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.
- W. Huang and A. Ribeiro, Persistent Homology Lower Bounds on High Order Network Distances, IEEE Transaction on Signal Processing, vol. 65, no. 2, pp. 319-334, January 2017.
- W. Huang and A. Ribeiro, Metrics in the Space of High Order Networks, IEEE Transaction on Signal Processing, vol. 64, no. 3, pp. 615-629, February 2016.
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.
- J. D. Medaglia, W. Huang, E. A. Karuza, S. T. Tompson-Schill, A. Ribeiro, D. S. Bassett, Functional Alignment with Anatomical Networks is Associated with Cognitive Flexibility, Nat Hum Behav., vol. (submitted), November 2016.
- W. Huang, L. Goldsberry, N.F. Wymbs, S.T. Grafton, D.S. Bassett and A. Ribeiro, Graph Frequency Analysis of Brain Signals, IEEE J. Special Topics Signal Process., vol. 10, no. 7, pp. 1189 - 1203, October 2016.