Evolutionary Model Merging in Complex Systems

• Evolutionary model merging is a dynamic process that combines diverse models using principles like reproduction, mutation, and recombination.
• It utilizes mathematical formalism and stochastic network effects to capture non-stationary dynamics and punctuated equilibria in model evolution.
• The approach underpins applications in cultural evolution, biological systems, and machine learning, balancing gradual adaptation with disruptive innovation.

Evolutionary model merging refers to a class of methodologies that draw on principles from evolutionary dynamics—including reproduction, mutation, selection, and recombination—to combine, adapt, and merge diverse models or strategies into new, fitter models. This concept has been developed in several domains, including computational biology, combinatorial optimization, cultural evolution, and, more recently, the merging of large-scale machine learning models. The defining feature is that model merging is treated as a dynamic process with populations of candidate models that interact (e.g., through crossover or copying), undergo variation, and are selectively retained based on performance or fitness.

1. Evolutionary Model Merging in the Context of Cultural Evolution

The individual-based model described in "Cultural Evolution as a Non-Stationary Stochastic Process" (Nicholson et al., 2016) formalizes evolutionary model merging as the dynamic fusion and adaptation of agent strategies, inspired by mechanisms from biological and social models. Agents are encoded by binary strings split into interaction genomes (defining a web of fixed pairwise interactions) and strategy/cultural genomes (encoding mutable, transmissible behaviors or beliefs). Evolution entails stochastic reproduction governed by networked fitness and a copying/recombination mechanism in which agents probabilistically merge parts of each other's strategies. Unlike traditional evolutionary models that feature gradual, steady adaptation, this framework exhibits punctuated dynamics—a progression through metastable stages (quasi-evolutionary stable states), abruptly punctured by "quakes" leading to the reorganization or merging of cultural patterns. This captures both gradual consolidation and the flurry of recombinative innovation following disruptive events.

The merging aspect is operationalized as a social copying process, where the probability for an agent to copy traits from a neighbor depends on the prevalence of the neighbor's strategy. This mechanism enables spontaneous formation, dissolution, and merging of cultural groups, providing a stochastic, network-structured substrate for evolutionary model merging.

2. Mathematical Formalism and Dynamical Properties

The model merging process is supported by explicit mathematical constructs. The fitness function for agent $a$ at time $t$ is

$$H_a(t) = -\mu N(t) + \sum_b j_{ab}(t),\qquad p_\text{off}(a) = \frac{1}{1 + \exp(-H_a(t))}$$

where $N(t)$ is the population, $\mu$ is a competition parameter, and $j_{ab}$ is a density-weighted interaction coupling. Crucially, the copying (model merging) probability is

$P(a, b) = \frac{C(b)}{C(a) + C(b)}$

with $C(x)$ denoting the number of agents with the same cultural genome as $x$. Copied traits are transferred over segments of the strategy string, and successful adoption—potentially followed by mutation—enables cross-pollination of strategies that, upon further innovation, become new centers of interaction (thus generating new models within the system).

The macroscopic consequence is a non-stationary stochastic process: the diversity and population size linger around temporary equilibria before abrupt transitions. These transitions—marked by culture merging and splintering—are characterized by heavy-tailed survival distributions. The survival probability $S(t_w)$ for a core group decays as a power law: $$S(t_w) \approx \left( \frac{t}{t_w} \right)^{-x},\qquad x \approx 0.2$$ yielding a distribution of lifetimes with no finite mean—reflecting the prevalence of both rapid turnovers and unexpectedly prolonged stabilities, consistent with real-world cultural evolution.

3. Connections to Biological and Social Models

The evolutionary model merging mechanism synthesizes features of the Tangled Nature Model (TNM) and the Axelrod model:

• The TNM provides a framework for reproduction with reproduction rates modulated by a complex interaction network, driving the system through sequences of long equilibria and sudden "quakes".
• The Axelrod model introduces social copying and trait mixing, where agents adopt traits from similar neighbors in a spatial domain. In the present context, copying is non-spatial and determined by frequency-dependent popularity—a variant more akin to nonlocal, internet-mediated social environments.

This hybridization creates a versatile null model for studying the evolutionary merging of strategies in both biological (e.g., gene flow, macroevolution) and cultural (e.g., innovation, societal paradigm shifts) domains. Punctuated equilibria and the emergence of hybrid forms (cultural or genetic) are emergent behaviors of the system.

4. Empirical Evidence and Validation

Empirical validation is provided by comparing simulation outputs to historical data. An illustrative example is the automotive industry: the introduction of the automobile is treated as a cultural innovation, precipitating a rapid proliferation in the number of car manufacturers (new "cultures"), followed by industry consolidation (merging of practices, extinction of unsuccessful approaches). The model captures this burst-contraction profile: a rapid increase in the number of distinct groups during a "quake", then consolidation into a handful of survivor strategies, consistent with observed macroeconomic and technological disruptions.

5. Broader Implications for Evolutionary Model Merging

Key implications for the broader theory and application of evolutionary model merging include:

• The co-occurrence of gradual adaptation and abrupt reorganization (merging/splitting events) is a generic feature across evolutionary domains once stochastic copying and network effects are present.
• The model demonstrates that merging dynamics are governed by nontrivial statistical laws (e.g., power-law lifetimes) that should be expected in diverse systems, from cultural innovation to technological consolidation and, by analogy, in ensembles of artificial models undergoing evolutionary merging.
• The interplay of copying, mutation, and non-stationary stochasticity is sufficient to induce spontaneous, large-scale reconfiguration—a foundational observation for understanding merging in both natural and artificial evolutionary systems.
• From a machine learning perspective, the process offers a principled blueprint for evolutionary model merging beyond deterministic weight averaging or optimization, favoring stochastic, population-level mechanisms capable of generating hybrid offspring with emergent properties.

6. Significance in Evolutionary Computation and Model Integration

The mechanistic and statistical insights from this work inform not only cultural evolution but also combinatorial optimization (where evolutionary algorithms hybridize and merge partial solutions), computational biology (merging alignments or gene families), and modern machine learning (ensemble merging, checkpoint fusion). Evolutionary model merging, as formalized here, provides a population-dynamic alternative to manual or purely optimization-based model combination—where the emergence, extinction, and merging of strategies unfold under frequency-dependent, network-driven stochastic processes.

A plausible implication is that in high-dimensional, fitness- or performance-landscaped spaces, introducing evolutionary merging mechanisms can balance exploitation of current solutions with the exploration and creation of unforeseen combinatorial innovations, potentially leading to improved adaptability and robustness in both natural and artificial learning systems.