Spectral Algorithm for Line Graphs to Find Overlapping Communities

in Social Networks

Camila P. S. Tautenhain and Mariá C. V. Nascimento

Instituto de Ciência e Tecnologia, Universidade Federal de São Paulo,

Keywords:

Overlapping Community Detection, Modularity Maximization, Spectral Method, Social Network Analysis.

Abstract:

A great deal of community detection communities is based on the maximization of the measure known as

modularity. There is a dearth of literature on overlapping community detection algorithms, in spite of the im-

portance of the applications and the overwhelming number of community detection algorithms yet proposed.

To this end, one of the suggestions in the literature consists of partitioning the set of edges into communities,

also known as link partitions, by applying community detection algorithms to line graphs. In line with this,

in this paper, overlapping vertex communities are obtained from link partitions by a method that selects the

communities of the edges that represent the highest modularity gain. We also introduce a spectral algorithm to

ﬁnd link partitions from line graphs. We show that the modularity of communities in line graphs is equivalent

to the adaptation of modularity of communities in the original graphs, when considering the non-backtracking

matrix instead of the adjacency matrix in its formula. The results of the experiments carried out with over-

lapping community detection algorithms showed that the proposed method is competitive with state-of-the-art

algorithms.

1 INTRODUCTION

Complex networks can model real systems such as

social, metabolic and citation networks. These net-

works can be partitioned into communities of densely

connected vertices for a better understanding of the

characteristics of the system. In the context of social

networks, the identiﬁcation of communities is of great

interest in applications such as targeting market cam-

paigns, prediction of interactions amongst users and

detecting non-human users.

The traditional approach to detect communities in

networks is to ﬁnd the best vertex partition according

to a given measure that assesses its quality. Examples

of measures are the modularity (Newman and Girvan,

2004), the statistical signiﬁcance (Lancichinetti et al.,

2011) and the map equation (Rosvall and Bergstrom,

2007).

The number of edges within the groups minus the

number of edges expected in a random graph with the

same degrees as the graph under the consideration is

virtually the deﬁnition of modularity. The higher the

modularity of a partition, the better the partition ac-

cording to this measure.

Because the modularity maximization problem is

NP-Hard (Brandes et al., 2008), its exact optimization

is unrealistic in scenarios where the networks have a

large number of vertices. Therefore, the literature has

been virtually focused on developing approximation

methods and heuristics. Spectral decomposition met-

hods in particular have presented good results in (Nas-

cimento and De Carvalho, 2011) and in (Newman,

2013b). Krzakala et al. (2013) introduced the non-

backtracking matrix which indicates whether or not

a pair of edges is adjacent. Based on computational

experiments, the authors suggested that the spectrum

of such matrix is more meaningful than the spectrum

of the adjacency matrix to detect communities in net-

works.

A characteristic of social networks is the existence

of overlapping communities meaning that vertices be-

long to more than one community (Xie et al., 2013).

A user of Facebook, for example, might join diffe-

rent groups. Nonetheless, many works are strict on

selecting only one community for each vertex, i.e.,

disjoint communities (Xie et al., 2013). Among the

methods capable of ﬁnding either disjoint or overlap-

ping communities, the ones presented in (Lancichi-

netti et al., 2011) and in (Xie et al., 2011) are worth

mentioning.

306

Tautenhain, C. and Nascimento, M.

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks.

DOI: 10.5220/0007403803060317

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 306-317

ISBN: 978-989-758-350-6

The partitioning of edges into communities to de-

ﬁne link partitions has also been studied in the lite-

rature as an approach to identify overlapping vertex

communities (Evans and Lambiotte, 2009). Evans

and Lambiotte (2009) suggested that link partitions

can be obtained by (vertex) community detection al-

gorithms applied to their respective line graphs. Zhou

et al. (2017), however, observed that the resulting par-

titions may present communities with an undesirable

large number of overlappings.

In this paper, we adapted the modularity mea-

sure to assess the adjacency relations between edges

instead of vertices. Moreover, we demonstrated that

this adaptation is equivalent to the classical modula-

rity in line graphs. As another contribution, we in-

troduce a strategy to identify disjoint and overlapping

vertex communities from link partitions and a spectral

divisive community detection algorithm, here called

SpecDiv. SpecDiv is a hybridization of the Leading

Eigenvalue algorithm proposed by Newman (2006)

with a local search heuristic. The proposed strategy

constructs the vertex communities according to the

link partitions found by SpecDiv and based on the

gain of modularity.

The computational experiments showed that this

strategy was competitive with state-of-the-art over-

lapping community detection algorithms. Further-

more, we obtain link partitions by applying classical

community detection algorithms to line graphs. The

experiments carried out with real and sparse LFR ar-

tiﬁcial networks showed that disjoint partitions obtai-

ned by the proposed strategy using line graphs are

closer to the expected partitions than those found by

traditional community detection methods. In both ca-

ses, SpecDiv is competitive with the reference algo-

rithms. These results suggest that link partitions can

also be advantageous for existing vertex community

detection algorithms. As part of the experiments, we

also illustrate a practical application of the overlap-

ping community detection in a Twitter network from

the literature.

2 RELATED WORKS

This section presents the community detection met-

hods related to the introduced algorithm. First, we

brieﬂy discuss the disjoint community detection met-

hods. After, we examine the overlapping community

detection methods and, in special, link community

methods.

2.1 Disjoint Community Detection in

Networks

This section brieﬂy describes disjoint community de-

tection methods used as reference in the experiments

performed and presented in this paper. They are: the

Louvain method (Blondel et al., 2008), Infomap (Ros-

vall and Bergstrom, 2007), Label Propagation Algo-

rithm (Raghavan et al., 2007) and Walktrap (Pons and

Latapy, 2005). We also discuss the Leading Eigenva-

lue method of Newman (2006) since our algorithm is

based on it and highlight a few other spectral-based

methods. As an extensive literature review on this

subject is out of the scope of this paper, we refer

the reader to the recent survey of Fortunato and Hric

(2016). We chose these methods due to their good

results to a wide variety of networks and consolida-

ted use as reference algorithms in the literature (Yang

et al., 2016).

There are two overlapping community detection

methods discussed in the next section which are also

capable of ﬁnding disjoint communities: the Or-

der Statistics Local Optimization Method (OSLOM)

(Lancichinetti et al., 2011) and the Speak Listener

Propagation Algorithm (SLPA) (Xie et al., 2011).

To ﬁnd the vertex partitions, the Louvain and the

Leading Eigenvalue methods aim at maximizing the

modularity measure. The Louvain method is a greedy

strategy that merges communities of vertices that in-

crease the modularity value. The Leading Eigenvalue

method is based on the observation that the signs of

components of the eigenvector associated with the lar-

gest eigenvalue of the modularity matrix can be used

to partition the set of vertices into two communities.

The method then hierarchically divides each commu-

nity into two new communities.

Infomap aims at optimizing the map equation

measure, which is inspired by the duality between

the community detection problem and random walks

paths in digraphs. Walktrap also relies on random

walks to form vertex partitions by inferring the proba-

bility of random walkers leaving communities to re-

ach vertices from other communities. The Label Pro-

pagation Algorithm (LP) is a strategy which propaga-

tes labels through the network to deﬁne the label of

a vertex based on the communities to which its neig-

hbors belong to.

Several other spectral-based methods for ﬁnding

disjoint communities in networks exists besides the

Leading Eigenvalue method. Zhang and Newman

(2015), for example, designed a heuristic based on the

spectral decomposition of the modularity maximiza-

tion matrix. As previously mentioned, Krzakala et al.

(2013) introduced the spectral decomposition of the

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks

307

non-backtracking matrix. Both Newman (2013a) and

Singh and Humphries (2015) studied variations of the

non-backtracking matrix to improve its representati-

veness regarding networks.

Closely related to the spectral decomposition met-

hods are the methods based on statistical inference

(Newman, 2013b; Ali and Couillet, 2017), which

Newman (2013b) showed to be equivalent to modula-

rity maximization decomposition methods under spe-

ciﬁc parameter settings. These methods detect com-

munities by ﬁtting the stochastic block models to the

networks using maximum likelihood methods.

2.2 Overlapping Community Detection

in Networks

As mentioned earlier, a reduced number of algorithms

to detect overlapping communities can be found in the

literature. Among them, OSLOM and SLPA perform

such task with satisfactory outcomes when applied to

LFR networks as pointed out in the literature review

presented by Xie et al. (2013).

SLPA is an extension of LP to detect overlapping

communities in networks. SLPA keeps the informa-

tion about the community labels propagated to each

vertex along the iterations. A post-processing strategy

is responsible for deﬁning the most frequent commu-

nity labels of the vertices as their communities, ba-

sed on a threshold parameter. OSLOM is a heuristic

method that employs a ﬁtness function to assess the

statistical signiﬁcance of the communities compared

to a null model, i.e., a network with no community

structure.

Evans and Lambiotte (2009) and Ahn et al. (2010)

suggested detecting communities of links instead of

vertices to deal with overlapping communities. In this

strategy, each vertex belongs to the communities of its

incident edges. Evans and Lambiotte (2009) recom-

mended the detection of link partitions through the

partitioning of the vertices of the line graphs. Ahn

et al. (2010), on the other hand, studied the hierarchi-

cal partitioning of links in graphs by measuring the

similarity of edges using a measure known as Jaccard

index.

Although Xie et al. (2013) have shown through ex-

periments that OSLOM and SLPA outperformed the

link partitioning method of Ahn et al. (2010), appro-

aching the problem by line graphs enables the use

of any vertex community detection algorithm to ﬁnd

overlapping communities. In this sense, methods that

aim at ﬁnding communities of links have been incre-

asingly studied in the last years. Some examples of

algorithms that ﬁnd overlapping communities based

on line graphs are the genetic algorithms proposed

by Pizzuti (2009), Shi et al. (2013) and Li and Liu

(2018).

Spectral methods have also been used to detect

overlapping communities from link partitions. The

pioneer spectral method for this task was the one in-

troduced by Zhang et al. (2007). The algorithm pro-

posed by the authors is based on the spectral decom-

position of an adaptation of the modularity measure

to deal with the soft assignment of vertices to com-

munities. The overlapping communities are achieved

by analyzing how much each vertex belongs to each

community. Jiang and McQuay (2012) also explored

the soft assignment of nodes to communities through

a spectral decomposition method.

Gui et al. (2018) designed a spectral method that

used the eigenvectors associated with the two largest

eigenvalues of the Laplacian matrix of a line graph to

project the graph into points in a plane. These points

were then used to calculate the similarity between the

vertices in order to hierarchically merge them into

communities.

As opposed to disjoint community detection, for

which several spectral clustering methods have been

studied (Nascimento and De Carvalho, 2011), the li-

terature still lacks spectral heuristics that maximize

the modularity measure in order to ﬁnd overlapping

communities. Even though Nicosia et al. (2009), for

example, introduced an adaptation of the modularity

measure to assess overlapping vertex communities,

they used a genetic algorithm to maximize it rather

than a spectral-based method.

3 MODULARITY

MAXIMIZATION

Let G = (V, E) be an undirected and simple graph

such that V and E are its set of vertices and edges,

respectively.

The set of vertices is represented by integer sca-

lars, such that V = {1, 2,. . . , |V|}. An edge is an unor-

dered tuple (i, j) that has end points in two vertices

i, j ∈ V . The adjacency matrix A = [a

i j

] ∈ R

|V |×|V |

re-

presents the pairwise relation between vertices of G.

An element a

i j

is 1 if edge (i, j) ∈ E, and 0, other-

wise. The total number of the edges is denoted by

m =

∑

i∈V

∑

j∈V

i j

. The degree of vertex i is given by

∑

j∈V

i j

Let the communities of a graph be deﬁned by

integer scalars called community labels. A ver-

tex partition into communities is deﬁned as P =

, c

, ..., c

|V |

}, where c

is the label of the commu-

nity that vertex i belong to. Two vertices are in the

same community when they share the same label.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

308

Considering the community labels of vertices i

and j, i.e. c

and c

, let δ

be 1 if c

= c

and

0, otherwise. The modularity matrix is given by

B = [b

i j

] ∈ R

|V |×|V |

, where each element is calcula-

ted according to Equation (1).

i j

= a

i j

−

(1)

The modularity of a partition P is given Q(P ) ac-

cording to Equation (2).

Q(P ) =

∑

i∈V

∑

j∈V

i j

(2)

According to Equation (2), higher values of modu-

larity are obtained when the number of edges within

the communities is greater than the expected number

of edges of a random graph with the same degree se-

quence as V .

3.1 Non-backtracking Matrix

An edge e

= (i, j) is said to be adjacent to anot-

her edge e

= (k, r) if and only if they share ex-

actly one end point. The set of edges adjacent to

= (i, j) is thereby given by N (i, j) = {(k, r) ∈

E such that (i, j) and (k, r) are adjacent in G}. More-

over, the degree of an edge e

= (i, j) is the number

of its adjacent edges and is given by d

(i, j)

= |N (i, j)|.

Because we only deal with undirected graphs, we

adapted the non-backtracking matrix introduced by

Krzakala et al. (2013) to ignore edge directions. Let

O = [o

(i, j),(k,r)

] ∈ R

|E|×|E|

be the non-backtracking

matrix, where (i, j), (k, r) ∈ E and each element is gi-

ven by Equation (3).

(i, j),(k,r)



1, if (i, j) is adjacent to (k, r)

0, otherwise

(3)

Therefore, o

(i, j),(k,r)

is 1 if edges (i, j) and (k, r)

share exactly one end vertex, i.e., they are adja-

cent. The non-backtracking matrix can be under-

stood as an adjacency matrix between edges. The

analogous deﬁnition of m to the non-backtracking

matrix is y =

∑

(i, j)∈E

∑

(k,r)∈E

(i, j),(k,r)

. Furthermore,

the non-backtracking modularity matrix is given by

H = [h

(i, j),(k,r)

] ∈ R

|E|×|E|

, where each element is cal-

culated according to Equation (4).

(i, j),(k,r)

= o

(i, j),(k,r)

−

(i, j)

(k,r)

(4)

We introduce an adaptation of the modularity me-

asure to consider the non-backtracking matrix instead

of the adjacency matrix in Equation (5).

Qo(P ) =

∑

(i, j)∈E

∑

(k,r)∈E

(i, j),(k,r)

(5)

By replacing the adjacency matrix by the non-

backtracking matrix, Equation (5) evaluates the mo-

dularity over the edges of G, instead of E: it calcu-

lates the number of edges within communities minus

the expected number edges in a random graph whose

edges have the same degree sequence as E.

3.2 Line Graphs

Let G

= (V

, E

) be the line graph of G. Its set of

vertices is V

= E, i.e., each edge of G is a vertex of

. The set of edges of G

is given by E

= {(e

, e

such that e

∈ E and e

∈ E are adjacent in G}. The

number of edges of G

is denoted by m

and coincides

with y, i.e., m

= y.

The adjacency matrix of G

is A

= [a

(i, j),(k,r)

] ∈

|E|×|E|

where a

(i, j),(k,r)

receives value 1 if and only

if edges (i, j), (k, r) ∈ E are adjacent in G. Thereby,

(i, j),(k,r)

= o

(i, j),(k,r)

and, consequently, A

= O. The

degree of a vertex e

= (i, j) ∈ V

is deﬁned by d

∑

∈V

= d

(i, j)

A vertex partition of G

represents a link parti-

tion of G. A link partition is here denoted by P

, c

, ..., c

|E|

}, where c

is the label of the commu-

nity to which e ∈ E belongs to.

Let the modularity of a partition P

of line graph

be deﬁned by Equation (6).

) =

∑

∈V

∑

∈V



−



= Q

) =

∑

(i, j)∈E

∑

(k,r)∈E

(i, j),(k,r)

(i, j)

(k,r)

(6)

On the one hand, Qo(P ) (Equation (5)) assesses

a vertex partition and, on the other, Q

) (Equation

(6)) evaluates a link partition.

Consider a pair of edges (i, j) ∈ E and (k, r) ∈ E.

The product δ

is equal to 1 if and only if i, j, k

and r are in the same community in vertex partition P

of G. In these cases, edges (i, j) and (k, r) will be in

the same community in the related link partition P

Consequently, Qo(P ) reduces to Q

Nevertheless, Q

) does not reduce directly into

Qo(P ). Two edges (i, j), (k, r) ∈ E might be in the

same community in a link partition P

. However, one

or more of their end points might be end point of ot-

her edges that belong to different communities in P

A vertex can be in any of the communities of its in-

cident edges in a vertex partition P . This observation

induces a natural overlapping of communities in P .

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks

309

Therefore, maximizing the non-backtracking mo-

dularity in a graph G is equivalent to maximize the

classical modularity in its line graph G

, provided that

the former accounts for overlapping communities.

4 SPECTRAL DECOMPOSITION

This section presents the spectral decomposition of

the modularity measure of a partition P

in an arbi-

trary graph F = (V

, E

) that can be either G or G

If F is a line graph, then the modularity is equivalent

to the non-backtracking modularity in G. The decom-

position of the modularity is based on the study per-

formed by Newman (2006). Assume that m

∈ R and

= [b f

i j

] ∈ R

|×|V

are the number of edges and

the modularity matrix of G

, respectively.

First, consider a bipartition of the vertices of F.

Let s = [s

] ∈ R

be a vector where each element

is either 1 or −1, depending on which of the com-

munities vertex i belongs to. The modularity can be

expressed in a matricial form in function of s, as pre-

sented in Equation (7).

Q(s) =

s (7)

Let λ

, λ

, . . . , λ

be the eigenvalues of B

de-

creasingly sorted by value, i.e., λ

≥ λ

≥ ··· ≥ λ

Moreover, let Λ ∈ R

|×|V

be a diagonal matrix

such that the diagonal element in row j and column

j is given by the j − th largest eigenvalue, λ

. The

eigenvectors of B

are arranged in matrix U = [u

i j

] ∈

|×|V

such that each column j is the eigenvector

associated with the eigenvalue λ

Matrix B

is symmetric because F is undirected

and thus admits an eigen-decomposition. The vector

s can be written as a linear combination of the eigen-

vectors: s =

∑

i∈V

, where α

= u

s ∈ R. Equa-

tion (7) can then be rewritten as shown in Equation

(8).

Q(s) =

∑

i∈V

∑

j∈V

∑

i∈V

(8)

Newman (2006) suggested approximating Equa-

tion (8) using the largest eigenvalue λ

. The decision

problem is thus to decide the values of each element

, i ∈ V

, that result in the largest value in Equation

(8). Because an element s

can be 1 or −1, each term

, i ∈ V

, is maximized by setting s

as 1 if

> 0 and as −1 if u

< 0. In case u

= 0, the choice

of s

does not change the value of the term. There-

fore, the resulting partition is P

= {c

, c

, . . . , c

where c

= 1 when s

= −1 and c

= 2 when s

= 1,

for i ∈ V

4.1 Hierarchical Division of

Communities

In order to hierarchically divide the pair of commu-

nities found through the spectral decomposition ex-

plained in the earlier section, Newman (2006) sug-

gested creating subgraphs induced by the vertices of

each community. This approach, however, requires

the recalculation of the modularity matrix of the in-

duced graphs to consider the degree sequence of the

original graph.

In this paper, instead of analyzing the modularity

matrix of the induced graphs, we deﬁne the modula-

rity matrix of each community by selecting only the

rows and columns of the vertices that belong to the

community.

To show that the introduced strategy is correct, let

Z be a set containing the indices of the vertices that

belong to an arbitrary community labeled l

. Equation

(9) isolates the contribution of the vertices in Z to the

modularity measure.

Q(P

) =

∑

i∈Z

∑

j∈Z

b f

i j

∑

i, j:{i, j}∈(V

×V

)−(Z×Z)

b f

i j

(9)

where (V

×V

) − (Z × Z) has as elements all pairs

of vertices i, j such that both j and i does not belong

to Z.

The ﬁrst term of Equation (9) is the contribution of

community labeled l

to the modularity. According

to this Equation, maximizing the ﬁrst term does not

change the value of the second term in Equation (9).

Let D be a mapping between the vertices from Z

and elements of {1, 2, . . . , |Z|}, i.e.: D = {(i, υ) such

that i ∈ Z and υ is selected without replacement from

{1, 2, .., |Z|}}. Let B

= [b f

υρ

] ∈ R

|Z|×|Z|

be the mo-

dularity matrix whose element b

υρ

has the value of

i j

, i : (i, υ) ∈ D and j : ( j, ρ) ∈ D.

Also, let z = [z

] ∈ R

|Z|

be the vector whose ele-

ments z

receive value 1 or −1 depending on which

community vertex i : (i, υ) ∈ D belong to. Thereby,

the modularity of the community labeled l

is given

by Q

(z) =

Because the spectral decomposition of Q

(z) is

analogous to the one performed on Q(s), the signs of

the eigenvectors associated with the leading eigenva-

lue of B

can be used to choose the values of z that

maximize Q

(z). After choosing z using the leading

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

310

eigenvalue approach, two new community labels are

selected for the vertices in Z.

5 VERTEX COMMUNITY

DETECTION IN LINE GRAPHS

The ﬁrst step of the introduced strategy is to ﬁnd a link

partition by applying a vertex community detection

algorithm to the corresponding line graph.

In addition to the introduced strategy to obtain ver-

tex partitions from link partitions, this paper also in-

troduces a divisive spectral community detection al-

gorithm.

5.1 Divisive Algorithm

The spectral divisive heuristic introduced in this paper

is strongly based on the leading eigenvalue algorithm

proposed by Newman (2006).

Algorithm 1 presents the heuristic, called Spe-

cDiv. It takes as input a graph F = (V

, E

), which

can be either the original graph G or its corresponding

line graph G

, the maximum number of communities

to be created, max_labels, and the minimum num-

ber of vertices required to further divide a community,

min_size. The algorithm returns a vertex partition for

the input graph F. Hence, in case F is the line graph,

the algorithm returns a link partition of the original

graph.

In line 1 of Algorithm 1 is constructed the mo-

dularity matrix B

according to Equation (1). Recall

that if F is a line graph, then the modularity matrix

of F is equivalent to the non-backtracking matrix of

the original graph. In line 3, a partition P

is created

wherein all vertices belong to the community labe-

led 0. In line 4 the list L which stores the labels of

the communities that can be further divided into two

communities is deﬁned. Line 7 sets the modularity

matrix of the initial iteration, B

(1)

, as B

From lines 8 to 29, SpecDiv repeatedly breaks in

two each community in L until either there is no com-

munity left in L or a predeﬁned maximum number of

communities is reached.

In lines 9 and 10, the set Z of indices of vertices

in the community labeled lr, chosen randomly from

L, is deﬁned. In case the number of vertices indicated

by Z is greater than min_size, in line 12, the pairwise

mappings (i, υ) between vertices i ∈ Z and sequential

numbers υ ∈ {1, 2, . . . , |Z|} are assigned to set D. This

set is used to map the vertices of Z to indices that

correspond to the appropriate components of B

and

u. In line 13, B

(it)

is constructed by selecting only the

Algorithm 1: SpecDiv.

Input : F = (V

, E

), max_labels, min_size

Output: P

∗

1 B

:= compute the modularity matrix of F

according to Equation (1)

2 lb := 0

3 P

:= {c

= 0, c

= 0, . . . , c

= 0}

4 L := {0}

5 P

∗ := P

6 it := 1

7 B

(1)

:= B

8 while L 6=

0 and number of communities of P

is less than max_labels do

9 lr := randomly chose a community label in

10 Z := {i such that c

= lr for i = 1, . . . , |V

11 if |Z| > min_size then

12 D := {(i, υ) such that i ∈ Z and υ is

selected without replacement from

{1, 2, .., |Z|}}

13 B

(it)

:= Select only the rows and

columns of B

which to vertices in Z

according to Section 4.1

14 L := L\{lr}

15 λ, u

:= Compute the largest eigenvalue

and associated eigenvector of B

(it)

16 U

:= {i ∈ Z such that u

υ1

≥ 0,

υ : (i, υ) ∈ D}

17 U

−

:= {i ∈ Z such that u

υ1

< 0,

υ : (i, υ) ∈ D}

18 c

= lb, for i ∈ U

19 c

= lb + 1, for i ∈ U

−

20 Q(P

) := Calculate modularity according

to Equation (2).

21 repeat

22 P

, ∆Q := move each i ∈ Z to the

community labeled lb or lb + 1 that

results in the largest modularity gain

23 until ∆Q ≤ 0;

24 if Q(P

) > Q(P

∗) then P

∗ = P

;

25 if communities labeled lb and lb + 1 are

not empty then L := L ∪ {lb} ∪ {lb + 1}

;

26 lb := lb + 2

27 it := it + 1

28 repeat

29 P

∗, ∆Q∗ := move each i ∈ V

to the

community that results in the largest

modularity gain

30 until ∆Q∗ ≤ 0;

31 return P

∗

rows and columns of B

that correspond to vertices in

Z, according to Section 4.1.

In line 14, the selected element lr from the list L

is removed. In line 15, the largest eigenvalue, λ and

the respective eigenvector, u

, of B

(it)

are calculated.

We used the implicitly restarted Arnoldi method from

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks

311

ARPACK library (Lehoucq et al., 1998) for this step

In lines 16 and 17, U

and U

−

are the sets of vertices

from Z whose corresponding signs of the components

of u

larger than or equal to zero are assigned to U

whereas those lower than 0 are assigned to U

−

. In

lines 18 and 19, vertices from U

and U

−

receive

two new community labels lb and lb + 1 in partition

, respectively. In case one of the sets is empty, the

community is not divided. Otherwise, the modularity

of partition P

is calculated according to Equation (2)

in line 21.

After that or when |Z| ≤ 8, a local search strategy

in line 23 moves each vertex in Z to the community

labeled lb or lb + 1 that results the largest modularity

gain. To avoid biased results, we select the vertices

to be moved in a random order. The local search is

repeated until there is no movement to improve the

modularity gain.

In line 25, the algorithm updates the best partition

found along the iterations, P

∗, to P

if Q(P

) is hig-

her than Q(P

∗). In line 27, the community labels lb

and lb +1 are inserted to the list L if both of them are

not empty, i.e., contain at least one vertex. In line 27,

the label of the next community to be created, lb, is

updated and in line 28 the number of iterations, it, is

incremented.

After deﬁning a vertex partition P

∗ in F, Spe-

cDiv employs a local search strategy to improve it,

until there is no move that improves the modularity

value in line 31. Different from the local search per-

formed in line 23, the vertices can be moved to any

community. Finally, Algorithm 1 returns P

∗ as out-

put.

The next sections explain the proposed strategies

to ﬁnd disjoint and overlapping communities.

5.2 Finding Disjoint Vertex Partitions

from Link Partitions

Algorithm 2 presents the strategy to ﬁnd disjoint ver-

tex communities having as input a link partition. It

takes as input a connected graph G and a link parti-

tion P

obtained by some vertex community detection

algorithm applied to the line graph of G, here called

In line 1, the algorithm assigns to CV

the commu-

nity labels of the edges incident to vertex i ∈ V in G,

considering the link partition P

. In line 2, to deﬁne a

We required that |Z| must be higher than min_size = 8

due to conditions imposed by the number of iterations se-

lected for the implicitly restarted Arnoldi method to con-

verge. The local search procedures are capable of impro-

ving small communities with respect to the modularity va-

lue.

Algorithm 2: Disjoint community detection.

Input : Connected graph G = (V, E), link

partition P

Output: P ∗

1 CV

:= {c

(i, j)

∈ P

such that (i, j) ∈ E}, for

i ∈ V

2 c

= mode(CV

), for i ∈ V

3 P := {c

, c

, . . . , c

|V |

}

4 P ∗ := move each i ∈ V from community

labeled c

to that from CV

that results the

largest modularity gain

5 return P ∗

Algorithm 3: Overlapping community detection.

Input : Connected graph G = (V, E), link

partition P

Output: P V

1 CV

:= {c

(i, j)

∈ P

such that (i, j) ∈ E}, for

i ∈ V

2 P V := {CV

,CV

, . . . ,CV

|V |

}

3 X := set of community labels of vertices

from V that does not present overlapping in

4 forall i ∈ V such that |CV

| > 1 do

5 if |{ci ∈ CV

∩ X }| > 0 then

X i := {ci ∈ CV

∩ X };

6 else X i := CV

;

7 CV

{ld ∈ X i|(

∑

j∈V such that c

=ld

i j

) > 0}

8 P V := {CV

,CV

, . . . ,CV

|V |

}

9 return P V

vertex partition, for each vertex i, mode(CV

) returns

to c

the most frequent label in CV

. The community

labels identiﬁed in line 2 deﬁne a vertex partition P

for G in line 3. In line 4, a local search procedure that

moves each vertex i ∈ V from community labeled c

to that from CV

that results in the largest modularity

gain is applied to P . Again, we select the vertices to

be moved in a random order. Finally, the algorithm

returns the reﬁned vertex partition P ∗ in line 5.

5.3 Finding Overlapping Vertex

Partitions from Link Partitions

In Algorithm 3, we describe the strategy to identify

overlapping vertex communities from a link partition

given as input, along with a connected graph G.

In line 1, |V | sets of community labels according

to the link partition given as input are created. In line

2, a list of overlapping communities with the vertex

community labels assigned to CV

is deﬁned.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

312

In line 3, set X with the community labels of the

vertices of G which are in only one community is con-

structed, i.e., vertices i ∈ V such that |CV

| = 1. We

shall refer to vertices that belong to more than one

community as overlapping vertices.

From lines 5 to 7, a post-processing strategy to de-

ﬁne the community labels of each overlapping vertex

i,CV

> 0, is applied to P V . In line 5, elements of X i

are deﬁned as the overlapping communities of vertex

i, CV

, in X . If CV

∩ X is empty, then X i := CV

line 6.

In line 7, the community labels of each overlap-

ping vertex i ∈ V are selected as those whose sum of

the modularity contributions between i and the verti-

ces of that community is higher than 0. We require

that a vertex is only moved to a community that exists

in X to avoid the creation of communities composed

only by overlapping vertices.

Finally, in line 10, the overlapping communities

P V with the community labels of each vertex deﬁned

according to line 7 are returned.

6 CASE STUDY IN A TWITTER

NETWORK

Before presenting the computational experiments, in

this section, we study a practical application of over-

lapping community detection in a Twitter retweet net-

work constructed by Ribeiro et al. (2018). The aut-

hors of this network also classiﬁed users that used ha-

teful speech in Twitter.

The network presented here is the main connected

component of a small subgraph induced by 5000 ver-

tices and 10 hateful users selected randomly from the

network. The order of the selected subgraph was due

to computational performance restrictions of the pro-

posed spectral divisive heuristic. The resulting num-

bers of vertices and edges after extracting the main

connected component of this subgraph are 2885 and

4874, respectively.

Figure 1

illustrates the overlapping vertex com-

munities P V found using Algorithm 3 having as in-

put a link partition, obtained by SpecDiv applied to

the line graph. In this ﬁgure, the colors of the vertices

and edges identify the vertex and link communities.

The overlapping vertices are multi-colored according

to their community labels. The large-sized vertices

deﬁned by the ‘*’ mark are the hateful users identi-

ﬁed by Ribeiro et al. (2018). The large-sized vertices

Some of the vertices and edges of the graph were sup-

pressed to improve its readability.

without marks corresponds to the top 1% vertices with

the highest degrees.

Figure 1: Overlapping communities found in a sample of a

Twitter network.

The non-backtracking modularity of P V , i.e., the

modularity of the line graph, is 0.8384. The modula-

rity of the partition obtained from the link partition by

Algorithm 2 is 0.6371.

Approximately 36.05% of the vertices belong to

overlapping communities. Among them, 62.31%

have 2-overlapping community labels and only 5.1%

have more than 4 communities. The average number

of overlapping community labels is 2.59, with a stan-

dard deviation from the average of 0.91.

According to Figure 1, most of the top 1% highest

degree nodes do not belong to overlapping commu-

nities and many of them belong to the largest com-

munities. On the other hand, 4 of the 9 hateful users

belong to overlapping communities. In particular, 2

of them belong to 4-overlapping communities. At le-

ast one hateful user in each community also belongs

to another community that includes another hateful

user. Overall, 44.44% of the hateful users belong to

overlapping communities.

To ensure the consistency of the results, we repli-

cated the aforementioned experiment on 5 different

random subgraph sampling drawn from the original

network. The average number of vertices and edges

of the networks are 2868.8 and 4953.2, respectively.

The average modularity values for the line graph and

for the partition obtained from the link partition are

respectively 0.6193 and 0.8359. The average num-

ber of overlapping communities is 2.6. On average,

37.99% of the vertices in the networks belong to over-

lapping communities. Among them, 62.45% are in

2-overlapping communities and only 5.23% belong

to more than 4-overlapping communities, on average.

These results are consistent with those presented in

Figure 1.

In only one subgraph sampling the hateful users

do not belong to any overlapping communities. The

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks

313

average percentage of hateful users that belong to

overlapping communities is 40.89% and presents the

large standard deviation value of 23.54%. In fact, this

experiment suggests that the subgraph sampling only

affects the distribution of hateful users in the commu-

nities.

An interesting application from these empiri-

cal observations based on overlapping communities

would be to study the inﬂuence of these overlapping

hateful users on other users of the network.

7 COMPUTATIONAL

EXPERIMENTS

This section presents the computational experiments

with sparse LFR networks with disjoint and overlap-

ping communities, generated using the software of

Lancichinetti et al. (2008).

All the LFR networks were generated having as

parameters: number of vertices equal to 1000, aver-

age and maximum degrees given by 3 and 10, respec-

tively and exponential distribution equal to 2. Table 1

presents the remaining characteristics that distinguish

the networks into sets: minimum and maximum num-

ber of vertices in the communities, minc and maxc,

respectively; number of overlapping vertices, OV n;

number of communities for the overlapping vertices,

OV m; and degree of mixture of the communities, µ.

Table 1: Characteristics of the generated LFR networks.

Set minc maxc OV n OV m µ

SD 10 50 0 - 0.1, 0.2, . . . , 0.8

LD 20 100 0 - 0.1, 0.2, . . . , 0.8

SOv 10 50 100 2 0.1, 0.2, . . . , 0.8

LOv 20 100 100 2 0.1, 0.2, . . . , 0.8

The disjoint networks are referred to as in sets SD

and LD, whereas the overlapping ones are said to be

in SOv and LOv sets. Networks from SD and SOv

are composed by small communities, whereas those

in LD and LOv are composed by large communities.

We generated 5 different networks for each set

and value of µ described in Table 1 and reported the

average results achieved over 5 independent executi-

ons. All the computational experiments were perfor-

med on a computer with a processor Intel Xeon E5-

1620 with 3.70Gz and main memory of 32 GB. The

disjoint and overlapping strategies were implemented

in R with igraph library (Csardi and Nepusz, 2006).

SpecDiv was implemented in C++ with ARPACK++

library. The parameter max_label of SpecDiv was set

to be 10% of the number of vertices in the graph and

the parameter min_size was set to 8.

7.1 Disjoint Communities

In this Section, we contrast the partitions found by

vertex community detection algorithms with those

obtained by the disjoint community detection strategy

that takes as input a link partition from the line graph

. To differentiate the strategies using line graphs

from those using the original graphs as inputs, we add

the sufﬁxes "−G

" and "−G" to the names of the al-

gorithms.

To assess a disjoint partition, we employed the

Normalized Mutual Information (NMI) measure (Da-

non et al., 2005), which indicates the correlation be-

tween the partitions in a scale from 0 to 1, where 1

indicates a perfect correlation.

First, Figure 2 present the average NMI values

achieved by SpecDiv-G, SpecDiv-G’ and by the refe-

rence disjoint community detection algorithms Lou-

vain method, LP, Infomap and Walktrap. According

to these ﬁgures, SpecDiv-G’ outperformed SpecDiv-

G and presented better results than the algorithms

with the exception of Infomap and Walktrap. Despite

being worse than SpecDiv-G’, SpecDiv-G still obtai-

ned better results than Louvain and LP for networks

with µ ≥ 0.3.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

SpecDiv−G

Louvain−G

LP−G

Infomap−G

Walktrap−G

NMI

(a) Networks from SD.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

SpecDiv−G

Louvain−G

LP−G

Infomap−G

Walktrap−G

NMI

(b) Networks from LD.

Figure 2: Average NMI values over 5 executions of the met-

hods for detecting disjoint communities.

Figure 3 displays the average NMI values achie-

ved by SpecDiv-G’ and by the disjoint community

detection strategy taking as input the link partition

obtained by the reference algorithms applied to the

line graphs. It also presents the results of OSLOM

and of the disjoint version of SLPA.

The results of SpecDiv, Louvain, LP, Infomap

and Walktrap improved on average 9.93%, 14.95%,

OSLOM identiﬁed overlapping in some of the disjoint

networks. We chose one community label from each over-

lapping vertex to present its results.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

314

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

Louvain−G'

LP−G'

Infomap−G'

Walktrap−G'

OSLOM

SLPA

NMI

(a) Networks from SD.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

Louvain−G'

LP−G'

Infomap−G'

Walktrap−G'

OSLOM

SLPA

NMI

(b) Networks from LD.

Figure 3: Average NMI values over 5 executions of the met-

hods for detecting disjoint communities using line graphs.

170.25%, 2.64% and 2.6%, respectively, when com-

pared to those for networks from SD presented in

Figure 2(a) and 16.62%, 20.32%, 258.57%, 4.64%

and 1.63%, respectively, when compared to those

for networks from LD presented in 2(b). LP, Info-

map and Walktrap were able to ﬁnd better results

than SpecDiv-G’ by using the introduced link stra-

tegy. Moreover, the disjoint communities obtained

from the link partitions, with the exception of Lou-

vain, were better than those found by OSLOM and

SLPA for µ ≥ 0.3.

The maximum standard deviation values from the

presented averages results of LP and OSLOM were

0.19 and 0.27, respectively. The remaining algo-

rithms found results consistently closer to the average,

reporting standard deviations values lower than or

equal to 0.05.

These results allow us to conclude that even

though the introduced link strategy is mainly designed

to detect overlapping communities, it can improve the

detection of disjoint communities in networks.

7.2 Overlapping Communities

The traditional NMI employed in the previous section

is not capable of comparing overlapping communi-

ties. In this sense, we employed the extension of the

NMI introduced by Lancichinetti et al. (2009), called

oNMI, to assess the communities with respect to the

ground truth provided by the LFR software. The im-

plementation is due to McDaid et al. (2011).

In this section, we contrast the oNMI values achie-

ved by the introduced overlapping strategy having

as input link partitions obtained by SpecDiv, Lou-

vain method, Infomap, LP and Walktrap with those

achieved by the state-of-the-art overlapping methods

OSLOM and SLPA

Figure 4 reports the oNMI values obtained by the

algorithms. According to Figure 4(a), for µ ≤ 0.3,

OSLOM obtained the communities with the highest

oNMI values. For µ ≥ 0.4, SLPA found the com-

munities with the highest oNMI values. SpecDiv-

G’, Louvain-G

and Walktrap-G

achieved better re-

sults than OSLOM for networks with µ ≥ 0.5 and

SpecDiv-G’ was better than Walktrap-G

in networks

with µ ≥ 0.6.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

Louvain−G'

LP−G'

Infomap−G'

Walktrap−G'

OSLOM

SLPA

oNMI

(a) Networks from SOv.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

●

SpecDiv−G'

Louvain−G'

LP−G'

Infomap−G'

Walktrap−G'

OSLOM

SLPA

oNMI

(b) Networks from LOv.

Figure 4: Average oNMI values over 5 executions of the

methods for detecting overlapping communities.

On the other hand, Figure 4(b) shows that in net-

works with large overlapping communities, OSLOM

found the communities with the highest oNMI values

for µ ≤ 0.2 and Louvain-G

method found the com-

munities with the best oNMI values for µ ≥ 0.3.

The standard deviations of OSLOM and SLPA

from the average values in the large community net-

works, however, were in the intervals [0.02,0.06] and

[0.02, 0.23], respectively. Thereby, SLPA might not

ﬁnd consistently good results in all its executions. On

the other hand, the standard deviations of the remai-

ning algorithms were lower than or equal to 0.05.

These values suggest the robustness of the algorithms

based on link partitions.

We, nonetheless, highlight that these networks are

very sparse and thus are different from the traditional

denser LFR networks considered in most of the expe-

riments performed in the literature.

7.3 Running Times

Table 2 summarizes the average running times of the

algorithms to detect disjoint and overlapping commu-

nities. The numbers presented between parentheses

are the standard deviation values from the averages.

The threshold parameter of SLPA was set to 0.1.

Spectral Algorithm for Line Graphs to Find Overlapping Communities in Social Networks

315

Table 2: Average running times of the algorithms in seconds.

Set SpecDiv-G’ Louvain-G

LP-G

Infomap-G

Walktrap-G

OSLOM SLPA

SD 370.88 (51.17) 32.05 (12.25) 45.92 (9.74) 45.17 (12.58) 38.49 (15.4) 37.86 (9.14) 1.74 (0.12)

LD 346.55 (38.36) 38.31 (10.53) 57.57 (4.89) 56.61 (5.6) 45.5 (13.2) 41.02 (4.34) 1.78 (0.16)

SOv 451.1 (70.05) 15.89 (3.59) 21.99 (2.03) 22.63 (3.31) 18.94 (4.64) 36.56 (8.24) 2.8 (0.22)

LOv 439.99 (30.43) 18.88 (4.06) 26 (2.52) 26.81 (2.57) 22.23 (4.9) 39.58 (3.79) 2.8 (0.28)

Despite the good results achieved by SpecDiv-G’ in

disjoint and overlapping communities, it is the most

time-consuming algorithm. Even though the fastest

algorithm to ﬁnd overlapping communities is SLPA,

it obtained disjoint partitions worse than SpecDiv-G’.

We did not present the results of SpecDiv-G,

Louvain-G, LP-G, Infomap-G and Walktrap-G be-

cause they presented worse results than SpecDiv-G’,

Louvain-G

, LP-G

, Infomap-G

and Walktrap-G

, re-

spectively, and are not capable of ﬁnding overlapping

communities. However, we must point out that, with

the exception of SpecDiv-G, the remaining algorithms

required less than 2s to ﬁnd disjoint communities.

SpecDiv-G required on average 48.8s and 47.55s to

ﬁnd disjoint communities for SD and LD, respecti-

vely.

8 CONCLUSIONS

Overlapping community detection in networks is a

challenging problem explored by a few studies in the

literature. In particular, a research direction to appro-

ach it consists of applying community detection algo-

rithms to the line graphs of the networks under inves-

tigation.

In this paper, we propose a strategy to detect

overlapping communities in networks founded on de-

tecting link partitions in line graphs and hence de-

ﬁning the overlapping communities by an algorithm

whose input is link partitions. As part of the strategy,

we suggest a spectral community detection algorithm

named SpecDiv.

The results of the experiments performed using

LFR networks showed that the overlapping commu-

nities found by the strategy based on line graphs

are competitive with state-of-the-art overlapping met-

hods.

Additional experiments were carried out using

disjoint LFR networks. As a result, the proposed stra-

tegy obtained better vertex partitions from the link

partitions than those obtained by SpecDiv and by the

reference algorithms applied to the original graphs.

In a case study using a Twitter network, we found

that most of the highest degree vertices do not belong

to overlapping communities. On the other hand, al-

most half of the users that use hateful speech belong

to overlapping communities. We have also observed

that hateful users belong to overlapping communities

which includes other hateful users. This observation

suggests the study of the inﬂuence of these users on

the rest of the network.

A major advantage of the introduced overlap-

ping strategy is its capability of obtaining overlapping

communities from any vertex community detection

algorithm.

The presented strategy is intended to be employed

in sparse networks when the number of vertices is ap-

proximately the number of edges. A lot of large-scale

networks have that characteristic. The high cost of the

methods based on line graphs makes them unsuitable

for large dense networks. To better approach this type

of networks, the strategy must be parallelized to be

more efﬁcient in multi-core computers.

We suggest as a future work to reduce substanti-

ally the number of edges of line graphs by conside-

ring edge contraction strategies in the original or the

line graph itself. Another research direction is the ex-

tension of the presented methods to support directed

graphs.

ACKNOWLEDGEMENTS

This work was funded by São Paulo Research Foun-

dation (FAPESP), grant numbers: 2016/22688-2 and

2015/21660-4; and by Conselho Nacional de Desen-

volvimento Cientíﬁco e Tecnológico (CNPq), grant

numbers: 448614/2014-6 and 308708/2015-6. The

second author is grateful to Leonardo V. Rosset for

giving her a hand.

REFERENCES

Ahn, Y.-Y., Bagrow, J. P., and Lehmann, S. (2010). Link

communities reveal multiscale complexity in net-

works. Nature, 466(7307):761.

Ali, H. T. and Couillet, R. (2017). Improved spectral

community detection in large heterogeneous net-

works. The Journal of Machine Learning Research,

18(1):8344–8392.

Blondel, V. D., Guillaume, J.-L., Lambiotte, R., and Le-

febvre, E. (2008). Fast unfolding of communities

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

316

in large networks. Journal of Statistical Mechanics:

Theory and Experiment, 2008(10):P10008.

Brandes, U., Delling, D., Gaertler, M., Gorke, R., Hoefer,

M., Nikolosk, Z., and Wagner, D. (2008). On modula-

rity clustering. IEEE Transactions on Knowledge and

Data Engineering, 20:172–188.

Csardi, G. and Nepusz, T. (2006). The igraph software

package for complex network research. InterJournal,

Complex Systems:1695.

Danon, L., Diaz-Guilera, A., Duch, J., and Arenas, A.

(2005). Comparing community structure identiﬁca-

tion. Journal of Statistical Mechanics: Theory and

Experiment, 2005(09):P09008.

Evans, T. S. and Lambiotte, R. (2009). Line graphs, link

partitions, and overlapping communities. Physical Re-

view E, 80:016105.

Fortunato, S. and Hric, D. (2016). Community detection in

networks: A user guide. Physics Reports, 659:1–44.

Gui, C., Zhang, R., Hu, R., Huang, G., and Wei, J. (2018).

Overlapping communities detection based on spectral

analysis of line graphs. Physica A: Statistical Mecha-

nics and its Applications, 498:50–65.

Jiang, J. Q. and McQuay, L. J. (2012). Modularity functions

maximization with nonnegative relaxation facilitates

community detection in networks. Physica A: Statisti-

cal Mechanics and its Applications, 391(3):854 – 865.

Krzakala, F., Moore, C., Mossel, E., Neeman, J., Sly, A.,

Zdeborová, L., and Zhang, P. (2013). Spectral re-

demption in clustering sparse networks. Proceedings

of the National Academy of Sciences, 110(52):20935–

20940.

Lancichinetti, A., Fortunato, S., and Kertész, J. (2009). De-

tecting the overlapping and hierarchical community

structure in complex networks. New Journal of Phy-

sics, 11(3):033015.

Lancichinetti, A., Fortunato, S., and Radicchi, F. (2008).

Benchmark graphs for testing community detection

algorithms. Physical Review E, 78(4):046110.

Lancichinetti, A., Radicchi, F., Ramasco, J. J., and Fortu-

nato, S. (2011). Finding statistically signiﬁcant com-

munities in networks. PLOS ONE, 6(4):1–18.

Lehoucq, R. B., Sorensen, D. C., and Yang, C. (1998). AR-

PACK users’ guide: solution of large-scale eigenvalue

problems with implicitly restarted arnoldi methods.

volume 6 of Software, Environments, Tools. SIAM.

Li, M. and Liu, J. (2018). A link clustering based me-

metic algorithm for overlapping community detection.

Physica A: Statistical Mechanics and its Applications,

503:410–423.

McDaid, A. F., Greene, D., and Hurley, N. (2011).

Normalized mutual information to evaluate overlap-

ping community ﬁnding algorithms. arXiv preprint

arXiv:1110.2515.

Nascimento, M. C. and De Carvalho, A. C. (2011). Spectral

methods for graph clustering–a survey. European

Journal of Operational Research, 211(2):221–231.

Newman, M. (2013a). Spectral community detection in

sparse networks. arXiv preprint arXiv:1308.6494.

Newman, M. E. (2013b). Spectral methods for community

detection and graph partitioning. Physical Review E,

88(4):042822.

Newman, M. E. J. and Girvan, M. (2004). Finding and

evaluating community structure in networks. Physi-

cal Review E, 69(2):026113.

Newman, M. E. J. (2006). Finding community structure in

networks using the eigenvectors of matrices. Physical

Review E, 74(3):036104.

Nicosia, V., Mangioni, G., Carchiolo, V., and Malgeri, M.

(2009). Extending the deﬁnition of modularity to di-

rected graphs with overlapping communities. Jour-

nal of Statistical Mechanics: Theory and Experiment,

2009(03):P03024.

Pizzuti, C. (2009). Overlapped community detection in

complex networks. In Proceedings of the 11th An-

nual conference on Genetic and Evolutionary Com-

putation, pages 859–866. ACM.

Pons, P. and Latapy, M. (2005). Computing communities in

large networks using random walks. In International

Symposium on Computer and Information Sciences,

pages 284–293. Springer.

Raghavan, U. N., Albert, R., and Kumara, S. (2007).

Near linear time algorithm to detect community struc-

tures in large-scale networks. Physical Review E,

76(3):036106.

Ribeiro, M. H., Calais, P. H., Santos, Y. A., Almeida,

V. A., and Meira Jr, W. (2018). Characterizing and

detecting hateful users on twitter. arXiv preprint

arXiv:1803.08977.

Rosvall, M. and Bergstrom, C. T. (2007). An information-

theoretic framework for resolving community struc-

ture in complex networks. Proceedings of the Natio-

nal Academy of Sciences, 104(18):7327–7331.

Shi, C., Cai, Y., Fu, D., Dong, Y., and Wu, B. (2013). A link

clustering based overlapping community detection al-

gorithm. Data & Knowledge Engineering, 87:394–

404.

Singh, A. and Humphries, M. D. (2015). Finding commu-

nities in sparse networks. Scientiﬁc Reports, 5:8828.

Xie, J., Kelley, S., and Szymanski, B. K. (2013). Overlap-

ping community detection in networks: The state-of-

the-art and comparative study. ACM Computing Sur-

veys, 45(4):43:1–43:35.

Xie, J., Szymanski, B. K., and Liu, X. (2011). SLPA: Un-

covering overlapping communities in social networks

via a speaker-listener interaction dynamic process. In

Data Mining Workshops (ICDMW), 2011 IEEE 11th

International Conference on, pages 344–349. IEEE.

Yang, Z., Algesheimer, R., and Tessone, C. J. (2016).

A comparative analysis of community detection al-

gorithms on artiﬁcial networks. Scientiﬁc Reports,

6:30750.

Zhang, S., Wang, R.-S., and Zhang, X.-S. (2007). Identiﬁ-

cation of overlapping community structure in complex

networks using fuzzy c-means clustering. Physica A:

Statistical Mechanics and its Applications, 374(1):483

– 490.

Zhang, X. and Newman, M. (2015). Multiway spectral

community detection in networks. Physical Review

E, 92(5):052808.

Zhou, X., Liu, Y., Wang, J., and Li, C. (2017). A density

based link clustering algorithm for overlapping com-

munity detection in networks. Physica A: Statistical

Mechanics and its Applications, 486:65–78.

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