Thresholdmodel

Threshold model is a framework used to study the diffusion of behaviors, opinions, or innovations in social networks. In its common linear threshold form, each agent i is assigned a threshold theta_i between 0 and 1, representing the fraction of its neighbors that must be active before i adopts the behavior. The network is G=(V,E) with N_i denoting the neighbors of i. The state s_i is 0 (inactive) or 1 (active). A set of seed nodes starts as active, and adoption propagates through the network according to the thresholds of the individuals.

Dynamics are typically discrete in time. A node becomes active if the fraction of its active neighbors among N_i is at least theta_i. Once active, a node remains active in the basic model, making the process monotone. Thresholds are drawn from a distribution F, and the cascade outcome depends on the network topology, degree distribution, and the threshold distribution. Variants include synchronous or asynchronous updates, deterministic or probabilistic adoption, and reversible thresholds where activation can fade.

Origins and theoretical context: The concept traces to Granovetter’s threshold model of collective behavior (1978). The network version, often called the linear threshold model, was formalized in the work of Watts (2002) on global cascades in random networks, which demonstrated how small seed sets could trigger large-scale adoption under certain combinations of thresholds and connectivity.

Applications and limitations: Threshold models are used to study diffusion of innovations, political mobilization, social influence, and information spread. Limitations include assumptions of complete knowledge of neighbors’ states, fixed thresholds, and irreversible adoption in the standard form; real-world processes may involve time delays, heterogeneity beyond simple thresholds, and nonmonotone dynamics.