44 research outputs found
A large deviation approach to super-critical bootstrap percolation on the random graph
We consider the Erd\"{o}s--R\'{e}nyi random graph and we analyze
the simple irreversible epidemic process on the graph, known in the literature
as bootstrap percolation. We give a quantitative version of some results by
Janson et al. (2012), providing a fine asymptotic analysis of the final size
of active nodes, under a suitable super-critical regime. More
specifically, we establish large deviation principles for the sequence of
random variables with explicit rate
functions and allowing the scaling function to vary in the widest possible
range.Comment: 44 page
Bootstrap percolation on the stochastic block model
We analyze the bootstrap percolation process on the stochastic
block model (SBM), a natural extension of the ErdH{o}s--R'{e}nyi random graph
that incorporates the community structure observed in many real systems.
In the SBM, nodes are partitioned into two subsets, which represent different communities,
and pairs of nodes are independently connected with a probability that depends on the communities they belong to.
Under mild assumptions on the system parameters, we prove the existence of a sharp phase transition
for the final number of active nodes and characterize the sub-critical and the super-critical regimes
in terms of the number of initially active nodes, which are selected uniformly at random in each community
Generalized Threshold-Based Epidemics in Random Graphs: the Power of Extreme Values
Bootstrap percolation is a well-known activation process in a graph,
in which a node becomes active when it has at least active neighbors.
Such process, originally studied on regular structures, has been recently
investigated also in the context of random graphs, where it can serve as a simple
model for a wide variety of cascades, such as the
spreading of ideas, trends, viral contents, etc. over large social networks.
In particular, it has been shown that in the final active set
can exhibit a phase transition for a sub-linear number of seeds.
In this paper, we propose a unique framework to study similar
sub-linear phase transitions for a much broader class of graph models
and epidemic processes. Specifically, we consider i) a generalized version
of bootstrap percolation in with random activation thresholds
and random node-to-node influences; ii) different random graph models,
including graphs with given degree sequence and graphs with
community structure (block model). The common thread of our work is to
show the surprising sensitivity of the critical seed set size
to extreme values of distributions, which makes some systems dramatically
vulnerable to large-scale outbreaks. We validate our results running simulation on
both synthetic and real graphs
Bootstrap percolation on the stochastic block model
We analyze the bootstrap percolation process on the stochastic block model
(SBM), a natural extension of the Erd\"{o}s--R\'{e}nyi random graph that allows
representing the "community structure" observed in many real systems. In the
SBM, nodes are partitioned into subsets, which represent different communities,
and pairs of nodes are independently connected with a probability that depends
on the communities they belong to. Under mild assumptions on system parameters,
we prove the existence of a sharp phase transition for the final number of
active nodes and characterize sub-critical and super-critical regimes in terms
of the number of initially active nodes, which are selected uniformly at random
in each community.Comment: 53 pages 3 figure
Performance analysis of non-stationary peer-assisted VoD systems
Abstract—We analyze a peer-assisted Video-on-Demand system in which users contribute their upload bandwidth to the redistribution of a video that they are downloading or that they have cached locally. Our target is to characterize the additional bandwidth that servers must supply to immediately satisfy all requests to watch a given video. We develop an approximate fluid model to compute the required server bandwidth in the sequential delivery case. Our approach is able to capture several stochastic effects related to peer churn, upload bandwidth heterogeneity, non-stationary traffic conditions, which have not been documented or analyzed before. We provide an analytical methodology to design efficient peer-assisted VoD systems and optimal resource allocation strategies. I