Cultural Evolution as a Non-Stationary Process

Updated 4 December 2025

Cultural evolution is a non-stationary stochastic process characterized by history-dependent, time-varying mechanisms governing trait transmission, innovation, and extinction.
Models like the Tangled Axelrod and continuous-trait Axelrod illustrate key phenomena such as metastability, punctuated equilibria, and persistent fragmentation through empirical and mathematical frameworks.
These frameworks offer actionable insights into phase transitions, power-law lifetime distributions, and dynamic diversity in cultural systems.

Cultural evolution as a non-stationary stochastic process refers to the class of theoretical models and empirical frameworks in which the transmission, innovation, and extinction of cultural traits or patterns are governed by stochastic—but inherently time-dependent and history-sensitive—dynamics. In contrast to the classical stationary Markovian perspective where transition probabilities or trait frequencies equilibrate to an invariant distribution, non-stationary stochastic models of cultural evolution explicitly incorporate variable environments, context-dependence, cumulative memory effects, or path dependency. These result in macrodynamics such as punctuated equilibrium, scale-free lifetime statistics, persistent fragmentation, metastability, and aging, mirroring key phenomena observed in anthropological, sociolinguistic, and technological histories (Nicholson et al., 2016, Campos et al., 17 Oct 2025, Wakano et al., 13 May 2025, McAvoy, 2015, Gabora, 2013).

1. Conceptual Foundations and Formal Definitions

Non-stationary stochastic modeling of cultural evolution extends classical evolutionary frameworks by abandoning assumptions of equilibrium, detailed balance, or fixed law of motion. Models typically define a population of agents (individuals, groups, or artifacts) characterized by discrete or continuous sets of traits, which evolve through probabilistic rules of vertical and horizontal transmission, mutation/innovation, and selection (McAvoy, 2015).

A generic non-stationary process is mathematically captured through time-inhomogeneous Markov chains or contexts where the transition kernel $K_t$ and possibly even the trait/state space $S_t$ depend on time:

$\mu_{t+1}(E) = \int_{\mathcal{S}_t} K_t(\sigma, E)\;d\mu_t(\sigma), \qquad E\subseteq \mathcal{S}_{t+1}$

Crucially, the kernels $K_t$ can encode not only parameter drift (e.g., changing rates of innovation or copying) but also changing interaction structures, payoff mappings, and even spontaneously emerging traits, ensuring that long-term behavior remains history-dependent and inherently non-stationary (McAvoy, 2015).

Non-stationarity is also central in context-driven models like the Context-driven Actualization of Potential (CAP) and its SCOP (State-Context-Property) formalism. Here, a cultural entity’s state $\Sigma_t$ at time $t$ is actualized into a (possibly novel or emergent) state by a time- and history-dependent context $e\in M_t$ , with context-driven stochastic transition probabilities $\mu(q, e, p)$ that need not obey the Kolmogorov axioms, reflecting interference and order effects (Gabora, 2013).

2. Mathematical Structures: Examples and Formalisms

Several concrete model classes exemplify the non-stationary stochastic process character of cultural evolution:

Tangled Axelrod Model (TAM) (Nicholson et al., 2016): Agents carry both an "interaction genome" and a "cultural genome" (binary strings); their interactions, copying, mutation, and death are governed by probabilistic rules. Trait and population dynamics self-organize into a sequence of metastable cultural patterns (QESS), separated by abrupt "cultural quakes." Survival probabilities of cultural patterns follow power-law statistics, implying the absence of a finite average lifetime and the emergence of scale-free, aging dynamics.
Continuous-trait Axelrod model (Campos et al., 17 Oct 2025): Each agent harbors a vector of continuous traits $x_i\in[0,1]^F$ . Interactions hinge on trait similarity and occur with probability determined by a similarity threshold $d$ . Perfect copying ( $\sigma=0$ ) leads to a genuine non-equilibrium phase transition at a critical $d_c$ , with the order parameter (domain density) scaling as $\mu\sim(d_c-d)^{1/3}$ and the largest cluster fraction exhibiting a discontinuous jump. For $\sigma>0$ , persistent copying noise prevents stabilization, leading to sustained fragmentation and nonstationary trait distributions.
Genealogical process models (Ancestral Learning Graph) (Wakano et al., 13 May 2025): Individual-based birth–death–innovation processes define evolving genealogies, where the total tree-length of extant ancestry shows sawtooth dynamics: linear growth punctuated by random collapses reflecting the extinction and rebirth of cultural lineages. These punctuated patterns propagate to the diversity and richness of cultural traits carried by the population, intrinsically linking macro-scale cultural diversity to the stochastic genealogical process.
Stochastic selection framework (McAvoy, 2015): A general setting, parameterized by time-dependent state spaces $S_t$ , population structure, aggregate payoff functions $F_t$ , and update rules $U_t$ , fully characterizes how cultural transmission, selection, mutation, or imitation processes can be analyzed in the context of non-stationary, inhomogeneous evolutionary Markov chains.

The table below summarizes key structural features of prominent non-stationary stochastic models of cultural evolution:

Model/Class	Trait Structure	Key Mechanism
Tangled Axelrod Model (TAM)	Binary strings	QESS, metastability
Continuous-axelrod model	Continuous ( $[0,1]^F$ )	Fragmentation, drift
Moran-type genealogy (ALG)	Trait sets, labels	Sawtooth lineage length
CAP/SCOP	Contextual state space	Context-dependent $\mu$
Stochastic selection processes	Arbitrary set $S_t$	Time-varying kernels

3. Dynamical Phenomena: Metastability, Fragmentation, and Aging

The departure from stationarity is marked by several robust dynamical phenomena:

Metastable States and Aging: Systems self-organize into quasi-stable configurations (metastable "cultural cores" or domain partitions), each possessing a lifetime that statistically increases with the age of the system. In the TAM, survival times follow a power law in the age ratio $t/t_w$ , $S(t_w, t)\sim (t/t_w)^{-x}$ with $x\approx 0.2$ . As such, no finite mean lifetime exists—a direct signature of "aging" analogous to spin glasses (Nicholson et al., 2016).
Punctuated Equilibria and Quakes: Both agent-based copying models and genealogy models display phases of slow, cumulative trait accretion and relatively abrupt collapse events, mirroring punctuated equilibrium in empirical cultural and technological change (e.g., rapid surges and subsequent declines in the number of automobile manufacturers after innovation shocks) (Nicholson et al., 2016).
Persistent Fragmentation and Non-freezing: Under conditions of sustained noise or drift, models such as the continuous-trait Axelrod variant never reach absorbing consensus; rather, trait distributions remain dynamic, with domains continuously forming and dissipating on all scales. The freezing time diverges with both system size and noise amplitude, ensuring persistent nonstationarity (Campos et al., 17 Oct 2025).
Sawtooth Dynamics in Trait Accumulation: In the genealogical framework, the accumulation of cultural innovations maps nearly linearly onto the total genealogical tree-length during metastable phases, only to collapse sharply upon lineage extinction, resulting in a characteristic sawtooth profile in the number of extant traits over time (Wakano et al., 13 May 2025).

4. Quantitative and Empirical Implications

Non-stationary stochastic processes offer distinct predictions and alignment with observed patterns in empirical datasets:

Scale-free lifetime distributions: The power-law lifespan distributions in both agent-based and genealogical models accord with the empirical absence of a characteristic "firm," technology, or cultural trait lifetime within real-world datasets (Nicholson et al., 2016).
Cumulative innovation and collapse: The dynamics of trait accumulation and extinction generate patterns of slow technological or cultural accretion interleaved with abrupt losses, echoing the archaeological record of tool cultures and the industrial history of innovation and bankruptcy cycles (Wakano et al., 13 May 2025).
Growth of Population and Diversity: Logarithmic growth in total population size $N(t)\simeq A+B\ln t$ and diversity $D(t)\simeq C+D\ln t$ in the TAM reflects the lack of equilibrium, with "cultural quakes" producing sudden macroscopic jumps (Nicholson et al., 2016).
Order parameters and phase transitions: For continuous-trait Axelrod models, critical phenomena include a continuous vanishing of mean domain density at a well-defined threshold $d_c\approx 0.0784$ with scaling exponent $\beta=1/3$ , and a discontinuous jump in the largest domain fraction, analytically and numerically captured via finite-size scaling ( $\nu=2$ ) (Campos et al., 17 Oct 2025).
Non-Kolmogorovian probabilistic structure: In CAP/SCOP models, non-stationarity is also manifest at the level of probability theory, with interference terms and non-commutativity emerging in transition measures, precluding Markovian or even Kolmogorovian stochastic models (Gabora, 2013).

5. Interpretative Frameworks and Theoretical Significance

The modeling of cultural evolution as a non-stationary stochastic process provides a unifying formalism and set of intuitions relevant for both mathematical and empirical investigations:

These models explain why cultural evolution resists characterization by a stable, time-invariant distribution—contrast with classical genetic fixation or business turnover models with exponential waiting times.
Fluctuations, cumulative memory effects, contextual actualization, and emergent collective properties dominate the macrodynamics, supporting interpretations that stress the role of history, innovation clustering, and cultural memory.
Non-stationary mathematical structures clarify why attempts to assign typical "lifespans" or "diversity levels" to cultural forms often fail in empirical analysis: the correct null models possess diverging variances and heavy-tailed distributions (Nicholson et al., 2016).
The CAP/SCOP and stochastic selection frameworks generalize across levels: from conceptual recombination (via quantum-like entanglement) and context-driven change in individual cognition, to agent-based models of meme and trait flow, and large, structured, or genealogically divergent populations (Gabora, 2013, McAvoy, 2015).

6. Extensions, Limitations, and Open Directions

Current non-stationary frameworks for cultural evolution admit several axes for further research:

Incorporation of explicit spatial structure, demography, and heterogeneous innovation/learning rates, as in the time-varying master equation formalism (McAvoy, 2015).
Empirical calibration using historical, linguistic, or technological datasets remains in early stages; models such as the TAM show qualitative matching to industrial diversification and collapse, but analytic matching at quantitative (parameter) levels is preliminary (Nicholson et al., 2016).
Quantum-like formalisms (CAP/SCOP) raise foundational questions regarding the representation of conceptual and cultural structure, especially concerning interference, entanglement, and contextuality in trait combination and higher-order innovation (Gabora, 2013).
Processes supporting the emergence and dissolution of "cultural attractors" under drift and noise—visible in continuous-trait models—can elucidate the stability and fragility of consensus formation in large societies (Campos et al., 17 Oct 2025).
Analysis of the stochastic genealogy (ancestral learning graphs) surfaces the deep link between trait diversity and the structure of cultural descent, providing a route to comparative analysis with population genetics, epidemiology, and network theory (Wakano et al., 13 May 2025).

These directions suggest that non-stationary stochastic process models will remain central to both the formal and empirical understanding of cultural evolution, providing tools for describing punctuated, scale-free, and fundamentally history-dependent patterns across domains.