Metapypulation: an overview

Metapypulation is a package to simulate the spread of culture in a metapopulation. It supplies a set of tools partly inspired by other frameworks such as Mesa which are general-purpose methods to perform agent-based modelling with Python. The main reason to create my own package for this purpose is to change the focus of the tool - population-based simulations instead of individual-based.

Structure

Metapypulation has three main classes:

an Individual class, which represents each individual in the metapopulation.
a Subpopulation class, representing the different discrete subpopulations that compose the metapopulation;
finally, a Metapopulation class, which puts Individual and Subpopulation together and have individuals interacting in this world.

In addition, I provide a Simulation class, which allows to run a simulation with several replicates, which outputs different measurements (see below). The class also provides some quick tools to plot the results of the simulation.

Cultural traits

Currently, each individual’s culture is represented by a set of \(N\) features. Each feature in turn can assume one of \(\nu\) traits. These features represent different (assumed) independent facets of culture: one could be language, burial tradition, boat building features, etc. So each individual is currently represented by a vector of integers.

Neutral model

The Neutral model follows loosely the set up selected by other authors working on the spread of cultural traits (e.g. Patterns in space and time: simulating cultural transmission in archaeology, Marko Porčić). While in most works the Neutral Model assumes some sort of mutation rate in the traits, in the form of errors in the process of copying traits, we assume no copying error and instead use the simplest model where one focal individual at random in each subpopulation and each generation copies one trait at random from another individual in the same sub-population. Because of the absence of mutations, given enough time, this process is expected to lead to uniformity in an isolated subpopulation.

Axelrod model

While we plan on adding several different ways for individuals to interact and change their traits, as of July 2024 the only such way to interact is shaped upon the Axelrod model of culture dissemination (The Dissemination of Culture: A Model with Local Convergence and Global Polarization, Robert Axelrod (1997), The Journal of Conflict Resolution, vol. 41, no. 2). In this model, at each generation an individual is chosen at random to copy a trait from a neighboring source on a lattice; the copy occurs with a probability proportional to the total similarity of the two random individuals. This mimicks homophily - the principle by which two individuals that resemble each other have a higher chance of having an exchange than two individuals that are completely different. In the metapopulation model that I developed, for each subpopulation we pick two random individuals that will act as target and source of the copy.

Diversity measures

Currently, there are two diversity measures implemented at the level of both the subpopulation and the whole metapopulation. The first is the Shannon diversity index, which for each feature is measured as

\[H^{\prime} = - \sum_{i = 1}^{\nu} p_i \ln p_i ,\]

where \(p_i\) is the frequency of trait \(i\) in the subpopulation / metapopulation. Then, the Shannon diversity index that we measure is the average of the index for each trait,

\[\bar{H} = H^{\prime} .\]

The second diversity measure that we calculate is the number of unique sets of traits. This is done through the function np.unique(..., return_counts = True).