Date of Award
Doctor of Philosophy (PhD)
Chair and Committee
We study phase transitions in spin glass type systems and in related computational problems. In the current work, we focus on the "community detection" problem when cast in terms of a general Potts spin glass type problem. We report on phase transitions between solvable and unsolvable regimes. Solvable region may further split into easy and hard phases. Spin glass type phase transitions appear at both low and high temperatures. Low temperature transitions correspond to an order by disorder type effect wherein fluctuations render the system ordered or solvable. Separate transitions appear at higher temperatures into a disordered: or an unsolvable) phases. Different sorts of randomness lead to disparate behaviors. We illustrate the spin glass character of both transitions and report on memory effects. We further relate Potts type spin systems to mechanical analogs and suggest how chaotic-type behavior in general thermodynamic systems can indeed naturally arise in hard-computational problems and spin-glasses. In this work, we also examine large networks: with a power law distribution in cluster size) that have a large number of communities. We infer that large systems at a constant ratio of q to the number of nodes N asymptotically tend toward insolvability in the limit of large N for any positive temperature. We further employ multivariate Tutte polynomials to show that increasing q emulates increasing T for a general Potts model, leading to a similar stability region at low T. We further apply the replica inference based Potts model method to unsupervised image segmentation on multiple scales. This approach was inspired by the statistical mechanics problem of "community detection" and its phase diagram. The problem is cast as identifying tightly bound clusters against a background. Within our multiresolution approach, we compute information theory based correlations among multiple solutions of the same graph over a range of resolutions. Significant multiresolution structures are identified by replica correlations as manifest in information overlaps. With the aid of these correlations as well as thermodynamic measures, the phase diagram of the corresponding Potts model is analyzed both at zero and finite temperatures. Optimal parameters corresponding to a sensible unsupervised segmentation correspond to the easy phase of the Potts model. Our algorithm is fast and shown to be at least as accurate as the best algorithms to date and to be especially suited to the detection of camouflage images.
Hu, Dandan, "Statistical Mechanics of the Community Detection Problem: Theory and Application" (2012). All Theses and Dissertations (ETDs). 959.