Visualisation of the evolution of ego-centered community structure using temporal personalised pagerank
At first, we say that a ego-community structure is a probability measure defined on the set of network nodes. Any subset of nodes may engender its own ego-community structure around. Many community detection algorithms can be modified to yield a result of this type, for instance, the personalized pagerank.
Next, we present a continuous version of Viard-Latapy-Magnien link streams (defined here), that we call "temporal density". Classical kernel density estimation is used to move from discrete link streams to their continuous counterparts. Using matrix perturbation theory we can prove that ego-community structure changes smoothly when the network evolves smoothly. This is very important, for example, for visualization purposes.
Combining the temporal density and personalized pagerank methods, we are able to visualize and study the evolution of the ego-community structures of complex networks with a large number of temporal links.
We illustrate and validate our approach using "Primary school temporal network data" provided by sociopatterns.org.
The lines in the following image corresponds to the students. A column at the timestamp t is the ego-community structure around one school student Υ, selected in advance as a center of the community. Cyan means in the community of Υ, black means out the community of Υ. Right x-axis is about sex (male/female). The left x-axis denotes the classes of participants (1A,1B,2A,2B,3A,3B,4A,4B,5A,5B,Teachers).
See also : Slides