Centrality and Partial Correlation Coefficient-Based Assortativity Analysis of Real-World Networks

2018 ◽  
Vol 62 (9) ◽  
pp. 1247-1264 ◽  
Author(s):  
Natarajan Meghanathan
Keyword(s):  
Correlation Coefficient ◽  
Real World ◽  
Betweenness Centrality ◽  
Partial Correlation ◽  
Closeness Centrality ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
First Order ◽  
Centrality Metrics ◽  
Partial Correlation Coefficient

Abstract The assortativity index (A. Index) of a complex network has been hitherto computed as the Pearson’s correlation coefficient of the remaining degree centrality (R-DEG) of the first-order neighbors (i.e. end vertices of the edges) in the network. In this paper, we seek to evaluate the assortativity of real-world networks with respect to prototypical centrality metrics (in addition to R-DEG) such as eigenvector centrality (EVC), betweenness centrality (BWC) and closeness centrality (CLC). Unlike R-DEG, the centrality values of the vertices with respect to these three metrics are influenced by the centrality values of the vertices in the neighborhood. We propose to use the notion of ‘Partial Correlation Coefficient’ to nullify the influence of the second-order neighbors (i.e. vertices that are two hops away) and quantify the assortativity of the first-order neighbors with respect to a particular centrality metric (such as EVC, BWC and CLC). We conduct an exhaustive assortativity analysis on a suite of 70 real-world networks of diverse degree distributions. We observe real-world networks to be more assortative (A. Index > 0) with respect to CLC and EVC and relatively more dissortative (A. Index < 0) with respect to BWC and R-DEG.

Centrality Metrics, Measures, and Real-World Network Graphs

2018 ◽  
pp. 1-33
Keyword(s):  
Real World ◽  
Betweenness Centrality ◽  
Closeness Centrality ◽  
Clustering Coefficient ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Centrality Metrics ◽  
Edge Betweenness ◽  
Correlation Measures ◽  
Local Clustering Coefficient

This chapter provides an introduction to various node and edge centrality metrics that are studied throughout this book. The authors describe the procedure to compute these metrics and illustrate the same with an example. The node centrality metrics described are degree centrality (DEG), eigenvector centrality (EVC), betweenness centrality (BWC), closeness centrality (CLC), and the local clustering coefficient complement-based degree centrality (LCC'DC). The edge centrality metrics described are edge betweenness centrality (EBWC) and neighborhood overlap (NOVER). The authors then describe the three different correlation measures—Pearson's, Spearman's, and Kendall's measures—that are used in this book to analyze the correlation between any two centrality metrics. Finally, the authors provide a brief description of the 50 real-world network graphs that are studied in some of the chapters of this book.

Centrality-Based Assortativity Analysis of Complex Network Graphs

2018 ◽  
pp. 66-77
Keyword(s):  
Complex Network ◽  
Real World ◽  
Betweenness Centrality ◽  
Closeness Centrality ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Centrality Metrics

In this chapter, the author analyzes the assortativity of real-world networks based on centrality metrics (such as eigenvector centrality, betweenness centrality, and closeness centrality) other than degree centrality. They seek to evaluate the levels of assortativity (assortative, dissortative, neutral) observed for real-world networks with respect to the different centrality metrics and assess the similarity in these levels. The author observes real-world networks are more likely to be neutral (neither assortative nor dissortative) with respect to both R-DEG and BWC, and more likely to be assortative with respect to EVC and CLC. They observe the chances of a real-world network to be dissortative with respect to these centrality metrics to be very minimal. The author also assesses the extent to which they can use the assortativity index (A.Index) values obtained with a computationally light centrality metric to rank the networks in lieu of the A.Index values obtained with a computationally heavy centrality metric.

Clique Size and Centrality Metrics for Analysis of Real-World Network Graphs

2019 ◽  
pp. 1183-1198
Author(s):  
Natarajan Meghanathan
Keyword(s):  
Real World ◽  
Random Network ◽  
Maximal Clique ◽  
Closeness Centrality ◽  
Node Degree ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Scale Free ◽  
Centrality Metrics ◽  
Free Network

The authors present correlation analysis between the centrality values observed for nodes (a computationally lightweight metric) and the maximal clique size (a computationally hard metric) that each node is part of in complex real-world network graphs. They consider the four common centrality metrics: degree centrality (DegC), eigenvector centrality (EVC), closeness centrality (ClC), and betweenness centrality (BWC). They define the maximal clique size for a node as the size of the largest clique (in terms of the number of constituent nodes) the node is part of. The real-world network graphs studied range from regular random network graphs to scale-free network graphs. The authors observe that the correlation between the centrality value and the maximal clique size for a node increases with increase in the spectral radius ratio for node degree, which is a measure of the variation of the node degree in the network. They observe the degree-based centrality metrics (DegC and EVC) to be relatively better correlated with the maximal clique size compared to the shortest path-based centrality metrics (ClC and BWC).

Clique Size and Centrality Metrics for Analysis of Real-World Network Graphs

2018 ◽  
pp. 6507-6521
Author(s):  
Natarajan Meghanathan
Keyword(s):  
Real World ◽  
Random Network ◽  
Maximal Clique ◽  
Closeness Centrality ◽  
Node Degree ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Scale Free ◽  
Centrality Metrics ◽  
Free Network

We present correlation analysis between the centrality values observed for nodes (a computationally lightweight metric) and the maximal clique size (a computationally hard metric) that each node is part of in complex real-world network graphs. We consider the four common centrality metrics: degree centrality (DegC), eigenvector centrality (EVC), closeness centrality (ClC) and betweenness centrality (BWC). We define the maximal clique size for a node as the size of the largest clique (in terms of the number of constituent nodes) the node is part of. The real-world network graphs studied range from regular random network graphs to scale-free network graphs. We observe that the correlation between the centrality value and the maximal clique size for a node increases with increase in the spectral radius ratio for node degree, which is a measure of the variation of the node degree in the network. We observe the degree-based centrality metrics (DegC and EVC) to be relatively better correlated with the maximal clique size compared to the shortest path-based centrality metrics (ClC and BWC).

Centrality-Based Connected Dominating Sets for Complex Network Graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 1-24 ◽  
Author(s):  
Natarajan Meghanathan
Keyword(s):  
Complex Network ◽  
Real World ◽  
Closeness Centrality ◽  
Degree Centrality ◽  
Dominating Sets ◽  
Eigenvector Centrality ◽  
Connected Dominating Sets ◽  
The Core ◽  
Centrality Metrics ◽  
Betweeness Centrality

The author proposes the use of centrality-metrics to determine connected dominating sets (CDS) for complex network graphs. The author hypothesizes that nodes that are highly ranked by any of these four well-known centrality metrics (such as the degree centrality, eigenvector centrality, betweeness centrality and closeness centrality) are likely to be located in the core of the network and could be good candidates to be part of the CDS of the network. Moreover, the author aims for a minimum-sized CDS (fewer number of nodes forming the CDS and the core edges connecting the CDS nodes) while using these centrality metrics. The author discusses our approach/algorithm to determine each of these four centrality metrics and run them on six real-world network graphs (ranging from 34 to 332 nodes) representing various domains. The author observes the betweeness centrality-based CDS to be of the smallest size in five of the six networks and the closeness centrality-based CDS to be of the smallest size in the smallest of the six networks and incur the largest size for the remaining networks.

Use of Centrality Metrics for Ranking of Courses Based on Their Relative Contribution in a Curriculum Network Graph

2018 ◽  
pp. 129-159
Keyword(s):  
Betweenness Centrality ◽  
Directed Acyclic Graph ◽  
Degree Centrality ◽  
Weighted Sum ◽  
Eigenvector Centrality ◽  
Network Graph ◽  
Curriculum Assessment ◽  
Acyclic Graph ◽  
Relative Contribution ◽  
Centrality Metrics

The author proposes a centrality and topological sort-based formulation to quantify the relative contribution of courses in a curriculum network graph (CNG), a directed acyclic graph, comprising of the courses (as vertices), and their pre-requisites (captured as directed edges). The centrality metrics considered are out-degree and in-degree centrality along with betweenness centrality and eigenvector centrality. The author normalizes the values obtained for each centrality metric as well as the level numbers of the vertices in a topological sort of the CNG. The contribution score for a vertex is the weighted sum of the normalized values for the vertex. The author observes the betweenness centrality of the vertices (courses) to have the largest influence in the relative contribution scores of the courses that could be used as a measure of the weights to be given to the courses for curriculum assessment and student ranking as well as to cluster courses with similar contribution.

Computationally Light vs. Computationally Heavy Centrality Metrics

2018 ◽  
pp. 34-65
Keyword(s):  
Betweenness Centrality ◽  
Clustering Coefficient ◽  
Degree Centrality ◽  
Strongly Correlated ◽  
Eigenvector Centrality ◽  
Correlation Measure ◽  
Centrality Metrics ◽  
Two Measures ◽  
Local Clustering Coefficient ◽  
Spearman’S Correlation

In this chapter, the authors analyze the correlation between the computationally light degree centrality (DEG) and local clustering coefficient complement-based degree centrality (LCC'DC) metrics vs. the computationally heavy betweenness centrality (BWC), eigenvector centrality (EVC), and closeness centrality (CLC) metrics. Likewise, they also analyze the correlation between the computationally light complement of neighborhood overlap (NOVER') and the computationally heavy edge betweenness centrality (EBWC) metric. The authors analyze the correlations at three different levels: pair-wise (Kendall's correlation measure), network-wide (Spearman's correlation measure), and linear regression-based prediction (Pearson's correlation measure). With regards to the node centrality metrics, they observe LCC'DC-BWC to be the most strongly correlated at all the three levels of correlation. For the edge centrality metrics, the authors observe EBWC-NOVER' to be strongly correlated with respect to the Spearman's correlation measure, but not with respect to the other two measures.

Centrality Metrics-Based Connected Dominating Sets for Real-World Network Graphs

2019 ◽  
pp. 1-29
Author(s):  
Natarajan Meghanathan ◽  
Md Atiqur Rahman ◽  
Mahzabin Akhter
Keyword(s):  
Shortest Path ◽  
Real World ◽  
Degree Centrality ◽  
Dominating Sets ◽  
Eigenvector Centrality ◽  
Connected Dominating Sets ◽  
The Real ◽  
Centrality Metrics ◽  
Local Clustering Coefficient ◽  
Node Size

The authors investigate the use of centrality metrics as node weights to determine connected dominating sets (CDS) for a suite of 60 real-world network graphs of diverse degree distribution. They employ centrality metrics that are neighborhood-based (degree centrality [DEG] and eigenvector centrality [EVC]), shortest path-based (betweenness centrality [BWC] and closeness centrality [CLC]) as well as the local clustering coefficient complement-based degree centrality metric (LCC'DC), which is a hybrid of the neighborhood and shortest path-based categories. The authors target for minimum CDS node size (number of nodes constituting the CDS). Though both the BWC and CLC are shortest path-based centrality metrics, they observe the BWC-based CDSs to be of the smallest node size for about 60% of the real-world networks and the CLC-based CDSs to be of the largest node size for more than 40% of the real-world networks. The authors observe the computationally light LCC'DC-based CDS node size to be the same as the computationally heavy BWC-based CDS node size for about 50% of the real-world networks.

Temporal Variation of the Node Centrality Metrics for Scale-Free Networks

2018 ◽  
pp. 78-97
Keyword(s):  
Temporal Variation ◽  
Closeness Centrality ◽  
The Other ◽  
Centrality Measures ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Scale Free ◽  
Centrality Metrics ◽  
Scale Free Networks ◽  
Free Network

Scale-free networks are a type of complex networks in which the degree distribution of the nodes is according to the power law. In this chapter, the author uses the widely studied Barabasi-Albert (BA) model to simulate the evolution of scale-free networks and study the temporal variation of degree centrality, eigenvector centrality, closeness centrality, and betweenness centrality of the nodes during the evolution of a scale-free network according to the BA model. The model works by adding new nodes to the network, one at a time, with the new node connected to m of the currently existing nodes. Accordingly, nodes that have been in the network for a longer time have greater chances of acquiring more links and hence a larger degree centrality. While the degree centrality of the nodes has been observed to show a concave down pattern of increase with time, the temporal (time) variation of the other centrality measures has not been analyzed until now.

scholarly journals Assortativity Analysis of Real-World Network Graphs based on Centrality Metrics

2016 ◽  
Vol 9 (3) ◽  
pp. 7 ◽  
Author(s):  
Natarajan Meghanathan
Keyword(s):  
Real World ◽  
Closeness Centrality ◽  
Degree Centrality ◽  
Font Size ◽  
Network Graph ◽  
Node Level ◽  
Centrality Metrics ◽  
Empirical Probability ◽  
Different Levels

<p><span style="font-size: 10.5pt; font-family: 'Times New Roman','serif'; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US">Assortativity index (<em>A. Index</em>) of real-world network graphs has been traditionally computed based on the degree centrality metric and the networks were classified as assortative, dissortative or neutral if the <em>A. Index</em> values are respectively greater than 0, less than 0 or closer to 0. In this paper, we evaluate the <em>A. Index</em> of real-world network graphs based on some of the commonly used centrality metrics (betweenness, eigenvector and closeness) in addition to degree centrality and observe that the assortativity classification of real-world network graphs depends on the node-level centrality metric used. We also propose five different levels of assortativity (strongly assortative, weakly assortative, neutral, weakly dissortative and strongly dissortative) for real-world networks and the corresponding range of <em>A. Index</em> value for the classification. We analyze a collection of 50 real-world network graphs with respect to each of the above four centrality metrics and estimate the empirical probability of observing a real-world network graph to exhibit a particular level of assortativity. We claim that a real-world network graph is more likely to be neutral with respect to the betweenness and degree centrality metrics and more likely to be assortative with respect to the eigenvector and closeness centrality metrics.</span></p>

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