7,267 research outputs found
Statistics of Epidemics in Networks by Passing Messages
Epidemic processes are common out-of-equilibrium phenomena of broad interdisciplinary interest. In this thesis, we show how message-passing approach can be a helpful tool for simulating epidemic models in disordered medium like networks, and in particular for estimating the probability that a given node will become infectious at a particular time. The sort of dynamics we consider are stochastic, where randomness can arise from the stochastic events or from the randomness of network structures. As in belief propagation, variables or messages in message-passing approach are defined on the directed edges of a network. However, unlike belief propagation, where the posterior distributions are updated according to Bayes\u27 rule, in message-passing approach we write differential equations for the messages over time. It takes correlations between neighboring nodes into account while preventing causal signals from backtracking to their immediate source, and thus avoids echo chamber effects where a pair of adjacent nodes each amplify the probability that the other is infectious. In our first results, we develop a message-passing approach to threshold models of behavior popular in sociology. These are models, first proposed by Granovetter, where individuals have to hear about a trend or behavior from some number of neighbors before adopting it themselves. In thermodynamic limit of large random networks, we provide an exact analytic scheme while calculating the time dependence of the probabilities and thus learning about the whole dynamics of bootstrap percolation, which is a simple model known in statistical physics for exhibiting discontinuous phase transition. As an application, we apply a similar model to financial networks, studying when bankruptcies spread due to the sudden devaluation of shared assets in overlapping portfolios. We predict that although diversification may be good for individual institutions, it can create dangerous systemic effects, and as a result financial contagion gets worse with too much diversification. We also predict that financial system exhibits robust yet fragile behavior, with regions of the parameter space where contagion is rare but catastrophic whenever it occurs. In further results, we develop a message-passing approach to recurrent state epidemics like susceptible-infectious-susceptible and susceptible-infectious-recovered-susceptible where nodes can return to previously inhabited states and multiple waves of infection can pass through the population. Given that message-passing has been applied exclusively to models with one-way state changes like susceptible-infectious and susceptible-infectious-recovered, we develop message-passing for recurrent epidemics based on a new class of differential equations and demonstrate that our approach is simple and efficiently approximates results obtained from Monte Carlo simulation, and that the accuracy of message-passing is often superior to the pair approximation (which also takes second-order correlations into account)
Recurrent Dynamic Message Passing with Loops for Epidemics on Networks
Several theoretical methods have been developed to approximate prevalence and
threshold of epidemics on networks. Among them, the recurrent dynamic
message-passing (rDMP) theory offers a state-of-the-art performance by
preventing the echo chamber effect in network edges. However, the rDMP theory
was derived in an intuitive ad-hoc way, lacking a solid theoretical foundation
and resulting in a probabilistic inconsistency flaw. Furthermore, real-world
networks are clustered and full of local loops like triangles, whereas rDMP is
based on the assumption of a locally tree-like network structure, which makes
rDMP potentially inefficient on real applications. In this work, for the
recurrent-state epidemics, we first demonstrate that the echo chamber effect
exits not only in edges but also in local loops, which rDMP-like method can not
avoid. We then correct the deficiency of rDMP in a principled manner, leading
to the natural introduction of new higher-order dynamic messages, extending
rDMP to handle local loops. By linearizing the extended message-passing
equations, a new epidemic threshold estimation is given by the inverse of the
leading eigenvalue of a matrix named triangular non-backtracking matrix.
Numerical experiments conducted on synthetic and real-world networks to
evaluate our method, the efficacy of which is validated in epidemic prevalence
and threshold prediction tasks. In addition, our method has the potential to
speed up the solution of the immunization, influence maximization, and
robustness optimization problems in the networks.Comment: Submitted, 14 pages, 7 figure
Predicting the epidemic threshold of the susceptible-infected-recovered model
Researchers have developed several theoretical methods for predicting
epidemic thresholds, including the mean-field like (MFL) method, the quenched
mean-field (QMF) method, and the dynamical message passing (DMP) method. When
these methods are applied to predict epidemic threshold they often produce
differing results and their relative levels of accuracy are still unknown. We
systematically analyze these two issues---relationships among differing results
and levels of accuracy---by studying the susceptible-infected-recovered (SIR)
model on uncorrelated configuration networks and a group of 56 real-world
networks. In uncorrelated configuration networks the MFL and DMP methods yield
identical predictions that are larger and more accurate than the prediction
generated by the QMF method. When compared to the 56 real-world networks, the
epidemic threshold obtained by the DMP method is closer to the actual epidemic
threshold because it incorporates full network topology information and some
dynamical correlations. We find that in some scenarios---such as networks with
positive degree-degree correlations, with an eigenvector localized on the high
-core nodes, or with a high level of clustering---the epidemic threshold
predicted by the MFL method, which uses the degree distribution as the only
input parameter, performs better than the other two methods. We also find that
the performances of the three predictions are irregular versus modularity
Variational approximations for stochastic dynamics on graphs
We investigate different mean-field-like approximations for stochastic
dynamics on graphs, within the framework of a cluster-variational approach. In
analogy with its equilibrium counterpart, this approach allows one to give a
unified view of various (previously known) approximation schemes, and suggests
quite a systematic way to improve the level of accuracy. We compare the
different approximations with Monte Carlo simulations on a reversible
(susceptible-infected-susceptible) discrete-time epidemic-spreading model on
random graphs.Comment: 29 pages, 5 figures. Minor revisions. IOP-style
Relevance of backtracking paths in recurrent-state epidemic spreading on networks
The understanding of epidemics on networks has greatly benefited from the recent application of message-passing approaches, which allow us to derive exact results for irreversible spreading (i.e., diseases with permanent acquired immunity) in locally treelike topologies. This success has suggested the application of the same approach to recurrent-state epidemics, for which an individual can contract the epidemic and recover repeatedly. The underlying assumption is that backtracking paths (i.e., an individual is reinfected by a neighbor he or she previously infected) do not play a relevant role. In this paper we show that this is not the case for recurrent-state epidemics since the neglect of backtracking paths leads to a formula for the epidemic threshold that is qualitatively incorrect in the large size limit. Moreover, we define a modified recurrent-state dynamics which explicitly forbids direct backtracking events and show that this modification completely upsets the phenomenology.Postprint (published version
Small-Coupling Dynamic Cavity: a Bayesian mean-field framework for epidemic inference
A novel generalized mean field approximation, called the Small-Coupling
Dynamic Cavity (SCDC) method, for Bayesian epidemic inference and risk
assessment is presented. The method is developed within a fully Bayesian
framework and accounts for non-causal effects generated by the presence of
observations. It is based on a graphical model representation of the epidemic
stochastic process and utilizes dynamic cavity equations to derive a set of
self-consistent equations for probability marginals defined on the edges of the
contact graph. By performing a small-coupling expansion, a pair of
time-dependent cavity messages is obtained, which capture the probability of
individual infection and the conditioning power of observations. In its
efficient formulation, the computational cost per iteration of the SCDC
algorithm is linear in the duration of the epidemic dynamics and in the number
of contacts. The SCDC method is derived for the Susceptible-Infected (SI) model
and straightforwardly applicable to other Markovian epidemic processes,
including recurrent ones. It exhibits high accuracy in assessing individual
risk on par with Belief Propagation techniques and outperforming heuristic
methods based on individual-based mean-field approximations. Although
convergence issues may arise due to long-range correlations in contact graphs,
the estimated marginal probabilities remain sufficiently accurate for reliable
risk estimation. Future work includes extending the method to non-Markovian
recurrent epidemic models and investigating the role of second-order terms in
the small coupling expansion of the observation-reweighted Dynamic Cavity
equations.Comment: 29 pages, 7 figures (including appendices
Networking - A Statistical Physics Perspective
Efficient networking has a substantial economic and societal impact in a
broad range of areas including transportation systems, wired and wireless
communications and a range of Internet applications. As transportation and
communication networks become increasingly more complex, the ever increasing
demand for congestion control, higher traffic capacity, quality of service,
robustness and reduced energy consumption require new tools and methods to meet
these conflicting requirements. The new methodology should serve for gaining
better understanding of the properties of networking systems at the macroscopic
level, as well as for the development of new principled optimization and
management algorithms at the microscopic level. Methods of statistical physics
seem best placed to provide new approaches as they have been developed
specifically to deal with non-linear large scale systems. This paper aims at
presenting an overview of tools and methods that have been developed within the
statistical physics community and that can be readily applied to address the
emerging problems in networking. These include diffusion processes, methods
from disordered systems and polymer physics, probabilistic inference, which
have direct relevance to network routing, file and frequency distribution, the
exploration of network structures and vulnerability, and various other
practical networking applications.Comment: (Review article) 71 pages, 14 figure
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