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I would like to do some cluster analysis. I created a clusters based on the distribution of points. Now I want to perform the next step, but I can't find the tool in ArcGIS Pro (name of the tool). I believe it is some kind of network analysis tool. Basically I want to know the distances and connections between each cluster. The example on the print screen shows what I mean.

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Sounds like you are looking for the Point Distance tool in the Analysis Toolbox. To get distances between points in the same layer, use that layer as both in features and near features.

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