Power laws detected in financial data, modeled with random multipliers.
problem Detecting power laws in financial data.
method Investigated data from financial instruments, proposed a model based on sums of Maxwell-Boltzmann distributions with random multipliers.
result Detected power laws with various exponents in financial data, proposed a universal model.
Bayesian inference models power-law graphs with efficient algorithms.
problem Modeling networks with heavy-tailed degree distributions.
method Constructs graphs using BFRY random variables and applies variational Bayesian inference.
result Automatic selection of power law behavior from data.
New SDE model from machine learning optimization with unique stationary distribution.
problem Stationary distribution of machine learning optimization models.
method Proved ergodicity and unique stationary distribution of power-law dynamic SDE.
result Power-law dynamic has a unique stationary distribution and is ergodic.
We introduce the stochastic multiplicative point process modelling trading activity of financial markets. Such a model system exhibits power-law spectral density S(f) ~ 1/f**beta, scaled as power of frequency for various values of beta between 0.5 and 2. Furthermore, we analyze the relation between the power-law autoco…
We use data on wealth of the richest persons taken from the "rich lists" provided by business magazines like Forbes to verify if upper tails of wealth distributions follow, as often claimed, a power-law behaviour. The data sets used cover the world's richest persons over 1996-2012, the richest Americans over 1988-2012,…
Sparse deep neural networks follow a power law in their connectivity.
problem Understanding the connectivity patterns in sparse deep neural networks.
method Experimentally tested multilayer perceptrons and convolutional neural networks, proposed an internal preferential attachment model.
result Sparse deep neural networks exhibit a power law in their connectivity, similar to biological neural networks.
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
Paper proves SVV model reproduces power-law skew in implied volatilities.
problem Reproducing power-law behavior in implied volatility skew.
method Analytical proof using Malliavin calculus and Volterra kernel selection.
result SVV model reproduces power-law skew under correct kernel choice.
Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1…
LLMs learn peaked distributions slowly due to power-law losses.
problem Slow convergence of loss in training large language models.
method Systematic analysis of toy models and empirical evaluation of LLMs.
result Power-law time scaling with an exponent of 1/3 for learning peaked distributions.
I consider the problem of the optimal limit order price of a financial asset in the framework of the maximization of the utility function of the investor. The analytical solution of the problem gives insight on the origin of the recently empirically observed power law distribution of limit order prices. In the framewor…
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…
Volatility models must be rough to match market skew.
problem Inconsistent non-rough volatility models with power law volatility skew.
method Asymptotic expansion and continuous price dynamics analysis.
result Volatility must be rough to align with market skew.
We show that there is a common mode of origin for the power laws observed in two different models: (i) the Pareto law for the distribution of money among the agents with random saving propensities in an ideal gas-like market model and (ii) the Gutenberg-Richter law for the distribution of overlaps in a fractal-overlap …
Study uses OT to simulate markets, revealing power-law returns are driven by informational effect.
problem Reproduce power-law returns in financial markets using realistic simulations.
method Constructed artificial markets, used optimal transport (OT) to measure similarity, incrementally introduced behavioral components.
result Informational effect of prices is dominant in reproducing power-law returns, and multiple components interact synergistically.
New statistical models capture double power-law behavior in data.
problem Capturing two-regime power-law behavior in datasets.
method Introducing completely random measures with double power-law behavior.
result Proposed models provide a better fit than Pitman-Yor process.
Power-law model outperforms logarithmic model in estimating stock and warrant price impacts.
problem Estimating immediate price impacts of stock and warrant orders.
method Used order flow data to estimate price impacts using power-law and logarithmic models.
result Power-law model outperforms logarithmic model in robustness and forecasting accuracy.
New models generate power law exponents from 0.5 to 2.5, enabling inference.
problem Inference difficulty in models generating power laws with exponents > 2.
method Design and implement inference algorithms for a new class of models.
result Inference possible for models generating power laws with exponents from 0.5 to 2.5.
Speculation leads to power law price changes.
problem Financial price changes follow a power law distribution.
method Identified speculation as the underlying mechanism.
result Price changes follow a random coefficient autoregressive process.
A new model corrects SBM's bias for power-law degree networks.
problem SBM's incapability to handle power-law degree distributions.
method Introducing degree decay variables to encode varying degree distributions.
result PLD-SBM approximately preserves the scale-free feature in real networks and corrects SBM's bias.
This study generalizes an econophysics model to account for trader heterogeneity, finding robust power-law exponents but sensitive prefactors.
problem The original Lillo-Mike-Farmer model assumed homogeneity in traders' order-splitting strategies, which this study generalizes.
method The study proposes a generalised Lillo-Mike-Farmer model and solves it exactly without heuristic assumptions.
result The power-law exponent in the order-sign ACF is robust for arbitrary heterogeneous intensity distributions, but the prefactor is sensitive to heterogeneity.
The study proves fractional-order differences and equations are key to modeling long and short memory in economics.
problem Modeling long and short memory in economic processes with discrete fractional differencing and integration.
method Proved discrete fractional differencing and integration are Grunwald-Letnikov fractional differences of non-integer order d. ARIMA and ARFIMA models are fractional-order difference equations. Proved exact fractional-order differences are needed for power law memory.
result Fractional differential equations are necessary for modeling continuous time long and short memory with power law.
Minimal model reveals power laws in financial markets.
problem Understanding universal behaviors in financial markets.
method Analytical solution of a minimal model based on symmetry constraints.
result Various power-law behaviors are interconnected, similar to critical exponents.
Large models follow power laws in performance with dataset size or parameters.
problem Understanding neural scaling laws in large language models.
method Joint generative data model and random feature model.
result Modeling and solving the dual limit reveals insights into scaling laws.
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit n−β for arbitrary power β>0. Critical volatility triggers log-normal to power-law transitions in interconnected systems.
problem Understanding the transition from log-normal to power-law distributions in interconnected systems.
method Analyzing an infinite option-on-option chain model, deriving a critical volatility threshold.
result A critical volatility threshold of approximately 250.66% for unconditional cases, dropping to 125.3% with selective survival.
Study compares exponential and power-law kernels in modeling high-frequency trading data.
problem Modeling high-frequency trading data with specific kernel types.
method Proposes and analyzes two bivariate Hawkes processes with exponential and power-law kernels.
result Identifies strengths and limitations of exponential and power-law kernels for high-frequency trading data.
We propose an explicit recursive method to approximate a power-law with a finite sum of weighted exponentials. Applications to moving averages with long memory are discussed in relationship with stochastic volatility models.
One of the first steps to understand and forecast economic downturns is identifying their frequency distribution, but it remains uncertain. This problem is common in phenomena displaying power-law-like distributions. Power laws play a central role in complex systems theory; therefore, the current limitations in the ide…
This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.
problem Understanding how neural network performance scales with key factors like data size and model complexity.
method Statistical mechanics techniques applied to one-pass stochastic gradient descent in a student-teacher framework.
result Derivation of analytical expressions for generalization error under power-law data spectra and identification of conditions for power-law scaling.
Power-law spectrum of random feature model is preserved in neural networks.
problem Preserving power-law spectrum in neural networks through random feature model.
method Characterized eigenvalues of population random-feature covariance using dyadic head-tail decomposition and Wick chaos expansions.
result Power-law exponent α is inherited from input covariance, modified by a logarithmic correction. We study the relaxation dynamics of a financial market just after the occurrence of a crash by investigating the number of times the absolute value of an index return is exceeding a given threshold value. We show that the empirical observation of a power law evolution of the number of events exceeding the selected thre…
We consider stochastic point processes generating time series exhibiting power laws of spectrum and distribution density (Phys. Rev. E 71, 051105 (2005)) and apply them for modeling the trading activity in the financial markets and for the frequencies of word occurrences in the language.
This study examines evolving networks of P2P lending relationships, revealing scale-free characteristics and the impact of interest rate and term.
problem Understanding the structural characteristics of evolving networks of debtor-creditor relationships in P2P lending.
method Modeling P2P lending networks as evolving networks with addition and deletion of nodes, analyzing attributes and factors affecting the scale-free exponent.
result P2P lending networks are scale-free with significant influence from interest rate and term on the exponent of power-law.
Theory explains power-law distributions without complex models.
problem Understanding power-law distributions in geometrically growing systems.
method Developed a theory of geometrically growing systems and applied it to explain various distributions.
result The geometrically growing system's distribution flattens over time, increasing relative size ratios.
Unified theory for neural scaling laws in hierarchically compositional data.
problem Understanding neural scaling laws in hierarchically compositional data.
method Probabilistic context-free grammars and power-law distributed production rules.
result Unified learning curve behavior for classification and next-token prediction tasks.
Network data appear in a number of applications, such as online social networks and biological networks, and there is growing interest in both developing models for networks as well as studying the properties of such data. Since individual network datasets continue to grow in size, it is necessary to develop models tha…
Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not…
An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is …
Study on KRR with power-law data, showing better sample complexity.
problem High-dimensional kernel ridge regression with anisotropic power-law covariance.
method Explicit characterization of kernel spectrum and asymptotic analysis of excess risk.
result Sample complexity is governed by effective dimension, not ambient dimension.
It is generally recognized that economical systems, and more in general complex systems, are characterized by power law distributions. Sometime, these distributions show a changing of the slope in the tail so that, more appropriately, they show a multi-power law behavior. We present a method to derive analytically a tw…
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
The daily volume of transaction on the New York Stock Exchange and its day-to-day fluctuations are analysed with respect to power-law tails as well long-term trends. We also model the transition to a Gaussian distribution for longer time intervals, like months instead of days.
The paper presents a multi-power law for predicting loss curves across different learning rate schedules.
problem Understanding and optimizing the relationship between model performance and hyperparameters, especially learning rates.
method Proposes a multi-power law that combines power laws based on the sum of learning rates and additional laws for loss reduction due to decay.
result The multi-power law accurately predicts loss curves for unseen learning rate schedules and finds a schedule that outperforms cosine learning rate.
The paper introduces new methods to measure cross-correlations between time series using power-law coherency.
problem Studying power-law cross-correlations between time series.
method Three estimators of the power-law coherency parameter H_ρ based on DCCA, DMCA, and HXA.
result DMCA-based method is the safest choice, HXA method is reasonable for long series, and DCCA-based method has unfavorable properties.
Study finds financial market data follows power-law exponents typical of stochastic processes.
problem Testing long-range memory in financial markets.
method Analyzed empirical return and trading activity time series from Forex.
result Power-law exponents of burst and inter-burst duration probability density functions are close to 3/2.
We provide an empirical investigation aimed at uncovering the statistical properties of intricate stock trading networks based on the order flow data of a highly liquid stock (Shenzhen Development Bank) listed on Shenzhen Stock Exchange during the whole year of 2003. By reconstructing the limit order book, we can extra…
We analyze the European transition economies and show that time series for most of major indices exhibit (i) power-law correlations in their values, power-law correlations in their magnitudes, and (iii) asymmetric probability distribution. We propose a stochastic model that can generate time series with all the previou…