Combinations of Affinity Functions for Different Community Detection Algorithms in Social Networks
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Combinations of Affinity Functions for Different Community Detection
Algorithms in Social Networks
Javier Fumanal-Idocin1 , Oscar Cordón2 , Marı́a Minárová3 , Amparo Alonso-Betanzos4 , Humberto Bustince1
1
Public University of Navarra, Pamplona, Spain, 2 Slovak University of Technology in Bratislava, Bratislava, Slovakia
3
University of A Coruña, A Coruña. Spain 4 University of Granada, Granada, Spain
javier.fumanal@unavarra.es
Abstract [12], a protein complex [13], and to discover routing
arXiv:2208.12874v1 [cs.SI] 26 Aug 2022
strategies [14]. Many algorithms have been proposed
Social network analysis is a popular discipline to solve community detection, a great deal of them
among the social and behavioural sciences, in which using the concept of modularity to measure how good
the relationships between different social entities are the sub-partitions detected [15]. The most popular
are modelled as a network. One of the most community detection algorithm is the Lovaine algorithm
popular problems in social network analysis is finding [16], which uses changes in modularity to obtain the
communities in its network structure. Usually, optimal partition. Other popular algorithms that use the
a community in a social network is a functional idea of modularity are the proposal in [15], which is a
sub-partition of the graph. However, as the definition greedy algorithm designed to quickly reach a solution,
of community is somewhat imprecise, many algorithms and the proposal in [17], in which the authors propose
have been proposed to solve this task, each of them to iteratively eliminate edges in a network using the idea
focusing on different social characteristics of the actors of edge betweenness. Further approaches include using
and the communities. In this work we propose to deep learning to generate an embedding of the graph
use novel combinations of affinity functions, which are and a reconstructed adjacency matrix in order to obtain
designed to capture different social mechanics in the a representation where the community structure can be
network interactions. We use them to extend already seen clearly [18]; spreading the labels in the network as
existing community detection algorithms in order to if it was an epidemic [19]; or using an equation, called
combine the capacity of the affinity functions to model the map equation, to model the information flows of the
different social interactions than those exploited by the network [20].
original algorithms.
Another proposal that does not include a modularity
optimization process is [21], in which the authors
1. Introduction propose to use a new kind of functions, the affinity
functions, that model the local interactions between
Social network analysis has become an important actors according to different social behaviours. They do
part of many disciplines such as physics [1], public so in order to propose a community detection algorithm,
health and administration [2, 3], and biology [4]. the Borgia Clustering, that simulates the dynamics
The idea behind studying complex systems using this of the conquests of Cesare Borgia in the Italian
technique is that the elements composing such systems Renaissance. However, the possibilities of using affinity
are part of multi-lateral interactions and processes, functions with other community detection algorithms
which affect deeply the system and each of its were not studied. This possibility is of spatial relevance
composing individuals. Many problems in computer because affinity functions could improve the network
science have been tackled using social network analysis features that some community detection algorithms
[5]. This is the case of information propagation [6], exploit. In fact, some algorithms compute different
leader detection [7], recommendation systems [8, 9] and representations of the network using computationally
decision making [10, 11]. One of the most popular expensive processes, like a convolution [18]. Affinity
problems of this kind is community detection. functions could work in a similar way in those algorithm
A community in a social network is a group of nodes that do not compute other representations for the
that present a significant interaction among them, and original network. Besides, although some combination
much less with the rest of the network. Community of affinity functions were proposed in [21], some
detection has been used to identify groups of friends mathematical properties of this combination could berelaxed in order to explore further results. • Best friend affinity: it measures the importance of
Taking these considerations into account, in this a relationship with an actor y for the actor x, in
work we aim to: relation to all the other relationships of x:
1. Extend the convex combination of affinity C(x, y)
functions to a more general expression. FCBF (x, y) = P (1)
a∈V C(x, a)
2. Explore the possibilities of using affinity
functions with community detection algorithms • Best common friend affinity: it measures the
other than the Borgia Clustering. importance of the relationship taking into account
how important are the common connections
In order to achieve our aims, we use different between the connected nodes to x and y, in
combinations of affinity functions without restricting relation to all other relationships of x in the
them to be convex combinations, and then we study network:
these combinations with three of the most popular
modularity-based community detection algorithms: maxa∈V {min{C(x, a), C(y, a)}}
FCBCF (x, y) = P
the Lovaine algorithm [16], the Greedy modularity a∈V C(x, a)
algorithm [15] and Girvan-Newman algorithm the [17], (2)
and compare the results with the Borgia Clustering
algorithm. • Machiavelli affinity: it computes how affine two
The rest of the work follows this structure: in actors x and y are based on how similar is the
Section 2 we describe the affinity functions used social structure that surrounds them:
and how we combine them, and we also discuss
the community detection algorithms used in our | Ix − Iy |
FCM ach (x, y) = 1 − , (3)
experimentation. In Section 3 we discuss the datasets max{Ix , Iy }
used in our experimentation and the results obtained. P
Finally, in Section 4 we draw our conclusions for this where Ia = z∈Z D(z), where Z is the set of
work and the future lines for our research. actors where C(a, z) > 0, ∀z ∈ Z, and D(z) is
the centrality degree of z.
2. Methods
Regarding the interpretation of the affinity value
In this section we recall the notion of affinity obtained, a 0 value means that no affinity has been
function, the functions studied and the algorithms tested found at all while an 1 value means that there is a
to perform community detection. perfect match according to the analyzed factors Since
affinities are not necessarily symmetrical, the strength of
2.1. Affinity functions this interaction depends on who the sender and receiver
are, as it happens in human interactions. We denote a
Affinity functions were proposed in [21] to measure network resulting from calculating all its edges with an
the strength of the relationship between a pair of actors affinity function an “affinity network”.
in a social network by capturing the nature of their local There are two different kinds of affinity functions:
interactions. They are defined as functions that take as personal and structural affinities. Personal affinities
input two different actors, x and y, and return a number are computed using the edges of the x, y actors, while
in the [0, 1] interval: the structural affinities use different centrality measures
of these actors. Two examples of personal affinity
FC : (x, y) → [0, 1] functions are the best friend and best common friend
affinities. The Machiavelli affinity is an example of a
where C is the adjacency matrix whose each entry structural affinity.
C(x, y) quantifies the relationship for the pair of actors Depending on the affinity function computed, the
x, y in a weighted network. The affinity value will resulting affinity network will have different structural
be higher or lower depending on which aspect of the properties. For example, for the case of the ratio
relationship we are taking into account, like the number of connections in the network with the respect to
of friends in common or the relative strength of the all the potential ones (commonly called density): the
interaction between x and y. Some affinity functions Machiavelli affinity will result in a network with density
are: equal to 1; the best friend affinity will preserve thesame density as in the adjacency network; and the best with α ∈ [0, 1], for the linear combination:
common friend affinity tends to increase the density with
respect to the adjacency network. FCLC = αFCBF (x, y) + βFCBCF (x, y) (5)
More affinity functions and further details about
with α, β ∈ [0, 1].
them can be found in [21].
However Eq. (4) might not be always an affinity
function, so we establish the constraint that:
2.2. Combinations of affinity functions
α + β 0 (7)
two affinity functions is also a valid affinity function.
However, some combinations of affinity functions In this way, if α + β = 1 we recover a convex
that are not convex are also affinity functions. As combination, and if not, we are obtaining a value lesser
long as we guarantee that the output is inside the than the one obtained using a convex combination, so it
[0, 1] interval, the resulting value is a valid affinity will never be above 1. We denote the expression Eq. (5),
function. We are interested in these expressions because as long as it holds Eq. (6) and Eq. (7), a less than-convex
some affinity functions tend to have a higher average combination.
value than others. Combining affinity functions can be It is straightforward that the less than-convex
problematic because the values of one of the functions combination of two affinity functions is another valid
can be irrelevant compared to the other in most cases. affinity function, because it will always be lower than
One possible solution to this problem is to extremely the convex combination of affinity functions (which is a
skew the value of the α to favour the irrelevant affinity. valid affinity function), and always bigger than 0.
However, this solution presents two problems:
2.3. Community detection algorithms
• This solution results not only in decreasing one
For our experimentation we have studied three
affinity, it also increases the other, which is not
very popular community detection algorithms based on
an indented behaviour. Also, finding the proper
different modularity optimization processes:
equilibrium between the mixing parameters can
require an expensive fine-tuning. • Lovaine community detection algorithm [16]:
this method optimizes the modularity of the
• The interpretability of this solution is community structure found as the algorithm
counter-intuitive. If one of the mixing parameters progresses. The algorithm is divided in two
is much higher than the other, we can interpret phases: (1) First, small communities are found
that one affinity function is much important than by optimizing modularity locally on each node.
the other in our analysis. However, this would not We do so by computing the changes in modularity
be the case, as we did not increase on purpose that when each node i is joined in the same community
parameter, we only tried to decrease the average as each of their neighbours, and choosing the
value of the other affinity function. biggest positive change in modularity for each
node. We do so for each node until no positive
Using a non-convex combination, might solve both change in modularity is possible. (2) Then, each
of these problems. By simply reducing one mixing of these communities is grouped into one node
parameter without affecting the other, we are able to and the process repeats iteratively.
scale both affinity functions to an appropriate average
value without performing an expensive fine-tuning or • Girvan-Newman community detection algorithm
affecting the interpretability of the combination. [17]: this algorithm progressively removes edges
To do so, we start by substituting the expression of a from the original network based on the idea
convex combination of two affinity functions: of how likely an edge is to be a bridge
between communities, which is called “edge
betweenness”. The algorithm repeats iteratively
FCCC = αFCBF (x, y) + (1 − α)FCBCF (x, y) (4) this process until there are no more edges: (1) Wecompute all the edge betweenness. (2) We remove 3.1. Performance metrics
the one with the highest value. Finally, we obtain
a dendrogram where the leaves are the individual Normalized Mutual Information (NMI) is a
nodes. normalization of the Mutual Information score to scale
the results between 0 (no mutual information) and 1
• Greedy modularity community detection (perfect correlation). We have used the NMI to evaluate
algorithm [15]: is a greedy algorithm designed the results obtained in community detection against
to reach a solution in few steps. It works by ground truth labels.
optimizing the modularity value of each partition 3.1.1. Normalized Mutual Information Given the
in the graph. Each node starts as the sole member conditional entropy of two populations:
of its community. Then, we select the edge from
the original graph that has the highest positive XX
modularity change. We do so iteratively, until H (Y |X) = − p (x, y) log p (y|x) (8)
x y
only all the nodes are connected. Then, we
choose the partition with better modularity as the
final result. then, the mutual information of two random discrete
variables can be defined as:
We chose these algorithms as they are very popular,
and the concept of modularity is one of the most popular I(X; Y ) = H(X) − H(X|Y ), (9)
ones in community detection in social network analysis.
The procedure to use them with affinity functions is the which measures the mutual dependence between X and
same for the three algorithms: we compute the affinity Y . So, the final expression for the NMI is:
function or the combination of affinity functions on the
original adjacency matrix, and then, we compute the 2I (X; Y )
N M I (X, Y ) = (10)
algorithm on the resulting affinity network. H (X) + H (Y )
For the sake of comparison, we have also compared
the results obtained with these algorithms with those 3.1.2. Modularity Modularity is a metric of the
obtained by the Borgia Clustering [21]. The Borgia structure of networks that measures the strength of
Clustering is a community detection algorithm that partition into different communities [26]. Networks
generalizes the gravitational clustering algorithm [22] with high modularity have dense connections between
using affinity functions, among other changes. This the nodes within communities but scarce connections
algorithm was explicitly designed to work with affinity between nodes in different ones.
functions, so we think that it is interesting to compare The expression of modularity, Q, is the following:
the results obtained using this algorithm with the rest
of the community detection algorithms using affinity c
X
functions. Q= (eii − a2i ) (11)
i=1
3. Experimental results
where c is the number of communities, where eu,v is the
In this section we have compared the results fraction of edge with one end in a community, u, and
performing community detection in three datasets: another end in community v, and a is the fraction of
edges with one end in the i-th community.
• Zachary Karate Club [23]: is a social network of
a university karate club that consists of 34 people 3.2. The effects of different affinity
and how they split into two communities. combinations in the networks
• Dolphin [24]: a social network of bottlenose
In this subsection we have tested the effects in each
dolphins, where each link represents the
dataset using different affinity functions compared to
frequency in which they played.
the original adjacency matrix. We expect the best
• Polbooks [25]: nodes represent books about friend affinity not to alter the degree in each node, but
US politics sold by Amazon. Edges represent other centrality measures may be affected. And for the
frequent co-purchasing of books by the same best common friend affinity, we expect it to increase
buyers, as indicated by the ”customers who the degree in each node, which might reduce other
bought this book also bought these other books” centrality measures, specially the betweenness, because
feature on Amazon. more edges will be present in the network.An example of three different affinity functions 3.3. Community detection experiments
applied to the Zachary karate network is shown in
Figure 1. We can see that adding the best common In this subsection we have evaluated the results
friend affinity did increase the number of edges and the obtained using the combinations of affinity functions
network, and the role that some actors present in the with community detection algorithms other than the
network varies significantly depending on the affinity Borgia Clustering, and then we have compared them
function used. We expect the results in community with those obtained using this algorithm. We use the
detection to change considerably since the centrality Modularity as internal index and the NMI as an external
measures of some actors change significantly. Specially one to evaluate our results.
relevant are the changes of actors “34” or “2”. It is In Table 1 we have displayed the results obtained for
also noteworthy how some actors do not change their each dataset and algorithm using the original adjacency
role in their network independently of the affinity used, matrix. We can see that the Greedy Modularity
like “17” or “8”. In this particular case, we can see algorithm obtained the best average results in the
that even though there are clearly two communities NMI index and the Lovaine algorithm did so for the
in three affinity networks, the border between them Modularity index.
varies significantly between them since some actors in We have studied how these indexes change when
between the communities seem to differ their position we apply the three algorithms using the affinity matrix
remarkably in the different affinity networks. resulting when using the combinations proposed in Eq.
(5) instead of the original adjacency matrix. We have
17
4 13
6
7
reported the modularity results in Figure 2 for the
11
8 22 5 modularity index, and in Figure 3 for the NMI index.
25
32 3 18 12
26
2
1 A summary of the best results can be found
28 29 20 14 in Table 2, alongside the results of the Borgia
24 10
9
Clustering for the same trials. We found that in most
34
31
30 27 15 cases the best friend affinity obtains the best results,
23 16
19 21
33
specially in the case of the Greedy Modularity and the
a) α = 0.25, β = 0.50 Girvan-Newman algorithms. However, in the case of the
1 Louvain algorithm, non-convex combinations of affinity
5
17
functions resulted best for the NMI index. Compared
2
22 12
18
11 6 to the results obtained using the adjacency matrix,
13
14 8 7 NMI index decreased dramatically for the Dolphin and
10 31 20 4 Polbooks datasets, but modularity seemed to improve in
9
23 most cases, specially for the Louvain algorithm.
27 21 15 28
16 19 32 3 All the studied algorithms seemed to obtain good
29
30
24
modularity results, both in the case of the adjacency
34 33 and the affinity matrices. In fact, these algorithms
26
25
seemed to over perform the Borgia Clustering in terms
b) α = 0.50, β = 0.10 of the modularity index, although the Borgia Clustering
6
presented much higher NMI values than the rest of
17
13
4 11
7
5
9 14 8 1 12 the algorithms, probably because the Borgia Clustering
31 2 22
19
15 3
18
computed correctly the number of real communities for
10 20
21
16
33
34 each dataset.
23 29
27
32
30
24
28
25
4. Results and Discussion
26
c) α = 0.75, β = 0.25
In this experimentation we studied the performance
Figure 1. Visual representation of the Zachary of different community detection algorithms using less
Karate club Network using three different than-convex combinations of affinity functions, in three
combinations of affinities, using the parameters different datasets. We found satisfactory results for the
indicated for each one. We can observe that two modularity metric in all datasets for all the algorithms
communities are clearly present in the three networks, tested, and for the two metrics studied in the Zachary
although the frontier between both is more clearly karate network. Also, we found the best friend affinity
present in (b) compared to (a) and (c). to be the most effective affinity function to performGreedy Modularity Girvan-Newman Louvain
value value value
value
value
value
Zachary
value value value
value
value
value
Dolphin
value value
0.0 0.1 0.25 0.5 0.75 0.9 1.0 0.0 0.1 0.25 0.5 0.75 0.9 1.0
value
0.0 0 0 0.067 0.19 0.28 0.31 0.34 0.0 0 0.55 0.55 0.55 0.55 0.55 0.55 0.5
0.30
0.1 0.069 0.054 0.039 0.026 0.024 0.021 0 0.1 0.42 0.42 0.42 0.42 0.42 0.42 0
0.25 0.4
0.25 0.07 0.064 0.057 0.047 0.041 0 0 0.25 0.42 0.42 0.42 0.42 0.42 0 0
0.20
0.3
value
value
0.5 0.07 0.068 0.064 0.058 0 0 0 0.5 0.42 0.42 0.42 0.42 0 0 0
value
0.15
0.75 0.073 0.072 0.069 0 0 0 0 0.75 0.42 0.42 0.42 0 0 0 0 0.2
0.10
0.9 0.074 0.073 0 0 0 0 0 0.9 0.42 0.42 0 0 0 0 0 0.1
0.05
1.0 0.075 0 0 0 0 0 0 1.0 0.42 0 0 0 0 0 0
0.00 0.0
Polbooks
Figure 2. Modularity results for the community detection algorithms using affinity networks. Best friend affinity
overperforms the combination with the best friend affinity in all cases. Results are generally better than those
obtained with the adjacency matrix according to this metric.
community detection. for sure affect the performance and execution time of the
The Louvain algorithm seems to be the most Girvan-Newman algorithm, that needs to recompute the
successful classical algorithm when using affinity edge betweenness in each iteration.
functions, according to the modularity index. The
Girvan-Newman algorithm also obtained a moderate 5. Conclusions
performance increase. On the contrary, the Greedy
modularity algorithm performed better with the original In this work we have proposed a new method
adjacency networks compared to the affinity functions to construct affinity functions using non-convex
tested. None of the studied algorithms performed better combinations of affinity functions. We have explained
than the Borgia Clustering in terms of real world labels, how these combinations work and how some centrality
but they did in terms of modularity. measures change based on the mixing factors of the
A possible explanation for the disparity between combination. Besides, we have used the affinity
the modularity and NMI indexes performance could be functions for the first time in classical community
due to the social interactions exploited by the affinity detection algorithms designed to work with the
functions, which could be different from the dynamics adjacency matrix. In order to test our proposal,
registered by the ground truth labels. Besides, the we have computed a series of experiments using
behaviour of some of these algorithms can be affected different combinations of affinity functions, and we have
dramatically by the affinity function chosen. For compared the best results obtained with the classical
example, the best common friend affinity leads to an algorithms with the Borgia Clustering.
increase in the number of edges in the network. This will We found that the use of affinity functions can haveGreedy Modularity Girvan-Newman Louvain
value value value
value
value
value
Zachary
value value value
value
value
value
Dolphin
value value
0.0 0.1 0.25 0.5 0.75 0.9 1.0 0.0 0.1 0.25 0.5 0.75 0.9 1.0
0.014 value
0.0 0 0.0022 0.012 0 0 0.00130.0086 0.0 0 0.014 0.014 0.014 0.014 0.014 0.014
0.010 0.012
0.1 0 0 0 0 0.00220.0022 0 0.1 0 0 0 0 0 0 0
0.008 0.010
0.25 0.00220.00220.00220.00220.0022 0 0 0.25 0 0 0 0 0 0 0
0.008
value
value
0.5 0.0022 0 0 0 0 0 0 0.006 0.5 0 0 0 0 0 0 0
value
0.006
0.75 0.0033 0 0 0 0 0 0 0.004 0.75 0 0 0 0 0 0 0
0.004
0.9 0.00540.0011 0 0 0 0 0 0.9 0 0 0 0 0 0 0
0.002 0.002
1.0 0.0022 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0
0.000 0.000
Polbooks
Figure 3. NMI results for the community detection algorithms usgin affinity networks. We found good results for
the Zachary karate network dataset, but underwhelming results for the Dolphin and Polbooks in all cases.
a positive impact on the modularity metric performance, 6. Acknowledgments
although evaluation with ground truth labels decreased
in most cases. However, in the case of the Borgia Javier Fumanal Idocin and Humberto
Clustering we found the opposite results. The ground Bustince’s re-search has been supported
truth labels evaluation was in general terms better than by the project PID2019-108392GBI00
the rest of the algorithms but the modularity values were (AEI/10.13039/501100011033).
lower. Maria Minárová research has been funded by the project
work was supported by the projects APVV-17-0066 and
APVV-18-0052.
Oscar Cordón’s research was supported by the Spanish
Future research shall study the relationship between Ministry of Science, Innovation and Universities under
the Borgia Clustering and the rest of the community grant EXASOCO (PGC2018-101216-B-I00), including,
detection algorithms. As the Borgia Clustering is European Regional Development Funds (ERDF).
very expensive to compute in large networks, we are
particularly interested in a possible hybrid algorithm References
that combines the Louvaine algorithm and the Borgia
Clustering, in order to surpass some of the limitations of [1] M. Newman, “The physics of networks,” Physics today,
vol. 61, no. 11, pp. 33–38, 2008.
the latter in terms of memory and execution time. Also,
we shall research the interaction of affinity functions [2] T. W. Valente, Social networks and health: Models,
methods, and applications, vol. 1. Oxford University
with specific community detection algorithms in order to Press New York, 2010.
develop specific functions to enhance the performance [3] A. P. Chaves, I. Steinmacher, and V. Vieira, “Social
for each individual algorithm. networks and collective intelligence applied to publicDataset Algorithm NMI Mod. [10] H. Zhang, F. Wang, Y. Dong, F. Chiclana, and
E. Herrera-Viedma, “Social trust-driven consensus
Greedy Mod. 0.45 0.38 reaching model with a minimum adjustment feedback
Zachary Girvan-Newman 0.35 0.41 mechanism considering assessments-modifications
willingness,” IEEE Transactions on Fuzzy Systems,
Lovaine 0.35 0.41 2021.
Greedy Mod. 0.41 0.49 [11] E. Herrera-Viedma, F. J. Cabrerizo, F. Chiclana, J. Wu,
M. J. Cobo, and S. Konstantin, “Consensus in group
Dolphin Girvan-Newman 0.44 0.51 decision making and social networks,” Studies in
Lovaine 0.33 0.51 Informatics and Control, vol. 26(3), pp. 259–268, 2017.
[12] S. Papadopoulos, Y. Kompatsiaris, A. Vakali, and
Greedy Mod. 0.49 0.50 P. Spyridonos, “Community detection in social media,”
Polbooks Girvan-Newman 0.48 0.51 Data Mining and Knowledge Discovery, vol. 24, no. 3,
Lovain 0.56 0.49 pp. 515–554, 2012.
[13] O. Mason and M. Verwoerd, “Graph theory and networks
Table 1. Results for community detection for the in biology,” IET systems biology, vol. 1, no. 2,
three datasets considered using three different pp. 89–119, 2007.
algorithms. In this case we used the adjacency matrix [14] N. P. Nguyen, T. N. Dinh, Y. Shen, and M. T.
to compute the network partition. Thai, “Dynamic social community detection and its
applications,” PloS one, vol. 9, no. 4, p. e91431, 2014.
[15] M. E. Newman, “Fast algorithm for detecting community
Dataset Algorithm NMI Mod. structure in networks,” Physical review E, vol. 69, no. 6,
p. 066133, 2004.
Greedy Mod. 0.31 0.23 [16] V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and
Zachary Girvan Newman 0.42 0.55 E. Lefebvre, “Fast unfolding of communities in large
Louvain 0.31 0.85 networks,” Journal of Statistical Mechanics: Theory and
Experiment, vol. 2008, oct 2008.
Borgia Clustering 0.83 0.36
[17] M. Girvan and M. E. Newman, “Community structure
Greedy Mod. 0.052 0.43 in social and biological networks,” Proceedings of
Dolphin Girvan Newman 0.038 0.55 the National Academy of Sciences, vol. 99, no. 12,
pp. 7821–7826, 2002.
Louvain 0.089 0.59
[18] L. Wu, Q. Zhang, C.-H. Chen, K. Guo, and D. Wang,
Borgia Clustering 1.00 0.37 “Deep learning techniques for community detection in
social networks,” IEEE Access, vol. 8, pp. 96016–96026,
Greedy Mod. 0.086 0.55 2020.
Polbooks Girvan Newman 0.014 0.55 [19] S. E. Garza and S. E. Schaeffer, “Community detection
Louvain 0.024 0.59 with the label propagation algorithm: A survey,” Physica
Borgia Clustering 0.56 0.49 A: Statistical Mechanics and its Applications, vol. 534,
p. 122058, 2019.
Table 2. A summary for the best results for [20] M. Rosvall, D. Axelsson, and C. T. Bergstrom, “The
community detection computed over affinity networks map equation,” The European Physical Journal Special
for the datasets and algorithms studied. Topics, vol. 178, no. 1, pp. 13–23, 2009.
[21] J. Fumanal-Idocin, A. Alonso-Betanzos, O. Cordón,
H. Bustince, and M. Minárová, “Community detection
and social network analysis based on the italian wars of
transportation systems: A survey,” Viii Sbsc, pp. 16–23, the 15th century,” Future Generation Computer Systems,
2011. vol. 113, pp. 25–40, 2020.
[4] S. Horvath, Weighted network analysis: applications [22] W. Wright, “Gravitational clustering,” Pattern
in genomics and systems biology. Springer Science & Recognition, vol. 9, no. 3, pp. 151 – 166, 1977.
Business Media, 2011. [23] W. W. Zachary, “An information flow model for conflict
[5] M. Newman, Networks. Oxford university press, 2018. and fission in small groups,” Journal of Anthropological
Research, vol. 33, no. 4, pp. 452–473, 1977.
[6] I. Lobel and E. Sadler, “Information diffusion
in networks through social learning,” Theoretical [24] D. Lusseau, K. Schneider, O. J. Boisseau, P. Haase,
Economics, vol. 10, no. 3, pp. 807–851, 2015. E. Slooten, and S. M. Dawson, “The bottlenose dolphin
community of doubtful sound features a large proportion
[7] S. M. H. Bamakan, I. Nurgaliev, and Q. Qu, “Opinion of long-lasting associations,” Behavioral Ecology and
leader detection: A methodological review,” Expert Sociobiology, vol. 54, no. 4, pp. 396–405, 2003.
Systems with Applications, vol. 115, pp. 200–222, 2019. [25] L. A. Adamic and N. Glance, “The political blogosphere
[8] W. Yu and S. Li, “Recommender systems based on and the 2004 us election: divided they blog,” in
multiple social networks correlation,” Future Generation Proceedings of the 3rd international workshop on Link
Computer Systems, vol. 87, pp. 312 – 327, 2018. discovery, pp. 36–43, ACM, 2005.
[9] J. Wu, S. Wang, F. Chiclana, and E. Herrera-Viedma, [26] M. E. Newman and M. Girvan, “Finding and evaluating
“Two-fold personalized feedback mechanism for community structure in networks,” Physical review E,
social network consensus by uninorm interval trust vol. 69, no. 2, p. 026113, 2004.
propagation,” IEEE Transactions on Cybernetics, 2021.You can also read
