The Iterated Local Model for tournaments

2026-01-30

AI Summary

This study addresses the generation of dense directed graphs—specifically tournaments—that exhibit realistic complex network properties by leveraging the principle of transitivity. To this end, the authors propose the Iterative Local Model with Transitivity (ILMT), which uniquely integrates transitivity rules with a node-cloning mechanism: at each iteration, a node and its adjacency relations are cloned, and selected arcs are directionally reversed to construct the tournament. The model produces quasi-random tournaments characterized by small diameter, high connectivity, and quasi-randomness across a range of parameters. The work further provides a systematic analysis of structural properties, including motif distributions and key graph invariants such as the cop number, domination number, and chromatic number, thereby revealing the universality and theoretical richness of the generated tournaments.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Transitivity is a central, generative principle in social and other complex networks, capturing the tendency for two nodes with a common neighbor to form a direct connection. We propose a new model for highly dense, complex networks based on transitivity, called the Iterated Local Model Tournament (ILMT). In ILMT, we iteratively apply transitivity to form new tournaments by cloning nodes and their adjacencies, and either preserving or reversing the orientation of existing arcs between clones. The resulting model generates tournaments with small diameters and high connectivity as observed in real-world complex networks. We analyze subtournaments or motifs in the ILMT model and their universality properties. For many parameter choices, the model generates sequences of quasirandom tournaments. We also study the graph-theoretic properties of ILMT tournaments, including their cop number, domination number, and chromatic number. We finish with a set of open problems and variants of the ILMT model for oriented graphs.

Problem

• transitivity
• tournaments
• complex networks
• quasirandom
• graph generation

Innovation

• Iterated Local Model
• Transitivity
• Tournament
• Quasirandom
• Motif