Adaptive networks have been recently introduced in the context of disease propagation on complex networks. They account for the mutual interaction between the network topology and the states of the nodes. Until now, existing models have been analyzed using low complexity analytical formalisms, revealing nevertheless some novel dynamical features. However, current methods have failed to reproduce with accuracy the simultaneous time evolution of the disease and the underlying network topology. In the framework of the adaptive susceptible-infectious-susceptible (SIS) model of Gross [Phys. Rev. Lett. 96, 208701 (2006)]10.1103/PhysRevLett.96.208701, we introduce an improved compartmental formalism able to handle this coevolutionary task successfully. With this approach, we analyze the interplay and outcomes of both dynamical elements, process and structure, on adaptive networks featuring different degree distributions at the initial stage.

Epidemics seldom occur as isolated phenomena. Typically, two or more viral agents spread within the same host population and may interact dynamically with each other. We present a general model where two viral agents interact via an immunity mechanism as they propagate simultaneously on two networks connecting the same set of nodes. By exploiting a correspondence between the propagation dynamics and a dynamical process performing progressive network generation, we develop an analytical approach that accurately captures the dynamical interaction between epidemics on overlay networks. The formalism allows for overlay networks with arbitrary joint degree distribution and overlap. To illustrate the versatility of our approach, we consider a hypothetical delayed intervention scenario in which an immunizing agent is disseminated in a host population to hinder the propagation of an undesirable agent (e.g., the spread of preventive information in the context of an emerging infectious disease).

Considerable attention has been paid, in recent years, to the use of networks in modeling complex real-world systems. Among the many dynamical processes involving networks, propagation processes-in which the final state can be obtained by studying the underlying network percolation properties-have raised formidable interest. In this paper, we present a bond percolation model of multitype networks with an arbitrary joint degree distribution that allows heterogeneity in the edge occupation probability. As previously demonstrated, the multitype approach allows many nontrivial mixing patterns such as assortativity and clustering between nodes. We derive a number of useful statistical properties of multitype networks as well as a general phase transition criterion. We also demonstrate that a number of previous models based on probability generating functions are special cases of the proposed formalism. We further show that the multitype approach, by naturally allowing heterogeneity in the bond occupation probability, overcomes some of the correlation issues encountered by previous models. We illustrate this point in the context of contact network epidemiology.

Epidemics occur in all shapes and forms: infections propagating in our sparse sexual networks, rumours and diseases spreading through our much denser social interactions, or viruses circulating on the Internet. With the advent of large databases and efficient analysis algorithms, these processes can be better predicted and controlled. In this study, we use different characteristics of network organization to identify the influential spreaders in 17 empirical networks of diverse nature using 2 epidemic models. We find that a judicious choice of local measures, based either on the network's connectivity at a microscopic scale or on its community structure at a mesoscopic scale, compares favorably to global measures, such as betweenness centrality, in terms of efficiency, practicality and robustness. We also develop an analytical framework that highlights a transition in the characteristic scale of different epidemic regimes. This allows to decide which local measure should govern immunization in a given scenario.

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Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.

Analytical description of propagation phenomena on random networks has flourished in recent years, yet more complex systems have mainly been studied through numerical means. In this paper, a mean-field description is used to coherently couple the dynamics of the network elements (nodes, vertices, individuals...) on the one hand and their recurrent topological patterns (subgraphs, groups...) on the other hand. In a SIS model of epidemic spread on social networks with community structure, this approach yields a set of ODEs for the time evolution of the system, as well as analytical solutions for the epidemic threshold and equilibria. The results obtained are in good agreement with numerical simulations and reproduce random networks behavior in the appropriate limits which highlights the influence of topology on the processes. Finally, it is demonstrated that, in the absence of degree correlation, our model predicts higher epidemic thresholds for clustered structures than for equivalent random topologies.

We introduce a mechanism which models the emergence of the universal properties of complex networks, such as scale independence, modularity and self-similarity, and unifies them under a scale-free organization beyond the link. This brings a new perspective on network organization where communities, instead of links, are the fundamental building blocks of complex systems. We show how our simple model can reproduce social and information networks by predicting their community structure and more importantly, how their nodes or communities are interconnected, often in a self-similar manner.PACS numbers: 89.75. Da, 89.75.Fb, 89.75.Hc, 89.75.Kd, 89.65.Ef A universal matter. Reducing complex systems to their simplest possible form while retaining their important properties helps model their behavior independently of their nature. Results obtained via these abstract models can then be transferred to other systems sharing a similar simplest form. Such groups of analog systems are called universality classes and are the reason why some models apply just as well to the sizes of earthquakes or solar flares than to the sales number of books or music recordings [1]. That is, their statistical distributions can be reproduced by the same mechanism: preferential attachment. This mechanism has been of special interest to network science [2] because it models the emergence of power-law distributions for the number of links per node. This particular feature is one of the universal properties of network structure [3], alongside modularity [4] and selfsimilarity [5]. Previous studies have focused on those properties one at a time [3][4][5][6][7][8], yet a unified point of view is still wanting. In this Letter, we present an overarching model of preferential attachment that unifies the universal properties of network organization under a single principle.Preferential attachment is one of the most ubiquitous mechanisms describing how elements are distributed within complex systems. More precisely, it predicts the emergence of scale-free (power-law) distributions where the probability P k of occurrence of an event of order k decreases as an inverse power of k (i.e., P k ∝ k −γ with γ > 0). It was initially introduced outside the realm of network science by Yule [9] as a mathematical model of evolution explaining the power-law distribution of biological genera by number of species. Independently, Gibrat [10] formulated a similar idea as a law governing the growth rate of incomes. Gibrat's law is the sole assumption behind preferential attachment: the growth rates of entities in a system are proportional to their size. Yet, preferential attachment is perhaps better described using Simon's general balls-in-bins process [11].Simon's model was developed for the distribution of words by their frequency of occurrence in a prose sample [12]. The problem is the following: what is the probability P k+1 (i + 1) that the (i + 1)-th word of a text is a word that has already appeared k times? By simply stating that P k+1 (i + 1) ∝ k · P k (i), Sim...