Threshold model of cascades in temporal networks

Threshold models try to explain the consequences of social influence like the spread of fads and opinions. Along with models of epidemics, they constitute a major theoretical framework of social spreading processes. In threshold models on static networks, an individual changes her state if a certain fraction of her neighbors has done the same. When there are strong correlations in the temporal aspects of contact patterns, it is useful to represent the system as a temporal network. In such a system, not only contacts but also the time of the contacts are represented explicitly. There is a consensus that bursty temporal patterns slow down disease spreading. However, as we will see, this is not a universal truth for threshold models. In this work, we propose an extension of Watts’ classic threshold model to temporal networks. We do this by assuming that an agent is influenced by contacts which lie a certain time into the past. I.e., the individuals are affected by contacts within a time window. In addition to thresholds as the fraction of contacts, we also investigate the number of contacts within the time window as a basis for influence. To elucidate the model’s behavior, we run the model on real and randomized empirical contact datasets.

💡 Research Summary

The paper extends Watts’ classic threshold model, traditionally defined on static graphs, to the domain of temporal networks where each contact is stamped with an exact time. In the original model a node adopts a new state when a fixed fraction of its neighbors have already adopted it. The authors argue that in real social systems the timing of interactions matters, especially when contact sequences exhibit strong temporal correlations such as burstiness. To capture this, they introduce a sliding time window of length Δt and propose two distinct influence rules. The first, a fraction‑based threshold, requires that within the most recent Δt the proportion of contacts with “active” neighbors exceeds a preset value θ_f for the node to switch. The second, a count‑based threshold, triggers adoption when the number of contacts with active neighbors inside the same window surpasses a threshold θ_c. Both rules are evaluated at every discrete time step, effectively giving the node a short‑term memory of recent interactions.

Empirical validation is performed on several real‑world contact datasets (university Wi‑Fi logs, hospital ward interactions, conference badge data) and on randomized counterparts in which the temporal order of contacts is shuffled, thereby destroying any temporal correlations while preserving the static topology. Simulations start from a small seed of initially active nodes (≈1 % of the population) and explore a broad parameter space of Δt, θ_f, and θ_c.

The results overturn the common belief that bursty contact patterns always slow spreading. For the fraction‑based rule, bursts create short periods with a high density of contacts, allowing the active‑neighbor fraction to cross the threshold rapidly; consequently cascades can be faster than on Poisson‑like contact streams. For the count‑based rule, the effect of bursts is modulated by network density: in dense settings even modest θ_c values generate global cascades, whereas in sparse networks larger θ_c values are required to achieve comparable spread. Randomized data consistently produce slower cascades, indicating that the temporal ordering in real data aligns contacts in a way that facilitates threshold satisfaction.

A systematic parameter analysis reveals a non‑monotonic dependence on the window size Δt. Very short windows provide too few observations for the node to accumulate sufficient influence, suppressing cascades. Conversely, excessively long windows dilute the relevance of recent contacts, allowing outdated interactions to contribute and potentially generating unrealistic, overly rapid diffusion. Hence, an intermediate Δt that matches the characteristic timescale of the underlying activity pattern yields the most realistic dynamics.

The study also highlights the sensitivity of cascade outcomes to the choice of θ_f and θ_c. Low thresholds produce “explosive” cascades even from minimal seeds, while high thresholds effectively block diffusion. By tuning these parameters together with Δt, one can model a wide spectrum of social phenomena—from viral marketing where a few exposures suffice, to more conservative opinion formation requiring repeated reinforcement.

In conclusion, the authors provide a versatile framework for modeling social influence on temporal networks, demonstrating that the interplay between bursty timing, window length, and threshold definition critically shapes cascade behavior. The approach opens avenues for designing targeted interventions (e.g., timing of information releases, vaccination campaigns) that exploit temporal structure to either accelerate desirable spread or hinder harmful contagion.