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.
Highlights
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
Bootstrap percolation can be essentially considered as an activation process on networks: (i) Nodes are either active or inactive; (ii) Once activated, a node remains active forever; (iii) Initially, each node is in an active state with a given probability p; (iv) Subsequently, inactive nodes become active if they have at least k active neighbors; (v) Nodes are activated in an iterative manner according to the condition in (iv), until no more nodes can be activated
Summary
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. Bootstrap percolation can be essentially considered as an activation process on networks: (i) Nodes are either active or inactive; (ii) Once activated, a node remains active forever; (iii) Initially, each node is in an active state with a given probability p; (iv) Subsequently, inactive nodes become active if they have at least k active neighbors; (v) Nodes are activated in an iterative manner according to the condition in (iv), until no more nodes can be activated This process has been investigated on different kinds of networks including lattices[4,5], trees[6,7], random networks[8,9,10,11], and so on. Hu et al.[27] suggested the optimization of information collection as a possible explanation for the origin of this spatial scaling law
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