In this paper, we analyze the evolution of a small-world network and its subsequent transformation to a
random network using the idea of link rewiring under the well-known Watts-Strogatz model for complex
networks. Every link u-v in the regular network is considered for rewiring with a certain probability and if
chosen for rewiring, the link u-v is removed from the network and the node u is connected to a randomly
chosen node w (other than nodes u and v). Our objective in this paper is to analyze the distribution of the
maximal clique size per node by varying the probability of link rewiring and the degree per node (number
of links incident on a node) in the initial regular network. For a given probability of rewiring and initial
number of links per node, we observe the distribution of the maximal clique per node to follow a Poisson
distribution. We also observe the maximal clique size per node in the small-world network to be very close
to that of the average value and close to that of the maximal clique size in a regular network. There is no
appreciable decrease in the maximal clique size per node when the network transforms from a regular
network to a small-world network. On the other hand, when the network transforms from a small-world
network to a random network, the average maximal clique size value decreases significantly
Distribution of maximal clique size of theIJCNCJournal
Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a branch-and-bound based exact algorithm that has been originally proposed to determine the maximum size clique for an entire network graph. We then run this algorithm on two categories of complex networks: One category of networks capture the evolution of small-world networks from regular network (according to the well-known Watts-Strogatz model) and their subsequent evolution to random networks; we show that the distribution of
the maximal clique size of the vertices follows a Poisson-style distribution at different stages of the evolution of the small-world network to a random network; on the other hand, the maximal clique size of the vertices is observed to be in-variant and to be very close to that of the maximum clique size for the entire network graph as the regular network is transformed to a small-world network. The second category
of complex networks studied are real-world networks (ranging from random networks to scale-free networks) and we observe the maximal clique size of the vertices in five of the six real-world networks to follow a Poisson-style distribution. In addition to the above case studies, we also analyze the correlation between the maximal clique size and clustering coefficient as well as analyze the assortativity index of the
vertices with respect to maximal clique size and node degree.
O N T HE D ISTRIBUTION OF T HE M AXIMAL C LIQUE S IZE F OR T HE V ERTICES IN ...csandit
The high-level contributions of this paper are as f
ollows: We modify an existing branch-and-
bound based exact algorithm (for maximum clique siz
e of an entire graph) to determine the
maximal clique size that the individual vertices in
the graph are part of. We then run this
algorithm on six real-world network graphs (ranging
from random networks to scale-free
networks) and analyze the distribution of the maxim
al clique size of the vertices in these graphs.
We observe five of the six real-world network graph
s to exhibit a Poisson-style distribution for
the maximal clique size of the vertices. We analyze
the correlation between the maximal clique
size and the clustering coefficient of the vertices
, and find these two metrics to be poorly
correlated for the real-world network graphs. Final
ly, we analyze the Assortativity index of the
vertices of the real-world network graphs and obser
ve the graphs to exhibit positive
assortativity with respect to maximal clique size a
nd negative assortativity with respect to node
degree; nevertheless, we observe the Assortativity
index of the real-world network graphs with
respect to both the maximal clique size and node de
gree to increase with decrease in the
spectral radius ratio for node degree, indicating a
positive correlation between the maximal
clique size and node degree.
Using spectral radius ratio for node degreeIJCNCJournal
In this paper, we show that the spectral radius ratio for node degree could be used to analyze the variation of node degree during the evolution of complex networks. We focus on three commonly studied models of complex networks: random networks, scale-free networks and small-world networks. The spectral radius ratio for node degree is defined as the ratio of the principal (largest) eigenvalue of the adjacency matrix of a network graph to that of the average node degree. During the evolution of each of the above three categories of networks (using the appropriate evolution model for each category), we observe the spectral radius ratio for node degree to exhibit high-very high positive correlation (0.75 or above) to that of the
coefficient of variation of node degree (ratio of the standard deviation of node degree and average node degree). We show that the spectral radius ratio for node degree could be used as the basis to tune the operating parameters of the evolution models for each of the three categories of complex networks as well as analyze the impact of specific operating parameters for each model.
A Proposed Algorithm to Detect the Largest Community Based On Depth LevelEswar Publications
The incredible rising of online networks show that these networks are complex and involving massive data.Giving a very strong interest to set of techniques developed for mining these networks. The clique problem is a well known NP-Hard problem in graph mining. One of the fundamental applications for it is the community detection. It helps to understand and model the network structure which has been a fundamental problem in several fields. In literature, the exponentially increasing computation time of this problem make the quality of these solutions is limited and infeasible for massive graphs. Furthermore, most of the proposed approaches are able to detect only disjoint communities. In this paper, we present a new clique based approach for fast and efficient overlapping
community detection. The work overcomes the short falls of clique percolation method (CPM), one of most popular and commonly used methods in this area. The shortfalls occur due to brute force algorithm for enumerating maximal cliques and also the missing out many vertices thatleads to poor node coverage. The proposed work overcome these shortfalls producing NMC method for enumerating maximal cliques then detects overlapping communities using three different community scales based on three different depth levels to assure high nodes coverage and detects the largest communities. The clustering coefficient and cluster density are used to measure the quality. The work also provide experimental results on benchmark real world network to
demonstrate the efficiency and compare the new proposed algorithm with CPM method, The proposed algorithm is able to quickly discover the maximal cliques and detects overlapping community with interesting remarks and findings.
EVOLUTIONARY CENTRALITY AND MAXIMAL CLIQUES IN MOBILE SOCIAL NETWORKSijcsit
This paper introduces an evolutionary approach to enhance the process of finding central nodes in mobile networks. This can provide essential information and important applications in mobile and social networks. This evolutionary approach considers the dynamics of the network and takes into consideration the central nodes from previous time slots. We also study the applicability of maximal cliques algorithms in mobile social networks and how it can be used to find the central nodes based on the discovered maximal cliques. The experimental results are promising and show a significant enhancement in finding the central nodes.
Distribution of maximal clique size of theIJCNCJournal
Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a branch-and-bound based exact algorithm that has been originally proposed to determine the maximum size clique for an entire network graph. We then run this algorithm on two categories of complex networks: One category of networks capture the evolution of small-world networks from regular network (according to the well-known Watts-Strogatz model) and their subsequent evolution to random networks; we show that the distribution of
the maximal clique size of the vertices follows a Poisson-style distribution at different stages of the evolution of the small-world network to a random network; on the other hand, the maximal clique size of the vertices is observed to be in-variant and to be very close to that of the maximum clique size for the entire network graph as the regular network is transformed to a small-world network. The second category
of complex networks studied are real-world networks (ranging from random networks to scale-free networks) and we observe the maximal clique size of the vertices in five of the six real-world networks to follow a Poisson-style distribution. In addition to the above case studies, we also analyze the correlation between the maximal clique size and clustering coefficient as well as analyze the assortativity index of the
vertices with respect to maximal clique size and node degree.
O N T HE D ISTRIBUTION OF T HE M AXIMAL C LIQUE S IZE F OR T HE V ERTICES IN ...csandit
The high-level contributions of this paper are as f
ollows: We modify an existing branch-and-
bound based exact algorithm (for maximum clique siz
e of an entire graph) to determine the
maximal clique size that the individual vertices in
the graph are part of. We then run this
algorithm on six real-world network graphs (ranging
from random networks to scale-free
networks) and analyze the distribution of the maxim
al clique size of the vertices in these graphs.
We observe five of the six real-world network graph
s to exhibit a Poisson-style distribution for
the maximal clique size of the vertices. We analyze
the correlation between the maximal clique
size and the clustering coefficient of the vertices
, and find these two metrics to be poorly
correlated for the real-world network graphs. Final
ly, we analyze the Assortativity index of the
vertices of the real-world network graphs and obser
ve the graphs to exhibit positive
assortativity with respect to maximal clique size a
nd negative assortativity with respect to node
degree; nevertheless, we observe the Assortativity
index of the real-world network graphs with
respect to both the maximal clique size and node de
gree to increase with decrease in the
spectral radius ratio for node degree, indicating a
positive correlation between the maximal
clique size and node degree.
Using spectral radius ratio for node degreeIJCNCJournal
In this paper, we show that the spectral radius ratio for node degree could be used to analyze the variation of node degree during the evolution of complex networks. We focus on three commonly studied models of complex networks: random networks, scale-free networks and small-world networks. The spectral radius ratio for node degree is defined as the ratio of the principal (largest) eigenvalue of the adjacency matrix of a network graph to that of the average node degree. During the evolution of each of the above three categories of networks (using the appropriate evolution model for each category), we observe the spectral radius ratio for node degree to exhibit high-very high positive correlation (0.75 or above) to that of the
coefficient of variation of node degree (ratio of the standard deviation of node degree and average node degree). We show that the spectral radius ratio for node degree could be used as the basis to tune the operating parameters of the evolution models for each of the three categories of complex networks as well as analyze the impact of specific operating parameters for each model.
A Proposed Algorithm to Detect the Largest Community Based On Depth LevelEswar Publications
The incredible rising of online networks show that these networks are complex and involving massive data.Giving a very strong interest to set of techniques developed for mining these networks. The clique problem is a well known NP-Hard problem in graph mining. One of the fundamental applications for it is the community detection. It helps to understand and model the network structure which has been a fundamental problem in several fields. In literature, the exponentially increasing computation time of this problem make the quality of these solutions is limited and infeasible for massive graphs. Furthermore, most of the proposed approaches are able to detect only disjoint communities. In this paper, we present a new clique based approach for fast and efficient overlapping
community detection. The work overcomes the short falls of clique percolation method (CPM), one of most popular and commonly used methods in this area. The shortfalls occur due to brute force algorithm for enumerating maximal cliques and also the missing out many vertices thatleads to poor node coverage. The proposed work overcome these shortfalls producing NMC method for enumerating maximal cliques then detects overlapping communities using three different community scales based on three different depth levels to assure high nodes coverage and detects the largest communities. The clustering coefficient and cluster density are used to measure the quality. The work also provide experimental results on benchmark real world network to
demonstrate the efficiency and compare the new proposed algorithm with CPM method, The proposed algorithm is able to quickly discover the maximal cliques and detects overlapping community with interesting remarks and findings.
EVOLUTIONARY CENTRALITY AND MAXIMAL CLIQUES IN MOBILE SOCIAL NETWORKSijcsit
This paper introduces an evolutionary approach to enhance the process of finding central nodes in mobile networks. This can provide essential information and important applications in mobile and social networks. This evolutionary approach considers the dynamics of the network and takes into consideration the central nodes from previous time slots. We also study the applicability of maximal cliques algorithms in mobile social networks and how it can be used to find the central nodes based on the discovered maximal cliques. The experimental results are promising and show a significant enhancement in finding the central nodes.
ABSTRACT
This paper introduces an evolutionary approach to enhance the process of finding central nodes in mobile networks. This can provide essential information and important applications in mobile and social networks. This evolutionary approach considers the dynamics of the network and takes into consideration the central nodes from previous time slots. We also study the applicability of maximal cliques algorithms in mobile social networks and how it can be used to find the central nodes based on the discovered maximal cliques. The experimental results are promising and show a significant enhancement in finding the central nodes.
Use of eigenvalues and eigenvectors to analyze bipartivity of network graphscsandit
This paper presents the applications of Eigenvalues and Eigenvectors (as part of spectral
decomposition) to analyze the bipartivity index of graphs as well as to predict the set of vertices
that will constitute the two partitions of graphs that are truly bipartite and those that are close
to being bipartite. Though the largest eigenvalue and the corresponding eigenvector (called the
principal eigenvalue and principal eigenvector) are typically used in the spectral analysis of
network graphs, we show that the smallest eigenvalue and the smallest eigenvector (called the
bipartite eigenvalue and the bipartite eigenvector) could be used to predict the bipartite
partitions of network graphs. For each of the predictions, we hypothesize an expected partition
for the input graph and compare that with the predicted partitions. We also analyze the impact
of the number of frustrated edges (edges connecting the vertices within a partition) and their
location across the two partitions on the bipartivity index. We observe that for a given number
of frustrated edges, if the frustrated edges are located in the larger of the two partitions of the
bipartite graph (rather than the smaller of the two partitions or equally distributed across the
two partitions), the bipartivity index is likely to be relatively larger.
Sector based multicast routing algorithm for mobile ad hoc networksijwmn
Multicast routing algorithms for mobile ad-hoc networks have been extensively researched in the recent
past. In this paper, we present two algorithms for dealing with multicast routing problem using the notion
of virtual forces. We look at the effective force exerted on a packet and determine whether a node could be
considered as a Steiner node. The nodes' location information is used to generate virtual circuits
corresponding to the multicast route. QoS parameters are taken into consideration in the form of virtual
dampening force. The first algorithm produces relatively minimal multicast trees under the set of
constraints. We improve upon the first algorithm and present a second algorithm that provides
improvement in average residual energy in the network as well as effective cost per data packet
transmitted. In this paper, the virtual-force technique has been applied for multicast routing for the first
time in mobile ad-hoc networks.
Erca energy efficient routing and reclusteringaciijournal
The pervasive application of wireless sensor networks (WNSs) is challenged by the scarce energy constraints of sensor nodes. En-route filtering schemes, especially commutative cipher based en-route filtering (CCEF) can saves energy with better filtering capacity. However, this approach suffer from fixed paths and inefficient underlying routing designed for ad-hoc networks. Moreover, with decrease in remaining sensor nodes, the probability of network partition increases. In this paper, we propose energy-efficient routing and re-clustering algorithm (ERCA) to address these limitations. In proposed scheme with reduction in the number of sensor nodes to certain thresh-hold the cluster size and transmission range dynamically maintain cluster node-density. Performance results show that our approach demonstrate filtering-power, better energy-efficiency, and an average gain over 285% in network lifetime.
Mathematics Research Paper - Mathematics of Computer Networking - Final DraftAlexanderCominsky
This Research Paper goes into the mathematics of computer networking hardware as well as encryption methods used to ensure data can safely and securely be transmitted from one point to another across a computer network and the web.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Performance of the Maximum Stable Connected Dominating Sets in the Presence o...csandit
The topology of mobile ad hoc networks
(
MANETs
)
change dynamically with time. Connected
dominating sets
(
CDS
)
are considered to be an effective topology for net
work-wide broadcasts
in MANETs as only the nodes that are part of the CD
S need to broadcast the message and the
rest of the nodes merely receive the message. Howev
er, with node mobility, a CDS does not exist
for the entire duration of the network session and
has to be regularly refreshed
(
CDS
transition
)
. In an earlier work, we had proposed a benchmarkin
g algorithm to determine a
sequence of CDSs
(
Maximum Stable CDS
)
such that the number of transitions is the global
minimum. In this research, we study the performance
(
CDS Lifetime and CDS Node Size
)
of the
Maximum Stable CDS when a certain fraction of the n
odes in the network are static and
compare the performance with that of the degree-bas
ed CDSs. We observe the lifetime of the
Maximum Stable CDS to only moderately increase
(
by a factor of 2.3
)
as we increase the
percentage of the static nodes in the network; on t
he other hand, the lifetime of the degree-based
CDS increases significantly
(
as large as 13 times
)
as we increase the percentage of static nodes
from 0 to 80
.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
Centrality Prediction in Mobile Social NetworksIJERA Editor
By analyzing evolving centrality roles using time dependent graphs, researchers may predict future centrality values. This may prove invaluable in designing efficient routing and energy saving strategies and have profound implications on evolving social behavior in dynamic social networks. In this paper, we propose a new method to predict centrality values of nodes in a dynamic environment. The proposed method is based on calculating the correlation between current and past measure of centrality for each corresponding node, which is used to form a composite vector to represent the given state of centralities. The performance of the proposed method is evaluated through simulated predictions on data sets from real mobile networks. Results indicate significantly low prediction error rate occurs, with a suitable implementation of the proposed method.
ENHANCING ENGLISH WRITING SKILLS THROUGH INTERNET-PLUS TOOLS IN THE PERSPECTI...ijfcstjournal
This investigation delves into incorporating a hybridized memetic strategy within the framework of English
composition pedagogy, leveraging Internet Plus resources. The study aims to provide an in-depth analysis
of how this method influences students’ writing competence, their perceptions of writing, and their
enthusiasm for English acquisition. Employing an explanatory research design that combines qualitative
and quantitative methods, the study collects data through surveys, interviews, and observations of students’
writing performance before and after the intervention. Findings demonstrate a beneficial impact of
integrating the memetic approach alongside Internet Plus tools on the writing aptitude of English as a
Foreign Language (EFL) learners. Students reported increased engagement with writing, attributing it to
the use of Internet plus tools. They also expressed that the memetic approach facilitated a deeper
understanding of cultural and social contexts in writing. Furthermore, the findings highlight a significant
improvement in students’ writing skills following the intervention. This study provides significant insights
into the practical implementation of the memetic approach within English writing education, highlighting
the beneficial contribution of Internet Plus tools in enriching students' learning journeys.
A SURVEY TO REAL-TIME MESSAGE-ROUTING NETWORK SYSTEM WITH KLA MODELLINGijfcstjournal
Messages routing over a network is one of the most fundamental concept in communication which requires
simultaneous transmission of messages from a source to a destination. In terms of Real-Time Routing, it
refers to the addition of a timing constraint in which messages should be received within a specified time
delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of
the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present
these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of
routing real-time messages on networks. We study whether it is possible to have an optimal on-line
algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message
routing’s algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual
examples. Next, we explore the Unidirectional Ring topology in finding the transmission’s
“makespan”.Lastly, we propose the same network modelling through the technique of Kinesthetic Learning
Activity (KLA). We will analyse the data collected and present the results in a case study to evaluate the
effectiveness of the KLA approach compared to the traditional teaching method.
More Related Content
Similar to DISTRIBUTION OF MAXIMAL CLIQUE SIZE UNDER THE WATTS-STROGATZ MODEL OF EVOLUTION OF COMPLEX NETWORKS
ABSTRACT
This paper introduces an evolutionary approach to enhance the process of finding central nodes in mobile networks. This can provide essential information and important applications in mobile and social networks. This evolutionary approach considers the dynamics of the network and takes into consideration the central nodes from previous time slots. We also study the applicability of maximal cliques algorithms in mobile social networks and how it can be used to find the central nodes based on the discovered maximal cliques. The experimental results are promising and show a significant enhancement in finding the central nodes.
Use of eigenvalues and eigenvectors to analyze bipartivity of network graphscsandit
This paper presents the applications of Eigenvalues and Eigenvectors (as part of spectral
decomposition) to analyze the bipartivity index of graphs as well as to predict the set of vertices
that will constitute the two partitions of graphs that are truly bipartite and those that are close
to being bipartite. Though the largest eigenvalue and the corresponding eigenvector (called the
principal eigenvalue and principal eigenvector) are typically used in the spectral analysis of
network graphs, we show that the smallest eigenvalue and the smallest eigenvector (called the
bipartite eigenvalue and the bipartite eigenvector) could be used to predict the bipartite
partitions of network graphs. For each of the predictions, we hypothesize an expected partition
for the input graph and compare that with the predicted partitions. We also analyze the impact
of the number of frustrated edges (edges connecting the vertices within a partition) and their
location across the two partitions on the bipartivity index. We observe that for a given number
of frustrated edges, if the frustrated edges are located in the larger of the two partitions of the
bipartite graph (rather than the smaller of the two partitions or equally distributed across the
two partitions), the bipartivity index is likely to be relatively larger.
Sector based multicast routing algorithm for mobile ad hoc networksijwmn
Multicast routing algorithms for mobile ad-hoc networks have been extensively researched in the recent
past. In this paper, we present two algorithms for dealing with multicast routing problem using the notion
of virtual forces. We look at the effective force exerted on a packet and determine whether a node could be
considered as a Steiner node. The nodes' location information is used to generate virtual circuits
corresponding to the multicast route. QoS parameters are taken into consideration in the form of virtual
dampening force. The first algorithm produces relatively minimal multicast trees under the set of
constraints. We improve upon the first algorithm and present a second algorithm that provides
improvement in average residual energy in the network as well as effective cost per data packet
transmitted. In this paper, the virtual-force technique has been applied for multicast routing for the first
time in mobile ad-hoc networks.
Erca energy efficient routing and reclusteringaciijournal
The pervasive application of wireless sensor networks (WNSs) is challenged by the scarce energy constraints of sensor nodes. En-route filtering schemes, especially commutative cipher based en-route filtering (CCEF) can saves energy with better filtering capacity. However, this approach suffer from fixed paths and inefficient underlying routing designed for ad-hoc networks. Moreover, with decrease in remaining sensor nodes, the probability of network partition increases. In this paper, we propose energy-efficient routing and re-clustering algorithm (ERCA) to address these limitations. In proposed scheme with reduction in the number of sensor nodes to certain thresh-hold the cluster size and transmission range dynamically maintain cluster node-density. Performance results show that our approach demonstrate filtering-power, better energy-efficiency, and an average gain over 285% in network lifetime.
Mathematics Research Paper - Mathematics of Computer Networking - Final DraftAlexanderCominsky
This Research Paper goes into the mathematics of computer networking hardware as well as encryption methods used to ensure data can safely and securely be transmitted from one point to another across a computer network and the web.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Performance of the Maximum Stable Connected Dominating Sets in the Presence o...csandit
The topology of mobile ad hoc networks
(
MANETs
)
change dynamically with time. Connected
dominating sets
(
CDS
)
are considered to be an effective topology for net
work-wide broadcasts
in MANETs as only the nodes that are part of the CD
S need to broadcast the message and the
rest of the nodes merely receive the message. Howev
er, with node mobility, a CDS does not exist
for the entire duration of the network session and
has to be regularly refreshed
(
CDS
transition
)
. In an earlier work, we had proposed a benchmarkin
g algorithm to determine a
sequence of CDSs
(
Maximum Stable CDS
)
such that the number of transitions is the global
minimum. In this research, we study the performance
(
CDS Lifetime and CDS Node Size
)
of the
Maximum Stable CDS when a certain fraction of the n
odes in the network are static and
compare the performance with that of the degree-bas
ed CDSs. We observe the lifetime of the
Maximum Stable CDS to only moderately increase
(
by a factor of 2.3
)
as we increase the
percentage of the static nodes in the network; on t
he other hand, the lifetime of the degree-based
CDS increases significantly
(
as large as 13 times
)
as we increase the percentage of static nodes
from 0 to 80
.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
Centrality Prediction in Mobile Social NetworksIJERA Editor
By analyzing evolving centrality roles using time dependent graphs, researchers may predict future centrality values. This may prove invaluable in designing efficient routing and energy saving strategies and have profound implications on evolving social behavior in dynamic social networks. In this paper, we propose a new method to predict centrality values of nodes in a dynamic environment. The proposed method is based on calculating the correlation between current and past measure of centrality for each corresponding node, which is used to form a composite vector to represent the given state of centralities. The performance of the proposed method is evaluated through simulated predictions on data sets from real mobile networks. Results indicate significantly low prediction error rate occurs, with a suitable implementation of the proposed method.
ENHANCING ENGLISH WRITING SKILLS THROUGH INTERNET-PLUS TOOLS IN THE PERSPECTI...ijfcstjournal
This investigation delves into incorporating a hybridized memetic strategy within the framework of English
composition pedagogy, leveraging Internet Plus resources. The study aims to provide an in-depth analysis
of how this method influences students’ writing competence, their perceptions of writing, and their
enthusiasm for English acquisition. Employing an explanatory research design that combines qualitative
and quantitative methods, the study collects data through surveys, interviews, and observations of students’
writing performance before and after the intervention. Findings demonstrate a beneficial impact of
integrating the memetic approach alongside Internet Plus tools on the writing aptitude of English as a
Foreign Language (EFL) learners. Students reported increased engagement with writing, attributing it to
the use of Internet plus tools. They also expressed that the memetic approach facilitated a deeper
understanding of cultural and social contexts in writing. Furthermore, the findings highlight a significant
improvement in students’ writing skills following the intervention. This study provides significant insights
into the practical implementation of the memetic approach within English writing education, highlighting
the beneficial contribution of Internet Plus tools in enriching students' learning journeys.
A SURVEY TO REAL-TIME MESSAGE-ROUTING NETWORK SYSTEM WITH KLA MODELLINGijfcstjournal
Messages routing over a network is one of the most fundamental concept in communication which requires
simultaneous transmission of messages from a source to a destination. In terms of Real-Time Routing, it
refers to the addition of a timing constraint in which messages should be received within a specified time
delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of
the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present
these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of
routing real-time messages on networks. We study whether it is possible to have an optimal on-line
algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message
routing’s algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual
examples. Next, we explore the Unidirectional Ring topology in finding the transmission’s
“makespan”.Lastly, we propose the same network modelling through the technique of Kinesthetic Learning
Activity (KLA). We will analyse the data collected and present the results in a case study to evaluate the
effectiveness of the KLA approach compared to the traditional teaching method.
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requirements for software developments. Software engineers applied software architectures for their
software system developments; however, they worry the basic benchmarks in order to select software
architecture styles, possible components, integration methods (connectors) and the exact application of
each style.
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weakness and benefits in order to select by the programmer during their design time. Finally, in this study,
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a design model of a business development (sales) system for professional service firms based on the Saudi
Arabian commercial market, which takes into account the new advances in technology while preserving
unique or cultural practices that are an important part of the Saudi Arabian commercial market. The
design model has combined a number of key technologies, such as cloud computing and mobility, as an
integral part of the proposed system. An adaptive development process has also been used in implementing
the proposed design model.
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role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy
relational inequality constraints is investigated. The feasible region is formed as the intersection of two
inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the
resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with
max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a
necessary and sufficient condition and three other necessary conditions are presented to conceptualize the
feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal
objective value. Also, it is proved that the optimal solution of the problem is always resulted from the
unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented
to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is
proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can
easily generate a feasible test problem and employ our algorithm to it.
LBRP: A RESILIENT ENERGY HARVESTING NOISE AWARE ROUTING PROTOCOL FOR UNDER WA...ijfcstjournal
Underwater detector network is one amongst the foremost difficult and fascinating analysis arenas that
open the door of pleasing plenty of researchers during this field of study. In several under water based
sensor applications, nodes are square measured and through this the energy is affected. Thus, the mobility
of each sensor nodes are measured through the water atmosphere from the water flow for sensor based
protocol formations. Researchers have developed many routing protocols. However, those lost their charm
with the time. This can be the demand of the age to supply associate degree upon energy-efficient and
ascendable strong routing protocol for under water actuator networks. During this work, the authors tend
to propose a customary routing protocol named level primarily based routing protocol (LBRP), reaching to
offer strong, ascendable and energy economical routing. LBRP conjointly guarantees the most effective use
of total energy consumption and ensures packet transmission which redirects as an additional reliability in
compare to different routing protocols. In this work, the authors have used the level of forwarding node,
residual energy and distance from the forwarding node to the causing node as a proof in multicasting
technique comparisons. Throughout this work, the authors have got a recognition result concerning about
86.35% on the average in node multicasting performances. Simulation has been experienced each in a
wheezy and quiet atmosphere which represents the endorsement of higher performance for the planned
protocol.
STRUCTURAL DYNAMICS AND EVOLUTION OF CAPSULE ENDOSCOPY (PILL CAMERA) TECHNOLO...ijfcstjournal
This research paper examined and re-evaluates the technological innovation, theory, structural dynamics
and evolution of Pill Camera(Capsule Endoscopy) technology in redirecting the response manner of small
bowel (intestine) examination in human. The Pill Camera (Endoscopy Capsule) is made up of sealed
biocompatible material to withstand acid, enzymes and other antibody chemicals in the stomach is a
technology that helps the medical practitioners especially the general physicians and the
gastroenterologists to examine and re-examine the intestine for possible bleeding or infection. Before the
advent of the Pill camera (Endoscopy Capsule) the colonoscopy was the local method used but research
showed that some parts (bowel) of the intestine can’t be reach by mere traditional method hence the need
for Pill Camera. Countless number of deaths from stomach disease such as polyps, inflammatory bowel
(Crohn”s diseases), Cancers, Ulcer, anaemia and tumours of small intestines which ordinary would have
been detected by sophisticated technology like Pill Camera has become norm in the developing nations.
Nevertheless, not only will this paper examine and re-evaluate the Pill Camera Innovation, theory,
Structural dynamics and evolution it unravelled and aimed to create awareness for both medical
practitioners and the public.
AN OPTIMIZED HYBRID APPROACH FOR PATH FINDINGijfcstjournal
Path finding algorithm addresses problem of finding shortest path from source to destination avoiding
obstacles. There exist various search algorithms namely A*, Dijkstra's and ant colony optimization. Unlike
most path finding algorithms which require destination co-ordinates to compute path, the proposed
algorithm comprises of a new method which finds path using backtracking without requiring destination
co-ordinates. Moreover, in existing path finding algorithm, the number of iterations required to find path is
large. Hence, to overcome this, an algorithm is proposed which reduces number of iterations required to
traverse the path. The proposed algorithm is hybrid of backtracking and a new technique(modified 8-
neighbor approach). The proposed algorithm can become essential part in location based, network, gaming
applications. grid traversal, navigation, gaming applications, mobile robot and Artificial Intelligence.
EAGRO CROP MARKETING FOR FARMING COMMUNITYijfcstjournal
The Major Occupation in India is the Agriculture; the people involved in the Agriculture belong to the poor
class and category. The people of the farming community are unaware of the new techniques and Agromachines, which would direct the world to greater heights in the field of agriculture. Though the farmers
work hard, they are cheated by agents in today’s market. This serves as a opportunity to solve
all the problems that farmers face in the current world. The eAgro crop marketing will serve as a better
way for the farmers to sell their products within the country with some mediocre knowledge about using
the website. This would provide information to the farmers about current market rate of agro-products,
their sale history and profits earned in a sale. This site will also help the farmers to know about the market
information and to view agricultural schemes of the Government provided to farmers.
EDGE-TENACITY IN CYCLES AND COMPLETE GRAPHSijfcstjournal
It is well known that the tenacity is a proper measure for studying vulnerability and reliability in graphs.
Here, a modified edge-tenacity of a graph is introduced based on the classical definition of tenacity.
Properties and bounds for this measure are introduced; meanwhile edge-tenacity is calculated for cycle
graphs and also for complete graphs.
COMPARATIVE STUDY OF DIFFERENT ALGORITHMS TO SOLVE N QUEENS PROBLEMijfcstjournal
This Paper provides a brief description of the Genetic Algorithm (GA), the Simulated Annealing (SA)
Algorithm, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm and attempts to
explain the way as how the Proposed Genetic Algorithm (GA), the Proposed Simulated Annealing (SA)
Algorithm using GA, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm can be
employed in finding the best solution of N Queens Problem and also, makes a comparison between these
four algorithms. It is entirely a review based work. The four algorithms were written as well as
implemented. From the Results, it was found that, the Proposed Genetic Algorithm (GA) performed better
than the Proposed Simulated Annealing (SA) Algorithm using GA, the Backtracking (BT) Algorithm and
the Brute Force (BF) Search Algorithm and it also provided better fitness value (solution) than the
Proposed Simulated Annealing Algorithm (SA) using GA, the Backtracking (BT) Algorithm and the Brute
Force (BF) Search Algorithm, for different N values. Also, it was noticed that, the Proposed GA took more
time to provide result than the Proposed SA using GA.
PSTECEQL: A NOVEL EVENT QUERY LANGUAGE FOR VANET’S UNCERTAIN EVENT STREAMSijfcstjournal
In recent years, the complex event processing technology has been used to process the VANET’s temporal
and spatial event streams. However, we usually cannot get the accurate data because the device sensing
accuracy limitations of the system. We only can get the uncertain data from the complex and limited
environment of the VANET. Because the VANET’s event streams are consist of the uncertain data, so they
are also uncertain. How effective to express and process these uncertain event streams has become the core
issue for the VANET system. To solve this problem, we propose a novel complex event query language
PSTeCEQL (probabilistic spatio-temporal constraint event query language). Firstly, we give the definition
of the possible world model of VANET’s uncertain event streams. Secondly, we propose an event query
language PSTeCEQL and give the syntax and the operational semantics of the language. Finally, we
illustrate the validity of the PSTeCEQL by an example.
CLUSTBIGFIM-FREQUENT ITEMSET MINING OF BIG DATA USING PRE-PROCESSING BASED ON...ijfcstjournal
Now a day enormous amount of data is getting explored through Internet of Things (IoT) as technologies
are advancing and people uses these technologies in day to day activities, this data is termed as Big Data
having its characteristics and challenges. Frequent Itemset Mining algorithms are aimed to disclose
frequent itemsets from transactional database but as the dataset size increases, it cannot be handled by
traditional frequent itemset mining. MapReduce programming model solves the problem of large datasets
but it has large communication cost which reduces execution efficiency. This proposed new pre-processed
k-means technique applied on BigFIM algorithm. ClustBigFIM uses hybrid approach, clustering using kmeans algorithm to generate Clusters from huge datasets and Apriori and Eclat to mine frequent itemsets
from generated clusters using MapReduce programming model. Results shown that execution efficiency of
ClustBigFIM algorithm is increased by applying k-means clustering algorithm before BigFIM algorithm as
one of the pre-processing technique.
A MUTATION TESTING ANALYSIS AND REGRESSION TESTINGijfcstjournal
Software testing is a testing which conducted a test to provide information to client about the quality of the
product under test. Software testing can also provide an objective, independent view of the software to
allow the business to appreciate and understand the risks of software implementation. In this paper we
focused on two main software testing –mutation testing and mutation testing. Mutation testing is a
procedural testing method, i.e. we use the structure of the code to guide the test program, A mutation is a
little change in a program. Such changes are applied to model low level defects that obtain in the process
of coding systems. Ideally mutations should model low-level defect creation. Mutation testing is a process
of testing in which code is modified then mutated code is tested against test suites. The mutations used in
source code are planned to include in common programming errors. A good unit test typically detects the
program mutations and fails automatically. Mutation testing is used on many different platforms, including
Java, C++, C# and Ruby. Regression testing is a type of software testing that seeks to uncover
new software bugs, or regressions, in existing functional and non-functional areas of a system after
changes such as enhancements, patches or configuration changes, have been made to them. When defects
are found during testing, the defect got fixed and that part of the software started working as needed. But
there may be a case that the defects that fixed have introduced or uncovered a different defect in the
software. The way to detect these unexpected bugs and to fix them used regression testing. The main focus
of regression testing is to verify that changes in the software or program have not made any adverse side
effects and that the software still meets its need. Regression tests are done when there are any changes
made on software, because of modified functions.
GREEN WSN- OPTIMIZATION OF ENERGY USE THROUGH REDUCTION IN COMMUNICATION WORK...ijfcstjournal
Advances in micro fabrication and communication techniques have led to unimaginable proliferation of
WSN applications. Research is focussed on reduction of setup operational energy costs. Bulk of operational
energy costs are linked to communication activities of WSN. Any progress towards energy efficiency has a
potential of huge savings globally. Therefore, every energy efficient step is an endeavour to cut costs and
‘Go Green’. In this paper, we have proposed a framework to reduce communication workload through: Innetwork compression and multiple query synthesis at the base-station and modification of query syntax
through introduction of Static Variables. These approaches are general approaches which can be used in
any WSN irrespective of application.
A NEW MODEL FOR SOFTWARE COSTESTIMATION USING HARMONY SEARCHijfcstjournal
Accurate and realistic estimation is always considered to be a great challenge in software industry.
Software Cost Estimation (SCE) is the standard application used to manage software projects. Determining
the amount of estimation in the initial stages of the project depends on planning other activities of the
project. In fact, the estimation is confronted with a number of uncertainties and barriers’, yet assessing the
previous projects is essential to solve this problem. Several models have been developed for the analysis of
software projects. But the classical reference method is the COCOMO model, there are other methods
which are also applied such as Function Point (FP), Line of Code(LOC); meanwhile, the expert`s opinions
matter in this regard. In recent years, the growth and the combination of meta-heuristic algorithms with
high accuracy have brought about a great achievement in software engineering. Meta-heuristic algorithms
which can analyze data from multiple dimensions and identify the optimum solution between them are
analytical tools for the analysis of data. In this paper, we have used the Harmony Search (HS)algorithm for
SCE. The proposed model which is a collection of 60 standard projects from Dataset NASA60 has been
assessed.The experimental results show that HS algorithm is a good way for determining the weight
similarity measures factors of software effort, and reducing the error of MRE.
AGENT ENABLED MINING OF DISTRIBUTED PROTEIN DATA BANKSijfcstjournal
Mining biological data is an emergent area at the intersection between bioinformatics and data mining
(DM). The intelligent agent based model is a popular approach in constructing Distributed Data Mining
(DDM) systems to address scalable mining over large scale distributed data. The nature of associations
between different amino acids in proteins has also been a subject of great anxiety. There is a strong need to
develop new models and exploit and analyze the available distributed biological data sources. In this study,
we have designed and implemented a multi-agent system (MAS) called Agent enriched Quantitative
Association Rules Mining for Amino Acids in distributed Protein Data Banks (AeQARM-AAPDB). Such
globally strong association rules enhance understanding of protein composition and are desirable for
synthesis of artificial proteins. A real protein data bank is used to validate the system.
International Journal on Foundations of Computer Science & Technology (IJFCST)ijfcstjournal
International Journal on Foundations of Computer Science & Technology (IJFCST) is a Bi-monthly peer-reviewed and refereed open access journal that publishes articles which contribute new results in all areas of the Foundations of Computer Science & Technology. Over the last decade, there has been an explosion in the field of computer science to solve various problems from mathematics to engineering. This journal aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals. Topics of interest include, but are not limited to the following:
Because the technology is used largely in the last decades; cybercrimes have become a significant
international issue as a result of the huge damage that it causes to the business and even to the ordinary
users of technology. The main aims of this paper is to shed light on digital crimes and gives overview about
what a person who is related to computer science has to know about this new type of crimes. The paper has
three sections: Introduction to Digital Crime which gives fundamental information about digital crimes,
Digital Crime Investigation which presents different investigation models and the third section is about
Cybercrime Law.
A STATISTICAL COMPARATIVE STUDY OF SOME SORTING ALGORITHMSijfcstjournal
This research paper is a statistical comparative study of a few average case asymptotically optimal sorting
algorithms namely, Quick sort, Heap sort and K- sort. The three sorting algorithms all with the same
average case complexity have been compared by obtaining the corresponding statistical bounds while
subjecting these procedures over the randomly generated data from some standard discrete and continuous
probability distributions such as Binomial distribution, Uniform discrete and continuous distribution and
Poisson distribution. The statistical analysis is well supplemented by the parameterized complexity
analysis
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
DISTRIBUTION OF MAXIMAL CLIQUE SIZE UNDER THE WATTS-STROGATZ MODEL OF EVOLUTION OF COMPLEX NETWORKS
1. International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.3, May 2015
DOI:10.5121/ijfcst.2015.5301 1
DISTRIBUTION OF MAXIMAL CLIQUE SIZE UNDER
THE WATTS-STROGATZ MODEL OF EVOLUTION OF
COMPLEX NETWORKS
Natarajan Meghanathan
Jackson State University, 1400 Lynch St, Jackson, MS, USA
ABSTRACT
In this paper, we analyze the evolution of a small-world network and its subsequent transformation to a
random network using the idea of link rewiring under the well-known Watts-Strogatz model for complex
networks. Every link u-v in the regular network is considered for rewiring with a certain probability and if
chosen for rewiring, the link u-v is removed from the network and the node u is connected to a randomly
chosen node w (other than nodes u and v). Our objective in this paper is to analyze the distribution of the
maximal clique size per node by varying the probability of link rewiring and the degree per node (number
of links incident on a node) in the initial regular network. For a given probability of rewiring and initial
number of links per node, we observe the distribution of the maximal clique per node to follow a Poisson
distribution. We also observe the maximal clique size per node in the small-world network to be very close
to that of the average value and close to that of the maximal clique size in a regular network. There is no
appreciable decrease in the maximal clique size per node when the network transforms from a regular
network to a small-world network. On the other hand, when the network transforms from a small-world
network to a random network, the average maximal clique size value decreases significantly.
KEYWORDS
Maximal Clique Size, Small-World Networks, Complex Networks, Random Networks, Link Rewiring,
Poisson Distribution, Network Evolution, Watts-Strogatz Model.
1. INTRODUCTION
Network Science is the field of analyzing complex real-world networks from a graph theoretical
standpoint. A complex network is abstracted as a graph wherein the nodes and links in the
network are modeled respectively as the vertices and edges of the graph. The graph theoretic
algorithms run on such complex network graphs (to determine one or more metrics characteristic
of the networks) need to be as efficient as possible. The graph theoretic metrics considered for
analysis of such complex networks [1] include centrality, clustering coefficient, diameter, clique
size, etc. Unlike the polynomial-time efficient algorithms that exist to determine most of the
above metrics, the maximum clique size problem is NP-hard [2]. A "clique" is a complete sub
graph of a graph such that any two vertices in the sub graph are connected with an edge.
Community detection algorithms (e.g., [3-5]) use cliques of various sizes as the basis to determine
closely-knit and overlapping communities in complex real-world networks as well as in networks
that are simulated using theoretical models.
For an n-vertex graph, the "maximum size clique" is the clique of the largest size k such that the
graph does not have any clique of size k+1 (where k ≤ n). Note that there may exist one or more
cliques of smaller size in the graph and for a particular vertex i, the largest size clique it is part of
need not be the maximum size clique for the entire network graph. Accordingly, we define the
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2
"maximal size clique for a vertex i" as the largest size clique the vertex is part of. The focus of
research in this area has been so far on developing efficient exact algorithms and heuristics (time
and space-wise) to determine respectively the maximum size cliques and approximations to the
same. Not much work has been conducted on determining the maximal size cliques for the
individual vertices of the graph. Specifically, we could not come across any work that has
analyzed the distribution of the maximal clique size for the individual vertices of graphs that
simulate the evolution of complex networks. In this pursuit, we choose a recently proposed
branch-and-bound strategy based efficient exact algorithm [6] to determine maximum size clique
for an entire network graph and modify it to determine the maximal size cliques for the
constituent vertices of the graph. We use the modified exact algorithm to analyze the distribution
of the maximal clique size of the network graphs that span two categories of complex networks:
small-world networks and random networks. We choose the well-known Watts-Strogatz model to
simulate the evolution of small-world networks (from a regular network) and their subsequent
transformation to a random network.
The rest of the paper is organized as follows: In Section 2, we first review and discuss the
recently proposed exact algorithm to determine maximum clique size for an entire graph and then
explain our modifications to the same to determine the maximal clique for a particular vertex of
the graph. Section 3 presents the evolution of small-world networks and their transformation to a
random network under the Watts-Strogatz model and describes the results of the diameter and
clustering coefficient of the networks that evolve during this transformation. Section 4 presents in
detail the distribution of the maximal clique size per node and the average maximal clique size as
a function of the probability of link rewiring and the initial number of links per node. Section 5
reviews the literature and discusses related work. Section 6 concludes the paper. In the entire
paper, the terms 'vertex' and 'node', 'edge' and 'link' have been used interchangeably. They mean
the same.
2. CLIQUE
A clique is a subset of the vertices of a graph such that any two vertices in the subset are
connected with an edge. The maximal clique size and maximum clique size problems are both
NP-hard [2]. There are several exact algorithms (that take exponential run-time at the worst case)
have been proposed to determine the maximum clique size for sparse graphs. With the surge of
research interests in the analysis of complex real-world networks from a graph theoretic
standpoint, we come across efficient algorithms (e.g., [6-9]) to extract the largest size cliques in
large/dense graphs. The "branch-and-bound" strategy is a commonly used strategy [10] behind
these algorithms and the idea is to explore the neighborhood of only those vertices that have
scope for being part of a clique that can be larger than the clique known until then during the
execution of the algorithm. The variation among the various exact algorithms is the pruning
strategy adopted to decide on the bounds for branching through the solution space and limiting
the search. In this section, we will first review a recently proposed exact branch-and-bound based
exact algorithm to determine maximum size clique for an entire network graph and then explain
our modifications to it to determine the maximal size cliques for the individual vertices of a
graph.
The pseudo code of the exact algorithm (proposed in [6]) to determine the maximum size clique
for a graph is outlined in Figure 1. The idea behind the algorithm is to keep track of the maximum
size clique found so far (variable max in the pseudo code, the initial value is 0) and update it
whenever an even larger clique is found. The algorithm proceeds in iterations, with iteration i
attempting to determine whether vertex vi could be part of a clique that is larger than the
maximum size clique known until then. The algorithm decides to explore the neighborhood of a
vertex only if the degree of the vertex is larger than the currently known maximum size clique;
3. International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.3, May 2015
3
moreover only the neighbors whose degree is at least the size of the currently known maximum
clique size are considered for further exploration (these neighbors form the set U passed to the
subroutine CLIQUE). The vertices are explored in the increasing order of their IDs (one iteration
per vertex).
Figure 1. Exact Algorithm to Extract the Maximum Size Clique in a Graph (adapted from [6])
The sub routine CLIQUE (called for a particular vertex vi and its set of neighbors U whose degree
is at least the size of the maximum clique, max, found so far) runs via a combination of iterations
and recursions. In each iteration, the sub routine randomly removes a node u in the set U, adds it
to the clique found so far and recursively calls the sub routine CLIQUE (with vertex u and the set
U comprising of vertices that are the neighbors of u as well as the neighbors of the vertices that
are part of the clique found so far). If the set U passed to the CLIQUE sub routine is empty, the
value of max is updated if it is less than the size of the clique found so far and the recursion is
terminated; otherwise, another recursive call to the sub routine CLIQUE is made with a new
vertex u randomly chosen/removed from the set U of vertices passed during the latest recursion.
The sub routine CLIQUE (called with vertex vi) proceeds in iterations until all the candidate
neighbor vertices in the set U (that was initially passed to it) are explored through a sequence of
recursive calls as explained above. At each stage, the sub routine CLIQUE (called for a particular
vertex vi) only explores those neighbors of vi and their individual neighborhoods that have scope
for being part of a clique that is larger than the maximum size clique found so far (whose size is
kept track of using the variable max).
A characteristic property of the exact algorithm described here is that the maximum size clique
for the entire graph is found during a particular iteration involving vertex vi such that vertices vj
whose neighborhood are explored after this iteration are not part of the clique. If the maximum
size clique for the entire graph is found in an earlier iteration itself, then the subsequent iterations
could run relative more quickly as all they will do is to simply prune the search space as much as
possible. Hence, the time efficiency of the exact algorithm is significantly influenced by the order
in which the vertices are considered to be explored for the iterations as explained in the previous
paragraphs and illustrated in Figure 1. Thus, if vertices are to be explored in the increasing order
of their IDs, the search time could significantly reduce only if a vertex with a smaller ID is part of
the maximum size clique. For guaranteed better performance, it would be more apt to run
MAXCLIQUE by exploring vertices in the decreasing order of their degree, rather than simply
based on the increasing order of their IDs (since the maximum clique size in a graph cannot be
larger than the largest value for the degree of any vertex in the graph).
4. International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.3, May 2015
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The exact algorithm (describe above to determine the maximum size clique for an entire graph)
can be easily modified to determine the maximal size clique for an individual vertex of the graph,
as illustrated in Figure 2. In order to determine the maximal clique size distribution of the vertices
of a graph, we have to now run the procedure MAXIMALCLIQUE for each vertex vi by
considering all the neighbors of vi for possible inclusion in the clique. This would increase the
overall run time of the algorithm; but, it is unavoidable. However, the strategies incorporated by
the exact algorithm in the procedure CLIQUE could be retained: the recursive calls in CLIQUE
are made only for those neighbors u that have the potential to be part of a clique that is larger than
the maximal size clique found until then for vertex vi. For guaranteed better performance of
procedure CLIQUE called with a vertex vi, it would be more apt to explore the neighbors of vi in
the decreasing order of their degree.
Figure 2. Exact Algorithm to Determine the Distribution of the Maximal Clique Size of the Vertices in a
Graph (adapted from [6])
3. SMALL-WORLD NETWORKS AND THEIR CHARACTERISTICS
Small-world networks are a category of complex networks that exhibit a smaller diameter
(maximum of the number of hops in the shortest paths connecting any two nodes) as well as a
larger clustering coefficient (the probability that two nodes that share a common neighbor are
connected). The other two categories of complex networks (random networks and scale-free
networks) exhibit a significantly lower clustering coefficient. Small-world networks are
characteristic of having a majority of the links in the local neighborhood of the vertices (helps to
sustain a larger clustering coefficient) and having very few links (but not negligible) that connect
vertices that do not have any common neighbors (still sufficient enough to ensure a smaller
diameter). As the clustering coefficient of a vertex is measured as the ratio of the number of links
among the neighbors of a vertex to that of the maximum possible number of links among the
neighbors of a vertex, the characteristic of possessing a majority of the links in the local
5. International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.3, May 2015
5
neighborhood of the vertices helps the small-world networks to incur a significantly larger
clustering coefficient, compared to the random networks and scale-free networks.
Initially, we start with a regular network wherein the number of neighbors per node (i.e., the
number of links per node, identified as Kregular) is the same (and typically, an even number of links
per node) as well as there is a particular pattern in the distribution of the links in the network
(typically dependent on the dimension of the network). The regular network envisioned in this
paper is a one-dimensional network with a ring as the underlying topological structure. Each node
is connected to at least two other nodes (i.e., to the two neighboring nodes that are each one hop
away in the ring): if there are more than 2 links per node, then the node is connected to nodes in
the increasing order of the hop count in the ring. In general, if the number of links per node is
Kregular, then a node is connected to neighbor nodes that are 1, 2, ..., Kregular/2 hops away from the
node on the ring. Figure 3 displays a 10-node regular network with four links per node (i.e.,
Kregular = 4) and each node is connected to nodes that are 1 and 2 hops away from it in the ring.
Figure 3. Example for an One-Dimensional Regular Network (Kregular = 4 Links per Node)
The WS model operates based on a tuning parameter called the probability of link rewiring
(Prewire). We rewire each link in the regular network with the probability Prewire. For each link u-v
in the regular network, we generate a random number (in the range 0 to 1) and if it is less than
Prewire, we decide to rewire the link. When a link u-v is chosen for rewiring, we choose a target
node w uniform-randomly among the nodes in the network (such that w is neither u nor v),
remove the link u-v and connect node u with node w (i.e., add the link u-w to the network). We
repeat the above procedure for every link in the initial regular network. Note that the newly added
links are not considered for rewiring.
We conduct simulations to transform a regular network to a small-world network and
subsequently to a random network according to the WS model. The simulations are conducted for
networks of 100 nodes and 200 nodes; the probability of rewiring is varied from values of 0.01 to
0.1, in increments of 0.01 (referred to as small-world network zone), and from values of 0.1 to
1.0, in increments of 0.1 (referred to as random network zone). The reasoning behind the above
distinction for the probability of rewiring is based on our observations from the simulation
results: for Prewire values of 0.01 to 0.1, the average diameter of any node in the network (average
of the maximum of the number of hops from a node to any other node) reduces significantly, but
with only a moderate reduction in the clustering coefficient - a phenomenon characteristic of
small-world networks. On the other hand, as we vary the probability of rewiring from 0.1 to 1.0,
the average diameter of any node in the network reduces only marginally, whereas the clustering
coefficient reduces significantly, indicating the transformation of the small-world network to a
random network. We also vary the initial number of links per node (Kregular) in the regular
network from 4 to 20, in increments of 2. The results presented in Figures 4-10 are the average of
6. International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.3, May 2015
6
the results observed for 100 network graphs, simulated for each value of the number of nodes
(100 and 200 nodes) and each value of the probability of rewiring as mentioned above.
Figure 4. Impact of the Probability of Link Rewiring and the Initial Number of Links per Node on the
Average Diameter per Node: Transition from Regular Network to Small-World Network and Random
Network
Figure 4 captures the absolute values of the average diameter of any node in the network as well
as the ratio of the average diameter with and without rewiring. For a given probability of
rewiring, we observe the absolute average value for the diameter to be smaller when we start with
a regular network with a larger number of links per node. As we do rewiring, within the small-
world zone, we observe the differences in the average diameter per node (for different values of
Kregular) to reduce significantly (in an exponential fashion); in the random network zone, the
average diameter per node for different values of Kregular does not vary appreciably. Based on the
results for the ratio of the average diameter per node with and without rewiring, we observe that
the percentage decrease in the average diameter per node is much higher for regular networks
with fewer numbers of initial links, indicating the effectiveness of rewiring in reducing the path
length. With increase in the number of nodes (from 100 nodes to 200 nodes), we observe the
network diameter to reduce further (for a given Prewire and Kregular), especially in the random
network zone.
Figure 5 illustrates the variation in the average clustering coefficient per node (averaged over all
the nodes in the network) in the small-world and random network zones. In the small-world zone,
we observe the percentage reduction in the clustering coefficient is by about only 25% (compared
to the value observed for the originating regular network, without any rewiring) and the rate of
decrease is the same for all values of Kregular. On the other hand, as we enter the random network
zone, the reduction in the clustering coefficient is significantly larger and specifically, the
clustering coefficient of networks that started with a lower Kregular value reduces much faster
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compared to networks that started with a larger Kregular value. As the number of nodes is increased
from 100 to 200, we observe the reduction in the clustering coefficient in the random network
zone (in terms of both the magnitude as well as the rate of decrease) to be larger. For a given
value of Kregular, the fraction of the number of links between any two neighbors of a node
(compared to the maximum number of links between the neighbors of a node) is bound to be
lower for networks with a relatively larger number of nodes, thus reducing the clustering
coefficient.
Figure 5. Impact of the Probability of Link Rewiring and the Initial Number of Links per Node on the
Average Clustering Coefficient per Node: Transition from Regular Network to Small-World Network and
Random Network
4. ANALYSIS OF THE DISTRIBUTION OF MAXIMAL CLIQUE SIZE
Figure 6 captures the variation in the average maximal clique size per node (average of the
maximal clique size of all the nodes, measured at the end of rewiring) for various values of the
probability of rewiring and the initial number of links per node in the originating regular network.
We observe that the small-world zone does not suffer any noticeable decrease in the average
maximal clique size per node and the ratio of the average maximal clique size per node with and
without rewiring is close to 1. As we transition from the small-world zone to the random network
zone, we observe the average maximal clique size to reduce relatively at a much faster rate, with
increase in the probability of rewiring. An interesting observation is that the average maximal
clique size of random networks that start with a larger Kregular value decreases at a much faster rate
compared to the rate of decrease of the average maximal clique size of random networks that start
with a lower Kregular value (though the absolute values for the average maximal clique size is still
larger for random networks that start with a larger Kregular value). This indicates that the larger
cliques present in the small-world networks that started with regular networks of larger Kregular
values are more likely to quickly get dismantled as the probability of link rewiring increases
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beyond the threshold value of Prewire for small-world network zone. Thus, the average maximal
clique size per nodes for different values of Kregular tend to get closer as we increase the
probability of rewiring in the random network zone, and such a convergence is relatively more
pronounced for networks with 200 nodes, compared to 100 nodes. Accordingly, for a given Prewire
and Kregular, the rate of decrease in the average maximal clique size per node is much more steeper
for networks with 200 nodes.
Figure 6. Impact of the Probability of Link Rewiring and the Initial Number of Links per Node on the
Average Maximal Clique Size per Node: Transition from Regular Network to Small-World Network and
Random Network
Figures 7-10 capture the variation in the maximal clique size for the nodes in the small-world
zones and random network zones. For a given value of Prewire and Kregular, we observe the
distribution of the maximal clique size is Poisson for both the zones. Figures 7 and 9 capture the
distribution of the maximal clique size in the small-world zone. For smaller values of Kregular (4
and 6 links per node), we observe the maximal clique size per node to be very close to the
average value for all the nodes; as we increase the value of Kregular, we observe the maximal clique
size per node to vary slightly, but not much different from the average value for the maximal
clique size - coinciding with the invariant nature of the average maximal clique size per node
observed in Figure 6. For a given value of Kregular, the average maximal clique size of a regular
network is 1+ Kregular/2 and the average maximal clique size per node in a small-world network is
very close to this value (with minimal variation) observed for its predecessor regular network. For
networks with larger Kregular values, the values for the maximal clique size per node is less than
the average value by at most 2 and greater than the average value by at most 1, and as observed in
Figures 7 and 9, these deviations occur with a vary small probability. The Poisson curve for the
maximal clique size per node shifts to the right in such a way that the peak for the curve increases
by a value of 1 as we increase the value of Kregular by 2. For a given value of Kregular, the Poisson
curve for the maximal clique size per node is more steep (minimal variation) and the tallest for a
Prewire value of 0.01 and becomes more spread out (shallow and more variations) as the Prewire
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value increases to 0.10 and above. Coinciding with the observations made in Figure 6, for a given
value of Kregular and Prewire, there is not much variation in the distribution of the maximal clique
size per node for networks of 100 nodes and 200 nodes.
Figure 7. Distribution of the Maximal Clique Size vs. Initial Number of Links per Node: Transition from
Regular Network to Small World Network [100 Node Network]
Figure 8. Distribution of the Maximal Clique Size vs. Probability of Link Rewiring: Transition from Small
World Network to Random Network [100 Node Network]
Figures 8 and 10 capture the variation in the maximal clique size for the nodes in the random
network zone for a given probability of rewiring and varying the initial number of links per node
with values of 4, 12 and 20 links - scenarios that exhibit minimal, moderate and maximum
variation in the maximal clique size per node as the probability of rewiring increases. For lower
values of the probability of rewiring (0.1 and 0.2; when the network is still in the small-world
zone), the distribution of the maximal clique size per node is taller for each value of Kregular and
the distributions are non overlapping (as the Kregular values are 4, 12 and 20, the average maximal
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clique size is around 3, 7 and 11 - vindicating the non-overlapping nature of the peaks and the
distribution of the maximal clique size for lower values of Prewire). With increase in Prewire, we start
observing the distributions of the maximal clique size for the three fairly different values of
Kregular to start overlapping; the distributions tend to shift to the left - coinciding with a decrease in
the average maximal clique value. With increase in Prewire, the shift towards lower values of the
maximal clique is more pronounced for networks with a larger Kregular value, vindicating the rapid
fall in the average maximal clique size; also for larger values of Kregular, the distributions for the
maximal clique become more spread out with increase in Prewire - lowering the probability of
finding the maximal clique size per node to be close to the average value. On the other hand, for
networks with lower values of Kregular, the distribution for the maximal clique size remains fairly
narrow (even with increase in Prewire), indicating that it is still possible to observe the maximal
clique size for any node to be close to the average value.
Figure 9. Distribution of the Maximal Clique Size vs. Initial Number of Links per Node: Transition from
Regular Network to Small World Network [200 Node Network]
Figure 10. Distribution of the Maximal Clique Size vs. Probability of Link Rewiring: Transition from Small
World Network to Random Network [200 Node Network]
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5. RELATED WORK
The focus of research in the literature so far has been on developing heuristics (both time and
space efficient) for determining the maximum size for the entire network graphs, based on
strategies (like branch and bound) that reduce the solution search space. For example, branch and
bound strategies based on node degree [6], vertex coloring [7] and vertex ordering [8] have been
proposed as potential strategies for effectively and efficiently pruning the solution search space.
In addition, there have been efforts to develop parallelized versions of branch and bound (e.g.,
[9]) that could be applied to determine cliques in large real-world network graphs with 1000 to
100 million nodes. In [12], the authors explore the use of maximal cliques of size at least k nodes
to identify overlapping communities based on percolation; two cliques of size k are said to
percolate into each other if they share k-1 nodes. With regards to analyzing the distribution of
clique-related metrics for complex networks, in [13], the authors have analyzed the distribution of
clique-degree of the nodes (the clique-degree of a node is the number of cliques of a particular
size the node is part of) in various real-world networks and have observed them to exhibit a
power-law distribution. As far as we know, there is no work that has analyzed the distribution of
the maximal clique size of the vertices for complex network graphs that evolve under any well-
known theoretical model (like the Watts-Strogatz model for small-world networks that eventually
transform to a random network with increase in the probability of link rewiring).
6. CONCLUSIONS
The following significant conclusions could be made from the research conducted in this paper.
with regards to the distribution for the maximal clique size per node for small-world networks and
random networks that evolve from a regular network. As we transform from a regular network
(with Kregular number of links per node) to a small-world network through link rewiring, the
maximal clique size of the nodes is invariant and very close to that of the average maximal clique
size per node as well as close to that of the average maximal clique size per node in the regular
network. As we transform from a small-world network to a random network (by increasing the
probability of rewiring), the distribution of the maximal clique size per node becomes more
broader and thereby the probability of observing a maximal clique size per node close to that of
the average maximal clique size is relatively much lower. Also, with increase in the probability of
rewiring, the distributions for the maximal clique size obtained for different Kregular values overlap
each other and shift towards a lower average value. Nevertheless, for all the scenarios/values for
the probability of rewiring and the initial number of links per node, the distribution for the
maximal clique size reflects that of a Poisson distribution. As the vertices in a random network
exhibit a Poisson style distribution for the node degree, we conjecture a high positive correlation
between the maximal clique size per node and node degree in small-world networks and random
networks, and this will be further analyzed in future work.
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Author
Dr. Natarajan Meghanathan is currently an Associate Professor of Computer Science at
Jackson State University, USA. His areas of research interests are Network Science and
Graph Theory, Wireless Ad hoc Networks and Sensor Networks, Cyber Security and
Machine Learning. He has published more than 150 peer-reviewed articles and obtained
grants from several federal agencies. He serves as the editor-in-chief of three
international journals as well as serves in the organizing committees of several
international conferences.