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

A STATISTICAL RESEARCH ON THE TYPICAL PATTERNS OF MODERN HOUSING FABRICS, CASE STUDY OF NANJING, CHINA

Proceedings of the XXV ISUF International Conference “Urban Form and Social Context: from Traditions to Newest Demands” (Krasnoyarsk, July 5–9, 2018)After nearly 20 years of massive social housing construction and another 20 years of housing real estate development, Chinese cities basically solved the citizen’s housing problem in the second decade of the 21st century. As a consequence, the major physical component of contemporary cities is modern housing fabrics, which cover over 30% urban land. It is generally believed this magnitude housing development is dominated by modernism residential building with a standard image of a slab apartment. However, as revealed in this research, the real situation is far more diversified and complicated, with various building types, like villas, slabs, towers, and different spatial arrangements, like parallel, zigzag, enclosure. How to classify these diversified realities and what are the typical patterns of different housing fabrics? To answer these questions, this research collected more than 200 housing fabric samples across the city of Nanjing. The latter is the Capital of Jiangsu Province, and a typical modern mega-city in Yangzi River Delta area. To get the reasonable categories of fabric types, a comprehensive classification system is applied. Different from the too simplified classification based on single parameter, building height, adopted in the national housing standard, this classification system is based on the matrix of various parameters, including building height, arrangement, and a building type. The various parameters and their intricate combinations guarantee the classification to be capable to seize and distinguish the formal features of different fabrics. Spacemate, a charting tool developed by B.M. Pont and et al. in TU Delft, is used to testify the classification. After the classification, the samples are divided into 21 categories. For each category, data samples, like spacing, dimension of building footprint, height, density, land coverage, and et al. are collected and a statistical analysis are conducted. Based on this qualitative sample studies, the typical patterns and their statistical models are built up. In the application part, a bioclimatic performance study of these typical patterns is presented. Due to the typicality and statistical precision, the complicated co-relation between urban fabric and bioclimatic performance could be discovered, efficiently and convincingly

Stability results for a hierarchical size-structured population model with distributed delay

In this paper we investigate a structured population model with distributed delay. Our model incorporates two different types of nonlinearities. Specifically we assume that individual growth and mortality are affected by scramble competition, while fertility is affected by contest competition. In particular, we assume that there is a hierarchical structure in the population, which affects mating success. The dynamical behavior of the model is analysed via linearisation by means of semigroup and spectral methods. In particular, we introduce a reproduction function and use it to derive linear stability criteria for our model. Further we present numerical simulations to underpin the stability results we obtained

Recollements induced by left Frobenius pairs

Let $T$ be a right exact functor from an abelian category $\mathscr{B}$ into another abelian category $\mathscr{A}$. Then there exists an abelian category, named comma category and denoted by $(T\downarrow\mathscr{A})$.In this paper, we construct left Frobenius pairs (resp. strong left Frobenius pairs) over $(T\downarrow\mathscr{A})$ using left Frobenius pairs (resp. strong left Frobenius pairs) over $\mathscr{A}$ and $\mathscr{B}.$ As a consequence, we obtain a recollement of (right) triangulated categories, generalizing the result of Xiong-Zhang-Zhang (J. Algebra 503 (2018) 21-55) about the recollement of additive (resp. triangulated) categories constructed from a triangular matrix algebra. This result is applied to the classes of flat modules and Gorenstein flat modules, the classes of Gorenstein projective modules and Gorenstein projective complexes, the class of Ding projective modules and the class of Gorenstein flat-cotorsion modules.Comment: 24 pages, all comments are welcom