Topic

# Connectivity

About: Connectivity is a research topic. Over the lifetime, 5418 publications have been published within this topic receiving 108611 citations.

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TL;DR: A distinctive feature of this work is to address consensus problems for networks with directed information flow by establishing a direct connection between the algebraic connectivity of the network and the performance of a linear consensus protocol.

Abstract: In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

11,658 citations

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TL;DR: It is shown that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs and is hence equivalent to variable selection for Gaussian linear models.

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.

3,793 citations

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TL;DR: A new supervised dimensionality reduction algorithm called marginal Fisher analysis is proposed in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizing the interclass separability.

Abstract: A large family of algorithms - supervised or unsupervised; stemming from statistics or geometry theory - has been designed to provide different solutions to the problem of dimensionality reduction Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called marginal Fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability We show that MFA effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions

2,339 citations

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TL;DR: This note shows that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected and for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound.

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

1,338 citations

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26 Jul 1999

TL;DR: This paper describes two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery, and proposes a new family of random graph models that point to a rich new sub-field of the study of random graphs, and raises questions about the analysis of graph algorithms on the Internet.

Abstract: The pages and hyperlinks of the World-Wide Web may be viewed as nodes and edges in a directed graph. This graph is a fascinating object of study: it has several hundred million nodes today, over a billion links, and appears to grow exponentially with time. There are many reasons -- mathematical, sociological, and commercial -- for studying the evolution of this graph. In this paper we begin by describing two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery. We then report a number of measurements and properties of this graph that manifested themselves as we ran these algorithms on the Web. Finally, we observe that traditional random graph models do not explain these observations, and we propose a new family of random graph models. These models point to a rich new sub-field of the study of random graphs, and raise questions about the analysis of graph algorithms on the Web.

1,116 citations