Random walks on activity-driven networks with attractiveness

(ω_{1},ω_{2})-temporal random hyperbolic graphs

We extend a recent model of temporal random hyperbolic graphs by allowing connections and disconnections to persist across network snapshots with different probabilities ω_{1} and ω_{2}. This extension, while conceptually simple, poses analytical challenges involving the Appell F_{1} series. Despite these challenges, we are able to analyze key properties of the model, which include the distributions of contact and intercontact durations, as well as the expected time-aggregated degree. The incorporation of ω_{1} and ω_{2} enables more flexible tuning of the average contact and intercontact durations, and of the average time-aggregated degree, providing a finer control for exploring the effect of temporal network dynamics on dynamical processes. Overall, our results provide new insights into the analysis of temporal networks and contribute to a more general representation of real-world scenarios.

Fixation probability in evolutionary dynamics on switching temporal networks

Population structure has been known to substantially affect evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. Research in the past two decades has found various families of amplifiers while suppressors still remain somewhat elusive. It has also been discovered that most networks are amplifiers of selection under the birth-death updating combined with uniform initialization, which is a standard condition assumed widely in the literature. In the present study, we extend the birth-death processes to temporal (i.e., time-varying) networks. For the sake of tractability, we restrict ourselves to switching temporal networks, in which the network structure deterministically alternates between two static networks at constant time intervals or stochastically in a Markovian manner. We show that, in a majority of cases, switching networks are less amplifying than both of the two static networks constituting the switching networks. Furthermore, most small switching networks, i.e., networks on six nodes or less, are suppressors, which contrasts to the case of static networks.

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.

Non-Markovian temporal networks with auto- and cross-correlated link dynamics

Many of the biological, social and man-made networks around us are inherently dynamic, with their links switching on and off over time. The evolution of these networks is often observed to be non-Markovian, and the dynamics of their links are often correlated. Hence, to accurately model these networks, predict their evolution, and understand how information and other relevant quantities propagate over them, the inclusion of both memory and dynamical dependencies between links is key. In this article we introduce a general class of models of temporal networks based on discrete autoregressive processes for link dynamics. As a concrete and useful case study, we then concentrate on a specific model within this class, which allows to generate temporal networks with a specified underlying structural backbone, and with precise control over the dynamical dependencies between links and the strength and length of their memories. In this network model the presence of each link is influenced not only by its past activity, but also by the past activities of other links, as specified by a coupling matrix, which directly controls the causal relations, and hence the correlations, among links. We propose a maximum likelihood method for estimating the model's parameters from data, showing how the model allows a more realistic description of real-world temporal networks and also to predict their evolution. Due to the flexibility of maximum likelihood inference, we illustrate how to deal with heterogeneity and time-varying patterns, possibly including also nonstationary network dynamics. We then use our network model to investigate the role that, both the features of memory and the type of correlations in the dynamics of links have on the properties of processes occurring over a temporal network. Namely, we study the speed of a spreading process, as measured by the time it takes for diffusion to reach equilibrium. Through both numerical simulations and analytical results, we are able to separate the roles of autocorrelations and neighborhood correlations in link dynamics, showing that not only is the speed of diffusion nonmonotonically dependent on the memory length, but also that correlations among neighboring links help to speed up the spreading process, while autocorrelations slow it back down. Our results have implications in the study of opinion formation, the modeling of social networks, and the spreading of epidemics through mobile populations.

Phase transition in the majority-vote model on time-varying networks

Social interactions may affect the update of individuals' opinions. The existing models such as the majority-vote (MV) model have been extensively studied in different static networks. However, in reality, social networks change over time and individuals interact dynamically. In this work, we study the behavior of the MV model on temporal networks to analyze the effects of temporality on opinion dynamics. In social networks, people are able to both actively send connections and passively receive connections, which leads to different effects on individuals' opinions. In order to compare the impact of different patterns of interactions on opinion dynamics, we simplify them into two processes, that is, the single directed (SD) process and the undirected (UD) process. The former only allows each individual to adopt an opinion by following the majority of actively interactive neighbors, while the latter allows each individual to flip opinion by following the majority of both actively interactive and passively interactive neighbors. By borrowing the activity-driven time-varying network with attractiveness (ADA model), the two opinion update processes, i.e., the SD and the UD processes, are related with the network evolution. With the mean-field approach, we derive the critical noise threshold for each process, which is also verified by numerical simulations. Compared with the SD process, the UD process reaches a larger consensus level below the same critical noise. Finally, we also verify the main results in real networks.

Impact of individual behavioral changes on epidemic spreading in time-varying networks

Changes in individual behavior often entangle with the dynamic interaction of individuals, which complicates the epidemic process and brings great challenges for the understanding and control of the epidemic. In this work, we consider three kinds of typical behavioral changes in epidemic process, that is, self-quarantine of infected individuals, self-protection of susceptible individuals, and social distancing between them. We connect the behavioral changes with individual's social attributes by the activity-driven network with attractiveness. A mean-field theory is established to derive an analytical estimate of epidemic threshold for susceptible-infected-susceptible models with individual behavioral changes, which depends on the correlations between activity, attractiveness, and the number of generative links in the susceptible and infected states. We find that individual behaviors play different roles in suppressing the epidemic. Although all the behavioral changes could delay the epidemic by increasing the epidemic threshold, self-quarantine and social distancing of infected individuals could effectively decrease the epidemic outbreak size. In addition, simultaneous changes in these behaviors and the timing of implement of them also play a key role in suppressing the epidemic. These results provide helpful significance for understanding the interaction of individual behaviors in the epidemic process.

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.

Dynamic hidden-variable network models

Models of complex networks often incorporate node-intrinsic properties abstracted as hidden variables. The probability of connections in the network is then a function of these variables. Real-world networks evolve over time and many exhibit dynamics of node characteristics as well as of linking structure. Here we introduce and study natural temporal extensions of static hidden-variable network models with stochastic dynamics of hidden variables and links. The dynamics is controlled by two parameters: one that tunes the rate of change of hidden variables and another that tunes the rate at which node pairs reevaluate their connections given the current values of hidden variables. Snapshots of networks in the dynamic models are equivalent to networks generated by the static models only if the link reevaluation rate is sufficiently larger than the rate of hidden-variable dynamics or if an additional mechanism is added whereby links actively respond to changes in hidden variables. Otherwise, links are out of equilibrium with respect to hidden variables and network snapshots exhibit structural deviations from the static models. We examine the level of structural persistence in the considered models and quantify deviations from staticlike behavior. We explore temporal versions of popular static models with community structure, latent geometry, and degree heterogeneity. While we do not attempt to directly model real networks, we comment on interesting qualitative resemblances to real systems. In particular, we speculate that links in some real networks are out of equilibrium with respect to hidden variables, partially explaining the presence of long-ranged links in geometrically embedded systems and intergroup connectivity in modular systems. We also discuss possible extensions, generalizations, and applications of the introduced class of dynamic network models.

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.

Epidemic spreading with awareness diffusion on activity-driven networks

Acknowledging the significance of awareness diffusion and behavioral response in contagion outbreaks has been regarded as an indispensable prerequisite for a complete understanding of epidemic spreading. Recent studies from the research community have accumulated overwhelming evidence for the incessantly evolving structure of the underlying networks. Thus there is an impelling need to capture the interplay between the epidemic spreading and awareness diffusion on time-varying networks. In this paper, we consider a behavioral model in which susceptible individuals become alert and adopt a preventive behavior under the local risk perception characterized by a decision-making threshold. The impact of awareness diffusion on the epidemic threshold is investigated under the framework of activity-driven network. Results show that the local epidemic situation in risk perception and the duration of preventive effect are crucial for raising the epidemic threshold. The analytical results are corroborated by Monte Carlo simulations.

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.

Generalized voterlike model on activity-driven networks with attractiveness

We study the behavior of a generalized consensus dynamics on a temporal network of interactions, the activity-driven network with attractiveness. In this temporal network model, agents are endowed with an intrinsic activity a, ruling the rate at which they generate connections, and an intrinsic attractiveness b, modulating the rate at which they receive connections. The consensus dynamics considered is a mixed voter and Moran dynamics. Each agent, either in state 0 or 1, modifies his or her state when connecting with a peer. Thus, an active agent copies his or her state from the peer (with probability p) or imposes his or her state to him or her (with the complementary probability 1-p). Applying a heterogeneous mean-field approach, we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1, which we use to evaluate the average time to reach consensus and the exit probability, defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state. We study a number of particular cases, finding an excellent agreement with numerical simulations of the model. Interestingly, we observe a symmetry between voter and Moran dynamics in pure activity-driven networks and their static integrated counterparts that exemplifies the strong differences that a time-varying network can impose on dynamical processes.

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.