47 research outputs found
A network reduction method inducing scale-free degree distribution
International audienceThis paper deals with the problem of graph reduction towards a scale-free graph while preserving a consistency with the initial graph. This problem is formulated as a minimization problem and to this end we define a metric to measure the scale-freeness of a graph and another metric to measure the similarity between two graphs with different dimensions, based on spectral centrality. We also want to ensure that if the initial network is a flow network, the reduced network preserves this property. We explore the optimization problem and, based on the gained insights, we derive an algorithm allowing to find an approximate solution. Finally, the effectiveness of the algorithm is shown through a simulation on a Manhattan-like network
Systemic Risk in a Unifying Framework for Cascading Processes on Networks
We introduce a general framework for models of cascade and contagion
processes on networks, to identify their commonalities and differences. In
particular, models of social and financial cascades, as well as the fiber
bundle model, the voter model, and models of epidemic spreading are recovered
as special cases. To unify their description, we define the net fragility of a
node, which is the difference between its fragility and the threshold that
determines its failure. Nodes fail if their net fragility grows above zero and
their failure increases the fragility of neighbouring nodes, thus possibly
triggering a cascade. In this framework, we identify three classes depending on
the way the fragility of a node is increased by the failure of a neighbour. At
the microscopic level, we illustrate with specific examples how the failure
spreading pattern varies with the node triggering the cascade, depending on its
position in the network and its degree. At the macroscopic level, systemic risk
is measured as the final fraction of failed nodes, , and for each of
the three classes we derive a recursive equation to compute its value. The
phase diagram of as a function of the initial conditions, thus allows
for a prediction of the systemic risk as well as a comparison of the three
different model classes. We could identify which model class lead to a
first-order phase transition in systemic risk, i.e. situations where small
changes in the initial conditions may lead to a global failure. Eventually, we
generalize our framework to encompass stochastic contagion models. This
indicates the potential for further generalizations.Comment: 43 pages, 16 multipart figure
Mean-field models for non-Markovian epidemics on networks
This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based compartmental modelling is just one of many different approaches used to model the spread of an infectious disease on a network; the major result of this paper is the rigorous proof that the edge-based compartmental model and the message passing models are equivalent for general independent transmission and recovery processes. This implies that the new model is exact on the ensemble of configuration model networks of infinite size. For the case of Markovian transmission themessage passing model is re-parametrised into a pairwise-like model which is then used to derive many well-known pairwise models for regular networks, or when the infectious period is exponentially distributed or is of a fixed length
A General Model of Dynamics on Networks with Graph Automorphism Lumping
In this paper we introduce a general Markov chain model of dynamical processes on networks. In this model, nodes in the network can adopt a finite number of states and transitions can occur that involve multiple nodes changing state at once. The rules that govern transitions only depend on measures related to the state and structure of the network and not on the particular nodes involved. We prove that symmetries of the network can be used to lump equivalent states in state-space. We illustrate how several examples of well-known dynamical processes on networks correspond to particular cases of our general model. This work connects a wide range of models specified in terms of node-based dynamical rules to their exact continuous-time Markov chain formulation
The Kuramoto model in complex networks
181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin
Statistical physics of vaccination
Historically, infectious diseases caused considerable damage to human societies, and they continue to do so today. To help reduce their impact, mathematical models of disease transmission have been studied to help understand disease dynamics and inform prevention strategies. Vaccination–one of the most important preventive measures of modern times–is of great interest both theoretically and empirically. And in contrast to traditional approaches, recent research increasingly explores the pivotal implications of individual behavior and heterogeneous contact patterns in populations. Our report reviews the developmental arc of theoretical epidemiology with emphasis on vaccination, as it led from classical models assuming homogeneously mixing (mean-field) populations and ignoring human behavior, to recent models that account for behavioral feedback and/or population spatial/social structure. Many of the methods used originated in statistical physics, such as lattice and network models, and their associated analytical frameworks. Similarly, the feedback loop between vaccinating behavior and disease propagation forms a coupled nonlinear system with analogs in physics. We also review the new paradigm of digital epidemiology, wherein sources of digital data such as online social media are mined for high-resolution information on epidemiologically relevant individual behavior. Armed with the tools and concepts of statistical physics, and further assisted by new sources of digital data, models that capture nonlinear interactions between behavior and disease dynamics offer a novel way of modeling real-world phenomena, and can help improve health outcomes. We conclude the review by discussing open problems in the field and promising directions for future research