Spatial Interaction Models: Historical Development and Formulations
Published:
This summary traces the chronological development and refinement of Spatial Interaction Models (SIM) based on Michael Batty’s lecture.
Origins: The Gravity Model
The intellectual roots of spatial interaction models lie in an analogy with Newtonian physics. The first to apply gravitational logic to human movement was Henry Carey (1858), who observed that human migration between cities followed patterns resembling gravitational attraction. This idea was formalised mathematically by Ernst Ravenstein (1885) in his Laws of Migration, where he established empirically that migration intensity decreases with distance.
The first explicitly gravitational formulation came from E.G. Young (1924) and was later popularised by George Zipf (1946), who proposed the P₁P₂/D rule:
\[T_{ij} = k \frac{P_i P_j}{d_{ij}}\]where $T_{ij}$ is the interaction between places $i$ and $j$, $P_i$ and $P_j$ are their populations, $d_{ij}$ is the distance separating them, and $k$ is a proportionality constant. This model treats population as the sole “mass” generating interaction, and assumes flows decay linearly with distance.
The astronomer John Stewart (1941) then introduced the concept of demographic potential — a measure of the cumulative gravitational pull exerted on one place by all others:
\[V_i = \sum_j \frac{P_j}{d_{ij}^\beta}\]Stewart’s contribution was to generalise the distance exponent to $\beta$, recognising that the rate of distance decay was an empirical question rather than a physical constant. This $\beta$ parameter — the distance decay parameter — became the central object of interest in all subsequent SIM development. A high $\beta$ means flows are very sensitive to distance (steep decay); a low $\beta$ means people are willing to travel further.
The General Gravity Model
Building on Stewart and Zipf, the general unconstrained gravity model takes the form:
\[T_{ij} = k \cdot V_i^\mu \cdot W_j^\alpha \cdot d_{ij}^{-\beta}\]| Term | Definition | Role |
|---|---|---|
| $T_{ij}$ | Predicted flow from $i$ to $j$ | Dependent variable |
| $k$ | Global scaling constant | Calibrates total flow volume |
| $V_i$ | Origin mass (e.g. population) | Measures emissiveness of origin |
| $W_j$ | Destination mass (e.g. store area, jobs) | Measures attractiveness of destination |
| $d_{ij}$ | Distance or cost between $i$ and $j$ | Deterrence term |
| $\mu, \alpha$ | Mass sensitivity parameters | How strongly origin/destination size drives flows |
| $\beta$ | Distance decay parameter | How rapidly flows diminish with distance |
The parameters $\mu$, $\alpha$, and $\beta$ are estimated from observed flow data. No marginal totals are constrained — both origin and destination flows are free to vary. This makes the model maximally flexible but less predictively accurate, since it cannot guarantee that predicted flows sum to known totals.
Typical applications: inter-regional trade flows; early migration studies; telephone communication patterns — any context where neither origin totals nor destination totals are independently known.
Wilson’s Entropy-Maximising Framework and the SIM Family (1967–71)
The most important theoretical advance came from Alan Wilson (1967, 1971), who derived the entire family of spatial interaction models from first principles using entropy maximisation. Rather than relying on the physics analogy, Wilson asked: given what we know about total flows, what is the most probable distribution of individual trips? Maximising entropy subject to constraints on observed totals yields exactly the gravity model family — but now with a rigorous statistical foundation.
Crucially, Wilson showed that the appropriate model form depends entirely on which marginal totals are known and should be treated as constraints. This gave rise to three distinct models.
Unconstrained Model
Used when neither $O_i$ nor $D_j$ are known. Wilson showed this corresponds to maximising entropy with only a constraint on total travel cost.
\[T_{ij} = k \cdot V_i^\mu \cdot W_j^\alpha \cdot f(c_{ij})\]Typical applications: exploratory analysis of interaction patterns; inter-regional trade; contexts where no marginal totals can be independently observed.
Production-Constrained Model
When the total flow leaving each origin $O_i$ is known (e.g. total shopping trips from a residential zone, derived from population), it should be held fixed. Wilson introduced a balancing factor $A_i$ to enforce this:
\[T_{ij} = A_i \cdot O_i \cdot W_j^\alpha \cdot f(c_{ij})\] \[A_i = \frac{1}{\displaystyle\sum_j W_j^\alpha \cdot f(c_{ij})}\]The constraint satisfied is:
\[\sum_j T_{ij} = O_i \quad \forall i\]| Term | Definition |
|---|---|
| $O_i$ | Known total outflow from origin $i$ (fixed) |
| $A_i$ | Balancing factor ensuring outflows are conserved |
| $W_j^\alpha$ | Destination attractiveness with sensitivity $\alpha$ |
| $f(c_{ij})$ | Cost/deterrence function (e.g. $e^{-\beta c_{ij}}$) |
| $\beta$ | Distance decay parameter (estimated from data) |
Here $\alpha$ and $\beta$ are the only free parameters — origin mass is already encoded in $O_i$. The destination inflows $\sum_i T_{ij}$ are a model output, not an input.
Typical applications: retail catchment modelling (flows from residential zones to supermarkets); journey-to-work where census commuter totals by origin are available; school choice modelling.
Attraction-Constrained Model
Symmetrically, when total inflow to each destination $D_j$ is known but origin totals are not:
\[T_{ij} = B_j \cdot V_i^\mu \cdot D_j \cdot f(c_{ij})\] \[B_j = \frac{1}{\displaystyle\sum_i V_i^\mu \cdot f(c_{ij})}\]The constraint satisfied is:
\[\sum_i T_{ij} = D_j \quad \forall j\]Typical applications: modelling flows to a fixed-capacity facility (hospital, airport) where total admissions or passengers are known; migration to cities where in-migration totals are recorded.
Doubly Constrained Model
When both $O_i$ and $D_j$ are known, both sets of balancing factors are applied simultaneously:
\[T_{ij} = A_i \cdot O_i \cdot B_j \cdot D_j \cdot f(c_{ij})\] \[A_i = \frac{1}{\displaystyle\sum_j B_j D_j f(c_{ij})}, \qquad B_j = \frac{1}{\displaystyle\sum_i A_i O_i f(c_{ij})}\]Since $A_i$ depends on $B_j$ and vice versa, these are solved iteratively (the Furness/IPF procedure) until convergence. Both constraints are satisfied simultaneously:
\[\sum_j T_{ij} = O_i \quad \forall i \qquad \text{and} \qquad \sum_i T_{ij} = D_j \quad \forall j\]The only free parameter is $\beta$ — both mass terms are fully determined by observed data, leaving only the distance decay to estimate.
Typical applications: transport planning trip distribution (four-step model); inter-regional migration where both in- and out-migration totals come from the census; freight movement between fixed origin and destination terminals.
The Role of the Cost Function
All three constrained models require specifying $f(c_{ij})$. Two standard forms are:
| Form | Equation | Properties |
|---|---|---|
| Power | $c_{ij}^{-\beta}$ | Heavy-tailed; suitable for long-distance flows |
| Negative exponential | $e^{-\beta c_{ij}}$ | Steep short-distance decay; preferred for urban trips |
For supermarket shopping — short, frequent, and urban — the negative exponential is standard, as most people are unwilling to travel more than a few kilometres and the deterrence effect is sharp.
Summary
| Model | Constraints | Free Parameters | Primary Use |
|---|---|---|---|
| Unconstrained | None | $k, \mu, \alpha, \beta$ | Exploratory, no known totals |
| Production-constrained | $O_i$ fixed | $\alpha, \beta$ | Retail, commuting (known origins) |
| Attraction-constrained | $D_j$ fixed | $\mu, \beta$ | Facility planning (known capacities) |
| Doubly constrained | $O_i$ and $D_j$ fixed | $\beta$ only | Transport planning, migration |
Wilson’s framework showed that these are not four separate models but one unified family, differentiated only by which empirical constraints are imposed. This theoretical unification, combined with the entropy-maximising derivation, transformed spatial interaction modelling from a physics analogy into a rigorous applied framework — and laid the foundation for every land use and transport model that followed.
References
Carey, H.C. (1858) Principles of Social Science. Philadelphia: J.B. Lippincott.
Fotheringham, A.S. and O’Kelly, M.E. (1989) Spatial Interaction Models: Formulations and Applications. Dordrecht: Kluwer Academic Publishers.
Furness, K.P. (1965) ‘Time function iteration’, Traffic Engineering and Control, 7(7), pp. 458–460.
Hansen, W.G. (1959) ‘How accessibility shapes land use’, Journal of the American Institute of Planners, 25(2), pp. 73–76.
Ravenstein, E.G. (1885) ‘The laws of migration’, Journal of the Statistical Society of London, 48(2), pp. 167–235.
Stewart, J.Q. (1941) ‘An inverse distance variation for certain social influences’, Science, 93(2404), pp. 89–90.
Wilson, A.G. (1967) ‘A statistical theory of spatial distribution models’, Transportation Research, 1(3), pp. 253–269.
Wilson, A.G. (1971) Entropy in Urban and Regional Modelling. London: Pion.
Zipf, G.K. (1946) ‘The P1 P2/D hypothesis: On the intercity movement of persons’, American Sociological Review, 11(6), pp. 677–686.