Spatial Interaction Models

Spatial interaction models

1. Introduction

Spatial interaction models try to simulate the flows between two points of origins and destination. These flows can be modelled with different conditions and be applied in “flows of raw materials, manufactured goods or capital in economics, or flows of particles in physic” (Dennett, 2012).

In this paper, the variables A (the economies of scale, an exponential function to determinate how attractive is, for example, a city), B (the friction parameter to determinate how efficient is a process, for example the transport system) and E (expansion/contraction parameter) will be changed to explore the flows between several cities.

The equation to simulate the number of interactions between two cities, following a maximum entropy model, is as follows:



Where T is transport between the two cities (i and j), B is the balance, E is employment and P is population in origin. Bcij is the friction, so the higher B is, the more important this factor is.

Next, the revenues to the location – Oi – will be calculated in order to balance the model from the disturbance caused in the interaction equation.

Where O (revenues) equals the sum of the transport to i and j, and the wages of j
How will this affect the population? This is updated by the next equation:



Basically, the new population equals the old population plus the difference between them. Finally, the next formula is used to update the changes caused by new population:



Where wages (O) and costs of living (C) result in a tension between money and population, modulated by the ɛ factor – importance that population give to wages.

In the next part, this paper will examine the changes in the flows between London, Manchester, Birmingham, Leeds, Glasgow and Liverpool by altering the variables A, B and E.


2. Analysis

2.1 Economies of scale (A)

The economy of scale parameter (A) shows how the larger cities will attract more population as its value increases. To explore this (see Fig. 7), the value of A will be altered to 1 (this factor has neutral impact in the analysis), 0.1 (this factor has less impact) and 2 (economies of scale have a great impact)

Fig. 7: Variations in economy of scale parameter (A) for different scenarios

The variation of population for these values are as follows:

Fig. 8 Variations in economy of scale parameter (A) for scenarios A1 (left), A2 (centre) and A3 (right).


In the scenario A1, where A=1 and transport costs (B) are steady as well as expansion/contraction parameter (E), the cities attract more population in a constant increase motivated by other factors, such as available jobs or wages.

In the scenario A2, where A=0.1, the size of the city has little impact in the flows of population. Therefore, big cities as London or Manchester have less growth than others as Birmingham, which becomes bigger than Manchester over the time. This simulation demonstrates that small cities receive population flows from bigger cities when A has a low value.

In the scenario A3, where A=2, the size of the city is a powerful factor of attraction and it becomes exponential (the bigger, the more population it attracts). London and Manchester concentrate all the new population while smaller cities remain constant or have a mild decrease in number of inhabitants.


2.2 Friction of transport (B)

The friction of transport costs parameter (A) shows how the cost of transport impacts in the population flows. To explore this (see Fig. 9), the value of B will be altered to 0.2, 0 and 1.

Fig. 9: Variations in friction parameter (B) for different scenarios



The variation of population for these values are as follows:

Fig. 8 Variations in friction parameter (B) for scenarios B1 (left), B2 (centre) and B3 (right).

In both scenarios B1, where B=0.2, and B3, where B=1, London presents a similar growth of population. As travelling is more expensive, London tends to accumulate more population due to other factors as employment or wages.

In the scenario B2, where B=0, London loses a small quantity of population over the time while Manchester and Birmingham are more attractive, as is inexpensive to travel and they are closer to London than the rest of the cities.


2.3 Contraction/expansion parameter (E)


The contraction/expansion parameter (E) shows how the importance of economy in the population flows. To explore this (see Fig.11), the value of E will be altered to 0.000001, 0.00000125 and 0.0000009.

Fig. 11: Variations in contraction/expansion parameter (E) for different scenarios


The variation of population for these values are as follows:

Fig. 12: Variations in rate for contraction/expansion parameter (E) for scenarios E1 (left), E2 (centre) and E3 (right).


The three scenarios (E1, where E=0.000001; E2, where E=0.00000125; and E3, where E=0.0000009) the result is similar: the cities growth exponentially accordingly to the increase of E value while A and B remain unaltered. The bigger ones attract more population over the time but the smaller cities keep also growing, so the E parameter acts as intensifier of the flows of populations: a more importance of economy expands the cities in any case.


3. Bibliography

Batty, M., 2007. Model cities. Town Planning Review 78, 125–151.

Torrens, PM, 2000. How Land Use Transportation model works [Online]. Available at: <>. [Accessed 10-04-2016]

Wilson, A., 2010a. Entropy in Urban and Regional Modelling: Retrospect and Prospect. 城市和区域建模中的熵: 回顾与展望. Geographical Analysis 42, 364–394.

Wilson, A., 2010b. The general urban model: Retrospect and prospect: The general urban model: Retrospect and prospect. Papers in Regional Science 89, 27–42. doi:10.1111/j.1435-5957.2010.00282.x

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