Asymptotic theory of time-varying social networks with heterogeneous activity and tie allocation

Modeling framework unifying contact and social networks

Temporal networks of face-to-face interactions between individuals are useful proxies of the dynamics of social systems on fast timescales. Several empirical statistical properties of these networks have been shown to be robust across a large variety of contexts. To better grasp the role of various mechanisms of social interactions in the emergence of these properties, models in which schematic implementations of such mechanisms can be carried out have proven useful. Here, we put forward a framework to model temporal networks of human interactions based on the idea of a coevolution and feedback between (i) an observed network of instantaneous interactions and (ii) an underlying unobserved social bond network: Social bonds partially drive interaction opportunities and in turn are reinforced by interactions and weakened or even removed by the lack of interactions. Through this coevolution, we also integrate in the model well-known mechanisms such as triadic closure, but also the impact of shared social context and nonintentional (casual) interactions, with several tunable parameters. We then propose a method to compare the statistical properties of each version of the model with empirical face-to-face interaction data sets to determine which sets of mechanisms lead to realistic social temporal networks within this modeling framework.

Macroscopic properties of buyer-seller networks in online marketplaces

Online marketplaces are the main engines of legal and illegal e-commerce, yet their empirical properties are poorly understood due to the absence of large-scale data. We analyze two comprehensive datasets containing 245M transactions (16B USD) that took place on online marketplaces between 2010 and 2021, covering 28 dark web marketplaces, i.e. unregulated markets whose main currency is Bitcoin, and 144 product markets of one popular regulated e-commerce platform. We show that transactions in online marketplaces exhibit strikingly similar patterns despite significant differences in language, lifetimes, products, regulation, and technology. Specifically, we find remarkable regularities in the distributions of transaction amounts, number of transactions, interevent times, and time between first and last transactions. We show that buyer behavior is affected by the memory of past interactions and use this insight to propose a model of network formation reproducing our main empirical observations. Our findings have implications for understanding market power on online marketplaces as well as intermarketplace competition, and provide empirical foundation for theoretical economic models of online marketplaces.

Interplay of simplicial awareness contagion and epidemic spreading on time-varying multiplex networks

There has been growing interest in exploring the dynamical interplay of epidemic spreading and awareness diffusion within the multiplex network framework. Recent studies have demonstrated that pairwise interactions are not enough to characterize social contagion processes, but the complex mechanisms of influence and reinforcement should be considered. Meanwhile, the physical social interaction of individuals is not static but time-varying. Therefore, we propose a novel sUAU-tSIS model to characterize the interplay of simplicial awareness contagion and epidemic spreading on time-varying multiplex networks, in which one layer with 2-simplicial complexes is considered the virtual information layer to address the complex contagion mechanisms in awareness diffusion and the other layer with time-varying and memory effects is treated as the physical contact layer to mimic the temporal interaction pattern among population. The microscopic Markov chain approach based theoretical analysis is developed, and the epidemic threshold is also derived. The experimental results show that our theoretical method is in good agreement with the Monte Carlo simulations. Specifically, we find that the synergistic reinforcement mechanism coming from the group interactions promotes the diffusion of awareness, leading to the suppression of the spreading of epidemics. Furthermore, our results illustrate that the contact capacity of individuals, activity heterogeneity, and memory strength also play important roles in the two dynamics; interestingly, a crossover phenomenon can be observed when investigating the effects of activity heterogeneity and memory strength.

Sideward contact tracing and the control of epidemics in large gatherings

Effective contact tracing is crucial to containing epidemic spreading without disrupting societal activities, especially during a pandemic. Large gatherings play a key role, potentially favouring superspreading events. However, the effects of tracing in large groups have not been fully assessed so far. We show that in addition to forward tracing, which reconstructs to whom the disease spreads, and backward tracing, which searches from whom the disease spreads, a third 'sideward' tracing is always present, when tracing gatherings. This is an indirect tracing that detects infected asymptomatic individuals, even if they have been neither directly infected by nor directly transmitted the infection to the index case. We analyse this effect in a model of epidemic spreading for SARS-CoV-2, within the framework of simplicial activity-driven temporal networks. We determine the contribution of the three tracing mechanisms to the suppression of epidemic spreading, showing that sideward tracing induces a non-monotonic behaviour in the tracing efficiency, as a function of the size of the gatherings. Based on our results, we suggest an optimal choice for the sizes of the gatherings to be traced and we test the strategy on an empirical dataset of gatherings on a university campus.

Self-initiated behavioral change and disease resurgence on activity-driven networks

We consider a population that experienced a first wave of infections, interrupted by strong, top-down, governmental restrictions and did not develop a significant immunity to prevent a second wave (i.e., resurgence). As restrictions are lifted, individuals adapt their social behavior to minimize the risk of infection. We explore two scenarios. In the first, individuals reduce their overall social activity towards the rest of the population. In the second scenario, they maintain normal social activity within a small community of peers (i.e., social bubble) while reducing social interactions with the rest of the population. In both cases, we investigate possible correlations between social activity and behavior change, reflecting, for example, the social dimension of certain occupations. We model these scenarios considering a susceptible-infected-recovered epidemic model unfolding on activity-driven networks. Extensive analytical and numerical results show that (i) a minority of very active individuals not changing behavior may nullify the efforts of the large majority of the population and (ii) imperfect social bubbles of normal social activity may be less effective than an overall reduction of social interactions.

Impact of environmental changes on the dynamics of temporal networks

Dynamics of complex social systems has often been described in the framework of temporal networks, where links are considered to exist only at the moment of interaction between nodes. Such interaction patterns are not only driven by internal interaction mechanisms, but also affected by environmental changes. To investigate the impact of the environmental changes on the dynamics of temporal networks, we analyze several face-to-face interaction datasets using the multiscale entropy (MSE) method to find that the observed temporal correlations can be categorized according to the environmental similarity of datasets such as classes and break times in schools. By devising and studying a temporal network model considering a periodically changing environment as well as a preferential activation mechanism, we numerically show that our model could successfully reproduce various empirical results by the MSE method in terms of multiscale temporal correlations. Our results demonstrate that the environmental changes can play an important role in shaping the dynamics of temporal networks when the interactions between nodes are influenced by the environment of the systems.

Stochastic sampling effects favor manual over digital contact tracing

Isolation of symptomatic individuals, tracing and testing of their nonsymptomatic contacts are fundamental strategies for mitigating the current COVID-19 pandemic. The breaking of contagion chains relies on two complementary strategies: manual reconstruction of contacts based on interviews and a digital (app-based) privacy-preserving contact tracing. We compare their effectiveness using model parameters tailored to describe SARS-CoV-2 diffusion within the activity-driven model, a general empirically validated framework for network dynamics. We show that, even for equal probability of tracing a contact, manual tracing robustly performs better than the digital protocol, also taking into account the intrinsic delay and limited scalability of the manual procedure. This result is explained in terms of the stochastic sampling occurring during the case-by-case manual reconstruction of contacts, contrasted with the intrinsically prearranged nature of digital tracing, determined by the decision to adopt the app or not by each individual. The better performance of manual tracing is enhanced by heterogeneity in agent behavior: superspreaders not adopting the app are completely invisible to digital contact tracing, while they can be easily traced manually, due to their multiple contacts. We show that this intrinsic difference makes the manual procedure dominant in realistic hybrid protocols.

Random walks in time-varying networks with memory

Random walks process on networks plays a fundamental role in understanding the importance of nodes and the similarity of them, which has been widely applied in PageRank, information retrieval, and community detection, etc. An individual's memory has been proved to be crucial to affect network evolution and dynamical processes unfolding on the network. In this work, we study the random-walk process on an extended activity-driven network model by taking account of an individual's memory. We analyze how an individual's memory affects random-walk process unfolding on the network when the timescales of the processes of the random walk and the network evolution are comparable. Under the constraints of long-time evolution, we derive analytical solutions for the distribution of walkers at the stationary state and the mean first-passage time of the random-walk process. We find that, compared with the memoryless activity-driven model, an individual's memory enhances the activity fluctuation and leads to the formation of small clusters of mutual contacts with high activity nodes, which reduces a node's capability of gathering walkers, especially for the nodes with large activity, and memory also delays the mean first-passage time. The results on real networks also support the theoretical analysis and numerical results with artificial networks.

Active and inactive quarantine in epidemic spreading on adaptive activity-driven networks

We consider an epidemic process on adaptive activity-driven temporal networks, with adaptive behavior modeled as a change in activity and attractiveness due to infection. By using a mean-field approach, we derive an analytical estimate of the epidemic threshold for susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) epidemic models for a general adaptive strategy, which strongly depends on the correlations between activity and attractiveness in the susceptible and infected states. We focus on strong social distancing, implementing two types of quarantine inspired by recent real case studies: an active quarantine, in which the population compensates the loss of links rewiring the ineffective connections towards nonquarantining nodes, and an inactive quarantine, in which the links with quarantined nodes are not rewired. Both strategies feature the same epidemic threshold but they strongly differ in the dynamics of the active phase. We show that the active quarantine is extremely less effective in reducing the impact of the epidemic in the active phase compared to the inactive one and that in the SIR model a late adoption of measures requires inactive quarantine to reach containment.

Spreading of computer viruses on time-varying networks

Social networks are the prime channel for the spreading of computer viruses. Yet the study of their propagation neglects the temporal nature of social interactions and the heterogeneity of users' susceptibility. Here, we introduce a theoretical framework that captures both properties. We study two realistic types of viruses propagating on temporal networks featuring Q categories of susceptibility and derive analytically the invasion threshold. We found that the temporal coupling of categories might increase the fragility of the system to cyber threats. Our results show that networks' dynamics and their interplay with users' features are crucial for the spreading of computer viruses.

Epidemic spreading and aging in temporal networks with memory

Time-varying network topologies can deeply influence dynamical processes mediated by them. Memory effects in the pattern of interactions among individuals are also known to affect how diffusive and spreading phenomena take place. In this paper we analyze the combined effect of these two ingredients on epidemic dynamics on networks. We study the susceptible-infected-susceptible (SIS) and the susceptible-infected-recovered (SIR) models on the recently introduced activity-driven networks with memory. By means of an activity-based mean-field approach, we derive, in the long-time limit, analytical predictions for the epidemic threshold as a function of the parameters describing the distribution of activities and the strength of the memory effects. Our results show that memory reduces the threshold, which is the same for SIS and SIR dynamics, therefore favoring epidemic spreading. The theoretical approach perfectly agrees with numerical simulations in the long-time asymptotic regime. Strong aging effects are present in the preasymptotic regime and the epidemic threshold is deeply affected by the starting time of the epidemics. We discuss in detail the origin of the model-dependent preasymptotic corrections, whose understanding could potentially allow for epidemic control on correlated temporal networks.

Epidemic spreading on time-varying multiplex networks

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 characterizing how these two properties affect contagion processes. To this end, we study susceptible-infected-susceptible epidemic models unfolding at comparable timescale with respect to the evolution of the multiplex network. We study both analytically and numerically the epidemic threshold as a function of the multiplexity and the features of each layer. We found that higher values of multiplexity significantly reduce the epidemic threshold especially when the temporal activation patterns of nodes present on multiple layers are positively correlated. Furthermore, when the average connectivity across layers is very different, the contagion dynamics is driven by the features of the more densely connected layer. Here, the epidemic threshold is equivalent to that of a single layered graph and the impact of the disease, in the layer driving the contagion, is independent of the multiplexity. However, this is not the case in the other layers where the spreading dynamics is sharply influenced by it. The results presented provide another step towards the characterization of the properties of real networks and their effects on contagion phenomena.

Dynamic topologies of activity-driven temporal networks with memory

We propose dynamic scaling in temporal networks with heterogeneous activities and memory and provide a comprehensive picture for the dynamic topologies of such networks, in terms of the modified activity-driven network model [H. Kim et al., Eur. Phys. J. B 88, 315 (2015)EPJBFY1434-602810.1140/epjb/e2015-60662-7]. Particularly, we focus on the interplay of the time resolution and memory in dynamic topologies. Through the random-walk (RW) process, we investigate diffusion properties and topological changes as the time resolution increases. Our results with memory are compared to those of the memoryless case. Based on the temporal percolation concept, we derive scaling exponents in the dynamics of the largest cluster and the coverage of the RW process in time-varying networks. We find that the time resolution in the time-accumulated network determines the effective size of the network, while memory affects relevant scaling properties at the crossover from the dynamic regime to the static one. The origin of memory-dependent scaling behaviors is the dynamics of the largest cluster, which depends on temporal degree distributions. Finally, we conjecture of the extended finite-size scaling ansatz for dynamic topologies and the fundamental property of temporal networks, which are numerically confirmed.

Epidemic spreading in modular time-varying networks

We investigate the effects of modular and temporal connectivity patterns on epidemic spreading. To this end, we introduce and analytically characterise a model of time-varying networks with tunable modularity. Within this framework, we study the epidemic size of Susceptible-Infected-Recovered, SIR, models and the epidemic threshold of Susceptible-Infected-Susceptible, SIS, models. Interestingly, we find that while the presence of tightly connected clusters inhibits SIR processes, it speeds up SIS phenomena. In this case, we observe that modular structures induce a reduction of the threshold with respect to time-varying networks without communities. We confirm the theoretical results by means of extensive numerical simulations both on synthetic graphs as well as on a real modular and temporal network.

Epidemic spreading on activity-driven networks with attractiveness

We study SIS epidemic spreading processes unfolding on a recent generalization of the activity-driven modeling 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, while the second defines the propensity to attract them. We derive analytically the epidemic threshold considering the time scale 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 confirmed via large-scale numerical simulations performed considering heterogeneous distributions and different correlations between the two variables. We find that heterogeneous distributions of attractiveness alter the contagion process. In particular, in the case of uncorrelated and positive correlations between the two variables, heterogeneous attractiveness facilitates the spreading. On the contrary, negative correlations between activity and attractiveness hamper the spreading. The results presented contribute to the understanding of the dynamical properties of time-varying networks and their effects on contagion phenomena unfolding on their fabric.

Random walks on activity-driven networks with attractiveness

Virtually all real-world networks are dynamical entities. In social networks, the propensity of nodes to engage in social interactions (activity) and their chances to be selected by active nodes (attractiveness) are heterogeneously distributed. Here, we present a time-varying network model where each node and the dynamical formation of ties are characterized by these two features. We study how these properties affect random-walk processes unfolding on the network when the time scales describing the process and the network evolution are comparable. We derive analytical solutions for the stationary state and the mean first-passage time of the process, and we study cases informed by empirical observations of social networks. Our work shows that previously disregarded properties of real social systems, such as heterogeneous distributions of activity and attractiveness as well as the correlations between them, substantially affect the dynamical process unfolding on the network.

Burstiness and tie activation strategies in time-varying social networks

The recent developments in the field of social networks shifted the focus from static to dynamical representations, calling for new methods for their analysis and modelling. Observations in real social systems identified two main mechanisms that play a primary role in networks' evolution and influence ongoing spreading processes: the strategies individuals adopt when selecting between new or old social ties, and the bursty nature of the social activity setting the pace of these choices. We introduce a time-varying network model accounting both for ties selection and burstiness and we analytically study its phase diagram. The interplay of the two effects is non trivial and, interestingly, the effects of burstiness might be suppressed in regimes where individuals exhibit a strong preference towards previously activated ties. The results are tested against numerical simulations and compared with two empirical datasets with very good agreement. Consequently, the framework provides a principled method to classify the temporal features of real networks, and thus yields new insights to elucidate the effects of social dynamics on spreading processes.