A NEW GENERALIZATION OF EDGE OVERLAP TO WEIGHTED NETWORKS

International Journal of Artificial Intelligence and Applications (IJAIA), Vol.11, No.1, January 2020
DOI : 10.5121/ijaia.2020.11108 93
A NEW GENERALIZATION OF EDGE OVERLAP TO
WEIGHTED NETWORKS
Ali Choumane
Lebanese University - Faculty of Sciences, LaRIFA Lab, Nabatieh – Lebanon
ABSTRACT
Finding the strength of an edge in a network has always been a big demand. In the context of social
networks, it allows to estimate the relationship strength between users. The best-known method to compute
edge strength is the Neighbourhood Overlap. It computes the ratio of common neighbours to all neighbours
of an edge terminal nodes. This method has been initially proposed for unweighted networks and later
extended for weighted ones. These two versions of the method are not mathematically equivalent: In fact,
an unweighted network is commonly considered as weighted with all edge weights equal to one. Using both
existent versions of Neighbourhood Overlap on such network produce completely different values. In this
paper, we tackle this problem and propose a new generalization for Neighbourhood Overlap that works
equally for unweighted and weighted networks. Experiment performed on networks with various
parameters showed similar performance of our measure to the existing measures.
KEYWORDS
Complex Networks, Edge Strength, Neighbourhood Overlap.
1. INTRODUCTION
A social network is a set of interconnected individuals, groups or organizations. For
computational purposes, it is naturally represented by graph where a set of nodes are connected
by edges. There are many analysis approaches that could be done to infer hidden information
from the topology of the network only, without knowing the details of interaction between the
nodes. One of the important analysis measures is the Neighbourhood Overlap (NO) [1]. NO
measure assigns a weight to each edge in the network representing its strength. The strength and
weakness in this context have special definitions [2, 3]. A strong edge is an edge in which its
terminals share relatively high number of their neighbours. A weak edge is a local bridge in
which its removal would make the shortest path between its terminals higher than 2, in other
terms, they don't share any neighbour. NO gives a value for strength by calculating the ratio of
common neighbours to all neighbours of both nodes, excluding themselves. Another approach is
to weight the edge by the number of hops of the shortest path between its terminals if the edge
were removed. This approach is computationally expensive, and it weights the edge according to
its weakness not its strength, in other words, two nodes having 100 neighbours in common and
two nodes with one common neighbour will have the same weight. NO proved to be useful in
many applications. One of the most important is community detection [4, 1, 5, 6] where nodes are
partitioned into subsets, each subset has a relatively larger internal edge density among its
members than with the nodes of different subsets.
The early proposed NO measure works for unweighted networks only. According to our
knowledge, only one extension to weighted networks has been proposed in [7]. As will be

94
explained in sections 2 and 3, this measure is not mathematically equivalent to the original
measure if applied to unweighted network, i.e. if we consider edge weights to be all 1s or if an
unweighted network is handled as weighted with weights being all 1s. In this case, the existing
extension produces different values than the original NO. In this paper we propose a new measure
that solves this point. It works equally for unweighted and weighted networks while having
similar performance in complex networks compared to the existing measures.
The article is structured as follows. In section 2, we present a literature review related to edge
strength computation in complex networks with focus on Neighbourhood Overlap and its
applications. Section 3 presents the proposed measure and its mathematical characteristics. In
section 4 we present an evaluation study that confirms the performance of our measure.
Conclusion and future work are reported in section 5.
2. RELATED WORK
The neighbourhood overlap of an edge (i,j) [1] is defined as the number of nodes who are
common neighbours of both i and j divided by the number of nodes who are neighbours of at least
one of the two nodes i or j (excluding each other). It is equal to:
𝑂𝑖𝑗 =
𝐶 𝑖𝑗
𝑘 𝑖 + 𝑘 𝑗−2−𝑐𝑖𝑗
(1)
where 𝑘𝑖 and 𝑘𝑗 are, respectively, the degrees of nodes i and j, and 𝑐𝑖𝑗 is the number of common
neighbours of both i and j. The factor of 2 in the denominator is to account for excluding vertices
i and j who are neighbours of each other. Low values of this measure indicate weak ties between
the corresponding nodes.
In [2], the authors defined strong and weak ties, and based on them, they used the formula above
to quantify the strength of an edge by a real number instead of just being weak or strong. In [8],
the authors confirmed the results by using a phone call data that allowed them to weight an edge
by the number of minutes spent on a call between two nodes in the network.
They realized that the curve of neighbourhood overlap as function of their percentile in the sorted
order of all edges is almost linear, which means that this measure could be actually used to infer
tie strength. In [9] authors applied the notion of strong and weak edges in social networks,
actually, they redefined the weak ties in a different way that suits better an online social network
like Facebook.
NO measure was used in various community detection algorithms, in which some of them
depends heavily on it. The algorithm proposed in [4] depends on the principle that low NO-
weighted edges represent local bridges between edges. It removes edges in increasing order of
their NO weight, thus removing the local bridges one after another, then finding the case that
gives highest quality function value (called modularity) [10]. Another algorithm based on NO
was introduced in [5], which uses the notion of weak ties to sort nodes into communities.

95
The existing NO measure (equation 1) computes the strength of an edge no matter the weight
associated to it in the input graph. In weighted graphs, the weight of an edge between two nodes
vehicles information about the strength of their relationship. Hence, computing NO in weighted
graphs should involve both the weight of the edge along with the proportion of common
neighbours of its terminal nodes. According to our knowledge, the only research that extended the
concept of NO to weighted networks is the work proposed in [7]:
𝑂𝑖𝑗
𝑤
=
∑ (𝑤 𝑖𝑘+𝑤 𝑗𝑘)𝑘∈𝑁 𝑖∩𝑁 𝑗
𝑆 𝑖+𝑆 𝑗−2𝑤𝑖𝑗
(2)
Where 𝑁𝑖 (𝑁𝑗) is the set of neighbours of node i (j), 𝑤𝑖𝑘 denotes the weight associated with the
edge between nodes i and k, and 𝑆𝑖 (𝑆𝑗) denotes the strength of node i (j), where 𝑆𝑖 = ∑ 𝑤𝑖𝑘𝑘∈𝑁 𝑖
.
The authors of this measure showed its performance in estimating edge strengths. The main
problem is that this measure is not equivalent to the original measure (equation 1) if all edge
weights are 1s, i.e. similar to unweighted network. Consider the following unweighted graph
where we consider edge weights to be all 1s.
The NO of edge [1,2] is computed as follows:
𝑂12
𝑤
=
𝑤10 + 𝑤20
𝑆1 + 𝑆2 − 2𝑤12
=
1 + 1
3 + 2 − 2
=
2
3
where 𝑂12 =
1
3 + 2 − 2 − 1
=
1
2
In this article we propose a new measure that solves this problem mathematically while
maintaining the performance of estimating edge strengths.
3. PROPOSED MEASURE
Our proposed measure (called NO) is calculated using the following equation:
𝑁𝑂𝑖𝑗
𝑤
=
∑ min(𝑤 𝑖𝑘,𝑤 𝑗𝑘)𝑘∈𝑁 𝑖∩𝑁 𝑗
∑ max(𝑤 𝑖𝑘,𝑤 𝑗𝑘)𝑘∈𝑁 𝑖∪𝑁 𝑗−{𝑖,𝑗}
(3)
Where 𝑁𝑖 ∩ 𝑁𝑗 is the set of common neighbors of nodes 𝑖 and 𝑗, and 𝑁𝑖 ∪ 𝑁𝑗 − {𝑖, 𝑗} is the union
of the set of neighbors of both nodes, without i and j that are neighbours to each other.

96
This formula was meant to be a generalization to the neighbourhood overlap measure 𝑂𝑖𝑗 on
unweighted graphs (cf. equation 1). To achieve the generalization, we represent the network as
sets of nodes, each set contains the neighbors of a node, the original neighbourhood overlap 𝑂𝑖𝑗
between nodes 𝑖 and 𝑗 can then be written as the size of intersection over the size of union of
neighbors of both nodes (excluding 𝑖 𝑎𝑛𝑑 𝑗 themselves):
|𝑁 𝑖∩𝑁 𝑗|
|(𝑁 𝑖−{𝑗} ∪ (𝑁 𝑗− {𝑗})|
(4)
Let us define a partial belonging set (a set that its elements partially belong to it) as a mapping
from the set of all nodes in the graph to the set of positive real numbers: 𝑀: 𝐺 → 𝑅+
that maps
each node to its membership in the set. Nodes outside the set are mapped to zero.
We define the size of a mapping to be ∑ 𝑀(𝑥)𝑥∈𝐺 , in our case it is the summation of all
memberships of any node x in the set. In an unweighted network, where all nodes have
memberships 1, this is equivalent to the cardinal of the set. In the case of a weighted network, we
represent the set of neighbours 𝑁𝑖 of a node 𝑖 as a mapping 𝑀 𝑁𝑖 defined as the
following: 𝑀 𝑁 𝑖
(𝑥) = 𝑤𝑖𝑥, thus, making the membership of the node 𝑥 to the set 𝑁𝑖 as the weight
of the edge connecting it to 𝑖.
Now the problem of generalization has transformed into a problem of defining the intersection
and union operators between two partial belonging sets, then it will be straight forward to replace
the size of intersection between the neighbours of two nodes in equation 4 to the size of
intersection between the partial belonging sets of the same two nodes, the same for union. In
regular sets, the union between sets A and B is the set of elements which are in A, in B, or in
both. In fact, it is the smallest set that is bigger than both. In the context of sets with partial
belongings, the mapping corresponding for the union of A and B is the maximum of the
mappings
corresponding to A and B, because it is the only mapping that matches being the smallest
mapping that is bigger than both:
∀𝑥 ∈ (𝐴 ∪ 𝐵), 𝑀𝐴∪𝐵(x) = max(𝑀𝐴(𝑥), 𝑀 𝐵(𝑥)) (5)
The intersection between sets A and B is the set that contains all the elements that are in both sets.
It is the largest set that is smaller than both. In partial belonging sets, it is the minimum of the two
mappings as it is the biggest mapping smallest than both, which is:
∀𝑥 ∈ (𝐴 ∩ 𝐵), 𝑀𝐴∩𝐵(x) = min(𝑀𝐴(𝑥), 𝑀 𝐵(𝑥)) (6)

97
Accordingly, the neighbourhood overlap for a weighted edge (i,j) is the fraction of the size of the
mapping 𝑀 𝑁 𝑖∩𝑁 𝑗
to the size of the mapping 𝑀 𝑁 𝑖∪𝑁 𝑗
that leads to the equation (3). The proposed
measure is a generalization of the original neighbourhood overlap (equation 1 and 4), i.e. it gives
same results in the case of unweighted graph (supposing the weight of each edge is 1). In fact,
when the weights of all edges are ones, the minimum of weights of the two edges to each
common neighbour will be also 1, thus the numerator will be equal to the count of common
neighbours. Likewise, the denominator will be equal to the number of all neighbours without i
and j themselves. In this case it gives the proportion of common neighbours to all neighbours
similarly to the original measure.
4. EVALUATION
In order to evaluate our measure, we compared it to the reference measure 𝑂𝑖𝑗
𝑤
, the only existing
extension on weighted networks (cf. section 2). We compared the values obtained by our measure
𝑁𝑂𝑖𝑗
𝑤
to 𝑂𝑖𝑗
𝑤
using the well-known Lancichinetti-Fortunato-Radicchi (LFR) benchmark. LFR
benchmark is a reference algorithm that generates benchmark synthetic networks that resemble
real-world networks [11]. This method allows to generate weighted networks and the underlying
community structure that satisfy the user’s parameters. Some of the parameters specify properties
of communities in the network: n (number of nodes), cmin and cmax (minimum and maximum
community size). The other parameters specify properties of the generated network: k (average
degree), kmax (maximum degree), μt (mixing parameter for the topology: each node shares a
fraction μt of its edges with nodes in other communities), and μw (mixing parameter for the
weights: each node shares a fraction μw of its total edge weights with nodes in other
communities). We consider it is important to evaluate NO on such networks with well-known
community structure as well as NO helps identifying communities as showed in section 2. In fact,
it has been shown that NO is a significant feature that allows to partition the network into
communities. NO is related to the network size and node degrees. Moreover, the distribution of
the edge weights and the topology of the network should have high influence on the obtained
values. Figures 1 and 2 show the results obtained by our measure and the reference measure 𝑂𝑖𝑗
𝑤
while varying the LFR parameters n, k, μt, μw. Each point is the plots is averaged over 50
networks. The main conclusion across all plots in both figures is that there is a linear relation
between our measure and 𝑂𝑖𝑗
𝑤
. Our measure is simply a linear translation of 𝑂𝑖𝑗
𝑤
and hence it can
replace it in all circumstances. Moreover, our measure shows a consistent profile for all network
sizes and mixing parameters where 𝑂𝑖𝑗
𝑤
seems to be sensitive for high mixing parameters (μt and
μw) in which it is hard to partition the network into communities. The obtained results confirm
the validity of our measure in estimating the neighbourhood overlap of edges in weighted
networks.

98
Figure. 1: The effects of network size, average degree and mixing parameter for the weights μw on LFR
weighted networks. Plots show average neighbourhood overlap of the network according to our method and
the method proposed in [7]. All results are from networks with mixing parameter for the topology = 0.3 and
maximum degree = 50. All results are averaged over 50 networks with each set of parameters.

99
Figure. 2: The effects of network size, average degree and mixing parameter for the topology μt on LFR
weighted networks. Plots show average neighbourhood overlap of the network according to our method and
the method proposed in [7]. All results are from networks with mixing parameter for the weights = 0.3 and
maximum degree = 50. All results are averaged over 50 networks with each set of parameters.
5. CONCLUSION AND FUTURE WORK
Neighbourhood Overlap measures edge strength in complex networks. In this article we proposed
a new generalization of this measure in the context of weighted networks. In contrast to the
existing measure 𝑂𝑖𝑗
𝑤
, our measure 𝑁𝑂𝑖𝑗
𝑤
can be applied to both weighted and unweighted
networks as it is mathematically equivalent to original Neighbourhood Overlap if the edge
weights are all 1s. Large experiment on the well-known LFR benchmark proved the validity of
our measure. It has similar performance compared to the existing measure. Furthermore, our
measure is stable across all network parameters while 𝑂𝑖𝑗
𝑤
is influenced by the topology and
weights distribution of the network. As part of future work, we plan to validate our method on
weighted directed networks and to use it in existing community detection algorithm.

100
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