Showing papers on "Spanning tree published in 2015"
TL;DR: A class of distributed Laplacian-gradient dynamics that are guaranteed to asymptotically find the solution to the economic dispatch problem with and without generator constraints are proposed.
Abstract: This paper considers the economic dispatch problem for a group of generator units communicating over an arbitrary weight-balanced digraph. The objective of the individual units is to collectively generate power to satisfy a certain load while minimizing the total generation cost, which corresponds to the sum of individual arbitrary convex functions. We propose a class of distributed Laplacian-gradient dynamics that are guaranteed to asymptotically find the solution to the economic dispatch problem with and without generator constraints. The proposed coordination algorithms are anytime, meaning that its trajectories are feasible solutions at any time before convergence, and they become better solutions as time elapses. In addition, we design the provably correct determine feasible allocation strategy that handles generator initialization and the addition and deletion of units via a message passing routine over a spanning tree of the network. Our technical approach combines notions and tools from algebraic graph theory, distributed algorithms, nonsmooth analysis, set-valued dynamical systems, and penalty functions. Simulations illustrate our results.
247 citations
TL;DR: A fast minimum spanning tree algorithm on the complete graph of N points that employs a divide-and-conquer scheme to produce an approximate MST with theoretical time complexity of O ( N 1.5) and the approximation is close to the exact MST so that in practical applications the performance does not suffer.
89 citations
01 May 2015
TL;DR: Experimental results indicate that the proposed MOPSO is worthy of being studied in the field of multi-objective optimization problems, and the proposed algorithm has a better tradeoff between the wire length and radius of the routing tree and has achieved a better delay value.
Abstract: Constructing a timing-driven Steiner tree is very important in VLSI performance-driven routing stage. Meanwhile, non-Manhattan architecture is supported by several manufacturing technologies and now well appreciated in the chip manufacturing circle. However, limited progress has been reported on the non-Manhattan performance-driven routing problem. In this paper, an efficient algorithm, namely, TOST_BR_MOPSO, is presented to construct the minimum-cost spanning tree with a minimum radius for performance-driven routing in Octilinear architecture (one type of the non-Manhattan architecture) based on multi-objective particle swarm optimization (MOPSO) and Elmore delay model. Edge transformation is employed in our algorithm to make the particles have the ability to achieve the optimal solution while Union-Find partition is used to prevent the generation of invalid solution. For the purpose of reducing the number of bends which is one of the key factors of chip manufacturability, we also present an edge-vertex encoding strategy combined with edge transformation. To our best knowledge, no approach has been proposed to optimize the number of bends in the process of constructing the non-Manhattan timing-driven Steiner tree. Moreover, the theorem of Markov chain is used to prove the global convergence of our proposed algorithm. Experimental results indicate that the proposed MOPSO is worthy of being studied in the field of multi-objective optimization problems, and our algorithm has a better tradeoff between the wire length and radius of the routing tree and has achieved a better delay value. Meanwhile, combining edge transformation with the encoding strategy, the proposed algorithm can significantly reduce nearly 20 % in the number of bends.
82 citations
TL;DR: The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms, and returns a spanning tree with cost at most OPT and Av - 1 ≤ dT(v) ≤ Bv + 1 for all v ∈ V.
Abstract: In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E) with a degree upper bound Bv on each vertex v ∈ V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this article we present a polynomial-time algorithm which returns a spanning tree T of cost at most OPT and dT(v) ≤ Bv + 1 for all v, where dT(v) denotes the degree of v in T. This generalizes a result of Furer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degree upper bound Bv, and returns a spanning tree with cost at most OPT and Av - 1 ≤ dT(v) ≤ Bv + 1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms.
81 citations
TL;DR: The connectivity controller is modified to use only local information and can be used in conjunction with artificial potential-based formation controllers and the performance is demonstrated in an experiment on a team of wheeled mobile robots.
Abstract: The preservation of connectivity in mobile robot networks is critical to the success of most existing algorithms designed to achieve various goals. The most basic method to preserve connectivity is to have each agent preserve its set of neighbors for all time. More advanced methods preserve a (minimum) spanning tree in the network. Other methods are based on increasing the algebraic graph connectivity, which is given by the second smallest eigenvalue $\lambda_{2}({\cal L})$ of the graph Laplacian ${\cal L} $ that represents the network. These methods typically result in a monotonic increase in connectivity until the network is completely connected. In previous work by the authors, a continuous feedback control method had been proposed which allows the connectivity to decrease, that is, edges in the network may be broken. This method requires agents to have knowledge of the entire network. In this paper, we modify the controller to use only local information. The connectivity controller is based on maximization of $\lambda_{2}({\cal L} ) $ and artificial potential functions and can be used in conjunction with artificial potential-based formation controllers. The controllers are extended for implementation on nonholonomic-wheeled mobile robots, and the performance is demonstrated in an experiment on a team of wheeled mobile robots.
79 citations
TL;DR: In this paper, a lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of a connected regular graph was derived in terms of the normalised Laplacian characteristic polynomial of the graph.
Abstract: Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).
75 citations
TL;DR: It is proved that the consensus problem is solvable in the asynchronous sampled-data setting without or with time-varying communication delays, if the union of the effective communication topology across any time interval with some given length contains a spanning tree.
Abstract: This paper addresses asynchronous consensus problems of multiple double-integrator agents with discontinuous information transmission, where each agent receives its neighbors' state information at discrete instants determined by its own clock. A novel consensus protocol is proposed based on continuous information of each agent itself and sampled information of each agent's neighbors. By using nonnegative matrix theory and graph theory, we prove that the consensus problem is solvable in the asynchronous sampled-data setting without or with time-varying communication delays, if the union of the effective communication topology across any time interval with some given length contains a spanning tree. Remarkably, the sampling intervals and communication delays are allowed to be arbitrarily large yet bounded. By proposing a modified protocol based on the idea of pinning control, we extend the existing result to the desired consensus problem. Numerical examples are finally provided to validate the theoretical results.
75 citations
27 May 2015
TL;DR: This work designs efficient approximation algorithms based on a transformation to the Directed Steiner Tree problem (DST) with a better time complexity and the same approximation factor compared to the state-of-the-art algorithm.
Abstract: The computation of Minimum Spanning Trees (MSTs) is a fundamental graph problem with important applications. However, there has been little study of MSTs for temporal graphs, which is becoming common as time information is collected for many existing networks. We define two types of MSTs for temporal graphs, MSTa and MSTw, based on the optimization of time and cost, respectively. We propose efficient linear time algorithms for computing MSTa. We show that computing MSTw is much harder. We design efficient approximation algorithms based on a transformation to the Directed Steiner Tree problem (DST). Our solution also solves the classical DST problem with a better time complexity and the same approximation factor compared to the state-of-the-art algorithm. Our experiments on real temporal networks further verify the effectiveness of our algorithms. For MSTw, our solution is capable of shortening the runtime from 10 hours to 3 seconds.
63 citations
TL;DR: Tree-based metrics showed linear and non-linear correlation across adulthood and are in close accordance with results from previous histopathological characterizations of the changes in white matter integrity in the aging brain.
61 citations
TL;DR: It is shown that allowing randomization in the protocol can be crucial for obtaining small extended formulations, and it is proved that for the spanning tree and perfect matching polytopes, small variance in the Protocol forces large size in the extended formulation.
Abstract: An extended formulation of a polyhedron $$P$$P is a linear description of a polyhedron $$Q$$Q together with a linear map $$\pi $$? such that $$\pi (Q)=P$$?(Q)=P. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis' factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441---466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $$P$$P equals the nonnegative rank of its slack matrix $$S$$S. Moreover, Yannakakis also shows that the nonnegative rank of $$S$$S is at most $$2^c$$2c, where $$c$$c is the complexity of any deterministic protocol computing $$S$$S. In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-$$2$$2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis' factorization theorem, this implies that the base-$$2$$2 logarithm of the smallest size of an extended formulation of a polytope $$P$$P equals the minimum complexity of a randomized communication protocol computing the slack matrix of $$P$$P in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.
59 citations
06 Jul 2015
TL;DR: In this article, it was shown that in both synchronous and partially synchronous networks, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree.
Abstract: We investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous networks, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree. Interestingly, the class of averaging algorithms, which have the benefit of being memoryless and requiring no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm.
We develop a proof strategy which for each positive result consists in a reduction to the nonsplit networks. It dramatically improves the best known upper bound on the decision times of averaging algorithms and yields a quadratic time non-averaging algorithm for approximate consensus in non-anonymous networks. We also prove that a general upper bound on the decision times of averaging algorithms have to be exponential, shedding light on the price of anonymity.
Finally we apply our results to networked systems with a fixed topology and benign fault models to show that with n processes, up to $$2n-3$$ of link faults per round can be tolerated for approximate consensus, increasing by a factor 2 the bound of Santoro and Widmayer for exact consensus.
TL;DR: It is shown that the average female connectome has more edges, is a better expander graph, has larger minimal bisection width, and has more spanning trees than the average male connectome, which shows that the female brain has better graph theoretical properties, in a sense, than the brain of males.
Abstract: Deep graph-theoretic ideas in the context with the graph of the World Wide Web led to the definition of Google’s PageRank and the subsequent rise of the most popular search engine to date. Brain graphs, or connectomes, are being widely explored today. We believe that non-trivial graph theoretic concepts, similarly as it happened in the case of the World Wide Web, will lead to discoveries enlightening the structural and also the functional details of the animal and human brains. When scientists examine large networks of tens or hundreds of millions of vertices, only fast algorithms can be applied because of the size constraints. In the case of diffusion MRI-based structural human brain imaging, the effective vertex number of the connectomes, or brain graphs derived from the data is on the scale of several hundred today. That size facilitates applying strict mathematical graph algorithms even for some hard-to-compute (or NP-hard) quantities like vertex cover or balanced minimum cut. In the present work we have examined brain graphs, computed from the data of the Human Connectome Project, recorded from male and female subjects between ages 22 and 35. Significant differences were found between the male and female structural brain graphs: we show that the average female connectome has more edges, is a better expander graph, has larger minimal bisection width, and has more spanning trees than the average male connectome. Since the average female brain weighs less than the brain of males, these properties show that the female brain has better graph theoretical properties, in a sense, than the brain of males. It is known that the female brain has a smaller gray matter/white matter ratio than males, that is, a larger white matter/gray matter ratio than the brain of males; this observation is in line with our findings concerning the number of edges, since the white matter consists of myelinated axons, which, in turn, roughly correspond to the connections in the brain graph. We have also found that the minimum bisection width, normalized with the edge number, is also significantly larger in the right and the left hemispheres in females: therefore, the differing bisection widths are independent from the difference in the number of edges.
17 Oct 2015
TL;DR: In this article, the integrality gap of the LP relaxation of the Asymmetric Traveling Salesman Problem is shown to be polyloglog(n), where n is a bounded degree polynomial of log log(n).
Abstract: We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n), where polyloglog(n) is a bounded degree polynomial of log log(n). We prove this by showing that any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree. Our main new ingredient is a procedure, albeit an exponentially sized convex program, that "transforms" graphs that do not admit any spectrally thin trees into those that provably have spectrally thin trees. More precisely, given a k-edge-connected graph G=(V, E) where k>= 7log(n), we show that there is a matrix D that "preserves" the structure of all cuts of G such that for a subset F of E that induces an O(k)-edge-connected graph, the effective resistance of every edge in F w.r.t. D is at most polylog(k)/k. Then, we use our recent extension of the seminal work of Marcus, Spiel man, and Srivastava [MSS13] to prove the existence of a polylog(k)/k-spectrally thin tree with respect to D. Such a tree is polylog(k)/k-combinatorially thin with respect to G as D preserves the structure of cuts of G.
21 Jul 2015
TL;DR: The results contradict the "folk theorem" noted in Awerbuch, et.al.
Abstract: In the CONGEST model, a communications network is an undirected graph whose n nodes are processors and whose m edges are the communications links between processors. At any given time step, a message of size O(log n) may be sent by each node to each of its neighbours. We show for the synchronous model: If all nodes start in the same round, and each node knows its ID and the ID's of its neighbors, or in the case of MST, the distinct weights of its incident edges and knows n, then there are Monte Carlo algorithms which succeed w.h.p. to determine a minimum spanning forest (MST) and a spanning forest (ST) using O(n log2 n/log log n) messages for MST and O(n log n) messages for ST, resp. These results contradict the "folk theorem" noted in Awerbuch, et.al., JACM 1990 that the distributed construction of a broadcast tree requires Ω(m) messages. This lower bound has been shown there and in other papers for some CONGEST models; our protocol demonstrates the limits of these models.A dynamic distributed network is one which undergoes online edge insertions or deletions. We also show how to repair an MST or ST in a dynamic network with asynchronous communication. An edge deletion can be processed in O(n log n /log log n) expected messages in the MST, and O(n) expected messages for the ST problem, while an edge insertion uses O(n) messages in the worst case. We call this "impromptu" updating as we assume that between processing of edge updates there is no preprocessing or storage of additional information. Previous algorithms for this problem that use an amortized o(m) messages per update require substantial preprocessing and additional local storage between updates.
TL;DR: A new algorithm is presented for generating a uniformly random spanning tree in an undirected graph that establishes a new connection between the effective resistance metric and the cut structure of the underlying graph.
Abstract: We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^{\omega}))$ -- that follows from the work of Kelner and Mądry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph.
TL;DR: In this paper, an event-triggered control scheme is proposed for synchronization based on the information from stochastic sampling, and sufficient conditions for cluster synchronization are presented according to the Lyapunov stability theory and linear matrix inequality technique.
Abstract: This paper is concerned with the issue of mean square cluster synchronization of non-identical nodes connected by a directed network. Suppose that the nodes possess nonlinear dynamics and split into several clusters, then an event-triggered control scheme is proposed for synchronization based on the information from stochastic sampling. Meanwhile, an equilibrium is considered to be the synchronization state or the virtual leader for each cluster, which can apply pinning control to the following nodes. Assume that a spanning tree exists in the subgraph consisting of the nodes belonging to the same cluster and the corresponding virtual leader, and the instants for updating controllers are determined by the given event-triggered strategy, then some sufficient conditions for cluster synchronization are presented according to the Lyapunov stability theory and linear matrix inequality technique. Finally, a specific numerical example is shown to demonstrate the effectiveness of the theoretical results.
TL;DR: This article modified the s-t path TSP algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree, which leads to an improved approximation ratio, surpassing the 20-year-old ratio set by the natural Christofides' algorithm variant.
Abstract: We present a deterministic (1+√5/2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old ratio set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this article can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.
TL;DR: It is demonstrated that the TAHC method can detect clusters in artificial trees, and also in MSTs of weighted social networks, for which the clusters are in agreement with the previously reported clusters of the original weighted networks.
Abstract: The identification of clusters or communities in complex networks is a reappearing problem. The minimum spanning tree (MST), the tree connecting all nodes with minimum total weight, is regarded as an important transport backbone of the original weighted graph. We hypothesize that the clustering of the MST reveals insight in the hierarchical structure of weighted graphs. However, existing theories and algorithms have difficulties to define and identify clusters in trees. Here, we first define clustering in trees and then propose a tree agglomerative hierarchical clustering (TAHC) method for the detection of clusters in MSTs. We then demonstrate that the TAHC method can detect clusters in artificial trees, and also in MSTs of weighted social networks, for which the clusters are in agreement with the previously reported clusters of the original weighted networks. Our results therefore not only indicate that clusters can be found in MSTs, but also that the MSTs contain information about the underlying clusters of the original weighted network.
TL;DR: It turns out that the parameter polynomial plays an important role in guaranteeing the consensus of linear multi-agent systems.
Abstract: This paper studies the mean square consensus of discrete-time linear time-invariant multi-agent systems with communication noises. A distributed consensus protocol, which is composed of the agent's own state feedback and the relative states between the agent and its neighbours, is proposed. A time-varying consensus gain a[k] is applied to attenuate the effect of noises which inherits in the inaccurate measurement of relative states with neighbours. A polynomial, namely ‘parameter polynomial’, is constructed. And its coefficients are the parameters in the feedback gain vector of the proposed protocol. It turns out that the parameter polynomial plays an important role in guaranteeing the consensus of linear multi-agent systems. By the proposed protocol, necessary and sufficient conditions for mean square consensus are presented under different topology conditions: 1 if the communication topology graph has a spanning tree and every node in the graph has at least one parent node, then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, ∑∞k = 0a2[k] < ∞ and all roots of the parameter polynomial are in the unit circle; 2 if the communication topology graph has a spanning tree and there exits one node without any parent node the leader–follower case, then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, limk → ∞a[k] = 0 and all roots of the parameter polynomial are in the unit circle; 3 if the communication topology graph does not have a spanning tree, then the mean square consensus can never be achieved. Finally, one simulation example on the multiple aircrafts system is provided to validate the theoretical analysis.
TL;DR: In this paper, the authors investigated the issue of mean square consensus for multiple agents connected by a directed network, where the graph of the network is supposed to have a spanning tree and each agent is taken as a Markovian jumping system, and sufficient conditions for consensus were presented whether the transition rates for the Markov chain being completely known or not.
Abstract: This study investigates the issue of mean square consensus for multiple agents connected by a directed network. The graph of the network is supposed to have a spanning tree and each agent is taken as a Markovian jumping system. By utilising the event-triggered strategy, some sufficient conditions for consensus are presented whether the transition rates for the Markov chain being completely known or not. Furthermore, the event-triggered function designed in this study is dependent on the stochastic sampled-data from neighbouring agents. Theoretical results are provided according to the graph theory, Lyapunov functional and linear matrix inequality approach. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical analysis.
TL;DR: Performance evaluation on the 27 Middlebury data sets shows that both the algorithms outperform the other two tree-based non-local methods, namely minimum spanning tree (MST) and segment-tree (ST).
TL;DR: This paper proposes a synchronization scheme relying on an intermittent information exchange protocol in the presence of time delays and possible packet dropouts and shows that its control objectives are achieved with a simple selection of the control gains.
21 Jul 2015
TL;DR: This work presents a near-optimal distributed algorithm for (1+o(1))-approximation of single-commodity maximum flow in undirected weighted networks that runs in (D+ √n)⋅ no(1) communication rounds in the Congest model.
Abstract: We present a near-optimal distributed algorithm for (1+o(1))-approximation of single-commodity maximum flow in undirected weighted networks that runs in (D+ √n)⋅ no(1) communication rounds in the Congest model. Here, n and D denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial O(m) time bound, and it nearly matches the Ω(D+√n) round complexity lower bound.The algorithm contains two sub-algorithms of independent interest, both with running time (D+√n)⋅ no(1): Distributed construction of a spanning tree of average stretch no(1). Distributed construction of an no(1)-congestion approximator consisting of the cuts induced by O(log n) virtual trees. The distributed representation of the cut approximator allows for evaluation in (D+√n)⋅ no(1) rounds.All our algorithms make use of randomization and succeed with high probability.
TL;DR: Several solution encodings as well as different strategies for some steps of the algorithm are explored and a BRKGA heuristic for the CMST problem is proposed.
TL;DR: The paper addresses distributed stabilization of a class of circular formations for steered kinematic particles in three-dimensional Euclidean space with distributed control design based on a hierarchical approach that relies on set stabilization and a reduction principle for asymptotic stability of closed sets.
TL;DR: In this article, the authors consider a system of qubits coupled via nearest-neighbor interaction governed by the Heisenberg Hamiltonian and show that if the underlying graph is a tree with more than two vertices then perfect state transfer does not happen.
Abstract: We consider a system of qubits coupled via nearest-neighbor interaction governed by the Heisenberg Hamiltonian. We further suppose that all coupling constants are equal to 1. We are interested in determining which graphs allow for a transfer of quantum state with fidelity equal to 1. To answer this question, it is enough to consider the action of the Laplacian matrix of the graph in a vector space of suitable dimension. Our main result is that if the underlying graph is a tree with more than two vertices, then perfect state transfer does not happen. We also explore related questions, such as what happens in bipartite graphs and graphs with an odd number of spanning trees. Finally, we consider the model based on the $XY$-Hamiltonian, whose action is equivalent to the action of the adjacency matrix of the graph. In this case, we conjecture that perfect state transfer does not happen in trees with more than three vertices.
18 May 2015
TL;DR: This article shows that, relative to common centrality measures, spanning centrality is more effective in identifying edges whose removal causes a higher disruption in an information propagation procedure, while being very resilient to noise, in terms of both the edges scores and the resulting edge ranking.
Abstract: The spanning centrality of an edge e in an undirected graph G is the fraction of the spanning trees of G that contain e. Despite its appealing definition and apparent value in certain applications in computational biology, spanning centrality hasn't so far received a wider attention as a measure of edge centrality. We may partially attribute this to the perceived complexity of computing it, which appears to be prohibitive for very large networks. Contrary to this intuition, spanning centrality can in fact be approximated arbitrary well by very efficient near-linear time algorithms due to Spielman and Srivastava, combined with progress in linear system solvers. In this article we bring theory into practice, with careful and optimized implementations that allow the fast computation of spanning centrality in very large graphs with millions of nodes. With this computational tool in our disposition, we demonstrate experimentally that spanning centrality is in fact a useful tool for the analysis of large networks. Specifically, we show that, relative to common centrality measures, spanning centrality is more effective in identifying edges whose removal causes a higher disruption in an information propagation procedure, while being very resilient to noise, in terms of both the edges scores and the resulting edge ranking.
05 Oct 2015
TL;DR: This paper shows that for any graph G with \(n \ge 7\) vertices, if the minimum degree of a vertex in G is at least \(n-k\), then there are completely independent spanning trees in G, and improves the lower bound of \(\frac{n}{2}\) on the Dirac’s condition for completelyIndependent spanning trees to \(\frac(n-1}{2) except for some specific graph.
Abstract: Completely independent spanning trees \(T_1,T_2,\ldots ,T_k\) in a graph G are spanning trees in G such that for any pair of distinct vertices u and v, the k paths in the spanning trees between u and v mutually have no common edge and no common vertex except for u and v. The concept finds applications in fault-tolerant communication problems in a network. Recently, it was shown that Dirac’s condition for a graph to be hamiltonian is also a sufficient condition for a graph to have two completely independent spanning trees. In this paper, we generalize this result to three or more completely independent spanning trees. Namely, we show that for any graph G with \(n \ge 7\) vertices, if the minimum degree of a vertex in G is at least \(n-k\), where \(3 \le k \le \frac{n}{2}\), then there are \(\lfloor \frac{n}{k} \rfloor \) completely independent spanning trees in G. Besides, we improve the lower bound of \(\frac{n}{2}\) on the Dirac’s condition for completely independent spanning trees to \(\frac{n-1}{2}\) except for some specific graph. Our results are theoretical ones, since these minimum degree conditions can be applied only to a very dense graph. We then present constructions of symmetric regular graphs which include optimal graphs with respect to the number of completely independent spanning trees.
TL;DR: This paper investigates the consensus seeking problem of mobile sensor networks (MSNs) with random switching topologies by introducing a global topology-aware linear quadratic (LQ) cost as the performance measure and transforms the consensus problem to the stabilization of a Markovian jump system with guaranteed cost.
Abstract: This paper investigates the consensus seeking problem of mobile sensor networks (MSNs) with random switching topologies. The network communication topologies are composed of a set of directed graphs (or digraph) with a spanning tree. The switching of topologies is governed by a Markov chain. The consensus seeking problem is addressed by introducing a global topology-aware linear quadratic (LQ) cost as the performance measure. By state transformation, the consensus problem is transformed to the stabilization of a Markovian jump system with guaranteed cost. A sufficient condition for global mean-square consensus is derived in the context of stochastic stability analysis of Markovian jump systems. A computational algorithm is given to synchronously calculate both the sub-optimal consensus controller gains and the sub-minimum upper bound of the cost. The effectiveness of the proposed design method is illustrated by three numerical examples.
TL;DR: This paper investigates the continuous-time multi-agent consensus with stochastic communication noises with Kalman-Bucy filtering based protocols and shows that if the communication graph has a spanning tree, then the consensus error is bounded above by O ( t - 1 ) in mean square and by O - 1 / 2 ( log log t ) almost surely.