Centrality Analysis of the United States Network Graph

Centrality Analysis of the United States Network Graph

Natarajan Meghanathan (Jackson State University, USA)
Copyright: © 2018 |Pages: 11
DOI: 10.4018/978-1-5225-2255-3.ch152
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Abstract

We model the contiguous states (48 states and the District of Columbia) of the United States (US) as an undirected network graph with each state represented as a node and there is an edge between two nodes if the corresponding two states share a common border. We determine a ranking of the states in the US with respect to the four commonly studied centrality metrics: degree, eigenvector, betweenness and closeness. We observe the states of Missouri and Maine to be respectively the most central state and the least central state with respect to all the four centrality metrics. The degree distribution is bi-modal Poisson. The eigenvector and closeness centralities also exhibit Poisson distribution, while the betweenness centrality exhibits power-law distribution. We observe a higher correlation in the ranking of vertices based on the degree centrality and betweenness centrality.
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Introduction

Network Science is one of the emerging fields of Data Science to analyze real-world networks from a graph theory point of view. Several real-world networks have been successfully modeled as undirected and directed graphs to study the intrinsic structural properties of the networks as well as the topological importance of nodes in these networks. The real-world networks that have been subjected to complex network analysis typically fall under one of these categories: social networks (Ghali et. al., 2012), transportation networks (Cheung & Gunes, 2012), biological networks (Ma & Gao, 2012), citation networks (Zhao & Strotmann, 2015), co-authorship networks (Ding, 2011) and etc. One category of real-world networks for which sufficient attention has not yet been given are the regional networks featuring the states within a country. In this chapter, we present a comprehensive analysis of a network graph of the states within a country with respect to the four commonly used centrality metrics in complex network analysis (Newman, 2010): degree, eigenvector, betweenness and closeness centralities.

We opine the chapter to serve as a model for anyone interested in analyzing a connected graph of the states within a country from a Network Science perspective. The approaches presented in this chapter could be useful to determine the states (and their cities) that are the most central and/or influential within a country. For example, the ranking of the vertices based on the shortest path centrality metrics (closeness and betweenness) could be useful to choose the states (and their cities) that could serve as hubs for transportation networks (like road and airline networks). We could identify the states that are most the central states as well as identify the states that could form a connected backbone and geographically well-connected to the rest of the states within a country and use this information to design the road/rail transportation networks. The degree centrality and eigenvector centrality metrics as well as the network-level metrics like minimum connected dominating set and maximal clique size could be useful to identify fewer number of venues (with several adjacent states to draw people) for political campaigns/meetings that would cover the entire country.

Table 1.
List of contiguous states (including DC) of the US in alphabetical order
IDState/District NameCodeIDState NameCodeIDState NameCode
1AlabamaAL18MaineME34OhioOH
2ArizonaAZ19MarylandMD35OklahomaOK
3ArkansasAR20MassachusettsMA36OregonOR
4CaliforniaCA21MichiganMI37PennsylvaniaPA
5ColoradoCO22MinnesotaMN38Rhode IslandRI
6ConnecticutCT23MississippiMS39South CarolinaSC
7DelawareDE24MissouriMO40South DakotaSD
8District of ColumbiaDC25MontanaMT41TennesseeTN
9FloridaFL26NebraskaNE42TexasTX
10GeorgiaGA27NevadaNV43UtahUT
11IdahoID28New HampshireNH44VermontVT
12IllinoisIL29New JerseyNJ45VirginiaVA
13IndianaIN30New MexicoNM46WashingtonWA
14IowaIA31New YorkNY47West VirginiaWV
15KansasKS32North CarolinaNC48WisconsinWI
16KentuckyKY33North DakotaND49WyomingWY
17LouisianaLA

Key Terms in this Chapter

Eigenvector Centrality: A measure of the number of neighbors of the node as well as the degree centrality of the neighbor nodes.

Degree Centrality: A measure of the number of neighbor nodes for the node.

Closeness Centrality: A measure of the number of hops of the shortest paths from the node to every other node in the network.

Betweenness Centrality: A measure of the number of shortest paths that go through the node between any two vertices in the network.

Centrality: A quantitative measure of the importance of the node on the basis of the topological structure of the network.

Correlation Coefficient: A measure of the statistical relationship between two or more datasets.

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