The preceding discussion is not readily applicable to the case of points on networks, such as accidents along roads or thefts in urban areas. In these instances one might attempt to compute the density by applying a two-dimensional kernel procedure as described in the preceding section, overlaying the network, and examining the density patterns along the network routes. However, this approach has been shown to produce biased density estimates (e.g. it suggests variable densities along routes when these are known to be uniform). To resolve this problem kernel density computations in situations where events occur along or adjacent to networks, should be calculated using shortest paths and adjusted kernel functions (the conventional functions described earlier are not suitable without modification. This question has been addressed by Okabe et al. (2009) and is being made available in a new release of their SANET software. They show that a non-continuous fixed bandwidth kernel function, which is unimodal and symmetric, can be used to compute unbiased kernel densities on large networks (e.g. the street network of Kashiwa, Japan, with 35,235 links, 25,146 nodes and 3109 point events). The function used is essentially a modified form of the linear or triangular kernel (Figure 4‑47F), applied to networks which may divide into multiple segments at junctions, but for which the integral of the kernel function is held at unity. The authors suggest that for problems of the type they have investigated (i.e. dense, urban networks, the bandwidth should be 100-300metres.