The society has a multi-layered structure, where the layers represent the different contexts resulting in a community structure with strong overlaps. To model this structure we begin with a single-layer weighted social network (WSN) model showing the Granovetterian correlations between link strength and topology. We find that when merging such WSN models, a sufficient amount of inter-layer correlation is needed to maintain these correlations, but they 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.

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. 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.

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