where $p(r,R)$ depends on how sub-patches are distributed within patches.

The total pair distribution number density is given by

$\displaystyle n(r)$
$\displaystyle=\sum_{k=1}^{L}b^{k}p(r,z(N/\theta)^{k/2}).$ (110)

An explicit form using the generalized pair distribution function model is

$\displaystyle n(r)$
$\displaystyle\propto\sum_{k=1}^{L}b^{k}{\frac{r}{R_{k}^{2}}}\exp\left(-\left(\frac{r}{R_{k}}\right)^{m}\right).$
(111)
$\displaystyle={\frac{r}{z^{2}}}\sum_{k=1}^{L}{\theta^{k}}\exp\left(-\left(\frac{r}{z}\right)^{m}\left(\frac{\theta}{b}\right)^{km/2}\right).$
(112)

We compute this equation numerically for varying $\theta$, $b$, and different
models for $p(r,R)$. Figure 24,illustrates examples for $b=4$ and
$\theta=0.8$, which leads to a scaling of approximately $\propto r^{0.67}$.

To derive the dependence of the exponent of the observed scaling law on the
parameters, we use the saddle-point approximation. First, we approximate the
sum over hierarchy layers with an integral over a constant hierarchy layer
density

$\displaystyle n(r)\propto
r\int\limits_{1}^{\infty}dk\exp\left[\underbrace{k\ln\theta-\left(\frac{r}{z}\right)^{m}\exp\left(\frac{mk}{2}\ln\frac{\theta}{b}\right)}_{=-w(k)}\right].$
(113)

Relying on the saddle-point method, we can approximate the integral to find

$\displaystyle n(r)\propto
r\frac{\exp\big{[}-w(k_{0})\big{]}}{\sqrt{w^{\prime\prime}(k_{0})}}$ (114)

where $k_{0}$ is the minimum of $w(k)$. We compute the derivatives

$\displaystyle w^{\prime}(k)$
$\displaystyle=-\ln\theta+\left(\frac{r}{z}\right)^{m}\frac{m}{2}\ln\frac{\theta}{b}\exp\left(\frac{mk}{2}\ln\frac{\theta}{b}\right)$
(115) $\displaystyle w^{\prime\prime}(k)$
$\displaystyle=\left(\frac{r}{z}\right)^{m}\left(\frac{m}{2}\ln\frac{\theta}{N}\right)^{2}\exp\left(\frac{mk}{2}\ln\frac{\theta}{b}\right)$
(116)

and with $w^{\prime}(k_{0})=0$ find the following equations for the minimum

$\displaystyle\exp\left(\frac{mk_{0}}{2}\ln\frac{\theta}{b}\right)$
$\displaystyle=\frac{\ln\theta}{\ln(\theta/b)}\left(\frac{z}{r}\right)^{m}\frac{2}{m}$
(117) $\displaystyle k_{0}$
$\displaystyle=\frac{2}{m}\frac{1}{\ln\theta/b}\left[\ln\left(\frac{\ln\theta}{\ln(\theta/b)}\right)+m\big{(}\ln
z-\ln r\big{)}+\ln\left(\frac{2}{m}\right)\right].$ (118)

Using the first of the two equations we note that for $w(k_{0})$, the
dependence on $r$ cancels out in the second term and so

$\displaystyle-w(k_{0}(r))=k_{0}(r)\ln\theta-W$ (119)

where $W$ is an irrelevant constant. Second, we use the same equation to see
that $w^{\prime\prime}(k_{0})$ does not depend on $r$ and therefore does not
concern us any further either. As a step in between, this means that

$\displaystyle n(r)\propto r\exp\big{[}k_{0}(r)\ln\theta\big{]}.$ (120)

Looking at Eq. (118), we see that the only $r$-dependent term gives the
minimum a structure of

$\displaystyle k_{0}=K-\frac{2}{\ln(\theta/b)}\ln r,$ (121)

and therefore we find

$\displaystyle n(r)$ $\displaystyle\propto
r\exp\left[-\frac{2\ln\theta}{\ln(\theta/b)}\ln r\right]$ (122)
$\displaystyle=r^{1-\frac{2\ln\theta}{\ln(\theta/b)}},$ (123)

i.e. the exponent of the sub-linear growth in the pair distribution function
is given as

$\displaystyle\alpha=1-\frac{2\ln\theta}{\ln(\theta/b)}.$ (124)

We can compare this estimation with numerical results. Above we used $N=4$ and
$\theta=0.8$ to find $\alpha\approx 0.67$. Eq. (124) yields $\alpha=0.72$, an
acceptable approximation. Notably, this result is also independent of the
parameter $m$, which represents the explicit form of the pairwise distance
distribution per patch. This explains why similar results are obtained despite
the use of different geometries.

We can perform the same analysis for the parabola pair distribution model Eq.
(59). We have

$\displaystyle n(r)$
$\displaystyle\propto\sum_{k=1}^{L}\frac{b^{k}}{R_{k}^{2}}\left(-\frac{3}{4}\frac{r^{2}}{R_{k}}+\frac{3r}{2}\right)$
(125)
$\displaystyle=\frac{r}{z^{2}}\sum_{k=1}^{L}\theta^{k}\left(-\frac{3}{4}\frac{r}{R_{k}}+\frac{3}{2}\right).$
(126)

Again, we use the saddle point method to approximate

$\displaystyle n(r)$ $\displaystyle\propto
r\frac{\exp\left[-w(k_{0})\right]}{\sqrt{w^{\prime\prime}(k_{0})}},$ (127)

with $k_{0}$ being the minimum of the function

$\displaystyle
w(k)=-k\ln\theta-\ln\left(-\frac{3}{4}\frac{r}{z}\frac{\theta^{k/2}}{b^{k/2}}+\frac{3}{2}\right).$
(128)

We arrive at

$\displaystyle n(r)\propto
r\frac{\theta^{\frac{2\ln{\left(\frac{4z\ln{\left(\theta\right)}}{r\left(2\ln{\left(\theta\right)}+\ln{\left(\frac{\theta}{b}\right)}\right)}\right)}}{\ln{\left(\frac{\theta}{b}\right)}}}3\sqrt{\pi}\ln{\left(\frac{\theta}{b}\right)}}{z^{2}\sqrt{\left(2\ln{\left(\theta\right)}+\ln{\left(\frac{\theta}{b}\right)}\right)\ln{\left(\theta\right)}}\left(2\ln{\left(\theta\right)}+\ln{\left(\frac{\theta}{b}\right)}\right)}.$
(129)

Leaving aside the irrelevant prefactors, we have

$\displaystyle\ln n(r)$ $\displaystyle\propto\ln r+\ln\theta\left(K-\frac{2\ln
r}{\ln(\theta/b)}\right)$ (130) $\displaystyle\propto\ln
r\left(1-\frac{2\ln\theta}{\ln(\theta/b)}\right),$ (131)

i.e. the same result as with the generalized pair distribution.

##  8 Studying mobility with the pair distance function

###  8.1 Comparison with other Mobility models

####  8.1.1 Gravity

The gravity model as originally introduced in [5], states that the number of
moves $T$ from area $i$ to $j$ is proportional to,

$\displaystyle T_{i\to j}\propto\frac{N_{i}N_{j}}{d_{i,j}}$ (132)

with $N_{i}$ the population at location $i$ and $d_{i,j}$ the distance between
locations $i$ and $j$.

Figure 25: Gravity model as in [5] for migration between (left) municipalities
in Denmark, (right) parishes in Denmark. The normalized flow represented on
the $y$-axis is the number of moves from $i$ to $j$ divided by the product of
population between $i$ and $j$, $T_{i,j}/(N_{i}N_{j})$.

####  8.1.2 Radiation

In this section, we derive a continuous radiation model and demonstrate how it
differs from our continuous gravity model. The radiation model of human
mobility is usually represented by the following equation [6],

$P_{ij}\propto\frac{m_{i}m_{j}}{(m_{i}+s_{ij})(m_{i}+m_{j}+s_{ij})}$ (133)

with $P_{ij}$ the predicted number of people traveling from location $i$ to
location $j$, $m_{k}$ the population at the origin location $k$, $s_{ij}$
represents the total population in a circle centered at $i$ with radius equal
to the distance between $i$ and $j$, excluding the populations of $i$ and $j$.

Following [6], the probability of having a single particle emitted from
location $i$ to location $j$ is,

$\displaystyle
P(1|m_{i},m_{j},s_{i,j})=\int_{0}^{\infty}\mathrm{d}zP_{m_{i}}(z)P_{s_{i}}(<z)P_{m_{j})(>z)}$
(134)

with,

$P_{m_{i}}(z)=\frac{\mathrm{d}P_{m_{i}}(<z)}{\mathrm{d}z}=m_{i}p(z)^{m_{i}-1}\frac{\mathrm{d}p(<z)}{\mathrm{d}z}$
(135)

if $m_{i}=1$, then it simplifies to $P_{1}(z)=\mathrm{d}P(<z)/\mathrm{d}z$.
Equation 134 simplifies to,

$\displaystyle P(1|1,1,s_{i,j})$
$\displaystyle=\int_{0}^{\infty}(1-P(<z))P(<z)^{s_{i,j}}\mathrm{d}P(<z)$ (136)
$\displaystyle=\frac{1}{(1+s_{i,j})(2+s_{i,j})}$ (137)

In the general case when $s_{i,j}\gg 1$, that is, when the number of houses in
the circle of center $i$ and radius the distance $i$ and $j$ is greater than
one, which is often the case when the distance is sufficiently large (see Fig.
5),

$P_{ij}=\frac{1}{(1+s_{ij})(1+1+s_{ij})}\sim s_{ij}^{-2}.$ (138)

However, the number of addresses in the circle $s_{i,j}$ of center $i$ and
radius $R(i,j)$ is closely related to the primitive of the pair distribution
function,

$\displaystyle s_{x,y}$
$\displaystyle=\int_{\Omega}dzp^{d}(z)\delta\\{R(x,z)<R(x,y)\\},$ (139)
$\displaystyle s(r)$
$\displaystyle=\int_{x}dx\int_{y}dy\,s_{x,y}p^{d}(x)p^{d}(y)\delta(r-R(x,y)),$
(140) $\displaystyle s(r)$
$\displaystyle=\int_{x}dx\int_{y}dy\,s_{x,y}p^{d}(x)p^{d}(y)\delta\\{r>R(x,y)\\}$
(141)

In the meantime, the cumulative distribution of pair distribution is (with $p$
the pair distribution function),

$\displaystyle\Phi(r)$ $\displaystyle=\int_{0}^{r}p(s)ds,$ (142)
$\displaystyle\Phi(r)$ $\displaystyle=\int_{0}^{r}ds\int_{\Omega}dx\
\int_{\Omega}dy\ ^{d}(x)p^{d}(y)\delta(s-R(x,y)),$ (143)
$\displaystyle\Phi(r)$ $\displaystyle=\int_{\Omega}dx\ \int_{\Omega}dy\
p^{d}(x)p^{d}(y)\delta\\{r>R(x,y)\\},$ (144) $\displaystyle\Phi(r)$
$\displaystyle=s(r).$ (145)

If we come back to the derivation of the pair distribution,

$f(x,y)\mathrm{d}^{2}x\mathrm{d}^{2}y=\pi(x,y)\varrho^{d}(x)\varrho^{d}(y)\mathrm{d}^{2}x\mathrm{d}^{2}y.$
(146)

In our continuous gravity law model, we assume that
$\pi(x,y)=\pi(R(x,y))=\pi(r)$. On the other hand, the radiation model assumes
that,

$\pi(x,y)=s(x,y)^{-2}=\Phi(x,y)^{-2}$ (147)

If we assume that on average,
$s(x,y)\simeq\int_{0}^{|x-y|}\mathrm{d}rp(r)=\Phi(r)$. Then the two models
will agree on average if,

$\Phi(r)^{2}=r$ (148)

The pair distribution function would be $p(r)=\pm r^{-0.5}$. While the
empirical pair distribution (Fig.2a) exhibits more complex patterns than this,
and the two models appear to be irreconcilable.

###  8.2 The intrinsic distance cost as the inverse of distance

Consider an individual situated at location $i$. The utility associated with
moving to location $j$ can be expressed as follows,

$U(i,j)=V(j)-C(r(i,j))$

where $V(j)$ represents the inherent values of location $j$ and $C(r(i,j))$ is
a function of the distance $r(i,j))$ between $i$ and $j$.

The probability of selecting location $j$ can be calculated using the
multinomial logit model, as outlined in [73],

$f(i,j)=\frac{e^{U(i,j)}}{\sum_{k}{e^{U(i,k)}}}$

The probability to move of a distance $r$ is the sum over all locations $k$
for which $r(i,k)=r$. This probability becomes independent of $i$. Thus, the
probability of moving of a distance $r$ aggregates over all locations $i$ as
$f(r)=\sum_{i}f(i,r)f(i)$. For simplification, let us that $f(i)$ is constant,
implying that the probability of movement from any given location is equal.
Furthermore, let us assume that for all $i,j\in\Omega$, $f(i,r)=f(j,r)$. In
this case $f(r)\propto Nf(i,r)$.

Moreover, the number of locations $k$ satisfying $r(i,k)=r$ corresponds to the
pair distribution function, $p(r)$, up to a constant, with this constant being
the total number of locations.

$f(r)=\frac{\sum_{i\in\Omega}\sum_{k|r(i,k)=r}e^{U(i,k)}}{\sum_{k}{e^{U(i,k)}}}$

Assuming a uniform value $V(k)$ across all locations, we derive:

$f(r)=\frac{\sum_{i\in\Omega}\sum_{k|r(i,k)=r}e^{V-C(r(i,k))}}{\sum_{k}{e^{U(i,k)}}}\\\
\sim{p(r)e^{V-C(r)}}$

with $p(r)$ the pair distribution function evaluated at $r$.

Given the human tendency to perceive distances logarithmically, especially for
larger magnitudes, we posit that the cost function $C(r)\propto\log(r)$. This
hypothesis is supported by the observation that humans might perceive
numerical differences in a logarithmic fashion, such as perceiving the
difference between 1 and 2 similarly to the difference between 10 and 20 or
even 100 and 200, as demonstrated in [74]. Consequently, upon normalization by
the pair distribution function, we derive the equation

$\frac{f(i,r)}{p(i,r)}\sim
e^{V-\log(r)}=e^{\log(V_{2})-\log(r)}=\frac{V_{2}}{r}$

where $V=\log(V_{2})$. This leads to

$\frac{f(i,r)}{p(i,r)}\sim\frac{1}{r}$

which is equivalent to equation (9) of the intrinsic distance cost, given
$\pi(r)=\frac{1}{r}$.

###  8.3 Reconciling Opportunity and Distance-Based Mobility Paradigms

As highlighted in [1, 28, 29], in classical mobility studies, two divergent
theories explain the movement of individuals. The first, inspired by Newton’s
law of gravity, suggests that mobility decreases as the physical distance
between locations increases, often modeled as a power law of distance [5]. The
second theory argues that mobility is not directly related to distance but to
intervening opportunities. It suggests that individuals prioritize locations
based on the availability of closer opportunities rather than distance itself,
leading to movements driven more by the opportunity distribution than by
travel constraints [27, 6].

The gravitational model describes how the flow of individuals between two
locations decreases as the distance between them increases, analogous to the
decrease in gravitational pull with distance in Newton’s theory. This decline
in mobility is modeled using functions such as exponential or power laws [75,
76, 77]. In this framework, the population sizes of the start and end
locations act like masses, attracting travel in direct proportion to their
size and inversely to the square of the distance between them. Known for its
simplicity and ease of computation, the gravity model is widely used in
various fields such as migration, intercity communication, and traffic flow
analysis. It can also be extended by integrating socio-economic
characteristics of locations to improve its accuracy and applicability [78].

Conversely, the intervening opportunity model emphasizes that mobility is
mainly driven by the availability of intervening opportunities rather than
distance [79]. It suggests that individuals are influenced more by the
availability of nearby opportunities than by the distance to distant
opportunities, with the spatial distribution of these opportunities dictating
destination choices, making distance a less critical factor. The radiation
model [6], an extension of this theory, streamlines the analysis by assuming
that the chosen opportunity is the optimal one. This model correlates
opportunity density with population density and provides a mathematical
formula for predicting trip endpoints. The model has also been extended to a
continuous spatial framework, providing a nuanced understanding of how social
factors influence movement patterns [80].

Our framework provides a unifying approach to the intervening opportunity
model and distance-based model In the equation 3,

$\displaystyle f(r)\propto\pi(r)p(r).$ (149)

The pair distribution function $p(r)$ captures the available opportunities to
move, when we divide the moving distance distribution by the pair
distribution, we normalize the empirical mobility traces by their intervening
opportunities. Consequently, the remaining quantity is found to be dependent
on the distance, with the result that $pi(r)=f(r)/p(r)\simeq 1$. This can be
interpreted as a distance term analogous to the gravity model. Furthermore,
our framework explain why intra-city movements are better fitted by a gravity
model with an exponential decay, which is similar to the exponential decay of
population density [57].

###  8.4 Local piece-wise power law

In this section, we investigate whether the continuous gravity model still
holds at the scale of cities. The observed moving distribution $f(r)$ can be
decomposed into the observed moving distribution of each city $c_{i}$ in the
set $\mathcal{C}$ of cities in Denmark,

$f(r)=\sum_{c_{i}\in\mathcal{C}}f_{c_{i}}(r),$ (150)

where $f_{c_{i}}(r)$ only counts moves originating from city $c_{i}$, i.e.
$f_{c}(r)=\int_{x\in c}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}yf(x,y)$. For
each city’s observed moving distance the appropriate normalization is not the
country-wide pair distribution but the relative pair distribution function,
which counts pairs with at least one location in the city of interest. We
define,

$p_{c}(r)=\int_{x\in
c}\mathrm{d}x\int_{y\in\Omega}\varrho^{d}(x)\varrho^{d}(y)\mathrm{d}x\mathrm{d}y.$
(151)

We can now compute the relative intrinsic distance cost,
$\pi_{c}(r)=f_{c}(r)/p_{c}(r)$. In the main manuscript Figures.3f-g show that
the relative intrinsic distance cost does not follow the continuous gravity
law and leads to a piece-wise intrinsic distance cost, Figures 28-32 show the
same process for many more cities. The piece-wise intrinsic distance cost is a
piece-wise power law distribution of the form:

$\displaystyle\pi_{c}(r)=\begin{cases}C_{1}(R_{c}/r)^{\beta},&0\leq r\leq
R_{c}\\\ C_{2}(R_{c}/r)^{\alpha},&r>R_{c}\end{cases}$ (152)

where $R_{c}$ is the mobility inferred city radius, $\alpha>1$,
$\beta\in\mathbb{R}$, and $x\in\mathbb{R}^{+}$. The three parameters
$\alpha,\,\beta,\,r_{c}$ are estimated by maximum-likelihood and compared to a
log-normal, Pareto, and exponential Pareto distribution [64]. The likelihood
ratios and p-values for all cities are reported in figures 33-35. We obtain
that on average, $\beta=0.60\pm 0.20$, $\alpha=2\pm 0.21$, $R_{c}$ follow a
heavy-tailed distribution between a power law and a log-normal distribution.

####  8.4.1 The remarkable case of islands

In the section, we highlight the remarkable point that the piece-wise two-step
intrinsic distance cost for cities still holds remarkably well for islands
(Fig.26). In the case of islands, the inter-city part of the piece-wise
process (when the exponent is on average 2), it mostly across the sea, and
therefore the relative pair distance distribution is null. However, if the
distance is large enough to reach the nearest landmass, the pair distance
distribution is positive again and the $2$ exponent can be interpolated
between the island and the nearest landmass (see Fig.26).

Figure 26: Piecewise intrinsic distance for the city of Rønne and Nexø both
situated on an island.

###  8.5 Simulation of piece-wise intrinsic distance cost

In this section, we try to recover the intrinsic distance cost, $\pi(r)=1/r$
from the aggregation of the piece-wise process of equation 152. We can
decompose the intrinsic distance cost as follows,

$\displaystyle\pi(r)=\frac{f(r)}{p(r)}=$
$\displaystyle\sum_{c\in\mathcal{C}}\frac{f_{c}}{p(r)}=\sum_{c\in\mathcal{C}}\pi_{c}(r)\frac{p_{c}(r)}{p(r)}$
(153)

The only term that is unknown in this equation is $p_{c}(r)$ the relative pair
distance distribution function for each city. However, we have already shown
that the pair distribution of cities follows a generalized gamma distribution
within the city boundary. Beyond the city boundary, we have shown that the
positions of cities follow an ideal gas (i.e. random position but no overlap),
which implies that we can model the relative pair distribution function with a
linear growth beyond the city radius. The relative pair distribution function
of a city, $c$, is thus,

$\displaystyle\hat{p}_{c}(r)=\begin{cases}C_{1}(r/R_{c}^{2})\exp(-(r/R_{c})^{m}),&0\leq
r\leq R_{c}\\\ C_{2}r,&r>R_{c}\end{cases}$ (154)

where $C_{1}$ and $C_{2}$ are two constants such that $\hat{p}_{c}$ is
continuous. To obtain the global pair distribution we need to weight each
relative pair distribution by its contribution, which is equal to the number
of addresses. We assume that the number of addresses is proportional to the
square of the radius and $p_{c}=R_{c}^{2}\hat{p}_{c}$. For simplicity, we also
assume that the general pair distribution function $p$ follows,

$\displaystyle p(r)=\begin{cases}C_{3}r^{2/3},&0\leq r\leq R_{\text{max}}\\\
C_{4},&r>R_{\text{max}}\end{cases}$ (155)

where $R_{\text{max}}=\max\\{R_{c}\,,c\in\mathcal{C}\\}$ the largest city
radius. The form of the pair distribution function closely follows the
empirical one of Figure 2a. Therefore, we obtain the intrinsic distance cost
when aggregating over the city scale,

$\displaystyle\pi_{c}(r)\frac{p_{c}(r)}{p(r)}=\begin{cases}C_{5}(r^{-1/6}/R_{c}^{2})\exp(-(r/R_{c})^{m}),&0\leq
r\leq R_{c}\\\ C_{6}r^{-5/3},&R_{c}<r\leq R_{\text{max}}\\\
C_{7}r^{-1},&r>R_{\text{max}}\end{cases}$ (156)

Remarkably, the inverse of distance scaling of the intrinsic distance cost for
$r\geq R_{\max}$emerges as a sole consequence of the random distribution of
city locations. The scaling $r\leq R_{\max}$ results from the aggregation over
the scale of cities, akin to our patch model SI. 7.6 or [7]. We obtain that
for $r\leq R_{\max}$,

$\displaystyle\pi(r)=\int_{0}^{r}C_{5}(r^{-1/6}/R_{c}^{2})\exp(-(r/R_{c})^{m})g(R)\mathrm{d}R+\int_{r}^{\infty}C_{6}r^{-5/3}g(R)\mathrm{d}R$
(157)

where $g$ is the distribution of the city radius. From the main text Fig.3,
$g$ can be modeled as a log-normal or a power-Pareto distribution. We solve
the equation 157 numerically by simulating over the city intrinsic distance
cost over an artificial geography of Denmark (eq.154,155). We recover the
intrinsic distance cost $\pi(r)=1/r$ (Fig.27).

Figure 27: For each city in Denmark, the distribution of observed moving
distance (orange) and the relative pair distribution function. Pairs are
restricted to those containing at least one address from the city of interest.
The second plot shows the intrinsic distance cost (black), as the ratio
between the observed moving distance and the relative pair distribution
function. The pink dashed line is the inferred mobility city radius, the light
teal corresponds to intra-city moves, and the dark teal to inter-city moves.
Figure 28: For each city in Denmark, the distribution of observed moving
distance (orange) and the relative pair distribution function. Pairs are
restricted to those containing at least one address from the city of interest.
The second plot shows the intrinsic distance cost (black), as the ratio
between the observed moving distance and the relative pair distribution
function. The pink dashed line is the inferred mobility city radius, the light
teal corresponds to intra-city moves, and the dark teal to inter-city moves.
Figure 29: For each city in Denmark, the distribution of observed moving
distance (orange) and the relative pair distribution function. Pairs are
restricted to those containing at least one address from the city of interest.
The second plot shows the intrinsic distance cost (black), as the ratio
between the observed moving distance and the relative pair distribution
function. The pink dashed line is the inferred mobility city radius, the light
teal corresponds to intra-city moves, and the dark teal to inter-city moves.
Figure 30: For each city in Denmark, the distribution of observed moving
distance (orange) and the relative pair distribution function. Pairs are
restricted to those containing at least one address from the city of interest.
The second plot shows the intrinsic distance cost (black), as the ratio
between the observed moving distance and the relative pair distribution
function. The pink dashed line is the inferred mobility city radius, the light
teal corresponds to intra-city moves, and the dark teal to inter-city moves.
Figure 31: For each city in Denmark, the distribution of observed moving
distance (orange) and the relative pair distribution function. Pairs are
restricted to those containing at least one address from the city of interest.
The second plot shows the intrinsic distance cost (black), as the ratio
between the observed moving distance and the relative pair distribution
function. The pink dashed line is the inferred mobility city radius, the light
teal corresponds to intra-city moves, and the dark teal to inter-city moves.
Figure 32: For each city in Denmark, distribution of observed moving distance
(orange) and the relative pair distribution function. Pairs are restricted to
those containing at least one address from the city of interest. The second
plot shows the intrinsic distance cost (black), as the ratio between the
observed moving distance and the relative pair distribution function. The pink
dashed line is the inferred mobility city radius, the light teal corresponds
to intra-city moves, the dark teal to inter-city moves.  Figure 33: For each
city of the piecewise fit, Figure.28-32, we compare the log-likelihood ratio
between the piecewise power law distribution of equation 152 and a Pareto
distribution. The left plot shows the distribution of p values with a pink
line at $p=0.05$. The right plot shows the distribution of the log-likelihood
ratio, with a pink line at the decision boundary $R=0$. A positive log-
likelihood ratio indicates that the piecewise models describe the data better,
the p-value give the significance of the result. Figure 34: For each city of
the piecewise fit, Figure.28-32, we compare the log-likelihood ratio between
the piecewise power law distribution of equation 152 and a lognormal
distribution. The left plot shows the distribution of p values with a pink
line at $p=0.05$. The right plot shows the distribution of the log-likelihood
ratio, with a pink line at the decision boundary $R=0$. A positive log-
likelihood ratio indicates that the piecewise models describe the data better,
the p-value give the significance of the result. Figure 35: For each city of
the piecewise fit, Figure.28-32, we compare the log-likelihood ratio between
the piecewise power law distribution of equation 152 and a piecewise
exponential-Pareto distribution. The left plot shows the distribution of p
values with a pink line at $p=0.05$. The right plot shows the distribution of
the log-likelihood ratio, with a pink line at the decision boundary $R=0$. A
positive log-likelihood ratio indicates that the piecewise models describe the
data better, the p-value give the significance of the result.

##  9 The definition of city

The precise demarcation of a city — where a city ends and its boundaries — is
a pressing issue in academic literature [81]. Depending on the definition
adopted, different results can be obtained [82]. To ensure that our results
are not just a byproduct of our chosen city definition and that they could
potentially be applied to other definitions, we evaluated the robustness of
the city boundaries.

First, we adopted the city definition provided by Danmarks Statistik [51].
According to this definition, Denmark consists of 1,473 cities.

To ensure more robust results, we compare the city definition to cities
defined by density-based clustering techniques. These methods distinguish
densely clustered data points that represent urban or urbanized areas from
sparse or noisy regions that typically correspond to rural or sparsely
populated areas. DBSCAN is particularly well-suited for this task because of
its innate ability to delineate clusters of different geometries, which is
critical for accommodating the non-uniform shapes of urban regions. This
algorithm sorts data points into core points, boundary points, and noise based
on surrounding data density. However, one of the main challenges is to
determine the optimal values for $\epsilon$ (neighborhood search radius) and
minimum data points, as these parameters can determine the level of detail of
the identified urban zones. [83] .

For data covering cities with different population densities, HDBSCAN stands
out. Building on the foundation laid by DBSCAN, HDBSCAN employs a hierarchical
tactic that makes the fixed $\epsilon$ value redundant. By examining the
density hierarchy, HDBSCAN can differentiate between densely populated cities
and smaller towns within the same data collection. This granularity, coupled
with the flexible clustering results, provides a detailed view of the
boundaries of cities [55].

Figure 36: Comparison between the Danmarks Statistik definition of cities (on
the left) and the cluster obtained from HDBSCAN (on the right). The colors are
ordered according to the size of the cluster/city.

Copenhagen, however, presents a unique scenario. Its urban influence extends
well beyond its administrative boundaries, as evidenced by phenomena such as
commuting patterns. As a result, neighboring cities have been merged with
Copenhagen to form the Copenhagen metropolitan area, or Hovedstadsområdet, as
defined by DST. Notably, the clustering results from HDBSCAN reflected this
merging, indicating the combined urban sprawl of the Copenhagen region.

Normalized Mutual Information (NMI) and Adjusted Mutual Information (AMI)
quantify the similarity between two clusterings. Both metrics measure the
information shared between the true labels and the labels assigned by a
clustering algorithm. Therefore, we can use them to evaluate the performance
of clustering algorithms in the absence of ground truth labels. The NMI is
defined as the mutual information between two clusterings divided by the
geometric mean of their entropies [84]. However, a limitation of the NMI is
that it ignores the random grouping of clusters, i.e. random cluster
assignments can produce a non-zero NMI value. The AMI, on the other hand,
corrects for this limitation by adjusting the score to account for chance,
ensuring that random cluster assignments result in an AMI score close to zero
[85]. Therefore, the AMI provides a more accurate representation of the
similarity between two clusterings in our case where the number of clusters is
not fixed.

The table 4 shows the value of the indices for the two algorithms. An NMI/AMI
score of 0.9 indicates that the two clusterings share a significant amount of
information. Such a high score typically indicates that the two clusterings
are almost identical, with only a few data points potentially clustered
differently. Therefore, the city definition from Danmarks Statistics is
substantially similar to the one we obtain using density-based clustering
techniques. Furthermore, table 3 shows that merging Copenhagen into the
Hovedstadsområdet is a better definition of a city according to the density-
based clustering techniques.

Algorithm | NMI | AMI
---|---|---
DBSCAN | 0.88 | 0.87
HDBSCAN | 0.91 | 0.89
Table 3: NMI and AMI values between empirical city clustering (Copenhagen merged) and algorithmic clustering (DBSCAN and HDBSCAN). Algorithm | NMI | AMI
---|---|---
DBSCAN | 0.71 | 0.67
HDBSCAN | 0.67 | 0.65
Table 4: NMI and AMI values between the empirical cities (Copenhagen not
merged) clustering and the algorithmic clustering (DBSCAN and HDBSCAN)

The parameters for HDSCAN were chosen to optimize the NMI between the
clustering and the real city labels. Figure 37 shows the values for different
values of minimum samples and minimum cluster size, the rest of the parameters
follow the default values of this implementation [55].

Figure 37: Heatmap of the NMI between the clustering of addresses locations
and their official city and the values for different combinations of HDSCAN
parameters.

Figure 36 shows all addresses colored by the cluster they belong to. The
background color is the official city border.

##  10 Maximum likelihood estimation of power laws

Fitting the statistical power-law model, to examine the distance distribution
of human movement, we fit a truncated power law with the form,

$\displaystyle p(x)\propto x^{-alpha}e^{-\lambda x}.$ (158)

where $\alpha$ is a constant parameter of the distribution (known as the
scaling parameter or exponent), $x$ is the travel distance ($x>x_{min}>0$),
and $\lambda$ is the parameter of the exponential distribution. $x_{min}$
represents the shortest distance above which the power law scaling
relationship begins. The scaling parameter $\alpha$ must be estimated before
finding the optimal values of $x_{min}$. The methods of [61] find $x_{min}$ by
generating a power-law fit starting from each unique value in the data set,
then selecting the one that gives the minimum Kolmogorov-Smirnov distance
between the data and the fit. For any given value of $x_{min}$, we estimated
the scaling parameter using maximum likelihood estimation. The goodness of fit
for the truncated power law distribution was considered in comparison to the
fit of other distributions (e.g., power law, exponential, and lognormal).

Although some of the power laws present a more complicated picture, we then
used the methods developed in this [64], which carefully extend [61]
procedures for a wider class of distributions.

Figure 38: Kolmogorov-Smirnov (KS) test for Houston data, the blue dots
represent the distribution of the test statistics over the sampled data. The
mean is represented by a dashed black line. The 5th and 95th percentiles are
represented by dashed green lines, indicating the range within which 90% of
the observed values fall. Figure 39: Kolmogorov-Smirnov test for Singapore
data, the blue dots represent the distribution of the test statistics over the
sampled data. The mean is represented by a dashed black line. The 5th and 95th
percentiles are represented by dashed green lines, indicating the range within
which 90% of the observed values fall. Figure 40: Kolmogorov-Smirnov test for
San Francisco data, the blue dots represent the distribution of the test
statistics over the sampled data. The mean is represented by a dashed black
line. The 5th and 95th percentiles are represented by dashed green lines,
indicating the range within which 90% of the observed values fall. Figure 41:
Kolmogorov-Smirnov test for France data, the blue dots represent the
distribution of the test statistics over the sampled data. The mean is
represented by a dashed black line. The 5th and 95th percentiles are
represented by dashed green lines, indicating the range within which 90% of
the observed values fall.

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