Given a set of points in three-dimensional space, which are the most common ways of measuring if they are co-localized or not? E.g., a graph-based approach would be to build a graph using points as vertices, connecting them by edges if they are below a distance threshold and them applying a connected components algorithms to see if the result is a single component. Continuous measures instead of discrete ones are welcome as well.
Two issues seem to intersect in your question. (1) are the points all in a similar local area; (2) is the graph constructed according to some proximity rule connected. I think if the answer to (2) is yes, the points are clustered. But the "proximity rule" that you use has to be carefully chosen ... if it is too weak, (large distance threshold) of course everything will be connected.
How about this -- if the data set is not HUGE develop a matrix of all inter-point distances.
(D(a,b) = some metric measure of distance between observation a and b.)
Then, apply a hierarchical agglomerative clustering scheme to this -- join the two closest points, replace that with their mean, join the next two etc. As you reach the end of the run, ask yourself are you needing to join points that are really outliers to the existing cluster; if yes, you probably have disconnected pieces. The resulting "dendrogram" will also allow you to slice across horizontally to see what observations are in the clusters.
You could also try a minimum weight spanning tree (can be done with a greedy algorithm) and see if you like the diameter of the resultant network? Just some ideas.
This sounds somewhat related to principal coordinate analysis. In R, we can compute a square matrix of pairwise euclidean distances between points (function dist). If the "coordinates" reflect some other sort of data rather than just points in space, other distance measures might be more appropriate (for example, for log-transformed gene expression values, manhattan distance should be used). Then, if we have some a priori hypothesis about how the points should be clustered (say, if the points correspond to measures taken under different conditions and we want to test whether there was a significant difference between these conditions), there is function adonis (package vegan) to test this. If there is no a priori hypothesis then all we can do is to find the most optimal way of clustering (no statistical testing, though). For this we can use, for example, k-means (function kmeans) or hierarchical clustering (function hclust) algorithms.
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