Chapter 6
Modelling asymmetric
relationships with
multiple types
We now turn to constructions that compose the layered models we have built so far.
For example, we have seen that the Florentine families data captures different kinds
of relationships (especially financial and social); but also directed relationships, representing family power or political leverage. Can we combine our approaches that
have modelled each of these aspects separately, to model both at once?
Our basic strategy is to separate the edge properties into layers, as before, but to
combine different sets of layers orthogonally. For example, if we create two versions
of each node to connect two different edge types, and two versions to connect the in
and out ends of directed edges, then we have four versions of each node. However, if
we keep in mind the property that each version represents, then it is straightforward
to connect the edges of the original social network to the right versions of each node.
In our example of the Florentine families, we would have four versions of each
node, whose meanings are social-out, social-in, financial-out, and financial-in, and
so a directed social edge would connect from a social-out node to a social-in node.
Of course, the “vertical” edges still need to be added. Here there are some
choices, depending on the details of exactly how we compose the constructions. For
example, we could imagine a clique connection between all four versions, or some
subset connecting only versions from the same subconstruction. We also need to
decide on principled ways to assign weights to these new edges.
6.1 Combining directed and typed embeddings
For simplicity, we continue to assume that there are two types of edges, which we
will think of as colors (say, red and green). Adding more types complicates the
explanation, but does not complicate the construction.
Each node of the original network is now replaced by four versions: red in, red
out, green in, and green out. The resulting graph contains two layers representing
the typed subgraphs, and two (orthogonal) layers representing the in and out layers.
69
Modelling asymmetric
relationships with
multiple types
We now turn to constructions that compose the layered models we have built so far.
For example, we have seen that the Florentine families data captures different kinds
of relationships (especially financial and social); but also directed relationships, representing family power or political leverage. Can we combine our approaches that
have modelled each of these aspects separately, to model both at once?
Our basic strategy is to separate the edge properties into layers, as before, but to
combine different sets of layers orthogonally. For example, if we create two versions
of each node to connect two different edge types, and two versions to connect the in
and out ends of directed edges, then we have four versions of each node. However, if
we keep in mind the property that each version represents, then it is straightforward
to connect the edges of the original social network to the right versions of each node.
In our example of the Florentine families, we would have four versions of each
node, whose meanings are social-out, social-in, financial-out, and financial-in, and
so a directed social edge would connect from a social-out node to a social-in node.
Of course, the “vertical” edges still need to be added. Here there are some
choices, depending on the details of exactly how we compose the constructions. For
example, we could imagine a clique connection between all four versions, or some
subset connecting only versions from the same subconstruction. We also need to
decide on principled ways to assign weights to these new edges.
6.1 Combining directed and typed embeddings
For simplicity, we continue to assume that there are two types of edges, which we
will think of as colors (say, red and green). Adding more types complicates the
explanation, but does not complicate the construction.
Each node of the original network is now replaced by four versions: red in, red
out, green in, and green out. The resulting graph contains two layers representing
the typed subgraphs, and two (orthogonal) layers representing the in and out layers.
69