sfunctions.eps


Followee recommendation in Twitter using fuzzy
link prediction

Fernando M. Rodŕıguez†, Luis M. Torres‡, Sara E. Garza∗

School of Mechanical and Electrical Engineering (FIME)

Universidad Autónoma de Nuevo León (UANL)

San Nicolás de los Garza, NL, Mexico
†fernando.aldape@gmail.com, ‡luis.torres.ciidit@gmail.com,

∗sara.garzavl@uanl.edu.mx

Abstract

In social networking sites, it is useful to receive recommendations about
whom to contact or follow. These recommendations not only allow to es-
tablish connections with people one might already know in real life, but also
with people or users that have similar interests or are potentially interesting.
We propose an approach that tackles contact (followee) recommendation in
Twitter by means of fuzzy logic. This fuzzy approach handles recommenda-
tion as a link prediction problem and uses three types of similarity between a
pair of users: tweet similarity, followee id similarity, and followee tweet sim-
ilarity. These similarities are calculated by extracting user profiles. These
profiles are, in turn, obtained by considering Twitter as a heterogeneous in-
formation network. To test our approach, we crawled a repository of 6,000
users and 2 million tweets, and we measured accuracy by comparing our re-
sults with the actual followee lists of the users. These results, which are also
compared against the results given by state-of-the-art methods, show a high
accuracy. Other advantages of the fuzzy system include a self-explanatory
capability and the ability to produce a non-binary friendship value.

Keywords: fuzzy systems, recommender systems, Twitter, link predic-
tion, expert systems, artificial intelligence

1 Introduction

Social networking sites (SNS’s) play a key role in our daily lives. These sites
not only enable users to publish and receive information they consider to
be of interest, but also allow them to keep in contact with friends, or even
to establish new relationships. Social networking sites, ultimately, serve as
a platform for personal or social expression and organization — hence their
importance for areas such as marketing, human-computer interaction, and
research.

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Due to the increasingly intensive use of SNS’s, it is common for users
to be overloaded with information (Benito-Ruiz, 2009; Gantz and Reinsel,
2010), e.g. status updates to read or possible new friends to contact. For this
reason, recommender systems have been adopted to filter content, activities,
products, and social connections of actual interest for the user (Park et al.,
2012).

An important representative of SNS’s nowadays — with 190 million
users, 60,000 daily comments, and ranking ten in Alexa’s Top Sites1 —
is Twitter2. This site offers a microblogging service where users are able to
publish short messages (140 characters maximum per post) called “tweets”
and receive the posts of other users that they decide to follow, where follow-
ing a user is similar to subscribing to an RSS news feed; the ones who follow a
user are called the user’s followers and the ones followed by a user are called
the user’s followees (in this work, we will use “contacts” and “friends” as
synonyms of this term). Other distinctive features of Twitter include post
forwarding, where the forwarded message is called retweet, user mentions in
messages (where each user account starts with “@”), and a special form of
message labeling, where the label is called hashtag and usually starts with
“#”.

Twitter, of course, is not exempt of the previously mentioned infor-
mation overload and crowding. Recommendation approaches for this SNS
so far include automatic suggestions of tweets, URL’s, hashtags, mentions,
retweets, and followees (Kywe et al., 2012). We focus on this last recom-
mendation item, since helpful followee recommendations aid users to receive
actual content of interest, to get acquainted with people of interest, and to
build stronger communities — either in Twitter or real life.

Because the information in Twitter tends to be imprecise, incomplete,
and could highly depend on the context where it is employed, we propose
a fuzzy hybrid recommendation system. The rules of this system create rec-
ommendations by predicting follower-followee relationships (which we shall
refer to as “friendships”) between pairs of users given the similarity of their
tweets, their followees, and the tweets of their followees. These similarities
are derived from user profiles whose information is retrieved by considering
Twitter as a heterogeneous information network. Our contributions thus
include:

- A fuzzy expert system that acts as a hybrid recommender system.

1Information available at: http://www.alexa.com/topsites. This list of Top Sites
was retrieved in March 2014.

2Available at: http://twitter.com.

2



This fuzzy system predicts the degree of friendship between a pair of
Twitter users and uses this degree as the basis for recommendation.

- A formal conceptual model for extracting user profiles by considering
Twitter as a heterogeneous network.

- Evidence for the effectiveness of the approach in different contexts
(Twitter, the Cit-HepTh citation network).

- A comparison with state-of-the-art methods.

The rest of this document is organized as follows: Section 2 reviews nec-
essary background and Section 3 discusses related work; Section 4 describes
our approach and Section 5 provides experiments and results; Section 6,
finally, presents conclusions and future work.

2 Background

This section introduces vocabulary and notation that will be used. It covers
basic notions for recommender systems, graph theory, information retrieval,
and fuzzy logic.

2.1 Recommender systems

In the midst of overwhelming amounts of information, the aim of a recom-
mender system is to suggest users those items that might be of interest, such
as movies, records, books, and other users. Most recommender systems are
personalized and build a profile to make suggestions based on the informa-
tion and particular preferences of the target user (i.e., the recipient of the
recommendations). Common approaches for this kind of recommendation
include content-based and collaborative filtering systems; while the former
find items that are similar to the ones highly rated by the user in the past,
the latter find items that were highly rated by users that are similar to the
target user (Ricci et al., 2011). In that sense, content-based systems are
item-oriented and collaborative filtering systems are user-oriented.

The usual output of a recommender system is a ranked list of items,
where the items at the top — ideally — are the most likely to achieve high
ratings by the target user. To build this list, several recommendation algo-
rithms start off with a set of candidate items and gradually refine this set.
When talking about friend or contact recommendations, the term “item” is
usually replaced by “user”. Consequently, it is common to refer to candi-
date users instead of candidate items and to recommended users instead of

3



recommended items; let us note that the latter are the ones included in the
(final) recommendation list. Another important aspect of friend recommen-
dation is that this task is sometimes stated as a link prediction problem, i.e.
the problem of determining if a connection will appear between a pair of
entities in a network (Getoor and Diehl, 2005).

2.2 Networks

Networks depict related entities and are formally represented with graphs.
A graph G = (V, E) consists of a set V = {v1, v2, . . . vn} of vertices and a
set E of edges. While the former set represents the entities of the network,
the latter stands for the relationships or connections among these entities;
E ⊆ V ×V , typically3. If E is symmetric, i.e. the existence of an edge (u, v)
implies the existence of (v, u) as well, then all edges run in both directions
and the graph is said to be undirected; being this the case, an edge can be
formally written as a 2-subset {u, v}. However, if the property of symmetry
does not hold, the graph is directed and (u, v) ∈ E depicts an edge that goes
out from u into v. In a directed graph, edges are commonly referred to as arcs
and are drawn as arrows. When the vertices or edges have labels, the graph is
said to be labeled, and an edge-labeled graph where the labels are numerical
values is said to be weighted (it is otherwise known as an unweighted graph
whose edges have indistinct values). A subgraph Gs = (Vs, Es) is a portion
of a graph, such that Vs ⊆ V and Es = {{u, v} : u, v ∈ Vs}.

The neighborhood of a vertex v (denoted by Γ(v)) is the set of vertices
which share an edge with v, i.e. Γ(v) = {u : {u, v}∈ E} . For a directed
graph, Γ(v) = Γ+(v)∪Γ−(v), where Γ+(v) is the out-neighborhood and Γ−(v)
is the in-neighborhood. While the former consists of the vertices whose arcs
go out from v, the latter includes vertices whose arcs go into v:

Γ+(v) = {u : (v, u) ∈ E} , (1)

Γ−(v) = {u : (u, v) ∈ E} . (2)

In several contexts, such as scientific coauthorship, transportation, and
recommendation (Newman, 2010), V contains vertices of different types (pa-
pers and authors / buyers and products / users, resources, and tags). This
can be formally represented with a k-partite graph, where V is composed of
k disjoint subsets and the edges in E join pairs of vertices that belong to
different subsets. When k = 2, the graph is said to be bipartite and thus

3Edges usually consist of vertex pairs, but three or more vertices can be simultaneously
joined by a single edge. In this case, the edge is referred to as hyperedge and the graph is
called hypergraph.

4



V = V1 ∪V2, (3)

V1 ∩V2 = ∅, and (4)

∀e = {u, v} , e ∈ E ⇒ (u ∈ Vi)∧ (v ∈ Vj), i 6= j, (5)

where Vi, Vj ∈{V1, V2}. Examples of bipartite graphs include author-paper,
actor-movie, metabolite-reaction, and buyer-product networks.

More complex situations can be modeled with heterogeneous information
networks (Sun et al., 2012, 2009; Wang et al., 2012; Ji et al., 2011), which
not only take into account different vertex types but also consider different
edge types (directed, undirected, weighted, . . . ) and connections between
arbitrary pairs of vertices (including the same type). Formally, for any
heterogeneous graph

V = V1 ∪ . . .∪Vn =
n
⋃

i=1

Vi and (6)

E ⊆{(u, v) : (u ∈ Vi)∧ (v ∈ Vj), i, j ∈{1, . . . n}} . (7)

An example of a heterogeneous graph is the roll call data network pre-
sented by Wang et al. (2012), which consists of two vertex types and three
edge types; while the vertices either stand for legislators or bills, the edges
either join legislator pairs (cosponsorship), bill pairs (semantic similarity),
or legislator-bill pairs (this last edge type is directed and represents votes).

2.3 The Vector Space Model

The vector space model is, so far, one of the central models for information
retrieval (Liu, 2007). This model views each document as a bag of words (a
representation where order is not important) and extracts a weight vector
from this bag. Each vector’s length is equal to the vocabulary (i.e. unique
words) of the whole document collection. The weight wi,j for a given word ki
in a particular document dj is usually awarded by a text frequency–inverse
document frequency or tf-idf score; this score indicates the importance of ki
in dj:

5



wi,j = tfi,j × idfi

idfi = log
N
ni

tfi,j =
fi,j

M
∑

x=1

fx,j

,

(8)

where fi,j is the number of times that ki appears in dj, M is the amount of
unique words in dj, N is the amount of documents in the collection, and ni
is the number of documents that ki appears in.

A common metric for calculating similarity between document vectors is
the cosine similarity (Baeza-Yates and Ribeiro-Neto, 1999), which is based
on the dot product of the vectors:

cosim(da, db) =
da � db
|da|× |db|

, (9)

where da and db are the documents, and da and db are the vectors. A
similarity of 0 indicates that the documents have no common words and a
similarity of 1 indicates that the documents are identical.

2.4 Fuzzy logic systems as linguistic processors

Fuzzy systems encode knowledge, which is commonly expressed as linguistic
information. These systems are based on fuzzy logic and were originally
proposed by Zadeh (1965). A fuzzy system maps a set of given inputs to
an output by means of an inference engine that uses a fuzzy rule base. To
perform inferences with this rule base, the inputs have to be fuzzified, and
the fuzzy result is defuzzified to obtain the corresponding output. This
process is illustrated in Figure 3.

The core of fuzzy system design is given by fuzzy sets, linguistic vari-
ables, and membership functions. A fuzzy set assigns a membership value
to every element, as opposed to a conventional set where elements are either
present or absent; e.g., in the set Old = {(45, 0.1), (65, 0.5), . . . (90, 0.9)}, the
element 90 has a membership value of 0.9. A membership value — whose
range is between 0 and 1, inclusive — is assigned by a membership function.
There are several forms of membership functions; however, the triangular
and trapezoidal forms are commonly used:

triangular(x, a, b, c) = max

(

min

(

(x − a)

(b − a + ǫ)
,

(c − x)

(c − b + ǫ)

)

, 0

)

, (10)

6



trapezoidal(x, a, b, c, d) = max

(

min

(

min

(

(x − a)

(b − a + ǫ)
, 1

)

,
(d − x)

(d − c + ǫ)

)

, 0

)

, (11)

where x is a given element of the fuzzy set and ǫ = 1 × 10−6 or another
suitable constant. Parameters a, b, c, and d are additionally illustrated in
Figures 1a and 1b.

(a) Triangular (b) Trapezoidal

Figure 1: Membership functions and parameters.

A linguistic variable is a variable associated with a numeric variable x
and does not take numbers as values, but instead takes linguistic values. For
example:

Volume = {“Low”, “High”, “Very high”, . . .} .

Each linguistic value is related to a fuzzy set or membership function. A
range of operation is defined for every linguistic variable and it is partitioned
considering the linguistic values used. Usually, an odd number of linguistic
partitions is used; three and five are the most common (see Figure 2).

As we previously mentioned, a fuzzy system is an integration of the logic
operations shown in Figure 3. Because we can use different membership
functions, fuzzifiers, inference engines, and defuzzifiers, there is a variety of
fuzzy systems. In this paper, as we will see later, we used a fuzzy system
with a singleton fuzzifier, trapezoidal and triangular membership functions,
a Mamdani max-min engine, and a centroid defuzzifier. Let us explain each
of these.

The singleton fuzzifier maps a value a into a fuzzy set A′ by setting
membership µA′(x) to 1 at x = a and 0 at any other value:

µA′(x) =

{

1 if x = a
0 otherwise.

(12)

7



Figure 2: (a) Five fuzzy sets or linguistic values: 1=“Very Low”, 2=“Low”,
3=“Medium”, 4=“High”, 5=“Very High”, (b) Three fuzzy sets or linguistic
values: 1=“Low”, 2=“Medium”, 3=“High”

Figure 3: Components of a fuzzy system.

With respect to the inference engine and the fuzzy rule base, in first
place, the fuzzy rule base is a matrix (which we shall denote by DB) where
every row is a rule and every column is an integer that represents the lin-
guistic value associated with that rule from antecedents to consequents. In
the case of three variables and five linguistic values, a maximum of 53 = 125
rules or rows and 4 columns (three inputs and one output) build up the ma-
trix. Second, considering a normalized interval for every linguistic variable,
a type1(x, n) function can be defined as follows:

type1(x, n) =























trapezoidal(x, 0, 0, 0.16, 0.33) if n = 1
triangular(x, 0.16, 0.33, 0.5) if n = 2
triangular(x, 0.33, 0.5, 0.66) if n = 3
triangular(x, 0.5, 0.66, 0.83) if n = 4
trapezoidal(x, 0.66, 0.83, 1, 1) if n = 5,

(13)

8



where x is an input value and n corresponds to a specific membership func-
tion of a linguistic variable. Considering a fuzzy system of three inputs, a
max-min inference engine is defined as follows:

I(r) = min(type1(x1, DB(r)), type1(x2, DB(r)), type1(x3, DB(r))) (14)

where r is the fuzzy rule number, DB is the fuzzy rule base (as already
stated), and I is the inference calculated. Finally, the output is defuzzified
as follows:

y =

∑R
r=1(I(r)×ym(DB(r)))

∑R
r=1 I(r)

, (15)

where ym is a precalculated center of mass of every membership function of
the output (the predefined values are ym =[0, 0.25, 0.5, 0.75, 1]) and R is
the number of rules. DB(r) establishes the correct value depending of the
fuzzy rule.

3 Related Work

As we will see throughout this section, related work is centered on followee
recommendation in Twitter and fuzzy recommendation. Twitter, in general,
has been a subject of study for some years now. For example, several works
analyze the “Twittersphere” as a complex network, including topologic and
geographic properties (Java et al., 2007); the impact of retweets, user in-
fluence, and the longevity of trending topics (Kwak et al., 2010); and how
users with similar interests are connected — a phenomenon known as ho-
mophily (Huberman et al., 2008). Other works include outcome prediction
for the American presidential election (Tumasjan et al., 2010), stock market
prediction (Sakaki et al., 2010), rumor propagation (Liu and Chen, 2011),
and epidemics detection (Culotta, 2010).

Regarding followee recommendations in Twitter, Hannon et al. (2010,
2011) introduce the Twittommender, a real-time system where users can
search for followees and receive followee recommendations as well. To gen-
erate these recommendations, a user profile of tf-idf weights is extracted and
presented as a query to the system. Nine profiling strategies are proposed;
five of these use a single information source and four combine two or several
sources — either of the same or different types (content vs. structure). The
strategies consist of: (1) user tweets, (2) followee tweets, (3) follower tweets,
(4) followee id’s, (5) follower id’s, (6) a combination of user, followee, and

9



follower tweets, (7) a combination of follower and followee id’s, (8) a combi-
nation of user tweets and follower id’s, and (9) an ensemble of strategies 1-7
where users which frequently have a high position in recommendations are
preferred over users whose frequencies or positions are low. These profiling
strategies were evaluated by measuring precision (overlap between the lists
of recommended users and the actual followee lists of the target users) and
ranking effectiveness (if relevant recommendations were placed in high po-
sitions of the list); a live trial where target users were asked if they would
follow the recommended users was also carried out. Collaborative filtering
strategies (4,5,7) gave, in general, better results than content-based strate-
gies (1-3,6), although the latter tended to place relevant recommendations
in higher positions.

Similar results are presented by Armentano et al. (2011a,b), who propose
two recommenders: purely based on content and purely based on structure.
For the content-based recommender, a set of users is randomly selected from
Twitter’s public timeline (a stream where the tweets of public accounts are
added) and represented with term-vector profiles; cosine similarity between
these users and the target user (whose profile is also a vector) is calculated
and users exceeding a given threshold are added into the recommendation
list. For the topology-based (structural) recommender, the followees of the
target user’s co-followers (i.e. the group of users who share followees with
the target user) are obtained and ranked according to a score that attempts
to balance popularity and resemblance to user preferences; this score takes
the product of a candidate recommendation’s proportion of repetitions (a
higher score is awarded to users recommended several times), proportion
of followers vs. followees, and proportion of mentions. Results were evalu-
ated using discounted cumulative gain, precision, and mean reciprocal rank;
even when no statistically significant differences were found between the two
recommenders, the content-based approach showed to position relevant rec-
ommendations at the top of the list. Other recommenders that make use of
structural information are given in the works by Hsu et al. (2006) and Silva
et al. (2010). The former describes a hybrid recommender that employs
graph proximity and logistic regression to suggest friends in the LiveJournal
weblog, and the latter uses complex network theory and genetic algorithms
to recommend friends of friends in the Oro-Aro SNS.

While several works treat recommendation as an information retrieval
problem, Tsourougianni and Ampazis (2013) present a followee hybrid rec-
ommender based on classification. The approach, which receives the target
user (or target user and followee) tweets as a query, locates the target user’s
friends-of-friends (FoF’s) and classifies their tweets as either positive or neg-

10



ative for recommendation; the users with the highest percentage of positives
are added to the recommendation list. The features for the classifier (a
Markovian classifier) consist of Orthogonal Sparse Bigrams, which are ex-
tracted per tweet. As in other cases, the approach was evaluated using
precision. While most of the results yielded a high precision, the authors
imply that the presence of bots in Twitter could cause outlier results.

Followee recommendations in Twitter are also tackled by Gavilanes et al.
(2013) with a system that, unlike others, is built on top of human-generated
recommendations. To show that these influence users’ behavior towards fol-
lowing new people, the authors study a 24-week corpus created with manual
recommendations from the #followfriday (or #ff ) trend, which consists of
users recommending their followees other users to follow (celebrities, au-
thorities in a certain topic, and the like); these recommendations are made
on Fridays, hence the name of the hashtag. To rank the human-generated
recommendations, several features are extracted and used with the Rota-
tion Forest algorithm to train a binary classifier. The features are grouped
into three categories: user, relation, and format. While the first category
includes user popularity and level of activity in Twitter, the second con-
siders level of communication between users and similarity (content-based
and geographical), and the third includes recommendation repetition (i.e.
how many times a user has been recommended) and context (e.g. day of
the week). The results were evaluated using mean average precision, and
the relation features were found to be the most useful; on the contrary, the
format features were the least useful.

Another recent approach concerns the Twilite system, which is intro-
duced by Kim and Shim (2013); preceded by a similar model-based ap-
proach named Twitobi (Kim and Shim, 2011), the Twilite system sits on
Latent Dirichlet Allocation and matrix factorization, which help to recover
the process of tweeting and following users. The resulting probabilistic gen-
erative model (whose parameters are learned via expectation maximization)
is used for followee and tweet recommendation.

Still within the context of recommendation in social media are the sem-
inal works by Chen et al. (2009, 2010) and Liu and Lee (2010). The first of
these works (Chen et al., 2009) proposes and evaluates four algorithms that
are applied on Beehive, an enterprise social networking site of IBM. One of
these algorithms is purely content-based, as it creates tf-idf vectors for user
profiles and generates the recommendation list based on cosine similarity;
the second algorithm (which is hybrid) extends the former by increasing
similarity if the users are linked. The third algorithm, purely structural,
is based on FoF’s and mutual friendships, and the last algorithm (named

11



SONAR) is leveraged by organizational information, such as patents, the or-
ganizational chart, and project wikis. It is important to note that, for every
recommendation, an explanation is provided (this feature being also used
in works by other authors). To assess the four algorithms, the authors con-
ducted a survey where users were asked a variety of questions regarding the
received recommendations; another form of evaluation consisted of a con-
trolled field study where a recommender widget was introduced in Beehive
and monitored for a sample of users. While all algorithms showed strengths
and weaknesses, in general SONAR was considered as a fair alternative, since
it takes advantage of more information. Posterior works by these authors
focus on Twitter, specifically on URL recommendation with topic relevance
and social voting (Chen et al., 2010). With respect to the work by Liu and
Lee (2010), this work — which is applied on the Cyworld SNS — shows that
collaborative filtering can be enhanced by combining nearest neighbors with
friends for item recommendation.

The approach by Li et al. (2011) suggests experts by using a fuzzy lin-
guistic method and fuzzy text categorization to analyze task documents.
Other items recommended using a fuzzy engine include electronic products
(Cao and Li, 2007), music (Park et al., 2006), and academic resources inside
universities (Porcel and Herrera-Viedma, 2010); fuzzy logic has been used,
as well, for learning and constructing user profiles (Castellano et al., 2010;
Xiang-Wei et al., 2009). More recently, a friend-to-friend recommendation
system is proposed by Yigit et al. (2015), where a classification of data in
four categories is made and a mathematical equation is used to generate
a recommendation; this approach performs better than extended FoF, a
graph-based system, and a conceptual fuzzy set based algorithm. Also, a
fuzzy system is used to extract useful interaction behavior of Twitter users
(Fu and Shen, 2014). Collective user behavior is analyzed to generate tweet
reply or not reply categories by means of fuzzy association rules that consider
activity time, number of friends/followers, and number of tweets as influen-
tial factors. Other fuzzy systems have been used for sentiment analysis on
social networks, for example, the analysis of opinions generated by Twitter
and Facebook with the use of fuzzy propagation modeling (Trung and Jung,
2014). In the approach by Bollen et al. (2011), a self-organized fuzzy neural
network is used to relate opinions generated in Twitter by mood tracking
tools with changes in Dow Jones industrial average over time.

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3.1 Discussion

Reviewing the aforementioned related approaches, we can see that none of
them does followee recommendation in Twitter using a fuzzy-based method
(hence the uniqueness of the solution we propose). The state of the art, to
the best of our knowledge, instead poses two kinds of approaches: followee
recommendation in Twitter using various methods and fuzzy approaches for
various tasks and contexts (other than followee prediction and recommen-
dation in Twitter). Let us discuss related work of each kind.

The followee recommendation task is not new and has been explored pre-
viously; however, a number of the proposed methods have been developed
over less complex (smaller, homogeneous, less noisy) contexts. Considering
that Twitter is by no means a simple context, the few works that actually
concentrate on this social network still have room for improvement — either
talking about precision, efficiency, or robustness. For example, the works by
Gavilanes et al. (2013) and Hannon et al. (2010) use a considerable number
of variables (the greater the number of variables, the more time it takes to
extract this information), while our approach proposes only three. On the
aspect of precision, the work by Armentano et al. (2011b) reports results
that could be further improved. On the aspect of robustness, Tsourougianni
and Ampazis (2013) state that bots (noise) affect the performance of the ap-
proach, and for this reason our approach is based on fuzzy logic. In summary,
our approach attempts to improve — in any aspect — the current state of
the art, or at least bring in new knowledge about followee recommendation
in Twitter.

With respect to fuzzy expert systems for various tasks and contexts, as
we have seen, these approaches so far have not been devoted to the task
of followee prediction and recommendation in Twitter; this is, therefore, an
opportunity area for the field of expert systems. Tackling this opportunity
area could bring novel perspectives on challenging problems and show that
the solution proposed by expert systems keeps fitting into new contexts (no
need of “reinventing the wheel”).

As a result, it is possible to state that our contribution lies within two
areas: fuzzy expert systems and followee recommendation in Twitter and
similar contexts. For the fuzzy expert systems area, our contribution consists
of a fuzzy system capable of predicting the degree of friendship between
a pair of users of a sui generis (massive, dynamic, heterogeneous) social
network; for the followee recommendation area, our contribution consists of
a robust, efficient approach that combines several information types (text,
structure) in a manner that has not been tried before.

13



4 Approach

To recommend followees, we designed a fuzzy system that takes as input
three similarity values between a target user ui and a candidate user uj:
comment (tweet) similarity, followee similarity, and followee tweet similarity.
The rules of the system produce the predicted degree of friendship (follower-
followee relation) between the users, and this value is used to determine if uj
is worth recommending. To calculate the similarities, we first extract user
profiles.

4.1 Generation of user profiles

As previously mentioned, a profile contains information that allows to char-
acterize the user and measure similarity with respect to other users, thus
enabling recommendations. We consider that a user profile can be con-
structed with (a) the comments of the user, (b) the comments of the user’s
followees, and (c) the id’s of the followees. This approach for generating
user profiles includes both content and structure: while the former is given
by comments that comprise topics and opinions, the latter is given by links
that reveal how the user is connected. It also — from the recommender
systems perspective — combines both content-based and collaborative fil-
tering strategies by employing information from the user and information
from contacts, respectively. In that sense, the approach is hybrid.

Upon generating a user profile, it is important to take into consideration
the amount of data per user; when this amount is not easily manageable, the
profile might as well be generated with a random sample of comments and
contacts. To the best of our knowledge, there is no standardized number of
followees to include or the criteria to select these.

4.1.1 Formal conceptual framework for profile generation

Let us formally depict the profile of a user ui with a triplet

pi = (ci, ei, fi)

of document vectors, where the vector ci represents the comments of ui,
the vector ei represents the comments of ui’s followees, and the vector fi
represents the id’s of ui’s followees; for later explanation, let us assume that
a function fp(ui) produces pi. All vectors consist of tf-idf weights that are
computed by conceiving Twitter as a heterogeneous information network
and constructing documents from this network.

14



Let G = (V, E) represent a heterogeneous information network of users
and comments. G is unweighted, vertex-labeled, and consists of two sub-
networks: a follower-followee directed network Gf = (Vf, Ef) and a user-
comment undirected bipartite network Gcf = (Vcf, Ecf). An example of this
kind of network is illustrated by Figure 4.

c6

c3

c4

c2

c7

c5

c1

u3

u1
u2

u5

u4

Figure 4: Example of a heterogeneous information network.

For Gf, each vertex v ∈ Vf stands for a user and each arc (a, b) ∈ Ef
represents a “who-follows-who” relationship where a follows b; these arcs
only join pairs of vertices in Vf, such that Ef ⊂ Vf × Vf. With respect
to Gcf, it represents a set of comments Vc generated by the set of users Vf
(note that Vc∪Vf = Vcf = V ); an edge {a, c}∈ Ecf where a ∈ Vf and c ∈ Vc
indicates that user a made comment c. From our example, we can see that
Vf = {u1 . . . u5}, Vc = {c1 . . . c7}, (u1, u2) ∈ Ef, and {u1, c1}∈ Ecf.

To retrieve followees and comments from users, let us introduce three
kinds of neighborhoods on the vertices of G: the f-neighborhood, the com-
ment neighborhood, and the extended comment neighborhood.

The f-neighborhood of a vertex i, which we will denote by Γ+
f
(i), is the

out-neighborhood of i in Gf and represents the set of followees for a given
user (see Eq. 1 in Section 2.2). In our example, Γ+

f
(u2) = {u3, u5}.

The comment neighborhood of a vertex i ∈ Vf, which we will denote by

15



Γc(i), is the neighborhood of i in the bipartite subgraph Gcf and represents
the set of comments (tweets) for a given user:

Γc(i) = {v : {v, i}∈ Ecf, i ∈ Vf} .

In our example, Γc(u2) = {c3, c4}.
The extended comment neighborhood of a vertex i ∈ Vf, which we will de-

note by Γe(i), corresponds to the comment neighborhoods of the f-neighbors
for i and represents the comments of the followees for a given user:

Γe(i) = {v : (i, j) ∈ Ef, v ∈ Γc(j)} .

Regarding our example, Γe(u2) = Γc(u3)∪Γc(u5) = {c5, c6}.
It is also important to introduce a function L(i) that maps a vertex i to

a label li, where the latter consists of a user id for v ∈ Vf or a comment
(text) for v ∈ Vc.

Taking the previous definitions into consideration, three types of doc-
uments can be generated per user: the comment document, the extended
comment document, and the followee document. These document types all
emerge from the same procedure: the concatenation of labels that belong to
a given neighborhood. Let us represent this procedure with a function

Φ(N) =
n

j∈N

L(j).

In this case ci = Φ(Γc(i)), ei = Φ(Γe(i)), and fi = Φ(Γ
+
f
(i)). For our exam-

ple, let us assume that L(u2) = “@newton”, L(c3) = “shoulders of giants”,
L(c4) = “action and reaction”, L(u3) = “@archimedes”, L(c6) = “eureka”,
L(u5) = “@galileo”, and L(c5) = “it moves”. Here,
cu2 = “shoulders of giants action and reaction”, fu2 = “@galileo @archimedes”,
and eu2 = “eureka it moves”.

By extending this concept, three corpora are constructed from G: a
comment corpus C, an extended comment corpus E, and a followee corpus
F. Each corpus consists of the union of its respective documents. Formally,

X =
⋃

i∈Vf

xi

where X ∈{C, E, F} and xi ∈{ci, ei, fi}.
Taking these considerations into account, the calculation of the tf-idf

scores for the document vectors ci, ei, and fi is straightforward from
Equation 8.

16



Given two profiles pi and pj, we create a similarity vector where each ele-
ment corresponds to the cosine similarity (Eq. 9) between document vectors
of the same type. Let us denote this vector with s(i,j) =

[

c(i,j), e(i,j), f(i,j)
]

,
where c(i,j) = cosim(ci, cj), e(i,j) = cosim(ei, ej), and f(i,j) = cosim(fi, fj).
For further explanation, let us assume that a function fsim(ui, uj) = s(i,j).

For supervised learning approaches, the similarity vectors can actually
be turned into a matrix S, where each row is an input vector. An additional
vector yD of desired outputs is needed and serves for the training and test
sets; each output o(i,j) ∈{0, 1} of this vector indicates whether the users are
contacts or not:

S =







c(1,2) e(1,2) f(1,2)
...

...
...

c(n−1,n) e(n−1,n) f(n−1,n)






and yD =

[

o(1,2) o(1,3) . . . o(n−1,n)
]T

.(16)

4.2 Fuzzy system design

As previously stated, our fuzzy system has three inputs and one output.
The linguistic variables for the inputs are tweet similarity, followee tweet
similarity, and followee similarity between a pair of users, while the output
is the predicted degree of friendship between these users; this result could
also be interpreted as the probability of a link (follower-followee relationship)
creating between the users. A diagram of the fuzzy system is shown in Figure
5.

Friendship
prediction
value

Fuzzy system

Followee id similarity

Tweet similarity

Followee tweet similarity

Figure 5: Architecture of the fuzzy system for recommendation prediction.

Each input variable has nine partitions, i.e. nine linguistic values (fuzzy
sets): Extremely low, Very low, Considerably low, Low, Medium, High, Con-
siderably high, Very high, and Extremely high. As we can see from Figure 6,
the fuzzy sets at both extremes are trapezoidal functions (half-trapezoids)
and the rest are triangular functions; these functions are not only common
but also computationally cheap. These functions are described in Eq. 17:

17



type1(x, n) =























































trapezoidal(x, 0, 0, 0.1, 0.2) if n = 1
triangular(x, 0.1, 0.2, 0.3) if n = 2
triangular(x, 0.2, 0.3, 0.4) if n = 3
triangular(x, 0.3, 0.4, 0.5) if n = 4
triangular(x, 0.4, 0.5, 0.6) if n = 5
triangular(x, 0.5, 0.6, 0.7) if n = 6
triangular(x, 0.6, 0.7, 0.8) if n = 7
triangular(x, 0.7, 0.8, 0.9) if n = 8
trapezoidal(x, 0.8, 0.9, 1, 1) if n = 9

(17)

 0

 0.2

 0.4

 0.6

 0.8

 1

 0  0.2  0.4  0.6  0.8  1

EL VL CL L M H CH VH EH

Figure 6: Linguistic values used (represented as memberships functions).
EL=“Extremely low”, VL=“Very low”, CL=“Considerably low”, L=“Low”,
M=“Medium”, H=“High”, CH=“Considerably high”, VH=“Very high”,
EH=“Extremely high”.

To fuzzify the inputs, the singleton method (Eq. 12) is used, as this is
the one usually employed and the input does not require further processing.
With respect to the fuzzy rule base, there are 93 = 729 rules possible; how-
ever, a smaller set was derived by manually analyzing 200 cases in which a
pair of users was connected (had a follower-followee relationship). Regard-
ing the inference engine, we use Mamdani’s max-min engine (Eq. 14). To
defuzzify the output, the centroid (center of mass) method (Eq. 15) is used.

18



4.2.1 Recommendation algorithm

To make recommendations for a target user, we select a set of candidate
followees (a suitable choice is to use a friends-of-friends or FoF approach)
and include in the recommendation list those candidates that exceed an
acceptance threshold τ for the prediction value obtained by the fuzzy system;
in our experiments, τ was set to 0.5. Additionally, only the top k elements
of the list may be chosen (in this paper, we do not apply this filter). This
procedure is shown in Algorithm 1.

Algorithm 1 Recommendation algorithm

Description: Receives a target user ui, a set C of candidate followees, an accep-
tance threshold τ, and a list size k. Returns a recommendation list Lk.

1: function recommend(ui, C, τ, k)
2: pi ← fp(ui)
3: L ← ()
4: for all uc ∈ C do
5: pc ← fp(uc)
6: s(i,c) ← fsim(pi, pc)
7: y(i,c) ← FS(s(i,c)) ⊲ Get prediction value from fuzzy system (FS).
8: if y(i,c) ≥ τ then
9: append(uc, L)

10: end if

11: end for

12: sort(L) ⊲ Sort L according to prediction values.
13: Lk ← L[1 : k] ⊲ Get the top k elements of L.
14: return Lk
15: end function

5 Experiments and Results

To test our approach, we created 30 datasets with information crawled from
Twitter and measured recommendation accuracy for each dataset, as it is
conventionally done for evaluating recommender systems; we compared our
results against a supervised variant of our approach, two state-of-the-art
methods (previously described in Section 3) and a baseline. With the in-
tent of evaluating our approach in other contexts, we also created a pair of
datasets for the Arxiv High-Energy Physics Theory citation network (Cit-
HepTh) and obtained accuracy for these datasets as well. Finally, to assess
the performance of input variables (followee similarity, tweet similarity, and

19



followee tweet similarity), we calculated the correlation of each input vari-
able with respect to the obtained output.

5.1 Experimental Setup

Sample networks. Because the Twitter network is very large to han-
dle, experiments were performed using smaller sample networks, where each
sample network consisted of a heterogeneous network such as the one de-
scribed in Section 4; each sample network had one target user, the followees
of this target user, and a number of non-contacts (i.e., users who were nei-
ther followees nor followers of the target user); the main idea was for the
fuzzy system to be able to recover the target user’s followees as recommen-
dations. As such, we evaluated the fuzzy system by calculating the accuracy
(percentage of matches) between the resulting recommendation list and the
actual followee list of the network’s target user. Since sample networks were
grouped into datasets, we obtained the average accuracy per dataset.

Twitter data. To generate the sample networks, a repository of users
and comments was first crawled from Twitter4. We collected a seed set of
100 users, which resulted from searching “Monterrey” (a northeastern city
of Mexico) in Twitter and selecting those users that complied with four
criteria: having Monterrey as their location, writing in Spanish, having at
least 30 followees, and having no more than 3,000 followers. The last two
criteria intended to discard companies and celebrities such as singers, actors,
and famous politicians; processing this kind of users is left as future work.
To have a larger repository, the seed set was expanded by including those
followees and followers that complied with the criteria previously mentioned;
the expansion was breadth-first, such that the initial level consisted of seed
users, the second level consisted of seed user contacts, the third level of
contact contacts, and so on. The current repository contains 18,499 users5.
From this repository, we randomly selected 6,000 users and approximately 2
million comments. These comments were filtered (stopwords were removed)
and spelling correction was performed.

Twitter datasets. With the data from Twitter, we constructed 3,000
heterogeneous sample networks, which were equally distributed to create 30

4All information was obtained using the Twitter API, available at https://dev.
twitter.com/.

5This repository has also been used for other studies involving Twitter, such as opinion
mining in Spanish (Rodŕıguez, 2013).

20



datasets with 100 networks each. While the number of users (size) of each
network was variable, on average, there were 24.44 users per network.

Cit-HepTh datasets. To assess our approach in a different context, we
also created datasets for the Arxiv High-Energy Physics Theory (Cit-HepTh)
citation network6, which is analogous to the Twitter heterogeneous network.
In this case, we are dealing with reference recommendation, i.e. recommend-
ing which paper to cite. In that sense, papers can be seen as equivalent to
users, a paper b that is cited by a paper a can be seen as a’s followee, and the
abstract of a is equivalent to the set of a’s tweets. For the case of Cit-HepTh,
we generated two datasets, where each dataset consisted of 30 heterogeneous
sample networks. Each network contained 100 papers on average.

Comparative experiments. With the intent of having a wider perspec-
tive on the effectiveness of the fuzzy approach, we performed a compari-
son against two state-of-the-art methods: Twittommender (Hannon et al.,
2010) and the co-followers structural strategy followed by Armentano et al.
(2011b); for Twittommender, we used the ensemble strategy. We addi-
tionally introduced a baseline, which consisted of a simple friends-of-friends
(FoF) approach — i.e., recommending the followees of the target user’s fol-
lowees (note that this would yield a high accuracy if the followee and FoF
lists had a considerable overlap).

Our supervised variant. Another point of comparison was given by the
supervised variant of our approach, which was implemented to further as-
sess the combination of our three input variables. This supervised variant
consisted of a multilayer perceptron neural network (we ran experiments us-
ing WEKA7 and its default parameters). We shall refer to this approach as
“NN variant” or simply “NN”.

5.2 Results and Discussion

A summary of our results is shown in Table 1 (note that this is the average
of the average results per dataset), and a box plot for the Twitter datasets is
depicted in Figure 7. As we can see from these results, our approaches (the
fuzzy system and the neural network) are highly competitive with regard

6This network is available as part of Stanford’s Large Network Dataset Collection.
URL: http://snap.stanford.edu/data/cit-HepTh.html.

7Available at: http://www.cs.waikato.ac.nz/ml/weka/.

21



to the rest of the methods; the neural network, in particular, clearly out-
performs all competitors (the difference with respect to Twittommender in
the Twitter datasets is, in fact, statistically significant with p = 1.16×10−6

using a two-sided paired t-test). This suggests that our three input variables
act as reasonable link predictors and are thus feasible for recommendation.
Moreover, let us note that almost all methods performed better on the Cit-
HepTh datasets, except for the co-followers strategy, which seems to be
better tailored for Twitter. This apparently suggests that Twitter’s nature
is more “messy” (e.g. user’s comments) and probably more chaotic; it could
be therefore convenient to employ the sui generis features of Twitter to try
to improve accuracy. However, it would be desirable to have more results
(more datasets) on the Cit-HepTh network to have a more confident point
of view on this perspective. This is left as future work.

Table 1: Average recommendation accuracy (30 Twitter datasets,
two Cit-HepTh datasets). FS=Fuzzy system, NN=supervised variant,
TM=Twittommender, CF=Co-followers, FoF=friends-of-friends (baseline).

FS NN TM CF FoF

Twitter 61.57% 71.75% 65.13% 29.99% 3.15%

Cit-HepTh 83.25% 86.77% 86.86% 10.84% 8.74%

Global 72.41% 79.26% 75.99% 20.41% 5.94%

As we can also see from the results, the fuzzy system is a strong com-
petitor both inside and outside Twitter. Although it ranks third, there is an
evident advantage over the baseline and the co-follower strategy (needless to
say, there is a statistically significant difference with these approaches, since
p < 2.2 × 10−16); in that sense, our approach remains competitive. With
respect to Twittommender, we can see that the fuzzy system’s accuracy lies
behind for approximately 3%, but let us also note that the fuzzy system is
only using three variables whereas Twittommender is using five (user tweets,
followee id’s, followee tweets, follower tweets, and follower id’s), i.e. it needs
to extract more information. Another disadvantage of Twittommender is
that it is more of a “hit or miss” approach — at least with our Twitter
results — and this can be appreciated in the box plot (Figure 7). On the
contrary, the fuzzy system tended to give more stable results.

With respect to the NN variant, the fuzzy system also stays behind.

22



In general, neural networks are more precise than fuzzy systems, but they
are also black boxes and no explanation can be extracted from them; on
the other hand, fuzzy systems explain the relationship between inputs and
outputs by means of linguistic variables and rules. For example, consider
the rule:

IF v1 is Low and v2 is Medium High and v3 is Extremely High THEN y is
Very High,

where v1 is user tweet similarity, v2 is followee tweet similarity, v3 is followee
id similarity, and y is the possibility of being contacts. In a linguistic manner,
this rule explains that if there is a low user tweet similarity and there is a
medium high followee tweet similarity and there is an extremely high followee
id similarity, then there is an extremely high possibility to be contacts.

We can choose any fuzzy rule and make the same interpretation. Fuzzy
systems have the disadvantage of not being very accurate against other
paradigms such as neural networks; however, because NN’s are black boxes,
it is very difficult to establish an interpretation of the relationship between
inputs and outputs. Moreover, a fuzzy system has a high interpretability
because it generates rules that explain in an explicit and linguistic way the
relationship between variables.

5.2.1 Individual variables

As we stated previously, with a fuzzy approach it is possible to know how
each input variable influences the output. To measure the individual influ-
ence of input variables, we calculated the correlation coefficient between each
input variable and the output. Our results for both Twitter and Cit-HepTh
are displayed in Table 2. As we can see from this table, for both cases, the
variable with the highest influence on the fuzzy system corresponds to fol-
lowee tweets. This result allows us to gain insight on the factors that affect
prediction, and thus make adjustments for future work.

5.2.2 Summary

In summary, our results show that the fuzzy approach is suitable for link
prediction in recommender systems, both in Twitter and in other contexts.
The approach is highly competitive with other methods. In some cases, it
clearly surpasses the accuracy obtained by its competitors; in other cases,
it stays behind by a small percentage but excels in other aspects, such as

23



Table 2: Correlation between input variables and prediction output.

Twitter Cit-HepTh

User tweets 0.12 0.43

Followee id’s 0.42 0.32

Followee tweets 0.85 0.65

amount of information used and capability of explanation. This last goal is
important for recommender systems in general (Bonhard and Sasse, 2006).

6 Conclusions and Future Work

A new fuzzy system for followee recommendation in Twitter has been pro-
posed in this paper. This system handles recommendation mainly as a link
prediction problem and uses three linguistic variables that express similar-
ity between users: tweet similarity, followee tweet similarity, and followee
similarity. Every similarity contributes in a different way for the prediction.
To calculate these similarities, a profile is built for each user. The profile
consists of vectors that contain information from comments and followee
relationships, and this information is extracted by considering the Twitter
social networking site as a heterogeneous information network.

From an objective point of view, the approach has both strengths and
weaknesses. On one hand, as with any expert system, creating the rules
requires knowledge and is an additional overhead of the approach; also,
as we could see from the results, accuracy is still below other methods and
could still be further improved. On the other hand, the system only requires
three types of information (on the contrary of other methods which require
more), is self-explanatory, and is capable of producing a degree instead of
only a yes/no binary answer. The self-explanatory capability, in particular,
allows the system to explain how variables interact to produce an output,
as opposed to “black-box” methods, such as neural networks or support
vector machines. With respect to the friendship degree, besides helping to
construct the recommendation list for a particular user, it can also be used
to carry out other tasks, such as discovering latent user communities or
clusters.

With respect to the impact of this work, our fuzzy approach brings in

24



a new perspective on the use and construction of expert systems for link
prediction and contact recommendation in microblog-oriented social net-
works. Our method, in particular, brings new knowledge on how to address
recommendation in a sui generis, heterogeneous, massive, dynamic environ-
ment, which is subject to inaccuracies (think, for example, about language
inconsistencies) and incomplete information. One of our main findings, in
particular, reveals that the combination of user tweet similarity, followee
id similarity, and followee tweet similarity yield accurate followee recom-
mendation results (these results are even better when the combination is
used in a supervised approach); in that sense, the use of only three infor-
mation types is promising — specially in the neural network case. Even
when the fuzzy approach can be further improved, we have taken the first
step towards a robust, efficient, accurate solution to the problem. Another
theoretical contribution that it is important to highlight and that might be
of interest is the conceptual model of Twitter as a heterogeneous network,
as this model not only facilitates the understanding of Twitter’s underlying
dynamics, but also serves as a basis for other data mining tasks, such as
cyberbullying detection and interest mining.

As future research directions, we propose several courses of action. One
of these consists of modifying the existing rule base to incorporate more
input variables (e.g. follower tweets, follower id’s, retweets, multimedia con-
tent, etc.) with the intent of improving accuracy and exploiting Twitter’s
unique features; of course, this implies an additional overhead and the trade-
off between quality and efficiency must be assessed. Now, beyond the sole
introduction of more variables, lies the issue of finding an optimal design
for the fuzzy system (fuzzification and defuzzification methods, membership
functions, partitions, etc.); in that sense, adaptive methods — such as ge-
netic algorithms, PSO, and similar — could be employed to find this optimal
design.

Another important future research direction concerns testing and adapt-
ing the fuzzy expert system for other contexts, both inside and outside Twit-
ter. In that sense, it seems valuable to discover how general the system could
be, and if it is possible to have a tailored version enhanced for Twitter (or
a specific language, such as Chinese or Arab, whose alphabet is different)
and a general version for heterogeneous networks analogous to Twitter (such
as citation networks). Consequently, there is still plenty of room to create
new “flavors” of the original fuzzy expert system. Related to this is the
hybridization of the fuzzy system with a supervised technique to create a
more powerful approach.

A final future research direction is related to community discovery and

25



graph clustering. Because the fuzzy system produces a friendship degree
for a pair of users, it is possible to view this degree as a network weighted
edge and use this information to construct a friendship network (or at least
parts of) and develop a local probabilistic graph clustering algorithm to
find groups of similar users in Twitter (or other complex networks). On
the contrary of the follow graph, which would be normally used for the
clustering task in Twitter, this friendship network constructed from the
fuzzy system could give new insights to community discovery in Twitter
and social networks in general.

26



Fuzzy NN TM CF FoF

0
2

0
4

0
6

0

Approach

A
cc

u
ra

cy
 (

%
)

Figure 7: Comparison using the Twitter datasets. Each box shows the min-
imum, first quartile, median, third quartile, and maximum for the accuracy
obtained from the 30 datasets (one box per approach). NN= neural network
(multilayer perceptron), TM= Twittommender, CF=co-followers strategy,
FoF=friends-of-friends strategy.

27



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