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Stochastically evolving graphs via edit semigroups.

Fan Chung1, Sawyer Jack Robertson1

  • 1Department of Mathematics, University of California San Diego, La Jolla, CA 92093.

Proceedings of the National Academy of Sciences of the United States of America
|November 26, 2025
PubMed
Summary
This summary is machine-generated.

We introduce a random process for evolving subgraphs within a host graph, creating a random walk on all possible subgraphs. This model offers a general stochastic approach for sampling random subgraphs, with applications in various fields.

Keywords:
Markov chainsleft regular bandsmixing timesrandom graphsspectral graph theory

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Area of Science:

  • Graph Theory
  • Stochastic Processes
  • Spectral Theory

Background:

  • Evolving graphs are crucial in diverse fields like deep learning, epidemic modeling, and social networks.
  • Existing models for evolving graphs often lack a unified stochastic framework for sampling subgraphs.
  • Understanding the dynamics of random subgraph evolution is essential for developing new graph-based algorithms.

Purpose of the Study:

  • To develop a general stochastic model for the random evolution of subgraphs within a host graph.
  • To analyze the spectral properties of the random walk generated by this evolving process.
  • To provide a theoretical framework for sampling random subgraphs.

Main Methods:

  • Utilizing spectral theory of semigroups, Tsetlin library, and hyperplane arrangements.
  • Defining a random walk on the space of all possible subgraphs.
  • Analyzing the eigenvalues and eigenvectors of the transition probability matrix.

Main Results:

  • Eigenvalues of the random walk can be indexed by subsets of edges of the host graph.
  • A closed-form formula for eigenvectors is derived for simple edits (adding/deleting edges).
  • A sharp bound for the rate of convergence of the random walk is established.

Conclusions:

  • The proposed random evolving process provides a general stochastic model for sampling random subgraphs.
  • The spectral analysis offers insights into the convergence properties of the random walk.
  • This framework unifies and extends existing models like the Moran forest and dynamic random intersection graphs.