Partitioning a graph into subgraphs with overlapping nodes
I never heard of any algorithm with the constraint of having an overlap between communities larger than a given threshold (4 here).
But I suggest the following: turn your graph into its line graph, use a classical node partitioning method that gives a hierarchy of communities, and then choose a partition that fits your requirements.
More details:

the line graph is the graph in which the nodes are the links of your original graph, and these nodes are linked together if corresponding links have an extremity in common, see wikipedia on line graph.
classical community detection algorithms partition nodes into disjoint sets; doing so on the line graph will produce a partition of the edge set, with some nodes in several communities (overlap).
many classical community detection algorithms actually produce a hierarchical decomposition, often encoded as a dendrogram, i.e. a tree of communities, sub-communities, sub-sub-communities, and so on, see wikipedia on dedrograms.
a partition into communities then is a choice of a place to cut this tree, wherever it fits your needs; for instance, one may maximize a quality function and a scale parameter, see e.g. Post-Processing Hierarchical Community Structures: Quality Improvements and Multi-scale View
therefore you may choose a cut maximizing a quality function while satisfying your requirement; you may for instance start with the optimal partition regarding the quality function, and then move the cut to fit your overlap requirement.

Good luck, please keep us informed of any progress :)