2,795 research outputs found
Coupling any number of balls in the infinite-bin model
The infinite-bin model, introduced by Foss and Konstantopoulos, describes the
Markovian evolution of configurations of balls placed inside bins, obeying
certain transition rules. We prove that we can couple the behaviour of any
finite number of balls, provided at least two different transition rules are
allowed. This coupling makes it possible to define the regeneration events
needed by Foss and Zachary to prove convergence results for the distribution of
the balls.Comment: 12 pages, 2 figure
First to Market is not Everything: an Analysis of Preferential Attachment with Fitness
In this paper, we provide a rigorous analysis of preferential attachment with
fitness, a random graph model introduced by Bianconi and Barabasi. Depending on
the shape of the fitness distribution, we observe three distinct phases: a
first-mover-advantage phase, a fit-get-richer phase and an innovation-pays-off
phase
Long-range percolation on the hierarchical lattice
We study long-range percolation on the hierarchical lattice of order ,
where any edge of length is present with probability
, independently of all other edges. For fixed
, we show that the critical value is non-trivial if
and only if . Furthermore, we show uniqueness of the infinite
component and continuity of the percolation probability and of
as a function of . This means that the phase diagram
of this model is well understood.Comment: 24 page
Percolation in an ultrametric space
We study percolation on the hierarchical lattice of order where the
probability of connection between two points separated by distance is of
the form . Since the distance is an
ultrametric, there are significant differences with percolation on the
Euclidean lattice. There are two non-critical regimes: , where
percolation occurs, and , where it does not occur. In the critical
case, , we use an approach in the spirit of the renormalization
group method of statistical physics and connectivity results of Erd\H{o}s-Renyi
random graphs play a key role. We find sufficient conditions on such that
percolation occurs, or that it does not occur. An intermediate situation called
pre-percolation is also considered. In the cases of percolation we prove
uniqueness of the constructed percolation clusters. In a previous paper
\cite{DG1} we studied percolation in the limit (mean field
percolation) which provided a simplification that allowed finding a necessary
and sufficient condition for percolation. For fixed there are open
questions, in particular regarding the existence of a critical value of a
parameter in the definition of , and if it exists, what would be the
behaviour at the critical point
Diffusion and Cascading Behavior in Random Networks
The spread of new ideas, behaviors or technologies has been extensively
studied using epidemic models. Here we consider a model of diffusion where the
individuals' behavior is the result of a strategic choice. We study a simple
coordination game with binary choice and give a condition for a new action to
become widespread in a random network. We also analyze the possible equilibria
of this game and identify conditions for the coexistence of both strategies in
large connected sets. Finally we look at how can firms use social networks to
promote their goals with limited information. Our results differ strongly from
the one derived with epidemic models and show that connectivity plays an
ambiguous role: while it allows the diffusion to spread, when the network is
highly connected, the diffusion is also limited by high-degree nodes which are
very stable
On a preferential attachment and generalized P\'{o}lya's urn model
We study a general preferential attachment and Polya's urn model. At each
step a new vertex is introduced, which can be connected to at most one existing
vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is
not disconnected, it joins an existing pioneer vertex with probability
proportional to a function of the degree of that vertex. This function is
allowed to be vertex-dependent, and is called the reinforcement function. We
prove that there can be at most three phases in this model, depending on the
behavior of the reinforcement function. Consider the set whose elements are the
vertices with cardinality tending a.s. to infinity. We prove that this set
either is empty, or it has exactly one element, or it contains all the pioneer
vertices. Moreover, we describe the phase transition in the case where the
reinforcement function is the same for all vertices. Our results are general,
and in particular we are not assuming monotonicity of the reinforcement
function. Finally, consider the regime where exactly one vertex has a degree
diverging to infinity. We give a lower bound for the probability that a given
vertex ends up being the leading one, that is, its degree diverges to infinity.
Our proofs rely on a generalization of the Rubin construction given for
edge-reinforced random walks, and on a Brownian motion embedding.Comment: Published in at http://dx.doi.org/10.1214/12-AAP869 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Universal Behavior of Connectivity Properties in Fractal Percolation Models
Partially motivated by the desire to better understand the connectivity phase
transition in fractal percolation, we introduce and study a class of continuum
fractal percolation models in dimension d greater than or equal to 2. These
include a scale invariant version of the classical (Poisson) Boolean model of
stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler
and Werner. The models lead to random fractal sets whose connectivity
properties depend on a parameter lambda. In this paper we mainly study the
transition between a phase where the random fractal sets are totally
disconnected and a phase where they contain connected components larger than
one point. In particular, we show that there are connected components larger
than one point at the unique value of lambda that separates the two phases
(called the critical point). We prove that such a behavior occurs also in
Mandelbrot's fractal percolation in all dimensions d greater than or equal to
2. Our results show that it is a generic feature, independent of the dimension
or the precise definition of the model, and is essentially a consequence of
scale invariance alone. Furthermore, for d=2 we prove that the presence of
connected components larger than one point implies the presence of a unique,
unbounded, connected component.Comment: 29 pages, 4 figure
Propagation of Chaos for a Balls into Bins Model
Consider a finite number of balls initially placed in bins. At each time
step a ball is taken from each non-empty bin. Then all the balls are uniformly
reassigned into bins. This finite Markov chain is called Repeated
Balls-into-Bins process and is a discrete time interacting particle system with
parallel updating. We prove that, starting from a suitable (chaotic) set of
initial states, as , the numbers of balls in each bin becomes
independent from the rest of the system i.e. we have propagation of chaos. We
furthermore study some equilibrium properties of the limiting nonlinear
process
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