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Many graph clustering quality functions suffer from a resolution limit, theinability to find small clusters in large graphs. So calledresolution-limit-free quality functions do not have this limit. This propertywas previously introduced for hard clustering, that is, graph partitioning.  We investigate the resolution-limit-free property in the context ofNon-negative Matrix Factorization (NMF) for hard and soft graph clustering. Touse NMF in the hard clustering setting, a common approach is to assign eachnode to its highest membership cluster. We show that in this case symmetric NMFis not resolution-limit-free, but that it becomes so when hardness constraintsare used as part of the optimization. The resulting function is strongly linkedto the Constant Potts Model. In soft clustering, nodes can belong to more thanone cluster, with varying degrees of membership. In this settingresolution-limit-free turns out to be too strong a property. Therefore weintroduce locality, which roughly states that changing one part of the graphdoes not affect the clustering of other parts of the graph. We argue that thisis a desirable property, provide conditions under which NMF quality functionsare local, and propose a novel class of local probabilistic NMF qualityfunctions for soft graph clustering.

Resolution-limit-free and local Non-negative Matrix Factorization quality functions for graph clustering