Phase transitions in spin glass type systems and, more recently, in related computational problems have gained broad interest in disparate arenas. In the current work, we focus on the "community detection" problem when cast in terms of a general Potts spin glass type problem. As such, our results apply to rather broad Potts spin glass type systems. Community detection describes the general problem of partitioning a complex system involving many elements into optimally decoupled "communities" of such elements. 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 (or noise). 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) phase. 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. The correspondence between the two types of transitions (spin glass and dynamic) is likely to extend across a larger spectrum of spin glass type systems and hard computational problems. We briefly discuss potential implications of these transitions in complex many body physical systems.

Abstract. We elaborate on a general method that we recently introduced for characterizing the "natural" structures in complex physical systems via a multi-scale network based approach for the data mining of such structures. The approach is based on "community detection" wherein interacting particles are partitioned into "an ideal gas" of optimally decoupled groups of particles. Specifically, we construct a set of network representations ("replicas") of the physical system based on interatomic potentials and apply a multiscale clustering ("multiresolution community detection") analysis using information-based correlations among the replicas. Replicas may be (i) different representations of an identical static system or (ii) embody dynamics by when considering replicas to be time separated snapshots of the system (with a tunable time separation) or (iii) encode general correlations when different replicas correspond to different representations of the entire history of the system as it evolves in space-time. Inputs for our method are the inter-particle potentials or experimentally measured two (or higher order) particle correlations. We apply our method to computer simulations of a binary Kob-Andersen Lennard-Jones system in a mixture ratio of A80B20, a ternary model system with components "A", "B", and "C" in ratios of A88B7C5 (as in Al88Y7Fe5), and to atomic coordinates in a Zr80Pt20 system as gleaned by reverse Monte Carlo analysis of experimentally determined structure factors. We identify the dominant structures (disjoint or overlapping) and general length scales by analyzing extrema of the information theory measures. We speculate on possible links between (i) physical transitions or crossovers and (ii) changes in structures found by this method as well as phase transitions associated with the computational complexity of the community detection problem. We briefly also consider continuum approaches and discuss the shear penetration depth in elastic media; this length scale increases as the system becomes increasingly rigid.

We apply a 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. Specifically, the problem is cast as identifying tightly bound clusters ("communities" or "solutes") against a background or "solvent." Within our multiresolution approach, we compute information-theory-based correlations among multiple solutions ("replicas") of the same graph over a range of resolutions. Significant multiresolution structures are identified by replica correlations manifest by information theory overlaps. We further employ such information theory measures (such as normalized mutual information and variation of information), thermodynamic quantities such as the system entropy and energy, and dynamic measures monitoring the convergence time to viable solutions as metrics for transitions between various solvable and unsolvable phases. Within the solvable phase, transitions between contending solutions (such as those corresponding to segmentations on different scales) may also appear. With the aid of these correlations as well as thermodynamic measures, the phase diagram of the corresponding Potts model is analyzed at both zero and finite temperatures. Optimal parameters corresponding to a sensible unsupervised segmentations appear within 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 camouflaged images.

[1] Nanometer-scale scanning electron microscopy was applied in visualizing the microscopic pores within shale kerogen. Geometrical information of all individual pores was extracted by image analysis. Image segmentation and separation showed that most of the intrakerogen pores are discrete and isolated from each other, having relatively spherical morphology. These isolated intrakerogen pores result in huge challenges in gas production, because they are not effectively connected to natural and hydraulic fractures. Statistical results showed that nanopores, which have diameters smaller than 100 nm, make up 92.7% of the total pore number, while they make up only 4.5% of the total pore volume. Intrakerogen porosity and specific surface area are 29.9% and 14.0 m 2 /g, respectively. Accurate visualization and measurement of intrakerogen pores are critical for evaluation of gas storage and optimization of hydraulic fracturing. By lattice Boltzmann simulations, permeabilities and tortuosities were simulated in the three principal directions. Long tails were observed in breakthrough curves, resulting from diffusion of solute particles from low-flow-velocity pores to larger conduits at late times. The long-tailing phenomena at the pore scale are qualitatively consistent with those observed in real productions. Understanding the porescale transport processes between microscopic pores within kerogen and large fracture systems is of great importance in predicting hydrocarbon production. Upscaling methods are needed to investigate largerscale processes and properties in shale reservoirs.

Recent decades have experienced the discovery of numerous complex materials. At the root of the complexity underlying many of these materials lies a large number of contending atomic- and largerscale configurations. In order to obtain a more detailed understanding of such systems, we need tools that enable the detection of pertinent structures on all spatial and temporal scales. Towards this end, we suggest a new method that applies to both static and dynamic systems which invokes ideas from network analysis and information theory. Our approach efficiently identifies basic unit cells, topological defects, and candidate natural structures. The method is particularly useful where a clear definition of order is lacking, and the identified features may constitute a natural point of departure for further analysis.

PACS 89.75.Fb -Structures and organization in complex systems PACS 64.60.Cn -Order-disorder transformations PACS 64.60.De -Statistical mechanics of model systems Abstract -We examine a global disorder transition when identifying community structure in an arbitrary complex network. Earlier, we illustrated [Phil. Mag. 92, 406 (2012)] that "community detection" (CD) generally exhibits disordered (or unsolvable) and ordered (solvable) phases of both high and low computational complexity along with corresponding transitions from regular to chaotic dynamics in derived systems. Using an exact generalized dimensional reduction inequality, multivariate Tutte polynomials, and other considerations, we illustrate how increasing the number of communities q emulates increasing the heat bath temperature T for a general weighted Potts model, leading to global disorder in the community structure of arbitrary large graphs. Dimensional reduction bounds lead to results similar to those suggested by mean-field type approaches. Large systems tend toward global insolvability in the limit of large q above a crossover temperature T× ≈ L|Je|/ [N ln q] where |Je| is a typical interaction strength, L is the number of edges, and N is the number of nodes. For practical system sizes, a solvable phase is generally accessible at low T . The global nature of the disorder transition does not preclude solutions by local CD algorithms (even those that employ global cost function parameters) as long as community evaluations are locally determined.

Inspired by a multi-resolution community detection (MCD) based network segmentation method, we suggest an automatic method for segmenting fluorescence lifetime (FLT) imaging microscopy (FLIM) images of cells in a first pilot investigation on two selected images. The image processing problem is framed as identifying segments with respective average FLTs against the background in FLIM images. The proposed method segments a FLIM image for a given resolution of the network defined using image pixels as the nodes and similarity between the FLTs of the pixels as the edges. In the resulting segmentation, low network resolution leads to larger segments, and high network resolution leads to smaller segments. Further, using the proposed method, the mean-square error (MSE) in estimating the FLT segments in a FLIM image was found to consistently decrease with increasing resolution of the corresponding network. The MCD method appeared to perform better than a popular spectral clustering based method in performing FLIM image segmentation. At high resolution, the spectral segmentation method introduced noisy segments in its output, and it was unable to achieve a consistent decrease in MSE with increasing resolution.

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