71 research outputs found
On the Spread of Viruses on the Internet
We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a non-vanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy: for it virus with effective spread rate λ, if the infection starts at a typical vertex, then it develops into an epidemic with probability λ^Θ ((log (1/ λ)/log log (1/ λ))), but on average the epidemic probability is λ^(Θ (1))
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
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Modelling Evolution in Structured Populations Involving Multiplayer Interactions
We consider models of evolution in structured populations involving multiplayer games. Whilst also discussing other models, we focus on the modelling framework developed by Broom and Rychtář (J Theor Biol 302:70–80, 2012) onwards. This includes key progress so far, the main gaps and limitations, the relationship and synergies with other models and a discussion of the direction of future work. In this regard as well as discussing existing work, there is some new research on the applicability and robustness of current models with respect to using them to model real populations. This is an important potential advance, as previously all of the work has been entirely theoretical. In particular, the most complex models will have many parameters, and we concentrate on considering simpler versions with a small number of parameters which still possess the key features which would make them applicable. We find that these models are generally robust, in particular issues that can arise related to small payoff changes at critical values and removal of pivotal vertices would have similar effects on other modelling system including evolutionary graph theory. These often occur where it can be argued that there is a lack of robustness in the real system that the model faithfully picks up, and so is not a problematic feature
Finding Rumor Sources on Random Trees
We consider the problem of detecting the source of a rumor which has spread
in a network using only observations about which set of nodes are infected with
the rumor and with no information as to \emph{when} these nodes became
infected. In a recent work \citep{ref:rc} this rumor source detection problem
was introduced and studied. The authors proposed the graph score function {\em
rumor centrality} as an estimator for detecting the source. They establish it
to be the maximum likelihood estimator with respect to the popular Susceptible
Infected (SI) model with exponential spreading times for regular trees. They
showed that as the size of the infected graph increases, for a path graph
(2-regular tree), the probability of source detection goes to while for
-regular trees with the probability of detection, say ,
remains bounded away from and is less than . However, their results
stop short of providing insights for the performance of the rumor centrality
estimator in more general settings such as irregular trees or the SI model with
non-exponential spreading times.
This paper overcomes this limitation and establishes the effectiveness of
rumor centrality for source detection for generic random trees and the SI model
with a generic spreading time distribution. The key result is an interesting
connection between a continuous time branching process and the effectiveness of
rumor centrality. Through this, it is possible to quantify the detection
probability precisely. As a consequence, we recover all previous results as a
special case and obtain a variety of novel results including the {\em
universality} of rumor centrality in the context of tree-like graphs and the SI
model with a generic spreading time distribution.Comment: 38 pages, 6 figure
Alarm Systems and Catastrophes from a Diverse Point of View
Using a chain of urns, we build a Bayesian nonparametric alarm system to predict catastrophic events, such as epidemics, black outs, etc. Differently from other alarm systems in the literature, our model is constantly updated on the basis of the available information, according to the Bayesian paradigm. The papers contains both theoretical and empirical results. In particular, we test our alarm system on a well-known time series of sunspot
AN URN MODEL FOR CASCADING FAILURES ON A LATTICE
A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial network
Herding model and 1/f noise
We provide evidence that for some values of the parameters a simple agent
based model, describing herding behavior, yields signals with 1/f power
spectral density. We derive a non-linear stochastic differential equation for
the ratio of number of agents and show, that it has the form proposed earlier
for modeling of 1/f^beta noise with different exponents beta. The non-linear
terms in the transition probabilities, quantifying the herding behavior, are
crucial to the appearance of 1/f noise. Thus, the herding dynamics can be seen
as a microscopic explanation of the proposed non-linear stochastic differential
equations generating signals with 1/f^beta spectrum. We also consider the
possible feedback of macroscopic state on microscopic transition probabilities
strengthening the non-linearity of equations and providing more opportunities
in the modeling of processes exhibiting power-law statistics
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