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Generic criticality of community structure in random graphs
Adam Lipowski1, Dorota Lipowska2
1Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2014
Summary
Community structure in random graphs reveals distinct behaviors based on link probability. For p=1 and p>1, community sizes exhibit power-law increases, with modularity decaying for larger p values.
Area of Science:
- Network Science
- Statistical Physics
- Computer Science
Background:
- Understanding community structure is crucial in analyzing complex networks.
- Random graphs provide a foundational model for network analysis.
Purpose of the Study:
- To investigate community structure in random graphs using modularity optimization.
- To analyze the impact of link probability (p/n) on community formation and size.
Main Methods:
- Newman greedy optimization of modularity.
- Nongreedy optimization of modularity for comparison.
- Numerical calculations and analysis of size distributions and modularity decay.
Main Results:
- For p<1, communities resemble clusters.
- For p=1 and p>1, community sizes (s(av), s(g)) show power-law increases (s(av)∼n(α'), s(g)∼n(α)).
- Modularity Q decays as Q∼p(-0.55) for large p.
Conclusions:
- Community structure in random graphs is sensitive to link probability.
- Power-law scaling of community sizes is observed at critical link probabilities.
- Both greedy and nongreedy modularity optimizations yield similar results for random graphs.
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