Summary: We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. A. Okabe and I. Yamada developed a ‘network \(K\) function’ [Geographical Anal. 33, No. 3, 271–290 (2001), doi:10.1111/j.1538-4632.2001.tb00448.x], analogous to Ripley’s \(K\) function, for analysis of such data. However, values of the network \(K\)-function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K-function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of \(K\), including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network \(K\)-function is the identity. The corrected estimator \(\hat{K}(r)\) is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of \(\hat{K}(r)\) under complete randomness. We extend these results to an ’inhomogeneous’ network \(K\)-function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).