Dynamic message-passing equations for models with unidirectional dynamics

Lokhov, Andrey Y.; Mézard, Marc; Zdeborová, Lenka

doi:10.1103/physreve.91.012811

Cited by 52 publications

(71 citation statements)

References 43 publications

(108 reference statements)

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“…It should be noted that the SIR epidemic of synchronous updating outbreaks more easily than asynchronous updating, only when the recovery rate is close to zero, the final state of synchronous updating tends to that of asynchronous updating. Moreover, there still exists a certain gap between the theoretical predictions and simulated results for some disassortative networks, and thus more accurate analytic approximation of the effective epidemic threshold (e.g., message-passing approach [39,40]) for SIR dynamics with arbitrary recovery rate remains an important problem.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Recovery rate affects the effective epidemic threshold with synchronous updating

Shu

Wang

Tang

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Accurate identification of effective epidemic threshold is essential for understanding epidemic dynamics on complex networks. The existing studies on the effective epidemic threshold of the susceptible-infected-removed (SIR) model generally assume that all infected nodes immediately recover after the infection process, which more or less does not conform to the realistic situation of disease. In this paper, we systematically study the effect of arbitrary recovery rate on the SIR spreading dynamics on complex networks. We derive the theoretical effective epidemic threshold and final outbreak size based on the edge-based compartmental theory. To validate the proposed theoretical predictions, extensive numerical experiments are implemented by using asynchronous and synchronous updating methods. When asynchronous updating method is used in simulations, recovery rate does not affect the final state of spreading dynamics. But with synchronous updating, we find that the effective epidemic threshold decreases with recovery rate, and final outbreak size increases with recovery rate. A good agreement between the theoretical predictions and numerical results are observed on both synthetic and real-world networks. Our results extend the existing theoretical studies, and help us to understand the phase transition with arbitrary recovery rate. 64.60.Ht How to accurately predict the effective epidemic threshold has attracted increasing attentions. The existing studies on the epidemic threshold generally suppose the recovery process with a constant recovery rate of 1, while the investigation on the effect of recovery rate is still insufficient. Considering the difference of recovery rate between different real diseases and the accompanying effects on the human health, it is very necessary to predict the effective epidemic thresholds with different recovery rates. In this work, the effect of recovery rate on the effective threshold of epidemic outbreak is systematically studied. We first develop a novel theoretical framework based on the edge-based compartmental theory. The developed theory predicts that recovery rate does not affect the spreading dynamics with asynchronous updating, but with synchronous updating, the effective epidemic threshold decreases with the recovery rate, and the final outbreak sizes increases with the recovery rate for a given effective transmission rate. It should be noted that the SIR epidemic of synchronous updating breaks more easily than asynchronous updating. To verify the accuracy of the theoretical predictions, we numerically predict the effective epidemic threshold using the variability measure on random regular networks, where the numerical results agrees well with the theoretical predictions. Moreover, we investigate how the recovery rate affects the epidemic outbreaks with synchronous updating on scale-free networks and real-world networks, and find the same variation trend of effective epidemic threshold. * Electronic address:

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Section: Conclusion and Discussionmentioning

confidence: 99%

Recovery rate affects the effective epidemic threshold with synchronous updating

Shu

Wang

Tang

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…(10) is only valid on a tree graph; only in this case θ k→i (t) are independent for all k ∈ ∂i, so that the corresponding probability is factorized as in (10). However, in practice the decorrelation assumption holds to a good precision even on general networks, even with small loops, see [42] for in-depth discussions and supporting numerical experiments. The quantities θ k→i (t) are updated as follows:…”

Section: A Dynamic Message-passing Equationsmentioning

confidence: 99%

“…We use the recently introduced Dynamic Message-Passing (DMP) equations [40][41][42] which provide the estimates (asymptotically exact on sparse graphs) of the probabilities P i σ (t) with a linear computational complexity in the number of edges and time steps. When applied to real-world loopy networks, the DMP algorithm typically yields a accurate prediction of the marginal probabilities as validated empirically [42] for a large class of spreading models on real-world networks. In the Methods section, we provide an intuitive derivation of the corresponding DMP equations for the generalized SIR model.…”

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confidence: 99%

Optimal deployment of resources for maximizing impact in spreading processes

Lokhov

Saad

2017

Proc. Natl. Acad. Sci. U.S.A.

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The effective use of limited resources for controlling spreading processes on networks is of prime significance in diverse contexts, ranging from the identification of "influential spreaders" for maximizing information dissemination and targeted interventions in regulatory networks, to the development of mitigation policies for infectious diseases and financial contagion in economic systems. Solutions for these optimization tasks that are based purely on topological arguments are not fully satisfactory; in realistic settings the problem is often characterized by heterogeneous interactions and requires interventions over a finite time window via a restricted set of controllable nodes. The optimal distribution of available resources hence results from an interplay between network topology and spreading dynamics. We show how these problems can be addressed as particular instances of a universal analytical framework based on a scalable dynamic message-passing approach and demonstrate the efficacy of the method on a variety of real-world examples.Spreading corresponds to omnipresent processes describing a vast number of phenomena in social, natural and technological networks [1][2][3][4] whereby information, viruses and failures propagate through their edges via the interactions between individual constituents. Spreading cascades have a huge impact on the modern world, be it negative or positive. An 11 minute power grid disturbance in Arizona and California in 2011 led to cascading outages and left 2.7 million customers without power [5]. As many as 579,000 people around the world could have been killed by the H1N1 influenza pandemic characterized by a rapid spreading through the global transportation networks [6]. The U.S. economy losses from the 2008 financial crisis resulted from cascading bankruptcies of major financial institutions are estimated at the level of $22 trillion [7]. Therefore, it is not surprising that efficient prediction and control of these undesired spreading processes are regarded as fundamental questions of paramount importance in developing policies for optimal placement of cascade-preventing devices in power grid, real-time distribution of vaccines and antidotes to mitigate epidemic spread, regulatory measures in interbanking lending networks and other modern world problems, such as protection of critical infrastructures against cyber-attacks and computer viruses [8].On the other hand, spreading processes can also be considered beneficial. The ice bucket challenge campaign in social networks raised $115 million donations to the ALS association fighting the Amyotrophic Lateral Sclerosis, in particular due to a significant involvement of celebrities acting as "influencers" [9]. In the context of political campaigning, there are already winners [10,11] and losers, and this division is likely to become more pronounced and critical in the future [12]. Winners are those who use communication and social networks effectively to set the opinions of voters or consumers, maximizing the impact of scarce...

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“…The core challenge is that even in cases where the microscopic processes guiding the dynamics are given, going from a very general statement like a master equation to a practical solution is usually unfeasible. This is due to the unavoidable difficulties of the exponential growth of the size of the state space with the number of particles and time intervals considered [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…Many concepts have been introduced in a natural analogy with the equilibrium theory, e.g. dynamical replica analysis [13,14], cavity method [15], dynamic message-passing algorithm [6,7,16,17], large deviation [18,19], TAP approaches [20] and extended Plefka expansion for continuous variables [21]. Despite all these advances, the issue is far from being settled and there is an active community searching for approximate methods that accurately reproduce numerical results from stochastic simulations [22].…”

Section: Introductionmentioning

confidence: 99%

A simple analytical description of the non-stationary dynamics in Ising spin systems

Vázquez¹,

Ferraro²,

Ricci-Tersenghi³

2017

J. Stat. Mech.

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The analytical description of the dynamics in models with discrete variables (e.g. Ising spins) is a notoriously difficult problem, that can be tackled only under some approximation. propose to use the same approximation based on the cluster variational method also for the non-stationary regime, which has not been considered up to now within this framework. We check the validity of this approximation in describing the non-stationary dynamical regime of several Ising models defined on Erdos-Rényi random graphs: we study ferromagnetic models with symmetric and partially asymmetric couplings, models with random fields and also spin glass models. A comparison with the actual Glauber dynamics, solved numerically, shows that one of the two studied approximations (the so-called 'diamond' approximation) provides very accurate results in all the systems studied. Only for the spin glass models we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states. Given the simplicity of the equations to be solved, we believe the diamond approximation should be considered as the 'minimal standard' in the description of the non-stationary regime of Ising-like models: any new method pretending to provide a better approximate description to the dynamics of Ising-like models should perform at least as good as the diamond approximation.2

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Dynamic message-passing equations for models with unidirectional dynamics

Cited by 52 publications

References 43 publications

Recovery rate affects the effective epidemic threshold with synchronous updating

Recovery rate affects the effective epidemic threshold with synchronous updating

Optimal deployment of resources for maximizing impact in spreading processes

A simple analytical description of the non-stationary dynamics in Ising spin systems

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