The Golden Ratio Emerges From Pure Randomness in Higher-Order Networks

On a single new vertex, the entire future of a growing structure can hinge. And because of that, some outcomes are vanishingly rare. For a network of 100 vertices built by a certain simple random rule, the chance it collapses into a bare, one-dimensional tree — with no triangles anywhere — is roughly $2^{-100}\sqrt{400\pi }$, a number so small it is effectively, impossibly, zero. Yet that same random recipe, run over and over, always produces something predictable. The trick of the mathematics in this paper is that wild individual randomness can still yield sharp, deterministic averages when you look at enough vertices.

Krapivsky and Lucas (2026) study a family of models called random recursive simplicial complexes (RRSC). They are the higher-dimensional cousins of the humble random recursive tree — the kind of tree you get by adding one new node at a time and attaching it to an existing node chosen at random. Their results are the abstract machinery of a beautiful, mostly unexplored corner of mathematics. But the ideas underneath — how genuine randomness can end in sharp collective law, and how structure can emerge from a single seed — reach far beyond topology into epidemiology, neural wiring, and the science of how complex systems assemble themselves.

The Science

Start with the building blocks. A 0-dimensional simplex is a point. A 1-dimensional simplex is a line segment. A 2-dimensional simplex is a triangle; a 3-dimensional simplex is a tetrahedron. A simplicial complex is a collection of these objects with one strict rule: if a simplex is in the complex, then every face of it — every edge of a triangle, every triangle of a tetrahedron — must also be there. One-dimensional simplicial complexes are just graphs, the familiar networks of points (vertices) and lines (edges). But real systems often involve interactions among more than two things at once: a conversation among three friends, a collaboration among five scientists, a neural spike co-activating a cluster of neurons. Simplicial complexes are the natural language for that higher-order structure (Bick et al., 2021).

The RRSC model is disarmingly simple. Begin with one "primordial" vertex, $v_1$. On each step, add a new vertex $v_n$ and join it to a single existing simplex chosen uniformly at random — a vertex, an edge, a triangle, depending on what's there. Add that new simplex and all its faces, and repeat.

The model comes in versions parameterized by a positive integer $m$. In RRSC($m$), each new vertex joins an existing simplex of dimension strictly less than $m$. So in RRSC(2), a new vertex attaches to either an existing vertex or an existing edge (dimension 0 or 1), but never directly to a triangle. The RRSC(1) model degenerates into the classic random recursive tree: every new vertex just attaches to an existing vertex, and the result is always a tree (Victor et al., 2022). The paper's job is to work out what happens for all the higher dimensions.

Importantly, this is a uniform choice — not preferential attachment, where hubs grow fatter as they get more popular. In preferential-attachment models, rich nodes get richer, which is why they produce scale-free networks with power-law degree distributions. Here every existing simplex has equal odds. And yet, as the authors show, in some ways the structure still concentrates: there is still a primordial vertex whose degree grows algebraically with the system size. The hub emerges anyway, even without the popularity contest (Barabási & Albert, 1999).

What They Found

The paper's most striking results concern what happens to the counts of simplices of each dimension as the number of vertices $N$ grows.

Let $S_d$ be the number of simplices of dimension $d$. Some things are pinned down immediately. The number of vertices is deterministic: $S_0=N$. So is the Euler characteristic — the alternating sum

$$\chi =\sum _{d=0}^m(-1{)}^d{S}_d=1,$$

a topological fingerprint that stays equal to one for every $N$ and every $m$, because every complex the rule can build is contractible to a point, topologically trivial (Hatcher, 2002).

The genuinely new results are the growth laws for the counts $S_d$. Because the $S_d$ are random — they fluctuate from one realization to the next — the authors need a way to make sense of them. They prove these variables are asymptotically self-averaging: the relative fluctuation $\mathrm{std}[S_d]/⟨S_d⟩$ shrinks like $N^{-1/2}$. In plain terms, the bigger the complex gets, the tighter each individual realization hugs the ensemble average. One run might have slightly more edges or slightly fewer than average, but the relative spread collapses to zero. This is the same phenomenon that makes a fair coin's observed fraction of heads converge to $1/2$ as you flip it a million times. It licenses the whole analysis: instead of tracking a messy distribution, you can study clean averages.

For RRSC(2), the authors obtain exact, closed-form amplitudes. The number of edges grows as

$$⟨S_1⟩\simeq a_{1}N,a_1=\frac{1+\sqrt{5}}{2}\approx \mathrm{1.618.}$$

That is the golden ratio — a surprise, and a lovely one. The number of triangles grows as

$$⟨S_2⟩\simeq a_{2}N,a_2=\frac{\sqrt{5}-1}{2}\approx 0.618,$$

and the grand total of all simplices satisfies $⟨S⟩\simeq (\sqrt{5}+1)N$. The golden ratio's two reciprocal parts split the edge and triangle counts, and the whole thing adds up to a clean, irrational rate of growth.

The formula reads like a found object, an unforced piece of beauty. And it is verified by simulation: the authors simulate thousands of independent realizations, and the distribution of $S_1/N$ concentrates precisely on $a_1$ as $N$ climbs from $10^2$ through $10^4$ (see

). The variance is also exactly derivable: $\mathrm{Var}[S_1]=N/3$, matching simulation to within noise.

For higher dimensions the tidy closed forms give way to algebraic systems that must be solved numerically, but the rigor holds. For RRSC(3), the amplitudes come out as

$$a_1\approx 2.089,a_2\approx 1.401,a_3\approx 0.312,A\approx 4.803,$$

and for RRSC(4):

$$a_1\approx 2.450,a_2\approx 2.188,a_3\approx 0.872,a_4\approx 0.134,A\approx \mathrm{6.644.}$$

Notice the pattern: higher-dimensional simplices are always rarer, their amplitudes shrinking as dimension rises. The paper tabulates amplitudes for all $m\le 10$ in an appendix. As $m$ grows, convergence slows — because building a $d$-dimensional simplex requires smaller ones to exist first, so the larger structures take longer to accumulate.

The bounds are also exact. Every RRSC(2) complex satisfies

$$N-1\le S_1\le 2N-3,0\le S_2\le N-2.$$

The lower bound is a bare tree (no triangles at all); the upper bound is an open book — a stack of triangles all sharing one common edge, like the pages of a book glued along their spines (see

). The gap between the two extremes is a full measure of just how much shape randomness is on the table.

The paper also computes extreme probabilities exactly. The chance of staying a one-dimensional tree after $N$ vertices is

$$T_N=\frac{(N-1)!}{(2N-3)!!},$$

which decays like $2^{-N}\sqrt{4\pi N}$. And the chance of the two most special trees — a star all centered on one vertex, or a pure linear chain — are each given by ${\sigma }_{N+1}=1/(2N-1)!!$, equally tiny. Even the probability of any linear graph $L_N$ is derived exactly. These are the kinds of exact, closed-form results that make combinatorial probability so satisfying.

Underlying all of this is an important structural fact about the degree distribution. In many growing network models the degree distribution has a power-law tail, $n_{\delta }\sim {\delta }^{-\nu }$. The authors deduce, by comparing the primordial vertex's average degree to the maximal degree, that the RRSC(2) model has decay exponent

$$\nu (2)=\frac{5+\sqrt{5}}{2}\approx 3.618,$$

again the golden-ratio pattern. The higher-dimension exponents are $\nu (3)\approx 2.716$ and $\nu (4)\approx 2.450$ — all comfortably above 2, meaning the distributions have finite variance. This is a remarkable outcome for a rule with no preferential attachment: the structure still develops hubs, and the hub—the primordial vertex—keeps growing algebraically with the system size.

Why This Changes Things

Why should anyone outside topology care? Because the RRSC models sit at the exact junction where three strands of modern science meet.

First, they are tractable models of higher-order structure — the geometry of relations involving more than two participants at once. Real systems — brains, social networks, collaboration patterns — are increasingly understood through higher-order interactions that reshape dynamics in ways pairwise links miss (Battiston et al., 2020). If a research group wants to know how shared triangles or tetrahedra of interaction grow and concentrate, they need exactly this kind of generative model with clean, provable growth laws. Before this paper, there was no parameter-free stochastic model in which you could prove the counts of every dimension grow linearly, with exact amplitudes.

Second, the results illuminate the radical tension between micro-randomness and macro-order. The rule is maximally egalitarian — every existing simplex has equal chance of hosting the next arrival — yet the outcome still concentrates: counts self-average, a privileged hub emerges, algebraic degree tails appear. This is the same lesson repeated across complex systems: robust collective structure does not require centralized design or rich-get-richer feedback. It can emerge from pure, unbiased randomness plus a rule of attachment. That reframes how we think about, say, the robustness of decentralized networks or the spontaneous emergence of hierarchy in biological aggregates.

Third, there is a methodological payoff. The proof that $S_d$ is self-averaging is not an afterthought; it is the load-bearing wall. Every average-based conclusion in the paper rests on it. The paper shows the general strategy — prove concentration, then extract averages — and applies it consistently from $m=2$ through $m=10$. For the wider community, the reusable tool is the style of argument: establish that a random count concentrates, then solve algebraic equations for its growth rate.

The open book is also a lovely concrete picture. Every RRSC complex lives between two extremes: the austere tree and the maximal stack of pages sharing a spine. The paper's bounds say the true random complex lives strictly between them, and the average counts tell you precisely where. The golden ratio appears as the answer for dimension two — a reminder that mathematics keeps handing us the same mysterious constant in genuinely new places.

There is a humility here worth naming. The origin of this model is itself random: it is the uniformly-chosen counterpart of the recursive random tree, the "higher-dimensional counterpart" as Krapivsky and Lucas put it. Preferential attachment gets all the fame, but the boring uniform rule turns out to be rich enough to support rigorous, surprising theorems. Sometimes the most productive model is the simplest one nobody thought to check.

What's Next

The paper is candid about what remains open. The full distribution of the degree — beyond the fraction of degree-one vertices, which they compute exactly as

$$n_1=\frac{7-\sqrt{5}}{22}$$

for RRSC(2) — is still a work in progress. The recursive scheme they outline grows rapidly cumbersome, and they have only pinned down the low-degree end.

A whole additional layer awaits: generalized degrees. In a simplicial complex you can count not just edges adjacent to a vertex, but $d$-dimensional simplices adjacent to a $D$-dimensional simplex, for any pair $0\le d<D$. The ordinary vertex degree is the $(0,1)$ case, but RRSC($m$) supports $m(m+1)/2$ distinct degree distributions, all unexplored. That is a small menu of future paper topics, each with its own open questions.

The authors also propose a sibling family, the HSC models, generating homogeneous simplicial complexes where every maximal simplex has the same dimension $m$. They report, tantalizingly, that these are more tractable than RRSC — an Appendix-length teaser for what might be a fuller theory of uniform-attachment higher-order growth.

Most fundamentally, the paper leaves a question hanging over the whole field: do these sharp, provable properties survive when you make the rule noisier — when attachment is no longer perfectly uniform, when a vertex can attach to several simplices, or when dimension is allowed to grow without bound? The RRSC(1) tree case and the uniform RRSC($m$) case both concentrate and self-average. The boundary where concentration breaks down, and randomness wins over law, is an open frontier. Knowing exactly where that boundary lies — how much disorder a system can take before its averages stop being reliable — is precisely the knowledge that transfers to real complex systems.

This is a paper about a model, not about any particular application. But it is a paper built on an idea that belongs to everyone: that from the most boring, uniform, unbiased random rule, a surprisingly structured world can be proven to emerge — one whose counts grow at exact, golden ratios, whose hubs appear without popularity contests, and whose largest hopes of staying small and simple fade exponentially fast. The structure that waits inside sheer chance is larger, and more orderly, than intuition suggests.

That is a comforting thought and a useful one, whether the vertices are neurons, scientists, or abstract points of topology.