Percolation Model Research Articles

Porosity effect on the thermal conductivity of sintered powder materials

In this work, the effective thermal conductivity of sintered powder materials is studied. The extensive literature related to the proposed models about this property in all kind of porous materials is reviewed, and a new equation is proposed as a function of the fully dense material conductivity, the porosity of the material and the tap porosity of the starting powder. This equation covers the porosity range of powder aggregates from the tap porosity to zero porosity, and also applies to sintered powders. The proposed equation has been experimentally validated by fitting to experimental data of metallic sintered powder materials measured at room temperature, resulting very good agreements. Also, alternative models proposed by other authors have been fitted to the same experimental data to check the relative goodness of the proposed model. The results allow to conclude that a percolation model can describe the behaviour of the effective thermal conductivity of sintered powder materials with low and medium porosity levels.

Physical Realizations of Interdependent Networks: Analogy to Percolation

Percolation on interdependent networks generalizes the well-studied percolation model in a single network to multiple interacting systems, unveiling spontaneous cascading failures, abrupt collapses, and high vulnerability. The main novelty of interdependent networks has been the introduction of two types of links, connectivity within networks and the dependency between them. The interplay between these two types of interactions results in novel critical phenomena and phase transitions. This abstract percolation paradigm was recently applied to magnetic networks, as an experimentally testable method for interdependent superconducting networks as well as to other systems like k-core percolation and overloaded networks. Here, we will review these physical applications and provide insights into several potential directions for the field of physically interdependent networks.

University of percolation at dynamic pseudocritical point

Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.

Taxonomy of high pressure vibration spectra of zincblende semiconductor alloys based on the percolation model

Due to their simple structure (two bond species randomly arranged on a cubic lattice), the zincblende A1−xBxC semiconductor alloys (zb-SCA) set a benchmark to explore how physical properties are impacted by disorder. A longstanding controversy was whether the lattice dynamics (phonons), governed by the bond force constant, i.e., a local physical property, is blind to the alloy disorder or actually sees it. Over the past two decades, we introduced the percolation model (PM) that distinguishes between like bonds depending on whether they vibrate in same (homo) or alien (hetero) environments (1-bond → 2-mode scheme). The PM seems to apply universally among zb-SCA, and hence would solve the controversy in favor of the second scenario. Here our aim is to take one step forward and complete in the main lines a PM-based taxonomy of high-pressure vibration spectra of zb-SCA. This might clarify how a disordered atomic alloy, seen by each bond species in terms of a homo/hetero composite (i.e., at the unusual mesoscopic scale) from the angle of the PM, behaves when the lattice shrinks under hydrostatic pressure. We focus on Cd1−xZnxTe as the last sensitive pending case. This tidying-up exercise is attractive at the fundamental level and useful for projecting phonon-based devices involving zb-SCA.

A Local–Transit Percolation and Clustering-Based Method for Highway Segment Importance Ranking

The impact of disturbances on a transportation network varies depending on the location and characteristics of the affected highway segments. Given limited resources, it is crucial to prioritize the protection and repair of highway segments based on their importance to maintaining overall network performance during disruptions. This paper proposes a novel method for ranking the importance of highway segments, leveraging a novel local–transit percolation and clustering-based method. Initially, the highway network is constructed by Graph theory, and the k-means clustering method is applied considering each segment’s transit and local traffic flows. Subsequently, a local–transit percolation model is constructed to generate an initial ranking of segments based on the size of the second-largest clusters during the percolation phase transition. A secondary ranking is performed by refining the results from the clustering phase. Results of a control experiment show that, compared to baselines, the proposed ranking approach demonstrates a significantly improved ability to sustain network demand and connectivity when high-ranked segments are moved. The model uncertainty analysis was conducted by adding noise to the gantry records, and the experiments demonstrated that the model exhibits robustness under noisy conditions. These findings highlight the effectiveness and superiority of the proposed method.

Reverse percolation models for growing real-world networks

Reverse percolation models for growing real-world networks

Triadic percolation in computer virus spreading dynamics

Abstract In recent years, the threats posed by computer viruses have become increasingly diverse and complex. While classic percolation theory provides a novel perspective for analyzing epidemics and information dissemination, it fails to capture the temporal dynamics of these systems and the effects of virus invasion and governmental regulation. Triadic percolation theory, a recent advancement, addresses these limitations. In this paper, we apply this new percolation mechanism to model the diffusion of computer viruses, deriving a precise mathematical formulation of the triadic percolation model and providing an analytical solution of the triadic percolation threshold. Additionally, we investigate the impact of nonlinear transmission probability characteristics on virus propagation. Numerical simulations demonstrate that reducing the network’s average degree (or the positive regulation) or increasing regulatory interventions raises the outbreak threshold for computer viruses while decreasing their final size. Moreover, the study reveals that nonlinear transmission probabilities result in an increased number of solutions for the final size of the computer viruses. Our findings contribute new insights into controlling the spread of computer viruses.

Widths of crossings in Poisson Boolean percolation

Abstract We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model contains a crossing of an $n\times n$ square, how wide is this crossing? The answer depends on whether we consider the critical, sub-, or super-critical regime, and is different for the occupied and vacant sets.

Investigating the role of lemon peel fiber in the casein gelation mechanism of low-fat yogurt

Incorporation of insoluble dietary fibers (IDFs) into yogurt can serve as both a nutritional supplement and texture modifier. However, the understanding of the interaction between IDFs and casein particles in yogurt, as well as their effect on the gelation mechanism of casein particles, remains limited. This study investigated the impact of varying contents of insoluble lemon peel fiber (LPF) on the casein gelation in low-fat yogurt. The gelation process of casein was monitored via oscillatory rheology, and the critical gel point was determined from time sweep curves. The post-gelation process was analyzed through the percolation model, and the topological structure after gelation was quantified using skeleton analysis. The incorporation of moderate content of LPF was found to reduce fermentation time (with the highest decrease being up to 1.2 h). This effect could be attributed to the enhanced main intermolecular forces (hydrophobic interactions and disulfide bonds), as well as the formation of casein-LPF-casein linked clusters with an elevated gelation rate exponent (α) value. These factors facilitated gel formation. Moreover, a moderate content of LPF facilitated cross-linking between caseins through a bridging effect, resulting in the development of a densely interconnected network with enhanced link density (88.25%) and reduced pore fraction (7.71%). Consequently, this led to a significant 17.6% inhibition in wheying off rate compared to low-fat yogurt without LPF during a 21-day storage time. However, the formation of weak or coarse gels was observed when the LPF content was low or high, the gelation rate was excessively rapid (exponent α = 3.05), or the presence of steric hindrance effect was present. The findings offered valuable insights for the application of insoluble particulate components in the food industry.

Effect of graphite fillers on electrical and thermal conductivity in epoxy-based composites: Percolation behavior and analysis

This work investigated a method for producing epoxy-based composites using various graphite fillers, such as natural, synthetic, and thermally expanded graphite. The study aimed to determine the impact of the filler type, size, and volume fraction on the composites’ thermal and electrical conductivity. Results indicate that the percolation model accurately represents electrical and thermal conductivity behavior, with graphite fillers forming conductive clusters in the polymer matrix. The percolation threshold for electrical conductivity varies between 2.8 and 8.5 vol.%, while for thermal conductivity, it ranges from 5.0 to 18.0 vol.%, which is twice that of electrical conductivity. This observation is due to both the electron transfer tunneling effect and the necessity of higher filler content to facilitating effective phonon transport. Notably, composites filled with thermally expanded graphite exhibit lower percolation thresholds. Understanding percolation behavior facilitates the prediction and optimization of composites with specific electrical and thermal properties for diverse applications.

Scaling behavior of cross-entropy loss in the identification of percolation phase transitions.

The cross-entropy loss function is widely used in machine learning to measure the performance of a classification model. Interestingly, our study find that this function has scaling behavior when deep neural networks are used to investigate percolation models. Specifically, we use convolutional neural networks with different pooling methods to study the site percolation on square lattices under two labeling methods (labeling based on spanning cluster and the exact solution of the critical point). Subsequently, graph convolutional neural networks (GCNs) with different pooling methods are utilized to do the same kind of experiment. Finally, the GCN with different pooling methods is used to study the percolation phase transitions on the Erdős-Rényi (ER) networks under labeling based on the critical point. The reliability of the classifiers is detected by the values of the critical point p_{c} and critical exponent ν which are obtained by the scaling behaviors of the percolation probability. The results demonstrate that the scaling exponent of cross-entropy ψ/ν depends on the labeling and pooling methods. Labeling based on critical points, which is equivalent to labeling based on spanning clusters in infinite systems, can be used to investigate the critical behaviors in finite systems. SAGPooling-Mean is an effective pooling method to study the scaling behavior of cross-entropy loss on two-dimensional square lattices and ER networks.

Detecting phase transitions based on siamese neural network

Machine learning has been widely applied in physics research. Although unsupervised learning can extract the critical points of phase transitions, the percolation model remains a challenge. Unsupervised learning using the raw configurations of the percolation model fails to capture the critical points. To capture the configuration characteristics of the percolation model, this paper proposes using the maximum cluster as input to the neural network. It is well understood that the order parameter of the percolation model is not simply the particle density, but rather the probability that a given site or bond belongs to the percolating cluster. Additionally, we introduce the use of a Siamese Neural Network (SNN) to detect percolation phase transitions. Unlike unsupervised dimensionality reduction methods or supervised binary classification outputs, the SNN produces a scalar output referred to as similarity. By combining the maximum cluster and the SNN, we not only successfully extract the critical value of the percolation model, but also calculate the correlation exponent via data collapse. We believe that the SNN has great potential in handling phase transition classification problems and can serve as a reference for studying other phase transition systems.

Development and Application of an Advanced Percolation Model for Pore Network Characterization by Physical Adsorption.

Physical adsorption is one of the most widely used techniques to characterize porous materials because it is reliable and able to assess micro- and mesopores within one approach. Challenges and open questions persist in characterizing disordered and hierarchically structured porous materials. This study introduces a pore network model aimed at enhancing the textural characterization of nanoporous materials. The model, based on percolation theory on a finite-sized Bethe lattice, includes all mechanisms known to contribute to adsorption hysteresis in mesoporous pore networks. The model accounts for delayed and initiated condensation during adsorption as well as equilibrium evaporation, pore blocking, and cavitation during desorption. Coupled with dedicated nonlocal-density functional theory kernels, the proposed method provides a unified framework for modeling the entire experimental adsorption-desorption isotherm, including desorption hysteresis scans. The applicability of the method is demonstrated on a selected set of nanoporous silica materials exhibiting distinct types of hysteresis loops (types H1, H2a, H1/H2a, and H5), including ordered mesoporous silica networks (KIT-6 and SBA-15/MCM-41 hybrid silica with plugged pores) and disordered mesoporous silica networks (hierarchical meso-macroporous monolith and porous Vycor glass). For all materials, a good correlation is found between calculated and experimental primary isotherms as well as desorption scans. The model allows us to determine key pore network characteristics such as pore connectivity and pore size distributions as well as a parameter correlated with the impact of pore network disorder on the adsorption behavior. The versatility and enriched textural insights provided by the proposed novel network model allow for a comprehensive characterization previously inaccessible and hence will contribute to further advancement in the textural characterization of novel nanoporous materials. It has the potential to provide important guidance for the design and selection of porous materials for optimizing various applications, including separation processes such as chromatography, heterogeneous catalysis and gas and energy storage.

Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

We investigate four types of percolation models — node, edge, bootstrap, and diffusion percolation — in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents (ν, γ, and β) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation dν=γ+2β is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension d is replaced by the Hausdorff dimension.

Improved electrical transport properties of La0.8Ba0.2MnO3 films via suppressed electron scattering behavior

La0.8Ba0.2MnO3 (LBMO) films were prepared by spin coating on LaAlO3 substrates with a (00 l) orientation. The effects of pre−sintering temperatures (Tps) on the electrical transport properties of LBMO films were investigated. The combination of microstructure, ionic valence, surface morphology, phenomenological percolation model, and small polaron hopping model revealed that LBMO films exhibited optimized electron−lattice coupling and improved electrical transport properties at higher Tps. In the ferromagnetic metal region, temperature−independent scattering and electron−lattice coupling were suppressed and optimized as the Tps rose, thereby improving the electrical transport properties of LBMO films. In the paramagnetic insulator region, LBMO films exhibited reduced activation energy as the Tps rose, thereby enhancing their delocalization and facilitating polaron hopping to neighboring sites. The provided experimental foundation and conduction mechanisms in different temperature regions were employed to investigate the causes of the improvement process. In sum, the facile control of the electron scattering mechanisms through Tps modulation can provide favorable conditions for application in near room−temperature bolometers.

Lattice model for percolation on a plane of partially aligned sticks with length dispersity.

A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks per unit area at the percolation threshold in terms of the distributions over length and orientational angle and are compared with findings from computer simulations. Consistent with findings from computer simulations, our model shows that the percolation threshold is (i) elevated by increasing degrees of alignment for a fixed length distribution, and (ii) lowered by increasing degrees of length dispersity for a fixed orientational distribution. The impact of length dispersity is predicted to be governed entirely by the first and second moments of the stick length distribution, and the threshold is shown to be quite sensitive to particulars of the orientational distribution function.

The resilience and determinants of global mineral resource supply chains: a network percolation perspective

Mineral resources are the basic materials for global economic development. Assessing mineral resource supply chain resilience is an important pillar of mineral resource supply chain stability management. The globality, heterogeneity and complexity of supply chain bring challenges to the resilience assessment of global mineral resource supply chain. To solve this problem, a method based on multi-region input-output model, network percolation model and econometric model is proposed, which is able to measure the resilience of global mineral resource supply chain and its influencing factors from the perspective of the whole system. The percolation phase transition is introduced to measure the critical state of global mineral resource supply chain system collapse facing external disruption. Using the proposed method, this paper conducts an empirical study on the evolution of global mineral resource supply chain resilience from 2005 to 2014. The results show that the resilience of global mineral resource supply chain declined by 39.6% in 2005–2014. Most of the critical links that caused the collapse of the global mineral resource supply chain network are the manufacturing sector and its upstream and downstream sectors. The structure of supply chain network plays a key role in network resilience. Increasing the number of linkages in upstream and downstream could improve network resilience, but the increase of linkage strength would deteriorate network resilience.

Critical dynamics of the sol-gel transition studied using particle-tracking microrheology.

The formation of tetra-PEG gels, a model network with a well-defined structure, was investigated using particle-tracking microrheology. The dynamic scaling and critical relaxation exponents in the sol-gel transition were determined by applying the time-cure superposition method. Some values of the exponents differed from those of the Rouse model theory. This was attributed to the failure of the theoretical assumption that the cluster structure formation follows the percolation model theory on a three-dimensional lattice.

How Fast do Rumours Spread?

We study a rumour propagation model along the lines of Lebensztayn and Rodriguez (Stat Probab Lett 78(14):2130–2136, 2008) as a long-range percolation model on Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}. We begin by showing a sharp phase transition-type behaviour in the sense of exponential decay of the survival time of the rumour cluster in the sub-critical phase. In the super-critical phase, under the assumption that radius of influence r.v. has 2+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2+\\epsilon $$\\end{document} moment finite (for some ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document}), we show that the rightmost vertex in the rumour cluster has a deterministic speed in the sense that after appropriate scaling, the location of the rightmost vertex converges a.s. to a deterministic positive constant. Under the assumption that radius of influence r.v. has 4+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4+\\epsilon $$\\end{document} moment finite, we obtain a central limit theorem for appropriately scaled and centered rightmost vertex. Later, we introduce a rumour propagation model with reactivation. For this section, we work with a family of exponentially decaying i.i.d. radius of influence r.v.’s, and we obtain the speed result for the scaled rightmost position of the rumour cluster. Each of these results is novel, in the sense that such properties have never been established before in the context of the rumour propagation model on Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}, to the best of our knowledge.

Anomalous power-law behavior in the electrical impedance of endothelial cellular networks

In this paper, we report on the electrical impedance measurement of human endothelial cellular networks and show the existence of emergent power law behavior in its admittance. In particular, we find that the admittance scales with the frequency ω as ωα, with the exponent that varies with the degree of the disruption caused by the inflammation in the endothelial cellular network. We demonstrate that the power law of the measured electrical admittance can be understood by a simple percolation model of a large R–C (resistor–capacitor) network, which allows us to relate quantitatively and the intensity of inflammation. Our results suggest that the electrical properties of heterogeneous biomaterials, like living tissues, behave as a complex microstructural network.