Yule Web: Coalescing Yule Processes

• Yule Web is a system of coalescing pure-birth Yule processes that model growth and merging, critical for understanding network clustering and heavy-tailed distributions.
• It arises as the edge-scaling limit of the Pólya Web, linking discrete urn models with continuous-time dynamics for robust probabilistic analysis.
• The framework features martingale limits converging to Gamma laws and negative association properties, offering practical insights for modeling real network evolution.

The Yule Web is a coalescing system of continuous-time Yule processes—a canonical stochastic model where elements (walks, vertices, or pages) exhibit both pure-birth growth and merging dynamics. It arises as a universal edge-scaling limit of systems of coalescing random walks, specifically as a local limit of the Pólya Web, and is the continuous-time analog to the studied web of coalescing lattice random walks. The Yule Web is characterized by processes that individually follow pure-birth (Yule) dynamics with state-dependent rates, but globally form a coalescent structure via collision and merging. It provides an analytically tractable and probabilistically rich framework for understanding phenomena such as network growth, clustering, and scaling limits in complex systems, particularly the formation of power-law tails, memory effects, and finite-time corrections relevant for network science and the empirical study of the World Wide Web (Lansky et al., 2016, Urbán, 17 Jan 2026).

1. Probabilistic Structure of the Yule Web

The Yule Web is constructed from a countable family of independent Poisson processes $\{A_k : k \in \mathbb{N}^+\}$ on the real line, where $A_k$ has rate $k$. For each pair $(k, s) \in \mathbb{N}^+ \times \mathbb{R}$, ladder times are recursively defined as

$$\mathcal T_{k,s}(0)=s, \quad \mathcal T_{k,s}(n+1) = \alpha_{k+n,\,\mathcal T_{k,s}(n)}$$

where $\alpha_{m,\,u}$ denotes the first jump of process $A_m$ after $u$. The process

$$V_{k,s}(t) = k + \max\{n \geq 0 : \mathcal T_{k,s}(n) \leq t\}, \quad t \geq s$$

is a piecewise constant pure-birth process whose rate at state $m$ equals $m$ (the classical Yule process). When two Yule processes occupy the same state at the same time, they coalesce and evolve together henceforth. The Yule Web is defined as the entire system of such coalescing $V_{k,s}$ processes.

Each $V_{k,s}$ exhibits the classical branching property: for $s\leq r\leq t$,

$$V_{k,s}(t) \stackrel{d}{=} \sum_{j=1}^{V_{k,s}(r)} \widetilde V_{1,r}^{(j)}(t),$$

where the $\widetilde V_{1,r}^{(j)}$ are i.i.d. Yule processes, emphasizing its Markov and branching structure (Urbán, 17 Jan 2026).

2. Edge-Scaling Limit from the Pólya Web

The Yule Web emerges as a scaling limit of the Pólya Web, a system of coalescing Pólya urn walks on $\mathbb{N}^2$. Near the "south-west edge," one holds $k$ fixed while taking the other coordinate and the time index to infinity. Under logarithmic time scaling $\tau = \log(nt)$ and appropriate rescaling,

$$U^{(n)}_{k,s}(\tau) = e^{-\tau} X_n\bigl(k, s, e^\tau\bigr)$$

(where $X_n$ is the $x$-coordinate) converges in finite-dimensional distributions to the Yule martingale

$U_{k,s}(\tau) = e^{-\tau} V_{k,s}(\tau).$

This construction precisely links the coalescing urn walks with continuous-time Yule dynamics, justifying the Yule Web as a universal local scaling object at the edge of the Pólya Web (Urbán, 17 Jan 2026).

3. Martingale Limits, Gamma Laws, and Strong Law of Components

Each $U_{k,s}(t) = e^{-(t-s)} V_{k,s}(t)$ is a nonnegative martingale. By standard convergence, $U_{k,s}(t)$ approaches a limiting Gamma random variable as $t\to\infty$:

$$\lim_{t \to \infty} U_{k,s}(t) = U_k \sim \Gamma(k, 1),$$

with density $g_k(x) = x^{k-1} e^{-x}/\Gamma(k)$.

For a collection of Yule walks started at time $s$, the number of non-coalesced components at infinity among $n$ walks is given by

$$C_n^Y = 1 + \sum_{k=1}^n \mathbf{1}\{\tau_{k-1,k} = \infty\},$$

where $\tau_{k-1,k}$ is the first meeting time of walks $k-1$ and $k$.

The marginal probability that two consecutive processes never coalesce is

$$\mathbb{P}[\tau_{k-1,k} = \infty] = \int_0^\infty \int_0^y \begin{vmatrix} g_{k-1}(x) & g_{k-1}(y) \ g_k(x) & g_k(y) \end{vmatrix} dx\,dy = \frac{1}{2^{2k} \binom{2k}{k}}.$$

Consequently,

$$\mathbb{E}\,C_n^Y = 1 + \sum_{k=1}^n \frac{1}{2^{2k} \binom{2k}{k}} \sim \sqrt{\pi n}$$

as $n\to\infty$. The variance satisfies $\operatorname{Var}(C_n^Y) = O(\sqrt n)$. A strong law holds:

$$\lim_{n \to \infty} \frac{C_n^Y}{\mathbb{E}\,C_n^Y} = 1 \quad \text{a.s.}$$

thus, $C_n^Y \sim \sqrt{\pi n}$ almost surely (Urbán, 17 Jan 2026).

4. Joint Gamma Densities and Determinantal Formulae

The collection of martingale limits $\{U_k\}$ determines the non-coalescent structure and joint behavior. Their joint density, for $\{U_{l_1}, U_{r_1}, ..., U_{l_n}, U_{r_n}\}$ in blocks, is given in determinantal form analogous to the Karlin–McGregor formula:

$$\frac{\partial^n}{\partial x_1 \cdots \partial x_n} \mathbb{P}[U_{l_i} = U_{r_i} \in dx_i, i=1..n] = \det[\mathcal{G}(x_1,...,x_n)],$$

where $\mathcal{G}$ is a block matrix of regularized Gamma CDF and density entries. This structure allows explicit calculation of all finite block statistics, including joint survival probabilities and higher-order anti-correlation features (Urbán, 17 Jan 2026).

5. Negative Association and BKR Inequality

Events corresponding to persistent separation, such as $\{\tau_{k-1,k} = \infty\}$, are negatively associated (NA), i.e., the probability of simultaneous separation for disjoint pairs is less than or equal to the product of their marginal probabilities. This is formalized via the van den Berg–Kesten–Reimer (BKR) inequality and a geometric separation argument involving dual walks. Harris’s lemma applies directly. These properties are significant: negative association enables sharp concentration bounds (Martingale Chernoff, large deviation estimates) for the component count and related observables. The entire hierarchy of limit theorems for $C_n^Y$ relies on these correlation inequalities (Urbán, 17 Jan 2026).

6. Connection to Generalized Nonlinear and Fractional Yule Models

The Yule Web, as above, is built from classical linear Yule processes, but considerable flexibility arises by moving to nonlinear or fractional Yule processes. In the context of network modeling, Polito, Lansky, and Sacerdote introduced the fractional nonlinear Yule model to allow persistent memory (fractionality, $\nu < 1$), nonlinear birth rates ($\lambda_k$ general in $k$), and saturation effects (finite possible link count per node or page) (Lansky et al., 2016).

The key evolution equation for node in-links (or analogous process in the Yule Web context) is

$$\frac{d^\nu}{dt^\nu} p_n^\nu(t) = -\lambda_n p_n^\nu(t) + \lambda_{n-1} p_{n-1}^\nu(t), \qquad n \geq 1,$$

with Caputo fractional derivative of order $0<\nu\leq1$. Explicit finite-time and asymptotic laws for in-link distributions are derived, with limiting power-law tails for linear and a wide class of nonlinear and saturating mechanisms. Finite-time corrections, obtained from incomplete beta and incomplete gamma functions, are essential in matching empirical in-link data, where network snapshots are always of finite duration.

Both classes of models—the coalescing Yule Web and the nonlinear/fractional Yule models—enable modeling and rigorous understanding of heavy-tailed distributions, finite-size effects, persistent memory, and clustering/coalescence phenomena in large networks such as the World Wide Web. The Yule Web’s determinantal structure and strong law are robust under these generalizations, with explicit formulas adapting to the richer nonlinear and memory-influenced dynamics (Lansky et al., 2016, Urbán, 17 Jan 2026).

7. Applications and Empirical Implications

The Yule Web provides a natural limiting object for analyzing component structure, scaling limits, and cluster statistics in large evolving systems. Its explicit martingale, Gamma limit, and negative association features facilitate parameter estimation and finite-sample inference in empirical studies of web-graph growth and network evolution. Both memory (fractionality) and nonlinear/saturating growth mechanisms are empirically observed in real networks—manifested in bursty link accrual, power-law degree distributions with finite cut-off, and clustering structures. The Yule Web framework and its extensions allow these observed features to be quantified and fitted, distinguishing between Markovian and non-Markovian growth and separating the effects of heavy-tailed memory from those of nonlinearity in degree growth, crucial for robust modeling of networked systems (Lansky et al., 2016, Urbán, 17 Jan 2026).