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I want to do a process where for each street segment, the network distance from the nearest point in a set of point features is calculated. The result is that when you click on a street segment using identify, one of the fields shows the network distance to the nearest point.

For euclidean straight line distance, this process is very simple using the Near(analysis) tool. I pretty much want to do the exact same thing except using network distance instead.

I have already created and built a network layer so no need for info on that.

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The task you have can be solved by using the Closest Facility solver in Network Analyst. This will let you find the closest features and report the travel time / distance from every input points to those features.

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