This function estimates mobility flows using different distribution laws. As described in Lenormand et al. (2016) , we propose a two-step approach to generate mobility flows by separating the trip distribution law, gravity or intervening opportunities, from the modeling approach used to generate the flows from this law. This function only uses the first step to generate a probability distribution based on the different laws.

## Usage

```
run_law(
law = "Unif",
mass_origin,
mass_destination = mass_origin,
distance = NULL,
opportunity = NULL,
param = NULL,
check_names = FALSE
)
```

## Arguments

- law
a character indicating which law to use (see Details).

- mass_origin
a numeric vector representing the mass at origin (i.e. demand).

- mass_destination
a numeric vector representing the mass at destination (i.e. attractiveness).

- distance
a squared matrix representing the distance between locations (see Details).

- opportunity
a squared matrix representing the number of opportunities between locations (see Details). Can be easily computed with

`extract_opportunities()`

.- param
a vector of numeric value(s) used to adjust the importance of

`distance`

or`opportunity`

associated with the chosen law. A single value or a vector of several parameter values can be used (see Return). Not necessary for the original radiation law or the uniform law (see Details).- check_names
a boolean indicating if the ID location are used as vector names, matrix rownames and colnames and if they should be checked (see Note).

## Value

An object of class `TDLM`

. A list of list of matrices containing for each
parameter value the matrix of probabilities (called `proba`

). If
`length(param) = 1`

or `law = "Rad"`

or `law = "Unif`

only a list of
matrices will be returned.

## Details

We compute the matrix `proba`

estimating the probability
\(p_{ij}\) to observe a trip from location \(i\) to
another location \(j\)
(\(\sum_{i}\sum_{j} p_{ij}=1\)). This
probability is based on the demand \(m_{i}\)
(argument `mass_origin`

) and the attractiveness
\(m_{j}\) (argument `mass_destination`

). Note that the population
is typically used as a surrogate for both quantities (this is why
`mass_destination = mass_origin`

by default). It also depends on the
distance \(d_{ij}\) between locations (argument `distance`

) OR
the number of opportunities \(s_{ij}\) between locations
(argument `opportunity`

) depending on the chosen law. Both the effect of the
distance and the number of opportunities can be adjusted with a parameter
(argument `param`

) except for the original radiation law or the uniform law.

In this package we consider eight probabilistic laws described in details in Lenormand et al. (2016) . Four gravity laws (Carey 1858; Zipf 1946; Barthelemy 2011; Lenormand et al. 2016) , three intervening opportunity laws (Schneider 1959; Simini et al. 2012; Yang et al. 2014) and a uniform law.

Gravity law with an exponential distance decay function (

`law = "GravExp"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`distance`

and`param`

will be used.Normalized gravity law with an exponential distance decay function (

`law = "NGravExp"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`distance`

and`param`

will be used.Gravity law with a power distance decay function (

`law = "GravPow"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`distance`

and`param`

will be used.Normalized gravity law with a power distance decay function (

`law = "NGravPow"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`distance`

and`param`

will be used.Schneider's intervening opportunities law (

`law = "Schneider"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`opportunity`

and`param`

will be used.Radiation law (

`law = "Rad"`

). The arguments`mass_origin`

,`mass_destination`

(optional) and`opportunity`

will be used.Extended radiation law (

`law = "RadExt"`

). The arguments`mass_origin`

,`mass_destination`

(optional),`opportunity`

and`param`

will be used.Uniform law (

`law = "Unif"`

). The argument`mass_origin`

will be used to extract the number of locations.

## Note

All the inputs should be based on the same number of
locations sorted in the same order. It is recommended to use the location ID
as vector names, matrix rownames and matrix colnames and to set
`check_names = TRUE`

to verify that everything is in order before running
this function (`check_names = FALSE`

by default). Note that the function
`check_format_names()`

can be used to control the validity of all the inputs
before running the main package's functions.

## References

Lenormand M, Bassolas A, Ramasco JJ (2016).
“Systematic comparison of trip distribution laws and models.”
*Journal of Transport Geography*, **51**, 158-169.

Carey HC (1858).
*Principles of Social Science*.
Lippincott.

Zipf GK (1946).
“The P1 P2/D Hypothesis: On the Intercity Movement of Persons.”
*American Sociological Review*, **11**(6), 677--686.

Barthelemy M (2011).
“Spatial Networks.”
*Physics Reports*, **499**, 1-101.

Schneider M (1959).
“Gravity models and trip distribution theory.”
*Papers of the regional science association*, **5**, 51-58.

Simini F, González MC, Maritan A, Barabasi A (2012).
“A universal model for mobility and migration patterns.”
*Nature*, **484**, 96-100.

Yang Y, Herrera C, Eagle N, González MC (2014).
“Limits of Predictability in Commuting Flows in the Absence of Data for Calibration.”
*Scientific Reports*, **4**(5662), 5662.

## Author

Maxime Lenormand (maxime.lenormand@inrae.fr)

## Examples

```
data(mass)
data(distance)
mi <- as.numeric(mass[, 1])
mj <- mi
res <- run_law(
law = "GravExp", mass_origin = mi, mass_destination = mj,
distance = distance, opportunity = NULL, param = 0.01,
check_names = FALSE
)
# print(res)
```