Time-Changed Coalescents

• Time-changed coalescents are a stochastic model that modifies classical coalescent processes by integrating a time-change function to capture non-uniform demographic and spatial effects.
• They offer explicit formulations for merger rates and coalescent times, accommodating varied population sizes, heavy-tailed offspring, and multiple mergers.
• The framework enables efficient computation of genealogical statistics, facilitating demographic inference and parameter identifiability in genetic data analyses.

A time-changed coalescent is a stochastic process describing the genealogy of a sample from a population where the "clock" of the underlying coalescent model (e.g. Kingman's or Λ-coalescent) is modified by a deterministic or random time-rescaling, often reflecting demography, environment, or spatial structure. In such processes, the merging dynamics follow those of the constant-parameter coalescent, but real time is replaced by an integral of a rate function, capturing effects such as fluctuating population size or non-uniform spatial mixing. The resulting coalescent is generally time-inhomogeneous, with merger rates and coalescence event distributions determined via the time-change.

1. Foundations and General Framework

The classical Kingman coalescent assumes constant population size and exchangeable reproduction, producing binary mergers at constant rate for every pair of lineages. The key extension in time-changed coalescents is to embed population or environmental signals as a deterministic, non-decreasing time-change $T(t)$, such that the standard coalescent run in $T$-time describes the ancestry in real time $t$. This principle operates across both discrete- and continuous-time settings, for single-type and multi-type populations, and both in classical and general $\Lambda$- or $\Xi$-coalescent frameworks.

For general $\Lambda$-coalescents (with measure $\Lambda$ on $[0,1]$), and time-dependent rate function $\zeta(t)>0$, the time-changed process is governed instantaneously by: $$\lambda_{b,k}(t) = \frac{1}{\zeta(t)} \binom{b}{k} \int_0^1 x^{k-2}(1-x)^{b-k}\Lambda(dx)$$ where $b$ is the current number of lineages and $k\geq2$ the merger size (Spence et al., 2015).

The time-change also arises from models with explicit population dynamics. In near-critical Galton–Watson trees, after conditioning on survival at large time $T$ and sampling $k$ individuals uniformly among survivors, the genealogy of the sample converges to a Kingman coalescent with merger times governed by a deterministic time-change reflecting population-size fluctuations (Harris et al., 2017).

Across models, the time-change is typically expressed as: $t \mapsto T(t) = \int_0^t \nu(s)^{-\gamma}\, ds$ where $\nu(s)$ encodes demographic or environmental scaling, and $\gamma$ is model-specific (Freund, 2019).

2. Construction in Evolving Populations

Galton–Watson and Branching Processes

For a continuous-time Galton–Watson process at branching rate $r$ with mean offspring $1 + \mu/T$ (near-critical), rescaling time by $1/T$ yields convergence to Kingman's coalescent with non-homogeneous rate $\lambda(t)$. Specifically,

• The ancestral tree topology matches Kingman's.
• Coalescence times have a non-trivial joint law, given via explicit formulas.
• The time-change $\phi(t)$ is:
  • Non-linear for non-critical ($\mu\neq0$) processes:

  $$\phi(t)=\ln\frac{e^{r\mu}-1}{e^{r\mu(1-t)}-1}/r\mu,$$

  with $$\lambda(t)=r\mu\frac{e^{r\mu(1-t)}}{e^{r\mu(1-t)}-1}$$. - Logarithmic in the critical case ($\mu=0$):

  $\lambda(t)=1/(1-t),\quad \phi(t)=-\ln(1-t)$

These formulas describe how random population-size fluctuations deterministically slow down or speed up Kingman dynamics (Harris et al., 2017).

A mixture-of-i.i.d. construction for merger times emerges: for the critical case, the coalescent times are distributed as the order statistics of transformed i.i.d. variables, capturing mild correlations induced by the time-change (Harris et al., 2017).

In multitype Galton–Watson processes, a similar deterministic time-change arises, with additional dependency of the genealogy's law on types through size-biasing and type-specific split rates; however, the underlying topology and time-change remain universal once types are discarded (Hernández et al., 7 Feb 2025).

Cannings and Discrete Models

Cannings models with varying population size $N_r$ produce discrete genealogies whose scaling limits are time-changed $\Lambda$-coalescents. Population-history enters as: $T(t)=\int_0^t \nu(s)^{-\gamma}\,ds$ with $\nu(s)$ rescaling population size and $\gamma$ determined by the merger regime, e.g. $\gamma=1$ for Kingman, $\gamma=2-\alpha$ for Beta-coalescents (Freund, 2019). These time-changed coalescents model arbitrary, but moderate, demographic changes and unify single- and multiple-merger scenarios.

3. Extensions: Heavy-Tailed Offspring and Multiple Mergers

When the offspring distribution is heavy-tailed with index $\alpha\in(1,2)$ ($\infty$ variance), the time-changed coalescent limit for $k$-sample genealogies is not Kingman but a universal $\Lambda$-coalescent allowing multiple mergers, with $$\Lambda_s(dx)\propto x^{1-\alpha}(1-s+x)^{\alpha-2}\,dx$$ (Harris et al., 2023). In this regime:

• Coalescent events correspond to giant birth events of order comparable to the population.
• The time rescaling remains deterministic, with $t\mapsto t/T$, but merger structures exhibit multiple block mergers and more complex partition-valued dynamics.
• The law of coalescent events and their sizes has explicit connections to Lauricella functions and Dirichlet distributions, offering new probabilistic universality classes as $\alpha$ varies (Harris et al., 2023).

This produces a phase transition: for $\alpha=2$, the process recovers the time-changed Kingman case; for $\alpha\in(1,2)$, true multiple-merger coalescents dominate.

4. Spatial and Structured Populations

Spatial models, such as coalescing random walks on discrete tori, also yield time-changed coalescents upon appropriate rescaling. The clock is determined by the typical meeting time of random walks: $$\theta_N \sim \begin{cases} \frac{N^d}{2 v_d} & (d\ge3) \ \frac{N^2 \log N}{\pi} & (d=2) \end{cases}$$ where $v_d$ is the escape probability from the origin. Under this time-change, the block-count process converges to Kingman’s coalescent (Beltrán et al., 2018). The same approach applies to other spatial graphs and non-reversible, non-homogeneous migration, provided local mixing ensures "mean-field" behavior on the time-changed scale.

5. Branching in Varying Environments and Point Process Coalescents

For Galton–Watson processes in a varying environment (GWVE), the coalescent point process $(A_i)$, recording successive coalescent times among adjacent sampled individuals, is not Markov but its distribution is explicitly determined by the survival probabilities of the underlying environment: $$P(A_i>n) = \prod_{m=1}^n P(\eta^{(-m)}=0) = e^{-\tau(n)},\quad \tau(n) = \sum_{m=1}^n -\ln P(\eta^{(-m)}=0)$$ Passing to a "coalescent time" $t = \tau(n)$, the point process becomes a standard constant-rate coalescent in $t$ (Blancas et al., 2022). In the linear-fractional case, the intervals $(A_i)$ become i.i.d., matching the classical coalescent structure.

6. Continuous-State and Non-exchangeable Coalescents

Continuous-state branching processes (CSBP) with Lévy branching mechanism $\Psi$ admit genealogical structures constructed via a time-change. The forward process is constructed as a flow of subordinators with Laplace exponent $v_t(\lambda)$, where $v_t(\lambda)$ solves

$\int_{v_t(\lambda)}^\lambda \frac{dz}{\Psi(z)} = t$

The time-changed genealogy of sampled individuals corresponds to inverse subordinators, with the coalescent process (the "consecutive coalescent") characterized by Markovian, but non-exchangeable, partition-valued processes whose merge rates are explicitly determined by $v_t(\lambda)$ and the Lévy measure. In particular, the time-inhomogeneity is understood via the subordinator flow, and explicit formulas determine the evolution of coalescent block sizes and rates (Foucart et al., 2018).

7. Algorithmic and Inference Implications

Time-changed coalescents enable exact and efficient computation of genealogical statistics, such as the site frequency spectrum (SFS). For time-inhomogeneous $\Lambda$- and $\Xi$-coalescents with rate-modulating function $\zeta(t)$, explicit, algorithmically tractable representations exist for the expected SFS via matrix decompositions involving the time-change: $$Q_{b,b-(k-1)}(t) = \frac{1}{\zeta(t)} \text{(rate as above)}$$ Algorithmic complexity is $O(n^3)$ for generic time-changed coalescents, and $O(n^2)$ for time-homogeneous special cases. The structure enables parameter identifiability, both for the coalescent measure and the time-change function $\zeta(t)$, provided sufficient sample size and regularity (Spence et al., 2015).

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In all these contexts, the essential role of the deterministic time-change is to encode the demographic, environmental, or spatial heterogeneity into the coalescent genealogy without altering the underlying merger kernel's structural form. This allows separate control and inference of demographic histories and underlying reproductive schemes, within a unified stochastic process framework. Time-changed coalescents therefore provide a rigorous probabilistic scaffolding for analyzing genealogies of populations subject to arbitrary, but suitably regular, temporal fluctuations and are a central object in modern mathematical population genetics and stochastic process theory.