82 research outputs found
Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks
Numerical simulation of continuous-time Markovian processes is an essential
and widely applied tool in the investigation of epidemic spreading on complex
networks. Due to the high heterogeneity of the connectivity structure through
which epidemics is transmitted, efficient and accurate implementations of
generic epidemic processes are not trivial and deviations from statistically
exact prescriptions can lead to uncontrolled biases. Based on the Gillespie
algorithm (GA), in which only steps that change the state are considered, we
develop numerical recipes and describe their computer implementations for
statistically exact and computationally efficient simulations of generic
Markovian epidemic processes aiming at highly heterogeneous and large networks.
The central point of the recipes investigated here is to include phantom
processes, that do not change the states but do count for time increments. We
compare the efficiencies for the susceptible-infected-susceptible, contact
process and susceptible-infected-recovered models, that are particular cases of
a generic model considered here. We numerically confirm that the simulation
outcomes of the optimized algorithms are statistically indistinguishable from
the original GA and can be several orders of magnitude more efficient.Comment: 12 pages, 9 figure
Griffiths phases in infinite-dimensional, non-hierarchical modular networks
Griffiths phases (GPs), generated by the heterogeneities on modular networks,
have recently been suggested to provide a mechanism, rid of fine parameter
tuning, to explain the critical behavior of complex systems. One conjectured
requirement for systems with modular structures was that the network of modules
must be hierarchically organized and possess finite dimension. We investigate
the dynamical behavior of an activity spreading model, evolving on
heterogeneous random networks with highly modular structure and organized
non-hierarchically. We observe that loosely coupled modules act as effective
rare-regions, slowing down the extinction of activation. As a consequence, we
find extended control parameter regions with continuously changing dynamical
exponents for single network realizations, preserved after finite size
analyses, as in a real GP. The avalanche size distributions of spreading events
exhibit robust power-law tails. Our findings relax the requirement of
hierarchical organization of the modular structure, which can help to
rationalize the criticality of modular systems in the framework of GPs.Comment: 14 pages, 8 figure
Activation thresholds in epidemic spreading with motile infectious agents on scale-free networks
We investigate a fermionic susceptible-infected-susceptible model with
mobility of infected individuals on uncorrelated scale-free networks with
power-law degree distributions of exponents
. Two diffusive processes with diffusion rate of an infected
vertex are considered. In the \textit{standard diffusion}, one of the
nearest-neighbors is chosen with equal chance while in the \textit{biased
diffusion} this choice happens with probability proportional to the neighbor's
degree. A non-monotonic dependence of the epidemic threshold on with an
optimum diffusion rate , for which the epidemic spreading is more
efficient, is found for standard diffusion while monotonic decays are observed
in the biased case. The epidemic thresholds go to zero as the network size is
increased and the form that this happens depends on the diffusion rule and
degree exponent. We analytically investigated the dynamics using quenched and
heterogeneous mean-field theories. The former presents, in general, a better
performance for standard and the latter for biased diffusion models, indicating
different activation mechanisms of the epidemic phases that are rationalized in
terms of hubs or max -core subgraphs.Comment: 9 pages, 4 figure
Stochastic spreading on complex networks
Complex interacting systems are ubiquitous in nature and society. Computational modeling of these systems is, therefore, of great relevance for science and engineering. Complex networks are common representations of these systems (e.g., friendship networks or road networks). Dynamical processes (e.g., virus spreading, traffic jams) that evolve on these networks are shaped and constrained by the underlying connectivity. This thesis provides numerical methods to study stochastic spreading processes on complex networks. We consider the processes as inherently probabilistic and analyze their behavior through the lens of probability theory. While powerful theoretical frameworks (like the SIS-epidemic model and continuous-time Markov chains) already exist, their analysis is computationally challenging. A key contribution of the thesis is to ease the computational burden of these methods. Particularly, we provide novel methods for the efficient stochastic simulation of these processes. Based on different simulation studies, we investigate techniques for optimal vaccine distribution and critically address the usage of mathematical models during the Covid-19 pandemic. We also provide model-reduction techniques that translate complicated models into simpler ones that can be solved without resorting to simulations. Lastly, we show how to infer the underlying contact data from node-level observations.Komplexe, interagierende Systeme sind in Natur und Gesellschaft allgegenwärtig. Die computergestützte Modellierung dieser Systeme ist daher von immenser Bedeutung für Wissenschaft und Technik. Netzwerke sind eine gängige Art, diese Systeme zu repräsentieren (z. B. Freundschaftsnetzwerke, Straßennetze). Dynamische Prozesse (z. B. Epidemien, Staus), die sich auf diesen Netzwerken ausbreiten, werden durch die spezifische Konnektivität geformt. In dieser Arbeit werden numerische Methoden zur Untersuchung stochastischer Ausbreitungsprozesse in komplexen Netzwerken entwickelt. Wir betrachten die Prozesse als inhärent probabilistisch und analysieren ihr Verhalten nach wahrscheinlichkeitstheoretischen Fragestellungen. Zwar gibt es bereits theoretische Grundlagen und Paradigmen (wie das SIS-Epidemiemodell und zeitkontinuierliche Markov-Ketten), aber ihre Analyse ist rechnerisch aufwändig. Ein wesentlicher Beitrag dieser Arbeit besteht darin, die Rechenlast dieser Methoden zu verringern. Wir erforschen Methoden zur effizienten Simulation dieser Prozesse. Anhand von Simulationsstudien untersuchen wir außerdem Techniken für optimale Impfstoffverteilung und setzen uns kritisch mit der Verwendung mathematischer Modelle bei der Covid-19-Pandemie auseinander. Des Weiteren führen wir Modellreduktionen ein, mit denen komplizierte Modelle in einfachere umgewandelt werden können. Abschließend zeigen wir, wie man von Beobachtungen einzelner Knoten auf die zugrunde liegende Netzwerkstruktur schließt
Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks
We analyze two alterations of the standard susceptible-infected-susceptible
(SIS) dynamics that preserve the central properties of spontaneous healing and
infection capacity of a vertex increasing unlimitedly with its degree. All
models have the same epidemic thresholds in mean-field theories but depending
on the network properties, simulations yield a dual scenario, in which the
epidemic thresholds of the modified SIS models can be either dramatically
altered or remain unchanged in comparison with the standard dynamics. For
uncorrelated synthetic networks having a power-law degree distribution with
exponent , the SIS dynamics are robust exhibiting essentially the
same outcomes for all investigated models. A threshold in better agreement with
the heterogeneous rather than quenched mean-field theory is observed in the
modified dynamics for exponent . Differences are more remarkable
for where a finite threshold is found in the modified models in
contrast with the vanishing threshold of the original one. This duality is
elucidated in terms of epidemic lifespan on star graphs. We verify that the
activation of the modified SIS models is triggered in the innermost component
of the network given by a -core decomposition for while it
happens only for , the
activation in the modified dynamics is collective involving essentially the
whole network while it is triggered by hubs in the standard SIS. The duality
also appears in the finite-size scaling of the critical quantities where
mean-field behaviors are observed for the modified, but not for the original
dynamics. Our results feed the discussions about the most proper conceptions of
epidemic models to describe real systems and the choices of the most suitable
theoretical approaches to deal with these models.Comment: 13 pages, 8 figure
Efficient sampling of spreading processes on complex networks using a composition and rejection algorithm
Efficient stochastic simulation algorithms are of paramount importance to the
study of spreading phenomena on complex networks. Using insights and analytical
results from network science, we discuss how the structure of contacts affects
the efficiency of current algorithms. We show that algorithms believed to
require or even operations per
update---where is the number of nodes---display instead a polynomial
scaling for networks that are either dense or sparse and heterogeneous. This
significantly affects the required computation time for simulations on large
networks. To circumvent the issue, we propose a node-based method combined with
a composition and rejection algorithm, a sampling scheme that has an
average-case complexity of per update for general
networks. This systematic approach is first set-up for Markovian dynamics, but
can also be adapted to a number of non-Markovian processes and can enhance
considerably the study of a wide range of dynamics on networks.Comment: 12 pages, 7 figure
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
Griffiths phases in infinite-dimensional, non-hierarchical modular networks
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs
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