Generalized Louvain Method for Community Detection in Large Networks
Pasquale De Meo
∗
, Emilio Ferrara
∗∗
, Giacomo Fiumara
∗
, Alessandro Provetti
∗
∗
Dept. of Physics, Informatics Section.
∗∗
Dept. of Mathematics. University of Messina, Italy
{pdemeo, eferrara, gfiumara, ale}@unime.it
Abstract—In this paper we present a novel strategy to
discover the community structure of (possibly, large) networks.
This approach is based on the well-know concept of network
modularity optimization. To do so, our algorithm exploits
a novel measure of edge centrality, based on the κ-paths.
This technique allows to efficiently compute a edge ranking
in large networks in near linear time. Once the centrality
ranking is calculated, the algorithm computes the pairwise
proximity between nodes of the network. Finally, it discovers
the community structure adopting a strategy inspired by the
well-known state-of-the-art Louvain method (henceforth, LM),
efficiently maximizing the network modularity. The experi-
ments we carried out show that our algorithm outperforms
other techniques and slightly improves results of the original
LM, providing reliable results. Another advantage is that its
adoption is naturally extended even to unweighted networks,
differently with respect to the LM.
Keywords-complex networks; community structure
I. INTRODUCTION
The investigation of the community structure inside net-
works has acquired a great relevance during the last years,
in particular in the context of Social Network Analysis
(SNA). This, also because of the unpredicted success of
Online Social Networks (OSNs). In fact, social phenomena
such as Facebook and Twitter amongst others, glue together
millions of users under a unique network whose features
are a goldmine for Social Scientists. Several works are
focused on the Social Network analysis of these OSNs;
others describe the strategies of analysis themselves.
In this paper we focus on the possible strategies of com-
munity detection. As to date, two paradigms exist to discover
the community structure of a network. The former is based
on the analysis of the global features of the network, for
example its topology. These approaches are characterized by
high computational complexity and high quality results. The
latter paradigm relies on exploiting local information, for
example those acquirable by nodes and their neighborhoods.
The computational cost of these techniques is lower than
those exploiting global features, but the reliability decreases.
In this work, we propose a novel strategy to discover
the inner community structure of a network. The main
characteristics of our approach are the followings: i) it
exploits global information of the network, establishing
which are the edges of the network that contribute to the
creation of the community structure; ii) to do so, it adopts
a novel measure of edge centrality, in order to rank all
the edges of the network with respect to their proclivity
to propagate information through the network itself; iii)
its computational cost is low, making it feasible even for
large network analysis; iv) it is able to produce reliable
results, even if compared with those calculated by using
more complex techniques, when this is possible; in fact,
because of the computational constraints, the adoption of
some existing techniques is not viable when considering
large networks, and their application is only limited to small
case-studies.
This paper is organized as follows: in the next Section we
provide some background information about the community
detection problem. Section III introduces the main objectives
of this work and describes an intuitive sketch about the novel
strategy of community detection we propose. In Section IV
the key concept of κ-path edge centrality is recalled, being it
a novel and efficient strategy of ranking edges with respect
to their centrality in the network. All the pieces are glued
together in Section V. We describe our strategy to detect
the community structure, inspired by the well-known state-
of-the-art LM [1], which is computationally suitable even
when large networks are analyzed. Experiments that have
been carried out are discussed in Section VI. Finally, Section
VII concludes, depicting some future directions of research.
II. BACKGROUND
Several techniques to investigate the community structure
of networks have been proposed in literature during last
years. There exist numerous comprehensive surveys to this
problem, such as [2], [3].
In its general formulation, the problem of finding commu-
nities in a network is intended as a data clustering problem.
In fact, it could be solved assigning each node of the network
to a cluster, in a meaningful way. Two approaches have been
widely investigated, i) spectral clustering based techniques,
and, ii) network modularity optimization strategies. The
former relies on the optimization of the process of cutting
the graph representing the given network. The latter is based
on the maximization of a benefit function, called network
modularity. We briefly recall them, separately.
The problem of minimizing the number of cuts in a
given graph has been proved to be NP-hard. To do so,
different approximate techniques have been proposed. An
example is by using the spectral clustering [4], exploiting
the eigenvectors of the Laplacian matrix of the network.
We recall that the Laplacian matrix L of a given graph has
components L
ij
= k
i
δ(i, j) −A
ij
, where k
i
is the degree of
a node i, δ(i, j) is the Kronecker delta (that is, δ(i, j) = 1
if and only if i = j) and A
ij
is the adjacency matrix
representing the graph connections. Another approach relies
on the strategy of the ratio cut partitioning [5], [6]. This
is a function that, if minimized, allows the identification
of large clusters with a minimum number of outgoing
interconnections. The principal issue of spectral clustering
based techniques is that one has to know in advance the
number and the size of communities comprised in the given
network. This makes this strategy unfeasible if the purpose is
to discover the unknown community structure of a network.
The strategy exploited in this paper adopts the second
paradigm, the one relying on the concept of network modu-
larity. It can be explained as follows: let consider a network,
represented by means of a graph G = (V, E), partitioned
into m communities; assuming l
s
the number of edges
between nodes belonging to the s-th community and d
s
is
the sum of the degrees of the nodes in the s-th community,
the network modularity Q is given by
Q =
m
X
s=1
"
l
s
|E|
−
d
s
2|E|
2
#
(1)
Intuitively, high values of Q implies high values of l
s
for
each discovered community; thus, detected communities are
dense within their structure and weakly coupled among each
other. Equation 1 reveals a possible maximization strategy:
in order to increase the value of the first term (namely, the
coverage), the highest possible number of edges should fall
in each given community, whereas the minimization of the
second term is obtained by dividing the network in several
communities with small total degrees.
The problem of maximizing the network modularity has
been proved to be NP complete [7]. To this purpose, several
heuristic strategies to maximize the network modularity Q
have been proposed as to date. Probably, the most pop-
ular one is called Girvan-Newman strategy [8], [9]. This
approach works in two steps, i) ranking edges by using the
betweenness centrality as measure of importance; ii) deleting
edges in order of importance, evaluating the increase of the
value of Q. In fact, it is possible to maximize the network
modularity deleting edges with high value of betweenness
centrality, based on the intuition that they connect nodes be-
longing to different communities. The process iterates until
a significant increase of Q is obtained. At each iteration,
each connected component of S identifies a community.
Unfortunately, the computational cost of this strategy is very
high (i.e., O(n
3
), being n the number of nodes in S). This
makes it unsuitable for the analysis of large networks. The
largest part of its cost is given by the calculation of the
betweenness centrality, that is itself very costly (even if the
most efficient algorithm [10] is adopted).
Several variants of this strategy have been proposed during
the years, such as the fast clustering algorithm provided by
Clauset, Newman and Moore [11], that runs in O(n log n) on
sparse graphs; the extremal optimization method proposed
by Duch and Arenas [12], based on a fast agglomerative
approach, with O(n
2
log n) time complexity; the Newman-
Leicht [13] mixture model based on statistical inferences;
other maximization techniques by Newman [14] based on
eigenvectors and matrices.
The state-of-the-art technique is called Louvain method
(LM) [1]. This strategy is based on local information and
is well-suited for analyzing large weighted networks. It is
based on the two simple steps: i) each node is assigned
to a community chosen in order to maximize the network
modularity Q; the gain derived from moving a node i into
a community C can simply be calculated as [1]
∆Q =
P
C
+k
C
i
2m
−
P
ˆ
C
+k
i
2m
2
−
P
C
2m
−
P
ˆ
C
2m
2
−
k
i
2m
(2)
where
P
C
is the sum of the weights of the edges inside
C,
P
ˆ
C
is the sum of the weights of the edges incident
to nodes in C, k
i
is the sum of the weights of the edges
incident to node i, k
C
i
is the sum of the weights of the edges
from i to nodes in C, m is the sum of the weights of all
the edges in the network; ii) the second step simply makes a
new network consisting of nodes that are those communities
previously found. Then the process iterates until a significant
improvement of the network modularity is obtained.
In this paper we present an efficient community detection
algorithm which represents a generalization of the LM.
In fact, it can be applied even on unweighted networks
and, most importantly, it exploits both global and local
information. To make this possible, our strategy computes
the pairwise distance between nodes of the network. To do
so, edges are weighted by using a global feature which
represents their aptitude to propagate information through
the network. The edge weighting is based on the κ-path
edge centrality, a novel measure whose calculation requires
a near linear computational cost [15]. Thus, the partition of
the network is obtained improving the LM. Details of our
strategy are explained in the following.
III. DESIGN GOALS
In this Section we briefly and informally discuss the ideas
behind our strategy. First of all, we explain the principal
motivations that make our approach suitable, in particular
but not only, for the analysis of the community structure of
Social Networks. To this purpose, we introduce a real-life
example from which we infer some features of our approach.
Let consider a social network, in which users are con-
nected among them by friendship relations. In this context,
we can assume that one of the principal activities could be
exchanging information. Thus, let assume that a “message”
(that, could be, for example, a wall post on Facebook or a
tweet on Twitter) represents the simplest “piece” of informa-
tion and that users of this network could exchange messages,
by means of their connections. This means that a user could
both directly send and receive information only to/from the
people in her neighborhood. In fact, this assumption will be
fundamental (see further), in order to define the concepts of
community and community structure. Intuitively, say that a
community is defined as a group of individuals in which the
interconnections are denser than outside the group (in fact,
this maximizes the benefit function Q).
The aim of our community detection algorithm is to
identify the partitioning of the network in communities, such
that the network modularity is optimal. To do so, our strategy
is to rank links of the network on the basis of their aptitude
of favoring the diffusion of information. In detail, the higher
the ability of a node to propagate a message, the higher
its centrality in the network. This is important because, as
already proved by [8], [9], we could ensure that the higher
the centrality of a edge, the higher the probability that it
connects different communities.
Our algorithm adopts different optimizations in order to
efficiently compute the link ranking. Once we define an
optimized strategy for ranking links, we can compute the
pairwise distances between nodes and finally the partitioning
of the network, according to the LM. The evaluation of the
goodness of the partitioning in communities is attained by
adopting the measure of the network modularity Q.
In the next sections we shall discuss how our algorithm is
able to incorporate these requirements. First of all, in Section
IV, we formally provide a definition of centrality of edges
in social networks based on the propagation of messages by
using simple random walks of length at most κ (called, here-
after, κ-path edge centrality). Then, we provide a description
of an efficient implementation of this algorithm, running in
O(κ|E|), where |E| is the number of edges in the network.
After this, in Section V we discuss the technical details of
our community detection algorithm.
IV. κ-PATH EDGE CENTRALITY
The concept of κ-path edge centrality has been recently
defined [15] as follows:
Definition 1: (κ-path edge centrality) For each edge e of
a graph G = (V, E), the κ-path edge centrality L
κ
(e) of
e is defined as the sum, over all possible source nodes s,
of the percentage of times that a message originated from
s traverses e, assuming that the message traversals are only
along random simple paths of at most κ edges.
The κ-path edge centrality is formalized, for an arbitrary
edge e, as follows
L
κ
(e) =
X
s∈V
π
κ
s
(e)
π
κ
s
(3)
where s are all the possible source nodes, π
κ
s
(e) is the
number of κ-paths originating from s and traversing the edge
e and π
κ
s
is the number of κ-paths originating from s.
A. Fast κ-path Edge Centrality Algorithm
In this section we recall the functioning of the strategy
adopted to efficiently compute the κ-path edge centrality.
The proposed algorithm [15] is called Weighted Edge Ran-
dom Walk κ-Path Centrality (or, shortly, WERW-Kpath).
It consists of two main steps: i) node and edge weights
assignment, and ii) simulation of message propagations
using random simple paths of length at most κ. In the
following, the two steps are discussed separately.
1) Step 1: Node and edge weights assignment
In the first stage, the algorithm assigns a weight to both
nodes and edges of the graph G = (V, E) representing the
given network. Node weights are exploited to choose the
source nodes from which the simulation of the message
propagations starts; edge weights represent initial values of
centrality and they are updated during the execution of the
algorithm. At the end of the execution of ρ simulations,
where the optimal value ρ = |E| − 1 has been proved in
[15], edge weights are exploited for the edge ranking.
To compute node weights, we recall the notion of local
effective density δ(v) of a node v, as follows:
Definition 2: Local effective density Given a graph G =
(V, E) and a node v, its local effective density δ(v) is
δ(v) =
|I(v)| + |O(v)|
2|E|
where I(v) and O(v) represent,
respectively, the number of ingoing and outgoing edges
incident on the node v.
This value intuitively represents how much a node con-
tributes to the overall connectivity of the graph. The higher
δ(v), the better v is connected in the graph.
As for edge weights, we recall the following definition:
Definition 3: Initial edge weight Given a graph G =
(V, E) and an edge e, its initial edge weight ω(e)
0
is
ω(e)
0
=
1
|E|
where |E| is the cardinality of E.
Intuitively, we initially manage a “budget” consisting of
|E| points; these points are equally divided among all the
nodes; the amount of points received by each edge represents
its initial rank.
2) Step 2: Simulation of message propagations
In the second step we simulate ρ simple random walks of
length at most κ on the network. In detail, at each iteration,
WERW-Kpath (Algorithm 1) performs these operations:
1) A node v of the graph G is selected with a probability
proportional to its local effective density δ(v)
P (v) =
δ(v)
φ
(4)
where φ =
X
v∈V
δ(v) is a normalization factor.
2) All the edges in G are marked as not traversed.
3) The procedure MessagePropagation is invoked.
Algorithm 1 WERW-Kpath(Graph G = (V, E), int κ)
1: Assign each node v: δ(v) =
|I(v)|+|O(v)|
|E|
2: Assign each edge e: ω(e) =
1
|E|
3: ρ ← |E| − 1
4: for i = 1 to ρ do
5: N ← 0 a counter to check the length of the κ-path
6: v ← a node chosen according to Eq. 4
7: MessagePropagation(v, N , κ)
Let us describe the procedure MessagePropagation (Al-
gorithm 2). This procedure carries out a loop until both the
following conditions hold true:
• The length of the path currently generated is no greater
than κ. This is managed through a length counter N .
• Assuming that the walk has reached the node v
n
, there
must exist at least an outgoing edge from v
n
which
has not been already traversed. In detail, we attached a
flag T (e) to each edge e; T (e) = 1 if the edge e has
already been traversed, 0 otherwise. If we call O(v
n
)
the set of outgoing edges from v
n
, it must hold that
|O(v
n
)| 6=
X
e∈O(v
n
)
T (e).
The former condition allows us to consider only paths up
to length κ. The latter condition, instead, avoids that the
message get trapped into a cycle.
If the conditions above are satisfied, the MessageProp-
agation procedure selects an edge e
n
with a probability
proportional to the edge weight ω(e
n
), given by
P (e
n
) =
ω(e
n
)
γ
(5)
where γ =
X
e
n
∈
ˆ
O(v
n
)
ω(e
n
) is a normalization factor, being
ˆ
O(v
n
) = {e
n
∈ O(v
n
) | T (e
n
) = 0}.
Let e
n
be the selected edge and let v
n+1
be the node
reached from v
n
by means of e
n
. The MessagePropagation
procedure awards a bonus (equal to β =
1
|E|
) to e
n
, sets
T (e
n
) = 1 and increases the counter N by 1. The message
propagation activity continues from v
n+1
.
At the end of all the processes of simulation of message
propagation, each edge e ∈ E is assigned a centrality value
L
κ
(e) (in the interval [
1
|E|
, 1]) equal to its final weight ω(e).
The time complexity of this algorithm is O(κ|E|). Our
community detection strategy, described in the following,
adopts this algorithm to weight edges of the network.
V. COMMUNITY STRUCTURE DETECTION
In the following, we present a novel algorithm to calculate
the community structure of a network. It is baptized Fast κ-
path Community Detection (or, shortly, FKCD). The strategy
Algorithm 2 MessagePropagation(Node v, int N , int κ)
1: while N < κ and
|O(v)| 6=
P
e∈O(v)
T (e)
do
2: e
vw
← e ∈ {O(v) | T (e) = 0}, according to Eq. 5
3: ω(e
vw
) ← ω(e
vw
) +
1
|E|
4: T (e
vw
) ← 1; v ← w; N ← N + 1
relies on three steps: i) ranking edges by using the WERW-
Kpath algorithm; ii) calculating the proximity (the inverse
of the distance) between each pair of connected nodes; ii)
partitioning the network into communities so to optimize
the network modularity [8], according to the LM [1]. The
algorithm is discussed as follows.
A. Fast κ-path Community Detection
First of all, our Fast κ-path Community Detection (hence-
forth, FKCD) needs a ranking criterion to compute the
aptitude of all the edges to propagate information through
the network. To do so, FKCD invokes the WERW-Kpath
algorithm, previously described. Once all the edges have
been labeled with their κ-path edge centrality, a ranking
in decreasing order of centrality could be obtained. This is
not fundamental, but could be useful in some applications.
Similarly, before to proceed, a first network modularity
esteem (hereafter, Q) could be calculated. This could help
in order to put into evidence how Q increases during next
steps. With respect to Q, we recall that its value ranges in the
interval [0, 1] and, the higher Q, the better the community
structure of the network appears evident. The computational
cost of this first step is O(κ|E|), with κ length of the κ-paths
and |E| cardinality of E.
The second step consists in calculating the proximity
among each pair of connected nodes. This is done by using
a L
2
distance (i.e., the Euclidean distance) calculated as
r
ij
=
v
u
u
t
n
X
k=1
(L
κ
(e
ik
) − L
κ
(e
kj
))
2
d(k)
(6)
where L
κ
(e
ik
) (resp., L
κ
(e
kj
)) is the κ-path edge cen-
trality of the edge e
ik
(resp., e
kj
) and d(k) is the degree
of the node. We put into evidence that, even though the L
2
measure would return a distance, in our case, the higher
L
κ
(e
ik
) (resp., L
κ
(e
kj
)), the more the nodes are near,
instead of distant. This important aspect leads us to consider
the results of Equation 6 as the pairwise proximities of
nodes. This step is theoretically computationally expensive,
because it should require O(|V |
2
) iterations, but in practice,
by adopting optimization techniques, its near linear cost is
O(d(v)|V |), where d(v) is the mean degree of all the nodes
of the network (and it is usually small in Social Networks).
The last step is the network partitioning. The main idea is
inspired by the LM [1] for detecting the community structure
of weighted networks in near linear time. The partitioning
is an iterative process. At each iteration, two simple steps
occur: i) each node is assigned to a community chosen in
order to maximize the network modularity Q; the possible
increase of Q derived from eventually moving a node i into
a community C is calculated according with Equation 2; ii)
the second step produces a meta-network whose nodes are
those communities previously found. The partitioning ends
when no further improvements of Q can be obtained.
This reflects in splitting communities connected by edges
with high proximity, which is a global feature, thus maxi-
mizing the network modularity. Its cost is O(γ|V |), where
|V | is the cardinality of V and γ is the number of iterations
required by the algorithm to converge (in our experience,
usually, γ < 5). The FKCD is schematized in Algorithm 3.
We recall that CalculateDistance computes the pairwise
node distance by using Equation 6, Partition extracts the
communities according to the LM descripted above and Net-
workModularity calculates the value of network modularity
by using Equation 1.
The computational cost of our strategy is near linear. In
fact, O(κ|E|+d(e)|V |+γ|V |) = O(Γ|E|), by adopting effi-
cient graph memorization in order to minimize the execution
time for the computation of Equations 1 and 6.
Algorithm 3 FKCD(Graph G = (V, E), int κ)
1: WERW-Kpath(G, κ)
2: CalculateDistance(G)
3: while Q increases at least of (arbitrarily small) do
4: P = Partition(G)
5: Q ← NetworkModularity(P)
VI. EXPERIMENTAL RESULTS
Our experimentation has been conducted both on synthetic
and real-world online social networks, whose datasets are
available online. All the experiments have been carried out
by using a standard Personal Computer equipped with a Intel
i5 Processor with 4 GB of RAM.
A. Synthetic Networks
The method proposed to evaluate the quality of the
community structure detected by using the FKCD exploits
the technique presented by Lancichinetti et al. [16]. We
generated the same synthetic networks reported in [16],
adopting the following configuration: i) N = 1000 nodes;
ii) the four pairs of networks identified by (γ, β) =
(2, 1), (2, 2), (3, 1), (3, 2), where γ represents the exponent
of the power law distribution of node degrees, β the expo-
nent of the power law distribution of the community sizes;
iii) for each pair of exponents, three values of average degree
hki = 15, 20, 25; iv) for each of the combinations above,
we generated six networks by varying the mixing parameter
µ = 0.1, 0.2, . . . , 0.6.
Figure 1 highlights the quality of the obtained results.
The measure adopted is the normalized mutual information
[17]. Values obtained put into evidence that our strategy
performs fair good results, avoiding the well-known effect
due to the resolution limit of the modularity optimization
[18]. Moreover, a classification of results as in Table I
(discussed later) is omitted because values of Q obtained
by using FKCD and the LM in the case of these quite small
synthetic networks are very similar.
Figure 1. Test of modularity optimization using the benchmark [16], for N
= 1000 nodes. The threshold value µ = 0.5 represents the border beyond
which communities are no longer defined in the strong sense, i.e., each
node has more neighbors in its own community than in the others [19].
B. Real-world Networks
Results obtained by analyzing several real-world networks
[20], [21] are summarized in Table I. This experimentation
has been carried out to qualitatively analyze the performance
of our strategy. Obtained results, measured by means of the
network modularity calculated by our algorithm (FKCD), are
compared against those obtained by using the original LM.
Our analysis puts into evidence the following obser-
vations: i) classic not optimized algorithms (for example
Girvan-Newman [8]) are unfeasible for large network anal-
ysis; ii) results obtained by using LM are slightly higher
than those obtained by using FKCD; on the other hand, LM
adopts local information in order to optimize the network
modularity, while our strategy exploits both local and global
information; this results in (possibly) more convenient iden-
tified community structures for some applications; iii) the
performance of FKCD slightly increase by using longer κ-
paths; iv) both the compared efficient strategies are feasible
even if analyzing large networks using standard resources
of calculus (i.e., a classic personal computer); this aspect
is important if we consider that there exist several Social
Network Analysis tools (e.g., NodeXL
1
) that require opti-
1
http://nodexl.codeplex.com/
![Figure 1. Test of modularity optimization using the benchmark [16], for N = 1000 nodes. The threshold value µ = 0.5 represents the border beyond which communities are no longer defined in the strong sense, i.e., each node has more neighbors in its own community than in the others [19].](/figures/figure-1-test-of-modularity-optimization-using-the-benchmark-3lby98ku.webp)