Abstract
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.
Highlights
Weighted networks capture the structure of complex systems where interaction strength is meaningful
Weighted networks have been less studied than their unweighted counterparts, especially for threshold driven processes, which play an essential role in systems of self-organised criticality[23–25], epidemiology[26], firing neurons[27–29], or social contagion[6,30]
We provide an analytical solution of the dynamics, for arbitrary degrees and weights, together with numerical simulations and combinatorial arguments to show that the speed of spreading depends non-monotonously on the extent of weight heterogeneity and may drastically accelerate or decelerate as compared to the unweighted case, even for fixed thresholds
Summary
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. We provide an analytical solution of the dynamics, for arbitrary degrees and weights, together with numerical simulations and combinatorial arguments to show that the speed of spreading depends non-monotonously on the extent of weight heterogeneity and may drastically accelerate or decelerate as compared to the unweighted case, even for fixed thresholds We observe this effect under more realistic synthetic scenarios, such as scale-free networks and lognormal weight distributions, as well as in data-driven simulations over large-scale empirical weighted networks. Our results may have broader implications as our methodology is not specific to threshold dynamics and may be extended to any binary state process, while our study and conclusions may be useful in accurately modelling other dynamical phenomena over weighted networks
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