2.2. Building layered models
13
(a) Within-layer connection
(b) Between-layer connection
Figure 2.1: Illustration for within-layer connection and between-layer connection
If R is the adjacency matrix of the red layer, G is the adjacency matrix of the
green layer, and T rg and T gr are the diagonal matrices representing the two vertical
edges that connect different versions of the same nodes, then the adjacency matrix
of the larger graph is
M =
R T rg
T gr G
.
(If the vertical edges are undirected then, of course, T rg and T gr are identical.)
Figure 2.2: The embedding of a simple graph with typed edges
Figure 2.2 shows what happens when this idea is worked out. The original
graph has 8 nodes. These nodes are connected by solid edges in a circle. They
are connected by dashed edges into two 4-cliques. In the embedding (whose details
we have not described yet) the solid layer distorts the 4-cliques of the dashed layer,
pulling them into trapezoids; and the dashed layer distorts the solid-layer circle into
an ellipse. There are two versions of each node, indicated by a subscript, The crosshatched lines are the embeddings of the added vertical edges. The vertical edge
joining the two versions of node 1, for example, is long, showing that the role of this
node in the two networks is quite different.
13
(a) Within-layer connection
(b) Between-layer connection
Figure 2.1: Illustration for within-layer connection and between-layer connection
If R is the adjacency matrix of the red layer, G is the adjacency matrix of the
green layer, and T rg and T gr are the diagonal matrices representing the two vertical
edges that connect different versions of the same nodes, then the adjacency matrix
of the larger graph is
M =
R T rg
T gr G
.
(If the vertical edges are undirected then, of course, T rg and T gr are identical.)
Figure 2.2: The embedding of a simple graph with typed edges
Figure 2.2 shows what happens when this idea is worked out. The original
graph has 8 nodes. These nodes are connected by solid edges in a circle. They
are connected by dashed edges into two 4-cliques. In the embedding (whose details
we have not described yet) the solid layer distorts the 4-cliques of the dashed layer,
pulling them into trapezoids; and the dashed layer distorts the solid-layer circle into
an ellipse. There are two versions of each node, indicated by a subscript, The crosshatched lines are the embeddings of the added vertical edges. The vertical edge
joining the two versions of node 1, for example, is long, showing that the role of this
node in the two networks is quite different.