# The Silent Signature: Behavior-based User Exposure in Mobility Data

Lucas G S Félix, Anne Josiane Kouam, Aline Carneiro Viana, Nadjib Achir, Jussara M Almeida

► To cite this version:

Lucas G S Félix, Anne Josiane Kouam, Aline Carneiro Viana, Nadjib Achir, Jussara M Almeida. The Silent Signature: Behavior-based User Exposure in Mobility Data. MDM 2025 - 26th IEEE International Conference on Mobile Data Management, IEEE, Jun 2025, Irvine (CA), United States. 10.1109/MDM65600.2025.00044. doi:10.1109/MDM65600.2025.00044

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# The Silent Signature:
## Behavior-based User Exposure in Mobility Data

Lucas G. S. Félix

Universidade Federal de Minas Gerais and Inria

Belo Horizonte, Brazil and Palaiseau, France

lucas.da-silva-felix@inria.fr

Anne Josiane Kouam

TU-Berlin

Berlin, Germany

kouam.djuigne@tu-berlin.de

Aline Carneiro Viana

Inria

Palaiseau, France

aline.viana@inria.fr

Nadjib Achir

Inria

Palaiseau, France

nadjib.achir@inria.fr

Jussara M. Almeida

Universidade Federal de Minas Gerais

Belo Horizonte, Brazil

jussara@dcc.ufmg.br

**Abstract**—Mobility is a fundamental aspect of human life, and mobility data offers valuable insights into user behavior. Yet, this data also exposes users to privacy risks given pattern unicity in their trajectories, i.e., the singularity in the displacements made by users. Existing strategies to quantify such user exposure either focus only on the sequences of places visited by each user, as the widely used uniqueness measure, or are tied to specific attack models. We here introduce MoBES, a novel, scalable, customizable and highly interpretable measure of user exposure in mobility data. MoBES leverages multiple existing metrics to build a multi-dimensional space, which in turn is used to capture each user's mobility signature behavior. MoBES quantifies user exposure based on how distinct a user's signature is from her neighbors in the defined metric space. As such, MoBES is designed to be a fundamental expression of user behavior, and not tied to any specific attack model. We evaluate MoBES on a real mobility dataset, showing that it effectively captures user exposure within the behavioral metric space. We also compare MoBES with the uniqueness measure, showing that MoBES is able to uncover users who, even though visiting the same places as others in the crowd, are still at risk of exposure due to the unicity of their mobility behavior.

**Index Terms**—mobility, privacy, vulnerability, exposure

## I. INTRODUCTION

Mobility is a fundamental aspect of human life, shaping both individual and societal well-being [1]. Mobility also serves as a human behavioral signature, and such signature can be modeled with related data from cell phones, GPS, and social media. By analyzing mobility patterns, we gain insights into daily routines or preferences of individuals, such as places visited and home/work locations. However, this understanding when exposed can also highlight their vulnerability to possible privacy violations. Prior research has shown that mobility datasets exhibit high user unicity [2], [3], i.e. the singularity in user patterns, which also expose the users vulnerability.

Measuring user exposure, however, is complex, as mobility patterns can be analyzed from multiple perspectives, each with the potential of revealing singular aspects that may expose users, potentially risking their privacy. A widely used exposure measuring approach is uniqueness [4], which quantifies how distinctive a user's mobility is based on time-ordered sequences of visited locations. While effective, uniqueness (and related adaptations [5]) has key limitations: (i) it is computationally expensive due to the need to analyze all temporal combinations of locations in users trajectories, and (ii) it provides a narrow view of user vulnerability, focusing solely on sequential visiting patterns. Furthermore, despite being a continuous measure, uniqueness is often applied as a binary classification (vulnerable vs. non-vulnerable), limiting its ability to assess multiple levels of exposure.

Human mobility – as an expression of human behavior – is inherently multifaceted, requiring a multiple perspectives analytical approach. Prior studies have characterized mobility using diverse metrics such as radius of gyration, diversity, regularity, stationarity, and entropy [6]–[9]. These metrics capture distinct aspects of mobility, revealing unique behavioral patterns beyond what uniqueness alone can detect [10]. While uniqueness extracts raw subsequences of visited locations, mobility metrics encapsulate processed insights about how individuals occupy space over time. For example, radius of gyration measures how confined a user's movements are, while diversity assesses how distinct the sub-sequences of visited locations are. Combining multiple metrics allows for a multi-dimensional behavioral space, uncovering new forms of user exposure in mobility patterns.

Consider a user who follows a highly predictable routine—commuting from home to work on weekdays and staying home on weekends. Even if her exact locations are non-unique due to densely populated areas, her low diversity and entropy make her mobility highly predictable, increasing exposure. Knowledge of such behavioral signature could compromise user privacy, even when the precise visited locations are obscured. This underscores the deep understanding of vul-

<footnote>This work is part of the Mob Sci-Dat Factory project (ANR-23-PEMO-0004) under the France 2030 program. It was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), by Fundação de Amostra à Pesquisa do Estado de Minas Gerais (FAPEMIG), by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and by the CAPES-STIC-AMSUD 22-STIC-07 LINT project.</footnote>
nerabilities in behavioral signatures in an individual basis is
crucial for tailored privacy protection..

Beyond uniqueness, prior work has explored mobility met-
rics to assess exposure [11], [12], often framing it as a classi-
fication task (vulnerable vs. non-vulnerable) using supervised
learning. However, to obtain the required training data (i.e.,
labeled users), they tie their methods to specific simulated
adversarial models (that may not reflect real-world threats)
and define vulnerability as the probability of re-identifying a
user in a dataset [11], [12]. This simulated attacks dependency
limits the applicability scope of such exposure works, as they
do not account for unforeseen attack methods and may not
capture actual adversarial strategies and capabilities. As a
result, uniqueness remains widely used despite its limitations,
as it provides a more straightforward measure of exposure.

This PhD research tackles the challenge of assessing hu-
man mobility exposure by exploring the unicity of multi-
perspective behavioral patterns, offering a more comprehen-
sive understanding and nuanced reflection of user vulnerability
in mobility data. We propose a new approach that: (i) goes
beyond a single-perspective metric, (ii) is independent of any
specific adversarial models, and (iii) captures deeper layers of
vulnerability fundamentally connected to user behavior, i.e.,
the user mobility signature. Notably, we aim to investigate
how this behavioral signature can represent a new, largely
overlooked dimension of user exposure in mobility data.

Building on this idea, we tackle two core research questions:

*   **RQ1:** How can we quantify user exposure in mobility data in a multi-perspective, customized, attack-model-independent, and controllable manner that is fundamentally tied to the users' mobility signatures?

*   **RQ2:** How can such exposure estimate be made fully observable and interpretable, enabling an understanding of which mobility patterns contribute to higher or lower exposure?

To address **RQ1**, we introduce MoBES (Mobility Behavior-
based user Exposure Score) – a novel, scalable, and customiz-
able measure leveraging multiple mobility metrics capturing
circadian behaviors, to quantify user exposure. Unlike prior
works [11], [12], MoBES is designed to be a fundamental
expression of user behavior. It is generic and flexible, allowing
for customization based on the desired exposure sensitivity.
Also, MoBES measure is nuanced, providing a continuous
exposure score instead of a binary classification (e.g., exposed
or not). This allows data custodians to prioritize privacy
interventions for the most exposed users or behavioral patterns,
helping them apply privacy techniques tailored to specific
data or, in resource-constrained situations, prioritize those at
greater risk of exposure.

For **RQ2**, we design MoBES to measure user exposure (cf. **RQ1**) that provides interpretable insights – i.e., allows understanding the assigned score levels and what contributes to them. To that end, we linked the design of MoBES to the definition of an multi-dimensional behavioral space, where each dimension corresponds to one of the considered mobility metrics. Within this multi-dimensional space, we assess user

exposure by measuring the distance between each user and her
nearest neighbors. We hypothesize that users at greater risk
of exposure exhibit a larger distance due to their unique or
distinguishable behaviors, as captured by the metrics. Such
approach is quite flexible, as it can be instantiated to capture
user exposure with respect to different metrics and precision
levels. Moreover, we employ a distance metric that allows for
an easy interpretation of the relative importance of each metric
to the final exposure score.

We evaluate MoBES using a real-world network-sourced
cellular anonymized dataset, covering almost 42,000 users and
15 days [13]. Our analysis reveals users who, despite not being
unique in terms of the sequences of places visited, do exhibit
mobility behavioral patterns that are quite distinct, making
them easily exposed. Comparing MoBES to uniqueness, we
demonstrate how our method uncovers hidden exposure risks
not detected by traditional approaches. Overall, MoBES con-
tributes to the state of the art by providing a novel, behavior-
driven analysis of user exposure in mobility data.

This paper is organized as follows. We discuss the back-
ground for our work (§ II) and introduce the design of MoBES
(§ III). We then discuss how MoBES is instantiated to analyze
user exposure in a real dataset (§ IV) and compare it with the
widely used uniqueness metric (§ V). Finally, we conclude the
paper, offering directions for future work.

II. ESTABLISHING THE GROUNDWORK

Mobility data sharing can be both a blessing and a curse.
On the one hand, it allows companies and researchers to
provide significant benefits to people's daily lives [14], [15].
On the other hand, it can expose individuals' personal sensitive
information (e.g., home, workplace, hospitals), or reveal per-
sonal affiliations, including political or religious beliefs. The
disclosure of sensitive information poses significant privacy
concerns, representing a major barrier in using and sharing
mobility datasets [16]. Previous works have shown that know-
ing just a few locations can reveal a user's trajectory. For
instance, [2] showed that individually knowing the 3 most
visited locations could identify 50% of 25 million users, while
[4] showed that 4 points temporally ordered is enough to
identify 95% of 1.5 million users. Lastly, [3] demonstrated
that the identification risk remains high (up to 93%), even in
a country-scale dataset (60 million users).

In practice, these findings imply that a minimal knowledge
of an individual's mobility can be sufficient to expose sensitive
user data within a large mobility dataset. The reason for this
vulnerability is the *intrinsic singularity or unicity of human
mobility*, usually influenced by several factors of geographical,
temporal, and social nature [1].

One of the earliest and most well-known techniques for
quantifying user exposure is uniqueness [4]. It measures how
distinct a trajectory or its ordered subsequences is within a
dataset. Computed over time windows of size $|t_w|$, a trajectory
is unique if it remains so in any window. Lower uniqueness
values indicate higher exposure. Yet, computing all possible
subsequences for each time window is computationally expensive. Thus, some studies simplify the process by considering only time-ordered points, disregarding combinations [5].

While uniqueness estimates exposure from raw trajectories mapped into a two-dimensional space of geographical location versus time, alternative approaches use mobility metrics from spatiotemporal visits such as [10]–[12]. In [10], the authors use entropy [8] to quantify exposure, linking predictability to mobility. More visited places mean higher exposure. However, it focuses solely on entropy, while a multi-dimensional approach provides broader insights. [11] and [12] use Supervised Machine Learning (SML) to quantify exposure, leveraging behavioral metrics. Since SML requires labeled data, exposure is tied to a probability of re-identification by a thread model. [11] measured user vulnerability with respect to 8 different types of threat models. They trained a tree-based ML model, taking a set of mobility metrics as input features, to predict the probability of being re-identified by one of these threat models. The bigger the probability, the bigger the user vulnerability. Similarly, [12] proposes EXPERT, a framework that explores mobility metrics to perform binary classification of user vulnerability (i.e., high or low) with respect to a location sequence attack. This approach analyzes metrics from complete trajectories and subsequences of visits of varying lengths, which brings a deeper (even though more costly) vulnerability analysis. Both approaches above show that the usage of mobility metrics, designed initially to characterize user behavior, has proven effective in quantifying exposure.

**Towards MoBES:** Considering the above, existing approaches assessing user vulnerability in mobility data have notable limitations. First, methods like uniqueness and EXPERT rely on binary assessments (e.g., unique or not, high or low vulnerability) [4], [12], which may limit the effectiveness of protection mechanisms. Continuous metrics [11] could offer more nuanced vulnerability assessments. Second, some techniques depend on specific adversarial models [11], [12], which may bring a false perspective of user protection as real-world attacks may differ from those modeled. Third, the computational cost is another challenge, as methods like uniqueness and EXPERT require extensive processing of spatiotemporal subsequences, making large-scale application difficult [11]. Finally, capturing only ordered location sequences may result in missing broader behavioral insights. In contrast, multi-metric approaches are able to consider different perspectives of user behavior and easily adapt to the data available.

### III. MOBILITY BEHAVIOR-BASED USER EXPOSURE

Our ultimate goal is to provide a more holistic understanding of how users may become exposed due to their behavior derived from mobility datasets. Thus, in this work we propose MoBES (Mobility Behavior-based user Exposure Score), a measure designed to quantify the level of exposure of a user in a mobility dataset. In broad terms, MoBES works by measuring how much distinguishable the signature (behavior) of a user is from its peers. In the following, we first introduce some notation, MoBES computation and how its value can be

interpreted. We then present the specific set of mobility metrics used here to capture user behavior. We finish with a summary of MoBES main strengths compared to the literature.

#### A. Definition

Let's consider a human mobility dataset consisting of a set of users $U = \{1, 2, \dots, i, \dots, n\}$, where each user $i$ is associated with a trajectory $Tr_i$. A trajectory $Tr_i$ is represented as a sequence of spatial-temporal points $(t, l)$, where $t$ is the timestamp of an observation and $l$ denotes user $i$'s location, expressed as a longitude-latitude pair, at time $t$. Let's also consider a set $M$ of $m$ metrics that can be computed given an input user trajectory. Specifically, given $Tr_i$, we can compute $M = \{m_1^i, m_2^i, \dots, m_r^i, \dots, m_m^i\} \in \mathbb{R}^+$ where each component $m_r^i$ represents the value of metric $m_r$ for user $i$. Thus, the multidimensional space defined by set $M$ encapsulates a model of user mobility behavior as captured by the selected $m$ metrics. Specifically, the mobility signature of a user $i$ is determined by computing the set $M$ from $i$'s trajectory, which effectively positions $i$ in the defined $m$-dimensional space. Such space serves as the foundation for computing the Mobility Behavior-based user Exposure Score – MoBES, as discussed next.

#### B. MoBES Computation

To quantify the level of exposure underlying a user's mobility signature, MoBES evaluates how distinct the user's signature is from the signatures of other users in the dataset. This is done by computing the distance between users in the $m$-dimensional space, defined as the average relative absolute difference across all dimensions. This measure yields an interpretable value that reflects the relative deviation of each dimension. By taking the average, we have an easily interpretable value that reflects the average deviation in the behavior from one user referred as target ($i$) to another user referred as neighbors ($j$), in the $m$-dimensional space defined by set $M$ of mobility metrics. First, we measure the distance between the user and its nearest neighbors:

$$D_{i,j} = \text{distance}(i, j) = \sum_{r=1}^{m} \left| \frac{m_r^i - m_r^j}{m_r^i} \right|, \quad (1)$$

where $m_r^i$ and $m_r^j$ are the values of the $r^{th}$ metric for users $i$ and $j$, respectively ($i, j \in U$). In broad terms, this distance captures how much user $i$ would need to adjust her behavior (the sum across all metrics) to “blend” with $j$. The idea is that if a user's behavior (as captured by the metrics) is very distinguishable (thus more exposed), she should make a larger adjustment to align with a nearby user. The values of $D$ are relative to the target user, being thus asymmetric, i.e., $D_{i,j} \neq D_{j,i}$. Note that, in Equation 1, all metrics are equally weighted. Yet, one could choose to favor specific dimensions of interest by weighting them differently. We leave the investigation of alternative metric weighting strategies to future work.

Based on the definition of $D_{i,j}$, we establish the Mobility Behavior-based user Exposure Score (MoBES), which quantifies the exposure of a given user based on her relative distance
to other users in the defined metric (i.e., behavioral) space. Specifically, the value of MoBES for a user i, MoBESi, is computed as the average distance from user i to her k-nearest neighbors in the m-dimensional space of metrics. We let k be an input parameter to accommodate various notions of exposure, from stricter (smaller values of k) to more relaxed ones (larger values of k). In other words, let NNi be the set of the k nearest neighbors of i, identified by using Equation 1 to compute the distances from i and every other user in the dataset. MoBESi is then defined as:

$$MoBES_i = \frac{\sum_{j \in NN_i^k} D_{i,j}}{k \times |M|} \quad (2)$$

Lastly, we define threshold $\tau$ as lower bound on MoBES score for a user to be considered exposed. Any user i with MoBESi ≥ $\tau$ is considered exposed. We also define the set of exposed users, denoted by MoBES*, as all users that meet this exposure criterion i.e., MoBES* = {i, $\forall i \in U$ such that MoBESi ≥ $\tau$}. The selection of $\tau$ is context dependent: higher values imply stricter notions of exposures as set MoBES* would include users that more strikingly distant from their closest neighbors in the metric space.

### C. MoBES interpretability

One of the key strengths of MoBES lies in its high interpretability. By leveraging an m-dimensional space of metrics, our approach allows for intuitive and straightforward analysis of a user's exposure due to her behavior, as captured in this space. This is done by examining the contribution of each individual metric to the overall distance of a user to her k-nearest neighbors. Specifically, for a user i and a metric $m_r$, the distance between i and another neighbor j can be calculated solely in dimension r, similarly to Eq. 1. Using this, we compute $I_{i,r}$, the contribution of metric value $m_r$ to the value of MoBESi as:

$$I_{i,r} = \frac{\sum_{j \in NN_i^k} \left| \frac{m_r^i - m_r^j}{m_r^i} \right|}{\sum_{j \in NN_i^k} D_{i,j}} \quad (3)$$

This fraction quantifies the percentage contribution of the metric $m_r$ to the user's exposure score MoBESi. Therefore, we obtain a percentage-based interpretation of each metrics' importance. Metrics with higher importance values indicate dimensions where the user's behavior is most distinct compared to their neighbors (i.e., captures facets of the user behavior that makes him more vulnerable to exposure). Thus, these metrics play a significant role in determining the user's overall exposure score in the behavioral space.

### D. Behavioral Metrics

To construct the m-dimensional space, we use a set of metrics that individually captures different perspectives of user mobility and, collectively, offers a holistic model of user mobility behavior, based on which MoBES can be computed [6].

We emphasize that it is not our goal to propose new mobility metrics. Instead, our focus is on how to capture, in a unique continuous (R) exposure score, multi-dimensional mobility behaviors of users. Indeed, as defined, MoBES computation is generic and can be instantiated for different sets of metrics. Even more, new metrics can be easily integrated as new dimensions, for refined modeling of user mobility.

For the sake of the present investigation, we select 13 metrics widely used in the literature for characterizing mobility behavior. We group these metrics into 3 categories, namely, Spatial, Temporal and Structural. Spatial metrics rely on geolocation, temporal metrics are derived from visiting times, while structural metrics depend on time-ordered visits, relying on both spatial and temporal information. Table I shows the metrics that were selected and gives an intuition about the behavior they capture.

Note that such metrics capture spatial-temporal routines, mobility preferences, and uncertainties in human behavior. Together, they can identify multifaceted behavior vulnerabilities. Note also that, as mentioned, the distance metric used captures the differences between users as percentages, eliminating the need for additional metric normalization.

### E. MoBES: Key Takeaways

This work presents a novel standpoint to quantifying and interpreting user exposure based on mobility behavior. We introduce an innovative measure – a user exposure score, which captures multiple perspectives of user mobility patterns. Our score is represented within an m-dimensional metric space and is independent of adversarial knowledge. As such, it captures fundamental behavioral properties that may contribute to user exposure. MoBES is generic and customizable, as it allows for different measure instantiations – by considering different sets of metrics and parameter values – to accommodate various scenarios of interest. By relying on simple and easy to compute calculations, MoBES quantifies user exposure in a scalable and controllable manner. By offering continuous scores (as opposed to binary user labels), it also ensures an intuitive and nuanced graduation of the user's degree of exposure.

Indeed, the flexibility to adjust parameters k and $\tau$ allows for capturing user exposure within the population at varying levels of precision. On one hand, k captures the size of the crowd with whom a target user may be "blended" not to be exposed. With k=1, it provides a localized view in the metric space, capturing the most similar user in terms of mobility behavior. With k=$|U|$, it captures broader patterns, identifying how a user is distinguishable from the whole population. The choice of $\tau$, in turn, allows for tuning the extent of behavioral difference, above which the user is considered exposed

Additionally, the distance metric (Eq. 1) is easy to interpret, allowing a continuous measure of how much a user's behavior deviates from their k nearest neighbors in each of the m dimensions. In other words, it allows for the quantification of the effort required from a user to reduce her exposure or to blend in with her neighbors. Indeed, MoBES provides each user the relative contribution of each behavioral metric to her
<table><caption>TABLE I: Mobility behavioral metrics.</caption><thead><tr><th>Type</th><th>Metrics</th><th>Reference</th><th>Intuition</th></tr></thead><tbody><tr><td rowspan="5">Spatial</td><td>Radius of gyration</td><td>RoG [9]</td><td>Captures the confinement zone characteristic of the life mobility of a user. Small values indicate high confined mobility. It is measured as the average distance to the center of mass (i.e. the central point) of the users trajectory.</td></tr><tr><td>2-Radius of Gyration</td><td>2kRoG [17]</td><td>Captures the displacement behavior relative to the strongest routine of the user, which is usually home and work, justifying the 2 in the metrics name.</td></tr><tr><td>Max. displacement</td><td>MDIST [18]</td><td>Captures the maximum distance displaced between two consecutive points.</td></tr><tr><td>Avg. and std. dev. jump lengths</td><td>AJL and SJL [9]</td><td>Captures the average distance between the consecutive displacements made by an user</td></tr><tr><td>Number of visits</td><td>NoV [6]</td><td>Captures how dynamic is the user in terms of amount of visits. High values indicate a high number of locations visited.</td></tr><tr><td rowspan="2">Temporal</td><td>Number of unique locations</td><td>NUL [9]</td><td>Captures the behavior concerning the diversification of visits per user</td></tr><tr><td>Avg. and std. dev. waiting times</td><td>AWT and SWT [8]</td><td>Captures activity in networking communication, what is reflected in frequency of recordings</td></tr><tr><td rowspan="4">Structural</td><td>Diversity</td><td>DIV [19]</td><td>Captures how diverse a user's visitation pattern is in terms of segments of visited locations. Given a trajectory as a time-ordered sequence of locations, diversity is defined by the number of distinct segments or locations sequence combinations that can be extracted from it.</td></tr><tr><td>Regularity</td><td>REG [20]</td><td>Captures if a user is prone to return to previously visited locations</td></tr><tr><td>Stationarity</td><td>STAT [21]</td><td>Quantifies if an individual stays continuously in the same location</td></tr><tr><td>Real entropy</td><td>ENT [8]</td><td>Captures the uncertainty in the user behavior. It builds upon the visitation frequency, the order in which the locations were visited, and the time spent at each location.</td></tr></tbody></table>

final score, enabling a *deeper understanding of the specific factors (i.e., metrics) driving the user exposure* (Eq. 3). This also brings *observability* advantages to MoBES.

As a powerful after-effect, MoBES enables the design of more *personalized protection mechanisms* tailored to (i) the user's most distinguishable behavioral metrics (as identified by Eq. 3) and (ii) the most at-risk users, rather than applying uniform measures across all metrics or the entire population. Thus, even less vulnerable users receive appropriate safeguards, allowing the design of more efficient, customized, precise, and sustainable approaches to privacy protection.

In the following section, we discuss results from one instan-
tiation of MoBES, where user exposure is estimated based on
the set of metrics introduced in Section III-D.

IV. INSTANTIATING MOBES

This section provides an overview of the dataset used to illustrate MoBES (§IV-A). We then deeply analyze user mobility behavioral patterns within each dataset (§IV-B) By understanding patterns on an individual user basis, we can appropriately parameterize MoBES (§IV-C) through its controllable capability. Finally, we assess the vulnerable behaviors captured by MoBES in the dataset (§IV-D), highlighting its interpretability, observability, and ability to customize to both multi-dimensional mobility patterns and at-risk users.

<table><caption>TABLE II: Datasets statistics (Confidence Interval (CI) 95% = 0 for all Avg).</caption><thead><tr><th>Metric</th><th>Shenzhen</th></tr></thead><tbody><tr><td><em>Total nb. of users</em></td><td>41159</td></tr><tr><td><em>Total nb. of days</em></td><td>15</td></tr><tr><td><em>Total nb. of unique locations</em></td><td>814</td></tr><tr><td><em>Avg. nb. of unique locations per user</em></td><td>7.81</td></tr><tr><td><em>Avg. nb. of unique locations per user per day</em></td><td>3.20</td></tr><tr><td><em>Avg. nb. of records/visits per user</em></td><td>158.33</td></tr><tr><td><em>Avg. nb. of records/visits per user per day</em></td><td>10.74</td></tr><tr><td><em>Avg. nb. of records/visits per hour per day</em></td><td>1.26</td></tr></tbody></table>

A. Data description and pre-processing

We focus on network-sourced dataset that capture mo-
bile communication activities (calls, SMS) within telecom
networks operators, such as Call Detail Records (CDRs).
CDRs typically cover large populations, extended periods
(e.g., months), and vast geographical areas (e.g., cities or
regions), offering valuable insights into mobility behaviors.
However, the spatial and temporal granularity of recorded
events influences how effectively CDRs serve as proxies
for mobility pattern analysis. In this paper, and to ensure
reproducibility, we evaluate MoBES using fully anonymized,
publicly available Shenzhen (China) CDR [13], [22].

Shenzhen CDRs capture mobile activities across three tele-
com networks in Shenzhen. With coverage spanning 1,090
base stations (1,952 km²) and an observation period of several
weeks, this dataset reflects diverse locations and complex
mobility patterns. As detailed in [7], users fall into three
main categories: *Scouters* (48%), who prioritize exploration;
*Routiners* (24%), with strong habitual patterns; and *Regulars*
(28%), who balance routine and exploration. This diversity
supports a comprehensive evaluation of *MoBES* performance.

Shenzhen dataset contains 414,271 individual trajectories. Each location in these trajectories correspond to connections to one of the cellular towers whenever an event occurs (call or SMS). While timestamps are recorded, all specific date information has been removed for anonymization, preventing the determination of whether records from different users were generated on the same day. To overcome this restriction, we adopt the method proposed in [7], assuming all users start on the same day, denoted as $d_1$. Timestamps are then assigned sequentially to days $d_x$ until a timestamp appears that is equal to or earlier than the previous one, marking the start of a new day, $d_{x+1}$. This leads to 949 days in total and to varying traffic days per user, a common scenario in public data initiatives (e.g., Geolife).

To mitigate spatial-temporal sparsity, we apply a 200m ×
<figure><img src="image_2.png" /><figcaption>Fig. 1: Shenzhen metrics distribution.</figcaption></figure>

200m tessellation based on OpenStreetMap [6]. We then filter out inactive users with fewer than 10 days of data and 100 records [19], [23], as their limited data could distort mobility behavior's exposure analysis. To address temporal gaps, we employ a data completion strategy that reconstructs missing records around home and work locations [7], [19], [23]. Finally, To mitigate the significant drop in user count over time, we focus on the first 15 days of the dataset, which also have the highest user coverage. This pruning helps prevent misleading uniqueness assessments on days with very few users. Table II summarizes resulting key dataset statistics, including averages and 95% Confidence Intervals.

## B. Extracted Metrics

This section explores the diversity of user mobility in the Shenzhen CDRs by analyzing trajectories and mobility metrics (cf. §III-D), as illustrated in Fig. 1. We start by examining the **spatial** metrics (RoG, 2kRoG, MDIST, AJL, SJL), we observe relatively low values, indicating small geographical displacement (i.e., short travel distances) and highly confined mobility (RoG below 20km, with most values under 5km). This limited movement is further reflected in the short distances between consecutive visits (AJL, SJL) and the low average number of unique locations per user (NUL) (Table II).

When analyzing the distribution of **temporal** metrics (AWT, SWT), a notable difference in behavior emerges. This discrepancy is largely due to the dataset's event-based data collection methodology. In Shenzhen CDRs, location updates depend on user activity (e.g., calls and SMS), which results in more frequent recordings for highly active users and fewer recordings for less active ones. Consequently, the observed averages and standard deviations of waiting time vary significantly, explaining the differences in Shenzhen's mobility patterns.

Finally, the temporal disparity also affects **structural** metrics (ENT, STAT, REG, DIV), as they depend on both time and space. Like temporal metrics, the detection of user stationarity in Shenzhen CDRs is highly sensitive to the data collection method, resulting in varying values while maintaining a balanced average. Although the observation period is short (15 days), the distribution of regularity (REG) values shows that, nature of human mobility, users tend to revisit the same locations frequently. However, high regularity does not always imply low diversity (DIV). Indeed, users frequently engage in exploratory behavior, diversifying their patterns while still maintaining elements of routine in their mobility. We can also observe that the entropy (ENT) – which measures the level of

randomness in visiting behaviors – remains low. Indeed, Shenzhen users exhibit highly confined mobility, strong regularity in their visits, and moderate stationarity despite high variability across the population. This means they spend varying amounts of time at locations but frequently move between familiar ones, forming predictable movement patterns contributing to low entropy values. In conclusion, Shenzhen CDRs reveal distinct characteristics across the population, enabling the analysis of different levels of user exposure.

## C. MoBES Parameterization

The MoBES approach is parameterized by two key values, as outlined in §III: *k*, the number of nearest neighbors used to capture a user's relative distance, and *τ*, the threshold that defines users as exposed. In this section, we analyze the impact of these parameters on MoBES estimation for our dataset. **Parameter k.** We evaluate MoBES using four different values of *k*: 1, 5, 10, and $\sqrt{|U|}$, where $|U|$ represents the total number of users in the dataset. These values are not arbitrarily. Indeed, in theory, a user with at least one sufficiently close neighbor could be considered as protected. However, if that neighbor changes behavior, exposure may arise. A large value of *k* provide a more stable estimation of MoBES and of user exposure. Finally, we also use $\sqrt{|U|}$, a commonly used empirical value for selecting *k* in nearest neighbors algorithms [24].

Fig. 2 illustrates the distribution of MoBES for different values of *k*. Note that as the size of *k* increases, the distribution shifts to the right, with MoBES values increasing accordingly. Indeed, larger values of *k* result in the inclusion of neighbors farther from the user *i*, which contributes to higher MoBES estimates. For each selected value of *k*, we obtain a ranking of users in *U*, where the top positions correspond to users with the highest exposure, while the bottom positions are occupied by users who are less exposed, i.e., closer to their neighbors. As shown in Fig. 2, users at the top of the ranking have a MoBES value of approximately 0.3. This indicates that the most exposed users need to adjust about 30% of their behavior to blend in with the overall population.

**Parameter τ.** A user's MoBES value does not need to be 0 (indicating identical metric values and behavior to their nearest neighbors) to be considered unidentifiable. τ serves as a tolerance threshold, representing a percentage of deviation. However, defining an appropriate τ is challenging, as it directly determines to what extent is the user exposed. In § V, we use three values for τ: {0.05, 0.10, 0.15}. With τ = 0.05, a user is considered as exposed if their behavior deviates
<figure><img src="image_3.png" /><figcaption>Fig. 2: Distribution of MoBES for different values of k.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Metrics importance for varying *k* and MoBES ranges. Insider values represent the importance average and *CI*<sub>95%</sub>. Hotter colors (closer to red) describe higher importance in MoBES measuring.</figcaption></figure>

by at least 0.05 across all dimensions from their *k*-nearest neighbors. $\tau = 0.05$ captures subtle behavioral variations within the neighborhood. Then, we increase it to a higher value, $\tau = 0.15$, to account for larger deviations.

### D. Patterns captured by MoBES

One of the advantages of MoBES is its high interpretability, which allows us to measure and rank the contribution of each metric to the risk score and properly address protection mechanisms according to the metrics that most impact user exposure. In this scenario, the impact of metrics can vary significantly

across users, showing the different behaviors observed. Indeed, among the 41,159 users in Shenzhen, almost 36,000 unique importance rankings, respectively, with *k* = 1, thus making it very hard to analyze the importance of the metrics manually. However, a general pattern emerges: certain metrics consistently rank at the top, while others tend to remain at the bottom. Thus, to make our further discussion easier, Fig. 3 shows our analysis on the metrics' importance, average and $CI_{95\%}$, when (i) varying *k* and (ii) grouping users according to three MoBES ranges: (low) $MoBES_i \le 0.05$, (moderate) $0.05 \le MoBES_i < 0.15$ and (high) $MoBES_i \ge 0.15$. Hotter colors (closer to red) represents metrics that most contributed to the user exposure, while metrics with the least contribution are in colder colors (closer to green).

We first examine the metrics that generally have minimal or no impact on capturing exposure: Regularity (REG), Diversity (DIV), Number of Visits (NoV), and Average Waiting Time (AWT). NoV exhibits a highly concentrated distribution, which explains its limited significance as a metric.

A similar pattern was observed with AWT but not with SWT, which is generally more significant. This difference arises from variations in the intervals between records, which better capture differences in users' cellular communication behaviors. While users' temporal behavior is smoothed out by the average, the standard deviation provides a more accurate representation of variability in data collection, making it a better indicator of such behavior. Similar to AWT, the REG and DIV distributions show little variation between users, contributing minimally. In some cases, these metrics could be excluded as their impact on capturing user exposure is negligible.

Next, we discuss the most important metrics in each scenario. Our analysis is supported by Table III, which presents the average and $CI_{95\%}$ of the top five most important metrics across different parameter settings, highlighting the varying behaviors captured in these configurations.

* **Low ($MoBES_i \le 0.05$):** This group consists of users with a low distance to their nearest neighbors, exhibiting at most a 5% variation from them. As shown in Fig. 3a, increasing the value of *k* reduces the number of users in this range due to the shift in the MoBES distribution. Consequently, when $k = \sqrt{|U|}$, only users with no variation and low mobility (less active in the network) remain (as indicated by the $CI_{95\%}$) across several metrics: Regularity (REG), Stationarity (STAT), Number of Unique Locations (NUL), Standard Deviation of Jump Lengths (SJD), Average Jump Lengths (AJL), Maximum Distance (MDIST), and Radius of Gyration (ROG). This means that all users in this group share identical metric values. These users represent the lower end of the distribution and visited only one location throughout the observation period. Since multiple users exhibit this same no-mobility behavior, they provide mutual protection within the group. The only varying metric among them is Waiting Time (WT), where the
<table><caption>TABLE III: Average and $CI_{95\%}$ for the five main behavioral metrics, when varying $k$ and MoBES ranges.</caption><thead><tr><th>Values</th><th>Grouping users w.r.t. MoBES range</th><th>k</th><th>ENT</th><th>MDIST</th><th>RoG</th><th>SWT</th><th>2KRoG</th></tr></thead><tbody><tr><td rowspan="4">Low</td><td rowspan="4">0 ≤ MoBES &lt; 0.05</td><td>1</td><td>0.86 ± 0.01</td><td>2.05 ± 0.05</td><td>0.72 ± 0.02</td><td>252.31 ± 1.08</td><td>0.62 ± 0.02</td></tr><tr><td>5</td><td>0.72 ± 0.01</td><td>1.17 ± 0.03</td><td>0.4 ± 0.01</td><td>253.98 ± 1.42</td><td>0.35 ± 0.01</td></tr><tr><td>10</td><td>0.6 ± 0.01</td><td>0.8 ± 0.02</td><td>0.26 ± 0.01</td><td>249.67 ± 1.81</td><td>0.24 ± 0.01</td></tr><tr><td>√|U|</td><td>0.19 ± 0.0</td><td>0.0 ± 0.0</td><td>0.0 ± 0.0</td><td>191.01 ± 3.61</td><td>0.0 ± 0.0</td></tr><tr><td rowspan="4">Moderated</td><td rowspan="4">0.05 ≤ MoBES &lt; 0.15</td><td>1</td><td>1.26 ± 0.01</td><td>7.1 ± 0.11</td><td>2.2 ± 0.04</td><td>237.69 ± 1.07</td><td>1.31 ± 0.03</td></tr><tr><td>5</td><td>1.21 ± 0.01</td><td>6.01 ± 0.09</td><td>1.91 ± 0.03</td><td>240.76 ± 0.91</td><td>1.22 ± 0.03</td></tr><tr><td>10</td><td>1.18 ± 0.01</td><td>5.42 ± 0.08</td><td>1.74 ± 0.03</td><td>243.14 ± 0.85</td><td>1.15 ± 0.03</td></tr><tr><td>√|U|</td><td>1.07 ± 0.01</td><td>2.91 ± 0.04</td><td>0.98 ± 0.02</td><td>253.24 ± 0.82</td><td>0.78 ± 0.02</td></tr><tr><td rowspan="4">High</td><td rowspan="4">MoBES ≥ 0.15</td><td>1</td><td>1.53 ± 0.11</td><td>16.15 ± 1.51</td><td>4.74 ± 0.53</td><td>251.05 ± 14.04</td><td>1.24 ± 0.35</td></tr><tr><td>5</td><td>1.46 ± 0.07</td><td>15.86 ± 0.97</td><td>4.51 ± 0.33</td><td>247.33 ± 8.74</td><td>1.37 ± 0.24</td></tr><tr><td>10</td><td>1.42 ± 0.05</td><td>15.82 ± 0.71</td><td>4.39 ± 0.24</td><td>243.66 ± 6.32</td><td>1.45 ± 0.17</td></tr><tr><td>√|U|</td><td>1.28 ± 0.01</td><td>11.35 ± 0.21</td><td>3.46 ± 0.07</td><td>229.41 ± 1.8</td><td>1.78 ± 0.06</td></tr></tbody></table>

avg. remains similar, but the std. dev. differs significantly, as it is influenced by individual user behavior.

* **Moderate** ($0.05 \le MoBES_i < 0.15$): This group consists of users with moderated similarity in behavioral patterns to their neighbors. Despite a wider range of $MoBES_i$ values (concentrating most of users), the $CI_{95\%}$ in Fig. 3b shows minimal variation, indicating consistent behavior among these users. They exhibit shorter mobility, as reflected in spatial metrics like RoG and MDIST (Table III). Additionally, they display low stationarity (STAT), meaning they frequently change locations, but these locations are familiar and regularly visited, as indicated by high regularity (REG) and low entropy (ENT). As $k$ increases, this shorter-mobility pattern becomes more pronounced. However, unlike the low MoBES group (which includes low-mobility users at $k = \sqrt{|U|}$), these users have higher AWT and SWT, suggesting their lower network activity that results in fewer recordings in the dataset. In summary, these users with moderate mobility exhibit regular movement patterns, visiting a limited set of locations and staying close to their center of mass.

* **High** ($MoBES_i \ge 0.15$): This group includes users with a wider variation in importance, indicating that high exposure patterns are not well-defined or consistent across all users. As $k$ increases, the $CI_{95\%}$ of importances decreases, indicating more stable and less variable behavior patterns. In this context, users with high mobility are predominantly exposed. They typically travel to more distant locations, visit a greater number of unique places, and are less stationary. Thus, they experience more diverse displacements compared to users with moderate MoBES, who tend to remain within the same region, closer to their center of mass.

In summary, this section demonstrates the high interpretability of MoBES. Our analysis reveals that some metrics, such as REG and DIV, contribute little to MoBES, emphasizing the need for an adaptable metric that can accommodate different mobility patterns within a dataset. Conversely, RoG and STAT play a crucial role in identifying mobility exposure levels, while ENT provides insights into movement predictability. Additionally, AWT and SWT are highly impacted by users'

<figure><img src="image_5.png" /><figcaption>Fig. 4: Uniqueness in different Time Windows ($t_w$).</figcaption></figure>

communication patterns, where higher values indicate less frequent network activity despite movement. These findings highlight the importance of an adaptable metric like MoBES in accurately capturing diverse mobility behaviors and their associated exposure risks.

## V. COMPARING MEASURES OF USER EXPOSURE

We now compare MoBES with the widely used uniqueness measure [4], despite its high computational cost. While MoBES identifies users at risk of exposure based on singularities in their mobility behavior signatures (captured in the metric space), uniqueness detects exposure through singularities in sequences of visited locations. Our goal is to analyze the overlap between the users identified as exposed by both methods and to discuss their similarities and differences.

### A. Uniqueness

Before comparing the two approaches, we first analyze the uniqueness values in Shenzhen CDRs. A lower uniqueness value indicates higher exposure, with users having a uniqueness of 1 (minimum) considered fully exposed, as their trajectories are unique within a given time period [4]. We refer
to this exposed user set as $Uniq^*$. Fig. 4 shows the uniqueness distributions for different aggregation windows $t_w$ (introduced in § II). Notably, the fraction of users in $Uniq^*$ is quite high, reaching nearly 80% of the population. Additionally, we observe a diminishing return effect as $t_w$ increases. Therefore, we set $t_w = 4$ for the following discussion.

## B. Comparing MoBES with Uniqueness

We now compare the sets of users identified as exposed by MoBES and uniqueness. In addition to $Uniq^*$, we define $MoBES^*$ as the set of exposed users according to MoBES and the selected values for $k$ and $\tau$. To capture different exposure scenarios, from strict to relaxed criteria, we compare $MoBES^*$ and $Uniq^*$ across a range of parameter values. Specifically, we vary $k = \{1, 5, 10, \sqrt{|U|}\}$ and $\tau = \{0.05, 0.1, 0.15\}$.

In our comparison of $Uniq^*$ and $MoBES^*$, we separately analyze three subsets: (i) $Uniq^* - MoBES^*$: users considered exposed by uniqueness but not by MoBES, (ii) $Uniq^* \cap MoBES^*$: users considered exposed by both uniqueness and MoBES, (iii) $MoBES^* - Uniq^*$: users considered exposed only by MoBES. Fig. 5 shows the fractions of users in each subset, for varying $k$ and $\tau$. As an example, for $k=1$ and $\tau=0.05$, around 31% of the users are considered exposed only by the uniqueness, 7.4% only by MoBES, and 61.5% by both measures. Note that, as our goal is to compare both measures, these fractions are computed with respect to all users considered exposed by any measure (i.e., $Uniq^* \cup MoBES^*$), and not by the total user population.

We make three key observations. **First and foremost,** regardless of the parameter values, there is a non-negligible fraction of users exposed only by one of the measures (and not by both). This corroborates that $MoBES$ and $uniqueness$ are indeed capturing different (and, for many users, complementary) notions of exposure. Notably, in all cases but those with very strict notions of exposure (small values of $k$ and very large values of $\tau$), there is a significant fraction of exposed users revealed only by MoBES, and not by uniqueness. This showcases the ability of $MoBES$ to uncover behavior singularities, which may expose a user, but are not made explicit simply by looking into the sequences of places visited.

**Second,** for a fixed $k$, increasing $\tau$ makes the MoBES exposure criteria stricter, naturally reducing the number of users in $MoBES^*$. Consequently, the fraction of users exposed only by uniqueness ($Uniq^*$) increases, while the fractions in the other subsets decrease.

**Third,** for a fixed $\tau$, increasing $k$ has the opposite effect, relaxing the MoBES exposure criteria. For instance, when $k = \sqrt{|U|} = 202$ and $\tau = 0.05$, all users exposed by uniqueness are also flagged by MoBES. Moreover, in this case, about 22% of users are identified as exposed exclusively by MoBES.

This highlights how varying $k$ and $\tau$ allows for identifying different groups at risk of exposure, depending on the chosen criteria of exposure, capturing users who would not be flagged using only the uniqueness measure.

<figure><img src="image_6.png" /><figcaption>(a) % of users considered exposed only by uniqueness</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(c) % of users considered exposed only by MoBES</figcaption></figure>

Fig. 5: Fractions of users considered exposed by MoBES and/or uniqueness, for various MoBES instantiations (parameters $k$ and $\tau$). Fractions computed over all exposed users according to either measure ($Uniq^* \cup MoBES^*$).

Next, we dive into the sets of users flagged as exposed only by uniqueness (i.e., $Uniq^* - MoBES^*$) shown in Fig. 5a, and only by MoBES (i.e. $MoBES^* - Uniq^*$), shown in Fig. 5c. For the sake of completeness, we also discuss users who are not considered exposed by either measure. This discussion naturally reflects characteristics of the analyzed dataset and builds on the results presented in §IV.

* **Exposed users according to uniqueness:** despite having unique displacement patterns, these users exhibit mobility behaviors (captured by the metrics) that are not very distinguishable from others (i.e., low MoBES). As discussed in §IV, these are users with short or low-mobility. Each user typically visits the same places, often in remote or isolated areas. While many of these places are distinct within a given time window, these users are considered exposed according to the uniqueness measure. However, in terms of mobility behavior, these users exhibit very similar patterns to one another.

* **Exposed users according to MoBES:** these are users who exhibit large displacements, often not confined to a single area, and keen to explore new locations. They have very distinct behavioral patterns (high MoBES). Yet, the locations visited are not unique in a given time period, and often are places highly frequented (e.g., malls).

* **Protected users according to both MoBES and uniqueness:** these are users who exhibit short and low mobility, whose visitation patterns are restricted to very populated
areas (e.g., large commercial centers). As such, they naturally blend with the crowd, according to both measures.

Our results demonstrate that MoBES complements the state-of-the-art uniqueness measure by enabling the assessment of user exposure across dimensions that the uniqueness measure may overlook. As shown, a significant number of users, while blending in with others in terms of the places visited during specific time periods, exhibit distinct behavior signatures that make them easily distinguishable and potentially identifiable.

## VI. CONCLUSIONS

This paper proposes a novel metric to quantify the exposure in mobility datasets called MoBES (Mobility Behavior-based user Exposure Score). MoBES works by leveraging behavioral mobility metrics to quantify variation from a user to its *k* nearest neighbors. Such metrics allow MoBES to capture the unicity in users mobility behavior in different perspective from the one captured by other metrics in the literature. MoBES joins several aspects seen in different metrics in the literature as *high-interpretability*, where it is possible to identify the individual contribution of each feature to the users exposure. *Continuous usability*, which allow to individually identify the degree of exposure of each user. *Independence on adversarial knowledge*, meaning that MoBes is not tied to any attack to quantify the users exposure. *Low computational cost*, which comes from a simple methodology based on *k*-NN, and lastly the possibility to *capture different perspectives of the exposure*, open the possibility to use side information. Hence, MoBES allows practitioners to quantify the overall exposure of users in their datasets, and properly address the vulnerability according to the characteristics and level of exposure.

We evaluated MoBES using CDR datasets from Shenzhen, observing distinct exposure patterns: highly-exposed users are explorers with diverse mobility and showed higher exposure, constrained by the city's size. Additionally, we compared MoBES with uniqueness, finding that it captures similar patterns at lower computational costs and identifies new, highly-exposed users who are missed by uniqueness.

As future work, we intend to develop an automatic way to select the values of *k* and *τ*, simplifying parameter selection. Besides, we plan to explore additional metrics that can capture new aspects of the user's behavior, and to explore distinct weights when considering these metrics. Finally, we intend to evaluate our approach with other datasets and assess how different defense mechanisms would perform in users with different ranges of MoBES.

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