The idea of layered networks is quite straight forward: the objects in your model are related to each other in more than one kind of way. This distinguishes a layered network from a multi-graph in which there are multiple connections of the same type. In some cases types of edges represent different relational features; for example, one could have a model with people as nodes and the ability to see each other (i.e. in line of sight) as being one kind of edge and the ability to hear each other (i.e. within natural hearing range) as another. Clearly these are different information paths with different properties on the kinds of information, speeds, distances possible, reciprocity (directedness), etc. Combining a city's road, power, and water networks on the same graph is another straightforward example. For these sorts of heterogeneous communication networks many of the common properties (such as path length and out-component) have already been adapted. But others (such as community structure and betweenness centrality) need a deeper look.

In the limited work that has been done so far (mostly what I've found is for computer connectivity over multiple channels) and the near-term extension that I've heard proposed the edge types are merely different for some property (like transmission speeds or transmission at work, home, school, train, etc.). In some cases they actually do something different or follow different rules (e.g. in network affect analyses of disease one can choose friendship relations but not genetic relationships) that affect the conclusions one draws from the analysis. But all these extensions do not require new methods and barely scratch the surface of what layered networks could be used for. Specifically, I think the biggest payoff will come from agent-based models that can use different networks to track different interaction forces and a higher order of heterogeneity in interaction structure.

The "problem" with adapting measures of network structure to layered networks is that there are multiple ways in which (for example) betweenness can be adapted to take account for the edge variety, and each one now measures something different and they might all be useful in some context. And we must consider the possibility that adapting methods for layered networks will inspire us to invent novel measures…structural properties that have never before been calculated.

Let's first consider the base case in which all our nodes are of the same type (e.g. people, so we're not yet going into k-partite territory with multiple edge types). Well, we already have some of these measures worked out for bipartite graphs, so if there were two types of edge we can take the dual of the graph (convert the nodes to edges, and edges to nodes) to convert our multilayered network into a network with multiple node types instead of edge types. That won't be a bipartite graph (usually) since we wouldn't expect the layered connections in a network to obey the connection restrictions of bipartitism, but some of the insights from bipartite measures may prove helpful. This dual conversion works for any number of layers (and types of nodes too), but there are lots of details to work out in such a transformation (e.g. dealing with self-edges, structure preservation, interpretation of measures taken on the converted graph for the original network), and that is the subject of a different post.

For now, I plan to build a Netlogo code example that includes a layered network. Probably I'll use something like the communication example because one type (vision) is directional and the other (hearing) is symmetric. And some other example in which different resources flow along different connections. Recommendations for research models with layered networks (existing ones to code up or an idea for a cool one to research) are welcome.