Local rules associated to k-communities in an attributed graph

We address the problem of finding local patterns and local rules in an attributed graph. A (global) closed pattern is the most specific attribute pattern shared by the vertices of the (possibly simplified) subgraph induced by some attribute pattern. A local closed pattern is the maximal attribute pattern associated to a particular dense region of this subgraph. As such local regions, we are in particular interested in k-communities of pattern subgraphs. In this case we show that there is a closure operator such that, given a pattern q subgraph and a k-community in this subgraph, returns the local closed pattern shared by all the members of the community. We then consider how to generate triples (c, e, l) where c is a (global) closed pattern whose subgraph contains e as a k-community, and l is the corresponding local closed pattern. This leads to implication rules expressing what new attributes are specific of the k-community e in the pattern c subgraph.