Bootstrapping on Undirected Binary Networks Via Statistical Mechanics
We propose a new method inspired from statistical mechanics for extracting geometric information from undirected binary networks and generating random networks that conform to this geometry. In this method an undirected binary network is perceived as a thermodynamic system with a collection of permuted adjacency matrices as its states. The task of extracting information from the network is then reformulated as a discrete combinatorial optimization problem of searching for its ground state. To solve this problem, we apply multiple ensembles of temperature regulated Markov chains to establish an ultrametric geometry on the network. This geometry is equipped with a tree hierarchy that captures the multiscale community structure of the network. We translate this geometry into a Parisi adjacency matrix, which has a relative low energy level and is in the vicinity of the ground state. The Parisi adjacency matrix is then further optimized by making block permutations subject to the ultrametric geometry. The optimal matrix corresponds to the macrostate of the original network. An ensemble of random networks is then generated such that each of these networks conforms to this macrostate; the corresponding algorithm also provides an estimate of the size of this ensemble. By repeating this procedure at different scales of the ultrametric geometry of the network, it is possible to compute its evolution entropy, i.e. to estimate the evolution of its complexity as we move from a coarse to a fine description of its geometric structure. We demonstrate the performance of this method on simulated as well as real data networks.
KeywordsBootstrapping Binary network Parisi matrix
This work was partially supported by National Science Foundation Grant DMS-1007219 (co-funded by Cyber-enabled Discovery and Innovation (CDI) program). P. Koehl acknowledges support from the NIH.
- 26.Herbert, S.: The architecture of complexity. Proc. Am. Philos. Soc. 106, 467–482 (1962)Google Scholar
- 27.Havlin, S., Cohen, R.: Complex Networks: Structure, Robustness, and Function. Cambridge University Press, Cambridge (2010)Google Scholar
- 29.Kim, J., Vu, V.: Generating random regular graphs. In: Proceedings of ACM Symposium on Theory of Computing (STOC), pp. 213–222 (2003)Google Scholar
- 40.Manly, B.: Randomization, Bootstrap, and Monte Carlo methods in biology. CRC Press, Boca Raton (2006)Google Scholar
- 64.Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977)Google Scholar