  The society has a multi-layered structure, where the layers represent the
different contexts. To model this structure we begin with a single-layer
weighted social network (WSN) model showing the Granovetterian structure. We
find that when merging such WSN models, a sufficient amount of inter-layer
correlation is needed to maintain the relationship between topology and link
weights, while these correlations destroy the enhancement in the community
overlap due to multiple layers. To resolve this, we devise a geographic
multi-layer WSN model, where the indirect inter-layer correlations due to the
geographic constraints of individuals enhance the overlaps between the
communities and, at the same time, the Granovetterian structure is preserved.
Furthermore, the network of social interactions can be considered as a
multiplex from another point of view too: each layer corresponds to one
communication channel and the aggregate of all them constitutes the entire
social network. However, usually one has information only about one of the
channels, which should be considered as a sample of the whole. Here we show by
simulations and analytical methods that this sampling may lead to bias. For
example, while it is expected that the degree distribution of the whole social
network has a maximum at a value larger than one, we get with reasonable
assumptions about the sampling process a monotonously decreasing distribution
as observed in empirical studies of single channel data. We analyse the
far-reaching consequences of our findings.
