Wednesday, November 12th 2014
Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
We compute spectra of large random stochastic matrices, i.e. Markovmatrices defined on random graphs, where each edge (i,j) in a(sparse) random graph is given a positive random weight Wij >0 insuch a fashion that the each column sum of the matrix W is normalized toone, Σi Wij= 1. We compute spectra of such matrices, both inthe thermodynamic limit, and for very large single instances. Thestucture of the graphs and the distribution of the non-zero weights
are largely arbitrary, as long as the mean degree remainsfinite in the thermodynamic limit, and the Wij
satisfy a detailedbalance condition. Knowing the spectra of stochastic matrices istantamount to knowing the complete spectrum of relaxation times ofstochastic processes described by them, so our results should havemany interesting applications for the study of relaxation in complexsystems. We discuss cell-signalling as a possible application of randomwalks in complex networks, and in particular signalling entropy(constructed in terms of Markov transition matrices) as a global measureof robustness of networks of signalling pathways.
Colloquium coordinators are Prof.dr. M. Aiello (e-mail :
Prof.dr. M. Biehl (e-mail:
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