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

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

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.

Use of Eigenvector Centrality for Mobile Target Tracking

The author proposes an eigenvector centrality (EVC)-based tracking algorithm to trace the trajectory of a mobile radioactive dispersal device (RDD) in a wireless sensor network. They propose that the sensor nodes simply sum up the strengths of the signals (including those emanating from a RDD) sensed in the neighborhood over a sampling time period and forward the sum of the signals to a control center (called sink). For every sampling time period, the sink constructs an adjacency matrix in which the entry for edge (i, j) is the sum of the signal strengths reported by sensor nodes i and j, and uses this adjacency matrix as the basis to determine the principal eigenvector whose entries represents the EVCs of the vertices with respect to the radioactive signals sensed in the neighborhood. The author proposes that the arithmetic mean (calculated by the sink) of the X and Y coordinates of the suspect sensor nodes (those with higher EVCs) be considered as the predicted location of the RDD at a time instant corresponding to the middle of the sampling time period.

Edge Centrality Metrics to Quantify the Stability of Links for Mobile Sensor Networks

In this chapter, we explore the use of neighborhood overlap (NOVER), bipartivity index (BPI) and algebraic connectivity (ALGC) as edge centrality metrics to quantify the stability of links for mobile sensor networks. In this pursuit, we employ the notion of the egocentric network of an edge (comprising of the end vertices of the edge and their neighbors as nodes, and the edges incident on the end vertices as links) on which the above three edge centrality metrics are computed. Unlike the existing approach of using the predicted link expiration time (LET), the computations of the above three edge centrality metrics do not require the location and mobility information of the nodes. For various scenarios of node density and mobility, we observe the stability of the network-wide data gathering trees (lifetime) determined using the proposed three edge centrality metrics to be significantly larger than the stability of the LET-based data gathering trees.

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

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.

Centrality Metrics, Measures, and Real-World Network Graphs

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

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.