arXiv subjects

Ilya Zaliapin

Publications and source records attributed to Ilya Zaliapin.

At least 19 recordsLinked to original sources

arXiv2022

Invariant Galton-Watson trees: metric properties and attraction with respect to generalized dynamical pruning

Invariant Galton-Watson (IGW) tree measures is a one-parameter family of critical Galton-Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are the only attractors of critical Galton-Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

math.PR↗

arXiv2021

Critical Tokunaga model for river networks

The hierarchical organization and self-similarity in river basins have been topics of extensive research in hydrology and geomorphology starting with the pioneering work of Horton in 1945. Despite significant theoretical and applied advances however, the mathematical origin of and relation among Horton laws for different stream attributes remain unsettled. Here we capitalize on a recently developed theory of random self-similar trees to introduce a one-parametric family of self-similar critical Tokunaga trees that elucidates the origin of Horton laws, Hack's laws, basin fractal dimension, power-law distributions of link attributes, and power-law relations between distinct attributes. The proposed family includes the celebrated Shreve's random topology model and extends to trees that approximate the observed river networks with realistic exponents. The results offer tools to increase our understanding of landscape organization under different hydroclimatic forcings, and to extend scaling relationships useful for hydrologic prediction to resolutions higher that those observed.

physics.geo-ph↗

arXiv2020

Invariance and attraction properties of Galton-Watson trees

We give a description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton-Watson tree and a one-parameter family of critical Galton-Watson trees with offspring distribution $\{q_k\}$ that has a power tail $q_k\sim Ck^{-(1+1/q_0)}$, where $q_0\in(1/2,1)$. Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the tail behavior of the initial Galton-Watson offspring distribution. The invariant measures satisfy the Toeplitz property for the Tokunaga coefficients and obey the Horton law with exponent $R = (1-q_0)^{-1/q_0}$.

math.PR↗

arXiv2019

Random Self-Similar Trees: A mathematical theory of Horton laws

The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding. Ubiquitous in hierarchical branching systems, the Horton laws have been rediscovered in many disciplines ranging from geomorphology to genetics to computer science. Attempts to build a mathematical foundation behind the Horton laws during the 1990s revealed their close connection to the operation of pruning -- erasing a tree from the leaves down to the root. This survey synthesizes recent results on invariances and self-similarities of tree measures under various forms of pruning. We argue that pruning is an indispensable instrument for describing branching structures and representing a variety of coalescent and annihilation dynamics. The Horton laws appear as a characteristic imprint of self-similarity, which settles some questions prompted by geophysical data.

math.PR↗

arXiv2018

Random self-similar trees and a hierarchical branching process

We study self-similarity in random binary rooted trees. In a well-understood case of Galton-Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts the tree leaves. This only happens for the critical Galton-Watson tree (a constant process progeny), which also exhibits other special symmetries. We extend the prune-invariance setup to arbitrary binary trees with edge lengths. In this general case the class of self-similar processes becomes much richer and covers a variety of practically important situations. The main result is construction of the hierarchical branching processes that satisfy various self-similarity definitions (including mean self-similarity and self-similarity in edge-lengths) depending on the process parameters. Taking the limit of averaged stochastic dynamics, as the number of trajectories increases, we obtain a deterministic system of differential equations that describes the process evolution. This system is used to establish a phase transition that separates fading and explosive behavior of the average process progeny. We describe a class of critical Tokunaga processes that happen at the phase transition boundary. They enjoy multiple additional symmetries and include the celebrated critical binary Galton-Watson tree with independent exponential edge length as a special case. Finally, we discuss a duality between trees and continuous functions, and introduce a class of extreme-invariant processes, constructed as the Harris paths of a self-similar hierarchical branching process, whose local minima has the same (linearly scaled) distribution as the original process.

math.PR↗

arXiv2018

Dynamical pruning of rooted trees with applications to 1D ballistic annihilation

We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree $T$, starting from the leaves, according to a pruning function defined on subtrees within $T$. The generalized pruning encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The main result is invariance of a finite critical binary Galton-Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. The second part of the paper examines the continuum 1-D ballistic annihilation model $A+A \rightarrow \emptyset$ for a constant particle density and initial velocity that alternates between the values of $\pm$1. The model evolution is equivalent to a generalized dynamical pruning of the shock tree that represents dynamics of sinks (points of particle annihilation), with the pruning function equal to the total tree length. The shock tree is isometric to the level set tree of the model potential (integral of velocity). This equivalence allows us to construct a complete probabilistic description of the annihilation dynamics for the initial velocity that alternates between the values of $\pm$1 at the epochs of a stationary Poisson process. Finally, we discuss several real tree representations of the ballistic annihilation model, closely connected to the shock wave tree.

math.PR↗

arXiv2018

Tokunaga self-similarity arises naturally from time invariance

The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested in this work. We define a geometric branching processes $\mathcal{G}(s)$ that generates self-similar rooted trees. The main result establishes the equivalence between the invariance of $\mathcal{G}(s)$ with respect to a time shift and a one-parametric version of the Tokunaga condition. In the parameter region where the process satisfies the Tokunaga condition (and hence is time invariant), $\mathcal{G}(s)$ enjoys many of the symmetries observed in a critical binary Galton-Watson branching process and reproduce the latter for a particular parameter value.

math.DS↗

arXiv2015

Horton Law in Self-Similar Trees

Self-similarity of random trees is related to the operation of pruning. Pruning $R$ cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex $v$ and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at $v$. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers $N_k[K]$ and $N_{ij}[K]$ are defined as the expected number of branches of order $k$, and the expected number of order-$i$ branches that merged order-$j$ branches, $j>i$, respectively, in a finite tree of order $K$. The Tokunaga coefficients are defined as $T_{ij}[K]=N_{ij}[K]/N_j[K]$. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: $T_k:=T_{i,i+k}[K]$. We show that for self-similar trees, the condition $\limsup(T_k)^{1/k}<\infty$ is necessary and sufficient for the existence of the strong Horton law: $N_k[K]/N_1[K] \rightarrow R^{1-k}$, as $K \rightarrow \infty$ for some $R>0$ and every $k\geq 1$. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.

cs.DM↗

arXiv2015

Horton self-similarity of Kingman's coalescent tree

The paper establishes a weak version of Horton self-similarity for a tree representation of Kingman's coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of branches of a given Horton-Strahler order in a tree that represents Kingman's N-coalescent process with a constant kernel, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman's tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.

math.PR↗

arXiv2014

Transport and Vulnerability in River Deltas: A Graph-Theoretic Approach

Maintaining a sustainable socio-ecological state of a river delta requires delivery of material and energy fluxes to its body and coastal zone in a way that avoids malnourishment that would compromise system integrity. We present a quantitative framework for studying delta topology and transport based on representation of a deltaic system by a rooted directed acyclic graph. Applying results from spectral graph theory allows systematic identification of the upstream and downstream subnetworks for a given vertex, computing steady flux propagation in the network, and finding partition of the flow at any channel among the downstream channels. We use this framework to construct vulnerability maps that quantify the relative change of sediment and water delivery to the shoreline outlets in response to possible perturbations in hundreds of upstream links. This enables us to evaluate which links hotspots and what management scenarios would most influence flux delivery to the outlets. The results can be used to examine local or spatially distributed delta interventions and develop a system approach to delta management.

physics.geo-ph↗

arXiv2014

Network robustness assessed within a dual connectivity perspective

Network robustness against attacks has been widely studied in fields as diverse as the Internet, power grids and human societies. Typically, in these studies, robustness is assessed only in terms of the connectivity of the nodes unaffected by the attack. Here we put forward the idea that the connectivity of the affected nodes can play a crucial role in properly evaluating the overall network robustness and its future recovery from the attack. Specifically, we propose a dual perspective approach wherein at any instant in the network evolution under attack, two distinct networks are defined: (i) the Active Network (AN) composed of the unaffected nodes and (ii) the Idle Network (IN) composed of the affected nodes. The proposed robustness metric considers both the efficiency of destroying the AN and the efficiency of building-up the IN. We show, via analysis of both prototype networks and real world data, that trade-offs between the efficiency of Active and Idle network dynamics give rise to surprising crossovers and re-ranking of different attack strategies, pointing to significant implications for decision making.

physics.soc-ph↗

arXiv2011

Tokunaga and Horton self-similarity for level set trees of Markov chains

The Horton and Tokunaga branching laws provide a convenient framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching. The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, and proven for two paradigmatic models: the critical Galton-Watson branching process with finite progeny and the finite-tree representation of a regular Brownian excursion. This study establishes the Tokunaga and Horton self-similarity for a tree representation of a finite symmetric homogeneous Markov chain. We also extend the concept of Horton and Tokunaga self-similarity to infinite trees and establish self-similarity for an infinite-tree representation of a regular Brownian motion. We conjecture that fractional Brownian motions are also Tokunaga and Horton self-similar, with self-similarity parameters depending on the Hurst exponent.

math.PR↗

arXiv2010

Another Look at Climate Sensitivity

We revisit a recent claim that the Earth's climate system is characterized by sensitive dependence to parameters; in particular, that the system exhibits an asymmetric, large-amplitude response to normally distributed feedback forcing. Such a response would imply irreducible uncertainty in climate change predictions and thus have notable implications for climate science and climate-related policy making. We show that equilibrium climate sensitivity in all generality does not support such an intrinsic indeterminacy; the latter appears only in essentially linear systems. The main flaw in the analysis that led to this claim is inappropriate linearization of an intrinsically nonlinear model; there is no room for physical interpretations or policy conclusions based on this mathematical error. Sensitive dependence nonetheless does exist in the climate system, as well as in climate models -- albeit in a very different sense from the one claimed in the linear work under scrutiny -- and we illustrate it using a classical energy balance model (EBM) with nonlinear feedbacks. EBMs exhibit two saddle-node bifurcations, more recently called "tipping points", which give rise to three distinct steady-state climates, two of which are stable. Such bistable behavior is, furthermore, supported by results from more realistic, nonequilibrium climate models. In a truly nonlinear setting, indeterminacy in the size of the response is observed only in the vicinity of tipping points. We show, in fact, that small disturbances cannot result in a large-amplitude response, unless the system is at or near such a point. We discuss briefly how the distance to the bifurcation may be related to the strength of Earth's ice-albedo feedback.

physics.ao-ph↗

arXiv2010

Predictability of extreme events in a branching diffusion model

We propose a framework for studying predictability of extreme events in complex systems. Major conceptual elements -- hierarchical structure, spatial dynamics, and external driving -- are combined in a classical branching diffusion with immigration. New elements -- observation space and observed events -- are introduced in order to formulate a prediction problem patterned after the geophysical and environmental applications. The problem consists of estimating the likelihood of occurrence of an extreme event given the observations of smaller events while the complete internal dynamics of the system is unknown. We look for premonitory patterns that emerge as an extreme event approaches; those patterns are deviations from the long-term system's averages. We have found a single control parameter that governs multiple spatio-temporal premonitory patterns. For that purpose, we derive i) complete analytic description of time- and space-dependent size distribution of particles generated by a single immigrant; ii) the steady-state moments that correspond to multiple immigrants; and iii) size- and space-based asymptotic for the particle size distribution. Our results suggest a mechanism for universal premonitory patterns and provide a natural framework for their theoretical and empirical study.

physics.geo-ph↗

arXiv2010

A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema

We consider a highly idealized model for El Nino/Southern Oscillation (ENSO) variability, as introduced in an earlier paper. The model is governed by a delay differential equation for sea surface temperature in the Tropical Pacific, and it combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform a theoretical and numerical study of the model in the three-dimensional space of its physically relevant parameters: propagation period of oceanic waves across the Tropical Pacific, atmosphere-ocean coupling, and strength of seasonal forcing. Phase locking of model solutions to the periodic forcing is prevalent: the local maxima and minima of the solutions tend to occur at the same position within the seasonal cycle. Such phase locking is a key feature of the observed El Nino (warm) and La Nina (cold) events. The phasing of the extrema within the seasonal cycle depends sensitively on model parameters when forcing is weak. We also study co-existence of multiple solutions for fixed model parameters and describe the basins of attraction of the stable solutions in a one-dimensional space of constant initial model histories.

physics.ao-ph↗

arXiv2009

Transport on river networks: A dynamical approach

This study is motivated by problems related to environmental transport on river networks. We establish statistical properties of a flow along a directed branching network and suggest its compact parameterization. The downstream network transport is treated as a particular case of nearest-neighbor hierarchical aggregation with respect to the metric induced by the branching structure of the river network. We describe the static geometric structure of a drainage network by a tree, referred to as the static tree, and introduce an associated dynamic tree that describes the transport along the static tree. It is well known that the static branching structure of river networks can be described by self-similar trees (SSTs); we demonstrate that the corresponding dynamic trees are also self-similar. We report an unexpected phase transition in the dynamics of three river networks, one from California and two from Italy, demonstrate the universal features of this transition, and seek to interpret it in hydrological terms.

physics.geo-ph↗

arXiv2007

Aftershock identification

Earthquake aftershock identification is closely related to the question "Are aftershocks different from the rest of earthquakes?" We give a positive answer to this question and introduce a general statistical procedure for clustering analysis of seismicity that can be used, in particular, for aftershock detection. The proposed approach expands the analysis of Baiesi and Paczuski [PRE, 69, 066106 (2004)] based on the space-time-magnitude nearest-neighbor distance $η$ between earthquakes. We show that for a homogeneous Poisson marked point field with exponential marks, the distance $η$ has Weibull distribution, which bridges our results with classical correlation analysis for unmarked point fields. We introduce a 2D distribution of spatial and temporal components of $η$, which allows us to identify the clustered part of a point field. The proposed technique is applied to several synthetic seismicity models and to the observed seismicity of Southern California.

physics.geo-ph↗

arXiv2007

A delay differential model of ENSO variability: Parametric instability and the distribution of extremes

We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $κ$, and propagation period $τ$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,τ)$ plane at constant $κ$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $κ$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperature'' ({\it i.e.}, thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.

physics.ao-ph↗