The spread of socially-learnt behaviours occurs in many animal species, and understanding how behaviours spread can provide novel insights into the causes and consequences of sociality. Within wild populations, behaviour spread is often assumed to occur as a "simple contagion". Yet, emerging evidence suggests behaviours may frequently spread as "complex contagions", and this holds significant ramifications for the modes and extent of transmission. We present a new framework enabling comprehensive examination of behavioural contagions by integrating social-learning strategies into network-based diffusion analyses. We show how our approach allows determination of the relationship between social bonds and behavioural transmission, identification of individual-level transmission rules, and examination of population-level social structure effects. We provide resources that allow general applications across diverse systems, and demonstrate how further study-specific developments can be made. Finally, we outline the new opportunities this framework facilitates, the conceptual contributions to understanding sociality, and its applications across fields.
For a broad range single-seed contagion processes acting on generalized random networks, we derive a unifying analytic expression for the possibility of global spreading events in a straightforward, physically intuitive fashion. Our reasoning lays bare a direct mechanical understanding of an archetypal spreading phenomena that is not evident in circuitous extant mathematical approaches.
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 cl…
In this paper we introduce a statistical inference framework for estimating the contagion source from a partially observed contagion spreading process on an arbitrary network structure. The framework is based on a maximum likelihood estimation of a partial epidemic realization and involves large scale simulation of contagion spreading processes from the set of potential source locations. We present a number of different likelihood estimators that are used to determine the conditional probabilities associated to observing partial epidemic realization with particular source location candidates. …
In this paper, we study the spreading speed of complex contagions in a social network. A $k$-complex contagion starts from a set of initially infected seeds such that any node with at least $k$ infected neighbors gets infected. Simple contagions, i.e., $k=1$, quickly spread to the entire network in small world graphs. However, fast spreading of complex contagions appears to be less likely and more delicate; the successful cases depend crucially on the network structure~\cite{G08,Ghasemiesfeh:2013:CCW}. Our main result shows that complex contagions can spread fast in a general family of time-ev…
Complex contagions describe diffusion of behaviors in a social network in settings where spreading requires the influence by two or more neighbors. In a $k$-complex contagion, a cluster of nodes are initially infected, and additional nodes become infected in the next round if they have at least $k$ already infected neighbors. It has been argued that complex contagions better model behavioral changes such as adoption of new beliefs, fashion trends or expensive technology innovations. This has motivated rigorous understanding of spreading of complex contagions in social networks. Despite simple …
Contact tracing can be thought of as a race between two processes: an infection process and a tracing process. In this paper, we study a simple model of infection spreading on a tree, and a tracer who stabilizes one node at a time. We focus on the question, how should the tracer choose nodes to stabilize so as to prevent the infection from spreading further? We study simple policies, which prioritize nodes based on time, infectiousness, or probability of generating new contacts.
Increasing levels of far-right extremist violence have generated public concern about the spread of radicalization in the United States. Previous research suggests that radicalized individuals are destabilized by various environmental (or endemic) factors, exposed to extremist ideology, and subsequently reinforced by members of their community. As such, the spread of radicalization may proceed through a social contagion process, in which extremist ideologies behave like complex contagions that require multiple exposures for adoption. In this study, I applied an epidemiological method called tw…
Many spreading processes in our real-life can be considered as a complex contagion, and the linear threshold (LT) model is often applied as a very representative model for this mechanism. Despite its intensive usage, the LT model suffers several limitations in describing the time evolution of the spreading. First, the discrete-time step that captures the speed of the spreading is vaguely defined. Second, the synchronous updating rule makes the nodes infected in batches, which can not take individual differences into account. Finally, the LT model is incompatible with existing models for the si…
A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion model on a network whose nodes are embedded in a manifold and which has both `geometric edges', which respect the geometry of the underlying manifold, and `nongeometric edges' that are not constrained by that geometry. Building on ideas from Taylor et al. \cite{Taylor2015}, we examine when a contagion propagates as a wave along a network whose nodes are embedded in a torus an…
Contagion models are a primary lens through which we understand the spread of information over social networks. However, simple contagion models cannot reproduce the complex features observed in real-world data, leading to research on more complicated complex contagion models. A noted feature of complex contagion is social reinforcement that individuals require multiple exposures to information before they begin to spread it themselves. Here we show that the quoter model, a model of the social flow of written information over a network, displays features of complex contagion, including the wea…
Social reinforcement mechanism, which characterizes the promoting effects when exposing to multiple sources in social contagion process, is ubiquitous in information-technology ecosystem and has aroused great attention in recent years. While the impacts of social reinforcement on single-layer networks are well-documented, extension to multilayer networks is needed to study how reinforcement from different social circles influences the spreading dynamics. To this end, we incorporate multilayer social reinforcement into ignorant-spreader-ignorant (SIS) model on multiplex networks. Our theoretica…
Prior social contagion models consider the spread of either one contagion at a time on interdependent networks or multiple contagions on single layer networks or under assumptions of competition. We propose a new threshold model for the diffusion of multiple contagions. Individuals are placed on a multiplex network with a periodic lattice and a random-regular-graph layer. On these population structures, we study the interface between two key aspects of the diffusion process: the level of synergy between two contagions, and the rate at which individuals become dormant after adoption. Dormancy i…
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 …
Social contagion is the process in which people adopt a belief, idea, or practice from a neighbor and pass it along to someone else. For over 100 years, scholars of social contagion have almost exclusively made the same implicit assumption: that only one belief, idea, or practice spreads through the population at a time. It is a default assumption that we don't bother to state, let alone justify. The assumption is so ingrained that our literature doesn't even have a word for "whatever is to be diffused", because we have never needed to discuss more than one of them. But this assumption is obvi…
It is commonly believed that information spreads between individuals like a pathogen, with each exposure by an informed friend potentially resulting in a naive individual becoming infected. However, empirical studies of social media suggest that individual response to repeated exposure to information is significantly more complex than the prediction of the pathogen model. As a proxy for intervention experiments, we compare user responses to multiple exposures on two different social media sites, Twitter and Digg. We show that the position of the exposing messages on the user-interface strongly…
Social network analysis is now widely used to investigate the dynamics of infectious disease spread from person to person. Vaccination dramatically disrupts the disease transmission process on a contact network, and indeed, sufficiently high vaccination rates can disrupt the process to such an extent that disease transmission on the network is effectively halted. Here, we build on mounting evidence that health behaviors - such as vaccination, and refusal thereof - can spread through social networks through a process of complex contagion that requires social reinforcement. Using network simulat…
Increasing evidence suggests that, similar to face-to-face communications, human emotions also spread in online social media. However, the mechanisms underlying this emotional contagion, for example, whether different feelings spread in unlikely ways or how the spread of emotions relates to the social network, is rarely investigated. Indeed, because of high costs and spatio-temporal limitations, explorations of this topic are challenging using conventional questionnaires or controlled experiments. Because they are collection points for natural affective responses of massive individuals, online…
Diffusion of innovation can be interpreted as a social spreading phenomena governed by the impact of media and social interactions. Although these mechanisms have been identified by quantitative theories, their role and relative importance are not entirely understood, since empirical verification has so far been hindered by the lack of appropriate data. Here we analyse a dataset recording the spreading dynamics of the world's largest Voice over Internet Protocol service to empirically support the assumptions behind models of social contagion. We show that the rate of spontaneous service adopti…
An outstanding problem of interdisciplinary interest is to understand quantitatively the role of social contacts in contagion dynamics. In general, there are two types of contacts: close ones among friends, colleagues and family members, etc., and ordinary contacts from encounters with strangers. Typically, social reinforcement occurs for close contacts. Taking into account both types of contacts, we develop a contact-based model for social contagion. We find that, associated with the spreading dynamics, for random networks there is coexistence of continuous and discontinuous phase transitions…
Understanding how contagions (information, infections, etc) are spread on complex networks is important both from practical as well as theoretical point of view. Considerable work has been done in this regard in the past decade or so. However, most models are limited in their scope and as a result only capture general features of spreading phenomena. Here, we propose and study a model of spreading which takes into account the strength or quality of contagions as well as the local (probabilistic) dynamics occurring at various nodes. Transmission occurs only after the quality-based fitness of th…
How does network structure affect diffusion? Recent studies suggest that the answer depends on the type of contagion. Complex contagions, unlike infectious diseases (simple contagions), are affected by social reinforcement and homophily. Hence, the spread within highly clustered communities is enhanced, while diffusion across communities is hampered. A common hypothesis is that memes and behaviors are complex contagions. We show that, while most memes indeed behave like complex contagions, a few viral memes spread across many communities, like diseases. We demonstrate that the future popularit…
Internet communication channels, e.g., Facebook, Twitter, and email, are multiplex networks that facilitate interaction and information-sharing among individuals. During brief time periods users often use a single communication channel, but then communication channel alteration (CCA) occurs. This means that we must refine our understanding of the dynamics of social contagions. We propose a non-Markovian behavior spreading model in multiplex networks that takes into account the CCA mechanism, and we develop a generalized edge-based compartmental method to describe the spreading dynamics. Throug…
The temporal dynamics of social interactions were shown to influence the spread of disease. Here, we model the conditions of progression and competition for several viral strains, exploring various levels of cross-immunity over temporal networks. We use our interaction-driven contagion model and characterize, using it, several viral variants. Our results, obtained on temporal random networks and on real-world interaction data, demonstrate that temporal dynamics are crucial to determining the competition results. We consider two and three competing pathogens and show the conditions under which …
The threshold model is a simple but classic model of contagion spreading in complex social systems. To capture the complex nature of social influencing we investigate numerically and analytically the transition in the behavior of threshold-limited cascades in the presence of multiple initiators as the distribution of thresholds is varied between the two extreme cases of identical thresholds and a uniform distribution. We accomplish this by employing a truncated normal distribution of the nodes' thresholds and observe a non-monotonic change in the cascade size as we vary the standard deviation.…
We study SIS epidemic spreading processes unfolding on a recent generalisation of the activity-driven modelling framework. In this model of time-varying networks each node is described by two variables: activity and attractiveness. The first, describes the propensity to form connections. The second, defines the propensity to attract them. We derive analytically the epidemic threshold considering the timescale driving the evolution of contacts and the contagion as comparable. The solutions are general and hold for any joint distribution of activity and attractiveness. The theoretical picture is…
In this work we study diffusion in networks with community structure. We first replicate and extend work on networks with non-overlapping community structure. We then study diffusion on network models that have overlapping community structure. We study contagions in the standard SIR model, and complex contagions thought to be better models of some social diffusion processes. Finally, we investigate diffusion on empirical networks with known overlapping community structure, by analysing the structure of such networks, and by simulating contagion on them. We find that simple and complex contagio…
It has recently become possible to study the dynamics of information diffusion in techno-social systems at scale, due to the emergence of online platforms, such as Twitter, with millions of users. One question that systematically recurs is whether information spreads according to simple or complex dynamics: does each exposure to a piece of information have an independent probability of a user adopting it (simple contagion), or does this probability depend instead on the number of sources of exposure, increasing above some threshold (complex contagion)? Most studies to date are observational an…
Most studies on social influence have focused on direct influence, while another interesting question can be raised as whether indirect influence exists between two users who're not directly connected in the network and what affects such influence. In addition, the theory of \emph{complex contagion} tells us that more spreaders will enhance the indirect influence between two users. Our observation of intensity of indirect influence, propagated by $n$ parallel spreaders and quantified by retweeting probability on Twitter, shows that complex contagion is validated globally but is violated locall…
Adoption of innovations, products or online services is commonly interpreted as a spreading process driven to large extent by social influence and conditioned by the needs and capacities of individuals. To model this process one usually introduces behavioural threshold mechanisms, which can give rise to the evolution of global cascades if the system satisfies a set of conditions. However, these models do not address temporal aspects of the emerging cascades, which in real systems may evolve through various pathways ranging from slow to rapid patterns. Here we fill this gap through the analysis…
Social networks have emerged as a critical factor in information dissemination, search, marketing, expertise and influence discovery, and potentially an important tool for mobilizing people. Social media has made social networks ubiquitous, and also given researchers access to massive quantities of data for empirical analysis. These data sets offer a rich source of evidence for studying dynamics of individual and group behavior, the structure of networks and global patterns of the flow of information on them. However, in most previous studies, the structure of the underlying networks was not d…
We complement the rich conceptual work on organizational corruption by quantitatively modelling the spread of corruption within organizations. We systematically vary four organizational culture-related parameters, i.e., organization structure, location of bad apple, employees propensity to become corrupted (corruption probability), and number of whistle-blowers. Our simulation studies find that in organizations with flatter structures, corruption permeates the organization at a lower threshold value of corruption probability compared to those with taller structures. However, the final proporti…
Physical contacts result in the spread of various phenomena such as viruses, gossips, ideas, packages and marketing pamphlets across a population. The spread depends on how people move and co-locate with each other, or their mobility patterns. How far such phenomena spread has significance for both policy making and personal decision making, e.g., studying the spread of COVID-19 under different intervention strategies such as wearing a mask. In practice, mobility patterns of an entire population is never available, and we usually have access to location data of a subset of individuals. In this…
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…
Increasing evidence suggests that, similar to face-to-face communications, human emotions also spread in online social media. However, the mechanisms underlying this emotion contagion, for example, whether different feelings spread in unlikely ways or how the spread of emotions relates to the social network, is rarely investigated. Indeed, because of high costs and spatio-temporal limitations, explorations of this topic are challenging using conventional questionnaires or controlled experiments. Because they are collection points for natural affective responses of massive individuals, online s…
The graph invariant examined in this paper is the largest eigenvalue of the adjacency matrix of a graph. Previous work demonstrates the tight relationship between this invariant, the birth and death rate of a contagion spreading on the graph, and the trajectory of the contagion over time. We begin by conducting a simulation confirming this and explore bounds on the birth and death rate in terms of well-known graph invariants. As a result, the change in the largest eigenvalue resulting from removal of a vertex in the network is the best measure of effectiveness of interventions that slow the sp…
Time-varying community structures widely exist in various real-world networks. However, the spreading dynamics on this kind of network has not been fully studied. To this end, we systematically study the effects of time-varying community structures on social contagions. We first propose a non-Markovian social contagion model on time-varying community networks based on the activity driven network model, in which an individual adopts a behavior if and only if the accumulated behavioral information it has ever received reaches a threshold. Then, we develop a mean-field theory to describe the prop…
Heterogeneous adoption thresholds exist widely in social contagions, but were always neglected in previous studies. We first propose a non-Markovian spreading threshold model with general adoption threshold distribution. In order to understand the effects of heterogeneous adoption thresholds quantitatively, an edge-based compartmental theory is developed for the proposed model. We use a binary spreading threshold model as a specific example, in which some individuals have a low adoption threshold (i.e., activists) while the remaining ones hold a relatively high adoption threshold (i.e., bigots…
Contacts' temporal ordering and dynamics are crucial for understanding the transmission of infectious diseases. We introduce an interaction-driven model of an airborne disease over contact networks. We demonstrate our interaction-driven contagion model, instantiated for COVID-19, over history-maintaining random temporal networks and real-world contacts. We use it to evaluate temporal, spatiotemporal, and spatial social distancing policies. We find that a spatial distancing policy is mainly beneficial at the early stages of a disease. We then continue to evaluate temporal social distancing, tha…
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-$i$ …
Security games model the confrontation between a defender protecting a set of targets and an attacker who tries to capture them. A variant of these games assumes security interdependence between targets, facilitating contagion of an attack. So far only stochastic spread of an attack has been considered. In this work, we introduce a version of security games, where the attacker strategically drives the entire spread of attack and where interconnections between nodes affect their susceptibility to be captured. We find that the strategies effective in the settings without contagion or with stocha…
We consider SIS contagion processes over networks where, a classical assumption is that individuals' decisions to adopt a contagion are based on their immediate neighbors. However, recent literature shows that some attributes are more correlated between two-hop neighbors, a concept referred to as monophily. This motivates us to explore monophilic contagion, the case where a contagion (e.g. a product, disease) is adopted by considering two-hop neighbors instead of immediate neighbors (e.g. you ask your friend about the new iPhone and she recommends you the opinion of one of her friends). We sho…
We derive analytic expressions for the possibility, probability, and expected size of global spreading events starting from a single infected seed for a broad collection of contagion processes acting on random networks with both directed and undirected edges and arbitrary degree-degree correlations. Our work extends previous theoretical developments for the undirected case, and we provide numerical support for our findings by investigating an example class of networks for which we are able to obtain closed-form expressions.
Network epidemiology has become a vital tool in understanding the effects of high-degree vertices, geographic and demographic communities, and other inhomogeneities in social structure on the spread of disease. However, many networks derived from modern datasets are quite dense, such as mobility networks where each location has links to a large number of potential destinations. One way to reduce the computational effort of simulating epidemics on these networks is sparsification, where we select a representative subset of edges based on some measure of their importance. Recently an approach wa…
The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent pr…
Social interactions are stratified in multiple contexts and are subject to complex temporal dynamics. The systematic study of these two features of social systems has started only very recently mainly thanks to the development of multiplex and time-varying networks. However, these two advancements have progressed almost in parallel with very little overlap. Thus, the interplay between multiplexity and the temporal nature of connectivity patterns is poorly understood. Here, we aim to tackle this limitation by introducing a time-varying model of multiplex networks. We are interested in character…
In this Chapter, we discuss the effects of higher-order structures on SIS-like processes of social contagion. After a brief motivational introduction where we illustrate the standard SIS process on networks and the difference between simple and complex contagions, we introduce spreading processes on higher-order structures starting from the most general formulation on hypergraphs and then moving to several mean-field and heterogeneous mean-field approaches. The results highlight the rich phenomenology brought by taking into account higher-order contagion effects: both continuous and discontinu…
In networks, multiple contagions, such as information and purchasing behaviors, may interact with each other as they spread simultaneously. However, most of the existing information diffusion models are built on the assumption that each individual contagion spreads independently, regardless of their interactions. Gaining insights into such interaction is crucial to understand the contagion adoption behaviors, and thus can make better predictions. In this paper, we study the contagion adoption behavior under a set of interactions, specifically, the interactions among users, contagions' contents…
In this chapter, we apply the theoretical framework introduced in the previous chapter to study how the modular structure of the social network affects the spreading of complex contagion. In particular, we focus on the notion of optimal modularity, that predicts the occurrence of global cascades when the network exhibits just the right amount of modularity. Here we generalize the findings by assuming the presence of multiple communities and an uniform distribution of seeds across the network. Finally, we offer some insights into the temporal evolution of cascades in the regime of the optimal m…
Diffusion of information, behavioral patterns or innovations follows diverse pathways depending on a number of conditions, including the structure of the underlying social network, the sensitivity to peer pressure and the influence of media. Here we study analytically and by simulations a general model that incorporates threshold mechanism capturing sensitivity to peer pressure, the effect of `immune' nodes who never adopt, and a perpetual flow of external information. While any constant, non-zero rate of dynamically-introduced spontaneous adopters leads to global spreading, the kinetics by wh…
