Predicting cascade size distribution on one-dimensional geographic networks

Abstract

We make rather specific assumptions about the distribution of agents across the geographic map and the likelihood of any pair of them being adjacent. We assume that the agents are nearly evenly spaced across the network. More precisely, we assume that if a network of width $w$ were divided into $N$ equal intervals of width W/N each interval would contain exactly one agent. The location of each agent within its interval follows a uniform distribution. We assume that each agent can only be adjacent to other agents within some radius of influence r of itself with r << w. As the cascade propagates across the spatial network, it may spontaneously terminate. This property is not found in locally treelike networks. This work examines the likelihood of such a termination and accounts for that possibility when estimating the CDF of the final cascade size distribution.

The approximation we develop is reasonably accurate for several response threshold distributions. Previous work already showed that, under these assumptions, the cascade will propagate at a constant overall speed. Our approach gets an estimate for that speed and the likelihood of spontaneous termination.

To address the main challenge of the possibility of a finite-time extinction, we find the mean number of new activations per unit time and the likelihood of a spontaneous termination. These statistics can be estimated by viewing the number of new activations per unit time as a Markov chain. Given the number of activations at some time tau - 1, we can estimate the number of spikes sent to inactive agents. Given that number, we can estimate the number of activations at time tau. We assume that the number of new activations follows a Poisson distribution, and use our estimate as the mean. This gives us an approximate one-step probability transition matrix of the number of new activations from time tau - 1 to time tau. Using this matrix, we find the mean number of new activations and the extinction probability.

Over the past few decades, there has been considerable research on the spread of various phenomena across networks. While the most general case of the cascade problem on an arbitrary network is too broad a question to address, the question has been studied under specific simplifying assumptions, both on the construction of the network and on the rules for the spread and adoption of the cascading phenomenon. In paricular, we are interested in studying cascades under the widely-used Centola-Macy threshold model on a class of spatial networks. The spatial aspects of these networks present challenges not found in other networks.