Random links between otherwise distant nodes can greatly facilitate the propagation of disease or information, provided contagion can be transmitted by a single active node. However we show that when the propagation requires simultaneous exposure to multiple sources of activation, called multiplex propagation, the effect of random links is just the opposite: it makes the propagation more difficult to achieve. We calculate analytical and numerically critical points for a threshold model in several classes of complex networks, including an empirical social network.
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…
Threshold cascade models have been used to describe spread of behavior in social networks and cascades of default in financial networks. In some cases, these networks may have multiple kinds of interactions, such as distinct types of social ties or distinct types of financial liabilities; furthermore, nodes may respond in different ways to in influence from their neighbors of multiple types. To start to capture such settings in a stylized way, we generalize a threshold cascade model to a multiplex network in which nodes follow one of two response rules: some nodes activate when, in at least on…
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…
Observed transmission line outage data is grouped into successive generations of events. The empirical distribution of the number of generations in the cascades follows a Zipf distribution that implies the increasing propagation as cascades progress. The slope of the Zipf distribution gives a System Event Propagation Slope Index (SEPSI). This new metric quantifies the cascade propagation, varies as expected, and determines the probabilities of small, medium, and large cascades.
The seniority of debt, which determines the order in which a bankrupt institution repays its debts, is an important and sometimes contentious feature of financial crises, yet its impact on system-wide stability is not well understood. We capture seniority of debt in a multiplex network, a graph of nodes connected by multiple types of edges. Here, an edge between banks denotes a debt contract of a certain level of seniority. Next we study cascading default. There exist multiple kinds of bankruptcy, indexed by the highest level of seniority at which a bank cannot repay all its debts. Self-intere…
Models of threshold driven contagion explain the cascading spread of information, behavior, systemic risk, and epidemics on social, financial and biological networks. At odds with empirical observation, these models predict that single-layer unweighted networks become resistant to global cascades after reaching sufficient connectivity. We investigate threshold driven contagion on weight heterogeneous multiplex networks and show that they can remain susceptible to global cascades at any level of connectivity, and with increasing edge density pass through alternating phases of stability and inst…
Rumor spreading on online social media is presenting a significant threat to society of post-truth epoch. Extensive efforts have been devoted to rumor identification and debunking, assuming that a specific rumor propagation is a single event network and neglecting possible interdependence between different rumor cascades. Here we study the collective propagation of multiple rumors, and surprisingly find a network of users that repeatedly participate in different rumor cascades. Though these repetitive users demonstrate minor difference at the level of single propagation network, they are found…
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）
Many real-world multilayer systems such as critical infrastructure are interdependent and embedded in space with links of a characteristic length. They are also vulnerable to localized attacks or failures, such as terrorist attacks or natural catastrophes, which affect all nodes within a given radius. Here we study the effects of localized attacks on spatial multiplex networks of two layers. We find a metastable region where a localized attack larger than a critical size induces a nucleation transition as a cascade of failures spreads throughout the system, leading to its collapse. We develop …
We study cascades on a two-layer multiplex network, with asymmetric feedback that depends on the coupling strength between the layers. Based on an analytical branching process approximation, we calculate the systemic risk measured by the final fraction of failed nodes on a reference layer. The results are compared with the case of a single layer network that is an aggregated representation of the two layers. We find that systemic risk in the two-layer network is smaller than in the aggregated one only if the coupling strength between the two layers is small. Above a critical coupling strength,…
We study the propagation of information in social networks. To do so, we focus on a cascade model where nodes are infected with {probability $p_1$ after their first contact with the information and with probability $p_2$ at all subsequent contacts.} The diffusion starts from one random node and leads to a cascade of infection. It is shown that first and {subsequent} trials play different roles in the propagation and that the size of the cascade depends in a non-trivial way on $p_1$, $p_2$ and on the network structure. Second trials are shown to amplify the propagation in dense parts of the net…
Motivated by online social networks that are linked together through overlapping users, we study the influence maximization problem on a multiplex, with each layer endowed with its own model of influence diffusion. This problem is a novel version of the influence maximization problem that necessitates new analysis incorporating the type of propagation on each layer of the multiplex. We identify a new property, generalized deterministic submodular, which when satisfied by the propagation in each layer, ensures that the propagation on the multiplex overall is submodular -- for this case, we form…
A fundamental problem in network science is to predict how certain individuals are able to initiate new networks to spring up "new ideas". Frequently, these changes in trends are triggered by a few innovators who rapidly impose their ideas through "viral" influence spreading producing cascades of followers fragmenting an old network to create a new one. Typical examples include the raise of scientific ideas or abrupt changes in social media, like the raise of Facebook.com to the detriment of Myspace.com. How this process arises in practice has not been conclusively demonstrated. Here, we show …
Distinct channels of interaction in a complex networked system define network layers, which co-exist and co-operate for the system's function. Towards realistic modeling and understanding such multiplex systems, we introduce and study a class of growing multiplex network models in which different network layers coevolve, and examine how the entangled growth of coevolving layers can shape the overall network structure. We show analytically and numerically that the coevolution can induce strong degree correlations across layers, as well as modulate degree distributions. We further show that such…
A general theory of top-down cascades in complex networks is described which explains two similar types of perturbation amplifications in the complex networks of business supply chains (the `bullwhip effect') and ecological food webs (trophic cascades). The dependence of the strength of the effects on the interaction strength and covariance in the dynamics as well as the graph structure allows both explanation and prediction of widely recognized effects in each type of system.
A more connected world has brought major consequences such as facilitate the spread of diseases all over the world to quickly become epidemics, reason why researchers are concentrated in modeling the propagation of epidemics and outbreaks in Multilayer Networks. In this networks all nodes interact in different layers with different type of links. However, in many scenarios such as in the society, a Multiplex Network framework is not completely suitable since not all individuals participate in all layers. In this paper, we use a partially overlapped Multiplex Network where only a fraction of th…
We present the mathematical analysis of generalized complex contagions in clustered multiplex networks for susceptible-infected-recovered (SIR)-like dynamics. The model is intended to understand diffusion of influence, or any other spreading process implying a threshold dynamics, in setups of interconnected networks with significant clustering. The contagion is assumed to be general enough to account for a content-dependent linear threshold model, where each link type has a different weight (for spreading influence) that may depend on the content (e.g., product, rumor, political view) that is …
The increasing integration of world economies, which organize in complex multilayer networks of interactions, is one of the critical factors for the global propagation of economic crises. We adopt the network science approach to quantify shock propagation on the global trade-investment multiplex network. To this aim, we propose a model that couples a Susceptible-Infected-Recovered epidemic spreading dynamics, describing how economic distress propagates between connected countries, with an internal contagion mechanism, describing the spreading of such economic distress within a given country. A…
We introduce the sandpile model on multiplex networks with more than one type of edge and investigate its scaling and dynamical behaviors. We find that the introduction of multiplexity does not alter the scaling behavior of avalanche dynamics; the system is critical with an asymptotic power-law avalanche size distribution with an exponent $τ= 3/2$ on duplex random networks. The detailed cascade dynamics, however, is affected by the multiplex coupling. For example, higher-degree nodes such as hubs in scale-free networks fail more often in the multiplex dynamics than in the simplex network count…
Complex contagion models have been developed to understand a wide range of social phenomena such as adoption of cultural fads, the diffusion of belief, norms, and innovations in social networks, and the rise of collective action to join a riot. Most existing works focus on contagions where individuals' states are represented by {\em binary} variables, and propagation takes place over a single isolated network. However, characterization of an individual's standing on a given matter as a binary state might be overly simplistic as most of our opinions, feelings, and perceptions vary over more tha…
Predicting cascade dynamics has important implications for understanding information propagation and launching viral marketing. Previous works mainly adopt a pair-wise manner, modeling the propagation probability between pairs of users using n^2 independent parameters for n users. Consequently, these models suffer from severe overfitting problem, specially for pairs of users without direct interactions, limiting their prediction accuracy. Here we propose to model the cascade dynamics by learning two low-dimensional user-specific vectors from observed cascades, capturing their influence and sus…
Nowadays, one of the challenges we face when carrying out modeling of epidemic spreading is to develop methods to control disease transmission. In this article we study how the spreading of knowledge of a disease affects the propagation of that disease in a population of interacting individuals. For that, we analyze the interaction between two different processes on multiplex networks: the propagation of an epidemic using the susceptible-infected-susceptible dynamics and the dissemination of information about the disease --and its prevention methods-- using the unaware-aware-unaware dynamics, …
In this work, we aim to contribute to the understanding of the human pro-social behavior by studying the influence that a particular form of social pressure "being watched" has on the evolution of cooperative behavior. We study how cooperation emerge in multiplex complex topologies by analyzing a particular bidirectionally-coupled dynamics on top of a two-layers multiplex network (duplex). The coupled dynamics appears between the Prisoner's Dilemma game in a network, and a threshold cascade model in the other. The threshold model is intended to abstract the behavior of a network of vigilant no…
With the emergence and rapid proliferation of social media platforms and social networking sites, recent years have witnessed a surge of misinformation spreading in our daily life. Drawing on a large-scale dataset which covers more than 1.4M posts and 18M comments, we investigate the propagation of two distinct narratives--(i) conspiracy information, whose claims are generally unsubstantiated and thus referred as misinformation to some extent, and (ii) scientific information, whose origins are generally readily identifiable and verifiable--in an online social media platform. We find that consp…
We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and for a given content (e.g., rumor, product, political view), each link type is associated with a content dependent parameter $c_i$ in $[0,\infty]$ that measures the relative bias type-$i$ links have in spreading this content. In this setting, we propose a linear threshold model of contagion where nodes switch state if their "perceived" proportion of active neighbors exceeds a threshold τ. Namely, a node connected to $m_i$ active neighbors and $k_i-m_i$ inactive neighbors via type-$i$ …
Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, $P(0)=0$. Above a critical value of a control parameter, the removal of a tiny fraction $Δ$ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discont…
Large cascades can develop in online social networks as people share information with one another. Though simple reshare cascades have been studied extensively, the full range of cascading behaviors on social media is much more diverse. Here we study how diffusion protocols, or the social exchanges that enable information transmission, affect cascade growth, analogous to the way communication protocols define how information is transmitted from one point to another. Studying 98 of the largest information cascades on Facebook, we find a wide range of diffusion protocols - from cascading reshare…
Exploiting the information about members of a Social Network (SN) represents one of the most attractive and dwelling subjects for both academic and applied scientists. The community of Complexity Science and especially those researchers working on multiplex social systems are devoting increasing efforts to outline general laws, models, and theories, to the purpose of predicting emergent phenomena in SN's (e.g. success of a product). On the other side the semantic web community aims at engineering a new generation of advanced services tailored to specific people needs. This implies defining con…
Accessing the network through which a propagation dynamics diffuse is essential for understanding and controlling it. In a few cases, such information is available through direct experiments or thanks to the very nature of propagation data. In a majority of cases however, available information about the network is indirect and comes from partial observations of the dynamics, rendering the network reconstruction a fundamental inverse problem. Here we show that it is possible to reconstruct the whole structure of an interaction network and to simultaneously infer the complete time course of acti…
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…
Most people simultaneously belong to several distinct social networks, in which their relations can be different. They have opinions about certain topics, which they share and spread on these networks, and are influenced by the opinions of other persons. In this paper, we build upon this observation to propose a new nodal centrality measure for multiplex networks. Our measure, called Opinion centrality, is based on a stochastic model representing opinion propagation dynamics in such a network. We formulate an optimization problem consisting in maximizing the opinion of the whole network when c…
Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）；Harvard Medical School（哈佛医学院）
Cascade models are central to understanding, predicting, and controlling epidemic spreading and information propagation. Related optimization, including influence maximization, model parameter inference, or the development of vaccination strategies, relies heavily on sampling from a model. This is either inefficient or inaccurate. As alternative, we present an efficient message passing algorithm that computes the probability distribution of the cascade size for the Independent Cascade Model on weighted directed networks and generalizations. Our approach is exact on trees but can be applied to …
We investigate cascades in networks consisting of strategic agents with interdependent security. We assume that the strategic agents have choices between i) investing in protecting themselves, ii) purchasing insurance to transfer (some) risks, and iii) taking no actions. Using a population game model, we study how various system parameters, such as node degrees, infection propagation rate, and the probability with which infected nodes transmit infection to neighbors, affect nodes' choices at Nash equilibria and the resultant price of anarchy/stability. In addition, we examine how the probabili…
University of Pisa（比萨大学）；ICAR-CNR（意大利国家研究委员会信息与高性能计算研究所）；Sapienza University（罗马大学）；ISI Foundation（ISI基金会）；CENTAI（CENTAI研究所）；Eurecat（加泰罗尼亚技术中心）
Despite echo chambers in social media have been under considerable scrutiny, general models for their detection and analysis are missing. In this work, we aim to fill this gap by proposing a probabilistic generative model that explains social media footprints -- i.e., social network structure and propagations of information -- through a set of latent communities, characterized by a degree of echo-chamber behavior and by an opinion polarity. Specifically, echo chambers are modeled as communities that are permeable to pieces of information with similar ideological polarity, and impermeable to in…
Networks provide a skeleton for the spread of contagions, like, information, ideas, behaviors and diseases. Many times networks over which contagions diffuse are unobserved and need to be inferred. Here we apply survival theory to develop general additive and multiplicative risk models under which the network inference problems can be solved efficiently by exploiting their convexity. Our additive risk model generalizes several existing network inference models. We show all these models are particular cases of our more general model. Our multiplicative model allows for modeling scenarios in whi…
