Topological data analysis of contagion maps for examining spreading processes on networks

Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth's surface; however, in modern contagions long-range edges -- for example, due to airline transportation or communication media -- allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct "contagion maps" that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.Comment: Main Text and Supplementary Informatio

Emergence of hysteresis loop in social contagions on complex networks

Understanding the spreading mechanisms of social contagions in complex network systems has attracted much attention in the physics community. Here we propose a generalized threshold model to describe social contagions. Using extensive numerical simulations and theoretical analyses, we find that a hysteresis loop emerges in the system. Specifically, the steady state of the system is sensitive to the initial conditions of the dynamics of the system. In the steady state, the adoption size increases discontinuously with the transmission probability of information about social contagions, and trial size exhibits a non-monotonic pattern, i.e., it first increases discontinuously then decreases continuously. Finally we study social contagions on heterogeneous networks and find that network topology does not qualitatively affect our results.This work was funded in part by the National Key Research and Development Program (Grant No. 2016YFB0800602), the National Natural Science the Foundation of China (Grant Nos 61472045,61573067), and the China Scholarship Council. (2016YFB0800602 - National Key Research and Development Program; 61472045 - National Natural Science the Foundation of China; 61573067 - National Natural Science the Foundation of China; China Scholarship Council)Published versio

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

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

Optimal modularity and memory capacity of neural reservoirs

The neural network is a powerful computing framework that has been exploited by biological evolution and by humans for solving diverse problems. Although the computational capabilities of neural networks are determined by their structure, the current understanding of the relationships between a neural network's architecture and function is still primitive. Here we reveal that neural network's modular architecture plays a vital role in determining the neural dynamics and memory performance of the network of threshold neurons. In particular, we demonstrate that there exists an optimal modularity for memory performance, where a balance between local cohesion and global connectivity is established, allowing optimally modular networks to remember longer. Our results suggest that insights from dynamical analysis of neural networks and information spreading processes can be leveraged to better design neural networks and may shed light on the brain's modular organization