Modeling the dynamical interaction between epidemics on overlay networks

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. Exploiting a correspondence between the propagation dynamics and a dynamical process performing progressive network generation, we develop an analytic 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).Comment: Accepted for publication in Phys. Rev. E. 15 pages, 7 figure

Percolation on random networks with arbitrary k-core structure

The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a Hard-core Random Network model that generates maximally random networks with arbitrary degree distribution and arbitrary k-core structure. We then solve exactly the bond percolation problem on the HRN model and produce fast and precise analytical estimates for the corresponding real networks. Extensive comparison with selected databases reveals that our approach performs better than existing models, while requiring less input information.Comment: 9 pages, 5 figure

Growing networks of overlapping communities with internal structure

We introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks. The model comprises two complementary mechanisms: One mechanism accounts for the evolution of the internal link structure of a single community, and the second mechanism coordinates the growth of multiple overlapping communities. The first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates. We demonstrate that this simple mechanism gives rise to an effective maximal degree within communities. This observation is related to the anthropological theory known as Dunbar's number, i.e., the empirical observation of a maximal number of ties which an average individual can sustain within its social groups. The second mechanism is based on a recently proposed generalization of preferential attachment to community structure, appropriately called structural preferential attachment (SPA). The combination of these two mechanisms into a single model (SPA+) allows us to reproduce a number of the global statistics of real networks: The distribution of community sizes, of node memberships and of degrees. The SPA+ model also predicts (a) three qualitative regimes for the degree distribution within overlapping communities and (b) strong correlations between the number of communities to which a node belongs and its number of connections within each community. We present empirical evidence that support our findings in real complex networks.Comment: 14 pages, 8 figures, 2 table

Exact solution of bond percolation on small arbitrary graphs

We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution of the node partitions (i.e., the constrained distribution of the size of each component) in undirected graphs. Besides opening the way to the theoretical prediction of percolation on arbitrary graphs of large but finite size, we show how our results find application in graph theory, epidemiology, percolation and fragmentation theory.Comment: 5 pages and 3 figure

Complex networks as an emerging property of hierarchical preferential attachment

Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties universally emerge. We propose that an important class of complex systems can be modeled as an organization of many embedded levels (potentially infinite in number), all of them following the same universal growth principle known as preferential attachment. We give examples of such hierarchy in real systems, for instance in the pyramid of production entities of the film industry. More importantly, we show how real complex networks can be interpreted as a projection of our model, from which their scale independence, their clustering, their hierarchy, their fractality and their navigability naturally emerge. Our results suggest that complex networks, viewed as growing systems, can be quite simple, and that the apparent complexity of their structure is largely a reflection of their unobserved hierarchical nature.Comment: 12 pages, 7 figure

Adaptive networks: coevolution of disease and topology

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 analytic 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 SIS model of Gross et al. [Phys. Rev. Lett. 96, 208701 (2006)], 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.Comment: 11 pages, 8 figures, 1 appendix. To be published in Physical Review

Field-Induced Magnetization Steps in Intermetallic Compounds and Manganese Oxides: The Martensitic Scenario

Field-induced magnetization jumps with similar characteristics are observed at low temperature for the intermetallic germanide Gd5Ge4and the mixed-valent manganite Pr0.6Ca0.4Mn0.96Ga0.04O3. We report that the field location -and even the existence- of these jumps depends critically on the magnetic field sweep rate used to record the data. It is proposed that, for both compounds, the martensitic character of their antiferromagnetic-to-ferromagnetic transitions is at the origin of the magnetization steps.Comment: 4 pages,4 figure

Propagation dynamics on networks featuring complex topologies

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 our model predicts higher epidemic thresholds for clustered structures than for equivalent random topologies in the case of networks with zero degree correlation.Comment: 10 pages, 5 figures, 1 Appendix. Published in Phys. Rev. E (mistakes in the PRE version are corrected here

A shadowing problem in the detection of overlapping communities: lifting the resolution limit through a cascading procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.Comment: 14 pages, 12 figures + supporting information (5 pages, 6 tables, 3 figures