Flexible model of network embedding

There has lately been increased interest in describing complex systems not merely as single networks but rather as collections of networks that are coupled to one another. We introduce an analytically tractable model that enables one to connect two layers in a multilayer network by controlling the locality of coupling. In particular we introduce a tractable model for embedding one network (A) into another (B), focusing on the case where network A has many more nodes than network B. In our model, nodes in network A are assigned, or embedded, to the nodes in network B using an assignment rule where the extent of node localization is controlled by a single parameter. We start by mapping an unassigned `source' node in network A to a randomly chosen `target' node in network B. We then assign the neighbors of the source node to the neighborhood of the target node using a random walk starting at the target node and with a per-step stopping probability $q$. By varying the parameter $q$, we are able to produce a range of embeddings from local ($q = 1$) to global ($q \to 0$). The simplicity of the model allows us to calculate key quantities, making it a useful starting point for more realistic models.