Incipient Percolation Cluster Research Articles

Superhighways and roads of multivariate time series shock transmission: Application to cryptocurrency, carbon emission and energy prices

Inferring the heterogeneous connection pattern of a networked system of multivariate time series observations is a key issue. In finance, the topological structure of financial connectedness in a network of assets can be a central tool for risk measurement. Against this, we propose a topological framework for variance decomposition analysis of multivariate time series in time and frequency domains. We build on the network representation of time–frequency generalized forecast error variance decomposition (GFEVD), and design a method to partition its maximal spanning tree into two components: (a) superhighways, i.e. the infinite incipient percolation cluster, for which nodes with high centrality dominate; (b) roads, for which low centrality nodes dominate. We apply our method to study the topology of shock transmission networks across cryptocurrency, carbon emission and energy prices. Results show that the topologies of short and long run shock transmission networks are starkly different, and that superhighways and roads considerably vary over time. We further document increased spillovers across the markets in the aftermath of the COVID-19 outbreak, as well as the absence of strong direct linkages between cryptocurrency and carbon markets.

The Dynamic Three-Dimensional Localization of Fields in Active Percolating Systems

We study a dynamic three-dimensional (3D) field localized states in a medium with percolation disorder, where the percolation cluster is filled by the active nanoemitters. In such a system, the incipient percolating cluster generates a fractal radiating structure in which the field is radiated and scattered by the anisotropic inhomogeneity. Our numerical 3D simulations show that such a nonlinear system with noninteger fractal dimension has well-defined localized solutions for fields (3D speckles). The statistics of speckles is studied too.

Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution.

We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability p≥p(c). We numerically demonstrate that the scaling limit of planar LERW(p) curves, for all p>p(c), can be described by Schramm-Loewner evolution (SLE) with a single parameter κ that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient κ. Several geometrical tests are applied to ascertain this. All calculations are consistent with SLE(κ), where κ=1.732±0.016. This value of the diffusivity coefficient is outside the well-known duality range 2≤κ≤8. We also investigate how the winding angle of the LERW(p) crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p(c). For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.

Loop-erased random walk on a percolation cluster: crossover from Euclidean to fractal geometry.

We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ≥ p_{c}, in two and three dimensions. We find that the fractal dimensions of LERW_{p} are close to normal LERW in a Euclidean lattice, for all p>p_{c}. However, our results reveal that LERW on critical incipient percolation clusters is fractal with d_{f}=1.217 ± 0.002 for d=2 and 1.43 ± 0.02 for d=3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior of the LERW_{p} crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p_{c}. For finite systems, two crossover exponents and a scaling relation can be derived. This work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.

Classification of transport backbones of complex networks

Transport properties in random and scale-free (SF) networks are studied by analyzing the betweenness centrality (BC) distribution P(B) in the minimum spanning trees (MSTs) and infinite incipient percolation clusters (IIPCs) of the networks. It is found that P(B) in MSTs scales as P(B)∼B(-δ). The obtained values of δ are classified into two different categories, δ≃1.6 and δ≃2.0. Using the mapping between BC and the branch size of tree structures, it is proved that δ in MSTs which are close to critical trees is 1.6. In contrast, δ in MSTs which are supercritical trees is shown to be 2.0. We also find δ=1.5 in IIPCs, which is a natural result because IIPC is physically critical. Based on the results in MSTs, a physical reason why δ≥2 in the original networks is suggested.

Topological-based bottleneck analysis and improvement strategies for traffic networks

A method is proposed to find key components of traffic networks with homogenous and heterogeneous topologies, in which heavier traffic flow is transported. One component, called the skeleton, is the minimum spanning tree (MST) based on the zero flow cost (ZCMST). The other component is the infinite incipient percolation cluster (IIC) which represents the spine of the traffic network. Then, a new method to analysis the property of the bottleneck in a large scale traffic network is given from a macroscopic and statistical viewpoint. Moreover, three effective strategies are proposed to alleviate traffic congestion. The significance of the findings is that one can significantly improve the global transport by enhancing the capacity in the ZCMST with a few links, while for improving the local traffic property, improving a tiny fraction of the traffic network in the IIC is effective. The result can be used to help traffic managers prevent and alleviate traffic congestion in time, guard against the formation of congestion bottleneck, and make appropriate policies for traffic demand management. Meanwhile, the method has very important theoretical significance and practical worthiness in optimizing traffic organization, traffic control, and disposal of emergency.

Walking on fractals: diffusion and self-avoiding walks on percolation clusters

We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in space dimensions d = 2, 3, 4. Our analysis yields estimates of universal exponents, governing the scaling laws for configurational properties of RWs and SAWs.

Monte Carlo study of the influence of the structural heterogeneity of the surface in the CO–NO reaction on Pd and its relation with experimental data

The reduction reaction of NO by CO over Pd is studied by means of a Monte Carlo simulation. A reaction mechanism similar to that established previously in the literature for Rh and a set of recently published kinetics parameters for Pd determined from experimental work are assumed. The heterogeneous effects of the catalytic substrate are analyzed by simulating various surfaces: a uniform square and hexagonal lattice, the incipient percolation cluster (IPC) fractal, and a lattice formed by a cluster distribution supported on a deterministic fractal. The results interpret reasonably the experimental data published recently in the case of crystals and supported catalysts.

OPTIMAL PATH AND MINIMAL SPANNING TREES IN RANDOM WEIGHTED NETWORKS

We review results on the scaling of the optimal path length ℓopt in random networks with weighted links or nodes. We refer to such networks as "weighted" or "disordered" networks. The optimal path is the path with minimum sum of the weights. In strong disorder, where the maximal weight along the path dominates the sum, we find that ℓopt increases dramatically compared to the known small-world result for the minimum distance ℓ min ~ log N, where N is the number of nodes. For Erdős–Rényi (ER) networks ℓ opt ~ N1/3, while for scale free (SF) networks, with degree distribution P(k) ~ k-λ, we find that ℓopt scales as N(λ - 3)/(λ - 1) for 3 < λ < 4 and as N1/3 for λ ≥ 4. Thus, for these networks, the small-world nature is destroyed. For 2 < λ < 3 in contrary, our numerical results suggest that ℓopt scales as ln λ-1 N, representing still a small world. We also find numerically that for weak disorder ℓ opt ~ ln N for ER models as well as for SF networks. We also review the transition between the strong and weak disorder regimes in the scaling properties of ℓopt for ER and SF networks and for a general distribution of weights τ, P(τ). For a weight distribution of the form P(τ) = 1/(aτ) with (τ min < τ < τ max ) and a = ln τ max /τ min , we find that there is a crossover network size N* = N*(a) at which the transition occurs. For N ≪ N* the scaling behavior of ℓopt is in the strong disorder regime, while for N ≫ N* the scaling behavior is in the weak disorder regime. The value of N* can be determined from the expression ℓ∞(N*) = apc, where ℓ∞ is the optimal path length in the limit of strong disorder, A ≡ apc → ∞ and pc is the percolation threshold of the network. We suggest that for any P(τ) the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter Z = ℓ∞/A where [Formula: see text] plays the role of the disorder strength and τc is defined by [Formula: see text]. In case P(τ) ~ 1/(aτ), the equation for A is reduced to A = apc. The relation for A is derived analytically and supported by numerical simulations for Erdős–Rényi and scale-free graphs. We also determine which form of P(τ) can lead to strong disorder A → ∞. We then study the minimum spanning tree (MST), which is the subset of links of the network connecting all nodes of the network such that it minimizes the sum of their weights. We show that the minimum spanning tree (MST) in the strong disorder limit is composed of percolation clusters, which we regard as "super-nodes", interconnected by a scale-free tree. The MST is also considered to be the skeleton of the network where the main transport occurs. We furthermore show that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality — number of times a node (or link) is used by transport paths. One component the superhighways, for which the nodes (or links) with high centrality dominate, corresponds to the largest cluster at the percolation threshold (incipient infinite percolation cluster) which is a subset of the MST. The other component, roads, includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the incipient infinite percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network, the superhighways.

Transport in Weighted Networks: Partition into Superhighways and Roads

Transport in weighted networks is dominated by the minimum spanning tree (MST), the tree connecting all nodes with the minimum total weight. We find that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality--the number of times a node (or link) is used by transport paths. One component, superhighways, is the infinite incipient percolation cluster, for which we find that nodes (or links) with high centrality dominate. For the other component, roads, which includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. The significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network, the superhighways.

Interpretation of the Experimental Data on the Reduction Reaction of NO by CO on Rhodium by Monte Carlo Simulations and by Solving the Kinetic Equations of the Reaction Mechanism

Some mechanisms of the reduction reaction of NO by CO on rhodium are analyzed and discussed, solving the kinetics equations and using Monte Carlo simulations, in terms of its ability to interpret the recent experiments of Zaera et al., who used a molecular beam method to study experimentally the kinetics of the reaction. Critical use is also made of the information on rate constants available for this system in the literature. Uniform catalytic surfaces and the statistical incipient percolation cluster (IPC) fractal are considered in the simulations.

Structural effects in the kinetic behavior of the monomer–dimer surface reaction over nondeterministic fractal surfaces

A study is made of the segregation effect of adsorbed species related to the inner structure of two bidimensional adsorbent fractals, the incipient percolation cluster (IPC) and the backbone of the IPC, and their connection with the rate-determining step of the kinetic mechanism of the monomer–dimer (MD) surface reaction on those fractals. Production, R AB , is proportional to the concentration of A in the gas phase y A and to the fraction of vacant superficial sites x E ( R AB = y A x E ), and it is shown that adsorption of the monomer is the rate-controlling step of the reaction mechanism.

Brosilow-Ziff model for the CO-NO surface reaction on disordered two- and three-dimensional substrates.

The Brosilow-Ziff model, which for a uniform lattice shows superficial poisoning by the "checkerboard mechanism" for simplified Langmuir-Hinshelwood kinetics of the reduction reaction of NO by CO, has been studied in the case of various disordered substrates: two-dimensional (2D) incipient percolation cluster, backbone, and diffusion-limited aggregate (2D) fractals, together with a deterministic fractal and a statistical substrate, both three-dimensional. Some additional effects such as diffusion, nearest-next-neighbor sites, and desorption are also considered. The systems were found to be nonreactive and pseudoreactive in all the cases which showed the checkerboard mechanism. Otherwise, a reactive zone (window) related directly to the geometric structure of the substrate is seen.

Time scaling in the reorganization of adsorbed species in surface reactions

The reorganization of the species adsorbed in the extreme zone of the phase diagram (y0+ = yA [Formula: see text] 0), close to the jamming state (yA = 0) of a surface reaction on a catalytic substrate, is studied for the monomer–dimer (MD), monomer – linear trimer (MLT), monomer – bent trimer (MBT), and dimer – linear trimer (DLT) systems on a uniform surface and on a fractal, the incipient percolation cluster (IPC). It is established that the evolution of the surface fraction xB of component B (D, LT, or BT) follows a scaling law xB = Atk in the intermediate time zone. PACS No.: 79.69Dp

Incipient infinite percolation clusters in 2D

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC. We also show that for any k≥1 the difference in size between the kth and (k+1)st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.

Self-avoiding walks on self-similar structures: finite versus infinite ramification

Self-avoiding walks (SAWs) generated by exact enumeration techniques are studied on the Sierpinski carpet (d = 2) and on the Sierpinski sponge (d = 3) (also called Sierpinski square lattices). A detailed comparison of the results for SAWs on these infinitely ramified fractals to SAWs on finitely ramified Sierpinski gaskets (Sierpinski triangular lattices), on regular lattices, and on the incipient percolation cluster is done, providing insight into the behaviour of SAWs on ordered and disordered structures. The SAWs on Sierpinski square lattices are found to display a kind of intermediate behaviour, sharing aspects of both SAWs on ordered and on fractal structures. As a consequence, a des Cloizeaux relation does not seem to hold for this structure, as opposed to its validity for SAWs on regular lattices, on Sierpinski triangular lattices and on the incipient percolation cluster.

The monomer–trimer and dimer–trimer superficial reactions on geometrically disordered substrates

A Monte Carlo study is made of monomer–linear trimer, monomer–bent trimer and dimer–linear trimer surface reactions on substrates made of percolation clusters having varying degrees of surface disorder and on a statistical fractal, the incipient percolation cluster. The phase diagrams show that the branching of the disordered substrate alters the width of the reactive zone, the production value and the character of the irreversible phase transitions. At the ends of the diagrams a linear decrease in the jamming fractions with increasing substrate branching is seen, together with a discontinuous transition between the jamming states and the neighboring absorbent zones.

Superlocalization of Waves and Electrons in Hopping Conductivity

We have found the two different densities of states d s = 4/3 and d v = 1.05 and two inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ(r) exp [-r do ] with values of d o1 = 1.73 and d o2 = 2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity a(t) exp [-(T 0 /T) β ] between impurities on a random fractal structure give the values of β = 2/5 for the scalar and of β = 1/2 for the vector interactions, respectively.

Self-avoiding walks on Sierpinski lattices in two and three dimensions.

The scaling properties of linear polymers on deterministic fractal structures, modeled by self-avoiding random walks (SAW) on Sierpinski lattices in two and three dimensions, are studied. To this end, all possible SAW configurations of N steps are enumerated exactly and averages over suitable sets of starting lattice points for the walks are performed to extract the mean quantities of interest reliably. We determine the critical exponent describing the mean end-to-end chemical distance (-)l(N) after N steps and the corresponding distribution function, P(S)(l,N). A des Cloizeaux-type relation between the exponent characterizing the asymptotic shape of the distribution, for l-->0 and N--> infinity, and the one describing the total number of SAW of N steps is suggested and supported by numerical results. These results are confronted with those obtained recently on the backbone of the incipient percolation cluster, where the corresponding exponents are very well described by a generalized des Cloizeaux relation valid for statistically self-similar structures.

Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.