Threshold models of cascades in the social sciences and economics explain the spread of opinion and innovation due to social influence. In threshold cascade models, fads or innovations spread between agents as determined by their interactions with other agents and their personal threshold of resistance. Typically, these models do not account for structure in the timing of interaction between the units. In this work, we extend a model of social cascades by Duncan Watts to temporal interaction networks. In our model, we assume friends and acquaintances influence agents for a certain time into the future. That is the influence of the past ages and becomes unimportant. Thus, our modified cascade model has an effective time window of influence. We explore two types of thresholds -- thresholds to fractions of the neighbors or absolute numbers. We try our model on six empirical datasets and compare them with null models.
In this study, we propose an extended Watts model to examine the effect of initiators on information cascades. The extended Watts model assumes that nodes with connections to initiators have low adoption thresholds than other nodes, due to the significant influence of initiators. We develop a tree approximation to describe the active node fraction for the extended Watts model in random networks and derive the cascade condition for a global cascade to occur with a small fraction of initiators. By analyzing the active node fraction and the cascade window of the extended Watts model on the Erdos-…
We study an extension of Duncan Watts' 2002 model of information cascades in social networks where edge weights are taken to be random, an innovation motivated by recent applications of cascade analysis to systemic risk in financial networks. The main result is a probabilistic analysis that characterizes the cascade in an infinite network as the fixed point of a vector-valued mapping, explicit in terms of convolution integrals that can be efficiently evaluated numerically using the fast Fourier transform algorithm. A second result gives an approximate probabilistic analysis of cascades on "rea…
Maintenance of cooperation was studied for a two-strategy evolutionary Prisoner's Dilemma game where the players are located on a one-dimensional chain and their payoff comes from games with the nearest and next-nearest neighbor interactions. The applied host geometry makes possible to study the impacts of two conflicting topological features. The evolutionary rule involves some noise affecting the strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the phase diagram as a function of noise leve…
We consider a dynamic network cascade process developed by Watts applied to a random networks with a specified amount of clustering, belonging to a class of random networks developed by Newman. We adapt existing tree-based methods to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node, and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a special case of a formula of Hackett et al. giving conditions for certain extinction of the cascade.
We consider in this paper the potential for ex ante prediction of the cascade size in a model of binary choice with externalities (Schelling 1973, Watts 2002). Agents are connected on a network and can be in one of two states of the world, 0 or 1. Initially, all are in state 0 and a small number of seeds are selected at random to switch to state1. A simple threshold rule specifies whether other agents switch subsequently. The cascade size (the percolation) is the proportion of all agents which eventually switches to state 1. We select information on the connectivity of the initial seeds, the c…
We present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced in [M.E.J. Newman, Phys. Rev. Lett. 103, 058701 (2009)]. A condition for the existence of global cascades is derived as well as a general criterion which determines whether increasing the level of clustering will increase, or decrease, the expected cascade size. Applications, examples of which are provided, include site percolation, bond percolation, and Watts' threshold model; in all…
Technical University of Munich, School of Computation Information and Technology, Department of Mathematics（慕尼黑工业大学计算信息与技术学院数学系）；Munich Data Science Institute（慕尼黑数据科学研究所）；Complexity Science Hub Vienna（维也纳复杂性科学中心）
We study the homogeneous symmetrical threshold model with independence (noise) by pair approximation and Monte Carlo simulations on Watts-Strogatz graphs. The model is a modified version of the famous Granovetter's threshold model: with probability $p$ a voter acts independently, i.e. takes randomly one of two states +/-; with complementary probability $1-p$, a voter takes a given state, if sufficiently large fraction (above a given threshold r) of individuals in its neighborhood is in this state. We show that the character of the phase transition, induced by the noise parameter p, depends on …
Elements of networks interact in many ways, so modeling them with graphs requires multiple types of edges (or network layers). Here we show that such multiplex networks are generically more vulnerable to global cascades than simplex networks. We generalize the threshold cascade model [D. J. Watts, Proc. Natl. Acad. Sci. U.S.A. 99, 5766 (2002)] to multiplex networks, in which a node activates if a sufficiently large fraction of neighbors in any layer are active. We show that both combining layers (i.e., realizing other interactions play a role) and splitting a network into layers (i.e., recogni…
We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so-called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and posses an up-down symmetry, which makes it similar to other binary opinion models with a single-flip dynamics, such as the majority-vote and the q-voter models. Because within the majority-vote model w…
Cascades on random networks are typically analyzed by assuming they map onto percolation processes and then are solved using generating function formulations. This approach assumes that the network is infinite and weakly connected, yet furthermore approximates a dynamic cascading process as a static percolation event. In this paper we propose a dynamic Markov model formulation that assumes a finite network with arbitrary average nodal degree. We apply it to the case where cascades follow a threshold rule, that is, that a node will change state ("flip") only if a fraction, exceeding a given thr…
Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy or engineering. Tipping points are critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change of the system. Many systems with tipping points can be modeled as networks of coupled multistable subsystems, e.g. coupled patches of vegetation, connected lakes, interacting climate tipping elements or multiscale infrastructure systems. In such networks, tipping events in one subsystem are able to induce tipping cascades via domino eff…
An Approximate-Master-Equation Formulation of the Watts Threshold Model on Hypergraphs
University of California, Los Angeles（加州大学洛杉矶分校）；Vermont Complex Systems Institute, University of Vermont（佛蒙特大学复杂系统研究所）；Korea University（韩国大学）；Santa Fe Institute（圣塔菲研究所）
Threshold models try to explain the consequences of social influence like the spread of fads and opinions. Along with models of epidemics, they constitute a major theoretical framework of social spreading processes. In threshold models on static networks, an individual changes her state if a certain fraction of her neighbors has done the same. When there are strong correlations in the temporal aspects of contact patterns, it is useful to represent the system as a temporal network. In such a system, not only contacts but also the time of the contacts are represented explicitly. There is a conse…
How big is the risk that a few initial failures of networked nodes amplify to large cascades that endanger the functioning of the system? Common answers refer to the average final cascade size. Two analytic approaches allow its computation: a) (heterogeneous) mean field approximation and b) belief propagation. The former applies to (infinitely) large locally tree-like networks, while the latter is exact on finite trees. Yet, cascade sizes can have broad and multi-modal distributions that are not well represented by their average. Full distribution information is essential to identify likely ev…
We propose a multistage version of the independent cascade model, which we call a multistage independent cascade (MIC) model, on networks. This model is parameterized by two probabilities: the probability $T_1$ that a node adopting a fad increases the awareness of a neighboring susceptible node, and the probability $T_2$ that an adopter directly causes a susceptible node to adopt the fad. We formulate a tree approximation for the MIC model on an uncorrelated network with an arbitrary degree distribution $p_k$. Applied on a random regular network with degree $k=6$, this model exhibits a rich ph…
Many complex networks are described by directed links; in such networks, a link represents, for example, the control of one node over the other node or unidirectional information flows. Some centrality measures are used to determine the relative importance of nodes specifically in directed networks. We analyze such a centrality measure called the influence. The influence represents the importance of nodes in various dynamics such as synchronization, evolutionary dynamics, random walk, and social dynamics. We analytically calculate the influence in various networks, including directed multipart…
Multilayer systems are coupled networks characterized by different contexts (layers) of interaction and have gained much attention recently due to their suitability to describe a broad spectrum of empirical complex systems. They are very fragile to percolation and first-neighbor failure propagation, but little is known about how they respond to non-local disruptions, as it occurs in failures induced by flow redistribution, for example. Acknowledging that many socio-technical and biological systems sustain a flow of some physical quantity, such as energy or information, across the their compone…
We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds --- a central problem in the study of network cascades. The majority of existing work on this problem, formally referred to as the influence maximization problem, is designed for submodular cascades. Despite the empirical evidence that many cascades are non-submodular, little work has been done focusing on non-submodular influence maximization. We propose a new heuristic for solving the influence maximization problem and show via simulations on real-world and synthetic …
In an effort to understand why individuals choose to participate in personally-expensive pro-environmental behaviors, environmental and behavioral economists have examined a moral-motivation model in which the decision to adopt a pro-environmental behavior depends on the society-wide market share of that behavior. An increasing body of practical research on adoption of pro-environmental behavior emphasizes the importance of encouragement from local social contacts and messaging about locally-embraced norms: we respond by extending the moral-motivation model to a social networks setting. We obt…
In this paper, we analyze dynamic switching networks, wherein the networks switch arbitrarily among a set of topologies. For this class of dynamic networks, we derive an epidemic threshold, considering the SIS epidemic model. First, an epidemic probabilistic model is developed assuming independence between states of nodes. We identify the conditions under which the epidemic dies out by linearizing the underlying dynamical system and analyzing its asymptotic stability around the origin. The concept of joint spectral radius is then used to derive the epidemic threshold, which is later validated …
