# Discovering and Predicting Temporal Patterns of WiFi-interactive Social
Populations ∗

Xiang Li, , Yi-Qing Zhang, , Athanasios V. Vasilakos, Xiang Li and Yi-Qing
Zhang are with Adaptive Networks and Control Lab., Department of Electronic
Engineering, Fudan University, Shanghai 200433, China,
(E-mail:{lix,,12110720032}@fudan.edu.cn). Athanasios V. Vasilakos is with
Department of Electrical and Computer Engineering, National Technical
University of Athens, Athens, Greece, (E-mail:th.vasilakos@gmail.com).
$\ast$ All correspondence should be addressed to Xiang Li.
The paper is the invited chapter of book “Opportunistic Mobile Social
Networks” Jie Wu and Yunsheng Wang (Eds.) published by CRC Press in Aug. 2014.

##  I Introduction

Since the publication of Moreno and Jennings’ sociometry book in 1934 [1],
network (graph) theory has become one of the most powerful tools to
characterize social interactions and population dynamics [2, 3, 4]. The
discoveries of small-world [5] and scale-free networks [6] in the late of
1990s arouse the world-wide attention on complex networks and network science,
and we have witnessed ten more years the fruitful and exciting advances to
understand the hidden patterns behind complex connectivity features and
characteristics of diverse large-scale networking systems. Such natural and/or
man-made examples range from the Internet, the World Wide Web, biological
brain networks, protein-to-protein interaction networks, power-grids, wireless
communication networks to categories of social, economic and financial
networks at different levels of human society. Extensive efforts and elegant
attempts have been devoted to answering a fundamental question: how does the
fascinating complex topological features affect or determine the collective
behaviors and performance of the corresponding complex networked system [7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. While this widely
concerned key question still remains open in the fast developing field of
network science, especially in the newly focused situation that the
information of temporal dimension as an explicit element defines the edges of
such a so-called temporal network [22] when they are active.

Nowadays, a great deal of incredible products of flourished information and
communication technologies (ICTs) are unobtrusively embedded into the physical
world of human daily activities: we communicate with our friends/colleagues
with emails and/or mobile phones, we make web-shopping and check out by credit
cards, and travel/commute in public transportation networks with the payment
of transit cards, and we surf online 24-hour everyday covered by WiFi or 3G
signals everywhere. Such creative digital devices/instruments not only reshape
our daily life, but also record tremendous digital traces produced by human
activities. These digital records offer unparalleled opportunities to
explosively digitize physical human interactions and provide fresh temporal
clues to throw sight into the human behavioral patterns.

In the literature, the successful stories mainly include the assistance from
the Bluetooth, the active Radio Frequency Identification (RFID), wireless
sensors and the WiFi technology [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36]. For instance, Eagle and his colleagues took the advantage of
Bluetooth embedded in mobile phones, and collected the proxy data of person-
to-person interactions in the program of Reality Mining [23, 24]. Barrat et
al. built a flexible framework based on the RFID technology, and recorded the
volunteers’ face-to-face interactions in different rendezvouses such as
conference, museum, and primary school [25, 26, 27]. Salathé et al. utilized
wireless sensors to trace person-to-person encounters among the members of a
high school, and further evaluated the respiratory disease transmission risk
in school campus [28]. Besides, many researchers explored the inter-contact
intervals of mobile users’ interactions by the Bluetooth and WiFi technology,
and designed more efficient algorithms to improve the performance of data
dissemination among a large-scale population of mobile users [29, 30, 31, 32].
Moreover, such fruitful researches push the desire to depict human social
networking populations with interdependent collective dynamics as well as the
behavioral patterns.

WiFi, as one of the most ubiquitous wireless accessing techniques, has been
widely deployed in human daily circumstances. Actually, the “WiFi-Free” sign
can be found in nearly every corner of urban areas, and the notion of “WiFi-
City” becomes reality. For example, since 2010, it has been reported the
significant increase of global number of WiFi certified product launches and
hotspots (both public and private), while more than $9,000$ product
certifications have been issued by the WiFi Alliance, and the global number of
hotspots is over 280 millions [37]. Therefore, the flood of commercial WiFi
systems comes as a powerful proxy tool to collect digital access traces of a
huge population equipped with of WiFi devices. As a snapshot of the modern
society, a university is in the coverage of WiFi signals, where the WiFi
system records the digital access logs of the authorized WiFi users when they
access to the campus wireless service. Such WiFi access records, as the
indirect proxy data of a large-scale population’s social interactions without
artificial interference, are the targets to explore in this chapter.

As the response of the above motivations, this chapter comes as a synthetical
review of our extensive efforts devoted to temporal networks and human
population dynamics in the past years [33, 34, 35, 36, 38, 39], and we target
a cyber-social population with the example of a university campus having the
WiFi coverage. We conduct the study based on a dataset from a Chinese
University, whose findings have also been verified with other open-source data
sets in different rendezvous, and this chapter only reviews this series of
works in detail with the focus of WiFi interactions. The left part of this
chapter is organized as follows. As a preliminary section, Section II
introduces the background of the WiFi data-collection setting in an involved
Chinese university, _i.e._ , Fudan University, and defines the related
concepts as well as the dataset preprocessing. Section III presents a
simplified temporal network version for such a cyber-social population with
the proxy of campus WiFi accesses, where the reachability of such individual-
level temporal networks affects the efficiency of information spreading.
Section IV defines the so-called event interaction to capture the concurrent
interactive patterns among a group of individuals, and constructs the
transmission graph to embed more temporal information, where both the vertex
and edge dynamics of transmission graphs present rich temporal patterns. In
Section V, we introduce the outlier performance of the involved low-degree
vertices’ role in such temporal networks, which has been underestimated in the
literature of static networks. We define a temporal quantity as the
participation activity potential to feature the role of individuals in a
population, which achieves the prediction accuracy of ranking the individual
centrality as high as 100% with the verification of the WiFi data set.
Finally, Section VI concludes the whole chapter.

##  II Background

Figure 1: The spatial distribution of wireless access points of six teaching
buildings.The vertical view of ‘Handan Campus’ of Fudan University, and the
internal structures of six teaching buildings with the detailed deployments of
wireless access points (red dots).

Fudan University covers four campuses, and _Handan Campus_ is the oldest and
also the largest campus, which accommodates around 75% of students of the
whole university. _Handan Campus_ is the first campus of Fudan university
covered by WiFi signal, and in 2009 there have been 517 wireless access points
(WAPs) distributed in all main buildings, which continuously provide dual-band
(802.11g/n (2.4-GHz) and 802.11a (5-GHz)) wireless accessing services to all
authorized campus members, _i.e._ , students, teachers, staffs, and visiting
scholars of Fudan University.

In the campus WiFi system, all WAPs share the same service set identifier
(SSID), which guarantee that wireless devices/clients (e.g., laptops, pads,
video game consoles, smart phones and digital audio players) can smoothly roam
from one WAP to another. Each WAP owns a unique IP address. When a client is
roaming, the WiFi system automatically records the access logs without
notifying the WiFi user. The assess logs of all authorized users are recorded
in a database administrated by the Informatization Office of Fudan University.
With the permission and assistance of the Informatization Office, we collected
the campus WiFi users’ access logs during the 2009-2010 fall semester
(10/18/09-01/09/10) as the so-called data set of “FudanWiFi09”, each piece of
which contains four parts: the Media Access Control (MAC) address of the
wireless device, the device’s connecting and disconnecting time stamps, and
the MAC address of the WAP that the device accesses to. Totally, the data set
of “FudanWiFi09” contains the records of 22,050 WiFi devices’ 423,422
behavioral trajectories in the period of nearly three months.

Since we target the proxy data of those wireless devices to feature the users’
interactive behaviors, we only focus on the WiFi data recorded in the
buildings which are open to public without safeguards for the WiFi users’
devices, i.e., when a user leaves such a place, he/she should bring the device
with himself/herself. Therefore, the data set of 18,715 individuals’ 262,109
behavioral trajectories from six teaching buildings with the WiFi coverage
have been employed in this work. Fig.1 presents the spatial distribution of
all 129 WAPs deployed in the six buildings from the vertical view of a part of
_Handan Campus_ of Fudan University.

For simplicity, we assume that every WiFi device with the MAC address recorded
in the WiFi system represents one WiFi user, neglecting the seldom cases that
one person may use several WiFi devices at the same time, and several WiFi
users may share the same WiFi device. Before transforming the access logs of
WiFi data as the proxy of a large-scale population of WiFi users’
interactions, we state our main assumption that the WiFi devices “seeing” the
same WAP imply the co-located interactions among these devices’ owners, which
is motivated by the assumption of ‘geographic coincidences’ proved in the work
of Crandall and et al. in 2010 [40]. They defined a spatiotemporal co-
occurrence between two Flickr users as an instance where they both took photos
at approximately the same place and at approximately the same time, which
infer the social ties in the population of humans.

Therefore, to verify the assumption of ‘geographic coincidences’ in the campus
WiFi system, the Euclidean distance between two WiFi users is critical. Due to
the difficulty of locating the WiFi devices’ indoor positions with a single
WAP, we can not exactly calculate the Euclidean distance between two WiFi
users in the campus. However, we can estimate the average Euclidean distances
by Voronoi decomposition with the detailed distribution of WAPs shown in Fig.
1 and the student number in every classroom from ‘Fudan University Curriculum
Schedule of the $1^{st}$ Semester in 2009-2010’. Since each building has
several floors, and the building materials between floors can dramatically
attenuate the wireless signals, which guarantee that the coverage regions of
two WAPs on different floors do not overlap. On each floor, the wireless
devices automatically connect to the closest WAP which offers strongest
signals. Only when the WAP is overloaded, the device will switch to another
adjacent WAP. In the campus WiFi system of Fudan University, each WAP can
serve 50 users at most, and all the involved WAPs were not overloaded in the 3
months. Therefore, we decompose each floor by the distribution of deployed
WAPs into the corresponding Voronoi tessellations. Given a Voronoi cell, we
apply the Monte-Carlo method that randomly locating the persons in the
classrooms of the cell to calculate the Euclidean distances of any two WiFi
users, and estimate the average Euclidean distances and the standard
deviations of any two WiFi users in every building, as summarized in Table I.
We conclude that the average Euclidean distances along with their standard
deviations do not change much in different buildings, indicating that the
assumption of ‘geographic coincidences’ is valid. Therefore, in such a close
indoor circumstance, people have a high probability to directly communicate
with each other or have some relationship via indirect interactions. In the
following sections, we utilize the collected data of campus WiFi assess logs
to explore categories of interactive patterns in a cyber-social population,
which is illustrated with the population of WiFi users in Handan campus of
Fudan University.

TABLE I: The estimated average Euclidean distances and standard deviations between any two WiFi users. Building | No. of WAPs |  $\langle d\rangle$(m) |  $\delta$(m)
---|---|---|---
the $2^{nd}$ bldg. | 16 | 6.26 | 1.96
the $3^{rd}$ bldg. | 21 | 7.14 | 0.96
the $4^{th}$ bldg. | 15 | 7.86 | 1.71
the $5^{th}$ bldg. | 28 | 6.98 | 1.37
the $6^{th}$ bldg. | 33 | 7.11 | 1.59
the west bldg. | 14 | 11.75 | 0.98

##  III Pairwise interactive patterns of temporal contacts and reachability

Figure 2: The illustration of temporal contact network and its aggregated
version. (a) WiFi accessing logs are translated into the corresponding
interaction events, where the black arrowed lines show the temporal contacts
between users in different interaction events. (b) The construction of three
temporal contact networks during the intervals [t0, t1], [t0, t2], [t0, t3],
respectively, where the edges are colored according to the latest time that
they are contacted. (c) The construction of the aggregated contact network
during the interval [t0, t3].

We start from a simplified version of WiFi assess logs as shown in Fig. 2 (a),
where five users ($a,b,c,d,e$) accessing to two different WAPs (AP 1 and AP 2)
during the interval of $[t_{0},t_{3}]$ are illustrated. Obviously, as time
evolves, user $a$ interacts with user $b$ before user $b$ interacts with users
$c$ and $d$. Since we assume that the WiFi devices “seeing” the same WAP
infers the co-located interactions among these devices’ owners, such
‘geographic coincidences’ indicate the temporal contacts among the involved
WiFi users co-located in the same WAP. Therefore, a temporal contact network
comes as a direct extension of contact networks with embedding the temporal
information.

To construct a temporal contact network, we define
${e_{i}}=({V_{i}},{t_{i1}},{t_{i2}})$ as an interaction event, where ${V_{i}}$
denotes the set of users connecting to the same WAP simultaneously, and any
user $v\in{V_{i}}$ interacts with other users in ${V_{i}}$ during the interval
from ${t_{i1}}$ to ${t_{i2}}$. We define a temporal contact (TC): user $i$
makes a temporal contact with user $j$ if there exists a sequence of
interaction events with non-decreasing time between them. Therefore, a
temporal contact network is constructed at the level of individual users,
_i.e._ , define a temporal contact network
$\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$ with $\mathcal{V}$ the set of
vertices (WiFi users) and $\mathcal{E}$ the collections of all temporal
contacts. Note that
$\mathcal{E}=\mathcal{E_{\rightarrow}}\bigcup\mathcal{E_{\rightleftharpoons}}$
is the union set of directed edges $\mathcal{E_{\rightarrow}}$ and
bidirectional edges $\mathcal{E_{\rightleftharpoons}}$, which stand for the
time-respecting paths and pairwise interactions, respectively. Given an
observed interval, as shown in Fig. 2 (b), there exist two time-respecting
paths: one path is from user $a$ to user $c$, and the other is from user $a$
to user $d$. Besides, there exist two pairwise interactions between two pairs
of users $b$, $c$, and users $b$, $d$. Therefore, we generate three different
temporal contact networks in different intervals
$[t_{0},t_{1}],[t_{0},t_{2}],[t_{0},t_{3}]$, respectively. As a comparison, we
aggregate all interaction events in the interval of $[t_{0},t_{3}]$, and
generate an aggregated contact network as shown in Fig. 2 (c), which clearly
tells the difference between two different contact networks even in the same
interval $[t_{0},t_{3}]$.

Figure 3: Network reachability. (a) The average network reachability of the
aggregated contact networks (ACNs) and temporal contact networks (TCNs) with
the time length $\Delta$t=30 days. (b) The maximum network reachability of the
ACNs and TCNs with the time length $\Delta$t=7 days.

Such different connectivity patterns between temporal contact networks and
aggregated contact networks are also reflected in the basic property of
network connectivity: reachability. Traditionally, in the case of aggregated
(static) networks, since the edges are continuous active, the size of the
component, $N_{i}$, that a given vertex $i$ belongs to, indicates an upper
bound of the number of vertices influenced by vertex $i$. Therefore, $N_{i}$
characterizes the reachability of vertex $i$ in the static contact network
aggregated within a given time interval. In the case of temporal (contact)
networks, however, the edges are no longer continuously active, whose active
durations and time stamps dominate the network transitivity. Since a temporal
contact network defined here is rather simplified, which only keeps the
contact order of time stamps, the reachability of vertex $i$ in a temporal
(contact) network is the number of vertices that can be temporally influenced
by $i$ during the time length $\Delta t$ [41]. More specifically, the
reachability of vertex $i$ of a temporal contact network equals to the number
of elements in the set
$\\{{\phi_{i,j}}(t_{2})\mid{\phi_{i,j}}(t_{2})>{t_{1}},\ j\in V\\}$, where
$\phi_{i,j}(t)$ is the time of the inception of the latest temporal contact
from $i$ to $j$ before $t$. The average (maximum) network reachability is the
mean (maximum) value of all vertices’ reachability.

Fig. 3 compares the network reachability (normalized by the network size)
between temporal contact networks and aggregated contact networks with the
same time length. Both the average and maximum values tell that the aggregated
contact networks’ reachability is much larger than those of temporal contact
networks, especially when the time length $\Delta t\rightarrow 0$. Besides,
the reachability of aggregated contact networks quickly attains its saturation
with increasing the time length $\Delta t$, while the reachability of temporal
contact networks saturates much slower. Since the saturated value of the
average reachability of temporal contact networks is much smaller than that of
aggregated contact networks, the temporal dimension provides a more precise
upper bound to the reachability of a network, which may avoid the
overestimation of the information efficiency over such networks.

Figure 4: The correlation of the temporal out-degrees and in-degrees. The
joint probability distribution of the temporal out-degrees dout and in-degrees
din with the given time length $\Delta$t as (a) 1 day, (b) 2 days, (c) 3 days,
(d) 5 days, (e) 7 days, (f) 8 days. When the time length $\Delta$t is larger
than 1 week (7 days), a set of data points always present at the right-up
corner.

In a temporal contact network, the out-degree $d_{out,i}$ of vertex $i$
quantifies the number of receptors temporally affected by $i$, while the in-
degree $d_{in,i}$ specifies the number of its potential inciters. In an
information spreading process on a temporal (contact) network, for example,
except the source/destination vertices as well as those leaf vertices, all
other vertices in the network play both roles of receiving and forwarding
information in the process. Therefore, we put more attention to the correlated
distribution of $d_{out,i}$ and $d_{in,i}$ of all vertices in the temporal
contact networks with different time lengths $\Delta t$. Fig.4 presents the
joint probability distributions $C^{\Delta t}(d_{out},d_{in})$ of out-degrees
and in-degrees with different time lengths $\Delta t=1,2,3,5,7,8$ days,
respectively. Each joint probability distribution is averaged by the generated
networks from the data set with a given $\Delta t$. In the case of the
smallest daily time length, _i.e._ , $\Delta t=1$ day, Fig.4(a) shows that
almost all data points reside in the lower triangular matrix, and many data
points are even sticking on the axes. That is to say, in a short observation
period, _e.g._ , 1 day, only a very limited number of users keep frequently
online during the whole day, while most individuals act according to their
curriculum schedules, which seldom cover the whole day. With the increase of
time length $\Delta t=2,3,5$ days, there are more and more vertices emerging
in the upper region of two axes (Figs.4(b)-(d)), which tell that the forward
and backward temporal influence of each user increases as time proceeds. When
the time length is more than one week (Figs. 4(e) and (f)), there are two
evidently nontrivial clusters of temporal hubs: one contains the vertices
along the diagonal, and the other contains the vertices anchoring in the upper
right corner. In the former cluster of hubs, the users balance to receive and
forward the influence from other users. While those users presented in the
upper right corner are more important due to their role of relay hubs, which
are critical to facilitate spreading processes on temporal networks.
Obviously, the weekly cycle of the curriculum schedules of all campus WiFi
users accelerates the emergence of such temporal patterns of relay hubs.

##  IV Concurrent interactive patterns of event interactions and temporal
transmission graphs

Temporal contact networks as a category of “reachability graph” are generated
based on the “time-respecting” paths and pairwise interactions of temporal
contacts, which focus on the temporal interactive sequence between a pair of
individuals while neglect the concurrent interactive dynamics of more involved
individuals. For example, recall Fig. 2 (b), the three users $b,c,d$
concurrently assess ‘AP 2’ in the overlapped interval $[t_{1},t_{2}]$, while
this concurrent interaction among three users has to be simplified to two
pairwise interactions ($bd$ and $bc$) with the definition of temporal
contacts. To characterize such concurrent interactions, we extend the temporal
contacts at the level of individual users to a group of users with the newly
defined ‘event interactions’ and ‘transmission graphs’ to embed more temporal
information.

Fig. 5 (a) gives an instance of five WiFi users’ access logs as the
corresponding individual behavioral traces, where the bold lines pertain to
their online durations, _i.e._ , the five users stay at the coverage of the
same WAP during [$t_{0},t_{9}$], and only these five users access to this WAP
during the interval. As we assume that the WiFi devices “seeing” the same WAP
infers the geographic co-located interactions among these devices’ owners
(_i.e._ , co-locating at the same time in the same small region), the five
users participate several access events when different users access and
disconnect the WAP at different time stamps. Therefore, we define an event
interaction as the concurrent interaction of multiple users accessing to the
same WAP within a given interval, _e.g._ , EI $E_{AB}^{t_{0}}$ characterizes
the concurrent interactions of users _A_ and _B_ at the beginning time
$t_{0}$. Fig. 5 (b) illustrates the process of translating the five users’
individual behavioral trajectories into event interactions (EIs). Since a user
only involves in one unique event interaction at any time, all illustrated
events in Fig. 5 (b) are involved with two users for simplicity, and an event
interaction with more users involved is still valid, which is reflected in the
later defined quantity of event size.

Figure 5: The illustration of event interactions and transmission graph.
(a)WiFi access logs as the proxy of the users’ behavioral trajectories. The
bold lines pertain to their online durations. (b) Each bold line possesses an
exclusive time interval, where the corresponding individuals are assembled
into a contact clique. (c) In a transmission graph, the vertices are event
interactions (EIs), and the edges between two EIs are the transmission paths
defined by the three rules. The aggregated version is derived from the
transmission graph, where no multiple edges and time labels are allowed.

Define an event interaction (EI) as a vertex, and two vertices (event
interactions) are connected in a ‘Transmission Graph’ with the following three
defined rules:

  1. 1.

At least one same user exists in both the source and sink EIs.

  2. 2.

In the time sequence, the beginning time of the source EI is the closest one
prior to the beginning time of the sink EI.

  3. 3.

Given a sink EI, when there exist several source EIs, the set of shared users
in a pair of source EI and sink EI does not intersect with those of other
pairs of source and sink EIs.

The first two rules follow the principle of temporal nearest adjacency, and
the third rule follows the principle of minimum number of transmission edges
(paths) generated. Therefore, we construct a transmission graph as illustrated
in Fig. 5 (c) with the above three rules, yielding five vertices (event
interactions of Fig. 5 (b)) and seven directed edges, where the time stamps
such as $t_{0},t_{6}$ over the directed edge between vertex $AB$ and vertex
$AC$ stand for the beginning time stamps of the two (source and sink) event
interactions. Also, we aggregate all event interactions following the three
rules of transmission edges with the same data set to generate an aggregated
transmission graph, which is illustrated in Fig. 5 (d).

To quantify the temporal information of event interactions (EIs) and the
transmission graph (TG), we further define the following quantities:

  * •

The size, _s_ , of a given EI is defined as the number of involved
individuals.

  * •

Given an EI active at the epoch of [$t^{begin},t^{end}$], its active duration
is defined as $\Delta t^{EI}=t^{end}-t^{begin}$.

  * •

Given a pair of source and sink EIs, the time-stamped label in the directed
edge is $(t_{source},t_{sink})$. The transmission duration $\delta$ is defined
as the interval between the beginning time of the source and sink EIs, _i.e._
, $\delta=t_{sink}-t_{source}$.

Since the vertices of a transmission graph are the event interactions, we
first present the vertex dynamics of transmission graphs to unveil the
statistical characteristics of concurrent interactions among WiFi users. As
shown in Fig. 6 (a), the probability distribution of all event interactions’
active durations collected during the three months falls into a truncated
power law, whose exponent of the power-law part approximates to 1. This
feature indicates that the long-lasting concurrent interactions can hardly
survive due to those newly emergent events with short active durations, while
the non-Poission distribution of concurrent interactive durations tells that
WiFi users do not randomly break their interactions with other users. On the
other hand, the number of event interactions with a given size is
exponentially distributed as shown in Fig. 6 (b), _i.e._ , the users randomly
co-locate with each other at the same time.

Figure 6: Temporal dynamics of event interactions. (a) The probability
distribution of event interactions’ active durations $\Delta$t${}^{EI}_{all}$.
(b) The size distribution of event interactions. (c) The probability
distributions of event interactions’ active durations $\Delta$t${}^{EI}_{s}$
with the given size s = 2,3,4,5, respectively. (d) The probability
distributions of event interactions’ active durations $\Delta$t${}^{EI}_{s}$
with the given size s=5,6,…11, respectively. The red bold line is the
distribution with s=5. The inset tells the dependence between size s, and the
exponent of exponential cutoff $\beta$.

Note that such an exponential distribution also implies that the large-size
event interactions are rare, and we may conjecture there exists some
interdependence between the probability distribution of event interactions’
active durations $\Delta t^{EI}$ and their sizes _s_ , which, however, does
not exist as shown in Figs. 6 (c)-(d). Although the probability distributions
of active durations $\Delta t^{EI}_{s}$ with size $s=2,3,4,5$ keep the
truncated power laws, whose power-law parts still obey the exponents close to
1, we observe that their exponential cutoffs gradually decay with the
increased event size from 2 to 5 (Fig. 6(c)). Furthermore, with the increased
event size from 6 to 11 as shown in Fig. 6(d), the probability distributions
of the active durations with size $s\geq 5$ present the same shape $\Delta
t^{-\gamma}\exp(-\frac{\Delta t}{\beta})(\gamma\approx 1)$. At the same time,
the inset of Fig. 6(d) tells that the exponent of exponential cutoff $\beta$
decreases with the growth of size _s_ from 2 to 5, which keeps invariant with
increasing _s_ from 5 to 11. Therefore, we conclude that the active durations
of event interactions own the _‘size free’_ feature, which, counter-
intuitively, is independent from the event sizes.

On the edge dynamics of transmission graphs, we focus on the transmission
durations to uncover the temporal relations between concurrent interactions.
As shown in Fig. 7 (a), the probability distribution of transmission durations
obeys a bifold power-law with two turning points: the first turning point is
at around 120 minutes, _i.e._ , the length of two teaching courses; the other
turning point is at around 24 hours, which indicates the existence of daily
circadian rhythms. To test the existence of periodic rhythms, we calculate the
integral days spent by each $\delta$ ([$\frac{\delta}{1440}$]), as shown in
Fig. 7(b), where there are two peaks at the 1st day and the 7th day. That is
to say, the daily burst behavior of ‘adjacent’ event interactions evolves with
the weekly rhythm. We further employ two (natural and artificial) de-seasoning
methods to remove the circadian and weekly rhythms. In the natural de-
seasoning method, we conserve the source and sink event interactions taking
place in the same day. In the artificial de-seasoning method, we randomly
select two event interactions and exchange their beginning active time (their
active durations are ignored), and check all individuals to ensure this round
of time-shuffle operation does not generate the event interactions with the
same involved individual which have the same beginning time. As shown in Fig.
7(a), the probability distributions of the transmission durations ‘filtered’
by two de-seasoning methods both fall into the truncated power-laws, and we
witness that such burst behaviors of populations take place _independently_ on
the daily and weekly rhythms.

Figure 7: Temporal dynamics of transmission durations. (a) The probability
distribution of transmission durations, $\delta$, from the whole dataset
(white circle), and ‘filtered’ by the natural de-seasoning method (gray
circle) and the artificial de-seasoning method (red square).(b) The
probability distribution of integral days spent by transmission duration
$[\frac{\delta}{1440}]$ (days).

##  V Temporal degrees and hubs: ranking and prediction

In Sec. 3, those users located in the upper right corner of Fig. 4 (f), i.e.,
users with both large in-degrees and out-degrees, function as the relay hubs
of temporal contact networks, which emerge with the weekly rhythms of WiFi
users’ curriculum schedules. In this section, we further explore the temporal
hubs with more temporal information embedded into event interactions and
transmission graphs as defined in Sec. IV. When neglecting temporal
information, traditionally, a static contact network with the same data set of
WiFi campus assess logs is generated, where the vertices are WiFi users, and
an edge connects two vertices if the two WiFi users have at least one round of
interaction (_i.e._ , geographic coincidence) during the whole 3 months. As a
comparison, we aggregate all transmission edges of Fig. 5 (c) (left) to
generate an aggregated transmission graph, as illustrated in Fig. 5 (c)
(right).

Figure 8: Outliers: temporal hubs in the aggregated transmission graph versus
leaf vertices in the static contact network. (a) A small sample of the
aggregated transmission graph consisting of several hubs that WiFi users
involved in. (b) The degrees versus their ranks of all WiFi users in the
aggregated transmission graph. The red circles represent the WiFi users who
have the top 10 temporal degrees.(c) The degrees versus their ranks of all
WiFi users in the static contact network with the same WiFi data set. The red
circles represent the same WiFi users in (b).

Both temporal and static aggregated networks present some differences. The
static contact network defines the vertices at the individual level of WiFi
users, which is topological homogeneous with the degrees’ coefficient of
variation (CV) as $1.59$. While the aggregated transmission graph defines the
vertices at the level of event interactions, which concurrently involve a
group of WiFi users, therefore, it is more heterogeneous with the degrees’
coefficient of variation as $7.13\gg 1$. Fig. 8 (a) visualizes a small sample
of the aggregated transmission graph. Notice the degree of event interactions
(vertices) in the aggregated tranmission graph is the sum of the out-degree
and in-degree in the corresponding transmission graph, we define the degree of
an event interaction as the degree of the event interaction’s involved users.
Generally, a (WiFi) user may participate in several event interactions with
different degrees, we define the temporal degree of a user as the maximal
degree of all the involved event interactions. Therefore, we rank the temporal
degrees of WiFi users as shown in Fig. 8(b), where the top 10 users with the
largest temporal degrees are identified with red circles. Interestingly, these
temporal-hub users are not so ‘attractive’ in the static contact network,
which, on the contrary, are almost the low-degree vertices as identified with
red circles in Fig. 8(c). That is to say, many low-degree vertices, which
generally were neglected in the literature of static contact networks, may
participate to play the dominant role as ‘hubs’ in the version of temporal
networks. Such difference also highlights the temporal significance of
concurrent interactions to characterize a typical cyber-social population with
the society example of a WiFi campus, and the outlier nontrivial performance
has also been verified in other public data sets [35].

Note that we can not directly identify the WiFi users who function as temporal
hubs from the traditional version of static contact networks, as we have
visualized that the low-degree individuals in the static human contact
networks gather together to form temporal ‘hubs’ in the aggregated
transmission graphs. Therefore, to identify (and even predict) such important
users who have high temporal degrees in the WiFi campus, a natural means is
constructing such an aggregated transmission graph before ranking all involved
users with their temporal degrees, which, however, is a task with very highly
computational load. For example, given a temporal data set, which yields $M$
event interactions with time stamps, we construct a transmission graph having
$H$ vertices and $P$ edges, whose computational complexity in the worst case
is $O(M^{2})$. While the process of aggregating such a transmission graph
requires the computational load of $O(H*P)$. Therefore, with the WiFi assess
data addressed in this chapter, we totally generate $M=260,925$ event
interactions and a transmission graph having $H=248,663$ vertices and
$P=260,513$ edges, whose computational load is as high as $O(1.32*10^{11})$ to
generate the aggregated transmission graph, excluding the search cost to rank
temporal degrees. Obviously, a new and efficient method to rank and predict
the temporal hubs without constructing such a transmission graph is desirable
and challenging, and we give a satisfactory solution as follows.

As shown in Fig. 9 (a), the vertices of an aggregated transmission graph are
event interactions(EIs), which contain several WiFi users followed by the
definitions of transmission graph in Section IV. The active duration of an
event interaction, $e_{i}$, is defined as $\Delta t^{EI}(e_{i})$. Given a
finite length of time period $\Delta T$ and an event interaction, $e_{i}$, the
total active (WiFi users’) number of the event interaction is $n(e_{i})$, and
we define the total active duration of the event interaction as

$\Delta t^{EI}_{sum}(e_{i})=\sum_{m=1}^{n(e_{i})}\Delta t^{EI}_{m}(e_{i}).$

Given the WiFi user $j$ participates event interaction $e_{i}$, we define the
_participation activity potential(PAP)_ of user $j$ involving event
interaction $e_{i}$ as:

$PAP(j,e_{i})=(\Delta
t^{EI}_{sum}(e_{i}))^{\alpha}(n(e_{i}))^{1-\alpha},\alpha\in[0,1]$ (1)

As we know, a WiFi user may involve several different event interactions,
therefore, given the set of event interactions involved by WiFi user $j$ as
$\Gamma(j)=\\{e_{i}|j\in e_{i}\\}$, we define the _maximum participation
activity potential (MPAP)_ of user $j$ as:

$MPAP(j)=\max_{e_{i}\in\Gamma(j)}(PAP(j,e_{i}))$ (2)

Figure 9: The illustration of predicting temporal hubs. (a) In an aggregated
transmission graph, each vertex is an event interaction (EI), and each EI
contains several Wi-Fi users. The temporal degree, $\kappa$, of a Wi-Fi user
is the maximum degree of the involved EIs, and the corresponding maximum
participation activity potential (MPAP) is the maximum value of all
participation activity potentials (PAPs) of the involved EIs. (b) The temporal
degree, $\kappa$, and MPAP are both in descending order. Given a Wi-Fi user,
its accuracy rate is the ratio of the other Wi-Fi users who are in the same
positions predicted by both $\kappa$ and MPAP. Figure 10: Predicting temporal
hubs of a transmission graph. (a) The weighted accuracy rate F($\alpha$) shows
that the accuracy rate of predicating temporal hubs achieves more than 96%
when randomly selecting a value of $\alpha$ from [0,1]. The red dot represents
that the maximum weighted accuracy rate can achieve 98.6% with a given optimal
nonlinear parameter $\alpha$=0.04. (b) The average accuracy rate
$\langle$AR($\kappa_{max}$)$\rangle$ of predicating temporal hubs versus the
rank of their temporal degrees $\kappa_{max}$ with the optimal $\alpha$=0.04.
The red bar represents the standard deviation $\delta$(AR($\kappa_{max}$)) of
the accuracy rate. (c) The average error between the accuracy rate given a
random nonlinear parameter, $\alpha$, and the optimal $\alpha$=0.04 to predict
top 1, 5, 10, 15 Wi-Fi users.

Different from temporal degrees to characterize WiFi users’ temporal role,
which require a heavy computation cost as we have discussed, the new proposed
quantity of maximum participation activity potential (MPAP) does not need any
preliminary information from the aggregated transmission networks. For
instance, in Fig. 9(a), WiFi user $v$ involves in six event interactions with
the corresponding degree $k_{i}$ and the participation activity potential
$PAP_{i}$, where $i=1,2,...6$. The temporal degree of user $v$ is the maximal
degree of the involved six event interactions,
$\kappa(j)=\max_{i=1,...,6}k_{i}$, which is dependent on constructing such an
aggregated transmission graph. To calculate the maximum participation activity
potential of user $j$, the maximum value of all PAPs of the involved event
interactions, $MPAP(j)=\max_{i=1,...,6}PAP_{i}$, is directly calculated from
the information of each event interaction itself.

Therefore, we propose the maximum participation activity potential (MPAP) of a
user as the candidate quantity of ‘temporal degrees’, with which we rank all
WiFi users and predict the temporal hubs in the campus population. As shown in
Fig. 9(b), we list all temporal degrees and all MPAPs of the WiFi users in the
descending order, where $`\kappa_{user}(1)>\kappa_{user}(2)>...^{\prime}$ and
$`MPAP_{user}(1)>MPAP_{user}(2)>...^{\prime}$. Given a WiFi user with temporal
degree $\kappa_{user}(i)$, all the other WiFi users in the population are
divided into the following three sets based on temporal degree
$\kappa_{user}(i)$:

  * •

set $S_{\kappa}(>)$: the users whose temporal degrees
$\kappa>\kappa_{user}(i)$.

  * •

set $S_{\kappa}(=)$: the users whose temporal degrees
$\kappa=\kappa_{user}(i)$.

  * •

set $S_{\kappa}(<)$: the users whose temporal degrees
$\kappa<\kappa_{user}(i)$.

Similarly, we divide all the same users into another group of three sets based
on $MPAP_{user}(j)$:

  * •

set $S_{MPAP}(>)$: the users whose $MPAP>MPAP_{user}(j)$.

  * •

set $S_{MPAP}(=)$: the users whose $MPAP=MPAP_{user}(j)$.

  * •

set $S_{MPAP}(<)$: the users whose $MPAP<MPAP_{user}(j)$.

Finally, we find the part of WiFi users who present in the following sets as
shown in Fig. 9(b):

  * •

the set of $S(>)$: the users present in $S_{\kappa}(>)\bigcap S_{MPAP}(>)$.

  * •

the set of $S(=)$: the users present in $S_{\kappa}(=)\bigcap S_{MPAP}(=)$.

  * •

the set of $S(<)$: the users present in $S_{\kappa}(<)\bigcap S_{MPAP}(<)$.

Therefore, the users ranked by their temporal degrees and the maximum
participation activity potential in the same positions describe the accuracy
rate predicted by the maximum participation activity potential defined as
follows:

$AR(v)=\frac{|S(>)|+|S(=)|+|S(<)|}{n-1}$

In more detail, given a rank of the temporal degree $r(\kappa)$, we define the
average predicative accuracy rate and its standard deviation as

$\langle
AR(r_{\kappa})\rangle=\frac{\sum_{r_{\kappa}(v)=r}AR(v)}{|\\{v|r_{\kappa}(v)=r\\}|}$
(3)

$\delta(AR(r_{\kappa}))=\sqrt{\frac{\sum_{r_{\kappa}(v)=r}(AR(v)-\langle
AR(r_{\kappa})\rangle)^{2}}{|\\{v|r_{\kappa}(v)=r\\}|}}$ (4)

Note that in the definition of participation activity potential $PAP$, Eq.
(1), an appropriate value of parameter $\alpha\in[0,1]$ influences the
accuracy rate. To find the optimal $\alpha$ and achieve the highest prediction
accuracy, we further define the following function $F(\alpha)$:

$F(\alpha)=\frac{\sum\kappa*\langle AR(\kappa)\rangle}{\sum\kappa}.$

As shown in Fig. 10 (a), the maximum accuracy ratio is $F(\alpha)=98.6\%$ with
the corresponding $\alpha=0.04$ as the optimum, and a random selection of
$\alpha\in(0,1)$ achieves the accuracy rate higher than 96%. Substituting the
optimum of $\alpha=0.04$ to the participation activity potential $PAP$ defined
in Eq. (1), the average accuracy rate (Eq. (3)) predicts the _top 15_ temporal
hubs by the maximum participation activity potential $MPAP$ as high as 100%
accuracy rate (Fig. 10(b)). Furthermore, to predict the different cases, for
example, of the top 1, 5, 10, and 15 temporal hubs, the performance difference
between a randomly selected $\alpha\in(0,1)$ and the optimal $\alpha=0.04$ is
as less as $2.5\%$ at most, which is visualized in Fig.10(c). Note that such
an optimal $\alpha=0.04$ is not universal, whose value is case by case
generally [34]. Therefore, a randomly selected $\alpha\in(0,1)$ may achieve a
sub-optimal performance of the prediction in practice.

##  VI Conclusion

To summarize this chapter, with the proxy data of WiFi assess logs collected
in a campus of Fudan University, we have presented the series of temporal
interactive patterns of the large-scale population of WiFi users as a
representative example of cyber-social populations in the networking era of
today. The efforts from the viewpoint of temporal networks offer sufficiently
powerful supports to discover human interactions, as we illustrated in this
chapter, from both pairwise and event interactions defined at the levels of
individual users and a group of concurrent users, respectively. Not limited to
the rich temporal patterns of the WiFi-interactive population, the outlier
performance of the temporal hubs along with the roles of leaf connectivity
status also sheds more light to the significance of temporal networks, which,
in return, help to rank and predict the important active users in the temporal
framework.

Nevertheless, the understanding of network science from the view points of
temporal networks as well as human population dynamics still leaves many
topics to explore, which is far beyond the previous addressed issues such as
time-varying, time-evolving, and time-switching networks. Especially, new
concepts from not only the network topological connectivity but also dynamical
processes and the collective performance need to bridge the gap between the
traditional (static/aggregated) version and the now focused temporal
dimension, for example but not limited to temporal degree, temporal path in
previous sections, and temporal clustering coefficient [38], temporal
betweenness[42], in and out-components[43], temporal small world[44], temporal
motif [45, 46, 36], and etc.

Return to the key question in the beginning of this chapter. The modeling of
such fascinating and yet temporal complex features of a network very recently
have reported several models as a combination of human dynamics and evolving
networks [47, 48, 49, 50], and the dynamical processes [51, 52, 53] as well as
the collective behaviors [54, 55] and even the structural controllability [39]
over such temporal networks have newly attracted more broad attention. When
finishing this chapter, we also notice the book entitled “Temporal networks”
[56] which covers a more extensive scope of temporal networks for the
interested readers’ further reference, while, on the other hand, this chapter
synthesizes this new emergent branch with a detailed WiFi-interactive social
population.

## Acknowledgment

Xiang Li and Yi-Qing Zhang were grateful to the Informatization Office of
Fudan University for the WiFi Data collection, and the Archives of Fudan
University for providing the blueprint of all six teaching buildings. This
work was partly supported by the National Key Basic Research and Development
Program (No.2010CB731403), the National Natural Science Foundation
(No.61273223), the Research Fund for the Doctoral Program of Higher Education
(No.20120071110029) and the Key Project of National Social Science Fund (No.
12&ZD18) of China.

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Xiang Li (M’05-SM’08) received the B.S. and Ph.D. degrees in control theory
and control engineering from Nankai University, China, in 1997 and 2002,
respectively, and he currently is a Professor and Head of the Electronic
Engineering Department, Fudan University. Prof. Xiang Li serves and served as
Associate Editor of IEEE Transactions on Circuits and Systems-I: Regular
Papers, IEEE Circuits and Systems Society Newsletters, Control Engineering
Practice, and Guest Associate Editor of Int. Journal of Bifurcations and
Chaos, and etc.. His main research interests include theories and applications
of complex network and network science, where he has (co-)authored 4 research
monographs and more than 150 peer-refereed publications in journals and
conferences. Prof. Xiang Li received the IEEE Guillemin-Cauer Best
Transactions Paper Award from the IEEE Circuits and Systems Society in 2005,
Shanghai Natural Science Award (1st class) in 2008, Shanghai Science and
Technology Young Talents Award in 2010, National Science Fund for
Distinguished Young Scholar of China in 2014, and other awards and honors.
---
Yi-Qing Zhang (S’14) received the B.S. degrees in electronics engineering from
Fudan University, Shanghai, China, in 2007. He is currently a Ph.D. candidate
in the Department of Electronic Engineering at Fudan University. His research
interests are mostly about dynamics on and of temporal networks, i.e.,
structure evolution of temporal networks, as well as dynamical processes such
as epidemics on temporal networks.
---
Athanasios V. Vasilakos (M’00-SM’11) is currently a Professor with the Kuwait
University. He served or is serving as an Editor or/and Guest Editor for many
technical journals,such as the IEEE TRANSACTIONS ON NETWORK AND SERVICE
MANAGEMENT; IEEE TRANSACTIONS ON CLOUD COMPUTING, IEEE TRANSACTIONS ON
INFORMATION FORENSICS AND SECURITY, IEEE TRANSACTIONS ON NANOBIOSCIENCE, IEEE
TRANSACTIONS ON CYBERNETICS; IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN
BIOMEDICINE; ACM Transactions on Autonomous and Adaptive Systems; the IEEE
JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He is also General Chair of the
European Alliances for Innovation(www.eai.eu).
---