The K-function method on a network and its computational implementation

This paper proposes two statistical methods, called the network K-function method and the network cross K-function method, for analyzing the distribution of points on a network. First, by extending the ordinary K-function method defined on a homogeneous infinite plane with the Euclidean distance, the paper formulates the K-function method and the cross K-function method on a finite irregular network with the shortest-path distance. Second, the paper shows advantages of the network K-function methods, such as that the network K-function methods can deal with spatial point processes on a street network in a small district, and that they can exactly take the boundary effect into account. Third, the paper develops the computational implementation of the network K-functions, and shows that the computational order of the K-function method is O(n(Q)(2) log n(Q)) and that of the network cross K-function is Q(n(Q) log n(Q)) where n(Q) is the number of nodes of a network.