Cascades and Myopic Routing in Nonhomogeneous Kleinberg's Small World Model

Kleinberg’s small world model [21] simulates social networks with both strong and weak ties. In his original paper, Kleinberg showed how the distribution of weak-ties, parameterized by γ, influences the efficacy of myopic routing on the network. Recent work on social influence by k-complex contagion models discovered that the distribution of weakties also impacts the spreading rate in a crucial manner on Kleinberg’s small world model [15]. In both cases the parameter of γ = 2 proves special: when γ is anything but 2 the properties no longer hold. In this work, we propose a natural generalization of Kleinberg’s small world model to allow node heterogeneity: instead of a single global parameter γ, each node has a personalized parameter γ chosen independently from a distribution D. In contrast to the original model, we show that this model enables myopic routing and k-complex contagions on a large range of the parameter space, improving the robustness of the model. Moreover, we show that our generalization is supported by realworld data. Analysis of four different social networks shows that the nodes do not show homogeneity in terms of the variance of the lengths of edges incident to the same node.