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I've been reading a little about neural networks in the brain, such as this paper.

Most models have a resolution parameter $γ$, and for each value of that parameter a network can be partitioned into communities. Let's say that high values correspond to smaller communities that are more highly correlated, while smaller values allow for larger and less correlated communities.

However, suppose that a group of neurons was a member of several communities. These communities would have a non-empty intersection that is highly correlated (we see it for high values of $γ$), but for lower values of $γ$ we would instead see two separate communities that are similarly correlated with the intersection but less correlated with each other (thus they aren't all a part of one big community).

As we reduce $\gamma$ further, we would still see both of these communities eventually merge into the union of all of them, so this split would only exist in a specific range.

The situation doesn't sound impossible, but neurons are always partitioned when talking about communities as far as I can tell, at least at specific resolutions. Is there a biological or other reason?

Shows two separate communities with an intersection

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Community means many different things; the answer depends on which of these you're talking about. Community often refers to the graph theory principle community structure, which often but not always involves partitioning nodes into discrete communities. Most of these methods do not allow for overlap, but some do; there's nothing in principle against it, these are all analysis techniques so you use or design the technique according to your goals.

"Community" can also refer to abstract measures of weighted graph organization; this sort of community does not actually involve any partitioning or defining community membership per se. For example, one might talk of "small worldedness" which is a feature of graphs where nodes are strongly connected locally but also have a short path between non-local nodes through hubs (nodes with high degree).

There's also the sense of community in terms of groups of neurons co-activated in response to a stimulus or in recollection of a memory. Usually the term used there is "ensemble" rather than "community". Many different ensembles can exist in an overlapping population; a single example . Some examples:

Wilson, M. A., & McNaughton, B. L. (1994). Reactivation of hippocampal ensemble memories during sleep. Science, 265(5172), 676-679.

Bathellier, B., Ushakova, L., & Rumpel, S. (2012). Discrete neocortical dynamics predict behavioral categorization of sounds. Neuron, 76(2), 435-449.

Harris, K. D. (2005). Neural signatures of cell assembly organization. Nature reviews neuroscience, 6(5), 399-407. (In this paper, Harris calls an ensemble an "assembly")

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