The Iterated Local Model for tournaments

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.

💡 Research Summary

**
The paper introduces the Iterated Local Model for Tournaments (ILMT), a deterministic construction that generates dense, highly connected directed graphs (tournaments) by repeatedly cloning vertices and manipulating the orientation of arcs between clones. Starting from a base tournament G₀ and an infinite binary sequence s (the generating sequence), each iteration t creates a new tournament Gₜ as follows: for every vertex x in Gₜ₋₁ add a clone x′; for each existing arc (u,v) add the arcs (u, v′) and (u′, v); always add the arc (x′, x) so that each clone points to its parent; finally, for each original arc (u,v) add either (u′, v′) if s(t)=1 or (v′, u′) if s(t)=0. Consequently the vertex set doubles at each step while preserving the tournament property.

The authors first establish structural properties. Theorem 1 shows that if the base tournament has no sink and the generating sequence contains at least one zero, then for sufficiently large t the diameter of ILMTₜ,ₛ(G₀) is at most three. The proof exploits the fact that a zero‑step introduces reverse arcs between clones, creating short directed cycles that guarantee any two vertices are connected by a path of length ≤3. Theorem 2 proves that the vertex‑connectivity doubles after one iteration: a k‑connected base tournament yields a 2k‑connected tournament after the first ILMT step. Hence connectivity grows without bound as the process iterates.

Motif analysis focuses on 3‑vertex subtournaments. Theorem 3 gives exact recurrences for the numbers aₜ of directed 3‑cycles (D₃) and bₜ of transitive triples (T₃). When s(t)=1 the clone‑clone arcs preserve orientation, so each existing D₃ spawns eight new D₃’s, yielding aₜ=8aₜ₋₁. When s(t)=0 the orientation flips, and a more intricate counting shows aₜ=5aₜ₋₁+bₜ₋₁+⌊2^{t‑1}n₀/2⌋, where n₀=|V(G₀)|. From these recurrences the authors derive asymptotic proportions: if the sequence consists entirely of ones, the proportion of D₃’s stabilises at the initial value µ; if the sequence has infinitely many zeros, the proportion converges to 1/4, meaning that in the limit a quarter of all 3‑vertex subsets form a directed cycle, the rest being transitive.

Quasirandomness is addressed in Section 3. A sequence of tournaments is quasirandom if the density of every fixed subtournament H converges to the density in a uniform random tournament, i.e., 2^{−|V(H)|\choose2}. The authors invoke the notion of a quasirandom‑forcing tournament (Theorem 5) and show that the linear order on four vertices (L₄) suffices to certify quasirandomness. Theorem 6 proves that if the generating sequence has infinitely many zeros, the ILMT sequence (Gₜ)ₜ≥0 is quasirandom. The intuition is that each zero‑step reverses the orientation of clone‑clone arcs, effectively randomising the relative order of clones; with infinitely many such steps the distribution of any fixed pattern approaches the uniform one.

Universality is established in Theorem 4 and Corollary 2. When the generating sequence contains infinitely many zeros, any finite tournament H appears as an induced subtournament of some ILMT₍ᵣ,ₛ₎(G₀). The constructive proof repeatedly uses zero‑steps to flip the direction of arcs incident to a chosen vertex, thereby embedding the desired orientation pattern. Consequently ILMT is a universal model for tournaments: it eventually contains every possible finite directed pattern.

Section 4 explores classic graph‑theoretic parameters. The authors note that ILMT tournaments are strong, so the cop number is at most 2; the presence of clone‑parent arcs yields small dominating sets, and the chromatic number (the minimum number of transitive subtournaments needed to partition the vertex set) can be bounded using the motif proportions derived earlier. While detailed bounds are not fully developed, the section sketches how the iterative cloning process influences these invariants.

The paper concludes with a discussion of open problems: (i) analysing spectral properties of ILMT tournaments as the generating sequence varies; (ii) extending the model to weighted or multi‑clone versions; (iii) fitting ILMT parameters to empirical data from social media platforms (Reddit threads, Twitter follow graphs, etc.); and (iv) studying dynamical processes (rumor spreading, consensus formation) on ILMT graphs. Overall, the work provides a novel deterministic framework that simultaneously captures small diameter, high connectivity, quasirandomness, and universality—features that are rarely co‑present in existing network models.