By Renfrey B. Potts
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Where E is the n x 1 node-link incidence matrix, f" is the 1 x 1 link flow vector with elements equal to the link flows of copy a, and g" the n x 1 copy flow vector with elements '1 u", g! = -via, 0, if i is the origin of copy flow a, if i is a destination of copy flow u, (23) otherwise. The total link flow and the total network flow are given by superposing the copy flows: 35 8. Flows and Conservation Laws f = Era, a a a i Copy flows have been labeled according to their origin. It is evident that we could just as easily label them by their destinations so that a particular copy flow would be a flow from all origins to a particular destination.
If a cordon line is a ring road encircling the Central Business District of a city, X can be chosen as the set of nodes within the CBD, X the set 24 11. Elements of Network Theory of nodes outside, and (X,x) is the set of roads crossing the ring road outwards from within the CBD. A traffic count on these roads at “external” stations on the ring road would enable the total traffic from within the CBD outwards to be calculated. (f) Undirected and Mixed Graphs For an undirected graph [ N ; L ] , the elements of L are unordered pairs of elements of N and are denoted by (i,j) or ( j ,i ) .
E ) Compressibility and Separability In some of the transportation network problems that we discuss in this text, other conservation equations besides those derived from Kirchhoff's law will be imposed. It frequently happens that the graph [ N ; L ] of a given transportation network is compressed or expanded into a new network [ N ' ; L ' ] . The compressing process may occur, for example, when several nodes and links in [ N ; L ] are combined to form [ A " ; L ' ] ; hence an analysis of flows on the larger network [ N ; L ] is replaced by one on the smaller network [ A " ; L'], where N' c N , L' c L.