Visualising empirical and simulated Agglomerations of (Imperial) Russia and the Soviet Union

The MARIUS Project

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.

The aim is to evaluate the degree of genericity of Soviet urban trajectories by growing articial systems of cities, from the most generic ones to the most particular ones.

The fitness of the model is evaluated against historical urban data, gathered within DARIUS Database.

Observed distribution of city sizes with DARIUS Database, 1897-2010

Inter-census urban growth

Urban growth corresponds to the average annual growth rate between the two selected dates.

Urban Statuses & Attributes

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.

The Cornerstone model

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.

Bonified Interurban Exchanges

To account for spillover effects in interurban exchanges, we model an interaction bonus for cities exchanging with many partners and large flows.

Costly Interactions

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.

Environmental resources

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.).

Redistributive regions

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.

Time-lags in regional transitions

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.

Contribution of mechanisms to the quality of simulation (closeness to data)

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.

Contribution of mechanisms' interactions to the quality of simulation (closeness to data)

This graph allows to explore the interaction of two mechanisms in reducing the distance between empirical and simulated trajectories of cities in the Former Soviet Union.

Optimal model after multi-calibration

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).

Mechanisms' combination:

This is the best set of mechanisms for the given period to minimize the distance between simulated and observed urban trajectories in the Former Soviet Union.

Parameter values:

Given the structure of the model, these are the values of parameter that perform best to reproduce the empirical urban evolution.

Evaluation Criteria:

These are the three measures used to evaluate the quality of the model (to be minimized).
To execute a model, either select one of the best calibrated models for a given period of time and mechanisms combination, or customize the parameter values yourself [model to run | customised model].

Model structures and parameterization

Generic parameters

Bonus parameter

Fixed Costs Transaction parameter

SubSurface Resource parameters

Double Redistribution parameters

Urban transition parameter

To execute the model with your parameterization, press
And wait a few seconds until the message 'Done' appears.


Rank-size distribution

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).

Dispersion of residuals

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 Trajectories

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.

Most Positive Residual trajectories

Most Negative Residual trajectories


Profiles of Residual cities

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.

Contribution of attributes' interactions to the variation of residuals



Who ?

  • DARIUS : Clémentine Cottineau
  • MARIUS : Paul Chapron, Guillaume Chérel, Clémentine Cottineau, Denise Pumain, Romain Reuillon
  • VARIUS : Clémentine Cottineau, Robin Cura, Romain Reuillon

  • Institutional Supports :

    About the MARIUS model :

    Clémentine Cottineau, Paul Chapron, Romain Reuillon, 2014, An incremental method for building and evaluating agent-based models of systems of cities.

    Model code :

    About DARIUS data :

    Clémentine Cottineau, 2014, DARIUS Database, figshare
    Clémentine Cottineau, 2015, Urban DARIUS Shape Files, figshare
    Clémentine Cottineau, 2015, Regional DARIUS Shape Files, figshare