  Network similarity measures quantify how and when two networks are
symmetrically related, including measures of statistical association such as
pairwise distance or other correlation measures between networks or between the
layers of a multiplex network, but neither can directly unveil whether there
are hidden confounding network factors nor can they estimate when such
correlation is underpinned by a causal relation. In this work we extend this
pairwise conceptual framework to triplets of networks and quantify how and when
a network is related to a second network directly or via the indirect mediation
or interaction with a third network. Accordingly, we develop a simple and
intuitive set-theoretic approach to quantify mediation and suppression between
networks. We validate our theory with synthetic models and further apply it to
triplets of real-world networks, unveiling mediation and suppression effects
which emerge when considering different modes of interaction in online social
networks and different routes of information processing in the brain.
