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Chapter 7. Modelling relationships that change over time
trajectory could exist.
We address these problems by defining a sequence of social networks over
time, treating each one as a layer in a way that, by now, should be familiar; binding
the layers together; and embedding the whole graph in a single space. The existence
of such a space makes it possible to compare, rigorously, the relationships of nodes
and edges between time periods. Because the networks at different times are all
embedded in the same space, it also becomes possible to define the concept of a
trajectory, and therefore to track nodes and relationships across time with a new
level of clarity.
A successful embedding makes it possible to ask and answer questions such
as: Do two nodes become closer or farther apart over time, and which node drives
this process? Are communities or other kinds of subgroups stable over time? If they
evolve, how do they change?
As before, we begin by considering the entire social network at each time step
as one of a set of layers in the sense of our previous constructions. Thus the network
at time t = 0 becomes the first layer, the network at t = 1 the second layer, and so
on. The versions of each node represent that same node of the social network at its
different moments of existence. The edges in each layer represent the strengths of the
relationship during a particular time period. Edges might appear and disappear from
one time to the next; but it is perhaps more common for their weights (the intensities
of the relationships) to vary with time. For simplicity, we create placeholder versions
of nodes that are not present during any time period but, of course, they will not be
connected to the rest of the graph when they are acting as placeholders. Thus each
layer contains versions of all nodes that are present in the network at any time.
The obvious approach to binding the subgraphs representing each time snapshot into a single graph would be to connect each node version in one subgraph to
its corresponding node version in the subgraph of the next time. Unlike the previous
versions of the layer construction, there is a natural ordering for the layers.
However, this does not work — the spectral embedding of a path is a curve in
two dimensions because the endpoints resemble each other more than they resemble
the internal nodes. This effect will happen to some extent to the path connecting each
particular node as it appears in subgraphs, creating a geometry where the effects of
the structure from the connections across time cannot be cleanly separated from the
structure inside each subgraph.
Instead, we connect all of the versions of each node together in a c-clique, as
we did for typed networks.
We also add a further refinement precisely because there is a natural ordering
of the layers. It may be appropriate and useful to allow the embedded structure of the
network at previous times to influence the embedded structure in the current period.
In other words we may want to force the structure at a particular time step to align
more strongly with the structure from earlier times to help with understanding what
is changing. This provides a mechanism for putting back one facet of the natural
ordering of the layers. Alignment also conceals small-scale changes from layer to
layer, making larger changes, or changes that are consistent over time, more easily
visible.
Chapter 7. Modelling relationships that change over time
trajectory could exist.
We address these problems by defining a sequence of social networks over
time, treating each one as a layer in a way that, by now, should be familiar; binding
the layers together; and embedding the whole graph in a single space. The existence
of such a space makes it possible to compare, rigorously, the relationships of nodes
and edges between time periods. Because the networks at different times are all
embedded in the same space, it also becomes possible to define the concept of a
trajectory, and therefore to track nodes and relationships across time with a new
level of clarity.
A successful embedding makes it possible to ask and answer questions such
as: Do two nodes become closer or farther apart over time, and which node drives
this process? Are communities or other kinds of subgroups stable over time? If they
evolve, how do they change?
As before, we begin by considering the entire social network at each time step
as one of a set of layers in the sense of our previous constructions. Thus the network
at time t = 0 becomes the first layer, the network at t = 1 the second layer, and so
on. The versions of each node represent that same node of the social network at its
different moments of existence. The edges in each layer represent the strengths of the
relationship during a particular time period. Edges might appear and disappear from
one time to the next; but it is perhaps more common for their weights (the intensities
of the relationships) to vary with time. For simplicity, we create placeholder versions
of nodes that are not present during any time period but, of course, they will not be
connected to the rest of the graph when they are acting as placeholders. Thus each
layer contains versions of all nodes that are present in the network at any time.
The obvious approach to binding the subgraphs representing each time snapshot into a single graph would be to connect each node version in one subgraph to
its corresponding node version in the subgraph of the next time. Unlike the previous
versions of the layer construction, there is a natural ordering for the layers.
However, this does not work — the spectral embedding of a path is a curve in
two dimensions because the endpoints resemble each other more than they resemble
the internal nodes. This effect will happen to some extent to the path connecting each
particular node as it appears in subgraphs, creating a geometry where the effects of
the structure from the connections across time cannot be cleanly separated from the
structure inside each subgraph.
Instead, we connect all of the versions of each node together in a c-clique, as
we did for typed networks.
We also add a further refinement precisely because there is a natural ordering
of the layers. It may be appropriate and useful to allow the embedded structure of the
network at previous times to influence the embedded structure in the current period.
In other words we may want to force the structure at a particular time step to align
more strongly with the structure from earlier times to help with understanding what
is changing. This provides a mechanism for putting back one facet of the natural
ordering of the layers. Alignment also conceals small-scale changes from layer to
layer, making larger changes, or changes that are consistent over time, more easily
visible.