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Critical cascades are found in many self-organizing systems. Here, we examine critical cascades as a design paradigm for logic and learning under the linear threshold model (LTM), and simple biologically inspired variants of it as sources of computational power, learning efficiency, and robustness. First, we show that the LTM can compute logic, and with a small modification, universal Boolean logic, examining its stability and cascade frequency. We then frame it formally as a binary classifier and remark on implications for accuracy. Second, we examine the LTM as a statistical learning model, studying benefits of spatial constraints and criticality to efficiency. We also discuss implications for robustness in information encoding. Our experiments show that spatial constraints can greatly increase efficiency. Theoretical investigation and initial experimental results also indicate that criticality can result in a sudden increase in accuracy.
Using graphene on hexagonal boron nitride (hBN) as an example, we introduce the concept of van der Waals heterostructures. First, we explain extraordinary high quality of graphene on hBN. Then we discuss the physics of formation of moiré patterns and a general problem of commensurability and incommensurability. We also discuss the basic consequences for electronic structure and electron transport properties, including a conductivity along zero-mass lines, formation of additional Dirac points and recently experimentally discovered new types of magneto-oscillation effects in graphene superlattuces.
In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin , as well as (more general) m-ary search trees, fragmentation trees, and median-of-(
) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.
Abundance thresholds are of fundamental importance in our attempts to understand the dynamics of wildlife infection. Identifying and manipulating these thresholds may also have substantial applied significance. The plague system in the Pre-Balkhash region of Kazakhstan has been extensively studied, including an unusually thorough investigation of the nature and importance of an abundance threshold for the infection. Great gerbils are the main reservoir host, with plague transmitted between them by a variety of flea species. Initial work identified such a threshold from time-series data, with great gerbil abundance being measured by level of occupancy (the proportion of the burrow systems in the landscape supporting an extended family group). However, this and other refinements of the threshold were better at predicting the absence of plague (below the threshold) than in guaranteeing its presence (above). Further analysis indicated that the threshold was a critical point in the percolation of plague across the landscape, rather than in a mass-action random mixing process. The performance of the threshold was also improved by incorporating both gerbil and flea abundance to generate a hyperbolic threshold curve.
A large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where, in addition, the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with respect to the network size.
We discuss percolation and random walks in a class of homogeneous ultrametric spaces together with similarities and differences in ultrametric and Euclidean spaces. We briefly outline the role of these models in the study of interacting systems. Several open problems are presented.
In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.
In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.
We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = pn ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter pn tending to 0.
We present in situ firn temperatures from the extreme 2012 melt season in the southwestern lower accumulation area of the Greenland ice sheet. The upper 2.5 m of snow and firn was temperate during the melt season, when vertical meltwater percolation was inefficient due to a ~5.5 m thick ice layer underlying the temperate firn. Meltwater percolation and refreezing beneath 2.5 m depth only occurred after the melt season. Deviations from temperatures predicted by pure conductivity suggest that meltwater refroze in discrete bands at depths of 2.0–2.5, 5.0–6.0 and 8.0–9.0 m. While we find no indication of meltwater percolation below 9 m depth or complete filling of pore volume above, firn at 10 and 15 m depth was respectively 4.2–4.5°C and 1.7°C higher than in a conductivity-only simulation. Even though meltwater percolation in 2012 was inefficient, firn between 2 and 15 m depth the following winter was on average 4.7°C warmer due to meltwater refreezing. Our observations also suggest that the 2012 firn conditions were preconditioned by two warm summers and ice layer formation in 2010 and 2011. Overall, firn temperatures during the years 2009–13 increased by 0.6°C.
While the terms ‘glacier’ and ‘ice cap’ have distinct morphological meanings, no easily defined boundary or transition distinguishes one from the other. Despite this, the exponent of the power law function relating volume to surface area differs sharply for glaciers and ice caps, suggesting a fundamental distinction beyond a smoothly transitioning morphology. A standard percolation technique from statistical physics is used to show that valley glaciers are in fact differentiated from ice caps by an abrupt geometric transition. The crossover is a function of increasing glacier thickness, but it owes its existence more to the nature of the underlying bedrock topography than to specifics of glacier mechanics: the crossover is caused by a switch from directed flow that is constrained by surrounding bedrock topography to unconstrained radial flow of thicker ice that has subsumed the topography. The crossover phenomenon is nonlinear and rapid so that few if any glaciers will have geometries or dynamics that blend the two extremes. The exponents of scaling relationships change abruptly at the crossover from one regime to another; in particular, the volume/area scaling exponent will switch from γ = 1.375 for glaciers to γ = 1.25 for ice caps, with few, if any, ice bodies having exponents that fall between these values.
We propose a class of random scale-free spatial networks with nested community structures called SHEM and analyze Reed–Frost epidemics with community related independent transmissions. We show that in a specific example of the SHEM the epidemic threshold may be trivial or not as a function of the relation among community sizes, distribution of the number of communities, and transmission rates.
A facile synthesis method was adapted to prepare Ortho-chloropolyaniline/graphite oxide (OPANI/GO) composites. The properties of these composites have been discussed based on different characterizations such as scanning electron microscopy, FTIR, and dielectric studies. The important peaks of benzenoid and quinoid rings in the spectrum confirm the formation of composites. Scanning electron microscopic studies indicate that the morphology of the prepared composites was influenced by different weight ratios of GO in OPANI and the resulting micrographs of these composites exhibited flaky and folded structures. The electrical conductivity study was carried out and found that 6 wt% of GO in PANI shows high conductivity value of 20 × 10−2 S/cm. The obtained results suggest an easy approach to prepare such novel nanocomposites with excellent dielectric properties which can be used in many technological applications.
Complex networks have recently attracted much interest due to their prevalence in nature and our daily lives (Vespignani, 2009; Newman, 2010). A critical property of a network is its resilience to random breakdown and failure (Albert et al., 2000; Cohen et al., 2000; Callaway et al., 2000; Cohen et al., 2001), typically studied as a percolation problem (Stauffer & Aharony, 1994; Achlioptas et al., 2009; Chen & D'Souza, 2011) or by modeling cascading failures (Motter, 2004; Buldyrev et al., 2010; Brummitt, et al. 2012). Many complex systems, from power grids and the Internet to the brain and society (Colizza et al., 2007; Vespignani, 2011; Balcan & Vespignani, 2011), can be modeled using modular networks comprised of small, densely connected groups of nodes (Girvan & Newman, 2002). These modules often overlap, with network elements belonging to multiple modules (Palla et al. 2005; Ahn et al. 2010). Yet existing work on robustness has not considered the role of overlapping, modular structure. Here we study the robustness of these systems to the failure of elements. We show analytically and empirically that it is possible for the modules themselves to become uncoupled or non-overlapping well before the network disintegrates. If overlapping modular organization plays a role in overall functionality, networks may be far more vulnerable than predicted by conventional percolation theory.
We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.
We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.
Real world networks typically have large clustering coefficients. The clustering coefficient can be interpreted to be the result of a triangle closing mechanism. We have here enumerated cliques and maximal cliques in multiple networks to show that real world networks have a high number of large cliques. While triangles are more frequent than expected, large cliques are much more over-expressed, and the largest difference between real world networks and their random counterpart occurs in many networks at clique sizes of 5–7, and not at a size of 3. This does not result from the existence of few very large cliques, since a similar feature is observed when studying only maximal cliques (cliques that are not contained in other larger cliques). Moreover, when the large cliques are removed, triangles are often under-expressed.
In all networks studied but one, all node members of large cliques produce a single connected component, which represent the central “core” of the network. The observed clique distribution can be explained by multiple models, mainly hidden variables model, such as the gravitation model, or the collapse of bipartite networks. These models can explain other properties of these networks, including the sub-graph distribution and the distance distribution of the networks. This suggests that node connectivity in real world networks may be determined by the similarity between the contents of the networks' nodes. This is in contrast with models of network formation that incorporate only the properties of the network, and not the internal properties of the nodes.
Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.
In this paper we estimate the expectation of the size of the largest component in a supercritical random geometric graph; the expectation tends to a polynomial on a rate of exponential decay. We sharpen the expectation's asymptotic result using the central limit theorem. Similar results can be obtained for the size of the biggest open cluster, and for the number of open clusters of percolation on a box, and so on.