## Example

The network is constructed in three steps. In step 1, an incidence matrix is obtained that shows the occurrence of hurricanes by regions. In step 2 an adjacency matrix is computed from the incidence matrix using matrix algebra. In step 3, the network graph is drawn from the symmetry of the adjacency matrix. To see how this works, consider the following hypothetical table of hurricane occurrences. Hurricane 1 (H1) affected regions 1 (R1) and 3 (R3). Hurricane 2 (H2) affected regions 1 and 2, and so on. We therefore have a 4 x 5 (hurricanes x region) incidence matrix called X. Then a 5 x 5 adjacency matrix A (Table 2) is computed by pre-multiplying the incidence matrix by its transpose.

 R1 R2 R3 R4 R5 H1 1 0 1 0 0 H2 1 1 0 0 1 H3 0 1 0 1 0 H4 1 0 0 0 0

Table 2 The adjacency matrix constructed from the hypothetical incidence matrix shown in Table 1. Here we see that region 1 is connected to regions 2, 3, and 5 since there was at least one hurricane to hit region 1 that went on to, or came from, these other regions. The diagonal elements of the matrix which consist of the frequency of hurricanes in each region are not used to construct the network

Table 2 The adjacency matrix constructed from the hypothetical incidence matrix shown in Table 1. Here we see that region 1 is connected to regions 2, 3, and 5 since there was at least one hurricane to hit region 1 that went on to, or came from, these other regions. The diagonal elements of the matrix which consist of the frequency of hurricanes in each region are not used to construct the network

 R1 R2 R3 R4 R5 R1 - 1 1 0 1 R2 1 - 0 1 1 R3 1 0 - 0 0 R4 0 1 0 - - R5 1 1 0 0 -

Fig. 5 Network graph based on the hypothetical set of hurricanes listed in Table 1. The network is constructed from the adjacency matrix shown in Table 2. Region

2 (R2) is connected to regions 1,4, and 5

Note that the adjacency matrix is symmetric with the value in row R1 and column R2 matching the value in column R1 and row R2 and so on. The network is constructed directly from the adjacency matrix where values of 1 indicate a link between the regions. The diagonal values are ignored. Figure 5 shows a graph of the network. Regions 1 and 2 each have three links; region 5 has two links and regions 3 and 4 each have one link. As mentioned, since we do not distinguish the time order of hits, the links are undirected.

### Full Network

The above example explains the steps we use to construct our U.S. hurricane network. Note that it is certainly possible to construct other networks with the same data, but here we limit ourselves to this straightforward approach. Figure 6 shows the incidence matrix, adjacency matrix, and network graph for the 275 hurricanes affecting the United States during the period 1851 through 2005. The incidence matrix has 275 rows and 23 columns while the adjacency matrix has dimensions 23 by 23. The network graph is plotted directly from the adjacency matrix. All the algebra, plots, and network analysis are done using the R language (R Development Core Team 2006).

The U.S. hurricane network shows the linkages between regions affected by the same hurricane. In small coastal states or regions a single hurricane can affect more than one region as is the case in the northeast. However, hurricanes affecting Florida frequently travel on to affect other non contiguous coastal regions.

As noted above, the network can be mapped in different ways. Figure 7 shows the U.S. hurricane network mapped onto a circle and onto the coastline. The circle map makes it easier to see the linkages resulting from traveling hurricanes. In particular we note relatively high number of links between northeastern Florida and the regions of New England.

Storm Number

Storm Number

Fig. 6 Incidence matrix (a), adjacency matrix (b), and network (c) of U.S. hurricanes. The storm number in the incidence matrix refers to the consecutive list of hurricanes since 1851. The black squares in the adjacency matrix indicate regions connected by at least one hurricane
Fig. 7 The U.S. hurricane network mapped onto a circle (a) and onto the coastal geography (b)