Long ties accelerate noisy threshold-based contagions

Network structure can affect when and how widely new ideas, products, and behaviors are adopted. In widely-used models of biological contagion, interventions that randomly rewire edges (generally making them "longer") accelerate spread. However, there are other models relevant to social contagion, such as those motivated by myopic best-response in games with strategic complements, in which an individual's behavior is described by a threshold number of adopting neighbors above which adoption occurs (i.e., complex contagions). Recent work has argued that highly clustered, rather than random, networks facilitate spread of these complex contagions. Here we show that minor modifications to this model, which make it more realistic, reverse this result: we allow very rare below-threshold adoption, i.e., rarely adoption occurs when there is only one adopting neighbor. To model the trade-off between long and short edges we consider networks that are the union of cycle-power-$k$ graphs and random graphs on $n$ nodes. Allowing adoptions below threshold to occur with order $1/\sqrt{n}$ probability along some "short" cycle edges is enough to ensure that random rewiring accelerates spread. Simulations illustrate the robustness of these results to other commonly-posited models for noisy best-response behavior. Hypothetical interventions that randomly rewire existing edges or add random edges (versus adding "short", triad-closing edges) in hundreds of empirical social networks reduce time to spread. This revised conclusion suggests that those wanting to increase spread should induce formation of long ties, rather than triad-closing ties. More generally, this highlights the importance of noise in game-theoretic analyses of behavior

Adaptive Random Walks on the Class of Web Graph

We study random walk with adaptive move strategies on a class of directed graphs with variable wiring diagram. The graphs are grown from the evolution rules compatible with the dynamics of the world-wide Web [Tadi\'c, Physica A {\bf 293}, 273 (2001)], and are characterized by a pair of power-law distributions of out- and in-degree for each value of the parameter $\beta$, which measures the degree of rewiring in the graph. The walker adapts its move strategy according to locally available information both on out-degree of the visited node and in-degree of target node. A standard random walk, on the other hand, uses the out-degree only. We compute the distribution of connected subgraphs visited by an ensemble of walkers, the average access time and survival probability of the walks. We discuss these properties of the walk dynamics relative to the changes in the global graph structure when the control parameter $\beta$ is varied. For $\beta \geq 3$, corresponding to the world-wide Web, the access time of the walk to a given level of hierarchy on the graph is much shorter compared to the standard random walk on the same graph. By reducing the amount of rewiring towards rigidity limit \beta \to \beta_c \lesss im 0.1, corresponding to the range of naturally occurring biochemical networks, the survival probability of adaptive and standard random walk become increasingly similar. The adaptive random walk can be used as an efficient message-passing algorithm on this class of graphs for large degree of rewiring.Comment: 8 pages, including 7 figures; to appear in Europ. Phys. Journal

Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks

The aim of the study was to compare the epidemic spread on static and dynamic small-world networks. The network was constructed as a 2-dimensional Watts-Strogatz model (500x500 square lattice with additional shortcuts), and the dynamics involved rewiring shortcuts in every time step of the epidemic spread. The model of the epidemic is SIR with latency time of 3 time steps. The behaviour of the epidemic was checked over the range of shortcut probability per underlying bond 0-0.5. The quantity of interest was percolation threshold for the epidemic spread, for which numerical results were checked against an approximate analytical model. We find a significant lowering of percolation thresholds for the dynamic network in the parameter range given. The result shows that the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7 +/- 1.4%, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small-world we observe suppression of the average epidemic size dependence on network size in comparison with finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to latency time of the disease.Comment: 13 pages, 6 figure

Epidemic processes in complex networks

In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.Comment: 62 pages, 15 figures, final versio