGH, a fully distributed, asynchronous, and localized protocol that uses distance estimation. k-NEIGH guarantees logarithmically bounded physical degree at every node, is the most efficient known protocol (requiring 2n messages in total, where n is the number of nodes in the network), and relies on simpler assumptions than existing protocols. Furthermore, we verify through simulation that the network topologies produced by k-NEIGH show good performance in terms of node energy consumption and expected interference

Quantum Query Complexity of Some Graph Problems

Abstract: Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

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: (i) do there exist control inputs for each agent to maintain network connectivity, and (ii) given desired controls for each agent, can one 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. Finally, we design a distributed "flow-control" algorithm to compute optimal connectivity-maintaining controls.

Hole detection or: "how much geometry hides in connectivity?"

Abstract: Wireless sensor networks typically consist of small, very simple network nodes without any positioning device like GPS. After an initialization phase, the nodes know with whom they can talk directly, but have no idea about their relative geographic locations. We examine how much geometry information is nevertheless hidden in the communication graph of the network: Assuming that the connectivity is determined by the well-known unit-disk graph model, we prove that using an extremely simple linear-time algorithm one can identify nodes on the boundaries of holes of the network. That is, there is enough geometry information hidden in the connectivity structure to identify topological features—in our example the holes in the network. While the theoretical analysis turns out to be quite conservative,an actual implementation shows that the algorithm works well under less stringent conditions.

Synchronization of networks with prescribed degree distributions

Abstract: We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building nonsynchronizing networks having a prescribed degree distribution.

Comparison of energy minimization algorithms for highly connected graphs

Abstract: Algorithms for discrete energy minimization play a fundamental role for low-level vision. Known techniques include graph cuts, belief propagation (BP) and recently introduced tree-reweighted message passing (TRW). So far, the standard benchmark for their comparison has been a 4-connected grid-graph arising in pixel-labelling stereo. This minimization problem, however, has been largely solved: recent work shows that for many scenes TRW finds the global optimum. Furthermore, it is known that a 4-connected grid-graph is a poor stereo model since it does not take occlusions into account. We propose the problem of stereo with occlusions as a new test bed for minimization algorithms. This is a more challenging graph since it has much larger connectivity, and it also serves as a better stereo model. An attractive feature of this problem is that increased connectivity does not result in increased complexity of message passing algorithms. Indeed, one contribution of this paper is to show that sophisticated implementations of BP and TRW have the same time and memory complexity as that of 4-connected grid-graph stereo. The main conclusion of our experimental study is that for our problem graph cut outperforms both TRW and BP considerably. TRW achieves consistently a lower energy than BP. However, as connectivity increases the speed of convergence of TRW becomes slower. Unlike 4-connected grids, the difference between the energy of the best optimization method and the lower bound of TRW appears significant. This shows the hardness of the problem and motivates future research.

Quantum walks with infinite hitting times

Abstract: Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choicemore » of coin. Hitting times are not very well defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.« less

Finding Four Independent Trees

Abstract: Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an $O(|V|^2)$ algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an $O(|V|^3)$ algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.

Node similarity in the citation graph

Abstract: Published scientific articles are linked together into a graph, the citation graph, through their citations. This paper explores the notion of similarity based on connectivity alone, and proposes several algorithms to quantify it. Our metrics take advantage of the local neighborhoods of the nodes in the citation graph. Two variants of link-based similarity estimation between two nodes are described, one based on the separate local neighborhoods of the nodes, and another based on the joint local neighborhood expanded from both nodes at the same time. The algorithms are implemented and evaluated on a subgraph of the citation graph of computer science in a retrieval context. The results are compared with text-based similarity, and demonstrate the complementarity of link-based and text-based retrieval.

Stochastic Graph Transformation Systems

Abstract: In distributed and mobile systems with volatile bandwidth and fragile connectivity, non-functional aspects like performance and reliability become more and more important. To formalize, measure, and predict such properties, stochastic methods are required. At the same time such systems are characterized by a high degree of architectural reconfiguration. Viewing the architecture of a distributed system as a graph, this is naturally modeled by graph transformations. To combine these two views, in this paper we introduce stochastic graph transformation systems associating with each rule its application rate - the rate of the exponential distribution governing the delay of its application. Beside the basic definitions and a motivating example, we derive continuous time Markov chains, establish a link with continuous stochastic logic, deal with the problem of abstraction through graph isomorphisms, and discuss the analysis of properties by means of an experimental tool chain.

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 While major progress has been achieved for several fundamental data sketching and statistics problems, there are many problems that seem to be hard in a streaming setting, including most classical graph problems Relevant examples are graph connectivity and shortest paths, for which linear lower bounds on the "space X passes" product are known Some recent papers have shown that several graph problems can be solved with one or few passes, if the working memory is large enough to contain all the vertices of the graph A natural question is whether we can reduce the space usage at the price of increasing the number of passes Surprisingly, no algorithm with both sublinear space and passes is known for natural graph problems in classical streaming modelsMotivated by technological factors of modern storage systems, some authors have recently started to investigate the computational power of less restrictive streaming models In a FOCS'04 paper, Aggarwal et al have shown that the use of intermediate temporary streams, combined with the ability to reorder them at each pass for free, yields enough power to solve in polylogarithmic space and passes a variety of problems, including graph connectivity They leave however as an open question whether problems such as shortest paths can be solved efficiently in this more powerful modelIn this paper, we show that the "streaming with sorting" model by Aggarwal et al can yield interesting results even without using sorting at all: by just using intermediate temporary streams, we provide the first 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 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

Minimal spanning forests

Abstract: Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In Z d this was proved by Alexander [Ann. Probab. 23 (1995) 87-104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are "thin" in a different sense, namely, on any graph, each component tree in the WMSF has p c = 1 a.s., where p c denotes the critical probability for having an infinite cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be "thick": on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We also show that the number of infinite clusters for Bernoulli(p u ) percolation is at most the number of components of the FMSF, where p u denotes the critical probability for having a unique infinite cluster. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.

Memory Capacity for Sequences in a Recurrent Network with Biological Constraints

Abstract: The CA3 region of the hippocampus is a recurrent neural network that is essential for the storage and replay of sequences of patterns that represent behavioral events. Here we present a theoretical framework to calculate a sparsely connected network's capacity to store such sequences. As in CA3, only a limited subset of neurons in the network is active at any one time, pattern retrieval is subject to error, and the resources for plasticity are limited. Our analysis combines an analytical mean field approach, stochastic dynamics, and cellular simulations of a time-discrete McCulloch-Pitts network with binary synapses. To maximize the number of sequences that can be stored in the network, we concurrently optimize the number of active neurons, that is, pattern size, and the firing threshold. We find that for one-step associations (i.e., minimal sequences), the optimal pattern size is inversely proportional to the mean connectivity c, whereas the optimal firing threshold is independent of the connectivity. If the number of synapses per neuron is fixed, the maximum number P of stored sequences in a sufficiently large, nonmodular network is independent of its number N of cells. On the other hand, if the number of synapses scales as the network size to the power of 3/2, the number of sequences P is proportional to N. In other words, sequential memory is scalable. Furthermore, we find that there is an optimal ratio r between silent and nonsilent synapses at which the storage capacity α = P//[c(1 + r)N] assumes a maximum. For long sequences, the capacity of sequential memory is about one order of magnitude below the capacity for minimal sequences, but otherwise behaves similar to the case of minimal sequences. In a biologically inspired scenario, the information content per synapse is far below theoretical optimality, suggesting that the brain trades off error tolerance against information content in encoding sequential memories.

On nonsingular trees and a reciprocal eigenvalue property

Abstract: Let G be a simple connected graph. Let A(G) be the adjacency matrix of G. We give a combinatorial description of A(G)−1 of a bipartite graph G with a unique perfect matching. As a corollary, we obtain the combinatorial description of the inverse of a nonsingular tree. A graph is said to have property (R) if is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and have the same multiplicity, for each eigenvalue λ then it is said to have the property (SR). We characterize all trees with property (R) and show that it is exactly the class of all trees with property (SR). The class of trees with property (R) is also identical to the class of corona trees, namely, the trees which can be obtained from other trees by adding pendants to all vertices. Other equivalent conditions for a tree to have property (R) are also given.

Connected treewidth and connected graph searching

Abstract: We give a constructive proof of the equality between treewidth and connected treewidth. More precisely, we describe an O(nk 3 )-time algorithm that, given any n-node width-k tree-decomposition of a connected graph G, returns a connected tree-decomposition of G of width < k. The equality between treewidth and connected treewidth finds applications in graph searching problems. First, using equality between treewidth and connected treewidth, we prove that the connected search number cs(G) of a connected graph G is at most logn + 1 times larger than its search number. Second, using our constructive proof of equality between treewidth and connected treewidth, we design an O(log n√log OPT)-approximation algorithm for connected search, running in time O(t(n) + nk 3 log 3/2 k + m log n) for n-node m-edge connected graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor O(√log OPT).

Connected treewidth and connected graph searching

Abstract: We give a constructive proof of the equality between treewidth and connected treewidth. More precisely, we describe an O(nk3)-time algorithm that, given any n-node width-k tree-decomposition of a connected graph G, returns a connected tree-decomposition of G of width ≤ k. The equality between treewidth and connected treewidth finds applications in graph searching problems. First, using equality between treewidth and connected treewidth, we prove that the connected search number cs(G) of a connected graph G is at most logn+1 times larger than its search number. Second, using our constructive proof of equality between treewidth and connected treewidth, we design an $O(log n\sqrt{log OPT}$)-approximation algorithm for connected search, running in time O(t(n)+nk3log3/2k+mlog n) for n-node m-edge connected graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor $O(\sqrt{log OPT}$).

Using the max-plus algorithm for multiagent decision making in coordination graphs

Abstract: Coordination graphs offer a tractable framework for cooperative multiagent decision making by decomposing the global payoff function into a sum of local terms. Each agent can in principle select an optimal individual action based on a variable elimination algorithm performed on this graph. This results in optimal behavior for the group, but its worst-case time complexity is exponential in the number of agents, and it can be slow in densely connected graphs. Moreover, variable elimination is not appropriate for real-time systems as it requires that the complete algorithm terminates before a solution can be reported. In this paper, we investigate the max-plus algorithm, an instance of the belief propagation algorithm in Bayesian networks, as an approximate alternative to variable elimination. In this method the agents exchange appropriate payoff messages over the coordination graph, and based on these messages compute their individual actions. We provide empirical evidence that this method converges to the optimal solution for tree-structured graphs (as shown by theory), and that it finds near optimal solutions in graphs with cycles, while being much faster than variable elimination.

Graph-Based Text Representation for Novelty Detection

Abstract: We discuss several feature sets for novelty detection at the sentence level, using the data and procedure established in task 2 of the TREC 2004 novelty track. In particular, we investigate feature sets derived from graph representations of sentences and sets of sentences. We show that a highly connected graph produced by using sentence-level term distances and pointwise mutual information can serve as a source to extract features for novelty detection. We compare several feature sets based on such a graph representation. These feature sets allow us to increase the accuracy of an initial novelty classifier which is based on a bag-of-word representation and KL divergence. The final result ties with the best system at TREC 2004.

BIOZON: a hub of heterogeneous biological data

Abstract: Biological entities are strongly related and mutually dependent on each other. Therefore, there is a growing need to corroborate and integrate data from different resources and aspects of biological systems in order to analyze them effectively. Biozon is a unified biological database that integrates heterogeneous data types such as proteins, structures, domain families, protein-protein interactions and cellular pathways, and establishes the relationships between them. All data are integrated on to a single graph schema centered around the non-redundant set of biological objects that are shared by each source. This integration results in a highly connected graph structure that provides a more complete picture of the known context of a given object that cannot be determined from any one source. Currently, Biozon integrates roughly 2 million protein sequences, 42 million DNA or RNA sequences, 32 000 protein structures, 150 000 interactions and more from sources such as GenBank, UniProt, Protein Data Bank (PDB) and BIND. Biozon augments source data with locally derived data such as 5 billion pairwise protein alignments and 8 million structural alignments. The user may form complex cross-type queries on the graph structure, add similarity relations to form fuzzy queries and rank the results based on analysis of the edge structure similar to Google PageRank, online at Biozon.org.

On optimal placement of relay nodes for reliable connectivity in wireless sensor networks

Abstract: The paper addresses the relay node placement problem in two-tiered wireless sensor networks. Given a set of sensor nodes in Euclidean plane, our objective is to place minimum number of relay nodes to forward data packets from sensor nodes to the sink, such that: 1) the network is connected, 2) the network is 2-connected. For case one, we propose a (6+e)-approximation algorithm for any e > 0 with polynomial running time when e is fixed. For case two, we propose two approximation algorithms with (24+e) and (6/T+12+e), respectively, where T is the ratio of the number of relay nodes placed in case one to the number of sensors. We further extend the results to the cases where communication radiuses of sensor nodes and relay nodes are different from each other.

Domain connected graph: the skeleton of a closed 3D shape for animation

Abstract: In previous research, three main approaches have been employed to solve the skeleton extraction problem: medial axis transform (MAT), generalized potential field and decomposition-based methods. These three approaches have been formulated using three different concepts, namely surface variation, inside energy distribution, and the connectivity of parts. By combining the above mentioned concepts, this paper creates a concise structure to represent the control skeleton of an arbitrary object.First, an algorithm is proposed to detect the end, connection and joint points of an arbitrary 3D object. These three points comprise the skeleton, and are the most important to consider when describing it. In order to maintain the stability of the point extraction algorithm, a prong-feature detection technique and a level iso-surfaces function-based on the repulsive force field was employed. A neighborhood relationship inherited from the surface able to describe the connection relationship of these positions was then defined. Based on this relationship, the skeleton was finally constructed and named domain connected graph (DCG). The DCG not only preserves the topology information of a 3D object, but is also less sensitive than MAT to the perturbation of shapes. Moreover, from the results of complicated 3D models, consisting of thousands of polygons, it is evident that the DCG conforms to human perception.