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Bootstrap percolation on spatial networks.

Gao J, Zhou T, Hu Y - Sci Rep (2015)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China.

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.

No MeSH data available.


Sgc, NOI and Sgc2 as a function of p for different α after k = 3 bootstrap percolation on undirected Kleinberg’s spatial networks.Two different types of Sgc(p) curves are observed, including a double phase transition (a) and a second-order one (b). When α ≥ −1, Sgc(p) curves behave alike and a double phase transition is present. Sgc abruptly jumps to 1 at , where NOI reaches its maximum (c). When α < −1, there is only a second-order phase transition with an increasing critical point as the decreasing of α, where Sgc2 reaches its maximum at different pc2 (d). Dash lines mark identification of critical points. Results are obtained by simulations on networks with fixed size L = 400 and averaged over 1000 realizations.
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f2: Sgc, NOI and Sgc2 as a function of p for different α after k = 3 bootstrap percolation on undirected Kleinberg’s spatial networks.Two different types of Sgc(p) curves are observed, including a double phase transition (a) and a second-order one (b). When α ≥ −1, Sgc(p) curves behave alike and a double phase transition is present. Sgc abruptly jumps to 1 at , where NOI reaches its maximum (c). When α < −1, there is only a second-order phase transition with an increasing critical point as the decreasing of α, where Sgc2 reaches its maximum at different pc2 (d). Dash lines mark identification of critical points. Results are obtained by simulations on networks with fixed size L = 400 and averaged over 1000 realizations.

Mentions: Figure 2 shows rich phase transition phenomena when taking Sgc as the order parameter. When α ≥ −1, the curves of Sgc(p) are well overlapped and the system undergoes a double phase transition, mixed of a hybrid phase transition and a second-order one as shown in Fig. 2a. Notice that Sgc has a continuous increase at (the second-order critical point), where the transition is of second-order. In contrast, Sgc has an abrupt jump directly from around 0.58 to almost 1 at (the first-order critical point), where there is a hybrid phase transition. Surprisingly, the two critical points seem to be constant when , as indicated by the four overlapped Sgc(p) curves in Fig. 2a. When α < −1, there is only a second-order phase transition with an increasing pc2 as the decreasing of α (see Fig. 2b). Specifically, when α = −2 and when α = −5. Although Sgc goes up sharper after p exceeds pc2 as α getting smaller, simulations justify that the curve of Sgc(p) is still continuous, meaning that the transition is indeed of second-order when α < −1.


Bootstrap percolation on spatial networks.

Gao J, Zhou T, Hu Y - Sci Rep (2015)

Sgc, NOI and Sgc2 as a function of p for different α after k = 3 bootstrap percolation on undirected Kleinberg’s spatial networks.Two different types of Sgc(p) curves are observed, including a double phase transition (a) and a second-order one (b). When α ≥ −1, Sgc(p) curves behave alike and a double phase transition is present. Sgc abruptly jumps to 1 at , where NOI reaches its maximum (c). When α < −1, there is only a second-order phase transition with an increasing critical point as the decreasing of α, where Sgc2 reaches its maximum at different pc2 (d). Dash lines mark identification of critical points. Results are obtained by simulations on networks with fixed size L = 400 and averaged over 1000 realizations.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4589777&req=5

f2: Sgc, NOI and Sgc2 as a function of p for different α after k = 3 bootstrap percolation on undirected Kleinberg’s spatial networks.Two different types of Sgc(p) curves are observed, including a double phase transition (a) and a second-order one (b). When α ≥ −1, Sgc(p) curves behave alike and a double phase transition is present. Sgc abruptly jumps to 1 at , where NOI reaches its maximum (c). When α < −1, there is only a second-order phase transition with an increasing critical point as the decreasing of α, where Sgc2 reaches its maximum at different pc2 (d). Dash lines mark identification of critical points. Results are obtained by simulations on networks with fixed size L = 400 and averaged over 1000 realizations.
Mentions: Figure 2 shows rich phase transition phenomena when taking Sgc as the order parameter. When α ≥ −1, the curves of Sgc(p) are well overlapped and the system undergoes a double phase transition, mixed of a hybrid phase transition and a second-order one as shown in Fig. 2a. Notice that Sgc has a continuous increase at (the second-order critical point), where the transition is of second-order. In contrast, Sgc has an abrupt jump directly from around 0.58 to almost 1 at (the first-order critical point), where there is a hybrid phase transition. Surprisingly, the two critical points seem to be constant when , as indicated by the four overlapped Sgc(p) curves in Fig. 2a. When α < −1, there is only a second-order phase transition with an increasing pc2 as the decreasing of α (see Fig. 2b). Specifically, when α = −2 and when α = −5. Although Sgc goes up sharper after p exceeds pc2 as α getting smaller, simulations justify that the curve of Sgc(p) is still continuous, meaning that the transition is indeed of second-order when α < −1.

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China.

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.

No MeSH data available.