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Polynomial growth in branching processes with diverging reproductive number
1Department of Physics and Center for Complex Network Research, University of Notre Dame, Notre Dame, Indiana 46556, USA.
Physical Review Letters
|February 21, 2006
Summary
This study reveals that graphs with power law degree distributions exhibit non-exponential spreading dynamics. Network growth becomes extensive and polynomial, not exponential, due to diverging moments.
Area of Science:
- Network science
- Graph theory
- Statistical physics
Background:
- Spreading dynamics on complex networks often follow exponential growth patterns.
- Power law degree distributions (pk ~ k^-gamma, 2
- The second moment of the degree distribution's divergence is key to understanding altered growth patterns.
Purpose of the Study:
- To investigate the spreading dynamics on graphs with power law degree distributions.
- To analyze the impact of a diverging reproductive number on network growth.
- To identify deviations from standard exponential growth patterns.
Main Methods:
- Modeling spreading dynamics as a branching process.
- Analyzing graphs with power law degree distributions (pk ~ k^-gamma, 2
- Examining the consequences of the divergence of the second moment of the degree distribution.
Main Results:
- Observed extensive population growth, where the number of reached vertices scales with graph size.
- Identified a vanishing time scale for reaching graph size in the large graph limit.
- Demonstrated polynomial growth in temporal evolution, with the exponent linked to graph characteristic distance.
Conclusions:
- The divergence of the second moment of the degree distribution fundamentally alters spreading dynamics.
- Network growth on these graphs is extensive and polynomial, not exponential.
- These findings offer new insights into network dynamics and open avenues for future research.
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