# Understanding mobility in networks: A node embedding approach

Matheus F. C. Barros

Carlos H. G. Ferreira

Bruno Pereira dos Santos

Universidade Federal de Ouro Preto  Universidade Federal de Ouro Preto
Universidade Federal de Minas Gerais
Politecnico di Torino  Universidade Federal de Ouro Preto  Lourenço A. P.
Júnior

Marco Mellia

Jussara M. Almeida

Instituto Tecnológico de Aeronáutica  Politecnico di Torino  Universidade
Federal de Minas Gerais

(2021)

###### Abstract

Motivated by the growing number of mobile devices capable of connecting and
exchanging messages, we propose a methodology aiming to model and analyze node
mobility in networks. We note that many existing solutions in the literature
rely on topological measurements calculated directly on the graph of node
contacts, aiming to capture the notion of the node’s importance in terms of
connectivity and mobility patterns beneficial for prototyping, design, and
deployment of mobile networks. However, each measure has its specificity and
fails to generalize the node importance notions that ultimately change over
time. Unlike previous approaches, our methodology is based on a node embedding
method that models and unveils the nodes’ importance in mobility and
connectivity patterns while preserving their spatial and temporal
characteristics. We focus on a case study based on a trace of group meetings.
The results show that our methodology provides a rich representation for
extracting different mobility and connectivity patterns, which can be helpful
for various applications and services in mobile networks.

###### keywords:

Network Mobility, Node Embedding, Network Management

††conference: Workshop on AI in Networks and Distributed Systems (WAIN) 2021
Milan,Italy

##  1 Introduction

Analyzing and modeling mobility patterns have fundamental roles in mobile
network prototyping, design, and deployment. To study such aspect, we can use
mobility data gathered directly from the devices or the infrastructure that
they are attached to. Examples of such data are GPS trace, call detail records
(from phones), location-based social media, etc [15]. With these data, one is
able to study the behavior of mobile entities and model their mobility
patterns, which can be used to evaluate the performance of newly designed
networking protocols and solutions [10, 19, 22].

One way to model mobility is through traditional temporal graph theory. In
this approach, nodes are the network entities, and the edges are interactions
(i.e., calls, message or data exchanges) over time. Conveniently, well-known
graph theory models [28, 3] are ready to use. Armed with this, several
proposals design solutions for, e.g., networking routing by exploring node
importance criteria such as centrality metrics, community structure, and
social metrics [24, 29, 44, 23, 13, 6, 3]. However, each metric captures a
specific notion of node importance, indicating which nodes are central
elements in the network structure. Although some earlier studies and solutions
on network mobility have explored mobility modeling based on such metrics,
they cannot generalize a network nodes’ importance criteria and, most
important, do not consider the temporal dimension [26, 11]. These limitations
are especially harmful, for example, to the validation of opportunistic
routing protocols since none of the existing modelings approaches fully cover
the nodes’ importance criteria on the network connectivity [18, 22].

Recently, node embedding techniques have been proposed as an alternative to
studying mobility rather than traditional graph theory. There exists a range
of mobility-related applications of embeddings such as Point of Interest (PoI)
prediction and recommendation [35, 36, 20, 1], and urban mobility modeling and
planning [42, 31, 33]. In a nutshell, node embedding encodes a graph structure
to low dimensional vector space (a.k.a. latent space), unveiling hidden
patterns that are hard to catch using only the graph topology [16]. Although
such features are desirable for exploring mobility patterns to build solutions
in a network context, few efforts use embeddings to understand mobility
patterns and capture the importance of connections established by mobile
nodes, specifically valuable to design routing and information dissemination
solutions in mobile networks [11].

Aiming to address the aforementioned issues, we propose using node embedding
to model and analyze the mobility pattern over spatial and temporal
dimensions. Our embedding-based methodology goes beyond capturing the
topological aspects of the network, the mobility, and it factors the evolution
of such patterns through a sequence of networks over time too. We present a
case study related to the regular meeting of groups of people that applies our
approach. The results show that we can extract and analyze mobility patterns,
identifying strategical nodes for network connectivity and tracking the
evolution of such properties over time. Our approach, albeit preliminary, can
be seen as a first alternative to support the development of communication
protocols, framework design, share and allocation of network resources, or
applications [28, 32] which considers node stability over time.

This work is organized as follows: Section 2 summarizes the main efforts in
the literature that employ embeddings approaches for studying network
mobility, Section 3 presents our case study, while Section 4 describe the
methodology. Next, the results are presented in Section 5. Finally, Section 6
discusses the main findings, offers conclusions, and future directions.

##  2 Related Work

Modeling mobility and interactions between entities has rich applicability.
For example, mobility-aware solutions were proposed focusing on routing
metrics [4, 22], protocols and actions for coordination [2], and analysis of
individuals’ behavior on a mobile network [14]. However, solutions based on
node embedding techniques have been little explored. Following, we review some
prior work that model mobility pattern using embedding methods.

Yu et al. [40] proposed inferring the relationship strength between users as
well as understanding the motivation behind them. To do this, they model a
contacts frequency network between two users whose edge weight is adjusted
based on contextual information from the contact’s location. Yang et al. [35]
focused on the tasks of predicting new friendships and predicting the location
of users. First, the authors proposed LBSN2Vec, an embedding approach that
receives as input a network of contacts modeled as a hypergraph from Location-
Based Social Networking (Location-Based Social Networking (LBSBN)) data. In
another work [36], the authors introduced contextual information made possible
by the creation of hyperedges considering not only users’ contacts but also
points of interest (PoI) and time.

Embeddings techniques in the context of mobility have been applied in
information retrieval tasks. Specifically, in the context of PoI
recommendation, Rahmani et al. [20] proposed CATAPE, a method that
simultaneously incorporates sequences of user locations and categories of
PoI’s visited by them to recommend potential new points of interest. Likewise,
Yuan et al. combined information from historical and contrasting users’
queries to find mobility similarity [41]. Such approach focuses on
recommending a PoI to a user through keywords in an incomplete query. Wang et
al. focused simultaneously on recommending and predicting users’ PoIs [31].

Motivated by urban mobility planning, Zhang et al.[43] used an embedding
strategy to extract mobility patterns on transport lines using user mobility
data in the transport system. Later, the authors considered data from PoI’s
for the construction of urban mobility networks for users in order to find
mobility patterns among users considering both data sources [42]. Similarly,
Wang et al. proposed a framework based on embedding whose focus is to analyze
the dynamics of communities formed by mobility between PoIs in different
periods (i.e., weekdays and weekends) [30].

Unlike some previous work [40, 35, 36], our approach does not consider
contextual information. We notice that such information offers greater
possibilities for pattern analysis and extraction. On the other hand, it poses
extra challenges such as data collection, storage, and processing. In
addition, collect such data demands financial, operational, privacy, and
security issues. Therefore, unlike these works, our approach is independent of
contextual information and is based exclusively on a sequence of contact
graphs that represent the topological organization of the network. This is
motivated by the problem addressed here, which consists of modeling the
importance of users for network connectivity considering their mobility.
Therefore, we only use network’ topological information in this initial
effort, although we can extend it to consider contextual information.

Finally, we discuss works related to improving urban mobility and networking
design [42, 31, 33]. Concerning these efforts, our proposal can be seen as an
initial attempt towards planning and designing applications inherent to
effective routing in opportunistic networks, which offers a range of
applications in smart cities such as dissemination/collection of information,
location, vehicular communication, among others [8, 25].

##  3 Dataset

One of the greatest challenges of performing mobility analysis in networks is
the lack of reliable and open data in such field. This is due to privacy
issues, scale (space, time, # entities) or costs needed to collect and store
such data (typically huge amounts). Fortunately, efforts have been done to
build mobility models able to generate synthetic trajectories for mobile
entities by using real data. Synthetic approaches offer a cheap and enabler
alternative for quick and scalable performance evaluation of networking
solutions in a near real environment with different scales of space, time, and
number of entities [38]. Also, synthetic models enable reproducibility and
fair comparison of different methods. Given this, in this work we check the
potential of node embeddings using synthetic mobility traces. In detail, we
use the Group Regularity Mobility Model (GRM) [19]. GRM models group meeting
dynamics and regularity in human mobility. It simulates the human mobility
pattern by considering the cyclical and regular behavior commonly exhibited by
humans, superposing the effect of sporadic groups meetings such as sport
events, presence of places for leisure, parties, etc. The results is a trace
where nodes moves, replicating the periodic behavior with the superposition of
those occasional meetings. These features differentiate GRM from other models
in the literature.

To produce a trace (nodes trajectories), GRM receives as input the area,
duration, number of nodes, and groups. Some statistical parameters offer the
ability to generate different scenarios: (i) Group Meeting Time that defines
the time interval between meetings; such interval is governed by a power-law
with exponential cut-off with parameters $\alpha_{gmt}$ and $\beta_{gmt}$;
(ii) Group Meetings Duration that defines the time for which nodes will spend
together; this duration follow a truncated power law distribution of
parameters $\alpha_{dur}$ and $\beta_{dur}$; (iii) Group Structure that
decides which nodes will be at each meeting; the grouping approach follows
also a power law with exponential cuts tuned by $\alpha_{size}$ and
$\beta_{size}$; (iv) Social Context is a network of social edges (“friends”)
between nodes; GRM consider social relationship into account because people
have higher probability to attend meetings in which friends’ nodes participate
into. For this, GRM assumes that each group has a regularity factor defined by
$K$. For example, groups with a factor $k=24h$ meet every 24, 48, or 72h. This
parameter works as a multiplier that generates the periodic behavior of the
traits. Here, we use the same parameters as the GRM’s authors to generate our
dataset trace (see Table 1). It is worthy to highlight that the choice of this
trace, in particular, does not prevent our approach from being used with other
traces.

Table 1: GRM input parameters.

Parameters | Values
---|---
# of Nodes | 100
# of Groups | 500
| Sim. Duration
---
87 days
K | 70% -24h; 15% -7 days; 15% - 6h
Grid | 30 x 30
Cell Size |  50m2
$\alpha_{gmt}$ | 3
$\beta_{gmt}$ | 30 days
$\alpha_{dur}$ | 3
$\beta_{dur}$ | 30 days
$\alpha_{size}$ | 2.24
$\beta_{size}$ | 30
Social Network |  Gaussian Random Partition [7]

##  4 Methodology

This section describes the methodology we propose in our study, starting with
our network model (Section 4.1) followed by the spatial-time aware node
embedding approach applied to model a latent space (Section 4.2).

###  4.1 Network Model

To model the nodes’ contact network observed in graphs, we first discretize
the period analyzed in non-overlapping time windows $\\{\Delta_{t}\\}$ of
duration 24 h111The choice of daily time windows is based on the meetings
periodicity presented in Table 1, but can be configured according to the
scenario. It should be noticed that 85% of meeting occurs on a daily scale.
represented by the set $T=\\{1,2,...,87\\}$. For each time window $\Delta_{t}$
we build an undirected graph $G_{\Delta_{t}}$, with
$G_{\Delta_{t}}=(V_{\Delta_{t}},A_{\Delta_{t}})$, such that $V_{\Delta_{t}}$ =
{$v_{1},v_{2},v_{3},...,v_{i}$} is the set of nodes representing individuals
at the instant of time $\Delta_{t}$; and $A_{\Delta_{t}}$ =
{$a_{1},a_{2},a_{3},...,a_{n}$} a set of edges connecting any two nodes
$v_{i}$ and $v_{j}$ ($v_{i}\neq v_{j}$). We assume two nodes are connected if
they are within a radius of 100 meters during the considered $\Delta_{t}$. In
practice, this could represent an exchange or a potential exchange of messages
between any two nodes for specific scenarios and applications such as those
provided by the Wi-Fi 6 [34, 39] technology. It is important to highlight that
such parameters can be configured according to the application.

With the definition of the graph, we consider some purely structural measures
typically in the literature in mobile network applications that provide
knowledge of node importance connectivity. Here, we can compare them to the
patterns captured from our methodology. Among the various possibilities, we
selected some of the primary measures of centrality (Degree, Betweenness,
Closeness, Eigenvector), and clustering coefficient, previously adopted in
various solutions in network management [24, 29, 44, 18, 23, 13]. For brevity,
we recommend reading [17, 3] for more details about such metrics.

###  4.2 DynamicNode2Vec

We use our prior DynamicNode2Vec technique to extract mobility patterns from
mobile nodes [11]. Such technique is able to temporally represent nodes from a
sequence of graph networks (as modeled in Section 4.1) in a low-dimensional
vector space (embeddings) while preserving the existing network properties
(e.g. neighborhood). DynamicNode2vec uses a biased random walk algorithm based
on a state-of-art method for static node embedding known as Node2Vec [12]. The
intuition behind those walks is to traverse the graph randomly and produce a
sequence (i.e., vectors) of visited nodes. The extraction of several random
sequences generates a sampling process, where the appearance of nodes depends
on the original graph. The algorithm defines two parameters: (i) $nw$ defined
as the number of walks (or extracted sequences) per node; (ii) $wl$ the number
of steps of each walk (or the length of each sequence).

To configure how the sampling will be performed, the algorithm can use Depth-
First Sampling (DFS) or Breadth-First Sampling (BFS) through the $p$ and $q$
parameters. In details, $p$ determines the probability of immediately
returning to an already visited node; $q$ controls whether the tour is close
to the origin node (for BFS) or it goes further away (for DFS). Here, we are
interested in BFS technique, which captures the proximity of nodes using a
neighborhood-first strategy. In a nutshell, by sampling preferably neighbor
nodes, the generated walks tend to include all nodes that belong to the same
portion of the graph. Therefore, we tend to encode the importance of
connections between nodes once mapped in the low-dimensional vector space. For
this, we set $p=1$ and $q=0.5$ to get this behavior (see authors
recommendations in [12].

Given we have a different graph at each time window $\Delta_{t}$, we generate
the paths (walks) for each of these graphs. As result, nodes that are closer
to each other in a given time window have a high number of shared neighbors,
and important edges for the network connectivity are more likely to appear
close together in the multiple paths sampled in the respective graph
$G_{\Delta_{t}}$. After, DynamicNode2Vec incorporates an optimization model
that efficiently observes the node association and their changes over time as
they appear on the sampled paths, mapping them into a latent space temporally
equivalent [37]. In summary, this is achieved as follows.

First, for each observed time window we compute a node co-occurrence matrix
through a sliding window (of size five as suggested in [37]) that runs over
the sampled paths. That way, it counts how many times two nodes are co-visited
within that sliding window. The greater the co-occurrence between these two
nodes, the closer they are in the network, and the more important is the
connection between them. This matrix is known as Positive Pointwise Mutual
Information (PPMI). At the next step, we create a compact representation in
$d$ dimensions from each PPMI matrix derived from the sequences of observed
networks. This step is done by a low-rank matrix factorization obtained by
solving an optimization problem. We consider two factors in the objective
function design to handle, respectively, overfitting ($\lambda$) and alignment
($\tau$). Their definition is based on a training process. For the sake of
brevity, we recommend reading the original reference for more detail [11]. For
our study, we set the parameters as follows: $nw=4$, $nw=8$, $\lambda=50$, $\
tau=15$ and $d=50$.

With the obtained embeddings, we can use two metrics to quantify the degree of
mobility of nodes on the network, and the importance of the connections
established by them: [11, 37]:

Cosine Distance: Quantifies the similarity between two vectors in the same
vector space. Given two vectors generated by a node (user) $v_{i}$ in any two
time windows $\Delta{t_{1}}$ and $\Delta{t_{2}}$, the cosine distance is
defined as $cos(x,y)=1-x\cdot y/(||x||||y||)$. The cosine distance ranges from
0 to 1. Values close to 0 indicate that the node is always close to the same
nodes in the two compared time windows; values close to 1 indicate that the
node has drastically changed its network of contacts in the period, with total
different neighbors.

Vector norm: By definition, the more a node appears in the sampled paths in
the graph, the greater is the vector norm of its vector, taking into account
all vector space of observed time. This way, it is possible to identify those
nodes that exhibit an important connectivity pattern since they are frequently
inserted in the sampled paths. The main advantage of using the vector norm
regarding the frequency with which nodes appear on the sampled paths is that
it is more robust to sporadic disturbances [37].

##  5 Results

In this section, we present the main results of our work. First, we
illustrates our embedding representation of the mobile network considering
space and time characteristics. Next, the node mobility and importance is
analyzed through the case study are presented.

###  5.1 Exemplifying the spatio-temporal model

To exemplify how our proposal works, we pick just the first six time windows.
Recall the DynamicNodeVec produce a node embedding representation of 50
dimensions. To be human-visible, we reduce from 50 to 2 dimensional
representation by using t-SNE [27], a dimensionality reduction technique for
data visualization. Furthermore, to illustrate how the node dynamics can be
captured in the latent space modeled, we select only two nodes that presented
the highest and lowest average day-to-day mobility (review Section 4.2)
according to the first six time windows. We compute the cosine distance day
after day for all nodes. Then, we compute the average of these distances for
each node.

Figure 1: t-SNE Two-dimensional representation of node embeddings for the
first six-time windows.

Figure 1 displays the nodes that have the highest and lowest average mobility
in the period. Their average cosine distance were 0.10 (min) and 0.43 (max),
respectively. The node with the lowest mobility presented stable (few
topological changes) behaviors in the first three time windows (Figure 1 top
plots). However, between the time windows $\Delta{t_{3}}$ and $\Delta{t_{4}}$,
a shifting is observed, indicating a change in connectivity of this node.
Afterward, the node remains stable until the window $\Delta_{t_{6}}$. On the
other hand, it is possible to notice a high day-to-day variability for the
node with the highest average mobility in the period analyzed. In practice,
this indicates that this node has significant neighborhood changes in the
network. In this way, we can distinguish nodes with different levels of
mobility and, if necessary, separate them into specific periods as needed by
several network applications.

###  5.2 Unveiling the mobility and connectivity importance

In order to understand the mobility patterns, we start by investigating the
mobility level of all nodes throughout the period observed on the latent space
modeled. Some of the possible applications of our approach include identifying
nodes with high mobility degrees for routing in opportunistic or delay-
tolerant networks [24, 18, 13]. Therefore, we analyze the nodes’ mobility
levels assuming that new connections - represented by topological changes in
the network and captured in the latent space - can occur with an acceptable
delay. Thus, we calculate the cosine distance of a time window $\Delta{t_{i}}$
for all subsequent time windows $\Delta{t_{j}}$, such that, $i<j$. In other
words, we evaluate the degree of mobility and the potential that a node has to
establish new connections throughout the analyzed period.

(a)

(b)

Figure 2: CDF of average and CV of mobility shifting of nodes (measured w.r.t.
cosine distance).

Figure 2(a) shows the nodes cosine distance average in a CDF. Looking at the
10% of the most stable nodes, in terms of connectivity, the average cosine
distance is at most 0.38. On the other hand, the 10% of the nodes that change
the most shows a mobility average greater than 0.54, exhibiting a 42% higher
mobility level. In order to check to what extent the mobility changes of the
nodes are stable or characterized by bursts in specific periods, we compute
the Coefficient of Variation (CV) as a measure of dispersion222CV is computed
by the ratio of the standard deviation to the mean and expressed in
percentage.. The CV distribution is shown in Figure 2(b). In practice, values
greater than 30 are considered high coefficients of variation [21]. Therefore,
it is possible to observe that more than 75% of the nodes in the network show
a greater dynamic in specific periods.

(a) Highest avg. cosine distance

(b) Lowest avg. cosine distance

Figure 3: Mobility shifting of the nodes with lowest and highest cosine
distance over consecutive days.

We then select nodes with the highest and lowest average cosine distance to
investigate mobility changes over consecutive days. Figures 3(a) and 3(b) show
heatmaps where one is able to compare day to day changes over the observed
period. The axes $x$ and $y$ indicate the cosine distance to two given time
windows. The color indicates the topological change level encoded in the
latent space, revealing their degree of mobility according to the cosine
distance. The more red and intense the cell, the greater the difference in
node connections over the two days observed. Conversely, the more blue and
intense, the smaller this difference. It is possible to notice that the node
with the high average cosine distance (Figure 3(a)) tends to present high
mobility levels for most pairs of days observed, which indicates constant
topological changes. We also investigated the node’s behavior according to the
highest and lowest average CV and found that they exhibit similar behavior at
mobility levels to the cosine distance average but with more dispersed changes
in even more specific periods.

(a)

(b)

Figure 4: CDF of average and CDF of the nodes’ connection importance (measured
w.r.t. nodes’ vector norm).

Recall that the cosine distance indicates node mobility in terms of
connectivity change. However, we can not claim that a node with a higher
mobility level provides the most important connections, for example, to allow
a message to reach the entire network. For this reason, we evaluate the vector
norm (see details in Section 4.2), which captures the importance of a node in
terms of connectivity. Figure 4(a) shows the CDF of the average vector norm
for all nodes. The 10% with the lowest value has an average norm smaller than
or equal to 4.45. Conversely, the 10% with the highest average value present
an average norm greater than or equal to 5.3, resulting in a difference of
almost 20%. Note that nodes with higher vector norm are more important and
central in terms of connectivity. We also evaluate the CV of the norms
obtained per node and present the distribution in Figure 4(b). Unlike cosine
distance, here, 98% of nodes have a high CV (greater than 30). This great
variability suggests that some nodes are more important for exchanging
information in the network at specific times as network connections evolve.

The vector norm of each node in each day can be used to understand nodes’
connectivity importance in the temporal aspect. Figure 5 presents a day-to-day
heatmap for all nodes’ norm vectors. Rows represent the nodes and columns
represent the days. Each day was normalized by using the z-score333Each value
is subtracted by the column average and divided by the column standard
deviation.. Thus, one can see how much node’s importance deviates relatively
from the average importance over days. The color of each cell reflects the
normalized norm value. Also, we compute a dendrogram to group nodes that
present similar importance on specific days. The Pearson correlation between
the node norms was employed as dendrogram grouping metric [5]. We can observe
that some nodes present greater importance than the others (indicated by red
cells) in specific periods, confirming our finding about the high CV of the
norm.

Figure 5: Heatmap of the node vector norm over the observed period.

As the cosine distance and the vector norm provide complementary analyses, we
investigate how these measures are related. Intuitively, we aim to understand
how the mobility of nodes is related to the importance of their connections.
To do so, we apply the Pearson’s correlation between the average and CV of
both measures, the result is summarized in Table 2. The strongest correlation
is negative and occurs between the average of the norm and the CV of the
cosine distance. This suggests that nodes with more important connections have
a stable mobility pattern. Then, we observe a moderate correlation between the
average cosine distance and the CV of the norm, which means that nodes who
consistently make abrupt connectivity changes tend to maintain connections
that are sometimes important and sometimes not. The other correlations are not
significant for $p=0.01$.

Table 2: Correlations between cosine distance and the vector norm. Metrics | Correlation
---|---
Avg. of Cosine Distance | CV of Cosine Distance | -0.04
Avg. of Cosine Distance | CV of Vector Norm | 0.70
Avg. of Vector Norm | CV of Cosine Distance | -0.95
Avg. of Vector Norm | CV of Vector Norm | -0.10

We also investigate the Person’s correlation between our approach and
traditional graph topological importance measures [9, 29, 44, 18, 13]. To do
it, we consider the main well-known topological measures used in these works.
Remember, they capture the most varied notions of nodes’ centrality and
connectivity in the network, and, therefore, different notions of importance.
Table 3444All observations are significant for $p=0.01$. depicts the
correlations between these measures.

Turning the focus on the centrality measures (Degree, Betweenness, Closeness,
and Eigenvector). It is possible to observe a moderate negative correlation
between traditional graph measures and the average of the cosine distance. It
suggests that the more dynamic a node is, the lower its centrality according
to different views. The average vector norm presents slightly positive
correlation with respect to traditional measures, a possible explanation is
that centrality measures typically capture unique notions of centrality and
importance, and do not consider the temporal factor [26]. Hence, they cannot
generalize a concept of centrality that is universally important in the graph.
Finally, we consider the clustering coefficient, which shows a low-to-moderate
positive and negative correlation, respectively, with both node metrics in
latent space. The positive case suggests that nodes with neighbors more
connected (high clustering coefficient) tend to change their connections
collectively (high avg. of cosine distance), which may reflect our case
study’s regular group meetings property. On the other hand, nodes with less-
connected neighborhoods tend to have essential links for network connectivity
(high avg. of vector norm), explained by bridges and peripheral connections in
the networks.

Table 3: Correlations between topological and embedding measurements. Embedding | Topological | Correlation
---|---|---
Avg. of Cosine Distance | Avg. of Degree | -0.64
Avg. of Cosine Distance | Avg. of Betweenness | -0.69
Avg. of Cosine Distance | Avg. of Closeness | -0.64
Avg. of Cosine Distance | Avg. of Eigenvector | -0.65
Avg. of Cosine Distance | Avg. of Clustering Coefficient | 0.51
Avg. of Vector Norm | Avg. of Degree | 0.33
Avg. of Vector Norm | Avg. of Betweenness | 0.50
Avg. of Vector Norm | Avg. of Closeness | 0.28
Avg. of Vector Norm | Avg. of Eigenvector | 0.32
Avg. of Vector Norm | Avg. of Clustering Coefficient | -0.55

##  6 Conclusion

Understanding mobility in mobile networks includes examining how entities move
along with their role for network connectivity. Those are patterns notably
relevant to design networking protocols and solutions. This work proposed a
methodology for modeling and analyzing mobile networks by employing a node
embedding approach capable of unveiling patterns related to network
connectivity, mobility, and their natural evolution. We noticed that most
solutions designs in the literature rely on graph theory metrics computer on
the network. Consequently, they tend to privilege particular characteristics
indicating which nodes are central points in the structure. Alternatively, our
approach offers a generalized notion of connectivity importance based on the
existing links in each node’s network tracking it while the connections in the
network change.

By applying our methodology in a case study as a proof of concept, we showed
that our approach could reveal nodes with different mobility and connectivity
importance levels, whereas the network topology evolves. Finally, we
contrasted the notion of connectivity importance provided by our methodology
with some topological network measurements used network solutions present in
the literature. We then observed that these measures could not generalize the
patterns of connectivity captured by our methodology.

As future works, our methodology offers a range of possibilities. An immediate
opportunity is to apply our methodology to design new solutions that require
an efficient node selection strategy for dissemination/collection of
information, location, vehicular communication, among other applications in
the context of smart cities. It also includes evaluating the performance of
such protocols, comparing purely topological measures to those proposed here.
Finally, another possibility is to extend the network modeling to include
contextual information about the nodes’ visitation locations. It would enable
new tasks to be performed, including prediction and recommendation of points
of interest to users.

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