Ultrametric Model Research Articles

Toward Ultrametric Modeling of the Epidemic Spread

An ultrametric model of epidemic spread of infections based on the classical SIR model is proposed. Ultrametrics on a set of individuals is introduced based on their hierarchical clustering relative to the average time of infection contact. The general equations of the ultrametric SIR model are written down and their particular implementation using the p-adic parametrization is presented. A numerical analysis of the p-adic SIR model and a comparison of its behavior with the classical SIR model are performed. The concept of hierarchical isolation and the scenario of its management in order to reduce the level of epidemic spread is considered.

Solvability of the p-adic Analogue of Navier–Stokes Equation via the Wavelet Theory

P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena.

Ultrametricity increases the predictability of cultural dynamics

A quantitative understanding of societies requires useful combinations of empirical data and mathematical models. Models of cultural dynamics aim at explaining the emergence of culturally homogeneous groups through social influence. Traditionally, the initial cultural traits of individuals are chosen uniformly at random, the emphasis being on characterizing the model outcomes that are independent of these (‘annealed’) initial conditions. Here, motivated by an increasing interest in forecasting social behavior in the real world, we reverse the point of view and focus on the effect of specific (‘quenched’) initial conditions, including those obtained from real data, on the final cultural state. We study the predictability, rigorously defined in an information-theoretic sense, of the social content of the final cultural groups (i.e. who ends up in which group) from the knowledge of the initial cultural traits. We find that, as compared to random and shuffled initial conditions, the hierarchical ultrametric-like organization of empirical cultural states significantly increases the predictability of the final social content by largely confining cultural convergence within the lower levels of the hierarchy. Moreover, predictability correlates with the compatibility of short-term social coordination and long-term cultural diversity, a property that has been recently found to be strong and robust in empirical data. We also introduce a null model generating initial conditions that retain the ultrametric representation of real data. Using this ultrametric model, predictability is highly enhanced with respect to the random and shuffled cases, confirming the usefulness of the empirical hierarchical organization of culture for forecasting the outcome of social influence models. These results appear to be highly independent of the empirical data source.

Cognitive processes of the brain: An ultrametric model of information dynamics in unconsciousness

We discuss differences in mathematical representations of the physical and mental worlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnected topological space. One of the basic models of such spaces is given by ultrametric spaces and more specially by m-adic trees. We use dynamical systems in such spaces to model flows of unconscious information at different level of mental representation hierarchy, for “mental points”, categories, and ideas. Our model can be interpreted as an unconventional computational model: non-algorithmic hierarchic “computations” (identified with the process of thinking at the unconscious level).

Models of p-adic mechanics

We segregate the class of ultrametric (p-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincare recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum p-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.

Ultrametric model of mind, I: Review

We mathematically model Ignacio Matte Blanco's principles of symmetric and asymmetric being through use of an ultrametric topology. We use for this the highly regarded 1975 book of this Chilean psychiatrist and pyschoanalyst (born 1908, died 1995). Such an ultrametric model corresponds to hierarchical clustering in the empirical data, e.g. text. We show how an ultrametric topology can be used as a mathematical model for the structure of the logic that reflects or expresses Matte Blanco's symmetric being, and hence of the reasoning and thought processes involved in conscious reasoning or in reasoning that is lacking, perhaps entirely, in consciousness or awareness of itself. In a companion paper we study how symmetric (in the sense of Matte Blanco's) reasoning can be demarcated in a context of symmetric and asymmetric reasoning provided by narrative text.

Scatter search with path relinking for phylogenetic inference

We propose the use of scatter search with path relinking for the inference of phylogenetic trees. Solutions are here represented as trees whose leaves span the set of species under study. These trees are evaluated using a minimum weight criterion under the ultrametric model. The main features of this approach are the utilization of a crossover-based schema for diversification generation, the use of path relinking for solution combination, and the utilization of an improvement method based on internal rotations of subtrees. The resulting algorithm is compared to other approaches such as evolutionary and memetic algorithms, using real data as benchmark. Scatter search provides better results for these instances.

Brachistochrone approach to the relaxation dynamics of complex hierarchical systems

We investigate the structure and the relaxation dynamics of complex hierarchical systems from a variational point of view. First, an additional argument for the use of the ultrametric caricature to describe disordered systems is provided. Focusing on ultrametric models, we show that two relevant dynamical limit behaviors of such models, the limit of convergence of the dynamics and the transition from compact to noncompact exploration, are in fact realizations of brachistochrone relaxation pathways. In turn, by making use of a rugged model whose conformation space topology deviates from ultrametricity under selective controllable conditions, we show that while the exponent of the resulting relaxation law behaves as ruggedness-dependent, its functional form is robust with respect to the introduction of ruggedness. Finally, within this rugged context, the relaxation dynamics of the two above-mentioned limit behaviors are shown to correspond to characteristic relaxation laws: Debye–Kohlrausch and power decay, respectively.

Multiple logarithmic oscillations for statistical fractals on ultrametric spaces with application to recycle flows in hierarchical porous media

The interference of logarithmic oscillations characteristic to statistical fractal and ultrametric structures is analyzed. We introduce an ultrametric model consisting in a hierarchy of branches for which the probability distribution ξ̃(n) of the total number, n, of branches has a long tail of the inverse power law type ξ̃(n) ∼ A(ln n)n -(1 + H) as n → ∞ where H is a fractal exponent characteristic for the ultrametric structure and A(ln n) is a periodic function of ln n. A comparison is made with a statistical fractal derived by means of the Shlesinger-Hughes stochastic renormalization approach. We consider a positive random variable, X, selected from a narrow unimodal probability density with finite moments. A process of stochastic amplification of the random variable is introduced which leads to a probability density of the amplified variable with a long tail: P̃(X) dX ∼ dXX-(1 + ℋ)B[ln (X/Xm)] as X → ∞ where ℋ is another fractal exponent, Xm is a cutoff value of the random variable X and B[ln (X/Xm)] is a periodic function of ln (X/Xm). Finally, the interaction between these two types of logarithmic oscillations is analyzed by assuming that the stochastic amplification of the random variable X takes place on an ultrametric structure. The final probability density of X, P*(X) dX, displays a phenomenon of amplitude modulation P*(X) dX ∼ d[ln (X/Xm)] [ln (X/Xm)](-1 + H)Ξ as X → ∞ where Ξ = Ξ(1) + Ξ(2) is made up of two additive contributions: a periodic function Ξ(1)[ln (X/Xm)] of ln (X/Xm) and a superposition Ξ(2) {ln (X/Xm), ln [ln (X/Xm)]} of periodic functions of ln (X/Xm) modulated by much slower periodic functions in ln [ln (X/Xm)]. The model is applied for the analysis of recycle flows in porous media. We assume that a hierarchical structure of pores exists which corresponds to an ultrametric space and that the recycle flow leads to a stochastic amplification of the residence time of fluid elements in the system. We show that the probability density of the residence time has an asymptotic behavior similar to that of P*(X) dX and investigate the possibilities of measurement of multiple logarithmic oscillations by means of tracer experiments. A physical interpretation of multiple logarithmic oscillations is given in the case of flow systems: they are generated by two delayed feedback processes occurring in two different logarithmic time scales, ln t and ln ln t. The fast delayed feedback process in ln t is given by the recycle flows corresponding to a given level of the porous structure whereas the slow delayed feedback process in ln ln t is due to the exchange of fluid among the different levels of the hierarchical structure of pores.

Fractal time, ultrametric topology and fast relaxation

An ultrametric model for very fast relaxation processes is suggested. We assume that a relaxation step is a succession of series- parallel elementary processes organized in a hierarchical structure formed from branches (relaxation channels) grouped in levels. The model is characterized by the following parameters: the average number 〈 n〉 0 of channels from the zeroth level of the hierarchy; the probability β that in a given level a new channel is generated, the frequency ω 0 of an elementary process and the probability of decay p attached to an overall relaxation step. The number of relaxation channels increases exponentially with the level index; as a result the survival probability of the model, l(t), decreases much faster than the usual exponential law: l(t) = exp { - 〈n〉 0p[( 1 β )−1] [ exp[ ω 0tβ (1−β) ]−s1]} . Due to this double exponential decay not only all positive moments of the lifetime probability density ϕ(t) = - ∂l(t) ∂(t) exist and are finite, but also all positive moments of the exponential of the lifetime, exp(μ t), μ> 0, exist and are finite. The relationships among the fast relaxation (1), the exponential (Markovian) relaxation (2) and the fractal time (slow) relaxation (3) are clarified: in the succession (1) → (2) → (3) each relaxation law can be obtained from the preceding one through a logarithmic transformation of the time variable. These three laws can be obtained by maximizing the informational entropy of the lifetime distribution; in the succession (1) → (2) → (3) the isoperimetric condition corresponding to a given law can be obtained from the isoperimetric condition of the preceeding law by a logarithmic transformation of time. The fast relaxation and the fractal time laws play symmetrical roles in comparison with the Markovian relaxation. The fast relaxation corresponds to an “antifractal” behavior.

Enhanced diffusion for time-disordered transport processes with ultrametric jump rates

This note deals with diffusion processes for which the transport operators depend on a set of parameters (mobility and diffusion coefficients) subject to random variations determined by a random jump frequency. The model assumes the existence of a complete temporal disorder: a succession of realizations of the random parameters corresponds to a succession of independent random variables selected from the same probability law. The average over the temporal disorder can be computed exactly: the state probability density obeys a linear integral equation which is similar to a continuous time random walk equation or to a system of age-dependent master equations. For time-homogeneous and translationally invariant systems the integral equation can be solved analytically. The behavior of the solution depends on the stochastic properties of the jump rate Ω. If the first two negative moments of Ω, ⟨Ω−1⟩ and ⟨Ω−2⟩, exist and are finite then the covariance matrix of the elements of the displacement vector obeys an Einstein like law: ⟨Δxi(t)Δxj(t)⟩ ∼ t as t → ∞. If ⟨Ω−1⟩ and ⟨Ω−2⟩ are infinite then a supradiffusive behavior ⟨Δxi(t)Δxj(t)⟩ ∼ ta with a ⩾ 1 may occur. A detailed analysis is made in the case if the distribution of jump rates is given by an ultrametric model. An ultrametric structure consisting in branches grouped in levels is generated by means of a stochastic chain process depending on two parameters: the probabilities α and β of a level and branch generation, respectively. The probability distribution ξ(n) of the total number of branches has a long tail ξ(n) ∼ n−(H +1) as n → ∞, where H = ln α/ln (1 − β) is a fractal exponent characteristic for the ultrametric structure. We show that the probability density of the jump rates may be approximated by a power law ∼ΩH−1 and that a = 2 for 1 > H > 0 and a = 1 for H > 2.

Hierarchical and overlapping structure representations of perceptions of professional service providers

Various professionals such as physicians, lawyers, architects and accountants have turned to marketing in recent years. In this paper we empirically examine individuals' perceptual structure with respect to these and six other such diverse types of professionals. Use of some state-of-the-art tree representation models enables us to accurately portray this structure. It is evident from this portrayal that professionals in this diverse group share several common features in addition to possessing some unique aspects. These features and aspects are discussed with respect to the hierarchical Ultrametric and Addtree models, and the overlapping or non-nested Extree model. Managerial implications for professional services marketing are discussed. These include implications for marketing mix strategy, and also diversification strategies of professional service firms.

Relaxational dynamics for a class of disordered ultrametric models.

The relaxational dynamics for a class of disordered, ultrametric spaces with arbitrary and irregular branchings $K$ is considered. It is shown that, if the transfer rates between sites depend upon their ultrametric distance and obey a certain constraint, the relaxational eigenvalues and eigenmodes can be obtained exactly for an arbitrary tree. The plausibility of the constraint is given on physical grounds. Approximate ensemble averagings are performed for the decay law leading to a power law similar to that found in regular models. This generalizes the exactly soluble regular $K=2$ model of Ogielski and Stein, and makes closer contact with real disordered systems.