243,939 research outputs found

    Potential Theory for boundary value problems on finite networks

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    We aim here at analyzing self-adjoint boundary value problems on finite networks associated with positive semi-definite Schrödinger operators. In addition, we study the existence and uniqueness of solutions and its variational formulation. Moreover, we will tackle a well-known problem in the framework of Potential Theory, the so-called condenser principle. Then, we generalize of the concept of effective resistance between two vertices of the network and we characterize the Green function of some BVP in terms of effective resistances

    Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls

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    This the text of a proceeding accepted for the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). We present some results of an ongoing research on the controllability problem of an abstract bilinear Schrodinger equation. We are interested by approximation of this equation by finite dimensional systems. Assuming that the uncontrolled term AA has a pure discrete spectrum and the control potential BB is in some sense regular with respect to AA we show that such an approximation is possible. More precisely the solutions are approximated by their projections on finite dimensional subspaces spanned by the eigenvectors of AA. This approximation is uniform in time and in the control, if this control has bounded variation with a priori bounded total variation. Hence if these finite dimensional systems are controllable with a fixed bound on the total variation of the control then the system is approximatively controllable. The main outcome of our analysis is that we can build solutions for low regular controls such as bounded variation ones and even Radon measures

    Social Stability and Extended Social Balance - Quantifying the Role of Inactive Links in Social Networks

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    Structural balance in social network theory starts from signed networks with active relationships (friendly or hostile) to establish a hierarchy between four different types of triadic relationships. The lack of an active link also provides information about the network. To exploit the information that remains uncovered by structural balance, we introduce the inactive relationship that accounts for both neutral and nonexistent ties between two agents. This addition results in ten types of triads, with the advantage that the network analysis can be done with complete networks. To each type of triadic relationship, we assign an energy that is a measure for its average occupation probability. Finite temperatures account for a persistent form of disorder in the formation of the triadic relationships. We propose a Hamiltonian with three interaction terms and a chemical potential (capturing the cost of edge activation) as an underlying model for the triadic energy levels. Our model is suitable for empirical analysis of political networks and allows to uncover generative mechanisms. It is tested on an extended data set for the standings between two classes of alliances in a massively multi-player on-line game (MMOG) and on real-world data for the relationships between countries during the Cold War era. We find emergent properties in the triadic relationships between the nodes in a political network. For example, we observe a persistent hierarchy between the ten triadic energy levels across time and networks. In addition, the analysis reveals consistency in the extracted model parameters and a universal data collapse of a derived combination of global properties of the networks. We illustrate that the model has predictive power for the transition probabilities between the different triadic states.Comment: 21 pages, 10 figure

    Graph Laplace and Markov operators on a measure space

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    The main goal of this paper is to build a measurable analogue to the theory of weighted networks on infinite graphs. Our basic setting is an infinite σ\sigma-finite measure space (V,B,ÎŒ)(V, \mathcal B, \mu) and a symmetric measure ρ\rho on (V×V,B×B)(V\times V, \mathcal B\times \mathcal B) supported by a measurable symmetric subset E⊂V×VE\subset V\times V. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jump-processes; and it extends earlier studies of infinite graphs G=(V,E)G = (V, E) which are endowed with a symmetric weight function cxyc_{xy} defined on the set of edges EE. As in the theory of weighted networks, we consider the Hilbert spaces L2(ÎŒ),L2(cÎŒ)L^2(\mu), L^2(c\mu) and define two other Hilbert spaces, the dissipation space DissDiss and finite energy space HE\mathcal H_E. Our main results include a number of explicit spectral theoretic and potential theoretic theorems that apply to two realizations of Laplace operators, and the associated jump-diffusion semigroups, one in L2(ÎŒ)L^2(\mu), and, the second, its counterpart in HE\mathcal H_E. We show in particular that it is the second setting (the energy-Hilbert space and the dissipation Hilbert space) which is needed in a detailed study of transient Markov processes.Comment: 72 page

    Early Stop Criterion from the Bootstrap Ensemble

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    This paper addresses the problem of generalization error estimation in neural networks. A new early stop criterion based on a Bootstrap estimate of the generlization error is suggested. The estimate does not require the network to be trained to the minimum of the cost function, as required by other methods based on asymptotic theory. Moreover, in constrast to methods based on cross-validation which require data left out for testing, and thus biasing the estimate, the Bootstrap technique does not have this disadvantage. The potential of the suggested technique is demonstrated on various time-series problems. 1. INTRODUCTION The goal of neural network learning in signal processing is to identify robust functional dependencies between input and output data (for an introduction see e.g., [3]). Such learning usually proceeds from a finite random sample of training data; hence, the functions implemented by neural networks are stochastic depending on the particular available training set. T..

    How Gibbs distributions may naturally arise from synaptic adaptation mechanisms. A model-based argumentation

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    This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials ("spike trains") produced by neuronal networks ? and; (ii) what are the effects of synaptic plasticity on these statistics ? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering "slow" synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.Comment: 39 pages, 3 figure
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