  Networks are a widely-used tool to investigate the large-scale connectivity
structure in complex systems and graphons have been proposed as an infinite
size limit of dense networks. The detection of communities or other meso-scale
structures is a prominent topic in network science as it allows the
identification of functional building blocks in complex systems. When such
building blocks may be present in graphons is an open question. In this paper,
we define a graphon-modularity and demonstrate that it can be maximised to
detect communities in graphons. We then investigate specific synthetic graphons
and show that they may show a wide range of different community structures. We
also reformulate the graphon-modularity maximisation as a continuous
optimisation problem and so prove the optimal community structure or lack
thereof for some graphons, something that is usually not possible for networks.
Furthermore, we demonstrate that estimating a graphon from network data as an
intermediate step can improve the detection of communities, in comparison with
exclusively maximising the modularity of the network. While the choice of
graphon-estimator may strongly influence the accord between the community
structure of a network and its estimated graphon, we find that there is a
substantial overlap if an appropriate estimator is used. Our study demonstrates
that community detection for graphons is possible and may serve as a
privacy-preserving way to cluster network data.
