Semantic Axelrod Model

• Semantic Axelrod Model is defined as an extension of Axelrod’s cultural model where traits are structured in prerequisite trees, modeling cumulative cultural evolution.
• The methodology simulates agents on a lattice who acquire traits via imitation, structured teaching, and innovation, governed by probabilities like Pℓ and Pm.
• Simulation results reveal phase transitions in trait diversity and tree depth driven by teaching rates, offering insight into the emergence of complex cultural repertoires.

The Semantic Axelrod Model is an extension of Axelrod's original model for cultural differentiation, generalized to represent not only the diversity and accumulation of independent cultural traits but also their structured, prerequisite dependencies. Instead of treating cultural features as independent loci, the Semantic Axelrod Model encodes a "trait space" as a forest of rooted trees, where each node corresponds to a skill or informational unit, and directed edges represent necessary prerequisite relationships. This approach allows rigorous investigation of how mechanisms such as imitation, structured teaching, and individual innovation interact to shape the depth, richness, and diversity of collective cultural repertoires, with direct relevance for explaining phenomena such as the increasing complexity of Paleolithic technologies (Madsen et al., 2014).

1. Model Structure and Extension to Prerequisite Dependencies

The foundational aim of the Semantic Axelrod Model is to address how cumulative culture emerges when cultural traits have ordered dependencies, in contrast to models where traits are independent. Axelrod’s 1997 model represents each agent as a vector of F features, each with q possible states, assuming independence between features. The Semantic Axelrod Model replaces this with a design space in which the set of all possible cultural traits is a forest of T rooted trees, each parameterized by a branching factor r and depth h. A trait is represented by a tree node, and possession of that trait by an agent is conditional on already possessing all of its parental ancestors in the tree, thus explicitly encoding prerequisite constraints.

Social learning processes are modified so that agents can only acquire a novel trait if all its prerequisites are met, or—importantly—if their teacher provides one missing prerequisite, modeling the effect of structured teaching or scaffolding. This framework enables direct modeling of how the ability to transmit prerequisite knowledge (with some probability) modulates the accumulation and diversity of complex cultural repertoires.

2. Formal Elements and Update Rules

Agents are arranged (typically, but not exclusively) on a square lattice with population size N (commonly 100, 225, or 400). The design space consists of T independent trees; each tree includes $\sum_{i=0}^h r^i$ nodes, and traits are defined as single nodes. The agent's cultural repertoire, $V(i)$, is the set of all traits currently possessed.

Social-learning proceeds via these steps per discrete time-iteration:

1. Select a focal agent $A$ at random.
2. Select a random neighbor $B$.
3. If $V(A) \cap V(B) = \varnothing$ or $V(B) \subseteq V(A)$: no social-learning occurs, proceed to innovation.
4. With probability $J(A, B) = \frac{|V(A)\cap V(B)|}{|V(A)\cup V(B)|}$ (the Jaccard index), attempt social learning. Otherwise, proceed to innovation.
5. In social learning, select a random trait $t \in V(B) \setminus V(A)$. If $A$ already possesses all prerequisites of $t$, $t$ is added to $A$’s repertoire. If $A$ lacks prerequisites, then with probability $\mathbb{P}_\ell$ the deepest missing prerequisite $p$ is taught to $A$ instead.
6. With probability $\mathbb{P}_m$, independently select a random agent $C$ and a new trait $u \notin V(C)$; $C$ acquires $u$ and all its prerequisites simultaneously (modeling trial-and-error innovation).

This process preserves the constraint of prerequisite order in learning and allows explicit tuning of both teaching fidelity ($\mathbb{P}_\ell$) and the innovation rate ($\mathbb{P}_m$).

3. Representation of Knowledge Structure

Each tree in the design space represents a self-contained domain of cultural knowledge, with the root corresponding to core, foundational skills and deeper nodes representing increasingly specialized competencies. Directed acyclic graphs (forests of rooted trees) ensure every trait's acquisition is contingent on possession of all ancestor traits along the path to the root. This implements the necessary "scaffolding" for the transmission of structured, hierarchical information, which earlier empirical and theoretical work suggested is favored by cultural selection under conditions of increasing technological complexity.

Learning-event rules enforce that a trait with missing prerequisites cannot be acquired directly via imitation; only structured teaching mechanisms (with probability $\mathbb{P}_\ell$) or innovation can bridge these gaps, modeling the emergence of apprenticeship and formal pedagogy as drivers of cumulative culture.

4. Simulation Protocol and Metrics

Key simulation parameters include:

Parameter	Values/Settings	Function/Role
Population size $N$	100, 225, 400	Number of agents on lattice
Number of trees $T$	4 or 16	Independent domains/design spaces
Branching factor $r$	3 or 5	Tree branching at each node
Depth $h$	3 or 5	Maximum hierarchical layers
Teaching rate $\mathbb{P}_\ell$	0.05, 0.1, …, 0.9	Probability of teaching deepest prerequisite
Innovation rate $\mathbb{P}_m$	$0$, $5\times 10^{-5}$, $10^{-4}$	Probability of idiosyncratic innovation

Simulation runs typically consist of $10^7$ elementary steps, with sampling every $10^6$ after a burn-in period. Twenty-five independent replicates are used per parameter combination to capture variability.

Metrics for quantifying emergent diversity and richness include:

• Number of distinct trait-forests (cultural configurations) present in the population.
• Mean normalized depth ($\textrm{radius}/h$) of learned trees.
• Average vertex degree (tree broadness).
• Remaining density: $|\bigcup_i V(i)|/$ (design-space size).
• Automorphism group size $|Aut(G)|$ and orbit count: algebraic measures of symmetry/class redundancy in learned subgraphs.

5. Simulation Results and Cultural Evolutionary Dynamics

Low values of the teaching rate ($\mathbb{P}_\ell \leq 0.05$) typically induce convergence to a near-monoculture, with the majority of the population sharing the same or very similar trait-sets, even in the presence of moderate innovation. As $\mathbb{P}_\ell$ increases (notably for $\mathbb{P}_\ell \geq 0.2$–$0.3$), the number and diversity of coexisting cultural configurations increase sharply.

A distinct phase transition is observed near $\mathbb{P}_\ell=0.3$–$0.4$ for mean normalized tree-depth, which rapidly increases to approximately $0.75$ of the maximal possible. The innovation rate $\mathbb{P}_m$ quantitatively shifts depth upward but does not change the critical threshold for the teaching effect. Population size has only a minor, second-order influence.

Analysis of automorphism group sizes and their distributions indicates that, under high teaching probabilities, learned trait-graphs have reduced symmetry and often converge to particular multimodal distribution shapes, reflecting the idiosyncratic combination of prerequisite substructures compared to the much higher symmetry of the full design space.

Larger design spaces—achieved by increasing $T$, $r$, or $h$—further amplify differentiation effects for any given $\mathbb{P}_\ell$, reflecting the combinatorial explosion of possible trait-set architectures.

6. Archaeological and Theoretical Significance

The Semantic Axelrod Model demonstrates that the ability to teach missing prerequisites creates a discontinuous, nonlinear increase in the acquisition of complex, interdependent skills. This provides a formal mechanism accounting for the archaeological record’s transition from simple Paleolithic toolkits to the diverse, highly specialized material assemblages of later prehistory.

High-fidelity structured teaching allows populations to explore and stabilize substantially larger and deeper design spaces than imitation alone would permit. This suggests that the emergence of teaching practices and apprenticeship may have constituted a key transition underlying behavioral modernity. The model thus systematically links the structure of cultural information to its modalities of transmission, bridging experimental studies of teaching with macro-historical patterns in cumulative technological evolution (Madsen et al., 2014).

7. Implications, Extensions, and Future Research Directions

A plausible implication is that faculty for structured teaching—and the institutionalization of cultural "scaffolding"—is a precondition for the emergence and maintenance of rich, hierarchically organized cultural repertoires. The Semantic Axelrod Model provides an extensible formalism for exploring the intersection of innovation, social learning, and teaching within complex design spaces. Future directions include: analytical characterization of the observed phase transition with respect to $\mathbb{P}_\ell$; extension to more general DAGs beyond tree-forests; and application to empirical data from archaeological and ethnographic settings, testing the model’s predictions on the dynamics and depth of real-world cultural traits with known prerequisite relationships.