73 research outputs found
Distributed Maintenance of Anytime Available Spanning Trees in Dynamic Networks
We address the problem of building and maintaining distributed spanning trees
in highly dynamic networks, in which topological events can occur at any time
and any rate, and no stable periods can be assumed. In these harsh
environments, we strive to preserve some properties such as cycle-freeness or
the existence of a root in each tree, in order to make it possible to keep
using the trees uninterruptedly (to a possible extent). Our algorithm operates
at a coarse-grain level, using atomic pairwise interactions in a way akin to
recent population protocol models. The algorithm relies on a perpetual
alternation of \emph{topology-induced splittings} and \emph{computation-induced
mergings} of a forest of spanning trees. Each tree in the forest hosts exactly
one token (also called root) that performs a random walk {\em inside} the tree,
switching parent-child relationships as it crosses edges. When two tokens are
located on both sides of a same edge, their trees are merged upon this edge and
one token disappears. Whenever an edge that belongs to a tree disappears, its
child endpoint regenerates a new token instantly. The main features of this
approach is that both \emph{merging} and \emph{splitting} are purely localized
phenomenons. In this paper, we present and motivate the algorithm, and we prove
its correctness in arbitrary dynamic networks. Then we discuss several
implementation choices around this general principle. Preliminary results
regarding its analysis are also discussed, in particular an analytical
expression of the expected merging time for two given trees in a static
context.Comment: Distributed Maintenance of Anytime Available Spanning Trees in
Dynamic Networks, Poland (2013
Triangle percolation in mean field random graphs -- with PDE
We apply a PDE-based method to deduce the critical time and the size of the
giant component of the ``triangle percolation'' on the Erd\H{o}s-R\'enyi random
graph process investigated by Palla, Der\'enyi and VicsekComment: Summary of the changes made: We have changed a remark about k-clique
percolation in the first paragraph. Two new paragraphs are inserted after
equation (4.4) with two applications of the equation. We have changed the
names of some variables in our formula
The structure of typical clusters in large sparse random configurations
The initial purpose of this work is to provide a probabilistic explanation of
a recent result on a version of Smoluchowski's coagulation equations in which
the number of aggregations is limited. The latter models the deterministic
evolution of concentrations of particles in a medium where particles coalesce
pairwise as time passes and each particle can only perform a given number of
aggregations. Under appropriate assumptions, the concentrations of particles
converge as time tends to infinity to some measure which bears a striking
resemblance with the distribution of the total population of a Galton-Watson
process started from two ancestors. Roughly speaking, the configuration model
is a stochastic construction which aims at producing a typical graph on a set
of vertices with pre-described degrees. Specifically, one attaches to each
vertex a certain number of stubs, and then join pairwise the stubs uniformly at
random to create edges between vertices. In this work, we use the configuration
model as the stochastic counterpart of Smoluchowski's coagulation equations
with limited aggregations. We establish a hydrodynamical type limit theorem for
the empirical measure of the shapes of clusters in the configuration model when
the number of vertices tends to . The limit is given in terms of the
distribution of a Galton-Watson process started with two ancestors
On Bootstrap Percolation in Living Neural Networks
Recent experimental studies of living neural networks reveal that their
global activation induced by electrical stimulation can be explained using the
concept of bootstrap percolation on a directed random network. The experiment
consists in activating externally an initial random fraction of the neurons and
observe the process of firing until its equilibrium. The final portion of
neurons that are active depends in a non linear way on the initial fraction.
The main result of this paper is a theorem which enables us to find the
asymptotic of final proportion of the fired neurons in the case of random
directed graphs with given node degrees as the model for interacting network.
This gives a rigorous mathematical proof of a phenomena observed by physicists
in neural networks
Mechanical mode dependence of bolometric back-action in an AFM microlever
Two back action (BA) processes generated by an optical cavity based detection
device can deeply transform the dynamical behavior of an AFM microlever: the
photothermal force or the radiation pressure. Whereas noise damping or
amplifying depends on optical cavity response for radiation pressure BA, we
present experimental results carried out under vacuum and at room temperature
on the photothermal BA process which appears to be more complex. We show for
the first time that it can simultaneously act on two vibration modes in
opposite direction: noise on one mode is amplified whereas it is damped on
another mode. Basic modeling of photothermal BA shows that dynamical effect on
mechanical mode is laser spot position dependent with respect to mode shape.
This analysis accounts for opposite behaviors of different modes as observed
Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels
The existence of self-similar solutions with fat tails for Smoluchowski's
coagulation equation has so far only been established for the solvable and the
diagonal kernel. In this paper we prove the existence of such self-similar
solutions for continuous kernels that are homogeneous of degree and satisfy . More precisely,
for any we establish the existence of a continuous weak
self-similar profile with decay as
Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
For critical bond-percolation on high-dimensional torus, this paper proves
sharp lower bounds on the size of the largest cluster, removing a logarithmic
correction in the lower bound in Heydenreich and van der Hofstad (2007). This
improvement finally settles a conjecture by Aizenman (1997) about the role of
boundary conditions in critical high-dimensional percolation, and it is a key
step in deriving further properties of critical percolation on the torus.
Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on
diameter and mixing time of the largest clusters. We further prove that the
volume bounds apply also to any finite number of the largest clusters. The main
conclusion of the paper is that the behavior of critical percolation on the
high-dimensional torus is the same as for critical Erdos-Renyi random graphs.
In this updated version we incorporate an erratum to be published in a
forthcoming issue of Probab. Theory Relat. Fields. This results in a
modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming
issue of Probab. Theory Relat. Field
Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance
Under the condition of detailed balance and some additional restrictions on
the size of the coefficients, we identify the equilibrium distribution to which
solutions of the discrete coagulation-fragmentation system of equations
converge for large times, thus showing that there is a critical mass which
marks a change in the behavior of the solutions. This was previously known only
for particular cases as the generalized Becker-D\"oring equations. Our proof is
based on an inequality between the entropy and the entropy production which
also gives some information on the rate of convergence to equilibrium for
solutions under the critical mass.Comment: 28 page
Near optimal configurations in mean field disordered systems
We present a general technique to compute how the energy of a configuration
varies as a function of its overlap with the ground state in the case of
optimization problems. Our approach is based on a generalization of the cavity
method to a system interacting with its ground state. With this technique we
study the random matching problem as well as the mean field diluted spin glass.
As a byproduct of this approach we calculate the de Almeida-Thouless transition
line of the spin glass on a fixed connectivity random graph.Comment: 13 pages, 7 figure
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