# Multiple gravity laws for human mobility within cities

Oh-Hyun Kwon  Department of Physics, Pohang University of Science and
Technology, Pohang, Republic of Korea  Inho Hong  Graduate School of Data
Science, Chonnam National University, Gwangju, Republic of Korea  Center for
Humans and Machines, Max Planck Institute for Human Development, Berlin,
Germany  Woo-Sung Jung  Department of Physics, Pohang University of Science
and Technology, Pohang, Republic of Korea  Department of Industrial and
Management Engineering, Pohang University of Science and Technology, Pohang,
Republic of Korea  Hang-Hyun Jo<EMAIL_ADDRESS>Department of Physics,
The Catholic University of Korea, Bucheon, Republic of Korea

###### Abstract

The gravity model of human mobility has successfully described the deterrence
of travels with distance in urban mobility patterns. While a broad spectrum of
deterrence was found across different cities, yet it is not empirically clear
if movement patterns in a single city could also have a spectrum of distance
exponents denoting a varying deterrence depending on the origin and
destination regions in the city. By analyzing the travel data in the twelve
most populated cities of the United States of America, we empirically find
that the distance exponent governing the deterrence of travels significantly
varies within a city depending on the traffic volumes of the origin and
destination regions. Despite the diverse traffic landscape of the cities
analyzed, a universal pattern is observed for the distance exponents; the
exponent value tends to be higher (lower) between regions with larger
(smaller) traffic volumes. This indicates that our method indeed reveals the
hidden diversity of gravity laws that would be overlooked otherwise. Our
findings are expected to give insights into urban travels for better
prediction and optimization.

##  I Introduction

Human mobility as the fabric of human societies has been studied for
understanding the mechanism of movements [1, 2, 3, 4, 5, 6, 7], diffusive
processes [8, 9], and its association with socioeconomic attributes [10, 11,
12, 13, 14, 15] at both individual and population levels. Taking advantage of
rich data, several population-level mobility models were developed to describe
travel patterns as a function of geographical factors, e.g., population
distribution and travel distance. The gravity model [16, 17, 18], intervening
opportunities model [19, 20], and radiation model [4, 21, 22, 23] have been
the leading concepts for the population-level mobility models [24].

Among such mobility models, the gravity model has remained the representative
population-level model, capturing the traffic in a simple form resembling
Newton’s gravity law; the traffic volume from one region to the other is
proportional to the product of population sizes of origin and destination
regions and inversely proportional to the distance between those regions [16,
17]. Precisely, its generalization is written as follows:

$\displaystyle T_{ij}=G\frac{m_{i}m_{j}}{r_{ij}^{\gamma}},$ (1)

where $T_{ij}$ ($r_{ij}$) is the traffic volume (geographical distance)
between the origin region $i$ and the destination region $j$. $m_{i}$ and
$m_{j}$ are their population sizes or traffic volumes, while $G$ is a
coefficient. The distance exponent $\gamma$ is to denote the deterrence of
mobility on distance, and it usually takes a value ranging from 0.5 to 3 [25].
Note that the deterrence function of the gravity model has been observed in
multiple forms, such as an exponential function [26], a combination of the
exponential and power-law forms [24], and the Hill function [27], besides the
common power-law form. The gravity model has been applied to systems of
various spatial interactions, including urban mobility [28, 29, 30, 31, 32,
33], intercity mobility [18, 34, 35], and general inter-regional interactions
[36, 37, 38, 39, 40, 41, 42, 43].

For most applications of the gravity model, the entire set of data for a given
geographical unit, e.g., cities or countries, has been analyzed to result in a
single distance exponent, effectively ignoring the heterogeneities within the
unit. Such heterogeneities can be accounted for by grouping the regions within
the unit and then by separately estimating the distance exponent for each pair
of origin and destination groups. This possibility of having multiple values
of the distance exponent within the same unit has recently been suggested in a
theoretical study considering heterogeneous population landscapes [44].

In our work, we test the validity of such “multiple” gravity laws on the data
of commuting in the twelve largest cities of the United States of America
(USA). For each city we first divide urban areas into 10 groups, and estimate
the value of the distance exponent between each pair of groups. We find the
varying patterns of the distance exponent for all cities analyzed and their
universality across different cities. Lastly, we compare the accountability of
the multiple gravity model with that of the conventional gravity model
concerning only a single distance exponent for the entire city. We also
discuss the origin of the multiple exponents in terms of travel costs and the
core-periphery structure of the city. Our findings suggest that the gravity
model on urban landscapes could be fine-tuned to incorporate the broad
spectrum of urban movements for better understanding, estimation, and
prediction.

##  II Data and method

We analyze the commuting dataset processed from the 2018 LODES dataset [45] of
the United States Census Bureau. The LODES dataset is a census survey
connecting homes (origins) and workplaces (destinations) at a Census Block
level, including origin-destination pairs and locations. In our work we choose
the twelve most populated Metropolitan Statistical Areas (MSAs) in the USA,
which we call cities hereafter. The entire area of each city is divided into
$1$ km $\times$ $1$ km square cells to aggregate the Census Block level data
of diverse scales into equivalent area units, i.e., cells. Table 1 shows the
number of cells and the number of trips for each city.

Table 1: Twelve most populated USA cities for the analysis with information on the number of cells and the number of trips within each city. City | Number of cells | Number of trips
---|---|---
New York City | 23,642 | 35,217,720
Los Angeles | 6,324 | 18,822,540
Chicago | 12,777 | 15,185,472
Dallas | 12,976 | 11,158,472
Houston | 9,488 | 9,222,488
Washington, D.C. | 10,875 | 9,454,492
Miami | 3,957 | 8,323,512
Philadelphia | 10,162 | 9,939,204
Atlanta | 14,691 | 8,383,040
Phoenix | 5,951 | 7,028,940
Boston | 8,077 | 8,237,372
San Francisco | 3,225 | 6,530,276
Figure 1: Data analysis framework (a–c) and its application to the case of
Chicago (d,e). (a) For a given city, cells in a square grid are clustered into
$10$ groups according to their traffic volumes $T_{i}$ in Eq. (3). The traffic
volume $T_{ij}$ and geographic distance $r_{ij}$ between a cell $i$ of group
$k$ (red) and a cell $j$ of group $k^{\prime}$ (blue) are identified. (b)
Then, the distance exponent $\gamma_{kk^{\prime}}$ is estimated using the
gravity model in Eq. (5). (c) Estimated values of the distance exponent, i.e.,
$\gamma_{kk^{\prime}}$ for $k,k^{\prime}\in\\{1,\ldots,10\\}$, form the
exponent matrix $\Gamma$. (d) Conventional estimation of the distance exponent
using the whole set of data for Chicago, resulting in a single distance
exponent $\gamma_{\rm s}\approx 0.53$ (dashed line) in Eq. (4). (e) Empirical
confirmation of multiple gravity laws within Chicago with different values of
$\gamma_{kk^{\prime}}$ for some cases with $k=k^{\prime}$ (dashed lines).

We describe the data analysis framework [see also Fig. 1(a–c)]. Each city has
$N$ cells, and the numbers of trips, $T_{i\to j}$, from the $i$th cell to the
$j$th cell for the pair of $i,j\in\\{1,\ldots,N\\}$ are given. Note that
$T_{i\to j}\neq T_{j\to i}$ in general, since the commuting dataset only
describes one-way trips from homes to workplaces. To illustrate the
bidirectional traffic flow, we symmetrize the traffic volume between two cells
as follows:

$\displaystyle T_{ij}\equiv T_{i\to j}+T_{j\to i}.$ (2)

Note that $T_{ii}=2T_{i\to i}$. The total number of trips for the $i$th cell
is obtained by summing $T_{ij}$ over all $j$s:

$\displaystyle T_{i}\equiv\sum_{j=1}^{N}T_{ij}.$ (3)

The conventional gravity model assumes that the rescaled traffic volume
between cells $i$ and $j$, defined as $T_{ij}/(T_{i}T_{j})$, decays with the
geographical distance between those cells, $r_{ij}$, as

$\displaystyle\frac{T_{ij}}{T_{i}T_{j}}\propto r_{ij}^{-\gamma_{\rm s}}\
\textrm{for}\ i,j\in\\{1,\ldots,N\\},$ (4)

where $\gamma_{\rm s}$ is the distance exponent for the entire set of pairs of
cells in the city. The value of $\gamma_{\rm s}$ is estimated using the
ordinary least squares linear regression. The fitting range of $r_{ij}$ is
limited to that showing the scaling behavior. Throughout the paper, we set the
lower and upper bounds of the fitting range to $1$ km and $50$ km,
respectively. We also calculate the $R^{2}$ value to quantify the quality of
the fitting. For the case of Chicago we estimate $\gamma_{\rm s}\approx 0.53$
($R^{2}\approx 0.17$) [Fig. 1(d)].

To investigate the possible variation of the gravity law within the city, we
sort the cells according to their traffic volumes, $T_{i}$, and then cluster
them into $10$ groups with an equal size of $N/10$ (one can find an
alternative clustering, e.g., in Ref. [11]). These groups are denoted by
$G_{k}$ for $k=1,\ldots,10$; the group of $k=1$ is for cells with the smallest
traffic volumes, while the group of $k=10$ is for cells with the largest
traffic volumes. We estimate the distance exponent for each pair of groups,
say $k$ and $k^{\prime}$, assuming the following functional form:

$\displaystyle\frac{T_{ij}}{T_{i}T_{j}}\propto r_{ij}^{-\gamma_{kk^{\prime}}}\
\textrm{for}\ i\in G_{k}\ \textrm{and}\ j\in G_{k^{\prime}}.$ (5)

Using the estimated values of $\gamma_{kk^{\prime}}$ for all possible pairs of
$k,k^{\prime}\in\\{1,\ldots,10\\}$, we obtain the exponent matrix
$\Gamma=[\gamma_{kk^{\prime}}]$, as depicted in Fig. 1(c). Note that
$\gamma_{kk^{\prime}}=\gamma_{k^{\prime}k}$ as we have symmetrized the traffic
volumes between cells such that $T_{ij}=T_{ji}$, leaving us with $55$ distinct
values of $\gamma_{kk^{\prime}}$. In the case of Chicago, different scaling
behaviors are observed with different values of $\gamma_{kk^{\prime}}$. The
results for several cases with $k=k^{\prime}$ are shown in Fig. 1(e); e.g.,
$\gamma_{1,1}\approx 0.10$ ($R^{2}\approx 0.014$) and $\gamma_{10,10}\approx
0.94$ ($R^{2}\approx 0.37$) among others. Our data analysis framework can be
applied to any mobility datasets as long as both $T_{ij}$ and $r_{ij}$ are
available.

##  III Result

Figure 2: Traffic landscapes and exponent matrices for the twelve most
populated cities in the USA. Each panel consists of the traffic landscape of
the city on the map (top) and the exponent matrix derived from the traffic
volumes between cells in the city (bottom). In the landscapes each cell is
colored according to the group it belongs to, following the color bar at the
bottom of the figure. A higher value of $k$ indicates the larger traffic
volume of the group. For comparison, we mark the estimated distance exponent
using the whole data of the city by a white bar in the color bar for exponent
values. Figure 3: $R^{2}$ values in the distance exponent estimation for the
twelve most populated cities in the USA. (a) The box-and-whisker plot for each
city describes a distribution of $R^{2}$ values for $55$ distance exponents in
the exponent matrix, $\Gamma=[\gamma_{kk^{\prime}}]$. The black thick vertical
line indicates the $R^{2}$ value for the distance exponent, $\gamma_{\rm s}$,
when using the whole set of data for each city. (b) The same results as in (a)
are presented but by separating the cases with $k+k^{\prime}>11$ (pink plots)
from the other cases with $k+k^{\prime}\leq 11$ (blue plots).

We apply the data analysis framework described in the previous Section to the
twelve most populated cities in the USA as listed in Table 1. We first observe
in Fig. 2 that cells with the largest traffic volumes are concentrated at one
or more centers of the cities. For example, cities such as Dallas and Houston
have a single center, while cities like Los Angeles apparently have more than
one center that are distributed over the cities. On the other hand, cells with
smaller traffic volumes are scattered over the cities.

The exponent matrices visualized in Fig. 2 show that for each city the
distance exponent $\gamma_{kk^{\prime}}$ has various values according to the
group indices $k$ and $k^{\prime}$, which clearly evidence the multiple
gravity laws within the city. For comparison, we mark as a white bar the value
of the distance exponent, $\gamma_{\rm s}$, i.e., when using the whole set of
pairs of cells in the city, in the color bar for the exponent values in Fig.
2. The finding of various values of $\gamma_{kk^{\prime}}$ is consistent with
our expectation that various scaling behaviors can be observed within the same
city, depending on the populations or traffic volumes of both origin and
destination cells [44]. Thus, our method can indeed reveal the hidden
diversity of gravity laws that would be overlooked otherwise. In addition, one
can say that the distance exponent $\gamma_{\rm s}$ might show an average
behavior of the multiple gravity laws.

Interestingly, we find a universal pattern in the exponent matrices of
different cities, despite the huge diversity of the population, geographical
constraints, and travel patterns of those cities; e.g., one can see a variety
of geographical constraints such as sea, lakes, mountains, and/or neighboring
cities in Fig. 2. To be precise, the value of the distance exponent
$\gamma_{kk^{\prime}}$ tends to be higher between groups of larger traffic
volumes. The pairs of groups with large traffic volume and small traffic
volume also tend to show higher values of $\gamma_{kk^{\prime}}$ than those
between groups with small traffic volumes. A possible explanation for such
observations will be discussed later.

We examine the quality of the distance exponent estimation in terms of $R^{2}$
values. In Fig. 3(a), we show the $R^{2}$ value distribution for
$\gamma_{kk^{\prime}}$ [Eq. (5)] of each city in a box-and-whisker plot. The
distribution is compared to the $R^{2}$ value for $\gamma_{\rm s}$ [Eq. (4)]
of the same city, depicted as a black thick vertical line. We observe that the
$R^{2}$ value for $\gamma_{\rm s}$ is not always better or worse than those
for $\gamma_{kk^{\prime}}$. To look at more details, we group $55$ exponent
values into two subsets; one is for the cases with $k+k^{\prime}>11$, and the
other is for the cases with $k+k^{\prime}\leq 11$. As shown in Fig. 3(b), it
turns out that the $R^{2}$ values for $\gamma_{kk^{\prime}}$ with
$k+k^{\prime}>11$ tend to be much larger than that for $\gamma_{\rm s}$ in
most cities, while the $R^{2}$ values for $\gamma_{kk^{\prime}}$ with
$k+k^{\prime}\leq 11$ show the opposite tendency. The larger $R^{2}$ values
for cases with larger $k$ and/or $k^{\prime}$ indicate that multiple gravity
laws can explain the mobility pattern for high traffic regions better than the
conventional gravity law characterized by a single distance exponent.

##  IV Discussion

In order to understand the observed universal pattern in the exponent
matrices, we consider two main factors, namely, the traffic landscape and the
travel cost. We find that the effects due to the traffic landscape seem to
explain the observed exponent matrices to some extent, which can also be
argued in terms of the travel cost.

As evident in the traffic landscapes of Fig. 2, all the cities analyzed might
be considered to have a so-called “core-periphery” structure whether the
number of central areas or centers is one or more [46]. We remark that in
contrast to cells with large traffic volumes, comprising the center(s), cells
with small traffic volumes are scattered over the cities; see a schematic
diagram for the core-periphery structure in Fig. 4. It implies that travel
distances between cells in the periphery tend to be more diverse than those
between cells in the center, thus possibly weakening the distance dependence
of traffic volumes between cells in the periphery. Indeed, as shown in Fig. 2,
the distance exponent $\gamma_{kk^{\prime}}$ tends to have smaller values
(e.g., $\lesssim 0.2$) for smaller $k$ and $k^{\prime}$, while
$\gamma_{kk^{\prime}}$ has an overall higher value when either $k$ or
$k^{\prime}$ gets larger.

Figure 4: Schematic diagram for understanding multiple gravity laws in a
centralized urban landscape. The dark shaded circle at the center and the
bright shaded ring at the periphery denote groups of $k=1$ and $k=10$,
respectively. Distance exponents $\gamma_{10,10}$, $\gamma_{1,1}$, and
$\gamma_{1,10}$ characterize the distance dependence of traffic volumes
between or within groups. Arrows visualize some possible trips between or
within groups. Figure 5: Standard deviations of travel distances between
groups [Eq. (6)] for the twelve most populated cities in the USA. The color
bar for the distance is in km. Note that cities have different ranges of
distance in the color bar. Figure 6: Average distance of travel distances
between groups [Eq. (7)] for the twelve most populated cities in the USA. The
color bar for the distance is in km. Note that cities have different ranges of
distance in the color bar.

Just because the possible travel distances are diverse does not necessarily
mean that the real travel distances are diverse. To examine this issue, we
calculate the standard deviation of travel distances between cells in two
groups of $k$ and $k^{\prime}$ which is defined as

$\sigma_{kk^{\prime}}\equiv\left[\frac{\sum_{i\in G_{k},j\in
G_{k^{\prime}}}T_{ij}\left(r_{ij}-\langle
r\rangle_{kk^{\prime}}\right)^{2}}{\sum_{i\in G_{k},j\in
G_{k^{\prime}}}{T_{ij}}}\right]^{1/2},$ (6)

where $\langle r\rangle_{kk^{\prime}}$ is an average travel distance between
cells in two groups of $k$ and $k^{\prime}$:

$\langle r\rangle_{kk^{\prime}}\equiv\frac{\sum_{i\in G_{k},j\in
G_{k^{\prime}}}{T_{ij}r_{ij}}}{\sum_{i\in G_{k},j\in
G_{k^{\prime}}}{T_{ij}}}.$ (7)

As shown in Fig. 5, $\sigma_{kk^{\prime}}$ tends to have higher values for
smaller $k$ and/or $k^{\prime}$ in all cities but Miami. This implies that the
diversity of the travel distances between groups seems to be anti-correlated
with the distance exponent values, as expected. As for the average travel
distance $\langle r\rangle_{kk^{\prime}}$, we find in Fig. 6 that the values
of $\langle r\rangle_{kk^{\prime}}$ for small $k$ and $k^{\prime}$ are overall
comparable to those for large $k$ and $k^{\prime}$, both of which are shorter
than those between groups of small traffic volumes and large traffic volumes.
This tendency is denoted by long and short arrows in Fig. 4. Considering the
fact that the dataset analyzed is for the commuting between homes and
workplaces, people living in the periphery do not travel so far from their
homes, while people going to work to the center from the periphery (or the
other way around) need to travel farther than others.

Next, we consider the travel cost for understanding the observed exponent
matrices. The larger value of the distance exponent implies the stronger
effect of the distance on the traffic volume, e.g., due to the higher travel
cost per distance traveled. Here the travel cost can be measured in terms of
elapsed time or transportation cost. In this sense, the larger values of the
distance exponent within central areas or between central and peripheral areas
could be due to the higher travel cost per distance. On the other hand, the
smaller values of the distance exponent between peripheral areas could be
partly due to the lower travel cost per distance. These arguments might be the
case considering various factors in the central areas such as congestion,
traffic signs, and speed limits that tend to increase the travel cost [47]. On
the other hand, the travels between peripheral areas tend to suffer from such
factors less often, e.g., by taking highways. Yet our explanation is
speculative at most, calling for more detailed empirical analyses in the
future.

##  V Summary and conclusion

In summary, we have devised the data analysis framework for urban mobility
patterns and applied it to the dataset of the twelve most populated cities in
the USA. We have found that the intra-city mobility patterns can be
successfully characterized by multiple gravity laws, by which it means that
the distance exponent value depends on the traffic volumes of the origin and
destination cells within the same city. This finding is in contrast to the
conventional gravity law characterized by a single distance exponent for a
given dataset or area of interest. The universal pattern in the distance
matrices of different cities is observed, implying some common mechanisms
behind such observations.

The dataset we have analyzed has some limits. First, it includes only trips
for commuting, but not other kinds of mobility such as shopping or tourism.
Second, it does not have any detailed information on the travel trajectory
and/or cost, hampering further analysis to study the mechanisms for the
multiple gravity laws.

Finally, we discuss about possible future works. To investigate the mechanisms
for the multiple gravity laws within the cities, one can study mathematical
models considering the heterogeneous core-periphery structure of urban
population and/or different travel costs depending on the mode of travel, etc.
Based on the understanding of the mechanisms, one is expected to enhance the
prediction and optimization of the mobility pattern within the city.

###### Acknowledgements.

O.-H.K. and W.-S.J. was supported by the National Research Foundation of Korea
(NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A106303012).
H.-H.J. acknowledges financial support by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIT) (No.
2022R1A2C1007358).

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