The Impact of Heterogeneous Thresholds on Social Contagion with Multiple Initiators

Control of cascading failures using protective measures

Cascading failures, triggered by a local perturbation, can be catastrophic and cause irreparable damages in a wide area. Hence, blocking the devastating cascades is an important issue in real world networks. One of the ways to control the cascade is to use protective me‌asures, so that the agents decide to be protected against failure. Here, we consider a coevolution of the linear threshold mo‌del for the spread of cascading failures and a decision-making game based on the perceived risk of failure. Protected agents are less vulnerable to failure and in return the size of the cascade affects the agent's decision to get insured. We find at what range of protection efficiency and cost of failure, the global cascades stop. Also we observe that in some range of protection efficiency, a bistable region emerges for the size of cascade and the prevalence of protected agents. Moreover, we show how savings or the ability of agents to repair can prevent cascades from occurring.

Stochastic modeling of cascade dynamics: A unified approach for simple and complex contagions across homogeneous and heterogeneous threshold distributions on networks

The COVID-19 pandemic has underscored the importance of understanding, forecasting, and avoiding infectious processes, as well as the necessity for understanding the diffusion and acceptance of preventative measures. Simple contagions, like virus transmission, can spread with a single encounter, while complex contagions, such as preventive social measures (e.g., wearing masks, social distancing), may require multiple interactions to propagate. This disparity in transmission mechanisms results in differing contagion rates and contagion patterns between viruses and preventive measures. Furthermore, the dynamics of complex contagions are significantly less understood than those of simple contagions. Stochastic models, integrating inherent variability and randomness, offer a way to elucidate complex contagion dynamics. This paper introduces a stochastic model for both simple and complex contagions and assesses its efficacy against ensemble simulations for homogeneous and heterogeneous threshold configurations. The model provides a unified framework for analyzing both types of contagions, demonstrating promising outcomes across various threshold setups on Erds-Rényi graphs.

Influence of heterogeneity of infection thresholds on epidemic spreading with neighbor resource supporting

The spread of disease on complex networks has attracted wide attention in physics, mathematics, and epidemiology. Recent works have demonstrated that individuals always exhibit different criteria for disease infection in a network that significantly influences the epidemic dynamics. In this paper, considering the heterogeneity of node susceptibility, we proposed an infection threshold model with neighbor resource support. The infection threshold of an individual is associated with the degree, and a parameter follows the normal distribution. Based on improved heterogeneous mean-field theory and extensive numerical simulations, we find that the mean and standard deviation of the infection threshold model can affect the phase transition and epidemic outbreak size. As the mean of the normal distribution parameter increases from a small value to a large value, the system shows a change from a continuous phase transition to a discontinuous phase transition, and the disease even stops spreading. The disease spreads from a discontinuous phase transition to continuous for the sizeable mean value as the standard deviation increases. Furthermore, the standard deviation also varies in the outbreak size.

Double transitions and hysteresis in heterogeneous contagion processes

In many real-world contagion phenomena, the number of contacts to spreading entities for adoption varies for different individuals. Therefore, we study a model of contagion dynamics with heterogeneous adoption thresholds. We derive mean-field equations for the fraction of adopted nodes and obtain phase diagrams in terms of the transmission probability and fraction of nodes requiring multiple contacts for adoption. We find a double phase transition exhibiting a continuous transition and a subsequent discontinuous jump in the fraction of adopted nodes because of the heterogeneity in adoption thresholds. Additionally, we observe hysteresis curves in the fraction of adopted nodes owing to adopted nodes in the densely connected core in a network.

Quantum Contagion: A Quantum-Like Approach for the Analysis of Social Contagion Dynamics with Heterogeneous Adoption Thresholds

Modeling the information of social contagion processes has recently attracted a substantial amount of interest from researchers due to its wide applicability in network science, multi-agent-systems, information science, and marketing. Unlike in biological spreading, the existence of a reinforcement effect in social contagion necessitates considering the complexity of individuals in the systems. Although many studies acknowledged the heterogeneity of the individuals in their adoption of information, there are no studies that take into account the individuals’ uncertainty during their adoption decision-making. This resulted in less than optimal modeling of social contagion dynamics in the existence of phase transition in the final adoption size versus transmission probability. We employed the Inverse Born Problem (IBP) to represent probabilistic entities as complex probability amplitudes in edge-based compartmental theory, and demonstrated that our novel approach performs better in the prediction of social contagion dynamics through extensive simulations on random regular networks.

Multilayer modeling of adoption dynamics in energy demand management

Due to the emergence of new technologies, the whole electricity system is undergoing transformations on a scale and pace never observed before. The decentralization of energy resources and the smart grid have forced utility services to rethink their relationships with customers. Demand response (DR) seeks to adjust the demand for power instead of adjusting the supply. However, DR business models rely on customer participation and can only be effective when large numbers of customers in close geographic vicinity, e.g., connected to the same transformer, opt in. Here, we introduce a model for the dynamics of service adoption on a two-layer multiplex network: the layer of social interactions among customers and the power-grid layer connecting the households. While the adoption process-based on peer-to-peer communication-runs on the social layer, the time-dependent recovery rate of the nodes depends on the states of their neighbors on the power-grid layer, making an infected node surrounded by infectious ones less keen to recover. Numerical simulations of the model on synthetic and real-world networks show that a strong local influence of the customers' actions leads to a discontinuous transition where either none or all the nodes in the network are infected, depending on the infection rate and social pressure to adopt. We find that clusters of early adopters act as points of high local pressure, helping maintaining adopters, and facilitating the eventual adoption of all nodes. This suggests direct marketing strategies on how to efficiently establish and maintain new technologies such as DR schemes.

Reentrant phase transitions in threshold driven contagion on multiplex networks

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 observations, 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 instability in the form of reentrant phase transitions of contagion. Our results provide a theoretical explanation for the observation of large-scale contagion in highly connected but heterogeneous networks.

Influence Maximization for Fixed Heterogeneous Thresholds

Influence Maximization is a NP-hard problem of selecting the optimal set of influencers in a network. Here, we propose two new approaches to influence maximization based on two very different metrics. The first metric, termed Balanced Index (BI), is fast to compute and assigns top values to two kinds of nodes: those with high resistance to adoption, and those with large out-degree. This is done by linearly combining three properties of a node: its degree, susceptibility to new opinions, and the impact its activation will have on its neighborhood. Controlling the weights between those three terms has a huge impact on performance. The second metric, termed Group Performance Index (GPI), measures performance of each node as an initiator when it is a part of randomly selected initiator set. In each such selection, the score assigned to each teammate is inversely proportional to the number of initiators causing the desired spread. These two metrics are applicable to various cascade models; here we test them on the Linear Threshold Model with fixed and known thresholds. Furthermore, we study the impact of network degree assortativity and threshold distribution on the cascade size for metrics including ours. The results demonstrate our two metrics deliver strong performance for influence maximization.

Competing contagion processes: Complex contagion triggered by simple contagion

Empirical evidence reveals that contagion processes often occur with competition of simple and complex contagion, meaning that while some agents follow simple contagion, others follow complex contagion. Simple contagion refers to spreading processes induced by a single exposure to a contagious entity while complex contagion demands multiple exposures for transmission. Inspired by this observation, we propose a model of contagion dynamics with a transmission probability that initiates a process of complex contagion. With this probability nodes subject to simple contagion get adopted and trigger a process of complex contagion. We obtain a phase diagram in the parameter space of the transmission probability and the fraction of nodes subject to complex contagion. Our contagion model exhibits a rich variety of phase transitions such as continuous, discontinuous, and hybrid phase transitions, criticality, tricriticality, and double transitions. In particular, we find a double phase transition showing a continuous transition and a following discontinuous transition in the density of adopted nodes with respect to the transmission probability. We show that the double transition occurs with an intermediate phase in which nodes following simple contagion become adopted but nodes with complex contagion remain susceptible.

Impact of directionality and correlation on contagion

The threshold model has been widely adopted for modelling contagion processes on social networks, where individuals are assumed to be in one of two states: inactive or active. This paper studies the model on directed networks where nodal inand out-degrees may be correlated. To understand how directionality and correlation affect the breakdown of the system, a theoretical framework based on generating function technology is developed. First, the effects of degree and threshold heterogeneities are identified. It is found that both heterogeneities always decrease systematic robustness. Then, the impact of the correlation between nodal in- and out-degrees is investigated. It turns out that the positive correlation increases the systematic robustness in a wide range of the average in-degree, while the negative correlation has an opposite effect. Finally, a comparison between undirected and directed networks shows that the presence of directionality and correlation always make the system more vulnerable.

Threshold driven contagion on weighted networks

Weighted networks capture the structure of complex systems where interaction strength is meaningful. This information is essential to a large number of processes, such as threshold dynamics, where link weights reflect the amount of influence that neighbours have in determining a node's behaviour. Despite describing numerous cascading phenomena, such as neural firing or social contagion, the modelling of threshold dynamics on weighted networks has been largely overlooked. We fill this gap by studying a dynamical threshold model over synthetic and real weighted networks with numerical and analytical tools. We show that the time of cascade emergence depends non-monotonously on weight heterogeneities, which accelerate or decelerate the dynamics, and lead to non-trivial parameter spaces for various networks and weight distributions. Our methodology applies to arbitrary binary state processes and link properties, and may prove instrumental in understanding the role of edge heterogeneities in various natural and social phenomena.

Balancing Speed and Coverage by Sequential Seeding in Complex Networks

Information spreading in complex networks is often modeled as diffusing information with certain probability from nodes that possess it to their neighbors that do not. Information cascades are triggered when the activation of a set of initial nodes – seeds – results in diffusion to large number of nodes. Here, several novel approaches for seed initiation that replace the commonly used activation of all seeds at once with a sequence of initiation stages are introduced. Sequential strategies at later stages avoid seeding highly ranked nodes that are already activated by diffusion active between stages. The gain arises when a saved seed is allocated to a node difficult to reach via diffusion. Sequential seeding and a single stage approach are compared using various seed ranking methods and diffusion parameters on real complex networks. The experimental results indicate that, regardless of the seed ranking method used, sequential seeding strategies deliver better coverage than single stage seeding in about 90% of cases. Longer seeding sequences tend to activate more nodes but they also extend the duration of diffusion. Various variants of sequential seeding resolve the trade-off between the coverage and speed of diffusion differently.

Assortative Mating: Encounter-Network Topology and the Evolution of Attractiveness

We model a social-encounter network where linked nodes match for reproduction in a manner depending probabilistically on each node's attractiveness. The developed model reveals that increasing either the network's mean degree or the "choosiness" exercised during pair formation increases the strength of positive assortative mating. That is, we note that attractiveness is correlated among mated nodes. Their total number also increases with mean degree and selectivity during pair formation. By iterating over the model's mapping of parents onto offspring across generations, we study the evolution of attractiveness. Selection mediated by exclusion from reproduction increases mean attractiveness, but is rapidly balanced by skew in the offspring distribution of highly attractive mated pairs.

Heuristic Strategies for Persuader Selection in Contagions on Complex Networks

Individual decision to accept a new idea or product is often driven by both self-adoption and others' persuasion, which has been simulated using a double threshold model [Huang et al., Scientific Reports 6, 23766 (2016)]. We extend the study to consider the case with limited persuasion. That is, a set of individuals is chosen from the population to be equipped with persuasion capabilities, who may succeed in persuading their friends to take the new entity when certain conditions are satisfied. Network node centrality is adopted to characterize each node's influence, based on which three heuristic strategies are applied to pick out persuaders. We compare these strategies for persuader selection on both homogeneous and heterogeneous networks. Two regimes of the underline networks are identified in which the system exhibits distinct behaviors: when networks are sufficiently sparse, selecting persuader nodes in descending order of node centrality achieves the best performance; when networks are sufficiently dense, however, selecting nodes with medium centralities to serve as the persuaders performs the best. Under respective optimal strategies for different types of networks, we further probe which centrality measure is most suitable for persuader selection. It turns out that for the first regime, degree centrality offers the best measure for picking out persuaders from homogeneous networks; while in heterogeneous networks, betweenness centrality takes its place. In the second regime, there is no significant difference caused by centrality measures in persuader selection for homogeneous network; while for heterogeneous networks, closeness centrality offers the best measure.

Optimal information diffusion in stochastic block models

We use the linear threshold model to study the diffusion of information on a network generated by the stochastic block model. We focus our analysis on a two-community structure where the initial set of informed nodes lies only in one of the two communities and we look for optimal network structures, i.e., those maximizing the asymptotic extent of the diffusion. We find that, constraining the mean degree and the fraction of initially informed nodes, the optimal structure can be assortative (modular), core-periphery, or even disassortative. We then look for minimal cost structures, i.e., those for which a minimal fraction of initially informed nodes is needed to trigger a global cascade. We find that the optimal networks are assortative but with a structure very close to a core-periphery graph, i.e., a very dense community linked to a much more sparsely connected periphery.

Local cascades induced global contagion: How heterogeneous thresholds, exogenous effects, and unconcerned behaviour govern online adoption spreading

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 and modelling of product adoption in the world’s largest voice over internet service, the social network of Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural thresholds, which appears to be independent of the degree of adopting egos. We show that the structure of real-world adoption clusters is radically different from previous theoretical expectations, since vulnerable adoptions—induced by a single adopting neighbour—appear to be important only locally, while spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern the global emergence of social spreading.

Contagion on complex networks with persuasion

The threshold model has been widely adopted as a classic model for studying contagion processes on social networks. We consider asymmetric individual interactions in social networks and introduce a persuasion mechanism into the threshold model. Specifically, we study a combination of adoption and persuasion in cascading processes on complex networks. It is found that with the introduction of the persuasion mechanism, the system may become more vulnerable to global cascades, and the effects of persuasion tend to be more significant in heterogeneous networks than those in homogeneous networks: a comparison between heterogeneous and homogeneous networks shows that under weak persuasion, heterogeneous networks tend to be more robust against random shocks than homogeneous networks; whereas under strong persuasion, homogeneous networks are more stable. Finally, we study the effects of adoption and persuasion threshold heterogeneity on systemic stability. Though both heterogeneities give rise to global cascades, the adoption heterogeneity has an overwhelmingly stronger impact than the persuasion heterogeneity when the network connectivity is sufficiently dense.