# The distorting lens of human mobility data

Riccardo Gallotti<EMAIL_ADDRESS>CoMuNe Lab, Fondazione Bruno Kessler, Via
Sommarive 18 38123 Povo (TN), Italy  Davide Maniscalco  CoMuNe Lab, Fondazione
Bruno Kessler, Via Sommarive 18 38123 Povo (TN), Italy  Marc Barthelemy
Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191,
Gif-sur-Yvette, France  CAMS (CNRS/EHESS) 190-198, avenue de France, 75244
Paris Cedex 13, France  Manlio De Domenico  CoMuNe Lab, Fondazione Bruno
Kessler, Via Sommarive 18 38123 Povo (TN), Italy  Department of Physics and
Astronomy “Galileo Galilei”, University of Padua, Italy

###### Abstract

The description of complex human mobility patterns is at the core of many
important applications ranging from urbanism and transportation to epidemics
containment. Data about collective human movements, once scarce, has become
widely available thanks to new sources such as Phone CDR, GPS devices, or
Smartphone apps. Nevertheless, it is still common to rely on a single dataset
by implicitly assuming that it is a valid instance of universal dynamics,
regardless of factors such as data gathering and processing techniques. Here,
we test such an overarching assumption on an unprecedented scale by comparing
human mobility datasets obtained from $7$ different data-sources, tracing over
$500$ millions individuals in $145$ countries. We report wide quantifiable
differences in the resulting mobility networks and, in particular, in the
displacement distribution previously thought to be universal. These variations
– that do not necessarily imply that the human mobility is not universal –
also impact processes taking place on these networks, as we show for the
specific case of epidemic spreading. Our results point to the crucial need for
disclosing the data processing and, overall, to follow good practices to
ensure the robustness and the reproducibility of the results.

## Introduction

In the past two decades, access to the rich high-resolution datasets was
exclusive to a rather limited research community, which nevertheless produced
a series of important results about universal features of mobility behavior
[1, 2, 3, 4, 5, 6, 7]. The potential impact of human mobility studies is
enormous, as exemplified by most COVID-19 studies which rely heavily on
mobility data [8, 9]. The recent need for tackling promptly the problems
associated with this pandemics both from the epidemiological and socio-
economical perspective has been answered by many IT companies, which activated
a number of ‘Data For Good’ programs and collaborations. This unprecedented
access to mobility data by a large number of research groups led to the
publication of a series of important results about human behavior and, more
recently, dynamics of interests such as the spreading of the COVID-19 [10, 11,
12, 13, 14, 15, 16, 17]. However, even if this sudden abundance of dataset
provided scientists with many research opportunities, it also comes with new
challenges. Without any doubt, an easier access to data favored the
possibility of performing innovative data-informed analysis, in some cases
across multiple countries and with a high temporal resolution. These recent
studies also allowed in many cases their full reproducibility by providing
their codes and datasets. However, for companies to be able to provide large
amounts of data, in almost real time, they had to overcome multiple
limitations, and since the methods used are proprietary, the data pipeline
cannot be openly released. As a consequence, important details might not be
disclosed, often leaving to the final user the task to use data under
uncertain, or even unknown, gathering methodology. Consequently, the lack of
full knowledge about the measurement and processing details might prevent
scientists from correctly interpreting the outcome of their analysis.
Moreover, not being able to be in control of the data elaboration from the raw
to their final, analyzable, form, limits the ability of researchers to tailor
the data to the research question they have in mind. Even more importantly,
each data provider extracts information using a different gathering and
processing methodology, raising the question to which extent different data
sets are in agreement with each other when used to model human mobility and
derived dynamical processes, such as traffic or epidemics. The implicit
assumption made is that the characteristics of the shared data is marginally
affected by the details of the underlying methodology. Such an assumption is
often explicitly acknowledged as a study limitation [18, 19, 11], but it has
been never systematically tested or verified, to date, and consequently, the
extent of these limitations underlying our understanding of human mobility and
their impact on modeling dynamics with high societal impact are essentially
unknown.

Here, we illustrate both the opportunities provided and the limitations faced
while studying mobility data coming from seven different providers (including
both openly accessible and closed data): Google, Facebook, GPS based
applications (Cuebiq, Safegraph), Mobile phones, vehicle GPS blackboxes and
public census. We first provide a short description of these datasets and more
details and discussions – in particular their limitations – can be found in
the Methods section.

## Results

### Overview of the data

We considered a diverse set of mobility datasets, accounting for the movements
of over 500 millions individual in 145 countries. Specifically, we used census
data which provide statistical information about the urban and inter-urban
commuters’ flows. We collected and analyzed official census data for the US at
the county level [20] and for Italy aggregated at the province level [21]. We
also used data coming from mobile phone data call detail records (CDR), used
to reconstruct the displacement distribution function for the US on the basis
of published fitting parameters [1], and those for Portugal, Spain and France,
from the data published for reproducibility purposes [22]. Another source of
mobility data comes from small blackboxes equipped with GPS trackers and
accelerometers that are installed in private vehicles for insurance reasons
and record trajectory data, and here we analyzed the urban and inter-urban
displacement statistics of Italian drivers as in [4]. We studied the Google
datasets that are elaborated from the opt-in service ‘Google location
history’, where the trajectory data captures the movement of Android
smartphone users between pairs of stop locations. We used the displacement
statistics released by the authors of [19] and which covers almost all
countries. Another important source of data is provided by Facebook and
obtained by tracking the movements of mobile phone users that opted-in to the
Location History and Background Location collection services [23]. Here, we
use the mobility network where the nodes represent administrative areas and
the edges correspond to the long-range mobility flows between these areas. The
flows are aggregated every 8 hours and we further aggregated them on a weekly
basis. If a flow is below a certain threshold (that we estimate being of order
10 users), it is not reported in the data. For epidemic simulations, we also
used, as a proxy for population, the baseline values describing the number of
Facebook active users in a given node, similarly averaged over a whole week.
The fifth dataset studied here comes from Cuebiq Inc., which collects mobility
data of anonymized mobile app users who opted-in to a large number of
different location based services in different countries including the US, the
UK, Italy, Spain, France and Germany. Here, we use Cuebiq-HDR mobility data
for Italy elaborated by Pepe and collaborators from the device-level data, and
aggregated weekly at the province scale [24]. We also use flows describing the
movements across the US, computed directly by Cuebiq and provided within the
framework of the Data4Good program. The flows are here also aggregated weekly
and at the county scale. We computed the baseline flows by aggregating them
over the weeks preceding the beginning of lockdowns in Italy and the US,
rescaled the flows by the country total population and used the associated
census population for epidemic modelling. Finally, the last dataset used in
our study is provided by Safegraph which collects statistics about visiting
patterns of different points of interests (PoI) by aggregating anonymized
location data from mobile applications. The data covers the US only, where the
home location is identified for each user at the level of census block group.
Here, we will use the flows aggregated weekly and at the county level [25],
and a two month period before the US lockdown to define the baseline flows.x

Figure 1:  The baseline mobility networks and displacement distributions for
the US and Italy. The baseline mobility networks captured for the US using (a)
Cuebiq mobility insight data, (b) Safegraph, or (c) official census commuting
flows display radically different structures. Whereas in the Cuebiq data the
data mostly comprises short-mid distance displacements characteristics of
daily mobility, both Safegraph and the Census data include a large fraction of
long trips that can be associated with the movements between different
regions. The baseline mobility network for the Italy using Cuebiq HDR data
(d), Facebook data (e) or official census commuting flows (f) also displays
remarkable differences: the Census data includes a larger fraction of short
trips while the network described by Facebook has a significantly smaller
number of connections. In the bottom panels we illustrate the Complementary
Cumulative Density Function (CCDF) of the displacement distribution $P(L)$ for
the US (g) and Italy (h), computed for the datasets discussed above, and
compare them with public flows derived from Google Location History data [19]
and to what is observed for the US using mobile phone CDR [1]. In Italy we use
measures provided by Octotelematics’ private vehicle’s GPS blackboxes [4].
Since these datasets have different granularity, all CCDF have been computed
only for $L>40$ km. The resulting distributions display a huge variability
between datasets, with Cuebiq and Mobile data characterised by a shorter tail
in the US data, while Google data displays a curve suggesting the presence of
two separate scales. In addition, Safegraph and the US census clearly display
a totally different mobility scale. In Italy, the shorter tail is instead
associated to Census and Octotelematics data, and Google, Cuebiq and Facebook
are displaying a similar long-tail trend.

### The weak universality of the displacement distribution

We first consider the distribution $P(L)$ of displacement distances $L$
computed with various datasets. This distribution has been intensively studied
during the last 15 years [1, 2, 4, 19], and represents one of the facets of
human behaviour which have been many times referred to as universal according
to available empirical evidence [1, 2, 26]. Moreover, studying $P(L)$ allows
for comparing our results with published data where the analysis has been done
on the raw data. [27, 4, 28]. The tail behaviour of $P(L)$ varies across
countries [19], as it is product of the combination of mobility acting at
multiple scales [4, 5] while also combining individual contributions that
differ by both by gender [5] and socio-economic group [6, 15]. These
differences reflected in the specific parameters of the functional forms taken
by these distributions, that are typically analysed in aggregated form (see
[4, 28, 19] and references therein).

In Fig.1, we show the mobility networks and displacement distributions for the
US and Italy computed using the different datasets described above. In panels
a-f we display the networks obtained for different datasets describing the
baseline mobility across US and Italy. Due to the processes used for creating
these datasets, the mobility networks are characterized by very different edge
densities (see Table 1 in Methods) which is visually manifest here (additional
measures and details can be found in the Supplementary Fig. 1). We then
compute the complementary cumulative (CCDF) of the displacement distribution
$P(L)$ (see Fig 1 g,h). This distribution has the advantage of capturing
simultaneously information about the distances covered and the associated
flows.

The CCDF of $P(L)$ is evaluated by progressively summing up the fraction of
flows up to a distance $\leq L$. Depending on the type of datasets, these
summations are made either by counting the number of individual displacements
shorter than $L$ or by summing up flows passing through edges connecting
distances shorter than $L$. The latter is the case of all the data analyzed
here, with the exception of the curves computed in other papers directly from
mobile phone CDR [1] or GPS blackboxes [4].

Figure 2:  Comparison of the displacement distributions across datasets. We
estimate the fitting parameters of $P(L)$ (scale and exponent) for various
countries using a truncated power law. We observe a wide variance of these
parameters. We include the results for the Figure 1 for USA –Cuebiq (cb),
Mobile (mb) [1], Google (gg), Safegraph (sg), Census (cs) – and Italy (as
illustrated in Figure 1 it also includes Octotelematics and Facebook (fb)
data), the values for the longitudinal weekly analysis of Cuebiq and Facebook
data, and the values estimated for the Google and Facebook baseline. We
observe that all data points fall along a single curve, which suggests strong
correlation between the free parameter and, ultimately, the existence of an
underlying law having a single degree of freedom. This curve (dashed line) is
estimated with a fit of the form $y=\frac{A}{1+x^{-1/B}}-C$ where $A\approx
22\pm 1$, $B\approx 1.7\pm 0.1$ and $C\approx 20\pm 1$. We note how the USA-
long data points (sg, cs) seem to deviate strongly from the average curve.
This is because as they are capturing a behaviour totally different from the
other sources of data.

These CCDFs, together with data coming from other aforementioned datasets, are
compared in panels Fig 1 (g,h), where we observe very large deviation
depending on the dataset and the processing methodology (see Supplementary
information for more details) used for analyzing the mobility of the same
country. Similar comparisons are made in Supplementary Fig. 2 for France,
Spain and Portugal using Facebook and Google data along with mobile phone data
coming from [22]), that also . Remarkably, the observed differences appear
larger than the changes associated with the changes in mobility due to
lockdowns in the US and Italy in 2020 (see Supplementary Fig. 4), which
represented, in some sense, a huge natural experiment and which we could
safely consider as ‘major disruptions’. We fit the curves for $P(L)$ to
compare Google and Facebook data across 58 countries and used different
functional forms for the fit (see Methods and Supplementary Fig. 3 where we
observe, again, large differences between the curves observed in different
datasets). We will discuss here the case of the truncated power law (TPL) form
with two parameters. In Fig. 2, we display these fitting parameters for 143
Google curves, 88 Facebook curves, the curves described in Fig 1, and the time
dependent curves of Supplementary Fig. 4. We see that the parameters of the
TPL (but also for the other fitting models, see Supplementary Fig. 5) are
distributed along a relatively narrow curve (that can be approximated by a
sigmoidal function), suggesting the possibility of an underlying weakly
universal behaviour, where the curve exponent is a function of the scale.
These results demonstrate the very large variability of mobility networks over
different countries and data providers, pointing out differences that might
bias models which are known to be sensitive to mobility, such as epidemics
spreading. Additionally, they point to limitations of the different data
design and the need for enhancing data collaborations to ensure the
reproducibility of scientific studies based on mobility networks data.

Furthermore, an international comparison based on Facebook and Google data
across 61 countries reveals that the expected universality of human
displacement is weak, as significantly different tail behaviours are observed
in the smaller and larger countries (see Supplementary Figure 6). While we are
still convinced that the underlying physical process driving individual
displacements is universal [4], bounding it to the geographical constraint of
national borders clearly influences the aggregated $P(L)$. This effect, added
to the data biases coming from both the socio-economical characteristics of
behaviour captured by the data, and the choices made in the processing of
these data, renders the characterization of individual mobility from the
distribution $P(L)$ of a single country not trivial.

### The large sensitivity of spreading processes

The long range mobility information captured in individual trajectories
datasets are used for a variety of applications, such as seasonal changes in
population distribution [29], migration [30], tourism [31], and international
mobility [32]. As a representative case study, we focus our attention on a
class of models where human mobility plays an important role: the spreading of
an infectious disease. To better understand the consequences of the
aforementioned variations observed on $P(L)$, we consider a standard
metapopulation SIR model (see [33] and the Methods section), where spatial
patches plays the role of nodes and human flows connect them. The choice of
this process is justified by the high societal impact it has, since human
mobility is a standard target for non-pharmaceutical interventions – such as
curfews and lockdowns – to mitigate the evolution of infectious disease
epidemics [34, 35, 36, 37]. Also, it is worth remarking that this class of
models is emblematic and widely adopted for practical applications: in
principle, other models could be considered as well, but they might depend on
a larger number of parameters and hyperparameters, thus making more difficult
the analysis of the impact of mobility data on the final results. The choice
of a metapopulation SIR keeps the number of parameters small while still
providing a valuable model used for realistic infectious diseases.

We start with the Facebook data for 71 countries where at least 10 nodes are
represented in the network. We first observe (see Supplementary Fig. 7a) that
the average degree of the networks appear to be a (sub-linearly) growing
function of the number of users, which means that more populated countries
have denser networks. This is most likely due to a pruning procedure that
removes edges with smaller flows for both privacy enhancing and disk space
reduction. Networks with higher average degree have a larger fraction of
shortcut nodes (as can be seen in Supplementary Fig. 7b, where shortcuts are
defined as links at least twice as long as the average distance between a node
and its closest neighbour). This biased pruning heavily influences the flows,
and imposes in particular strong cutoffs on the longer connections (see more
discussions about this point in Supplementary Fig. 8, where we use a toy model
in order to analyse the relation between the density of edges and the
displacement distribution). As a consequence of tampering the displacement
tails, the spreading behavior on mobility networks ranges from lattice-like to
a small-world behaviour, depending on the dataset processing details. Also,
with less shortcuts, the spreading dynamics is heavily penalized in networks
captured with smaller user-bases and smaller average degrees as it is indeed
observed in simulations (see again Supplementary Fig. 7).

The differences between the spreading dynamics in the US and Italy computed
with different datasets is shown in Fig. 3. We first observe that for the US
Safegraph and Census data, the results are exactly the same (figures shown in
Supplementary Fig. 9). In contrast, simulations done with the Cuebiq data,
being at a shorter scale, produce a more complex spreading behaviour with two
peaks. In Supplementary Fig. 10, we show how this first peak is characterised
by a localized behaviour reflecting a ‘lattice like’ spreading [38]. Italian
data displays less differences, with the peak that appears not delayed but
rather lowered for shorter scale Facebook and Census data.

To further investigate the differences observed between the US and Italy, we
use the Cuebiq dataset and create a set of networks by pruning flows below an
increasing threshold. In Fig. 3(a,c), we show that the displacement
distribution remains unchanged. However, despite the small effect on $P(L)$,
the pruning has a very important impact on the spreading dynamics (Fig.
3(b,d)) that appears to be progressively delayed as the pruning is increased
[38]. This is particularly true for the US case, while the Italian case -
being at a smaller spatial scale - is not as sensitive to the pruning which
induces a drop in the peak but not a delay. The differences observed in Fig. 3
as a consequence of the pruning are, again, smaller with respect to the
differences observed when comparing across datasets and countries. This fact
can be better appreciated in Supplementary Fig. 11, showing how the relative
incidence at peak simulated with the SIR model are determined by the
characteristics of the $P(L)$.

(a) |  (b)
---|---
(c) |  (d)
Figure 3: Effect of flow pruning on the spreading dynamics. In these figures,
the notation $p_{x}$ indicates the edge pruning with flow smaller than $x$. In
(a) and (c) we show by examining the CCDF of the displacement distribution
that the pruning of flows does not affect significantly $P(L)$. Despite the
small effect on the displacement distribution, we observe very different
dynamics for disease spread on the corresponding mobility networks, in
particular in the US case (b), while in the Italian case (d), the impact on
the dynamics is marginal, probably due to the smaller size of this country.

## Discussion

While aware that data gathering and processing techniques used for human
mobility are different, rather unexpectedly we have found extremely wide
differences between datasets of human movements which are commonly used
interchangeably as a proxy for universal human behavior, and genuinely
supposed to give comparable results. On the one hand, our empirical findings
from the analysis of multiple datasets call for a more transparent access to
data and its protocols. On the other hand, our findings demonstrate that many
of the limitations that were only supposed to affect results do, in practice,
significantly affect the insights about human behavior that can be obtained
from mobility data.

Specifically, our study demonstrates that it is not sufficient to analyze
human mobility and its multifaceted aspects, such as universal dynamics, by
means of a single dataset: ensembles are needed to allow for a careful
evaluation of potential biases in data gathering and processing protocols. The
first step is to become more aware than before of such a critical impact: in
particular, we noticed a series of technical aspects about these various
datasets, that are often overlooked by both their producers and by scientists
using them (see the SI for more details). In this respect, our analysis
provides an overview of the limitations inherent to empirical studies and
point towards the urgent need for setting shared standards in the design and
elaboration of human mobility datasets. The focus was recently on how to model
complex socio-technical systems [39], but their complexity also affects their
empirical analysis, implying a special care is needed for their preparation
due to their high sensitivity to specific processing.

Mobility datasets being passively collected from ICT data records, are
necessarily influenced by the behaviour that produces the record itself [40,
41, 42, 43]. The method used is often undisclosed and results can be very
sensitive to parametric choices. For instance, this is the case when
aggregating data over any areal units which influences the observed results
(see Supplementary Fig. 12), or when stop locations are defined according to
radically different methodologies [44]. Also, pruning flows for anonymity
reasons is a critical point and can have a dramatic large-scale impact as we
saw for the Google or Facebook data. In particular, scale choice and pruning
lead to different displacement distributions and affects particularly its tail
behavior. Spreading processes over wide areas is more than the sum of the
spreading processes over its parts and the tail of the displacement
distribution is critical here, as we have shown with our application to
epidemic dynamics.

The differences in methodologies might render meta-analysis and comparing
different countries difficult, if not impossible. In the worst case, results
obtained for a specific region might not be straightforwardly ported to
another geographic area. Without verifying that such results are robust to
changes in the protocols for data processing, the knowledge gained from one
place – under the implicit assumption of universal behavior, which we have
only partially reproduced with our extensive analysis – could not agree with
the one gained from another place, jeopardizing global policies.

Most of the problems we have outlined in this study could however be solved
and we propose a series of good practices that largely align with what already
largely discussed in the Mobile Phone Data Community [42, 43]. A few
contributions [15, 6, 45, 46] indeed already tested, or controlled for, some
of the issues we illustrate in the Supplementary Materials, but this kind of
testing is however often unfortunately neglected or impossible. Certainly,
encouraging a direct access to device-level data, although naturally strictly
regulated to the higher anonymization and privacy-preserving standards, as
Cuebiq and Mobile phone providers did, will enforce a high level data quality.
Also, more transparency in the data production and pre-elaborating baselines
and metadata (as Facebook did) is needed. Building of communities as
encouraged by Cuebiq and Safegraph, creating open platforms and a share
framework for mobility data [47] as proposed by initiatives such as
DataCollaboratives [48] (especially associated to emergencies like COVID) are
also measures that will improve the quality of data.

On a more general perspective, it is always desirable that the study of
complex systems relies on more than a single instance [49]. Even trivial
operations on the data, like thresholding, can be very delicate [50] and
require a robustness analysis that, either for lack of generalized access to
multiple datasets or other plausible reasons, too often is overlooked. In
fact, results might be platform dependent [51] and data quantity cannot
compensate for lack of data quality [52]. Therefore, a quantitative approach
to social sciences aiming at the highest standards of reproducibility should
go beyond single datasets, requiring by design to be tested with comprehensive
meta-analyses [53].

## Methods

### Datasets

#### Census data

Census data provide statistical information about the urban and inter-urban
commuters’ flows using a widely statistical representative sample. The data
can in principle be collected at the street address level of granularity, but
is often aggregated at different administrative levels (district,
municipality, provinces, counties, …). Census data typically covers the whole
adult population in a country. However, the quality of these data as proxies
for mobility may be limited by the fact that the home (or work) location in
the national registers needs to be regularly updated as individuals move to
new homes and are still registered at their original homeplace. This mismatch
may vary strongly between countries as it depends on how enforced or rewarded
is the update of the public register. Moreover, these data fail to capture
other types of mobility besides home-work commuting of adults, such as the
movements of tourists seasonal or freight traffic. At a urban level, census
survey data are reasonably similar to the estimates coming from Mobile Phone
or Twitter data [27] and allow for the opportunity of studying epidemic
spreading considering the movements of different age groups [54, 55], although
the use at a national scale [55, 56] can be influenced by the aforementioned
limitations, as discussed in this paper. Here, we collected and analyzed
official 2015 census data describing aggregated commuting patterns for the US
[20], at the county level, and the official 2011 Census data for Italy [21],
aggregated at the province level.

#### Mobile phone data

Call detail records (CDR) from mobile phone data were at the root of the first
ICT data-informed attempts at modelling human mobility [1, 2]. The information
provided by the CDR are records of when and where a user exchanges information
(audio, text or data) with an antenna. Unless some triangulation methods are
applied, the spatial distancing between the antennas represent the
characteristic granularity of the mobile phones trajectory data. In the
earlier days of mobile communication, when it was limited to calls and SMS,
the data temporal sampling was uneven and sparse, which would cause biases in
how the trajectories were reconstructed [44], with an over-representation of
long trips. This is most likely not the case with modern data, since smart-
phones are continuously connected to the mobile network for data exchange.
Thanks to its wide coverage and the fact that it includes a wide range of
mobility behaviors, mobile phone data represent one of the more important
assets that can be used for studying human interaction [57] and as proxies of
human mobility for epidemic modelling [22, 58] with applications in the study
of the spread of a wide range of diseases [59, 60, 61, 62, 35, 63, 64]. The
access to these data is usually limited to a small number of research groups
that build tight partnerships with the mobile industry (with the notable
exception of D4D challenges [65, 66]), but the active collaboration between
industry and academia in principle grants that the data processing can be
custom built for the scientific use and the underlying assumption shared for
reproducibility purposes. Here, we will reconstruct the displacement
distribution function for the US on the basis of published fitting parameters
[1], and those for Portugal, Spain and France, from the data published for
reproducibility purposes [22].

#### GPS data

Small blackboxes equipped with GPS trackers and accelerometers are installed
in private vehicles for insurance reasons and record trajectory data. This
type of data is naturally limited to a single transportation mode, but is able
to capture locations with a great spatial accuracy (typically of order $10m$),
and the dynamics of the trip with stops and velocity, which allows the
reconstruction with high accuracy of a driver’s displacements. The scientific
use of a dataset describing the vehicular movements in Italy, provided by
Octotelematics, has given the opportunity for exploring different facts of
human mobility behaviour [67, 68, 69, 70, 3, 4]. However, with the exception
of a small dataset provided for a data challenge [71], their access has been
limited to a limited number of groups who had granted access to the raw data
by an industrial partner. Here, we analyze the displacement statistics of
Italian drivers in various cities as described in [4].

#### Google datasets

Google data are elaborated from the opt-in service ‘Google location history’
and the trajectory data, capturing the movement of Android smartphone users,
has been segmented using a machine learning technique [72] into a series of
displacements between pairs of stop locations. The displacement have been then
aggregated over a grid and on a weekly bases. The grid size is been described
as a $5km\times 5km$ grid in [19] and as a $5km^{2}$ cell grid in other
publications on similar datasets. To ensure the users’ privacy, a network
pruning has been applied where flows between cells is smaller than $100$
users/week. The strength of this dataset is its wide international coverage.
However, since longer connections have smaller flows, the pruning procedure
systematically cuts inter-urban mobility and is a clear limitation of these
datasets. In particular, this leads us to think that some results derived from
Google data using this procedure are biased towards an over-representation of
short movements. While the analysis at a urban scale [73, 74, 75, 13] is
probably marginally affected by this problem, the relative weight of long
range connections is certainly biased. This bias might be one of the factors
leading to the observation [19] of a steeper decline in the displacement
distribution in less developed countries where, coincidentally, smartphones
experience also a smaller market penetration and thus the effect of the
threshold-based pruning of small long range flows can be naturally stronger.
As we will see (see text and Fig. 3), this pruning effect also influences all
analysis describing the disease spreading process over the mobility network at
a country scale, such as those discussed in [11, 76]. The Google data analysed
here has been shared through a Data4Good programme. We do not however have
access to the data and a reproducibility request associated to two published
papers [19, 11], was made in October 2020 is still pending at the time of the
submission of this paper. We will therefore use the displacement statistics
released by the authors of [19] and which have been computed using bins of
$1km$. The data covers almost all countries for any displacement larger than
$1km$ but we restricted our analysis to displacement larger than $5km$ taking
into account the cut-off induced by the underlying cell dimension.

#### Facebook

Facebook data is obtained by tracking the movements of the mobile phone
application users that opted-in to the Location History and Background
Location collection services [23]. The access is granted through a Data4Good
platform which is accompanied by a limited description of the dataset and very
limited methodological description. These datasets provided by the Facebook
Data4Good can describe, in different forms, the mobility at different scales
in correspondence to natural disasters (disaster maps) or the COVID-19
epidemics. The COVID-19 DiseaseMaps dataset included three descriptors of
human mobility that have been used for research: i) small scale mobility flows
between small tiles of size approx $1km\times 1km$ (the dimension of the tiles
varies between datasets [77]) and internal to administrative areas [12, 16,
77]; ii) long range mobility flows between administrative areas [10, 78]; iii)
co-location matrices computed at an administrative level [23, 47, 79]. Co-
location matrices being intrinsically different from OD matrices, are not
considered here. Similarly to what is happening to Google datasets, Facebook
performs the pruning of small flows and long distances, thus introducing an
important bias and might severely affect results at a large scale (and less at
a urban level), and leading to national origin-destination matrices that are
visibly sparse as observed in [10], Figure 1e and Supplementary Fig. 1. (This
effect of pruning appears as reduced in co-location matrices.) Also, we note
that Facebook also provides ‘baseline’ values that represent a typical flow
prior to the event (disaster or disease) characterising the dataset. The
strength of the Facebook data is its wide international coverage, coupled with
the possibility of providing metadata such as user gender. Limitations here
come from the data processing under the form of small flow pruning and the
fragmentation of the network into smaller sub-networks. For instance, U.S.A.
flows for the COVID-19 disease maps are provided only at state level, with no
information about the movements between states. Here, we will use the mobility
network where the nodes represent administratives areas and the edges the long
range mobility flows between these areas. The flows are aggregated every 8
hours and we obtain temporal networks. If the flow is below a certain
threshold (that we estimate it to be of about 10 users), it is not reported in
the data. For these temporal networks, we aggregate flows on a weekly basis.
Similarly, we also define an aggregated baseline network combining all
baseline flows reported for different 8 hours bins and day of the week.
Baseline flows are present however only for origin-destination pairs and time
bins where it the flow reported is above the filtering threshold in at least
one of the weeks comprised in the dataset, thus some information is
necessarily lost also in this case. For epidemic simulations, we also used, as
a proxy for population, the baseline values describing the number of facebook
active users recorded in a given node. Similarly to flows, active users are
reported in bins of 8 hours and for each day of the week and have been
accordingly averaged over a whole week.

#### Cuebiq

Cuebiq collect mobility data of anonymized mobile app users who opted-in to a
large number of different location based services in different countries
including U.S.A., U.K., Italy, Spain, France and Germany. Access to this data
has been provided under data governance frameworks to address the COVID-19
emergency via a Data4Good program that constitute a clear attempt at building
a collaborative environment around the data. Cuebiq provided to the members of
the program access to device-level privacy-enhanced data, where noise is added
to home and work locations at the census block group level, and stops
associated with privacy-sensitive locations are removed entirely from the
dataset, in order to preserve privacy. This data has been used in the early
days of the pandemic to produce (non peer reviewed) mobility reports [80, 24,
81], often accompanied by interactive visualisation of the mobility reduction
patterns. A clear advantage of these datasets is the access to the device-
level trajectory data at GPS precision, from which it has been possible to
study the behavioural changes associated to lockdowns [82, 83, 84]. The
limitation coming with this abundance of data is the need for building a
strong preprocessing pipeline in order to analyse this data, but nevertheless
allows for a tailored and transparent design of the segmentation algorithm.
Here, we use Cuebiq-HDR mobility data for Italy derived from the device-level
data and aggregated weekly at the province scale by Pepe and collaborators
[24]. We also use flows describing the movements across the U.S.A., computed
directly by Cuebiq and provided within the framework of the Data4Good program.
The flows are here also aggregated weekly and at the county scale, based on a
proprietary segmentation method. We computed the baseline flows aggregating
those of the weeks preceding the beginning of lockdowns in Italy and U.S.A
respectively, rescaled the flows to the total country population and used the
associated census population for epidemic modelling.

#### Safegraph

Safegraph collects statistics about visiting patterns of different points of
interests (PoI) by aggregating anonymised location data from mobile
applications. The data covers the US only, where the home location is
identified for each user at the level of census block group. Safegraph data
activated a Data4Good program that gives free access to Academics for non-
commercial work. Safegraph actively attempted at building a community around
their data by organising regular seminars and a platform where results and
issues can be exchanged. The limitation of this dataset is that, since it is
focusing on PoIs, mobility flows are not directly available from the data
[15], as the flows recorded are between home locations and visited PoI. This
means that two subsequent visits to two location A and B would be recorded not
as a movement between A and B but as two movements from Home to A and from
Home to B. Deriving the mobility network at the urban level requires a rather
complex procedure [15], which is likely not suitable for being extended at a
country scale. Nevertheless, these mobility flows have been released as OD
matrices to describe population flows during the COVID-19 pandemic [25] and
used to reconstruct mobility at long range [85]. The advantages of this data
is that there is a great abundance of detailed information about the users
activity at PoI level, including social distancing estimates that have been
also subject to scientific use [86, 87]. Here, we will use the aggregated
flows published in Nature Scientific Data [25], aggregated weekly and at
county level, and used a two month period before the U.S.A. lockdown do define
the baseline flows.

#### Network features

We recap in Table 1, the main features of the baseline networks obtained from
the various datasets used for Italy and the US.

Supplementary Table 1: Baseline Networks characteristic numbers. We display here the dimensions of the networks shown in Fig. 1. We compute the number of nodes $N$, the number of directed Edges $E$, the average (in- or out-) degree $\langle k\rangle=E/N$ and the edge density $D=E/(N(N-1))$. The Italian Cuebiq (ITA Cuebiq) network has been aggregated over different provinces which contains different numbers of nodes. Having neglected self-loops, the USA Cuebiq dataset has $31$ disconnected nodes. Dataset | $N$ | $E$ | $\langle k\rangle$ | $D$
---|---|---|---|---
ITA Cuebiq | 107/107 | 3,735 | 34.9 | 0.329
ITA Facebook | 110/110 | 926 | 8.4 | 0.077
ITA Census | 110/110 | 4,019 | 36.5 | 0.335
USA Cuebiq | 3077/3108 | 60,181 | 19.6 | 0.006
USA Safegraph | 3108/3108 | 1,574,604 | 506.6 | 0.163
USA Census | 3108/3108 | 131,391 | 42.3 | 0.014

### Curve fitting as analysis method

In this paper we use the distribution of displacements $P(L)$ as one of the
analysis tools to inspect at the same time the flows and the distance covered
in a flow network/origin destination matrix. In the past years, there has been
an open discussion about the functional form of this distribution, that we
know to be governed by the multi-scale characteristics of human mobility [4,
5]. As in many similar cases, we use curve fitting as an analytic tool for
extracting information from the data collected. Curve fitting is clearly a
method limited by the functional form chosen. A large number of functional
forms have been used for mobility datasets, and the most common ones are:

  1. 1.

a truncated power law (TPL) with three [1] or two [34] free parameters;

  2. 2.

a lognormal (LN) [28];

  3. 3.

model-driven generalized gamma functions [4].

An exponential form is also sometimes considered a ta shorter scale, but can
be captured by the exponential cutoff using TPL and model-driven generalized
gamma functions. In our analysis, we attempted fits using 2 and 3 parametric
TPL, LN and three different model-driven forms: the Random Uncorrelated
Accelerations (RUA, 1 parameter), the Random Acceleration Kicks (RAK, 2
parameters) and the Weibull (WB, 2 parameters).

The RUA follows the functional form $P(L)\propto
L^{-\frac{3}{4}}\exp(-(L/L_{s})^{\frac{1}{2}})$, and is what expected when a
traveller speed follows a brownian motion [4], the RAK is in practice a
generalisation of RUA, as follows the similar form $P(L)\propto
L^{-\gamma}\exp(-(L/L_{s})^{\frac{1}{2}})$, where $\gamma$ is a free
parameter, and represents the saddle points approximation for a model where a
traveller accelerates and decelerates at fixed increments following a poisson
process [4].

The plots were technically carried out by fitting directly the CCDF function,
without any data binning, using the python _l_ mfit library, which performs a
(non-linear) least square fit. This procedure clearly weights more the short
distances in the distribution, where data is more dense, and basically ignores
the tail. Since the detailed characterisation of tail for the curves at hand
was essential, we compensated by fitting the logarithm of the y-axis of the
empirical curves. The results are satisfactory, as illustrated in
Supplementary Fig. 13 and in the collection of figures attached to the
Supplementary Materials.

### Metapopulation SIR model

In order to understand the geographical diffusion of diseases, one has to
combine the microscopic contagion processes with the long-range disease
propagation due to human mobility across different spatial scales. In order to
tackle this problem, epidemic modeling has relied on reaction-diffusion
dynamics in metapopulations [88]. Metapopulations can be thought as nodes of a
complex network of spatial patches, where links encode human flows from one
place to another and are responsible for between-patch transmission [89].

We denote by $M$ the number of patches, $N$ the total number of agents and
$N_{i}$ the population of the i-th patch. At any time, we have
$\sum_{i}N_{i}=N$ and, if the system is closed (i.e. there are no births and
deaths), this number $N$ is conserved. The mobility of the agents between the
patches is ruled by a weighted adjacency matrix $\mathbf{W}$, whose entry
$W_{ij}$ is the flux from patch $i$ to patch $j$. The probability $P_{ij}$
that an agent placed in $i$ moves to $j$ must be proportional to the flux
$W_{ij}$ and reads (as in [88])

$P_{ij}=\frac{W_{ij}}{\sum_{j=1}^{M}W_{ij}}$ (1)

At this point one has to introduce the reaction dynamics, that takes place
independently within the patches. A very wide-used and simple model for this
is the so called SIR-model: agents belongs either to the susceptible (S), the
infected (I) or the recovered (R) compartment; therefore, in each patch $i$ we
have that $S_{i}+I_{i}+R_{i}=N_{i}$. The allowed reactions are the following:
one agent that is in patch $i$ can move from the $S_{i}$ to the $I_{i}$ state
by getting the infection from another infected agent with infection rate
$\beta$ [88]

$S_{i}+I_{i}\xrightarrow{\beta}I_{i}+I_{i}$ (2)

and one agent (in patch $i$) can move from the $I_{i}$ to the $R_{i}$ state by
healing from the infection with recovery rate $\mu$ [88]

$I_{i}\xrightarrow{\mu}R_{i}$ (3)

therefore, the continuous-time equations for the infection dynamic are

$\begin{cases}\frac{dS_{i}}{dt}=-\beta I_{i}\frac{S_{i}}{N_{i}}\\\
\frac{dI_{i}}{dt}=\beta I_{i}\frac{S_{i}}{N_{i}}-\mu I_{i}\\\
\frac{dR_{i}}{dt}=\mu I_{i}\end{cases}$ (4)

Now, if we work under the assumption that the state of an agent does not
affect its diffusive behavior, the mobility for all the agents is described by
a unique mobility matrix $\mathbf{P}$, and the continuous-time equation
relative to the mobility for a generic compartment $X$ is

$\frac{dX_{i}}{dt}=\sum_{j=1}^{M}\mathcal{P}_{ji}X_{j}-\sum_{j=1}^{M}\mathcal{P}_{ij}X_{i}$
(5)

Therefore by summing up Eqs.(4,5) one obtains the system of $3M$ differential
equations of the model (that basically is the same of [90] with $\epsilon=1$)

$\begin{cases}\frac{dS_{i}}{dt}=-\beta
I_{i}\frac{S_{i}}{N_{i}}+\sum\limits_{j=1}^{M}\mathcal{P}_{ji}S_{j}-\sum\limits_{j=1}^{M}\mathcal{P}_{ij}S_{i}\\\
\frac{dI_{i}}{dt}=\beta I_{i}\frac{S_{i}}{N_{i}}-\mu
I_{i}+\sum\limits_{j=1}^{M}\mathcal{P}_{ji}I_{j}-\sum\limits_{j=1}^{M}\mathcal{P}_{ij}I_{i}\\\
\frac{dR_{i}}{dt}=\mu
I_{i}+\sum\limits_{j=1}^{M}\mathcal{P}_{ji}R_{j}-\sum\limits_{j=1}^{M}\mathcal{P}_{ij}R_{i}\end{cases}$
(6)

This system of equations cannot be solved analytically but only numerically;
notice that for the basic reproductive ratio we simply have $R_{0}=\beta/\mu$
(see for example [90] for details).

### Simulation details

All the simulations were done by choosing the free parameters $\beta$ and
$\mu$ in order to have $R_{0}=2.6$, building the mobility matrix from the data
and initializing the population of the patches with the data. The local
population was set as proportional to the reported baseline userbase for
Facebook data in countries different from Italy, while for Facebook Italy,
Cuebiq, Safegraph, and Census data the resident population has been used. The
other free parameter to be chosen (over the total duration of the simulation,
that was selected to be $t=200$ days just to be sure to reach the end of the
epidemic) is the time step, that was chosen as $dt=0.5$ days. The effect of
this parametric choice has been tested by performing simulations with $dt=0.1$
days, which yielded exactly the same numerical results. Given the initial
conditions and the parameters, the code for the simulations solves Eq.(6) by
using the built-in ode function of the software R, providing the time
evolution for $S$,$I$,$R$ individuals in all the patches at each time step
starting from $t=0$. All the simulations have at $t=0$ a fully susceptible
population but one infected (the seed), whose location was fixed, in each
network, in the node with the larger population. From the time evolution, the
code trivially calculates the total incidence (namely the total number of new
infected) at each time step, the number of recovered at the end of the
epidemic in each patch and the attack rate in each patch. Notice that by the
moment that in some networks unconnected components are present, in order to
calculate the attack rate the recovered were not divided by the total
population of the network, but only by the population of the component of the
network in which the initial seed was placed.

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## Acknowledgments

We thank Cuebiq and Facebook for providing us free access to their Flow
Network Data through their Data for Good programme.

## Supplementary Text

### Technical limitations affecting human mobility datasets

In this manuscript, we observe great differences in how mobility datasets
coming from different providers are able to describe mobility flows at a
country level. In this section of the Supplementary Materials, we briefly
discuss here a series of possible causes for the differences. This list does
not however represent a comprehensive text illustrating all biases associated
with mobility data collection and elaboration, for which we suggest further
readings [40, 41, 42, 43].

#### Behaviour captured

Most datasets discussed are being passively collected from ICT data records.
They are therefore necessarily influenced by the type of behaviour that
produces the records. Issues associated with the behaviour captured cannot in
general be solved, but awareness of the consequences of passive collection of
movement behaviour can limit the errors introduces in the analysis.

Mobile Phone Call Detail records used to be associated with phone calls and
SMS messages, but as the use of the “data” connection is becoming widespread,
the data captured are now less coupled with the communication behaviour, but
can still be limited by the closure of the data connection by users.

Providers aggregating data from several mobile apps, such as Cuebiq or
Safegraph, and on a broader sense also Google and Facebook, capture data when
the device GPS connection is used by one of the location based applications
associated with the providers, which may require the user to be actively using
that app or not, depending on the specifics of the app.

Other GPS data such those of Octotelematics coming from devices installed to
cars for insurance reason, but also data from Tomtom data coming from single
apps specifically gathering the movements of bikers or runners, are
necessarily limited to movements associated with a specific mode of transport.

Finally, the accuracy of census data can be hindered by the tendency of users
to not update their records in case of relocation. This tendency may depend
upon the incentives set by each country for this update (for example, in the
case of Italy, citizens are highly incentivised to update their public records
to have access to the Public Health System).

#### Segmentation and sampling

Mobility data is generally gathered in form of raw trajectories formed by a
sequence of position where a user has been observed at a certain time. For
understanding the user movement behaviour, these trajectories are usually
segmented into a sequence of movements in space and stops where the user stays
in a particular place for a certain duration in time. There are several
alternative methods for this segmentation. Many data provider designed their
own optimised for their data which details are often undisclosed private. The
observed segmented results can be very sensitive to algorithmic and parametric
choices, as discussed in [44]. The impact of this issue can be reduced for
academic use by allowing to operate directly on the raw trajectories using
open source methods with explicit parametric choice.

#### Spatial Granularity

Segmented trajectories are, in most cases, further aggregated over
(administrative) areal units to obtain the so-called Origin-Destination (OD)
matrices. The choice of those areal units clearly influences the observed
results, a problem known in literature as “Modifiable areal unit problem”.
Many datasets circulating are influenced by this problem. In Supplementary
Figure 12, we illustrate this with an example using it on our main indicator,
the displacement distribution $P(L)$. In panel (a) we show how the distances
estimated between the centroids of the unit areas may differ from the average
distance of users moving between these areas, while in panel (b) we show how
different spatial granularity yields to different observed $P(L)$. While this
problem is intrinsic with the choice of aggregating data for obtaining a flow
matrix, allowing researchers to use the granularity most adapt to the analysis
performed instead to adapting to a standard one chosen by the data provider
can without any doubt limit the biases introduced.

#### Upscaling

All data gathered, with the exception of Census, describe only a fraction of
the total population. Often, for ICT-based datasets, the representativity of
this sample is also biased towards high-income, more urban and mostly male
populations. To estimate the real mobility flows of the whole population
between different areas, that is the information captured in the OD matrices,
it is necessary to “upscale” the observed flows. Several methods can be used
for upscaling, in most case they are based on a proportionality with respect
to an estimate of the local observed population. The knowledge of the active
populations in each areal unit, which could also be a function of time,
becomes therefore a crucial piece of information to rebalance the observed
flows to what is expected to be the mobility of the total population. Lack of
knowledge about active population can conversely limit the analysis due to the
need of having more imprecise upscaling. This issue can be easily solved by
the company, by providing explicit information about the user-base population
as a function of time (as Facebook and Cuebiq do in the data analysed in this
paper), when possible with data disaggregated by income and gender in order to
allow for alleviating the potential biases in the data using post-
stratification methods.

#### Self-loops

For many applications, knowing how large is the fraction of people who exited
an areal unit with respect to those who circulated only locally is very
relevant. This information is captured, in an OD matrix, in the diagonal
elements that describe, from a network perspective, self-loops of the mobility
flows network. This self loops are in general present in most aggregated
datasets. However, it is often not very clear what they do represent. It can
encompass at the same time information on the number users observed in the
dataset, i.e. those activate the capturing device but never left their home
location. Or alternatively users who performed a local movement (as identified
by the segmentation algorithm), but never arriving at leaving the areal unit.
Unless specified, it is in principle not clear if users who belong to the self
loop are also counted it outwards flows. All this info information can be very
relevant for mobility analysis and epidemic modelling and we advise, as a
minimum, to describe clearly what is represented by self loops and to further
include any other “non movement” information available among the information
accompanying the OD matrices.

#### Pruning

In all datasets, small flows between two areas can be unobserved because of
the local user-base population is too small or because the behaviour captured
or the segmentation method has a limited efficiency at identify the users’
movement. This makes the observed OD matrices systematically sparser than the
real ones. This effect is further worsened by an active “pruning” of the
mobility network done to limit the size of the datasets shared and to ensure
K-Anonymity. As discussed in the paper, in the Facebook data we estimated that
the pruning happens for flows below (10 users)/(8 hours), while Google declare
cuts below 100 users/week, and this pruning heavily influences our possibility
of using the data collected for studying long-range mobility. This major
problem can be solved, again, by providing to researcher a way of elaborating
directly raw data without applying any heavily lossy pre-elaboration such as
pruning.

#### Fragmentation

Again for limiting the size of data archives, the datasets provided are often
divided into large administrative ares (countries, US states, …). Similarly to
pruning, also this process forcefully cuts long range mobility. This problem
can be illustrated by the data provided by Facebook for USA, where they used
state-level fragmentation, not allowing for any analysis about the spreading
behaviour across the continental USA. Spreading processes over wide areas is
more than the sum of the spreading processes over its parts. The issue of
fragmentation is unfortunately not easy to solve, as in many cases the data
providers only has data associated to a single country. This makes the problem
of studying international flow one of the hardest we face. Only a few sources
have international coverage, but aviation data are limited to a single mode of
transport that can underestimate mobility between contiguous countries. Google
data, as we saw, is heavily influenced by pruning cutting long range flows and
Facebook offers only fragmented data. We believe that this great limitation in
the opportunity of analysing international flows, information nowadays so
vital for epidemic containment, represent a paradigmatically example
illustrating the need of a shared platform where to share, elaborate and
analyse mobility data for academic purposes.

## Supplementary Figures

(a) |  (b) |  (c)
---|---|---
(d) |  (e) |  (f)
Supplementary Fig. 1:  Visual comparison of baseline flow maps selecting a
fixed number of high flow edges. Here we display the same networks displayed
in Fig.1, but limiting each network to having at max the number of edges of
the sparser network of the same country by removing edges with smaller flows.
For U.S.A. we have Cuebiq (a) which is the sparser, Safegraph (b) and Census
(c). Here networks are visibly different even if one compensate for the
different edge density. For Italy we have Cuebiq (d), Facebook (e) and Census
(f). In this case, the differences between networks are less pronounced, with
only notable difference the lack of connectivity between the peninsula and the
islands of Sardigna and Sicily for Census data (panel c)

(a) |  (b)
---|---

(c)
---

Supplementary Fig. 2:  The Displacement Distributions of France (a), Spain (b), Portugal (c). Here, we rescaled the mobile data coming from [22] to become representative of the underlying population similarly to what done with Cuebiq and Facebook data. The light purple lines represent the evolution in time of the Facebook dataset, while the thick line represents the baseline. The differences associated with the COVID-19 lockdown, one of the more impactful events in world history, is smaller than the observed difference between datasets.  (a) |  (b) |  (c)
---|---|---
Supplementary Fig. 3:  Comparison of the scale parameters fitted for Facebook and Google datas with different models. We illustrate here the difference between the fitted parameters for a set of 58 countries that are present in both the Facebook and the Google datasets. In panels a) and b) we represent the scale parameter of two bi-dimensional fits (truncated power law $P(L)\propto L^{-\alpha}e^{\frac{L}{L_{s}}}$ and the saddle point approximation of the random acceleration kicks model $P(L)\propto L^{-\alpha}e^{\left(\frac{L}{L_{s}}\right)^{0.5}}$ ). Due to the very sensible degree of freedom associated with the scaling parameter, the differences observed in the distance scale $L_{s}$ are way larger than what observed in c), where the scale is estimated using the monoparametric random uncorrelated acceleration model ($P(L)\propto L^{-0.75}e^{\left(\frac{L}{L_{s}}\right)^{0.5}}$ )  (a) |  (b)
---|---
Supplementary Fig. 4:  The evolution in time of the Cuebiq data in the USA (a)
and Italy (b) Again, the differences associated with the COVID-19 lockdown is
smaller than the observed difference between datasets. There the CCDF is
analyzed for $L>10$ and not limited to the continental U.S.A. The black dashed
line represents the baseline curve.

(a) |  (b)
---|---

(c)
---

Supplementary Fig. 5:  Comparison of the displacement distributions across the
globe. We observe a wide variance in the fit parameters estimated for the
$P(L)$. In panels (a), (b), and (c) we represent the the scale and exponent
parameter for the Random Acceleration Kicks models, a Lognormal fit, and a
Weibull fit respectively. As in Fig. 2, we include the curves of Figure 1 for
USA –Cuebiq (cb), Mobile (mb) [1], Google (gg), Safegraph (sg), Census (cs) –
and Italy (that, as illustrated in Figure 1 includes also Octotelematics and
Facebook (fb) data). The values for the longitudinal weekly analysis of Cuebiq
and Facebook data, and the values estimated for the Google and Facebook
baseline. We observe how all data points align around a curve, which suggests
strong correlation between the free parameter and, ultimately, the existence
of an underlying law having a single degree of freedom.

(a) |  (b)
---|---

(c)
---

Supplementary Fig. 6:  Comparison of the displacement distributions across the
globe. Using the Google dataset, we observe how the tail behaviour of $P(L)$
is radically different if one consider small or insular countries (a), medium
size countries (b),or large countries or archipelagos (c). These subsets of
countries have been identified using the RAK fits as large countries have an
exponent $>2$, the medium size countries have been selected between those with
exponent between 0 and 1 and the small countries have exponent $<-3$. In the
smaller countries we observe an exponential-like cutoff, while in the larger
countries the curves are more stretched and suggest the existence of multiple
scales. This difference leads to a wide variance in the fit parameters
estimated for the $P(L)$ in panels (d) and (e), where we represent the the
scale and exponent parameter for the Truncated Power Law and Random
Acceleration Kicks models respectively. We include the curves of Figure 1 for
USA –Cuebiq (cb), Mobile (mb) [1], Google (gg), Safegraph (sg), Census (cs) –
and Italy (that, as illustrated in Figure 1 includes also Octotelematics and
Facebook (fb) data). The values for the longitudinal weekly analysis of Cuebiq
and Facebook data, and the values estimated for the Google and Facebook
baseline. We observe how all data points align around a curve, which suggests
strong correlation between the free parameter and, ultimately, the existence
of an underlying law having a single degree of freedom.

(a) |  (b)
---|---

(c)
---

Supplementary Fig. 7:  Small world effect in epidemic diffusion. (a) Focusing
on a single dataset, the Facebook baseline mobility, we observe how the
average number of edges per node grows as the number of users increases (the
fit suggest a scaling sub-linear behaviour). This is most likely due to a
pruning procedure that removes edges with smaller flows for both privacy
enhancing and disk space reduction. However, introducing a threshold to
minimal flow (we estimate is about 10 users/8 hours for these data)
systematically cuts longer edges where movement costs are higher, and in
particular between marginal areas. (b) This reflects into a great change into
the shape of the network. Networks with higher average degree has a larger
fraction of shortcut nodes (where shortcuts are defined as links at least 2
times as long as the average distance between a node and the closest
neighbour). The fit here suggests a logistic behaviour. (c) The fraction of
shortcut nodes, in turns, determines the behaviour of spreading processes on
the network, here again we illustrate it over the Facebook baseline mobility
networks. As we did also in Fig. 3 and Supplementary Fig. 11, we simulate the
spreading of epidemics using a SIR model starting from a single seed in the
node with the larger userbase. We see how when the fraction of shortcut nodes
passes a relatively small threshold, we have consistently a faster spreading
and thus a higher incidence at peak.
---
Supplementary Fig. 8:  Displacement distribution of a pruned gravity-generated network. Starting with resident population of Italy at province level, we generated the flow with a toy Gravity model as $\phi_{i,j}\propto P_{i}P_{j}L_{i,j}^{-2}$ where $P_{i}$ is the population at node $i$ and $L_{i,j}$ the distance between nodes $i$ and $j$. Then, we progressively removed edges having the smallest flows until reaching an edge density $D$ indicated in the figure legend. This process strongly influences the observed $P(L)$ when $D<0.8$. Now, as seen in Table 1, empirical edge densities are lower than $0.35$ in all three Italian datasets, suggesting that a significant part of long flows might be missing in the data captured.  (a) |  (b)
---|---
Supplementary Fig. 9:  Effect of different datasets on spreading dynamics.
Here we represent the differences between the spreading dynamics in U.S.A. and
Italy using the SIR model described in the paper. We present the evolution
over time of the incidence, that accounts for the new infected in that unit
time. In panel (a), we see how for U.S.A. Safegraph and Census (solid orange
and dashed green lines), that as we discussed are characterized by an
significantly larger weight of long range flows, yield exactly the same result
that likely is equivalent to homogeneous mixing. Since Cuebiq data does not
display this over-abundance of long range connections, the epidemic diffusion
is mostly local (see Supplementary Fig. 10) and produces a more complex
behaviour where the incidence curve has two peaks. In panel (b), Italian data
display instead smaller differences, probably because of the smaller spatial
scale of the Italian peninsula with respect to the spatial scales at hand.
Here the peak appear not delayed but rather lowered for Facebook and Census
data, that are in this case characterized by an under-representation of long-
range connections.
---
Supplementary Fig. 10:  Mapping of the attack rate across U.S.A. at the first
epidemic peak simulated with Cuebiq data. The map represents the epidemic
state in correspondence of the first peak in the U.S.A. Cuebiq simulation (day
40). The epidemics, starting in Manhattan (NY), appear geographically
contained and spreading only locally. On the other hand, the equivalent
Safegraph and Census plot would show a map of the U.S.A. homogeneously red.
---
Supplementary Fig. 11:  Understanding the small world effect in epidemic diffusion by fitting the $P(L)$ The relative incidence at peak, that is the largest fraction of population that become infected in a single day during the epidemic simulation, can be reconnected to the property of the $P(L)$. We use here the scale parameter of the $P(L)$ estimated using the RUA model, that being characterized by a single parameters yields the more stable estimate for the spatial scale (see Supplementary Fig. 3). This relative scale parameter is, for each country, obtained by dividing the RUA scale parameter by the maximal length of the continental landmass, estimated as the longest side of a minimum rotated rectangle shape of the continental outline of the country). We analyse in this plot not only the Facebook baseline, but also the networks illustrated in Figure 1 and observe clearly how to larger relative values of the relative RUA scale, that we can associated to a larger tendency to small world behavior and, similarly to what observed in Supplementary Fig. 7 (c), we have that larger scale mobility dictates significantly larger epidemic effects. At the same time, as observed in Fig. 3, we also observe great variation in the incidence at peak in the same country depending on the different source of data used.  (a) |  (b)
---|---
Supplementary Fig. 12:  Analysis of the effects of granularity on Facebook baseline data. a) the differences in the $P(L)$ distribution function in Italy changes if one consider as displacement lengths the distance between the centroids of the administrative areas or the average length indicated by Facebook. From this graph, we can deduce that the real displacements are largerly smaller than what estimated using the centroid distance. b) the differences in the $P(L)$ distribution function in Hungary changes if one consider two different areal units lengths. The Hungarian datasets comes replicated with two different granularities, one consistent with the other countries (here indicated as Large grain), having 20 nodes, another with a smaller areal unit, having 821 nodes. We observe a wide difference between the displacement distribution estimated using the average lengths over these two datasets, with the larger grain dataset yielding larger distances estimates over the same dataset.  (a) |  (b) |  (c)
---|---|---
(d) |  (e) |  (f)
Supplementary Fig. 13:  Visual comparison of baseline flow maps selecting a
fixed number of high flow edges. We illustrate here the alternative fits
executed on Google data using the models described in the Methods. We can
appreciate how the fits appear to be well converged. For mid-size countries
(UK and Argentina), the alternative fit functions give very similar results.
For smaller (Republic of Korea, Jamaica) or larger (U.S.A. or Austraila)
countries, the observed behavior diverges from what it can be fit with the
models proposed, especially for the monoparametric RUA fit.

Provider | Source | Area | Elaboration | Precision | Stop time | NW Aggregation | Self-loops | Pruning | Availability
---|---|---|---|---|---|---|---|---|---
Teléfonica[35] | Mobile Phone | Germany | Authors | Cell tower | 15 min | NUTS3 / weekly | Yes | $\phi<5$ | Restricted
Facebook [10, 78] | Social Media Access | Several countries | Company | GPS | ? | Administrative / 8 hours | Yes | Yes | Data4good
Orange [63] | Mobile Phone | France | Authors | Cell tower | 1 hour | Municipality / weekly | Yes | ? | Restricted
Baidu [91, 34] | Location Based Service | China | Company | ? | ? | Province / daily | Yes | ? | Restricted
Baidu [11] | Location Based Service | China | Company | ? | ? | Prefecture / daily | Yes | ? | Restricted
Cuebiq HDR [24] | Mobile Apps | Italy | Authors | GPS | 5 min | Province / daily | Yes | No | Public
Teléfonica [92] | Mobile Phone | Colombia | Authors | Cell tower | day | Department / daily | Yes | No | Restricted
Google [11] | Location History | Worldwide | Company |  5km2 area | ? | Municipality / weekly | (Yes) | Yes | Restricted
Google [11] | Location History | Italy | Company |  5km2 area | ? | Province / weekly | (Yes) | Yes | Restricted
Google [14] | Location History | Europe | Company |  (5km2 area) | ? | Country / beweekly | (Yes) | Yes | Restricted
Google [73, 74, 13] | Location History | Intra-Urban | Company |  (5km2 area) | ? | Weekly | Yes | Yes | Restricted
Safegraph [15] | Mobile Apps | USA Metro areas | Authors | PoI | ? | Census Block/Weekly | ? | Yes | Data4Good
Safegraph [25] | Mobile Apps | USA | Authors | PoI | ? | Census Block/Daily | Yes | Yes | Public
Supplementary Table 2: Characteristics of the datasets used for modeling the
epidemic spreding of COVID-19. Note: the Google dataset is in principle also
part of a Data4good initiative and accessible upon request, however we started
the request procedure more than a year and a half ago and our demand is still
“under consideration”. Information between parenthesis has been found in a
different paper on the same dataset.