Community learning in location based social networks 2

55
Chapter 3
Overview of Dataset
Since there are no standard datasets available for research in LBSNs, we have
designed a crawling approach to obtain a sampled, large-scale, representative and
real world dataset. Among all dominating LBSNs, Foursquare has been reported to
have the highest number of active users with most frequent users’ daily activities.
Thus, we choose Foursquare as the testbed in the thesis.
In this chapter, we first give an overview of Foursquare in Section 3.1 and
introduce our crawling method in Section 3.2. We then present an overview of the
data structure of Foursquare dataset in Section 3.3. We next report the first- order
analysis on the obtained sampled dataset in Section 3.4. Finally, we describe the
two sub-datasets for evaluations in Section 3.5 and Section 3.6.
3.1 Foursquare
Foursquare describes their service as “an application that helps you and your friends
make the most of where you are.” It is a friend-finder, a social city guide and a game
that challenges users to experience new things and rewards them for doing so by
various badges. As of September, 2013, there are more than 4 0 million Foursquare
56
Upload
Photos
Add
accompanying
friends
Share in
Twitter and
Facebook
Upload
Photos
Write Tips
Figure 3.1: Additional activities other than checking in in Foursquare. (Screen

captured on 30th December, 2 013)
users worldwide and more than 4.5 billion accumulative check-ins with millions
more every day
1
.
Similar to other LBSNs, Foursquare lets users check in to a place when theyre
there, tell friends where they are and track the history of where they have been and
who they have been there with. When doing a check-in, Foursquare examines the
users’ current location and shows a list of nearby places. Users can also register
new places. Location is based on GPS hardware in the mobile device or network
location provided by the application. Each check-in awards the user points and
sometimes “badges”. The user who checks in the most oft en to a venue becomes
the “mayor” and users r egula r ly vie for “mayorships”.
Foursquare lets people connect to friends, which are equivalent to the concept
of friends on other online social networks. Besides finding friends directly using
name search or importing from phone contacts or other social networks, Foursquare
1
/>57
Step 1: User sends his physical location
Step 2: Foursquare sends candidate venues
Step 3: User sends the venue and optionally push to Twitter
Figure 3.2: Steps of check-in activity in Foursquare.
users can also add friends who are currently in the same venues. The “Here Now”
section in the venue page shows a list of users who are currently at the particular
venue. In this way, users a re able to find causal friends with similar interests.
As mentioned in Chapter 1, users in Foursquare are prompted for providing
additional multimedia information together with their check-ins. As Figure 3.1
shows, users can upload photos and write comments about the current venue. In
addition, users can share their check-ins in Twitter or Faceb ook and tag friends
who are currently together with him/her.

3.2 Data Crawling
We aim to obtain a set of active users and their activities in Foursquare. The ac-
tivities of interest include checking in at venues, posting tips and uploading photos,
where the most popular and dominant activity is checking in. Here, we seek to first
obtain users who frequently perfor m check-ins and then crawl the other activities
of these active users.
Foursquare have provided rich endpoints for data access. However, to pro tect
users’ privacy, Foursquare limits the access to users’ check-in records, which are
only available for the current acting user. Fortunately, the connection between
LBSNs and microblogging services provides an alternative way to access such data.
Figure 3.2 shows the check-in process. When a user tries t o check in at a certain
58
Figure 3.3: Check-in page in Foursquare.
venue, he/she first sends the current exact physical location in terms of la t itude
and longitude (Step 1). Then Foursquare compares the r eceived location with their
huge venue database and suggests a few names of places in the order of decreasing
geographical distance (Step 2). After that, the user selects one place name for
his/her current locat ions and sends it back to Foursquare and optionally he/she
may push the check-in informat ion through Twitter (Step 3). In addition, Twitter
provides a glance into its millions of users and billions of tweets t hr ough a Streaming
API
2
which provides a sample of all tweets matching some keywords selected by
the API user.
We monitor Twitter streams with the key words specified as “4sq.com” in
two time periods: January to March 2012 and August to November 2012. Each
sampled check-in message contains a short link (F igure 3.2) to the original check-in
page (Figure 3.3), where the details of the check-in, such as user ID, venue ID,
2
/>59

Table 3.1: Key information retrieved for users.
ID First Name Last Name Gender Profile Photo Home City
6512 Xina B. Female Surry Hills, NSW
check-in time, etc are available.
When we have obtained the list of users who share their check-in activities
through Twitter during the crawling periods, we are able to retrieve t heir other
activities, such as tip posting
3
, photo uploading
4
and friendship information
5
using
the Foursquare APIs.
3.3 Data Structures
Table 3.1 lists the key information retrieved for users, including user IDs, user
names, users’ profile photos
6
and t heir home cities.
Table 3.2 lists the key information retrieved for venues, including venue IDs,
venue names a nd the physical geogr aphical location. Each venue may be associated
with one or more semantic catego ries, such as restaurant, mall, etc. Foursquare
maintains a hierarchically organized venue categories
7
, where there ar e nine pre-
defined root categories, including art & enterta inment, food, nightlife spot, etc.
Figure 3.4 shows a par t of the hierarchy and there are tota lly 437 categories a s
for now. In the sampled dataset, 95.23% of venues are tagged with one category,
0.05% of venues are tagged with more than one categories and 4.72% venues are
not tagged with any categories. Venues that are not tagged with any categories are

3
/>4
/>5
/>6
Due to privacy reasons, we have blurred the photo
7
/>60
Table 3.2: Key information retrieved for venues.
ID Name Address Location City Country Category
4b053 VivoCity Harbour Walk (1.26,103.8) Singapore Singapore Mall
usually unimp ortant venues with very few check-ins or meaningless venues.
Figure 3.4: Venue category hierarchy in Foursquare (selected).
Table 3.3 lists the key information retrieved for check-ins, including user IDs,
venue IDs and the times of the check-ins. All times are recorded in Greenwich Mean
Time (GMT), which is also referred to as the Universal Time Coordinated (UTC).
Table 3.3: Key info rmation retrieved for check-ins.
ID User ID Venue ID Time
5004f 5062890 26de4 2012- 09-17T05:04 :46Z
Table 3.4 and Table 3.5 list the key information retrieved for tips and images,
respectively. We can regard tips and photos as special check-ins by enriching the
check-ins with multimedia contents. Similarly, all times a re recorded in GMT.
61
Table 3.4: Key information retrieved for tips.
ID User ID Venue ID Time Text
4e3474 33693 12 4b59e1 2012-12-05-26 T00:24:37Z Love the sky park
Table 3.5: Key information retrieved for photos.
ID User ID Venue ID Time Image
4e7d3 1 2250919 4a73e8 2012-08-19T03:04:46Z
In data preprocessing, we remove two kinds of suspicious check-ins. First, we
remove check-ins from users who have performed more than ten check-ins within a

minute. Second, we remove “sudden moves” where the two check-ins implies that
a user is travelling at a speed faster than 1, 000km/hour (Faster than the speed of
normal commercial jet airplanes). In addition, we notice that certain venues are
deleted by Foursquare in the housekeeping process. We remove all check-ins which
were performed on these deleted venues.
Finally, Table 3.6 shows the statistics of the sampled Foursquare dataset. We
regard users’ declared “homecity” in Foursquare as users’ tr ue home city, where we
remove a user if more than 50% of his/her check-is are not in his/her declared home
city.
62
Table 3.6: Statistics of the Foursquare dataset.
January – Mar ch, 2012 August – November, 2012
New York City Singapore New York City Singapore Chicago London
Number of users 131,494 60,244 350,652 160,063 91,478 58,143
Number of venues 138,627 155,191 369,673 413,843 52,107 61,906
Number of check-ins 1,127,699 313,269 13,023,087 6,082,394 4,390,944 3,230,865
Number of tips 657,470 298,210 1,341,471 789,218 651,310 389,201
Number of photos 178,975 165,191 N.A. N.A. N.A. N.A.
3.4 Data Analysis
In this section, we present the chara cteristics of the sampled, large-scale dataset
with an investigation of the temporal/ geographical characteristics and the distri-
bution of the frequency of activities per user.
Time distribution of the check-ins
First of all, we visualize t he global distribution of sampled Foursquare venues visited
from January to March 2012 and August to November 2012 in Figure 3.5, where
colors represent the popularity of venues with “red”: number of check-ins > 100,
“green”: 50 ≤ number of check-ins ≤ 100 and “blue”: 10 ≤ number of check-ins <
50. We see that while check-ins are globally distributed, the density of check-ins
is highest in U.S., especially in New York, where Foursquare was launched. Other
hot areas include cities in West Europe, South East Asia, Japan and South Korea.

Though most ar eas in China are currently blank, we see an increasing trend of
using Foursquare check-in services in the east coast, especially in big cities, such as
Shanghai.
While Figure 3.5 convey the scale and density o f the sampled Foursquare
dataset, we can furt her explore the nature of these check-ins by a ggregating venue
categories a cross all check-ins. The aggregated view in Figure 3.6 shows that the
63
Figure 3.5: Global distribution of sampled Foursquare venues visited from Jan-
uary to March 2012 and August to November 2012. Colors represent the
popularity of venues with “red”: number of check-ins > 100, “green”: 50 ≤
number of check-ins ≤ 100 and “blue”: 10 ≤ number o f check-ins < 50.
most popular check-in venues are restaurants, homes, shops/stores/malls.
Geographical distribution of the check-ins
Considering the t emporal distribution of check-ins, we show both the aggregate
daily patterns and weekly patterns of users’s check-ins. To resolve the time differ-
ences across geographical regions, we first obtain the time zones of all venues by
EarthTools
8
and then convert each check-in time t o the corresponding lo cal time
according t o the geographical location.
Figure 3.7a shows the aggregated check-in pattern per day. This patt ern
provides a glimpse into the globally daily “heartbeat”, with two major peaks: one
around 12pm and one around 6pm, where people are out at restaurants or food
courts fo r lunch/dinner. This correlate with the o bservation that most check-ins
are performed at venues, which belong to “Food” categories. The similar patterns
8
/>64
Figure 3.6: Venue category cloud for check-ins.
(a) Daily check-in pattern. (b) Weekly check-in pattern.
Figure 3.7: Check-in patterns.

were repor t ed in [79].
Figure 3.7b shows the aggregated check-in pattern per week. As expected,
weekdays clearly show two peaks during lunch time and dinner time, while over
the weekends these two peaks blend, reflecting a fundamentally different weekend
schedule for most Foursquare users.
Finally, Figure 3.8 shows the distribution patterns of user behaviors in the
sampled dataset. In Figures 3.8a, 3.8b, 3.8c and 3.8d, we report the proportion
of users v.s. the number of check-ins performed, t he number of venues visited, the
number of photos uploaded and the number of tips written. Similar to previous
reported observations, the four distributions exhibit similar trend, where only a few
65
users ar e extremely active in various activities, while a large number of users have
only few activities.
(a) Number of check-ins performed pe r user. (b) Number of venues visited per user.
(c) Number o f photos uploaded p e r user. (d) Number of tips written per user.
Figure 3.8: Distribution patterns of user behaviors in the sampled dataset.
3.5 Dataset for Community Unde r standing
We construct the dataset based on the data crawled from January to March 2012.
The task of community understanding aims to mine communities, which are in-
terpretable and exhibit clear community profiles in terms of multimedia contents.
Thus we aim to select users who contribute mor e check-ins, tips and photo s. We
66
Table 3.7: Dataset for Community Understanding.
Users Checkins Tips Images
Global 13,068 86,302 33 5,877 69,510
Singapore 8,736 3 2,156 156,761 9,775
New York City 9,918 51,043 213,302 22,135
select the candidate users as follows. First, we compute the activeness score for
user u as:
activeness(u) = a × C(u) + b ×T(u) + c ×P(u), (3.1)

where C(u), T (u) and P(u) are the number of check-ins, tips and photos of u, re-
spectively. a, b and c adjust the importance of each contribution and we empirically
set a = b = c =
1
3
. We then select the top 80% of users ranked by the users’ active-
ness scores at the global scale and two city scales, respectively. Users at the global
scale include but not limited to users in Singapore and users in New York City.
They also include active English speaking Foursquare users from other cities, such
as San Francisco, Chicago, London, etc. Table 3.7 summarizes the dataset. We
regard users’ actual behaviors as the ground truth for evaluations. For example, in
evaluating the prediction accuracy of users’ visiting preferences, the actual venues
that are visited by the testing users are regarded as the ground truth.
3.6 Dataset for Community Matching
We co nstruct the dataset based on the data crawled from August to November 2012.
Since our focus is to match communities across geographical regions for locally
interesting venue recommendation to tourists, we select users from Chicago (CHI),
London (LD N), New York City (NYC) and Singapore (SG) and aim to recommend
venues to them when they are in a city other than their home cities. We locate
67
Table 3.8: Properties of sampled popular regions: N is the number of active users,
M
L
is the number of local venues, C
L
is the number of local check-ins, M
F
is the
number of foreign venues and C
F

is the number o f foreign check-ins.
N M
L
C
L
M
F
C
F
New York City 26, 411 64, 24 9 448, 072 301, 782 810, 545
Chicago 7, 138 36, 164 353, 290 120, 940 34 1, 651
Singapore 8, 033 50, 722 406, 490 20, 940 36, 874
London 6, 320 25, 031 258 , 605 66, 031 158, 605
venues in different cities based on the geographical bounding boxes returned by
Goog le’s geocoding API
9
. Table 3.8 lists the properties of t he four cities that we
perform experiments on. Active users are those who have performed at least ten
check-ins during the crawling periods. Foreign venues for a user are those that
locate outside the user’s home city and foreign check-ins f or a user are check-ins
performed at foreign venues. We regard users’ actual visit of local and for eign
venues as the g round tr uth for evaluations.
9
/>68
69
Chapter 4
Community Understanding
In LBSNs, users implicitly interact with each other by visiting places, issuing com-
ments and/or uploading photos. These heterogeneous interactions convey the latent
information for identifying meaningful and interpretable so cial communities, which

exhibit unique location-oriented characteristics.
In this chapter, we present an approach to simultaneously detect and under-
stand interest communities in LBSNs by representing the heterogeneous interactions
with a multi-modal non-uniform hypergraph. Here the vertices of the hypergraph
are users, venues, comments or photos and the hyperedges characterize the k-partite
heterogeneous intera ctio ns such as posting certain comments or uploading certain
photos while visiting certain places. We then view each detected social community
as a dense subgraph within the heterogeneous hypergraph, where the user commu-
nity is constructed by the vertices and edges in the dense subgraph and the profile
1
of the community is characterized by the vertices related with venues, comments
and pho t os and their inter-r elat ions. We present an efficient algor it hm to detec-
t the overlapped dense subgraphs, where the profile of each social community is
guaranteed to be available by constraining the minimal number of vertices in each
1
The profile of a community shows the common characteristics of its community member s.
70
modality. Extensive experiments on the selected subset of our sampled Fo ursquare
dataset well validated the effectiveness of the proposed framework in terms of de-
tecting meaningful social communities and uncovering their underlying profiles in
LBSNs.
The rest of the chapter is organized a s follows. Section 4.1 int r oduces the
motivation and challenges of community understanding in LBSNs. Section 4.2 re-
views the relat ed work on community understanding. Section 4.3 gives an overview
of the proposed framework for community understanding in LBSNs. Section 4.4 de-
tails the hypergraph construction. Section 4.5 formulates the community detection
and understanding task as a dense subgraph detection over heterogeneous hyp er-
graph problem and int r oduces an effective algorithm to solve the pro blem. We
then report the empirical evaluation in Section 4.6, which gives both qualitative
and quantitative results. Finally, Section 4.10 gives the concluding remarks.

4.1 Motivation and Challenges
In the er a of Web 2.0, social networking has emerged to be a popular way for peo-
ple to connect, communicate and share information with each other, and this has
led to an explosion of multimedia informatio n. Users in social networks interact
with each other by contributing and consuming multimedia data (photos, locatio ns,
texts, etc.) and there exist var ious typ es of objects and heterogeneous relations in
the networks. Specifically, with the high penetration of GPS-enabled smart phones
in recent years, we have witnessed the boom o f LBSNs, where users can check in
at venues, write tips a nd upload photos while these information can be immedi-
ately disseminated via social graphs to their friends and public. While tips may
cover a variety of diverse t opics related to venue activities and recommendations.
Photos, on the other hand, visually present the interesting aspects of venues visit-
ed. These location-tagged multimedia data generated by Foursquare users provide
71
us unprecedented opportunities to understand the collective user behaviours on a
large scale. The voluminous amount and heterogeneity of user generated contents
and fast expansion of network diameter challenge us to perform community detec-
tion and community understanding on hundreds of t housands to even millions of
entities.
One of the fundamental tasks in social media network analysis is to under-
stand human collective behaviours by identifying people’ social positions based on
the detected cohesive subgroups whose group members interact with each other
more frequently than those outside the group [47, 93, 37]. While previous research
efforts have reported promising results on clustering communities from traditional
social networks [1 43, 47], the heterogeneous user behaviours in LBSNs bring togeth-
er both “virtual” and “physical” int eractio ns, which makes it very challenging to
develop new fr ameworks to model the network in a natural and unified manner for
community detection and understanding. Figure 4.1 shows a snapshot o f typical
user behaviours in Foursquare, which might correspond to a certain overlapping
communities we aim to detect and understand.

In LBSNs, t he heterogeneous interactions among multiple types of entities
such as users, tips, venues and photos naturally form a multi-modal and non-
uniform hypergraph, where each modality corresp onds to one type of entities and
each hyperedge connects a varying number of entities from different modalities. For
example, check-ins connect venues to users while tips/photos connect textual top-
ics/visual concepts to users and venues. These kinds of interactions are naturally
represented by hyperedges with non-uniform affinity relations. In addition, there
also exist latent r elations among venues. For example, grocery stores are more
similar to supermarkets than to parks. These interactio ns and relations natur ally
converge into a non-uniform and heterogeneous hypernetwork (Figure 4.2d). How-
ever, the state-of-the-art community mining approaches usually handle standard
72
Upload a photo
Post a tip
Perform a check-in
Figure 4.1: Overview of the community understanding problem. (L eft) Heteroge-
neous users’ behaviours in Foursquare, where users can check in at venues (black
arrows), write t ips (blue arrows) and upload photos (red arrows) at various venues.
(Right) The detected overlapping communities, where each ellipse represents a pro-
filable community whose characteristics are showed through the tripartite inter-
entity relation graph. Each user may belong to one or mo r e communities. (Best
view in color).
network types, such as those shown in Fig ures 4.2a,4.2b and 4.2c, while we aim
to tackle community mining and understanding problem from heterogeneous and
non-uniform hypergraphs ( Figure 4.2d). If we would utilize traditional community
detection approaches such as modularity optimization [93] or other heuristics driven
methods [47, 31], we need t o reduce the complex hypergraph to simpler bipartite
or one-modal g raph thro ugh processes such as “flattening” or “proj ection” with
possible information lo ss [90, 157].
In addition, as is also mentioned in Chapter 1, most of the state-of-the-art

techniques extract community structures by minimizing certain objective functions
while ignoring the equally important task of “understanding” the characteristics of
the groups [67]. However, as pointed out by Fortunato [37], there is no guarantee
that these approaches can provide good quality detection. Guimera et al. also re-
vealed that a maximum modularity may not imply that t r ue community structure
is discovered, since random networks may also contain high modularity partition-
s [52]. Though Tang and Liu have attempted t o profile the mined communities
73
(a) (b)
Modality 1
Modality 2
Modality 3
(c)
A
Modality 1
Modality 2
Modality 3
1
2
3
4
5
A
B
C
D
E
I
II
III

IV
V
(d)
Figure 4.2: A summary of the different network types (a) A one-modal single-
edge graph. (b) A one-modal multi-edge graph. (c) A t r ipartite three-unifor m
hypergraph. (d) A heterogeneous non-uniform hypergraph. In (d), there exist the
following hyp eredges: (1,A), (2,C,III), (3,C),(4,B,I),(5 ,D,II),(5,D,II),(E,IV),(I,II),
(II,III,IV) and (IV,V) (Best view in color).
by extracting descriptive features by using some heuristics [129], it remains unclear
what are the underlying reasons to bind the members tog ether and how to interpret
the community profiles in terms of the extra cted features.
4.2 Related Work on Community Understanding
Some works attempts to understand the group formation based on statistical struc-
tural analysis. Backstrom et al. studied pro minent online groups in the digital
domain, aiming at answering some basic questions about the evolution of groups,
like what are the structural features that influence whether individuals will join
communities [5]. They found that the number of friends in a group is the most
important factor to determine whether a new user would join the group. Their
findings provide a global level of structural analysis to help understand how com-
munities attract new members. However, more efforts are required to understand
the fo r ma t ion of a particular community.
Some works try t o extract the annota tions from relational data from text.
For example, Roy et al. constructed a hierarchical structure as well as corresponding
annotations based on a complicated generative process [109]. The model complexity
74
and scalability hinder its application to group pro filing in large scale networks.
Chang et al. proposed NUBBI to infer description of its entities and of relationships
between these entities from a text corpora [22]. The probabilistic topic model
assumes the words are generated based on the topics associated with an entity or
the topics of the pairwise relationship of entities.

Some other works extract communities based on relation and text infor-
mation together where each topic represents a distribution of wor ds and can be
considered as the words associated with a group. Link-LDA treats the citations
the same way as norma l words, i.e., the citation generated ba sed on a multinomial
distribution over the documents [34]. Pairwise Linked-LDA [88] essent ially com-
bines the topic model [12 ] and the mixed membership stochastic block model [1]
via forcing the latent mixture of communities to be the same for both word to pics
and relation topics. Mei et al. enforced the connected documents to share similar
topics and used the network information as regularization while extracting the top-
ics of texts. All these methods extracted the topics a ssociated with text instead of
capturing the corresponding text associated with a given community [82].
According to the concept of Homophily [81], a connection occurs at a higher
rate between similar people than dissimilar people. Homophily is one of the first
chara cteristics studied by early social science researchers and holds for a wide vari-
ety of relationships and the phenomenon is also observed in social media [3 6, 133].
In order to understand the formation of a community, some works aim to investi-
gate the inverse problem: given a group of users, can we figure out why they are
connected? And what are their shared similarities? Tang et al. proposed a group
profiling approach by extracting shared a t t ributes of community members [132].
One example of shared attribute could be a set of topics commonly contributed by
the community members. Since a group consists of people with shared interests, one
intuitive way of group profiling is to clip a community with “some topics” shared
75
Figure 4.3: Group profile of a community of people who enjoy night life.
by most members in the community. For example, in blogosphere
2
, blo ggers upload
blog posts; in content sharing sites such as Digg and Del.icio.us, users post news or
bookmarks and provide tags on the shared contents. These content information and
tags essentially represents the latent interests of individuals and can be regarded as

topics. For example, Figure 4.3 shows an example profile of people who enjoy night
life by both textual and visual “topics” in Foursquare. However, the aggregation-
based approaches separate the community detection and understanding into two
steps, while our proposed framework does both simultaneously.
4.3 Overview
To tackle these challenges, we propose a novel and unified framework which per-
forms both community detection and community understanding in LBSNs. To de-
tail the process, we first construct a heterogeneous, multi-modal and non-uniform
hypergraph which naturally captures various kinds of interactions, such as check-in
2
This term was first coined by Brad L. Graham on 10 September , 1999 and implies that blogs
exist tog ether as a connected community or as a social networ k in which everyday authors can
publish their opinions.
76
actions or tip-posting actions in LBSNs. We then propose an efficient alg orithm to
discover multiple overlapping communities by constraining the minimum number
of entities in each modality.
The advantages of our proposed method are multifold: (1) the method is
general, which can be used in any community detection and understanding tasks as
long as the network structure is represented as a graph or hypergraph; (2) it allows
new types o f interactions and modalities to be easily added with the technology
advancement and emerging of new services; (3) the approach is able to automati-
cally determine the number of interest communities with overlapping entities; and
(4) community understanding is straightforward since the final computed commu-
nity contains both users and the “reasons” why they a re put in that particular
community. In the context of LBSNs, the “r easons” are the combination of venues
they visit, tips they post and photos they upload as well as the strengths of the
inter-relations a mo ng entities from different modalities.
4.4 Heterogeneous Hypergraph Const r uction
The heterogeneity of activities in LBSNs naturally brings multiple types of entities

and interactions into the same network, which we call a multi-modal hypernetwork
where each modality corresponds to a typ e of entity. In Foursquare network, there
are four modalities: user, venue, tip and photo, and three types of interaction-
s: a user checks in at a venue, posts a tip and uploads a photo at a venue. In
addition, ent it ies of same modality may also be related. Figure 4.4 illustrates the
typical interactions and inter-venue connections in Foursquare. These heterogenous
interactions naturally lead to the construction of a non-uniform and multi-modal
hypergraph. In this section, we first introduce the different types of vertices (Sec-
tion 4.4.1) and hyperedges (Section 4.4.2) involved in the hypergraph before giving
details on how we construct each type of hyperedges (Section 4.4.3, 4.4.4, 4.4.5 and
77
4.4.6). For clarity and convenience, we list the variables used in this section in
Table 4.1.
Table 4.1: List of notations of variables used in Chapter 3.
Notation Explanation
V the set of all vertices
V
i
the ith subset of vertices
E the set of all hyperedges
E
i
the set of hyperedges with n
i
-ary affinity re-
lation
u
i
the ith user
l

i
the ith venue category
t
i
the ith tip topic
d
i
the ith photo concept
(u
i
, l
j
) (hyperedge) The ith user visits the j’s venue
category
(u
i
, d
k
, l
j
) (hyperedge) The ith user cont r ibutes the k’s
image concept at the j’s venue category
(u
i
, t
k
, l
j
) (hyperedge) The ith user cont r ibutes the k’s
tip t opic at the j’s venue category

x the probability vector, with each x
i
repre-
senting the probability of choosing the ith
vertex of V in a community
w
i
the weight of hyperedge i involving vertices:
(x
i
1
, ··· , x
i
n
i
)
C a community of vertices and C ⊆ V
Continued on next page
78
Table 4.1 – continued from previous page
Notation Explanation
ε the variable to control community size
c
i
the lower bound of the existence probability
of modality i within the final detected com-
munities
λ
i
, α

i
, β
i
and π
i
the Lagrangian multipliers
U the set of pairs (x
i
, x
j
) which can increase the
objective function
K the number of initializations
User
Venue
Tip
Photo
Figure 4.4: Heterogeneous entities and intera ctio ns in Foursquare. Red line and
arrow represent interactions among user, photo and venue; Blue line and arrow
represent interactions among user, tip and venue; Black line and arrow represent
interactions between user and venue; Green circle connects venue categories with
similar functions.
4.4.1 Vertices
There are four types o f vertices involved in interactio ns in LBSNs: user, venue,
tip and photo, as is shown in Figure 4.4. Formally, let V be the vertex set, which
can be divided into g subsets, i.e., V =

g
a=1
V

a
. In Foursquare network, g = 4
79
and each subset V
a
corresponds to set of vertices of modalities: user, venue, tip
and photo, respectively. Let V
1
= U = {u
1
, ··· , u
n
u
}, V
2
= L = {l
1
, ··· , l
n
l
},
V
3
= T = {t
1
, ··· , t
n
t
}, and V
4

= D = {d
1
, ··· , d
n
d
} be sets of users, venues, tips
and pho t os respectively, where n
a
is the number o f vertices in V
a
, a ∈ {1, 2, 3, 4}.
4.4.2 Hyperedges
There are three types o f interactions involved in the Foursquare network: a user
checks in at a venue, posts a tip and uploads a photo at a venue. Thus, the first three
types of hyperedges correspond to each interaction, respectively. Formally, let E be
the hyperedge set, which can be divided into s subsets, i.e. , E =

s
b=1
E
b
, with each
hyperedge representing a n
b
-ary affinity relation. In Foursquare network, s = 4. We
build three sets of hyperedges corresponding to three interactions: E
1
= {(u
i
, l

j
)}
representing a check-in perfor med by user u
i
at venue l
j
, E
2
= {(u
i
, t
j
, l
k
)} repre-
senting tip t
j
is posted by user u
i
at venue l
k
and E
3
= {(u
i
, d
j
, l
k
)} representing

photo d
j
is uploaded by user u
i
at venue l
k
. In addition, we also want to model
inter-venue similarities and define hyperedge set E
4
= {(l
1
, ··· , l
h
)}. We can then
denote the hypergraph as G = {V, E, w}, where w : E → R is a weighting function
which associates a real value with each hyperedge, with larger weights represent -
ing stronger affinity relations. Figure 4.5 illustrates the process for hypergraph
constructions for Foursquare hypernetwork.
Given the enormous number of entities in each modality, the hypergraph is
extremely sparse, which unavoidably weakens the structure inf ormation. Thus, to
better char acterize different types of interactions, we seek to first group semanti-
cally similar entities together in each modality to construct a denser hypergr aph.
The following subsections describe the constructions of each type of hyperedges in
details.