243,939 research outputs found
Potential Theory for boundary value problems on finite networks
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
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
has a pure discrete spectrum and the control potential is in some sense
regular with respect to 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 . 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
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
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
-finite measure space and a symmetric measure
on supported by a measurable
symmetric subset . 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 which are endowed with
a symmetric weight function defined on the set of edges . As in the
theory of weighted networks, we consider the Hilbert spaces and define two other Hilbert spaces, the dissipation space
and finite energy space . 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 , and, the second, its counterpart in . 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
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
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
- âŠ