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The Iterated Local Model for tournaments

Anthony Bonato, MacKenzie Carr, Ketan Chaudhary, Trent G. Marbach, Teddy Mishura

TL;DR

The paper presents the Iterated Local Model for Tournaments (ILMT), a transitivity-based growth framework that evolves a base tournament $G_0$ along a binary generating sequence $s$ to produce a sequence of tournaments $G_t = \mathrm{ILMT}_{t,s}(G_0)$ via node cloning and orientation choices. ILMT yields dense, well-connected tournaments with small diameter, enables precise motif counts, and exhibits universality of subtournaments when $s$ has infinite support, with many ILMT sequences converging to quasirandom behavior; in particular, $d^*(T_4) \to 1/64$ and motif densities approach random-tournament benchmarks. The quasirandom regime is underpinned by a Markov-chain description of 4-node types, yielding a stationary distribution $\pi=[3/8,1/8,1/8,3/8]$ and continuum-many non-isomorphic quasirandom ILMT families. The paper also analyzes cop-number, domination numbers, and chromatic number, showing that 1-steps preserve certain parameters while 0-steps can drive unbounded chromatic growth, thereby outlining future directions including probabilistic variants and extensions to sparser oriented graphs.

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.

The Iterated Local Model for tournaments

TL;DR

The paper presents the Iterated Local Model for Tournaments (ILMT), a transitivity-based growth framework that evolves a base tournament along a binary generating sequence to produce a sequence of tournaments via node cloning and orientation choices. ILMT yields dense, well-connected tournaments with small diameter, enables precise motif counts, and exhibits universality of subtournaments when has infinite support, with many ILMT sequences converging to quasirandom behavior; in particular, and motif densities approach random-tournament benchmarks. The quasirandom regime is underpinned by a Markov-chain description of 4-node types, yielding a stationary distribution and continuum-many non-isomorphic quasirandom ILMT families. The paper also analyzes cop-number, domination numbers, and chromatic number, showing that 1-steps preserve certain parameters while 0-steps can drive unbounded chromatic growth, thereby outlining future directions including probabilistic variants and extensions to sparser oriented graphs.

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.
Paper Structure (5 sections, 20 theorems, 15 equations, 3 figures)

This paper contains 5 sections, 20 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Diameter, Connectedness, and Motifs
  3. Quasirandomness
  4. Graph-theoretical properties
  5. Discussion and Future Work

Key Result

theorem thmcountertheorem

If the tournament $G_0$ has no sink and $s$ has nonempty support, then for sufficiently large $t$, the diameter of $\mathrm{ILMT}_{t,s}(G_0)$ is at most 3.

Figures (3)

  • Figure 1: The first three time-steps of ILMT tournaments, where $G_0$ is a directed edge, and the sequence $s$ has values $s(1) = 1$ and $s(2) = 0.$ In $G_2$, $x"$ denotes the clone of $x$ while $(x')'$ denotes the clone of $x'$, for $x=a,b.$
  • Figure 2: Tournaments $G$, $H$, and $T$, each with chromatic number 2, for which a 0-step increases the chromatic number by 0, 1, and 2, respectively.
  • Figure 3: A sequence of oriented graphs formed by a 1-step and then a 0-step.

Theorems & Definitions (32)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • corollary thmcountercorollary
  • proof
  • theorem thmcountertheorem
  • proof
  • corollary thmcountercorollary
  • ...and 22 more