Network Comparison: Persistent Homology

We live in an era of networks. How different is the network of your friends on Facebook different from the network of Barack Obama's friends on Facebook? This interesting question traces to the network comparison problems, i.e. how to quantify the difference between a pair of networks.

Networks have been traditionally compared using features, e.g. size of the networks, degree distribution of the networks. Such methods have two main drawbacks: (i) tend to be domain specific, (ii) are likely to yield conflicting judgement -- both networks A and B are similar to C but A is highly different from B.

The problem can be solved by defining a proper measure of distance in the space of networks. We did so, and showed it can be closely approximately using a homology tools named persistent homology. The methods succeed in distinguishing collaboration networks of math community from that of engineering community.

Link for the package. The toolbox Javaplex is provided in the package and needs to be installed by calling load_javaplex.

Link for the dataset of coauthorship networks for 22 journals.

References:

Brain Signal Analysis: Aligned & Liberal

Human brain activities can be ideally modeled using networks and signals supported on networks. Each node in the network represents a brain region of similar functionality, signals denote the level of activity of each brain region, and the network describes anatomical connectivity or functional coherence.

Some part of the signals is aligned with the network: brain regions with high anatomical connection possess similar activities. Some part of the signals is liberal with the network: brain regions with high anatomical connection posses quiet distinguish activities.

We designed a systematic method to separate measurements of human brain activities into aligned and liberal parts. This package contains the self-explained and well-commented functions. Example dataset for the anatomical network and time series measurements for and individual is also provided as complement.

Link for the package.

References: