## Brief Introduction to Networks

Network analysis is the practical application of graph theory. Graph theory is the study of mathematical structures used to model pair-wise relations between objects. Networks (or graphs) have been constructed and studied for individuals, groups, transportation, or occurrences from a wide range of disciplines including computer science, biology, economics, political science, and sociology. In fact, an early application of network analysis was in the area of social interaction. Network analysis has recently been introduced into the study of climate by Tsonis et al. (2006; 2007). Because it is rather new to climatology and has not yet, to our knowledge, been applied to hurricanes, we begin with an introduction using concepts from social network analysis (Scott 1991; Wasserman and Faust 1994).

Consider authors publishing in the field of hurricane climatology. Authors can be represented as nodes with links to other authors established through a scientific citation. If author A is cited by authors B and C then a network is established between the authors. The connections between authors are called vertexes or nodes and the links connecting them are called edges. Figure 1 is a hypothetical example of a social network of authors linked by citation. If author B cites author A at least once then an arrow from B to A is drawn. If two authors cite each other a double arrow is used.

First note that the network is aspatial meaning that the absolute and relative positions of the nodes and links in the graph on the page are arbitrary. Instead what is important are the number of nodes and their linkages. Here our hypothetical network consists of 4 nodes and 5 links. The network is a concise way to examine relations. For instance the network shows that author A is cited by the three other authors so its node has the highest in-node value. While author A generates citations, he tends not to give them out. In contrast, author D is the only one that cites the other three authors so its node has the highest out-node value. Also author B does not cite author C and vice versa. But authors B and C are connected through authors A and D since B cites D who cites C and since B cites A who is cited by and cites C. This is an example of a directed graph since the links have arrows. In an undirected graph all links point both ways so no arrows are used. This is the case when the relationship between nodes is transitive. For example, if the network

Fig. 1 Hypothetical social network of authors publishing in the field of hurricane climate. The authors (A, B, C, and D) are represented with circles (nodes) and the links indicating at least one citation are indicated with arrows represents scientists who author papers and the links are co-authorships then all relations are transitive and the links do not have arrow heads.

The configuration of links among the network nodes reveals the network structure. A path connecting two nodes is a sequence of distinct nodes and links beginning with the first node and terminating with the last. For the example, above node B is connected to node C through A or through D, so that the path is BAC or BDC. If there is a path between two nodes then the nodes are said to be reachable. The length of the path is the number of links. So the length of the path from A to C is one and from B to C is two. However, another path from B to C is through A and D in which case the length between B and C is three. A shortest path between two nodes is called a geodesic. The diameter of the network is the length of the longest geodesic between all pairs of nodes in the graph. Therefore the maximum geodesic distance between any pair of nodes is the diameter. Interestingly, although the network is aspatial, many of the terms used in network analysis suggest spatial or geometric representations, including centrality, distance, isolation, and diameter. 