314 Network Data Structures

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Network data structures for geographic informationscience (GISci) are methods for storing network datasets in a computer in order to support a range of net-work analysis procedures. Network data sets areamong the most common in GISci and include trans-portation networks (e.g., road or railroads), utilitynetworks (e.g., electricity, water, and cable networks),and commodity networks (e.g., oil and gas pipelines),among many others. Network data structures muststore the edge and vertex features that populate thesenetwork data sets, the attributes of those features, and,most important, the topological relationships amongthe features. The choice of a network data structurecan significantly influence one's ability to analyze theprocesses that take place across networks. This entrydescribes the mathematical basis for network datastructures and reviews several major types of networkdata structures as they have been implemented in geo-graphic information systems (GIS).

Graph Theoretic Basisof Network Data Structures

The mathematical subdiscipline that underlies networkdata structures is termed graph theory. Any graph ornetwork (the terms are used interchangeably in thiscontext) consists of connected sets of edges and ver-tices. Edges may also be referred to as lines or arcs, andvertices may be termed junctions, points, or nodes.Within graph theory there are methods for measuringand comparing graphs and principles for proving theproperties of individual graphs or classes of graphs.

Graph theory is not concerned with the shape of thefeatures that constitute a network, but rather with thetopological properties of those networks. The topo-logical invariants of a graph are those properties thatare not altered by elastic deformations (such as astretching or twisting). Therefore, properties such asconnectivity, adjacency, and incidence are topologicalinvariants of networks, since they will not changeeven if the network is deformed by a cartographicprocess. The permanence of these properties allowsthem to serve as a basis for describing, measuring, andanalyzing networks.

Graph theoretic descriptions of networks caninclude statements of the number of features in the net-work, the degree of the vertices of the graph (where the

degree of a vertex is the number of edges incident to it),or the number of cycles in a graph. Descriptions of net-works can also be based on structural characteristics ofgraphs, which allow them to be grouped into idealizedtypes. Perhaps the most familiar type is tree networks,which have edge "branches" incident to nodes, but nocycles are created by the connections among thosenodes. River networks are nearly always modeled astree networks. Another common idealized graph type isthe Manhanan network, which is made up of edgesintersecting at right angles. This creates a series of rec-tangular "blocks" that approximate the street networkscommon in many U.S. cities. Other idealized typesinclude bipartite graphs and hub-and-spoke networks.

If one wishes to quantitatively measure propertiesof graphs rather than simply describe them, there is aset of network indices for that purpose. The simplestof these is the Beta index, which measures the con-nectivity of a graph by comparing the number ofedges to the number of vertices. A more connectedgraph will have a larger Beta index ratio, since rela-tively more edges are connecting the vertices. TheAlpha and Gamma indices of connectivity compareproven properties of graphs with observed properties.The Alpha index compares the maximum possiblenumber of fundamental cycles in the graph to theactual number of fundamental cycles in the graph.Similarly, the Gamma index compares the maximumpossible number of edges in a graph to the actualnumber of edges in a graph. In each case, as the lattermeasure approaches the former, the graph is morecompletely connected. Other measures exist forapplied instances of networks and consequentlydepend on nontopological properties of the network.The reader is directed to textbooks on the topicof graph theory for a more comprehensive review ofthese and other more advanced techniques.

lmplementations ofNetwork Data Structures in GtS

No nto po I og I ca I D ata Structu resWhile the graph theoretic definition of a network

remains constant, the ways in which networks arestructured in computer systems have changed dramat-ically over the history of GISci. The earliest com-puter-based systems for automated cartography storednetwork edges as independent records in a database.Each record contained a starting and ending point for

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Table 1 Star Data Structure Vertex List

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The DIME data structure evolved into the structureemployed for the Topologically Integrated GeographicEncoding and Referencing (TIGER) files that are stillused by the Census Bureau to delineate population tab-ulation areas. There are many advantages of the dual-incidence data structure, and the wide acceptance of thedata sfructure combined with the comprehensive natureof the TIGER files led to its status as the de facto stan-dard for vector representations in GIS. Two elements ofthis advance profoundly influenced the ability to con-duct network analysis in GIS. First, the DIME structuecaptures incidence, which is one of the primary topo-logical properties defining the structure of networks. Ascan be seen in Figure 1, all edges that are incident to agiven point can be determined with a simple databasequery. Second, many of the features captured by theCensus Bureau were streets or other transportation fea-tures. Since the Census Bureau has a mandate that cov-ers the entire United States, this meant that a nationaltransportation database was available for use in GIS,and this database was captured in a structure that couldsupport high-level network analysis.

However, the dual-incidence topological datamodel also imposes some difficult constraints on net-work analysts. The Census Bureau designed the datastructure in order to well define polygons with whichpopulations could be associated. To do so, the datamodel had to enforce planarity. Planar graphs arethose that can be drawn in such a way that no twoedges cross without a vertex at that location. Thus, atevery location where network features cross, a pointmust exist in the database. This is true regardless ofwhether or not a true intersection exists between thenetwork features, and it is most problematic whenmodeling bridges or tunnels. While network featurescertainly cross each other at bridges, there is no inci-dence between the features, and network analysisshould not permit flow between features at that point.Moreover, since planar enforcement demands thatnetwork features (such as roads) be divided at every

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