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# Diffusion and synchronization dynamics reveal the multi-scale patterns of
spatial segregation

Aleix Bassolas  Departament d’Enginyeria Informatica i Matematiques,
Universitat Rovira i Virgili, 43007 Tarragona, Spain  Sergio Gómez
Departament d’Enginyeria Informatica i Matematiques, Universitat Rovira i
Virgili, 43007 Tarragona, Spain  Alex Arenas  Departament d’Enginyeria
Informatica i Matematiques, Universitat Rovira i Virgili, 43007 Tarragona,
Spain

###### Abstract

Urban systems are characterized by populations with heterogeneous
characteristics, and whose spatial distribution is crucial to understand
inequalities in life expectancy or education level. Traditional studies on
spatial segregation indicators focus often on first-neighbour correlations but
fail to capture complex multi-scale patterns. In this work, we aim at
characterizing the spatial distribution heterogeneity of socioeconomic
features through diffusion and synchronization dynamics. In particular, we use
the time needed to reach the synchronization as a proxy for the spatial
heterogeneity of a socioeconomic feature, as for example, the income. Our
analysis for 16 income categories in cities from the United States reveals
that the spatial distribution of the most deprived and affluent citizens leads
to higher diffusion and synchronization times. By measuring the time needed
for a neighborhood to reach the global phase we are able to detect those that
suffer from a steeper segregation. Overall, the present manuscript exemplifies
how diffusion and synchronization dynamics can be used to assess the
heterogeneity in the presence of node information.

##  I Introduction

The expansion of urbanization and progressive increase of the population in
cities has intensified the concern over the many dimensions of segregation
—i.e., school, economic or ethnics— that have a tangible impact in the health,
education and equal opportunities of citizens [1, 2, 3, 4, 5, 6, 7, 8]. In
fact, quantifying the extent of segregation and the identification of
economically and socially isolated neighborhoods has been a topic of wide
interest that first led to the development of global metrics, and which were
later extended to spatial metrics [9, 10, 11, 12, 13, 14]. Most of the initial
spatial measures were limited to first neighbour indices, which facilitated
the development of multi-scalar indices that provide a more nuanced picture of
segregation [15, 16, 17, 18, 19, 20, 21], yet understanding the role played by
each of the scales and their interplay still remains a challenge.

Dynamical processes in general, and in particular diffusion [22, 23, 24, 25,
26, 27, 28, 29, 30] and synchronization [31, 32, 33, 34, 35] dynamics, have
been widely studied in complex networks on account of their relation with the
spread of diseases and information [36, 37] and real-world phenomena in social
or economic systems [38, 39, 40]. Interestingly, they provide insights on the
topological scales and structure of networks and reveal the existence of
functional meso-scale structures [27, 30, 32, 34, 41].

Here we use previous knowledge on diffusion and synchronization dynamics to
assess the multi-scale patterns of residential segregation. By moving the
focus from the network topology and organization to the node states, we are
able to measure how well distributed a population with a certain
characteristic is using the time needed to reach the absorbing state. Our
framework requires thus the implementation of a population dynamic to drive
the system towards the homogeneous state, in our case diffusion and
synchronization dynamics. None of them constitute here an attempt to model or
predict the changes in the spatial distribution of a population characteristic
but are highly stylized simplifications of their evolution that allow us to
measure the time needed to attain the homogeneous state, which we consider to
be the non-segregated scenario. Dynamical approaches are thus introduced here
not because they provide a realistic approximation to the evolution of
population dynamics but because they offer a significant advantage to measure
multi-scale correlations as they do not require to take distance explicitly
into account. Moreover, the assumption that cities converge towards uniformity
is rather unrealistic without a heavy external driver, and is only a means to
construct our measures.

As case studies we provide an analysis on the distribution of citizens of a
certain income category in cities from the United States, and the distribution
of a set of socioeconomic indicators in the city of Paris throughout an
average day (see Supplementary Material Section 2 and Supplementary Figs.
S8-S10). The analysis on the spatial organization of income categories reveals
that the most deprived and affluent sectors display higher diffusion and
synchronization times linked to a higher heterogeneity, and allow us to split
the cities in two groups depending on the difference on the level of
segregation. Finally, we evaluate the level of synchronization at the
neighborhood level which allow us to spot the more sensitive places in a city.

##  II Results

###  II.1 Diffusion dynamics and income segregation

Citizens exhibit a huge diversity of characteristics usually captured by
socioeconomic indicators such as education level, income or ethnicity, and
they are often heterogeneously distributed in space: those individuals with
similar characteristics tend to live close between them. To assess the
heterogeneity of a population with a characteristic $k\in K$, we consider a
graph $G(V,E)$ with adjacency matrix $A=\\{a_{ij}\\}$ in which the spatial
units are represented as a set of nodes $V$ connected by a set of edges $E$.
The adjacency matrix $A$ we have considered takes $a_{ij}=1$ when spatial
units $i$ and $j$ are adjacent and $a_{ij}=0$ otherwise, which is the
traditional connectivity matrix used to capture residential segregation.
Still, other types of (weighted) matrices could be considered to assess, for
example, the impact of mobility in segregation. The state of a node
$x_{i}^{k}$ is given by the fraction of citizens living in node $i$ that
belong to socioeconomic category (or class) $k$, written as

$x_{i}^{k}=\frac{n_{i}^{k}}{\sum\limits_{k^{\prime}}n_{i}^{k^{\prime}}},$ (1)

where $n_{i}^{k}$ is the total number of citizens in unit $i$ that belong to
category $k$. As extreme cases, $x_{i}^{k}=0$ when there are no citizens of
category $k$ living in $i$, and $x_{i}^{k}=1$ when all the citizens in node
$i$ belong to category $k$. Of course, the normalization condition

$\sum_{k\in K}x_{i}^{k}=1$ (2)

is fulfilled for all nodes $i$.

To measure the multi-scalar patterns of segregation, our assumption is that
cities suffering from stronger residential segregation are further from the
stationary state where the citizens of category $k$ are homogeneously
distributed in space. Although cities are in continuous change and most likely
far from equilibrium, similar approaches such as the long-standing Schelling
and the Alonso-Muth-Mills models have been able to draw relevant conclusions
from the equilibrium state [42, 43].

By adopting diffusion dynamics we do not refuse the high complexity of
population dynamics influenced by a wide variety of demographic, economic,
political, and behavioral factors [44, 45, 46, 47] but avoid introducing
further parameters and factors that could hinder our aim of characterizing the
segregation of a particular population category. Bear in mind that our final
goal is by no means to assess real-world migration processes but to construct
a multi-scalar measure of segregation that does not explicitly include the
distance and the use of more complex and realistic approaches that would
complicate the interpretation of the results. Diffusion constitutes one of the
most basic approximations to how information, or any other characteristic, is
transmitted through a system. Although far from the real behavior, it provides
one of the simplest scenarios where the flow of population follows a gradient.

In fact, we focus on one of the best-case scenarios where the values of
$x_{i}^{k}$ converge towards equilibrium following a gradient, which could be
interpreted as the change of residence of citizens of category $k$ to regions
where they are less abundant.

Class | Income ($)
---|---
1 | Less than 10,000
2 | 10,000 – 14,999
3 | 15,000 – 19,999
4 | 20,000 – 24,999
5 | 25,000 – 29,999
6 | 30,000 – 34,999
7 | 35,000 – 39,999
8 | 40,000 – 44,999
9 | 45,000 – 49,999
10 | 50,000 – 59,999
11 | 60,000 – 74,999
12 | 75,000 – 99,999
13 | 100,000 – 124,999
14 | 125,000 – 149,999
15 | 150,000 – 199,999
16 | 200,000 or more
Table 1: Income range (in US dollars) corresponding to each category (or
class). Figure 1: Diffusion dynamics as a measure for income segregation. (A)
Synchronization time for each of the $16$ income categories in Boston,
Cleveland, Denver and Detroit. (B) Median value of $\widetilde{\tau}_{\rm
diff}(k)$ across income categories as a function of its variance. (C) Ranking
for the median value of $\widetilde{\tau}_{\rm diff}(k)$ for the studied set
of US cities.  Figure 2: Average temporal evolution of the abundance of
households within the lowest and highest income. Temporal evolution of
centroids after performing a k-means clustering on the normalized abundance of
households with category $k$, $P(x_{i}^{k},t)$, as a function of time $t$ for
the lower (A) and higher (B) income categories. Figure 3: Synchronization time
as a measure for income segregation. (A) Synchronization time for each of the
$16$ income categories in Boston, Cleveland, Denver and Detroit. (B) Ranking
for the median value of $\widetilde{\tau}_{\rm sync}(k)$ for the studied set
of US cities. (C) Average value of $P(\widetilde{\tau}_{\rm sync}(k))$ as a
function of each income category $i$ for the two main clusters detected. (D)
Location and cluster assignment for each of the analyzed cities.

We focus on the economic segregation in the metropolitan areas of the United
States with more than $1$ million inhabitants and analyze a dataset containing
the number of households within an income interval $k$ residing in each census
tract (see Table 1).

Once we have the set of initial node states $x_{i}^{k}$, their evolution
through time is determined by the diffusion dynamics

$\frac{dx_{i}^{k}}{dt}=\frac{1}{s_{i}}\sum\limits_{j=1}^{N}a_{ij}(x_{j}^{k}-x_{i}^{k}),$
(3)

where

$s_{i}=\sum\limits_{j=1}^{N}a_{ij}$ (4)

is the degree of node $i$. For simplification purposes, we have opted to use a
normalized diffusion dynamic, with diffusion strength equal to 1. Note that we
have independent diffusion processes for each category $k$.

The diffusion dynamic lasts until the stationary state, $x_{i}^{k}={\langle
x^{k}\rangle}$, $\forall i$ is reached, and we denote the spanned time as
$\tau_{\rm diff}(k)$. Since the time to reach the stationary state can be
infinitely large, we have considered that it is reached when the variance of
$x_{i}^{k}$, in time, becomes lower than $0.0001$. We hypothesize that lower
values of $\tau_{\rm diff}(k)$ are related to a more homogeneous distribution
of the population within a category $k$, and the other way around when it is
higher. In the extreme case in which all units have the same initial value of
$x_{i}^{k}$, the diffusion time $\tau_{\rm diff}(k)$ would attain its minimum
value. As we aim to compare cities with different characteristics, we control
for confounding factors such as the particular distribution of $x^{k}$ or the
topology of the graph by running the same diffusion dynamics on the same graph
but where the values of $x^{k}$ have been reshuffled, thus defining the
average null-model diffusion time $\tau^{\rm null}_{\rm diff}(k)$ calculated
over $500$ reshuffling realizations. The relative diffusion time we will use
throughout this manuscript can then be written as

$\widetilde{\tau}_{\rm diff}(k)=\frac{\tau_{\rm diff}(k)}{\tau^{\rm null}_{\rm
diff}(k)}.$ (5)

A relative diffusion time equal to one means that it is compatible with the
null model, i.e., there are no remarkable spatial dependencies, while a
greater value suggests that spatial heterogeneities delay the arrival to the
stationary state.

We analyze the normalized diffusion times $\widetilde{\tau}_{\rm diff}(k)$ by
running simulations for all US cities above $1$ million of inhabitants and
each of the $16$ income categories $k$ as a proxy for how heterogeneously
distributed is the population; we have excluded New York City, whose adjacency
network does not provide an accurate picture of residential segregation due to
the particular geography of Manhattan. In Fig. 1(A) we display
$\widetilde{\tau}_{\rm diff}(k)$ in Boston, Cleveland, Detroit and Denver
observing a common qualitative behavior: smaller values for middle-income
categories, and higher ones for the categories in the extremes of the income
distribution. Our results suggest that the wealthier and most deprived
citizens suffer from stronger segregation and display a more clustered spatial
distribution. More interestingly, category $9$ seems to be the more
homogeneously distributed across space, in agreement with the results observed
in [48] and with the mean and standard deviation of $x^{k}$ as well as the
Moran’s I (see Supplementary Material Section 1 and Supplementary Fig. S1).
Still, there are strong quantitative differences, with Cleveland and Detroit
displaying higher values for most of the categories, in contrast to Boston and
Denver.

Since $\widetilde{\tau}_{\rm diff}(k)$ takes a set of $16$ values for each
city, we calculate their median and variance values over all categories to
ease the comparison between the set of cities studied. While the median value
provides information on the segregation across all economic categories, its
variance reports the variability among them. Figure 1(B) shows this median
value of $\widetilde{\tau}_{\rm diff}(k)$, $\mbox{med}(\widetilde{\tau}_{\rm
diff}(k))$ as a function of its variance, $\mbox{var}(\widetilde{\tau}_{\rm
diff}(k))$. The prior cities appear ordered as Detroit, Cleveland, Boston and
Denver, although the variance is very similar for Cleveland and Boston, likely
due to the high values observed for low-income categories in Boston. Finally,
we provide in Fig. 1(C) the ranking of the selected US cities according to
$\mbox{med}(\widetilde{\tau}_{\rm diff}(k))$, as a measure of the overall
segregation in cities. On top of it, we find cities such as Milwaukee or
Detroit, which have been reported to suffer from economic and ethnic
segregation [49, 50, 51].

By applying diffusion dynamics we implicitly assume that $x^{k}$ evolves
homogeneously towards consensus, which more than a realistic scenario, it is a
means to calculate the time needed to reach consensus and obtain a measure of
segregation. To further inspect the actual change of $x_{i}^{k}$ between 2011
and 2019 in each of the spatial units $i$, we first construct the normalized
time-series for each spatial unit across those years as

$P(x_{i}^{k},t)=\frac{x_{i}^{k}(t)}{\sum\limits_{t^{\prime}}x_{i}^{k}(t^{\prime})},$
(6)

and then cluster, for each category $k$, the temporal profiles of all the
nodes. For the clustering, we have made use of the k-means algorithm [52, 53],
grouping together those units with a similar temporal evolution, and setting
the number of clusters to 3. The resulting time-series of the corresponding
centroids for the highest and lowest income categories are depicted in Fig. 2,
where a non-monotonic behavior is observed in most of the cases, with
oscillatory behaviors through time of varying amplitude.

Figure 4: Local synchronization time as a measure for income segregation.
Normalized synchronization time for each census tract in Denver (A–C) and
Detroit (E–G) for three different income categories: (A, E) class 1 (low
income); (B, F) class 8 (middle income); (C, G) class 16 (high income). We
provide as a reference the median income of each census tract in (D) Denver
and (H) Detroit.

###  II.2 Synchronization dynamics and income segregation

According to the oscillations in the temporal evolution of $x^{k}_{i}$ (Fig.
2), diffusion dynamics appear to be a rather simplistic approach to assess the
time needed to converge. Even thought we do not aim to mimic the real
evolution of $x^{k}_{i}$, we seek for a dynamic that at least can resemble its
real behaviour in a qualitative way. Thus, despite still constituting a
stylized approximation, a dynamical process with an oscillatory behavior, like
a system of coupled Kuramoto oscillators, appears to be a better way to assess
the spatial heterogeneity of socioeconomic indicators across cities. To
analyze segregation in terms of synchronization dynamics, we treat each of the
spatial units $i$ as an individual Kuramoto oscillator, with an initial phase
$\theta_{i}^{k}(0)$ that is set by distributing the fraction of population in
node $i$ that belongs to a category $k$ within the range $[0,\pi]$ as

$\theta_{i}^{k}(0)=x_{i}^{k}\pi.$ (7)

The interaction between spatial units is given by the Kuramoto model

$\dot{\theta}_{i}^{k}(t)=\omega_{i}^{k}+\frac{1}{s_{i}}\sum\limits_{j=1}^{N}a_{ij}\sin\left(\frac{\theta^{k}_{j}(t)-\theta_{i}^{k}(t)}{2}\right),$
(8)

where we have modified the traditional interaction term between oscillators by
dividing the angle difference by two, allowing for the interaction between
regions displaying extreme values of $x_{i}^{k}$. Additionally, to facilitate
the global synchronization of the system, we set all the individual natural
frequencies of the oscillators to the same value, i.e., $\omega_{i}=1,\
\forall i$. In order to account separately the segregation of each category
$k$, our approach assumes that there is no interaction between categories and,
thus, $x_{i}^{k}$ synchronize independently of $k$.

Figure 5: Synchronization time and median income. Normalized synchronization
time as a function of the median income averaged over bins of $5,000 for class
1 (A), class 8 (B) and class 16 (C).

We use the standard order parameter $|z^{k}|$ to assess the global level of
synchronization for a category $k$ in a city, where

$z^{k}=\frac{1}{N}\sum\limits_{j=1}^{N}e^{i\theta_{j}^{k}},$ (9)

and $N$ is the total number of spatial units or Kuramoto oscillators [35]. We
consider that a city has reached the synchronized state when $|z^{k}|>0.999$.
As in the case of diffusion, we assess how the distribution of initial phases
determines the synchronization of the system, a city in our case, by measuring
the time $\tau_{\rm sync}(k)$ required to reach the synchronized state. The
more heterogeneously distributed the initial phases are, the higher the time
the system requires to synchronize. To distinguish between the effect produced
by the spatial distribution $x_{i}^{k}$ from its overall distribution as well
as the topology of the graph, we also measure the average time the system
needs to synchronize when the same phases are redistributed at random,
$\tau^{\rm null}_{\rm sync}(k)$. The normalized synchronization time of the
system is then given by the ratio

$\widetilde{\tau}_{\rm sync}(k)=\frac{\tau_{\rm sync}(k)}{\tau^{\rm null}_{\rm
sync}(k)}.$ (10)

Like for diffusion, a synchronization time close to one means that the spatial
distribution of phases is compatible with the null model, and a larger value
indicates that spatial heterogeneities delay the appearance of a synchronized
state.

In Fig. 3(A) we inspect the normalized synchronization time in Boston,
Cleveland, Detroit and Denver when spatial units interact through Kuramoto-
like dynamics. All four of them share similar features, with central classes
displaying smaller synchronization times compared to the most disadvantaged
and wealthier ones. An expected result since those individuals in the extremes
of the income distribution tend to be more isolated and clustered together
compared to middle-income citizens. Despite sharing qualitative features, the
cities shown display sharp quantitative differences. Almost all categories
appear to be significantly more isolated in Detroit and Cleveland compared to
Denver and Boston, where $\widetilde{\tau}_{\rm sync}(k)$ looks much flat.
Overall, the synchronization results are compatible with the diffusion ones,
likely because both dynamical processes share common features. We have further
checked that the mean ${\langle x^{k}\rangle}$ does not determine directly the
normalized synchronization times in Supplementary Fig. S1.

Likewise with diffusion, we calculate the median and variance of
$\widetilde{\tau}_{\rm sync}(k)$ over all categories to be able to compare
between analyzed cities (see Supplementary Fig. S2 for the individual rankings
of $\widetilde{\tau}_{\rm sync}(k)$ for the categories $k=1$ and $k=16$). The
ranking is shown in Fig. 3(B) and has cities such as Detroit, Cleveland,
Milwaukee or Memphis close to the top, which are well-known for being among
the most economically segregated cities in the United States. The location in
the ranking of the cities in Fig. 3(A) is consistent with our observations,
with Boston and Denver on the bottom of the ranking and Detroit and Cleveland
on the top of it.

Our index is given by the median value of the normalized synchronization
times, yet depending on the dimension of segregation we aim to capture, we can
also construct an index based on a population-weighted average. Whereas the
median gives equal weight to each economic category focusing on the
segregation suffered by residents of category $k$, the weighted average
provides an overall picture of segregation taking the population of each
category into account. We show the ranking obtained for the weighted average
index and its relation with $\mbox{med}(\widetilde{\tau}_{\rm sync}(k))$ in
Supplementary Figs. S6 and S7. Additionally, we show in Supplementary Figs. S3
and S4 how $\widetilde{\tau}_{\rm sync}(k)$ significantly correlates with the
traditional Moran’s I [54] as well as a multi-scale quantity based on class
mean first passage times developed in [20, 21], reinforcing the idea that
synchronization (and diffusion) dynamics indeed capture the patterns of
residential segregation. Despite the dynamics we have used are stylized
versions of the real behavior of the quantity $x^{k}_{i}$ and do not capture
the full complexity of its temporal evolution, it is able to capture
segregation with values comparable to other segregation indicators.

Although $\widetilde{\tau}_{\rm sync}(k)$ is larger for extreme categories in
most of the cities, some of them like Denver display smaller variations than
others such as Detroit and, therefore, it might be of interest to group cities
according to the change in synchronization times. By running a k-means
algorithm on the normalized value of $P(\widetilde{\tau}_{\rm sync}(k))$ so
that $\sum_{k}P(\widetilde{\tau}_{\rm sync}(k))=1$, we can split the cities of
study between those with higher and smaller differences in
$\widetilde{\tau}_{\rm sync}(k)$, see Fig. 3(C). In Fig. 3(D), we display the
cluster assigned to each metropolitan area, where no strong spatial pattern is
observed. Still, the cities in the Midwest, which are known for being
economically segregated, fall into the red cluster, together with other cities
such as Baltimore or Los Angeles. If, instead, we focus on the blue cluster,
we have cities such as Sacramento or Washington D.C. Among the cities
discussed in Fig. 3(A), Denver falls into the group with more homogeneous
segregation (in blue) and the rest into the one with more unequal segregation
patterns (in red).

Beyond the global quantification of segregation, we can also evaluate the
local level of segregation of a concrete census tract $i$ at a given time step
$t$ by computing

$\rho^{k}_{i}(t)=\cos(\theta^{k}_{i}(t)-\Phi^{k}(t)),$ (11)

where $\theta^{k}_{i}(t)$ is the phase of unit $i$ at time $t$ and
$\Phi^{k}(t)$ is the average phase of all the oscillators in a city in a given
time $t$ [32]. When $\rho^{k}_{i}(t)>0.999$ we consider that oscillator $i$
has synchronized, from which we can obtain $\tau^{\rm loc}_{i}(k)$. However,
given that $\rho_{i}(\tau^{\rm loc}_{i}(k))$ can oscillate through time, we
only consider that a unit $i$ has reached the global synchronized state at a
time $\tau^{\rm loc}_{i}(k)$ when $\rho_{i}(t>\tau^{\rm loc}_{i}(k))$ does not
go below $0.999$ anymore, otherwise our methodology could fail to capture
long-range correlations. In order to provide a metric for each spatial unit,
simulations last until each of the spatial units have fullfiled the
synchronization criteria. Normalizing $\tau^{\rm loc}_{i}(k)$ by its null
model counterpart, it yields $\widetilde{\tau}^{\rm loc}_{i}(k)$, a measure of
the local synchronization time.

Figure 4 displays the normalized synchronization times for each of the census
tracts in Denver and Detroit, focusing on three very distinct income
categories: low income, Fig. 4(A,E); middle income, Fig. 4(B,F); and high
income, Fig. 4(C,G). To ease the comparison between income categories, the
range of values is common for all the maps, evincing the strong differences
between Detroit and Denver, especially for the low and high-income categories.
The shape of the segregation in Detroit can be outlined by the lower-income
downtown and the richer suburbs, being the most segregated parts, and a less-
segregated region in-between. In the case of Denver, we only slightly see high
values for the low-income category in the North of the city and the high-
income category in the South.

As we detail in Supplementary Fig. S5, the spatial patterns of segregation
product of the synchronization dynamics are significantly different to those
obtained from first-neighbor quantities such as the Moran’s I. Instead of
focusing on those regions whose proportion of citizens is high (or low)
compared to its neighbors, our methodology highlights those with a ratio of
population within a category $k$ distinct than the average, either because it
is high or low, and spatially isolated from those regions with average values.
In other words, a region with a high proportion of residents of category $k$
might not show a large local spatial correlation if their neighbors have
similar values but could, instead, produce high values of
$\widetilde{\tau}^{\rm loc}_{i}(k)$ if it is isolated from those regions
displaying a proportion of citizens closer to the city average. As the
majority of spatial measures, our approach can also suffer the so-called
modifiable areal unit problem [55] in a similar fashion. However, given that
our methodology captures mid and long-range correlations instead of local
differences, it might be less affected by such small local changes.

Finally, we inspect if the synchronization time of a region displays any type
of connection with its actual income. To do so, we plot in Fig. 5 the
normalized local synchronization time as a function of the median income
averaged over all the census tracts within bins of $5,000 in four US cities.
Again we see that segregation is much stronger in Detroit followed by
Cleveland and Boston. High-income regions are more segregated in Boston
compared to Cleveland. In general terms, the census tracts with a median
income between $50,000 and $80,000 seem to be the less segregated ones as they
synchronize faster for both low and high-income categories. These results are
in agreement with the cluster assignment of the previous cities, with Detroit,
Cleveland and Boston in the red cluster where low and high-income categories
need more time to synchronize, and Denver in the blue cluster where only the
high-income categories need more time to synchronize.

##  III Discussion

Traditional spatial segregation indicators that focus on local scale of
segregation fail in most cases to capture the presence of long-range
correlations, thus highlighting the need of multi-scale indices [15, 16, 17,
18, 20, 19, 21]. Our framework does not consider any specific scale, but uses
a dynamical approach that captures the patterns of segregation across the
multiple scales. We have revealed how categories in the extreme of the income
distribution are more heterogeneously distributed in space compared to middle
classes, displaying larger diffusion and synchronization times. This approach
has also allowed us to group together those cities that display common
features of segregation. In this context, it is important to note that our
work does not attempt to model the evolution of income segregation nor can be
used as a forecasting tool, but takes modeling assumptions to assess the level
of segregation that a distribution of population exhibits.

Despite the main manuscript focuses on the economic segregation, our
methodology can be used to assess the heterogeneity in the spatial
distribution of any characteristic. Moreover, it can go beyond the spatial
component of segregation by including in the analysis other types of graphs,
e.g., the daily mobility network of citizens. In this way, we could assess how
citizens of diverse socioeconomic environments interact through mobility [56,
57, 58, 59].

Summarizing, we show how diffusion and synchronization dynamics can be used in
some systems to assess the heterogeneity in the distribution of node features.
While the present work focuses on the initial phases of oscillators and their
synchronization time, node metadata could also be understood as an internal
frequency and provide further insights on feature correlation across
topological scales.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

A.B. performed the research. A.B. S.G. and A.A. designed the research and
wrote the manuscript.

## Acknowledgments

A.B. acknowledges financial support from the Ministerio de Ciencia e
Innovación under the Juan de la Cierva program (FJC2019-038958-I). We
acknowledge support by Ministerio de Economía y Competitividad
(PGC2018-094754-BC21, FIS2017-90782-REDT and RED2018-102518-T), Generalitat de
Catalunya (2017SGR-896 and 2020PANDE00098), and Universitat Rovira i Virgili
(2019PFR-URV-B2-41). A.A. acknowledges also ICREA Academia and the James S.
McDonnell Foundation (220020325).

## Data Availability Statement

The income data analyzed in the present text can be found at [60].

##  Section S1 Supplementary results for diffusion and synchronization
dynamics and economic segregation

We provide here supplementary results related to the study of income
segregation in US cities. Figure S1 reports (A) the mean and (B) standard
deviation of $x_{i}^{k}$ in Boston, Cleveland, Detroit and Denver. Both of
them reach minimum values between 8-10.

Supplementary Figure S1: (A) Mean of $x^{k}$ for each income category in
Boston, Cleveland, Detroit and Denver. (B) Standard deviation $\sigma(x^{k})$
for each income category in Boston, Cleveland, Detroit and Denver. (C) Moran’s
I for each income category in Boston, Cleveland, Detroit and Denver. (D)
Scatter plot of the mean of $x^{k}$ as a function of $\widetilde{\tau}_{\rm
sync}(k)$.

The fact that classes 8-10 appear to be the less segregated is also supported
by the Moran’s I as Fig. S1(C) shows. To further assess that the mean
$<x^{k}>$ does not strongly determine the values of $\widetilde{\tau}_{\rm
sync}(k)$, we plot both quantities in Fig. S1(D), where no strong pattern is
observed. Categories with low $<x^{k}>$ display high variability in
$\widetilde{\tau}_{\rm sync}(k)$ and vice-versa.

In Fig. S2 we provide the ranking of the selected US cities according to the
value of $\widetilde{\tau}_{\rm sync}(k)$ for the lowest and highest income
categories 1 and 16, respectively. As can be seen, there are significant
variations in the ranking depending on which economic category is shown; for
example, Cleveland is close to the top for category 1 but far apart for 16,
and the other way around for Seattle.

A Class 1
---
B Class 16
Supplementary Figure S2: Ranking of the selected US cities according to the
value of $\widetilde{\tau}_{\rm sync}(k)$, for income class 1 (A) and income
class 16 (B).

##  Section S2 Comparison with other segregation measures

In this section we assess how the normalized synchronization time
$\widetilde{\tau}^{k}_{\rm sync}$ relates to other segregation measures. In
particular we focus on the widely used Moran’s I [54], which focuses on local
correlations, and one obtained from class mean first passage times (CMFPT)
developed in [20, 21], which captures long-range spatial correlations.

For each city and category $k$ the Moran’s I can be written as

$I^{k}=\frac{\displaystyle\frac{1}{W}\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(x^{k}_{i}-\bar{x}^{k})(x^{k}_{j}-\bar{x}^{k})}{\displaystyle\frac{1}{n}\sum_{i=1}^{n}(x^{k}_{i}-\bar{x}^{k})^{2}},$
(S1)

where $x^{k}_{i}$ is the fraction of population in $i$ that belongs to
category $k$, $\bar{x}^{k}$ is its mean across all spatial units, the weights
$w_{ij}$ correspond in our case to the spatial adjacency matrix $a_{ij}$, and
$W=\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}$ is the total weight.

As an index to assess the long-range correlations in the spatial distribution
of the income categories, we will use the class mean first passage times
between classes. In this methodology [20, 21], random walkers start from each
of the spatial units in a system and move through the spatial adjacency graph
until they have visited the $16$ classes at least once. For this, each
location is assigned to a class with probability proportional to its
corresponding fraction of population. By averaging the number of steps that a
walker needs to reach class $j$ across all the units that belong to category
$i$ and for multiple realizations, we can obtain the class mean first passage
times $\tau_{ij}$, which encapsulate the average number of steps needed to
reach a unit of category $j$ when a walker departs from a unit of category
$i$. After normalizing by a null-model in which colors are uniformly
reshuffled at random to compensate for uneven class abundances, we finally
obtain the normalized class mean first passage times $\widetilde{\tau}_{ij}$.
The quantity $\widetilde{\tau}_{ij}$ provides thus information on how much
time you need to reach category $j$ when a walker departs from a unit of
category $i$ as compared to the null-model, values below $1$ mean that two
categories are closer than in the null-model and vice-versa for values above
$1$. To summarize the segregation of category $k$ in a city we will use the
CMFPT index, i.e., the $\mbox{med}(\widetilde{\tau})_{k}$ given by the median
value of $\tau_{jk}$ $\forall j$.

For each city included in our analysis, we measure the Pearson correlation
coefficient $r_{p}$ between each of the additional segregation quantities and
$\widetilde{\tau}^{k}_{\rm sync}$ for all the $16$ categories $k$. More
specifically, for each city $r_{p}$ is calculated over a set of $16$ points.
The distribution of $r_{p}$ across cities is shown in Fig. S3 for the Moran’s
I (A) and $\mbox{med}(\widetilde{\tau})_{k}$ (B), where a skewness towards
high values is clearly observed. Most of the cities display correlations above
$0.8$ with the Moran’s I and $0.7$ with the CMFPT index. Additionally, we also
show in Fig. S4 the significance of the correlations observed in each of the
cities, which are also below $0.001$ in most of the cases.

Supplementary Figure S3: Correlation between $\widetilde{\tau}^{k}_{\rm sync}$
and the additional segregation indicators. For each city in our study, we
calculate the Pearson correlation coefficient $r_{p}$ between
$\widetilde{\tau}^{k}_{\rm sync}$ and the additional segregation metrics over
the $16$ income categories. The correlation coefficient for a city is thus
obtained from a set of $16$ points, one per category. (A) Distribution of
$r_{p}$ between Moran’s I and $\widetilde{\tau}^{k}_{\rm sync}$ across cities.
(B) Distribution of $r_{p}$ between the segregation calculated through
normalized CMFPT $\mbox{med}(\widetilde{\tau})_{k}$ and
$\widetilde{\tau}^{k}_{\rm sync}$ across cities.

Supplementary Figure S4: Significance of the Pearson correlation coefficients
between $\widetilde{\tau}^{k}_{\rm sync}$ and the other segregation
indicators. For each of the additional indices, we display the significance of
the correlations across cities. (A) Significance of correlations between
Moran’s I and $\widetilde{\tau}^{k}_{\rm sync}$. (B) Significance of
correlations between the segregation calculated through normalized class mean
first passage times $\mbox{med}(\widetilde{\tau})_{i}$ and
$\widetilde{\tau}^{k}_{\rm sync}$ . The correlation coefficient and
significance for each city is obtained by comparing the segregation values for
the $16$ income categories. The significance values are depicted as * for
$\mbox{p-value}<0.05$, ** for $\mbox{p-value}<0.01$, and *** for
$\mbox{p-value}<0.001$.

In the main text we discuss the potential of our methodology to assess the
multiscale patterns of segregation in front of traditional first-neighbor
approaches. In Fig. S5 we further investigate this fact by plotting for
Boston, Cleveland, Denver and Detroit the local normalized synchronization
times, the local Moran’s $I^{\rm loc}_{i}(k)$, and the raw ratio of population
of category $k$ in each of the census tracts.

Although the segregation hotspots detected by our methodology and the local
Moran’s I seem similar, the patterns detected are significantly different.
Whereas $I^{\rm loc}_{i}(k)$ captures strong differences between neighboors,
$\widetilde{\tau}_{i}^{\rm loc}(k)$ highlights isolated regions even if the
differences with their first-neighboors is low; most likely, this is because
they are far apart from regions displaying ratios of population closer to the
city average and require more time to reach the global synchronized state. In
fact, the areas highlighted by synchronization dynamics have a larger scale
and allow us to identify common mesoscale patterns of segregation across
cities: a downtown that displays high values, a ring around it with low
values, and finally the suburbs with high values again. By focusing on
Detroit, we can see that not only the poorer downtown appears highlighted but
also the suburbs due to their very low ratio of population of category $1$.
Similar patterns can also be observed in Cleveland and Denver.

Supplementary Figure S5: Comparison of local segregation indicators in Boston,
Cleveland, Denver and Detroit. (A) Normalized synchronization time, (B) Local
Moran correlation, and (C) proportion of citizens for each census tract and
income class 1 (most deprived).

The segregation index developed in the main text is calculated as the median
of $\widetilde{\tau}^{k}_{\rm sync}$ which confers an equal weight to each of
the income categories, disregarding the amount of population in each category.
However, we can also construct a weighted index $\bar{\widetilde{\tau}}_{\rm
sync}$ that can be built as

$\bar{\widetilde{\tau}}_{\rm
sync}=\frac{\displaystyle\sum_{k}P_{k}\widetilde{\tau}_{\rm
sync}(k)}{\displaystyle\sum_{k}P_{k}},$ (S2)

where $P_{k}$ is the total number of citizens that belong to category $k$ in a
given city. The ranking of cities according to the value of
$\bar{\widetilde{\tau}}_{\rm sync}$ (Fig. S6) displays only slight changes
with, for example, Philadelphia and Los Angeles closer to the top of the
ranking. We test the relation between both indices in Fig. S7, where a clear
relationship between both quantities is revealed.

Supplementary Figure S6: Segregation in US cities according to an index
calculated through a weighted average. Ranking of cities according to the
weighted index of segregation $\bar{\widetilde{\tau}}_{\rm sync}$.
Supplementary Figure S7: Comparison between segregation indicators obtained
through synchronization dynamics. Comparison between the weighted index of
segregation $\bar{\widetilde{\tau}}_{\rm sync}$ and the index
$\mbox{med}(\widetilde{\tau}_{\rm diff}(k))$ used in the main text.

##  Section S3 Beyond economic segregation: Paris around the clock

Besides only economic segregation, our methodology can be used to assess the
spatial heterogeneity of any other quantity, and to exemplify it, we assess in
this section the segregation of the population in Paris according to a wide
set of socioeconomic indicators. The data compiles the fraction of population
per district within a certain category at each hour of the day in French
cities; in this work, we focus on Paris [61, 62, 63]. The list of indicators
and categories analyzed can be found in Table Supplementary SI.

Indicator | Categories |  |  |  |
---|---|---|---|---|---
Activity type | At home | At work | Studying | Shopping | Leisure
Age | 16-24 | 25-34 | 35-64 | 65 and more |
Educational level | Low | Middle-low | Middle-high | High |
Socioprofessional status | Inactive | Low | Middle-low | Middle-high | High
Last travel mode | Pub. trans. | Private motor | Soft mobility |  |
Occupational status | Active | Student | Unemployed | Retired | Inactive
Sex | Male | Female |  |  |
Table Supplementary SI: Socio-economic indicators and activity types analyzed
for Paris.

For each indicator or category, we have a certain distribution of population
per spatial unit and hour of the day, thus we can compute how the quantity
$\widetilde{\tau}_{\rm sync}(k)$ varies during the day, as we show in Fig.
S8(A,B) for the five activity types, and the five socio-professional status;
the patterns of synchronization through time turn out to be very distinct. For
example, the level of synchronization remains basically constant throughout
the day for low, middle and high socio-professional status, while it increases
(decreases) between 8am and 8pm for inactive (high) socio-professional status.
If we focus instead on the ranking of $\widetilde{\tau}_{\rm sync}(k)$ at 10am
and 10pm, see Fig. S8(C), the lower occupational and socio-professional status
seem to be the most segregated indicators as they are on top of the ranking at
both times of the day. Other categories that should be uniformly distributed
across the city, such as sex, are very close to $1$, thus indicating no
segregation.

Supplementary Figure S8: Synchronization around the clock in Paris. (A)
Normalized synchronization time for the distribution of population performing
each of the five types of activities. (B) Synchronization time for the
distribution of population of each socio-professional status. (C) Change of
synchronization times for all of the indicators at 10am (green) and 10pm
(red). Supplementary Figure S9: Clustering analysis of segregation around the
clock in Paris. (A) Pattern of synchronization times $P$ for each of the four
main groups detected with the K-Means algorithm. (B) Cluster assignment for
each of the indicators analyzed. Supplementary Figure S10: Local
synchronization around the clock in Paris. (A, B) Normalized synchronization
time for each Paris district for the population performing leisure activities
at 10am and 10pm. (C, D) Normalized synchronization time for each Paris
district for the population with inactive socio-professional status at 10am
and 10pm. For visualization purposes the color range is common to all four
maps.

The hourly patterns of each metric allow for the grouping of indicators
behaving similarly as we did for US cities. As before, we focus more on the
time-series profile rather than the specific values taken by bon
$\widetilde{\tau}^{h}_{\rm sync}(k)$, thus analyzing the normalized
$P(\widetilde{\tau}^{h}_{\rm sync}(k))$ for each hour of the day $h$. The
k-means clustering reveals four distinct clusters (see Fig. S9) which
correspond to: those increasing during workings, those decreasing, those
remaining almost constant, and those with a more characteristic behavior with
a peak during midday and at the end of the day, roughly around the lunch and
dinner times.

Finally, we assess the local segregation of districts by measuring their local
normalized synchronization time. In particular, we show an example in Fig. S10
for the population performing leisure activities and those with inactive
socio-professional status. In agreement with the temporal pattern shown in
Fig. S8, the segregation is much higher at 10pm compared to 10am, especially
concentrated in the centre of the city; a not so surprising result given that
most of the leisure activities are concentrated in that part of the city. In
the case of the population with inactive socio-professional status, the
hotspots seem to be concentrated in the northern part of the city, a region
known for suffering a thriving inequality.

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