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"Percolation"
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Recent Progress on Hybrid Percolation Transitions
Percolation describes the formation of a giant cluster once the average degree of a network exceeds a critical value. A hybrid percolation transition (HPT) denotes a phenomenon in which a discontinuous jump of the order parameter and the critical behavior, a basic pattern of a continuous transition, appear together at the same threshold. Such HPTs have been reported in many different systems. In this review, we present several representative examples of HPTs and classify them into two categories: global suppression-induced HPTs and cascading failure-induced HPTs. In the former class, critical behavior manifests itself in the distribution of cluster sizes, whereas in the latter it emerges in the distribution of avalanche sizes. We further outline the universal scaling relations shared by both types.
Journal Article
Stationary cocycles and Busemann functions for the corner growth model
We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface.
Journal Article
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria
We analyze a class of nonlinear partial differential equations (PDEs) defined on
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Possible origin for the similar phase transitions in k-core and interdependent networks
The models of k -core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in k -core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering k -core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, d f ′ , and correlation length exponent, ν ′ ) observed in d -dimensional IN spatial networks also exist with the same values in k -core percolation. That is, the fractality of the k -core giant component fluctuations is manifested by a fractal fluctuation dimension, d ˜ f = 3 / 4 , within a correlation size N ʹ that scales as N ′ ∝ ( p − p c ) − ν ˜ , with ν ˜ = 2 . Here we define, ν ˜ ≡ d ⋅ ν ′ and d ˜ f ≡ d f ′ / d . This implies that both models, IN and k -core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k -core we find here that for k = 1 and k = 2 the influences are SR and in contrast for k ⩾ 3 the influence is LR. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the k -core giant component and for IN as well as for any mixed-order transition are established. Our analysis enhances the comprehension of k -core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.
Journal Article
Resilience of networks with community structure behaves as if under an external field
Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic– paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.
Journal Article
The dynamic nature of percolation on networks with triadic interactions
Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical simulations. We find that triadic percolation on real network topologies reveals a similar phenomenology. These results radically change our understanding of percolation and may be used to study complex systems in which the functional connectivity is changing in time dynamically and in a non-trivial way, such as in neural and climate networks.
Triadic interactions are higher-order interactions relevant to many real complex systems. The authors develop a percolation theory for networks with triadic interactions and identify basic mechanisms for observing dynamical changes of the giant component such as the ones occurring in neuronal and climate networks.
Journal Article
No exceptional words for Bernoulli percolation
Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as percolation of words . We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z^3 with parameter p = 1/2 . We also extend this result in various directions, proving the same result on Z^d , d 3 , for any value p ın (p_c^site(Z^d), 1 - p_c^site(Z^d)) , and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.
Journal Article
Percolation phase transition in weight-dependent random connection models
We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.
Journal Article