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(ProQuest: ... denotes formulae omitted.)
Introduced in its contemporary form in 1946 (ref. 1), but with roots that go back to the eighteenth century2, the gravity law1,3,4 is the prevailing framework with which to predict population movement3,5,6, cargo shipping volume7 and inter-city phone calls8,9, as well as bilateral trade flows between nations10. Despite its widespread use, it relies on adjustable parameters that vary from region to region and suffers from known analytic inconsistencies. Here we introduce a stochastic process capturing local mobility decisions that helps us analytically derive commuting and mobility fluxes that require as input only information on the population distribution. The resulting radiation model predicts mobility patterns in good agreement with mobility and transport patterns observed in a wide range of phenomena, from long-term migration patterns to communication volume between different regions. Given its parameter-free nature, the model can be applied in areas where we lack previous mobility measurements, significantly improving the predictive accuracy of most of the phenomena affected by mobility and transport processes11-23.
In analogy with Newton's law of gravity, the gravity law assumes that the number of individuals Tij that move between locations i and j per unit time is proportional to some power of the population of the source (mi) and destination (nj) locations, and decays with the distance rij between them as
... (1)
where a and b are adjustable exponents and the deterrence function f(rij) is chosen to fit the empirical data. Occasionally Tij is interpreted as the probability rate of individuals travelling from i to j, or an effective coupling between the two locations24. Despite its widespread use, the gravity law has notable limitations:
Limitation one, we lack a rigorous derivation of (1). Whereas entropy maximization25 leads to (1) with a5b51, it fails to offer the functional form of f(r).
Limitation two, lacking theoretical guidance, practitioners use a range of deterrence functions (power law or exponential) and up to nine parameters to fit the empirical data5,7,8,11,14.
Limitation three, as (1) requires previous traffic data to fit the parameters [a, b, ...], it is unable to predict mobility in regions where we lack systematic traffic data, areas of major interest in modelling of infectious diseases.
Limitation four, the gravity law has systematic predictive discrepancies. Indeed, in Fig....