Taxonomy and survey of community

International Journal of Computer Science & Engineering Survey (IJCSES) Vol.5, No.4, August 2014
TAXONOMY AND SURVEY OF COMMUNITY
DISCOVERY METHODS IN COMPLEX NETWORKS
Ahlem Drif1 and Abdallah Boukerram2
1Computer Science Department, University of Setif1, Algeria
2Computer Science Department, University of Abderrahmane Mira, Bejaia, Algeria
ABSTRACT
The community detection in complex networks has attracted a growing interest and is the subject of several
researches that have been proposed to understand the network structure and analyze the network
properties. In this paper, we give a thorough overview of different community discovery strategies, we
propose taxonomy of these methods, and we specify the differences between the suggested classes which
helping designers to compare and choose the most suitable strategy for the various types of network
encountered in the real world.
KEYWORDS
Community detection methods, complex networks, quality function, graph, social networks.
1. INTRODUCTION
In complex networks, the communities are groups of nodes which share probably a common
proprieties and/or similar functions. The communities may be correspond, for example, to groups
of Web pages accessible over the Internet that have the same subject [19], functional modules as
cycles and pathways in metabolic networks [25], [47], a set of people or groups of people with
some pattern of contacts or interactions between them [24], [38], and subdivisions in the food
webs [49], [32]. The main objective of community detection is to discover a pertinent community
structure. See Fig. 1 for a toy example with this kind of a structure which shows the network
structure of the Web site of Ferhat Abbas University (Algeria). As we can see from the figure,
this network contains six communities that identify groups of users with similar behaviour for
which personalized versions of the Web site may be created. The principle idea for community
discovery domain proposed in [24], [46] was to focus on a different measure of the quality of a
division other than the simple cut size or its variants [43].
The community discovery methods have attracted considerable interest and curiosity from the
science community in recent decades. As the field of community identification has grown quite
popular and the number of published proposals for community discovery algorithms as well as
reported applications is high, we do not even pretend to be able to give an exhaustive survey of all
the methods, but rather an explanation of the methodologies commonly applied and pointers to
some of the essential publications related to each research branch in order to provide a taxonomy
for community discovery methods. We begin by given basic definitions of community structure
and the essential elements of this thematic in section 2. In section 3, we propose taxonomy of the
detection community methods and provide a survey defining the basic idea of each one, the
measure and/or process used to identify the community structure and to which class each method
belongs. Conclusion is given in section 4.
DOI:10.5121/ijcses.2014.5401 1

2
Figure 1. (a) The network structure of the Web site of Ferhat Abbas University (Algeria). The structure
identifies the users’ session and all the sessions (the nodes represent resources and edges represent the
browsing sequences of users during each session).
(b)We have obtained 6 communities according to the users’ behavior. Communities, labeled by colors,
were detected by applying Fast algorithm [45].
2. COMMUNITIES DESCRIPTION
Despite the large amount of study in this area, a consensus on what is the definition of community
has not been reached. Conceptually, the definitions can be separated into two main categories,
self-referring and comparative definitions. Central to all such definitions is the concept of sub
graph.
2.1. Graph concepts
We consider a graph G = (V,E), with V is the set of vertices and E is the set of edges. Vertices are
also known as nodes, points and (in social networks) as actors, agents or players. Two vertices u
and v are adjacent if there exist an edge (u, v) that connects them. In an undirected graph, each
edge is an unordered pair v,w. In a directed graph, edges are ordered pairs. The number of nodes
in a graph is usually denoted n while the number of edges is usually denoted m. Graph is used to
represent all types of complex networks. In this survey, the presented algorithms divide the graph
top down into communities, or also work bottom up merging singleton sets of nodes iteratively
into communities.
2.2. Comparative definitions
The comparative definitions consist to compare the number of internal links to the number of
external links. Authors sometimes use a criterion of similarities between nodes to discover the
community structure. The intuitive notion of criterion of similarities derives both from the
relative strength, frequency, density, or closeness of links within a subgroup, and the relative
weakness, infrequency, sparness, or distance of links from subgroup members to non-members. A
community may be defined as LS set that is a subgraph definition that compares links within the
subgroup to links outside the subgroup by focusing on the greater frequency of links among
subgroup members compared to the links from subgroup members to outsiders [59]. Comparative

definitions include also that of strong community, and that of weak community that introduced by
Radicchi et al [52].
2.3. Self-referring definitions
The self-referring definitions consider the subgraph alone. It identifies classes of subgraphs like
cliques, n-cliques, k-plexes, etc... They are maximal subgraphs, which cannot be enlarged with
the addition of new vertices and edges without losing the property which defines them. Self-referring
definitions can include the definition introduced by Newman in [41] which defines a
community as indivisible subgraph.
2.4. Quality Functions
How can we know if detected communities are good or no and how to value such partitions?
What is the better partition for the network in question? When we break the dendrogram to obtain
the level of adequate partition of the network or else a large number of appropriate communities?
To answer to those questions, Newman and Girvan [46] introduce a measure of quality of a
particular partition which they called modularity. The modularity is based on assortative mixing
measure proposed by Newman [44]. The modularity is defined in the following way: Given a
particular division of a network into k communities, let e denote a kxk matrix whose element is
the fraction of all edges in the network which connects vertices in community i to those in
community j. The trace of this matrix gives the fraction of edges in the network that connect
vertices in the same community:
3
ܶݎ݁ = ෍݁௜௜
௜
(1)
A good community partition should have a high value of the trace, but the trace on its own is not
a good indicator of the quality of the division since, for example, placing all vertices in a single
community would give the maximal value of Tr = 1 without giving any information about the
community structure. The row (column) sum represents the fraction of edges that connect to
vertices in community i, is then defined as
ܽ௜ = ෍ ݁௜௝
௝
(2)
If the network does not exhibit community structure, or if the partitions are allocated without any
regard to the underlying structure, the expected value of the fraction of links within partitions can
be estimated. It is simply the probability that a link begins at a node in i, ai, multiplied by the
fraction of links that end at a node in i, ai. So the expected number of intra-community links is
just aiai. On the other hand we know that the real fraction of links exclusively within a partition is
eii. So, we can compare the two directly and sum over all the partitions in the graph.
ଶ൯ = ܶݎ݁ − ‖݁ଶ‖
ܳ = ෍൫݁௜௜ − ܽ௜
௜
(3)
It measures the fraction of edges in a community, minus the expected value of the same quantity
in a network with the same community divisions but random connections between the nodes. If a
particular division gives no more with in community edges that would be expected by random
chance the modularity is zero. Values other than 0 indicate deviations from randomness, and
values above 0.3 indicate a modular structure [45]. In practice, values above 0.7 are rare, and

indicate a very clear structure. However, it is possible that the partition of high modularity don’t
suit to more pertinent community partitions [20].
3. PROPOSITION OF TAXONOMY OF COMMUNITY DISCOVERY METHODS
The problem of community detection goes to traditional approaches in computer science which
were the graph partitioning and the data clustering approaches. However, these traditional
approaches are not adapted to discover a structure community. In the traditional algorithm of
partitioning graph [31] [51] [18], the number and the size of groups must be specified in input. In
data clustering methods, the graphs considered have not the specific proprieties of complex
networks. Moreover, in hierarchical clustering when many partitions recovered how can we know
which one is the best? Therefore, several approaches were proposed in order to deal with the
incapacity of classical methods and to provide a significant community structure.
In this section, we survey the research in the area of communities identification, suggest a
classification, and we situate some of the research with respect to the proposed taxonomy.
According to our analysis, we have proposed taxonomy for community discovery methods in
order to well locate these different approaches (see Fig.2 and Fig.3). It seems that this taxonomy
allows us to explicit the different choices of one of these community discovery approaches,
decides which is the most suitable for a given type of complex networks and for which reasons
are. These appropriate choices will give us a considerable flexibility in choice of the division
quality functions. Our classification is based on the three following points of view:
A. Agglomeration versus divisive:
According to manner of grouping the nodes in groups, Jain and Dubes [28] defined two distinct
approaches to do: agglomerative and divisive. As the same, we conclude that all the community
discovery methods use an agglomerative or divisive approach, depending on whether the partition
is refined or coarsened during each iteration:
 Bottom-up or agglomerative algorithms that start with each node in its own singleton
community or another set of small initial communities, iteratively merging these
communities into larger ones.
 Top-down or divisive algorithms that split the network iteratively or recursively into
4
smaller and smaller communities.
B. Stochastic versus deterministic:
In the context of evaluation of performance of community discovery methods, we have noted that
the deterministic and the stochastic methods differed on function of time execution and partition
quality. Therefore, we think that is fort important to distinguish between the deterministic method
and the stochastic ones.
C. Various computer implementations of community discovery methods:
In many approaches, the communities are characterised and detected, directly or indirectly, by
some global proprieties of graph, such intermediary, betweeness, etc..., or by some process as
random walks, synchronisation, etc..., communities also may be interpreted as a topological
organization form. All of these remarks allow us to classify the existent methods according to the
technical details or the process used during the discovery of communities.

3.1. Agglomerative Methods
Agglomerative methods start with a state in which each node (or a small set of nodes) is the sole
member (or members) of one of n communities, they repeatedly join communities together in
pairs until obtain one community which corresponding to all complex network. In this section, we
present the agglomerative approaches, we give a classification based on which the method is
deterministic or stochastic, and then we give a grouping of methods in function of the process
used during the discovery of communities.
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Figure 2.Community discovery methods taxonomy: Agglomerative methods.
3.1.1 Stochastic methods
In this section, we present the agglomerative approaches that use a stochastic process in which
future states only depend on the current state, not the past, taking values from some countable
state space.
3.1.1.1 Algorithms based on a dynamic process
This Section describes methods employing processes running on the graph and focusing on spin-spin
interactions, random walk and synchronization.
3.1.1.1.1 Random walk techniques
The method suggested by Zhou and Lipowsky [66] is based on the average number of stages so
that a Brownian particle (random movement of a particle) reaches a given node since another
source node. The authors used the Brownian movement to introduce the Netwalk algorithm (NW)
which is based on the proximity index. The proximity index for each nearest-neighboring pair of
nodes i and j is defined
Λ(i, j) =
ටΣ ൣ݀݅݇ − ݆݀݇൧2 ܰ݇≠݅,݆
(ܰ − 2) (4)

If two nearest-neighboring nodes ݅ and ݆ belong to the same community, then the mean-first-passage-
time ݀௜௞ from ݅ to any another node ݇ will be approximately equal to that from ݆ to ݇. So,
it will be small if ݅ and ݆ belong to the same community and large if they belong to different
communities. Initially, the Netwalk algorithm considers each node as a single community. Then it
merge the two communities with the lowest proximity index into a single community and then
update the proximity index between this new community and all the other remaining communities
that are connected to it. This merging process is continued until all the nodes are merged into a
single community corresponding to the whole network. This algorithm runs in time ܱ(݊ଷ) which
make the application of these approaches on large graphs inadmissible.
Pons et al. [50] proposed the Walktrap algorithm by using the intuitive report that the random
walks will be made trap in zones dense, in other words when a walker is in a community it has a
strong probability of remaining in the same community at the following stage. The authors define
metric of distance which is related to the spectral approaches which are based on the fact that two
close nodes belonging to the same community have similar components on the principal
eigenvectors. The algorithm computes the connected components, and applies then an
agglomerative algorithm which discovered communities on connected subgraphs. This algorithm
gives better performance in ܱ(݊. ݉. log (݊)) but when the length of the steps becomes important,
the quality of the results decreases.
3.1.1.1.2 Synchronization techniques
Using the synchronization process to discovery communities based on the idea that there is a
relationship between topological scales and dynamic time scales in complex networks. Several
studies show [4], [7] that high densely interconnected sets of oscillators - where oscillators are
placed at the nodes - synchronize more easily that those with sparse connections. This scenario
suggests that for a complex network with a non-trivial connectivity pattern, starting from random
initial condition that highly interconnected oscillators forming local clusters will synchronize
first, whereas a full synchronization requires a longer time. This process occurs at different time
scales if a clear community structure exists. In [7], the algorithm scales in a timeܱ(݊. ݉), or (݊ଶ)
on sparse graphs, and gives good results on practical examples. However, synchronization-based
algorithms may not be reliable when communities are very different in size.
3.1.1.1.3 System of spins
System of spins is another promising dynamic process to reveal communities in complex
networks. Reichardt et al. [53] proposed a discovered community algorithm which is based on the
Potts models with ܳ states. The Potts model is one of the most models used in statistical physics
in order to describe the behaviour of the magnetic bodies [30]. It corresponds to model these
bodies like the spins with ܳ states located at the nodes of a network and which are in interaction
between neighbours in order to align its for a ferromagnetic body, or with being well in
opposition for an antiferromagnetic body, according to the sign of the constant of coupling. Fu
and Anderson [23] showed by analogy that there is a relation between the energy of the physical
systems, which is represented by the Hamiltonian and the cost function in an optimization
combinative problem. The Hamiltonian of a spin is given by
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ܪ = −ܬ ෍ ߜఙ೔ఙೕ + ߛ ෍
݊௦(݊௦ − 1)
2
௤
(௜,௝)∈ா ௦ୀଵ
(5)
Here ܧ is the set of edges, ߪ௜(݅ = 1. . ݊) denotes the individual spins which are allowed to take ݍ
values. ݊௦ indicates the number of spins that have ݏ spin such that Σ ݊௦ = ܰ ௤௦
ୀଵ , ݆ is the

ferromagnetic interaction strength, ߜ: is a positive parameter, and ߪ is the Kronecker delta. Each
node is characterized by a spin which can have ݍ possible values. The first sum is the standard
ferromagnetic Potts term which represents a homogeneous distribution of the spins in the
network, and is minimized by ܪ௙௘௥௥ = −ܬܯ. The second term sums up all the possible pairs of
spins which have equal value. It represents the diversity of the configuration of spins or the
existing classes of spins. We find the system fundamental state in order to define the community
structure. The communities correspond to the classes of nodes having equal values of spin. The ݍ
number of possible spins corresponds to the maximum number of communities which we can
detected and it must be selected so that it is higher than the communities effective number. The
authors use Monte Carlo single spin flip heat-bath algorithm to determine the community
structure. The energy optimization system, which is represented by the second term of the
Hamiltonian, is found by simulated annealing algorithm [54]. This minimization of energy
corresponds to support the edges intra-community and to optimize the edges inter-community.
The algorithm has a non deterministic and non hierarchical nature, so it is able to detect the
affiliation of the nodes which belong to the overlapped communities, and it allows the
quantification of the communities’ stability. However, simulated annealing is not a method of
total optimization effective and the algorithm is not convenient to be applied to wide area
networks.
3.1.1.2 Algorithms based on modularity optimization
Using the modularity optimization approaches to discover communities comes from the idea that
a great value of modularity (ܳ) represents a good partition in communities. Several studies
proposed to optimize this value of all the possible partitions in order to find the best modularity.
3.1.1.2.1 Simulated annealing
Guimer et al. [26] show that finding the modularity of a network is analogous to finding the
ground-state energy of a spin system. They demonstrate that, due to fluctuations, stochastic
network models give rise to modular networks. The authors were employed a simulated annealing
procedure for modularity optimization. This optimization is based on local moves, where a single
node is shifted from one cluster to another, taken at random, and a global moves, consisting of
merges and splits of communities. Simulated annealing converges generally more closely towards
the optimal solution but it can be used only for small networks.
3.1.2 Deterministic methods
In this section, we describe the agglomerative approaches that use deterministic methods in order
to reveal correctly all the communities.
A great value of modularity represents a good division of a network into communities, and then
one should be able to find such good divisions by searching through the possible candidates for
ones with high modularity. While finding the global maximum modularity over all possible
divisions seems hard in general, reasonably good solutions can be found with approximate
optimization techniques. In this section, we survey the algorithms which are based on a greedy
optimization.
3.1.2.1.1 Greedy techniques
The fast algorithm proposed by Newman [45] uses a greedy optimization in which, starting with
each node being the sole member of each community, we repeatedly join together the two
7

communities whose amalgamation produces the largest increase in ܳ but don’t join the pair of
communities whose there are no edges between them. For a network of ݊ nodes, after (݊ − 1)
such joins, the algorithm stops when the results of merging process is a single community. Thus
the total running time is (݉݊) , or ܱ(݊)ଶ on a sparse graph. The output of the algorithm can be
represented in the form of a dendrogram and the optimal cross-section of the dendrogram found
by looking for the optimal value of ܳ. Clauset et al [10] proposed an improvement of the fast
algorithm which performs well in ܱ(݈݊݋݃ଶ݊), the proposed method computes the change in
modularity and finds a pair of communities ݅; ݆ with the largest Δܳ௜௝. These algorithms based its
decisions on local information of the various communities. However, the structure of community
isn’t a local quantity, whereas the algorithms that based on local information such the GN
algorithm [24] finds this structure more correctly.
3.1.2.2 Algorithms based on topological organization
Several approaches of discovered communities are based on the observation that a community
can be interpreted as the union of a set of sub graphs which share nodes between them. This
observation based on such a topological organization allows expressing the tendency of the nodes
to gather in communities. In this section, we describe the methods that belong, in the diagram of
our classification, to the topological organization class which are by nature agglomerative
methods.
3.1.2.2.1 Clique
Palla et al [47], [12] have defined a new method that employed a percolation process to detect the
overlapped communities in the networks. The clique percolation method (CPM) uses the high
density of the internal edges which seem to form cliques. So, two k-cliques are adjacent if they
share (݇ − 1) nodes, and a community is defined as a union of all the k-cliques that can be
reached by chains of adjacent k-cliques. Such communities can be better visualized using a
k-clique template which is an isomorphic object for a complete graph of k-nodes. This object can
be placed on a k-clique of the network and be moved towards the adjacent k-clique by changing
one of its nodes and keeping its others (݇ − 1) nodes. Thus, the community (k-clique percolation
cluster) is the largest connected subgraph which can be entirely explored while rolling the object
on the k-clique.
An extension of CPM algorithm was proposed for the weighted networks [17] and the directed
networks [48]. To detect communities, from k-cliques, the maximal cliques should first of all be
computed. The algorithm complexity is exponential and directly proportional to the size of the
network, nevertheless the authors proved that the algorithm can be carried out in an acceptable
short time on real networks with up to 10ହ nodes. However, CPM method supposes that the graph
has a large number of cliques, thus it can fail to detect significant partitions for graphs containing
just a few cliques.
Lehmann et al [34] have addressed the problem of community detection on bipartite networks and
it has been shown that if the bipartite information is available, the proposed biclique community
detection algorithm retains all of the advantages of the k-clique algorithm [12], but avoids
discarding important structural information when performing a one-mode projection of the
network.
In [56], Shen et al. presented an algorithm to detect at the same time the hierarchy of the
communities and their overlapping, using the whole of maximal cliques. The maximal cliques
whose the nodes belong to others largest maximal cliques are called subordinates maximal cliques
and the majority of them have small sizes. The subordinate maximal cliques may degrade the
8

performance of the algorithm and should be eliminated. Thus, subordinate maximal cliques that
are lower than a threshold k are eliminated. In real-world networks, the threshold value ݇ is
typically between 3 and 6. After the elimination phase, some nodes that don’t belong to any
remaining maximal cliques are called subordinate nodes. Initially, the algorithm finds all the
maximal cliques by using Bron-Kerbosch algorithm [9]. Then, the subordinate maximal cliques
are neglected, and each of the maximal cliques and each of the subordinate nodes are determined
as initial communities. Then, it repeatedly computes the similarity between each pair of
communities and join together the two communities whose amalgamation produces a maximum
similarity. The similarity between two communities ܥଵ and ܥଶ is defined as
9
ܯ =
1
2݉
෍ [ܣ௩௪ −
݇௩ ݇௪
2݉
]
௩∈஼భ,௪∈஼మ,௩ஷ௪
(6)
Where ܣ௩௪ is the adjacency matrix, ݇௩ is the degree of the node ݒ, and ݉ = ଵ
ଶ
Σ௩௪ ܣ௩௪ is the
total number of the edges in the network. The authors of the same paper [56] introduced an
extended modularity to determine the quality of division in communities
ܧܳ =
1
2݉
෍ ෍
1
ܱ௩ܱ௪
[ܣ௩௪ −
݇௩݇௪
2݉
]
௜ ௩∈஼೔,௪∈஼೔
(7)
Where ܱ௩ is the number of communities containing node ݒ.The algorithm determines the suitable
cut of the dendrogram according to the maximum value of ܧܳ. EAGLE algorithm don’t have
only the capacity to detect sub communities until none of them can be divided (indivisible graph),
but also it detects the communities which overlap without loss of their hierarchical details.
However, the algorithm is time-consuming due to the research of all the maximum cliques in the
network.
3.1.2.2.2 Motif
Arenas et al. [3] redefined the communities as set of classes which contains motifs. The motifs
make it possible to represent the suitable structure of the network. The authors developed a
modularity extensions such the motif modularity which was defined as the fraction of motifs
inside the communities minus the fraction in a random network which preserves the nodes
strengths. Then a generalization of modularity was proposed by relaxing the condition that all
nodes of the motif should be fully inside the modules. The authors proposed path modularity and
envisaged the length of paths in order to give the best partitions in communities according to the
characteristics of the studied networks.
3.1.2.3 Algorithms based on spectral analysis
The spectral methods consist in embedding graph in an Euclidean space such that the nodes
strongly connected are represented in the same part of space and the nodes without or with few
connections are represented remotely.
3.1.2.3.1 Spectral clustering
Donetti et al. [13] proposed an approach based on the spectral properties of the Laplacien matrix
of the graph. The coordinate ݅ and ݆ of the eigenvectors corresponding to the smallest non null
eigenvalues are correlated when nodes ݅ and ݆ are in the same community. An Euclidean distance
or an angular distance between nodes is computed starting from these eigenvectors, then this

distance being used in a hierarchical clustering algorithm. The number of eigenvectors that have
to be taken into account is a priori not known. At every step of the clustering process the
modularity is computed. Once the whole dendrogram is completed, the splitting with the
maximum modularity is chosen as the output for the corresponding number of eigenvectors. An
improvement of this approach has been proposed by using of a normalized Laplacien matrix
version [14]. In [29], Jiang et al have reformulated the modularity and employed the spectral
clustering in order to maximize the modularity and in consequence to correctly detect the
community structure in the network. These spectral methods generate good results but the
eigenvectors calculations are time-consuming which give a limited performance.
3.1.2.4 Algorithms based on global, local, or clustering proprieties
This section describes the algorithms that deal with some proprieties and adopt an agglomerative
framework. Methods that use local proprieties are based on analysis of local connection patterns
taking into consideration the internal connectivity within the clusters. The other methods that use
global proprieties look not only for the inter-connectivity between clusters but also internal
connectivity within the clusters and capture large-scale network structure.
3.1.2.4.1 Algorithm based on clustering proprieties
Eckmann et al. [16] defined a method based on the clustering proprieties. The key idea of the
algorithm is that the high curvature region of a network will belong to the same community. In
the first phase the method regroups nodes which are directly related and illuminated the
connective links that arise from physical proximity in order to leaving the more relevant remote
links. If two nodes ܣ and ܤ are congruent and ܤ and ܥ are congruent, then ܣ and ܥ will be
congruent with high probability, forming a triangle. To quantify the aggregation of triangles with
congruent edges, the method defines a local curvature at a node݊. This curvature defines a World
Wide Web landscape whose connected regions of high curvature characterize a common topic.
3.1.2.4.2 Algorithm based on local proprieties
Bagrow et al [5] have presented detection communities method which uses local information.
They introduced the concept of shell that is defined as a set of nodes of geodesic distance from a
starting node. The first shell includes the closest neighbors of the starting node and the second
one includes the next neighbors of the closest neighbors of this starting node and so on until the ݈
shell. Initially the ݈ shell ݈ = 0 starts from a node ݆ which is then added to the list of community
members. Repetitively, the algorithm extends the number of shell by adding nodes which are
from ݈ hop from node ݆ in the list of community members. When the constraint of threshold is
reached or the total connected component is obtained the process is stopped. The authors defined
an adhesion matrix which gathers the vectors representing the communities of each starting node
in order to obtain an idea about the total structure of the network. Then, according to the distance
between these vectors, a process is carried out in order to permute the lines which have a short
distance between them in order to gather the sub communities belonging to the same community.
However, the founded communities depends closely on the localization of the starting node in
particular if it is close from nodes that do not belong to its community. The algorithm generates a
high cost in ܱ(݊ଷ).
3.1.2.4.3 Algorithm based on global proprieties
An iterative community detection algorithm based on a measure of information discrepancy
(MID) was proposed by Zhang et al. [62]. The authors defined the profile of any node based on
the shortest path (SP) between it and all the other nodes in the network; if the profiles of a set of
10

nodes are similar, then they are possible in the same community, because the profile of each node
characterizes its overall connection information in the network. The principle idea of the
algorithm is that if two nodes ݅ and ݆ have similar SP profiles, they must have a very close link
relationship. In this algorithm, the nodes which have a larger degree and have a proportion less
than some predetermined value ܥ to all the nodes in the network are called hubs. If node in a
community structure is assembled around some hubs it is called the hub community, otherwise, it
is called the non-hub community.
2.2. Divisive methods
The divisive methods split the graph into several communities by removing gradually the edges
which connect two distinct communities. Thus, the network is divided into several components
representing communities. This process of suppression of edges can be stopped at any stage
according to constraint used in each divisive method. In this section, we go on to describe the
various divisive methods using the main classification proposed in section (3).
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Figure 3. Community discovery methods taxonomy: Divisive methods
3.2.1 Stochastic Methods
Several community detection approaches have been developed dealing with stochastic methods
and perform a divisive process in order to well reveal communities.
3.2.1.1 Algorithms Based on a Dynamic Process
3.2.1.1.1 Random Walk Techniques
Zhou et al. [64] [65] introduced the concept of network random walking and distance measure (as
described in section 3.1.1.1.1). The Authors had proposed a divisive algorithm which based on
dissimilarity index between nearest neighbors of a network. Initially, the approach [65] considers
the whole network as just a single community. For each community, a threshold parameter is
introduced and takes the initial value upp (the upper dissimilarity threshold) of that community.
Repetitively, the algorithm carries out the necessary changes of the dissimilarity threshold and
examines all the edges in the community to see whether two nearest neighbors are friends; if

(i, j)  (see eq. 4) nodes i and j are marked as friends. Different friends’ sets are then formed.
A node which does not have any friends is moved to the friends set that has the strongest
interaction whit it. In each phase, the nodes of the community are distributed into a number of
disjoined communities. After all the communities are processed, the dendrogram is drawn to
show the relationship between the various communities as well as the upper and lower
dissimilarity thresholds of each community.
The search for the optimal modularity value is an NP-hard problem that means [8] that the space
of possible partitions grows faster than any power of the system size.
3.2.1.2.1 Extremal optimization
Duch et al. [46] proposed a heuristic search method to restrict the search space and find the
optimal modularity value. They consider that the total modularity Q is the sum of the local
modularity i q on each node such as i q is normalized in the interval 1,1 and defined
1 in case of self-edges). This approach defines a
12
k i
( ) a (i)
k
q
i    (8)
k
r
i
r
i
i
Where : i 
fitness of node is i , ki : is the degree of node i and kr (i) is the number of edges between
node i and the node which belongs to the same community r . Initially, the nodes of the network
are divided into two random partitions containing the same number of nodes. In each iteration, the
system self- organizes by moving the node that has the lower fitness to another partition and the
fitness must be recalculated. Therefore the edges between the partitions are removed and we
proceed recursively with each new resultant component. The process is repeated until the
modularity Q cannot be improved. The algorithm gives an optimal modularity Q which is better
than several existing algorithms of this kind and runs in O(n2 log n) but the final partition depends
closely on the initialization phase of the random partition in the network.
3.2.1.3 Algorithms based on statistic inference
3.2.1.3.1 Expectation maximization
Newman et al [47] have used the probabilistic mixture models and the expectation maximization
algorithm to inferring module assignments and to identifying the optimal number of modules in a
complex network by dividing its nodes into classes such that the members of each class have
similar patterns of connection to other nodes. However, expectation maximization algorithms are
known to converge to local maxima of the likelihood but not always to global maxima, and hence
it is possible to get different solutions from different starting nodes. The used method almost
converges to the same solution or to another similar one, whereas for others it is necessary to
perform several runs with different initial conditions to find a good maximum of the likelihood.
Ball et al. [55] have proposed a probabilistic model of link communities for detecting overlapping
and non overlapping communities. The type of links is a main factor in determining all
communities, i .e, it is the edges that are partitioned, where the expected number of edges of color
z that lie between nodes i and j is iz jz (or
2iz jz
generative model and uses an expectation-Maximization (EM) algorithm to find maximum log
likelihood, it solves the following equations

 (9)
( ) (10)
13
 
A q z
j ij ij
( )
iz A q z

ij ij ij
( )
The value of qij (z) in Eq. (10) is the probability that an edge between i and j has color z , which
is precisely the quantity needed in order to infer link communities in the network.
 
 
iz jz
z iz jz
qij z
 
The algorithm start with random initial conditions and iterate until (i)
iz converge. The approach
can be applying to networks of millions of nodes and gives results competitive with several
existing algorithms. Nonetheless, it doesn’t precise a criterion for determine the parameter k that
represent the number of communities in the network.
3.2.2 Deterministic Methods
3.2.2.1 Algorithms Based On Spectral Analysis
3.2.2.1.1 Spectral Graph Partitioning
Newman and Leicht [48] have proposed an extension of the spectral optimization method and
have adapted the modularity [13] to the oriented complex network. To rewrite the function of
modularity for oriented networks, Newman and Leicht [48] proceeded as follows: Consider two
nodes, i and j . Node i has high out-degree but low in-degree while node j has the reverse
situation. This means that a given edge is more likely to run from i to j than vice versa. Then the
probability that a link from a node i is directed to i is:
in
i where in
m
k k out
j
ki and out
k j are the in- and
out-degrees of the nodes. This proposal defines the equivalent of Eq.3 as follows
(11)


1    
Q  
,
ij
k k
ij m
A
m
 
The elements of modularity matrix are defined as
(12)
out
j
in
i
in
i
k k
m
B A
out
j
ij  ij 
To obtain the symmetric modularity matrix, Q is written as
(13)
c i c j
1 s B B s v s
m
Q  T  T  
T
( ) ( )2
4
i i
Where i is the eigenvalues of (B  BT ) corresponding to eigenvector vi , and si is +1 if node i is
assigned to community 1 and -1 if it is assigned to community 2. Then to divide the network into
two communities, we calculate the eigenvector corresponding to the largest positive eigenvalue of
the symmetric modularity matrix (B  BT ) and the communities are identified based on the signs of
the elements of the eigenvector. Thus to discover communities, a generalization of the modularity
matrix is

(14)
g
Bij B  B ( )
ij ij ik
Where B(g) is the modularity matrix of the subgraph. These spectral optimization methods extract
oriented information of links which result in identifying a significant community structure in time
O(n2 log n) .
In [56], the authors focus on the spectral properties of the adjacency and modularity matrices
using random matrix methods. This approach is built based on the stochastic block model and
find the spectrum of eigenvalues of the modularity matrix. The spectrum illustrate the presence of
a sharp transition between a regime in which there is a structure of community and a regime in
which there is none. The method is a poor method of community detection of real world networks
but it is an optimal method in the sense that no method can detect communities in the regime
where the modularity method fails [57].
3.2.2.2 Algorithms Based on Global, Local, or Clustering Proprieties
3.2.2.2.1 Algorithm Based on Local Proprieties
Shen et al. [50] have proposed a filtration recursive method using a random model for networks in
order to simultaneously carry out the suppression of several edges in each operation of filtration.
To quantify the quality of division, the algorithm apply a recursive community coefficient (CRC)
such as if the CRC of under network is smaller than that of its father network, then it consider
under network as an indivisible built local community. The method of Shen et al. [50] offers
complexity gain O(m2  (c  1)m) , for a network of m edges and c communities. Moreover, the method
can detect the local communities according to densities' of their external edges in increase order
in particular in the wide-area networks. Nevertheless, this method becomes slow and vague when
the density of edges between the communities is proximate to the density of the edges within the
communities.
The approaches proposed by Newman et al. [4] [8] are inspired from Freeman works [58]. The
intuitive design of a central point in the communication, which based on the structural property
betweeness, allows defining this point as being connecting between other points along their
shortest paths of communication [58]. Then, the betweeness centrality of a node i is the number of
the set of all geodesic paths that pass through i [9]. Newman et al. [8] defined three measures:
shortest-path betweeness, current-flow betweeness and random walk betweeness. The algorithm
can calculate the shortest paths between a particular pair of nodes using the breadth-first search in
time O(mn2 ) [59], [60]. Newman has proposed [61] a powerful algorithm which finds all edge
betweeness in time O(mn) . After that, the detection of the communities is carried out by the
removal of the edges with largest betweeness. The algorithm produces high quality partitions for
networks which have small size. However, at each step, when removing an edge, both version of
the shortest path betweeness algorithm and random walk betweeness algorithm update all
computations which is very expensive in computation and need to be carried out in O(n3 ) . The
current-flow betweeness algorithm runs in O(n4 ) on a sparse graph which makes it not convenient
for larger network.
Sales-Pardo et al [52] have observed that the hierarchical structure gives a very significant
knowledge of the dynamics of several complex networks such biological networks [62] [63],
determines the organization of complex systems, and extracts the relevant information at each
level. The authors of this paper employ the proximity concept in the hierarchy between all pairs
14

 
k g

of nodes, this measure, which called node affinity, based on Newman-Girvan modularity. First of
all, the algorithm performs the search of partitions that are local maxima in the modularity
landscape, next it find the affinity matrix using those partitions and their basin of attraction [64],
and ,at last, the hierarchical tree obtained by using box clustering method. This method reveals
the hierarchical organization of the network by the nested-box pattern along the diagonal of
affinity matrix which is block-diagonal. The algorithm gives meaningful partitions for some
social, technological and biological networks but it is quite slow.
Fortunato et al [51] have suggested a divisive algorithm that uses a centrality measure [65] which
is based on the concept of efficient propagation of information over the network. The efficiency
in the communication between two nodes i and j is equal to the inverse of the shortest path
length, and the average efficiency of the graph G is defined as the average of the individual
efficiencies over all n(n 1) ordered pairs of distinct nodes. Information centrality is the relative
drop in the network efficiency cause by removal of the edge from the graph G. The algorithm
finds and removes iteratively the edge with the highest information centrality. The major
drawback of this algorithm is the computational cost. Its running time is O(n4 ) . In [66], Fortunato
discussed in some detail the different centrality measure proposed in literature. Partitions obtained
with these techniques are consistent, mainly because information centrality has a strong
correlation with edge betweeness.
3.2.2.2.3 Algorithm based on clustering coefficient proprieties
Watts and Strogatz [67] have proposed a simple model which represents several proprieties in
social networks such as the clustering coefficient and the degree distribution. The clustering
coefficient quantifies how well connected are the neighbors of a node in a network. Radicchi et al
[12] have proposed a divisive algorithm for detecting community using the concept of edge
clustering coefficient. They define the edge clustering coefficient in analogy with the node
clustering coefficient, it explain the fraction between the number of triangles to which a given
edge belongs and the number of triangles that might potentially include it. Formally, for the edge-connecting
15
node i to node j , the edge-clustering coefficient is defined as
(15)
Where (3)
(3) 
1
i j
z
(3) .
,  
i j k k
min[( 1),( 1)]

i j
C
zij the number of triangles is built on the edge (i, j) and min[(ki 1), (k j 1)] is the maximal
possible number of these triangles. To address the problem of lacking of triangles in the network
structure, Radicchi et al [12] have defined cycles of a higher order g as
(16)
Where (g)
g
( )
z
g i j
i j s

g
i j
( )
,
( ) ,
,
1
C

zij is the number of cyclic structures of order g that contain the edge (i, j) and (g)
sij is the
number of all possible cyclic structures of order g . The algorithm removes at each step the
smallest edge clustering coefficient and each removal operation requires only a local update of
clustering coefficients, so the algorithm is much faster than several algorithms.
Lind et al [68] studied the clustering coefficient of bipartite networks in which there are no cycles
of size three, and therefore, the standard definition of clustering coefficient given in [69] cannot
be used. Thus, the coefficient is defined as the number of existing square over the total number of
all possible squares. Zhang et al. [49] have proposed a community detection algorithm for the

bipartite networks whose principle idea is to remove at each step the edge which have smallest
value of edge clustering coefficient. The authors define the coefficient LC4 and LC3 . The edge
clustering coefficient LC4 is defined as
LC (18)
16
LC q
iX
 (17)
iX k k k k
4, ( (2) (2) i  1)( X  1)  i 
X
Where qiX is the number of squares to which a given edge liX belongs. ki is the degree of node i
and (2)
ki is the number of the second neighbours of node i except the nodes which are the first
neighbors of node X. In bipartite networks, triples are the basic unit which includes two nodes
from the same set. It is the basic unit which gives the relationship of two nodes in the same set.
The clustering coefficient LC3 of edge liX is the average of edges similarity of all the triples to
which edge liX belongs. Here node i and X belong to different sets. It is given as
t
3,  
NX
mi
( )
2
1

iX    
K k t
   
2 2

k X i
m
k
N N X NX
m i mi
i X
t
k k t
k k
Where nodes m and i are from the same set and i and X are not in the same set. tmi is the number
of triples which contain node m and i . t NX is the number of triples which contain node N and X .
The divisive algorithm of bipartite structure produced a community structure more significant
than that produced by applying existing algorithms for non-bipartite networks. However, the
number of detected communities has to be determined in advance and there is not a stopping
criterion during the partition in communities.
4. CONCLUSIONS
The discovery of a significant structure of communities is an important mechanism for
understanding the structures and functions of complex networks. That’s why several meaningful
methods have been proposed in the literature. In this article, we have presented a survey on
community discovery methods. We have described some of the essential definitions and
techniques of community identification and have proposed taxonomy of them. For future work
we should applied community detection methods on different type of complex networks
according to their classification in the proposed taxonomy in order to conclude which class of
methods is most appropriate for each type of complex networks.
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Authors
Ahlem DRIF received an engineering degree in computer science from University of
Sétif in Algeria (UFAS) in 2002 and magister degree in computer science in 2006.
Currently, she is assistant professor and has 9 years teaching experience. She is a Ph.D.
student and a member at the Laboratory of Network and Distributed System (LRSD) at
the UFAS and has 10+ papers in international conferences. Her main research interests
lie in the areas of social networks, community discovery methods, dynamic topology,
ad hoc networks, power-aware protocols for wireless networks
Abdallah BOUKERRAM is professor of Computer Science at University of Bejaia
in Algeria. He obtained the PhD degree in Computer Science from University of Louis
Pasteur Strasbourg 1991 (France). He is director of Laboratory of Network and
Distributed System (LRSD) at the UFAS. His current research domains are social
networks, network security, distributed systems, and grid computing.

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