4.2. Typed edge spectral embedding
33
between roles. How this works out is easiest to see from an influence or information
flow perspective. For example, an individual might hear a joke at work. If the joke
is to reach that individual’s social domain, they have to remember and repeat it in a
social context, and there is a certain resistance or cost associated with making that
happen.
Our association of the weight of the vertical edge to the total weight of the
individual in a subgraph also reflects the assumption that an individual who is central
or powerful in one subgraph plausibly is in a better position to be central or powerful
in other subgraphs. If they hear more jokes at work (so to speak), they are more
likely to disseminate jokes into their social domain.
We have constructed a cn × cn network in which all of the edges are now of the
same type, so we can embed it in the standard way using spectral embedding, and
project it into an appropriate number of dimensions.
In the embedding, distances represent dissimilarity (equivalently, geometric
closeness represents relationship closeness), but there are now multiple nodes corresponding to each individual. For nodes of the same color, distance represents similarity within the subgraph of that color — for example, the distance between two
individuals in a layer representing friendship represents how close they are, or could
be, as friends, but in the context of all of the other relationships of other types in
which they, and their entire group, participate. It is not meaningful to consider the
distance between the embedded point representing the friend role of one individual
and the embedded point representing the colleague role of another.
The length, in the embedding, of each vertical edge reveals the magnitude of
the difference between the role an individual plays in one social network and the role
they play in another. For example, someone who is a key person in both a friendand work-related network will be placed centrally in both, and so the vertical edge
connecting those roles will be short. On the other hand, someone who is key in
a friend network but peripheral in the work-related network is being pulled towards
the center of the friend layer, but towards the outside of the work-related layer, and so
his or her vertical edge will tend to be long. The information revealed by the vertical
nodes is only available because of the layered approach to representing typed edges.
Edge prediction uses proximity of two unconnected nodes as the basis for suggesting that the two individuals “should” or “might” have a relationship. Having
multiple versions of nodes corresponding to each individual in the embedding makes
it possible to do typed edge prediction.
If the friend layer versions of two nodes are close but they are not connected,
then we could predict for each of them that “this might be a potential friend” while if
the work-related layer versions are close, we could predict “this might be a potential
colleague”. Such predictions are both more accurate and richer than those that could
be generated from an untyped version of the same social network.
The layer approach imposes costs for creating and embedding a larger matrix,
and the analyses of the embedded network are complicated by the multiple copies
of each node, but there are benefits as well that arise from access to these embedded
multiple versions and, in particular, the similarities and differences between them.
33
between roles. How this works out is easiest to see from an influence or information
flow perspective. For example, an individual might hear a joke at work. If the joke
is to reach that individual’s social domain, they have to remember and repeat it in a
social context, and there is a certain resistance or cost associated with making that
happen.
Our association of the weight of the vertical edge to the total weight of the
individual in a subgraph also reflects the assumption that an individual who is central
or powerful in one subgraph plausibly is in a better position to be central or powerful
in other subgraphs. If they hear more jokes at work (so to speak), they are more
likely to disseminate jokes into their social domain.
We have constructed a cn × cn network in which all of the edges are now of the
same type, so we can embed it in the standard way using spectral embedding, and
project it into an appropriate number of dimensions.
In the embedding, distances represent dissimilarity (equivalently, geometric
closeness represents relationship closeness), but there are now multiple nodes corresponding to each individual. For nodes of the same color, distance represents similarity within the subgraph of that color — for example, the distance between two
individuals in a layer representing friendship represents how close they are, or could
be, as friends, but in the context of all of the other relationships of other types in
which they, and their entire group, participate. It is not meaningful to consider the
distance between the embedded point representing the friend role of one individual
and the embedded point representing the colleague role of another.
The length, in the embedding, of each vertical edge reveals the magnitude of
the difference between the role an individual plays in one social network and the role
they play in another. For example, someone who is a key person in both a friendand work-related network will be placed centrally in both, and so the vertical edge
connecting those roles will be short. On the other hand, someone who is key in
a friend network but peripheral in the work-related network is being pulled towards
the center of the friend layer, but towards the outside of the work-related layer, and so
his or her vertical edge will tend to be long. The information revealed by the vertical
nodes is only available because of the layered approach to representing typed edges.
Edge prediction uses proximity of two unconnected nodes as the basis for suggesting that the two individuals “should” or “might” have a relationship. Having
multiple versions of nodes corresponding to each individual in the embedding makes
it possible to do typed edge prediction.
If the friend layer versions of two nodes are close but they are not connected,
then we could predict for each of them that “this might be a potential friend” while if
the work-related layer versions are close, we could predict “this might be a potential
colleague”. Such predictions are both more accurate and richer than those that could
be generated from an untyped version of the same social network.
The layer approach imposes costs for creating and embedding a larger matrix,
and the analyses of the embedded network are complicated by the multiple copies
of each node, but there are benefits as well that arise from access to these embedded
multiple versions and, in particular, the similarities and differences between them.