MARIUS is an incremental family of agent-based models of systems of cities. It
was built to simulate the demographic trajectories of cities in the Former Soviet Union.
The rules of the models are designed to range from generic interurban interactions to specific mechanisms characterizing Soviet cities and their geographical environment. The structure of the model family is modular enough to allow various sets of mechanisms playing together or alternatively.
MARIUS is built as a modular family of models. Each model contains the cornerstone assumptions about city growth (the cornerstone model), as well as a combination of supplementary mechanisms corresponding to different hypotheses regarding the explanation of the diversity of urban trajectories in the modelled system of cities.
Generic mechanisms included in the cornerstone model consist in scaling laws relating city population and city wealth, supply and demand, as well as a gravity model of interurban interactions. Cities are supposed to generate larger economic output per capita as they grow in size, and to interact more intensely with the cities that are larger and closer to them.
To account for spillover effects in interurban exchanges, we model an interaction bonus for cities exchanging with many partners and large flows.
Transaction costs and entry costs are supposed to reduce the pool of potential partners to those with which cities can expect profitable transactions. This mechanism aims at eliminating interactions between very distant and very small cities.
Because of their location in an anysotropic environment, cities benefit from uneven opportunities of local resource extraction (for example : coal mining or oil extraction). The meaning of resource here can also include negative resources affecting city growth (polluted sites, etc.).
Regions as political and administrative territories can involve redistribution of wealth among cities. This mechanism consists in a mutualisation of wealth at the regional and national levels, a possible capture of some of this resource by the capital city, and the redistribution of the rest to every city according to its needs, measured by total population.
As the different parts of the territory differ in their position in the urban transition, this mechanism takes into account the time lag of regions in terms of potential rural migration to cities.
Models with different combination of mechanisms have been calibrated intensively against empirical data, using generic algorithms for more than 100000 generations. This plot shows the results of a regression explaining one measure of the quality of models (a small difference between simulated and empirical urban trajectories) by their mechanisms composition (the fact that any of the supplementary mechanisms is activated or not" Each bar represents the value of the estimated coefficient for each activated mechanism, in comparison with the same model structure without this mechanism, everything else being equal.
Given the calibration of all model structures against the same data and evaluation criteria, these tables describe the best calibrated model for a given period and a selected level of parsimony (the number of mechanisms included in the model structure).
This rank-size representation is common to study the hierarchical structure of systems of cities and their evolution towards equalisation or differenciation of city sizes. Blue dots indicate simulated cities over time, in comparison with empirical observations (in grey).
This graph represents the value of each city's population, observed (x axis) and simulated (y axis) at the last step of the simulation. A perfect model would exhibit a distribution of cities along the orange line.
Residuals represent the difference between observed and simulated population for each city (in logs). Positive residuals mean that cities grew faster in reality than what we were able to simulate, whereas negative residuals indicate that we over-estimated the growth of such cities. Residuals help trigger how and where model needs to be improved.
Profiles of residual cities are obtained after a regression on the value of residual population. We plot the coefficient values of some available urban attributes (status of capital, resources, location and past growth). This regression helps profiling the type of cities most over- and under-estimated by the model, that is the topical areas where the model needs to be improved.