Bootstrap percolation on spatial networks

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• Bootstrap percolation models networked activation processes, explaining social phenomena like information propagation.
• Online social networks exhibit complex spatial structures influencing these processes.

Purpose of the Study:

• Investigate bootstrap percolation on spatial networks with power-law distributed long-range links.
• Analyze the impact of the power-law exponent on phase transitions and network behavior.

Main Methods:

• Utilized bootstrap percolation theory on undirected spatial networks.
• Modeled long-range link lengths using a power-law probability density function with a tunable exponent.
• Employed the size of the giant active component as the order parameter to identify phase transitions.

Main Results:

• A parameter-dependent critical value for the power-law exponent was identified.
• Above this critical value, a double phase transition (second-order and hybrid) occurs.
• Below this value, only a second-order phase transition is observed.
• A parameter-independent critical value around -1 was discovered, where double phase transition critical points remain stable.
• This critical value of -1 closely matches exponents found in real online social networks (e.g., LiveJournal).

Conclusions:

• The study clarifies the self-organization of spatial structures in online social networks.
• Findings offer insights into the mechanisms of information spreading within these networks.
• The critical exponent of -1 appears to be a significant universal feature for information diffusion in real-world networks.