  The spread of new beliefs, behaviors, conventions, norms, and technologies in
social and economic networks are often driven by cascading mechanisms, and so
are contagion dynamics in financial networks. Global behaviors generally emerge
from the interplay between the structure of the interconnection topology and
the local agents' interactions. We focus on the Linear Threshold Model (LTM) of
cascades first introduced by Granovetter (1978). This can be interpreted as the
best response dynamics in a network game whereby agents choose strategically
between two actions and their payoff is an increasing function of the number of
their neighbors choosing the same action. Each agent is equipped with an
individual threshold representing the number of her neighbors who must have
adopted a certain action for that to become the agent's best response. We
analyze the LTM dynamics on large-scale networks with heterogeneous agents.
Through a local mean-field approach, we obtain a nonlinear, one-dimensional,
recursive equation that approximates the evolution of the LTM dynamics on most
of the networks of a given size and distribution of degrees and thresholds.
Specifically, we prove that, on all but a fraction of networks with given
degree and threshold statistics that is vanishing as the network size grows
large, the actual fraction of adopters of a given action in the LTM dynamics is
arbitrarily close to the output of the aforementioned recursion. We then
analyze the dynamic behavior of this recursion and its bifurcations from a
dynamical systems viewpoint. Applications of our findings to some real network
testbeds show good adherence of the theoretical predictions to numerical
simulations.
