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Efficient orbit-aware triad and quad census in directed and undirected graphs.

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  • 1Department of Computer & Information Science, University of Konstanz, Box 67, Konstanz, 78457 Germany.

Applied Network Science
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We introduce a novel method for counting four-node subgraphs (quad census) in networks, essential for understanding network effects and stochastic models. This unified approach is computationally efficient and extends to directed graphs.

Keywords:
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Area of Science:

  • Graph theory
  • Network analysis
  • Computational mathematics

Background:

  • Subgraph statistics are crucial for network analysis, social network analysis, and systems biology.
  • Assessing subgraph prevalence is vital for MCMC sampling and estimation in stochastic network models.
  • Existing methods for subgraph counting are often limited in scope or generality.

Purpose of the Study:

  • To develop a unified approach for counting all induced and non-induced four-node subgraphs (quad census).
  • To enable per-node and per-edge counting of these subgraphs, categorized by their non-automorphic roles.
  • To extend the quad census methodology to directed graphs.

Main Methods:

  • A novel approach based on a clique-listing subroutine.
  • Unified counting of induced and non-induced four-node subgraphs.
  • Extension of the quad census to directed graphs, inspired by the triad census.

Main Results:

  • The first unified method for comprehensive quad census.
  • Demonstrated computational efficiency and superior performance compared to previous methods.
  • Development of the asymptotically fastest triad census algorithm as a byproduct.

Conclusions:

  • The proposed quad census method offers a simple yet powerful tool for network analysis.
  • This approach enhances the study of network effects and stochastic network models.
  • The findings provide significant advancements in both subgraph enumeration and network algorithm efficiency.