Bootstrap percolation on spatial networks

Abstract

Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of long-range links’ lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameter-dependent critical value for the power-law exponent, above which there is a double phase transition, mixed of a second-order phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a second-order phase transition. We further find a parameter-independent critical value around −1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value −1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the self-organization of spatial structure of online social networks, in terms of the effective function for information spreading.

Maximizing the collective learning effects in regional economic development

“…In the network: (i) Nodes are in either active or inactive status; (ii) Nodes remain active once activated. For the activation process [19]: (i) A given ratio of nodes…”

“…The establishment of spatial links works as follows. First, for each region, a random distance r between 2 and 2 / D is generated with probability Q r r P 5 ) (  [19]. The distance D is the maximum neighboring distance between that region and all other regions.…”

Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

“…The BP and DP have been studied on lattices [10,11,12,13,6,14,15,16,17,18,8], trees [1,19,20], and complex networks [21,7,22,9]. A few facts are known about the BP and DP.…”

Polluted bootstrap percolation with threshold two in all dimensions

“…This model, called polluted bootstrap percolation, was introduced by Gravner and McDonald [GM] in 1997. In the intervening peroid, rigorous progress on growth processes in random environments has been limited, but see [DEKMS,BDGM1,BDGM2,GMa,GZH,JLTV] for some examples of work on related models.…”