Global and Local Metrics of the Network

Next we examine various structural properties (local and global) of the network. We use the R routines developed by Butts (2006) under the sna package and by Csardi (2007) under the igraph package. We consider three measures of nodal centrality. Where centrality is loosely defined as being in the "middle" of the network. Middle nodes are nodes that are connected to many other nodes in the network. They are considered structurally important to the network. The three measures we consider are degree, closeness, and betweenness.

The "degree" (prestige) of a node (vertex) is its most basic structural property, the number of links (edges) connected to it. In the hurricane network the degree of the node is the number of regions that have been affected by a hurricane affecting the particular location. Figure 8 shows a bar plot of the nodal degree. Here we see that the south Texas node has degree of 6 since it is linked to 6 other regions including central Texas, north Texas, Louisiana, northwest Florida, southwest Florida, and southeast Florida. In comparison, the central Texas node has degree 2 being linked only to south and north Texas. Nodes with the largest degree include southwest Florida and North Carolina.

Paths through the network are the successive links between the nodes. One path from south Texas to Maine is constructed by starting in south Texas and following the link to northwest Florida. Since northwest Florida is linked to North Carolina, which is linked to Maine a path of length 3 links south Texas with Maine. The shortest path between any two nodes is called the geodesic. The shortest path between south Texas and Maine is 2 (through southwest Florida). A node's "closeness'' provides an index for the extent to which it has short paths to all other nodes in the graph. Mathematically it is defined as

where d(i,j) is the geodesic distance between nodes Ï and j and I V(G)I is the number of nodes in the network. Figure 9 shows the closeness by region.

Another important property of network nodes is called "betweenness." Betweenness is defined as the number of geodesic paths that pass through a node.

ATX BTX CTX LA MS AL AFL BFL ÇFL OFL OA SC NC VA MD DE NJ NY CT HI MA NH ME

Fig. 8 Node degree. The node degree is the number of links connected to the node. Here the node degree represents the number of regions affected by hurricanes that have affected the particular region. For instance, south Texas (ATX) has degree 6 meaning that 6 other regions have been affected by hurricanes affecting south Texas

ATX BTX CTX LA MS AL AFL BFL ÇFL OFL OA SC NC VA MD DE NJ NY CT HI MA NH ME

Fig. 8 Node degree. The node degree is the number of links connected to the node. Here the node degree represents the number of regions affected by hurricanes that have affected the particular region. For instance, south Texas (ATX) has degree 6 meaning that 6 other regions have been affected by hurricanes affecting south Texas

Fig. 9 Node closeness. The node closeness is an index that quantifies the number of paths through the node that are geodesies
Fig. 10 Node betweenness. Node betweenness is defined as the number of geodesic paths that pass through a node

It is the number of "times" that any node needs to go through a given node to reach any other node by the shortest path. Conceptually, nodes with high betweenness lie on a large number of non-redundant shortest paths between other nodes; thus these nodes can thus be thought of as "bridges." A redundant path is one in which the path is traversed by more than one hurricane. Figure 10 shows betweenness values for each of the nodes in the hurricane network.

Global properties of the network may also be of interest. For instance, the diameter of the network can be defined as the maximum geodesic distance over the network. Here we find that this distance is 5 for the U.S. landfall network. Thus the maximum shortest path between any two nodes is 5 links. This path connects south Texas with New Hampshire and runs through central Texas, Alabama, New York, and Rhode Island. Note that these intermediate nodes tend to have small values of betweenness.

Another global property is the clustering coefficient. Returning to our example from section 2 where we considered the citation network of hurricane researchers, two authors are adjacent in the network if they site each other's work. Consider an author having two adjacent authors if these adjacent authors cite each other then we have a cluster or clique. The clustering coefficient of the entire network can be defined as the probability that adjacent nodes of a node are connected. The clustering coefficient for the U.S. hurricane network is 0.46 indicating that slightly less than half of all regions that are linked to a specific region are also linked together.

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