“Takeover times for a simple model of network infection”
The work of Ottino-Löffler, Scott, and Strogatz investigates the probability density function associated with a simple disease propagation model in various networks. The disease propagation model is as follows. Originally a single node is infected. At each iteration, a random node in the network is chosen and if that node is infected, a random node connected to that node also becomes infected. Therefore if the node randomly chosen in that iteration is not infected nothing happens. Also, if the node randomly chosen and the neighbor it is supposed to infect are already infected, there is no net affect either. As a result, this makes the initial infection growth and the final infection growth quite slow meaning these two stages of the process dominate the infection time. The dynamics of the network infection is inherently statistical due to the random nature of the infection process. Naturally statistical methods were employed to analyze the system.
For a network ring (1D lattice), the infection time follows a normal distribution. For a star graph, the infection time follows a Gumbel distribution (a graph where all edges connect a single node to the rest of the nodes but no other edges exist). For a complete graph, the infection time follows a convolution of two Gumbel distributions. For a d-dimensional lattice, or a graph where each node connects to the 2*d closest nodes to it, the limiting case of high dimension is a Gumbel distribution, and for most values of d the result is somewhere between a Gumbel distribution and the 1D lattice’s normal distribution, however, for the 2d case, the distribution is also purely a normal distribution rather than a mix of a Gumbel and normal distribution: an unexpected result. Besides the obvious application to spreading disease in an interacting group of humans, the analysis appears to show similar probability density functions to that of real observations of disease growth within a culture. Namely, experimental observations of the incubation time of some culture growths show the same sort of skewed probability density function as those seen in the analysis. The explanation given is that the invasion of the healthy culture is a random process that occurs sort of one switch at a time at the interface between the diseased and non diseased culture, and the analysis shows that for such a model of disease spread, the right skewed distribution is automatically generated. That means it could be possible to have a theoretical and statistically precise understanding of these disease growths that underlies the observed phenomena which could lead to better disease treatment and medical procedures.