Statistical Mechanics of the Community Detection Problem: Theory and Application

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

Crowdfunding To Generate Crowdsourced R&D: The Alternative Paradigm Of Societal Problem Solving Offered By Second Generation Innovation And R&D

In a global context of resource scarcity few incentives exist for firms to pursue innovations that provide social externalities if these are not inherently profitable. The purpose of this article is to present an alternative paradigm of societal problem solving entirely premised on ‘second generation innovation’ processes. Further, a theoretical model of multidimensional, or three dimensional, knowledge creation is offered, together with the notion of a ‘multiplier effect’ that relates to how knowledge creation can increase exponentially when knowledge is not constrained by proprietary requirements. Second generation innovation is based on probabilistic processes that utilize and maximize economies of scale in pursuit of problem solving. Two processes that contribute to the potential of second generation innovation to solve societal problems are crowdfunding and crowdsourcing. It is argued that the processes required to enable a new paradigm in societal problem solving already exist. A further model is developed based on potential synergies between crowdfunding and crowdsourced research and development. This theoretical model predicts that R&D productivity can be accelerated significantly, and if applied in fields such as proteomics or medical research in general can accelerated increases in research output and therefore benefits to society

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200

2013 Conference Abstracts: Annual Undergraduate Research Conference at the Interface of Biology and Mathematics

URC Schedule and Abstract Book for the Fifth Annual Undergraduate Research Conference at the Interface of Biology and Mathematics Date: November 16-17, 2013Plenary Speaker: Mariel Vazquez, Associate Professor of Mathematics at San Francisco State UniversityFeatured Speaker: Andrew Liebhold, Research Entomologist for the USDA Forest Servic