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Culturomics meets random fractal theory: Insights into long-range correlations of social and natural phenomena over the past two centuries
Jianbo Gao, Jing Hu, Xiang Mao, Matjaz Perc
TL;DR
The paper asks what drives irregular patterns in culturomic word-use trajectories. It applies fractal analysis to yearly frequencies from books and finds that social and natural phenomena have different scaling properties, with natural phenomena showing persistent long-range correlations and social phenomena fitting other process types.
Problem
Irregular patterns in observations can arise from different processes, while strict criteria for identifying deterministic chaos are rarely met in observed data.
Source: §1 · p. 1Method
The authors apply detrended fluctuation analysis and adaptive fractal analysis to yearly usage-frequency trajectories for words describing social and natural phenomena.
Source: §1 · pp. 1–2Results
Natural terms had AFA H=0.55–0.85 (mean 0.69), while social terms had AFA H=0.74–1.33 (mean 1.11), indicating fundamentally different process properties.
Source: §3 · p. 5Takeaways & Limitations
Natural phenomena show properties typical of persistent long-range correlations, whereas social phenomena are indicative of nonstationary processes, on-off intermittency, or Lévy walks.
Source: §3 · p. 5For peaks in “war” usage, only additional future data can distinguish an on-off intermittent process from a Lévy walk.
Source: §2.2 · p. 5
Abstract
from arXiv · showhide
Culturomics was recently introduced as the application of high-throughput data collection and analysis to the study of human culture. Here we make use of this data by investigating fluctuations in yearly usage frequencies of specific words that describe social and natural phenomena, as derived from books that were published over the course of the past two centuries. We show that the determination of the Hurst parameter by means of fractal analysis provides fundamental insights into the nature of long-range correlations contained in the culturomic trajectories, and by doing so, offers new interpretations as to what might be the main driving forces behind the examined phenomena. Quite remarkably, we find that social and natural phenomena are governed by fundamentally different processes. While natural phenomena have properties that are typical for processes with persistent long-range correlations, social phenomena are better described as nonstationary, on-off intermittent, or Levy walk processes.
1. INTRODUCTION
The paper extends random fractal analysis to culturomic word-frequency trajectories, using the Hurst parameter to characterize their correlations and inform interpretations of the phenomena they describe.
1. INTRODUCTION
- Strict criteria for identifying deterministic chaos, including stationarity, determinism, and exponential divergence, are rarely met in observed data, motivating attention to noise and random processes.Random fractal theory is used to analyze stochastic processes that produce scale-invariant data.Source: §1 · p. 1
- Culturomics supplies yearly n-gram counts from a corpus covering over five million books, about 4% of all books ever published, enabling usage-frequency trajectories.The corpus also provides annual total n-gram counts, from which yearly frequencies can be obtained.Source: §1 · p. 1
- The study applies random fractal theory to culturomic trajectories to estimate H accurately and characterize whether their irregular time series contain memory.The authors aim to go beyond earlier trajectory interpretations using scaling parameters, particularly the Hurst parameter.Source: §1 · p. 1
- For H, values below 0.5 indicate antipersistence, 0.5 indicates memorylessness or short-range correlations, and values between 0.5 and 1 indicate persistent long-range correlations.Values above 1 can characterize nonstationary or certain special stationary processes, including on-off intermittency and Lévy walks.Source: §1 · p. 2
- Because trajectories span only about 200 annual points from 1770 to 2007 and often contain trends, the authors caution that scaling estimates require care.They use traditional detrended fluctuation analysis and adaptive fractal analysis, which is based on nonlinear adaptive multiscale decomposition.Source: §1 · p. 2
2. RESULTS
Adaptive fractal analysis finds persistent long-range correlations across the examined natural phenomena, while most examined social phenomena have H > 1 and fit other process types. Detrending and short trajectories constrain how precisely those patterns can be interpreted.
2.1. Natural phenomena
- Earthquake (detrended H = 0.65; original H = 0.75) and hurricane (detrended H = 0.70; original H = 0.85) both show persistent long-range correlations.The hurricane trend is more pronounced, and its raw and detrended estimates differ more than those for earthquake.Source: §2.1 · p. 2, Figure 1 · p. 2, Figure 2 · p. 3
- Upward trends in natural-phenomena terms may reflect growing data availability and awareness, so raw H > 1 cases require accounting for those trends.The authors caution that discussion of such raw estimates is largely baseless until the trends are properly considered.Source: §2.1 · p. 3
2.1. Natural phenomena; 2.2. Social phenomena
- Detrended natural terms averaged H = 0.69 within the persistent-correlation range, whereas the large majority of social terms had H > 1 (AFA).The paper characterizes natural phenomena as persistent long-range processes and social phenomena as more likely nonstationary, on-off intermittent, or Lévy-walk-like.Source: Table I · p. 3, §2.1 · p. 3, §2.2 · p. 4
2.2. Social phenomena
- War has H = 1.09 after detrending and H = 1.15 in the original data, classifying it as nonstationary, on-off intermittent, or Lévy-walk-like.World Wars I and II produce prominent peaks in the usage trajectory.Source: §2.2 · pp. 3–4, Figure 3 · p. 4
- Unemployment has H = 1.32 after detrending and H = 1.39 in the original data, which the authors interpret as reflecting nonstationarity.Its usage was absent or unmentioned before 1900; broad peaks begin around 1930 and 1970.Source: §2.2 · p. 4, Figure 4 · p. 4
- Across thirteen additional social terms, the large majority had H > 1; detrended AFA averaged H = 1.11, below the original-data average of H = 1.26.The lower detrended average likely reflects removal of some nonstationarity.Source: §2.2 · p. 4, Table II · p. 5
- Short social trajectories and H alone cannot distinguish nonstationarity from on-off intermittency or Lévy walks; some social terms do not have H > 1.The authors cite famine and Christian as exceptions and note that additional future data are needed to distinguish process types.Source: §2.2 · pp. 4–5
3. DISCUSSION
Fractal analysis of culturomic trajectories finds different scaling laws for natural and social phenomena. Natural phenomena show persistent long-range correlations, while social phenomena are associated with nonstationary or intermittent processes, or Lévy walks.
3. DISCUSSION
- The study applies DFA and AFA to two centuries of culturomic trajectories for 1-grams describing social and natural phenomena.The authors report that the resulting scaling laws align with existing theory and offer interpretations of possible drivers.Source: §3 · p. 5
- Natural phenomena have H = 0.55–0.85 (mean 0.69), consistent with processes generating persistent long-range correlations.The authors conjecture that occurrences of the phenomena in nature predominantly drive the long-range correlations in the corresponding word frequencies.Source: §3 · p. 5
- Social phenomena have H = 0.74–1.33 (mean 1.11), indicating nonstationary processes, on-off intermittency, or Lévy walks, unlike natural phenomena.The analysis cannot distinguish among those three process types; the authors suggest social phenomena do not follow the same scaling laws as natural phenomena.Source: §3 · p. 5
- Culturomics data combined with advanced analysis offer opportunities to explore human culture broadly.Source: §3 · p. 5
A.1. Nonlinear adaptive multiscale decomposition
The decomposition divides a time series into overlapping segments and fits each with a polynomial. It blends neighboring fits across their overlap to produce a continuous, smooth trend estimate that can handle arbitrary trends.
A.1. Nonlinear adaptive multiscale decomposition
- The time series is partitioned into segments of length w = 2n + 1, with neighboring segments overlapping by n + 1 points.Source: §A.1 · p. 6
- Each segment is fitted with a polynomial of order M; M = 0 gives piecewise-constant fitting and M = 1 gives piecewise-linear fitting.Source: §A.1 · p. 6
- Overlapping segment fits are blended with distance-dependent weights to ensure continuity and smoothness while eliminating boundary jumps.The weights decrease linearly with distance from each segment’s center; the procedure can remove nonstationarity, including baseline drifts and motion artifacts.Source: §A.1 · p. 6
A.2. Fractal analysis
Adaptive fractal analysis estimates scaling by detrending a random-walk-like series at different window sizes and measuring residual fluctuations. It agrees with detrended fluctuation analysis and may scale better for H > 1, especially on short nonstationary series.
A.2. Fractal analysis
- Integrating an increment process produces a random-walk process; applying that step to data already random-walk-like shifts the final H estimate by +1 without penalty.The increment-process and random-walk spectra are given as 1/f^(2H−1) and 1/f^(2H+1), respectively.Source: §A.2 · p. 6
- AFA estimates H from how residual fluctuations around a window-specific trend scale with window size.It uses the random-walk process, or the original data if already random-walk-like, and relates the residual variance to w^H.Source: §A.2 · p. 6
- The residual-variance scaling relation is justified for an increment process with Hurst parameter H when the window frequency is much lower than the data’s highest frequency.The paper derives the relation by equating residual variance with spectral power over the frequency range from 1/w to the cutoff.Source: §A.2 · p. 6
- AFA and DFA yield consistent scaling results, while AFA may scale better for H > 1 because its window trends are smooth rather than abruptly changing between segments.The paper notes that this may be favorable for short nonstationary time series.Source: Figure 5 · p. 6, §A.2 · p. 7