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Depolymerizable Olefinic Polymers Based on Fused-Ring Cyclooctene Monomers
Published on: December 16, 2022
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Cyclization in bipartite random graphs
1Geophysical Center of Russian Academy of Science, 3, Molodezhnaya street, 119296 Moscow, Russia.
Physical Review. E
|April 16, 2020
Summary
This study models the evolution of finite bipartite graphs by adding edges. It reveals how graph structure and the emergence of a giant component depend on component size and total graph size.
Area of Science:
- Graph theory
- Statistical physics
- Network science
Background:
- Studies the time evolution of finite bipartite graphs.
- Graphs evolve by edges being added sequentially, leading to component coalescence or cycling.
- Graph states are defined by occupation numbers of components based on vertex types and edge counts.
Purpose of the Study:
- To analyze the time evolution of finite bipartite graphs.
- To reformulate the master equation for graph evolution using a generating functional.
- To investigate the emergence of a giant component in evolving graphs.
Main Methods:
- Modeling graph evolution through edge addition.
- Utilizing a master equation to describe probability distribution over graph states.
- Employing a generating functional for exact solutions.
- Analyzing the behavior of linked components in the large graph limit.
Main Results:
- Developed an exact solution for the evolution equation of the generating functional.
- Derived average population numbers for linked components.
- Showed that in the large graph limit, the distribution factorizes into two multipliers.
- Identified that both multipliers contain information about the giant component's emergence.
Conclusions:
- The study provides a framework for analyzing the dynamics of finite bipartite graphs.
- The emergence of a giant component is linked to critical thresholds in graph evolution.
- The factorization of the distribution in the large graph limit offers insights into universal properties of random graphs.
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