Chapter 8
Modelling positive and
negative relationships
Almost all social networks model relationships as properties that are either nonexistent, or have a positive intensity. Now we turn to the situation where the relationship between two nodes might also be a negative one — the two individuals or
organizations are in opposition, or have a mutual antipathy.
At one level, this is an easy extension. In social networks so far, adjacency
matrix entries are either zero or positive; a negative relationship can be modelled by
a negative edge weight. The difficulty arises in trying to model the effects of transitivity. For social networks with positively weighted edges, transitivity has a natural
interpretation: if A has a close relationship with B, and B has a close relationship
with C, it seems natural that A and C might be considered to be similar, or might be
likely to have a positive relationship if they met, because of their mutual connection,
B.
However, what is the relationship between A and C, if A and B have a negative
relationship and so do B and C? Conventionally, “the enemy of my enemy is my
friend” but this cannot be a standard pattern, especially in an environment with a
large number of actors [22].
In this chapter, we develop an embedding approach that also enables modelling
social networks with negatively weighted edges and, along the way, resolves the issue
of how to think about the enemy of my enemy.
8.1 Signed Laplacian
We derive spectral graph embeddings for signed graphs that create embeddings with
the obvious desirable properties: nodes connected by positive edges are placed close
together, while those connected by negative edges are placed far apart. The difficulty, of course, is how to balance the “pull” of positive edges against the “push” of
negative ones to produce appropriate global solutions that approximate optima. We
derive an unnormalized and two normalized Laplacian matrices for signed graphs,
97
Modelling positive and
negative relationships
Almost all social networks model relationships as properties that are either nonexistent, or have a positive intensity. Now we turn to the situation where the relationship between two nodes might also be a negative one — the two individuals or
organizations are in opposition, or have a mutual antipathy.
At one level, this is an easy extension. In social networks so far, adjacency
matrix entries are either zero or positive; a negative relationship can be modelled by
a negative edge weight. The difficulty arises in trying to model the effects of transitivity. For social networks with positively weighted edges, transitivity has a natural
interpretation: if A has a close relationship with B, and B has a close relationship
with C, it seems natural that A and C might be considered to be similar, or might be
likely to have a positive relationship if they met, because of their mutual connection,
B.
However, what is the relationship between A and C, if A and B have a negative
relationship and so do B and C? Conventionally, “the enemy of my enemy is my
friend” but this cannot be a standard pattern, especially in an environment with a
large number of actors [22].
In this chapter, we develop an embedding approach that also enables modelling
social networks with negatively weighted edges and, along the way, resolves the issue
of how to think about the enemy of my enemy.
8.1 Signed Laplacian
We derive spectral graph embeddings for signed graphs that create embeddings with
the obvious desirable properties: nodes connected by positive edges are placed close
together, while those connected by negative edges are placed far apart. The difficulty, of course, is how to balance the “pull” of positive edges against the “push” of
negative ones to produce appropriate global solutions that approximate optima. We
derive an unnormalized and two normalized Laplacian matrices for signed graphs,
97