23 research outputs found

    Consensus State Gram Matrix Estimation for Stochastic Switching Networks from Spectral Distribution Moments

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    Reaching distributed average consensus quickly and accurately over a network through iterative dynamics represents an important task in numerous distributed applications. Suitably designed filters applied to the state values can significantly improve the convergence rate. For constant networks, these filters can be viewed in terms of graph signal processing as polynomials in a single matrix, the consensus iteration matrix, with filter response evaluated at its eigenvalues. For random, time-varying networks, filter design becomes more complicated, involving eigendecompositions of sums and products of random, time-varying iteration matrices. This paper focuses on deriving an estimate for the Gram matrix of error in the state vectors over a filtering window for large-scale, stationary, switching random networks. The result depends on the moments of the empirical spectral distribution, which can be estimated through Monte-Carlo simulation. This work then defines a quadratic objective function to minimize the expected consensus estimate error norm. Simulation results provide support for the approximation.Comment: 52nd Asilomar Conference on Signals, Systems, and Computers (Asilomar 2017

    Uniform existence of the IDS on lattices and groups

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    We present a general framework for thermodynamic limits and its applications to a variety of models. In particular we will identify criteria such that the limits are uniform in a parameter. All results are illustrated with the example of eigenvalue counting functions converging to the integrated density of states. In this case, the convergence is uniform in the energy.Comment: 30 pages, 6 figures, proceedings of the conference Analysis and Geometry on Graphs and Manifolds 2017 at the University of Potsda

    The roles of random boundary conditions in spin systems

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    Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result

    Mating of trees for random planar maps and Liouville quantum gravity: a survey

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    We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for 8/3\sqrt{8/3}-LQG which allow one to prove its equivalence with the Brownian map.Comment: 68 pages, 12 figure

    Dagstuhl News January - December 2011

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    "Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

    Dynamic processes on networks and higher-order structures

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    Higher-order interactions are increasingly recognized as a critical aspect in the modeling of complex systems. Higher-order networks provide a framework for studying the relationship between the structure of higher-order interactions and the function of the complex system. However, little is known about how higher-order interactions affect dynamic processes. In this thesis, we develop general frameworks of percolation aiming at understanding the interplay between higher-order network structures and the critical properties of dynamics. We reveal that degree correlations strongly affect the percolation threshold on higher-order networks and interestingly, the effect of correlations is different on ordinary percolation and higher-order percolation. We further elucidate the mechanisms responsible for the emergence of discontinuous transitions on higher-order networks. Moreover, we show that triadic regulatory interaction, as a general type of higher-order interaction found widely in nature, can turn percolation into a fully-fledged dynamic process that exhibits period doubling and a route to chaos. As an important example of dynamic processes, we further investigate the role of network topology on epidemic spreading. We show that higher-order interactions can induce a non-linear infection kernel in a pandemic, which results in a discontinuous phase transition, hysteresis, and superexponential spreading. Finally, we propose an epidemic model to evaluate the role of automated contact-and-tracing with mobile apps as a new containment measure to mitigate a pandemic. We reveal the non-linear effect on the reduction of the incidence provided by a certain fraction of app adoption in the population and we propose the optimal strategy to mitigate the pandemic with limited resources. Altogether, the thesis provides new insights into the interplay between the topology of higher-order networks and their dynamics. The results obtained may shed light on the research in other areas of interest such as brain functions and epidemic spreading
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