What if fossil-fuel consumption is not just a trend, but a system with memory? This study finds that consumption patterns across thirty-eight countries retain unusually strong long-term structure. Fossil fuel is a type of hydrocarbon or its derivatives, and it is a critical energy resource supporting the development of the global economy.

In modern society, fossil energy accounts for more than eighty percent of global disposable energy consumption. Over the past half-century, fossil-fuel consumption has increased about eight times since nineteen fifty, while reliance has shifted from simple coal toward oil and natural gas.

Today, coal consumption is declining over the world, but oil and gas are still multiplying. Because fossil fuels significantly affect economic development and environmental protection, regulating and protecting their rational extraction and utilization has legal significance.

The block points to legislation for natural-gas power generation and legal regulations for offshore oil and gas safety. It also describes legal guarantees for rational resource use and the importance of establishing mining rights, mining order, and a long-term security-prevention mechanism through law.

The visibility graph is presented as a novel analytical framework for detecting, interpreting, and predicting complex dynamics in fossil-fuel consumption. The study mainly analyzes internal characteristics of fossil-fuel consumption and provides a research basis for examining its relationship with the economic environment.

Its stated advantages include mapping time series to complex networks to find hidden information and representing time-series characteristics through fractal characteristics. The visibility graph method proposed by Lacasa and colleagues is used to construct a complex network for characterization.

First, each discrete time-series value is mapped into one network node; links are then built according to a visibility rule between points in the series. A time series with fractal characteristics has long correlation and self-similarity.

The Hurst index is used to determine whether the time series is fractal. When the Hurst index is zero point five, the time nodes have a random degree series, and current trends will not affect future trends. When the Hurst index is above zero point five and up to one, the degree series is positively correlated, indicating that current trends affect future trends.

When the Hurst index is from zero up to below zero point five, the degree series is negatively correlated, and the block also states that current trends affect future trends. Coal, oil, and gas consumption data were obtained from Our World in Data.

Because some countries had insufficient data, the study excluded countries with missing data and finally obtained fossil-fuel consumption data for thirty-eight countries. Global coal, oil, and gas consumption increased gradually during the given period, but all fossil-fuel consumption decreased significantly in twenty twenty.

The block also describes greater use of clean energy, replacement of oil with natural gases, and coal consumption positioned between oil and natural-gas consumption. Figure three charts global coal, oil, and gas consumption from nineteen seventy through twenty twenty, measured in units of ten to the power of four terawatt-hours.

All three series generally rise over the period, while the chart shows a pronounced drop in twenty twenty. This visual establishes the consumption history used in the study, providing the empirical time series that the authors later map into a visibility network for analyzing temporal structure.

Figure 5 presents coal, oil, and gas consumption from nineteen eighty-six to twenty twenty for thirteen countries, and consumption increased in these thirteen countries. The countries show significant fuel-consumption variation because of differences in economy, policies, and resource reserves.

China has the largest growth in coal consumption in the world, while the United States is the largest consumer of oil and gas. China’s total energy consumption exceeded that of the United States in two thousand four, and the block relates this to rapid economic development in China.

Figure five presents four polar charts tracking coal, oil, gas, and combined fossil-fuel consumption across thirteen countries from nineteen sixty-five to twenty twenty. The expanding and differing country traces make visible how consumption changed over time, with China’s coal growth especially highlighted by the authors.

This matters because the figure establishes the fuel-use patterns and cross-country variation that motivate analyzing long-term trends and memory in these time series. Figure 6 shows the visibility graph of coal consumption, with maps for average degree, average cluster coefficient, diameter, and average path length across thirty-eight countries.

India, Indonesia, Malaysia, and Pakistan are the top four countries in average degree, and all exceed twenty. In these four countries, average path length is less than two; the sentence begins to interpret this as indicating a correlation between their coal-consumption network elements.

Figure six maps four visibility-graph measures for coal consumption across the thirty-eight countries: average degree, average cluster coefficient, diameter, and average path length. The color scales show the corresponding ranges, while the authors identify India, Indonesia, Malaysia, and Pakistan as the top four countries for average degree, each exceeding twenty.

Their average path length is below two, highlighting how the network measures summarize relationships among coal-consumption time series. Figure 7 presents the visibility graph of oil consumption, using average degree, average cluster coefficient, diameter, and average path length across thirty-eight countries.

China and India have average degree above twenty, while Italy, Czechia, Slovakia, and Mexico have diameters above ten. Iran, Taiwan China, Argentina, Romania, and Pakistan have average cluster coefficients above zero point seventy-five; Iran’s exceeds zero point eight, indicating stronger correlation with its neighbors in oil consumption.

Italy, Mexico, Spain, and Czechia have average path lengths above three point five. Figure seven maps four visibility-graph measures for oil consumption across thirty-eight countries: average degree, average cluster coefficient, diameter, and average path length.

The legends show ranges of five point zero to twenty-three point zero three eight, zero point six two five to zero point eight one zero, three to twelve, and one point five nine six to four point six eight nine, respectively. The authors note that China and India have average degrees above twenty, while Italy, Czechia, Slovakia, and Mexico have diameters exceeding ten.

Table one reports Hurst indices and degree Hurst values for coal, oil, and gas consumption across countries. The authors use Hurst values above zero point five to identify persistence, and most entries exceed zero point nine, indicating long-term memory in fossil-fuel consumption.

Bold degree-Hurst values mark cases between zero point four and zero point five, such as coal in Indonesia at zero point four two zero and gas in Japan at zero point one eight zero, highlighting consumption patterns with a stochastic influence on future time points. Table one reports Hurst indices for coal, oil, and gas consumption, alongside the Hurst index of each visibility graph’s degree distribution, for the United States and Venezuela.

The authors use these values to examine long-term memory and stochastic behavior: values between four-tenths and six-tenths are considered stochastic, including the United States coal degree Hurst of zero point five four six and Venezuela’s gas degree Hurst of zero point five two four.

This matters because the degree-based measure reveals additional structure beyond the original consumption series. The Hurst indices of coal, oil, and natural-gas consumption data are all greater than zero point five. Most fossil-fuel consumption Hurst indices exceed zero point nine, indicating long-term memory in consumption of all three fossil fuels.

A Hurst index equal to zero point five corresponds to a randomized time series, and values closer to zero point five indicate stronger stochasticity in the series’ effect on the future. The study considers a series stochastic when its Hurst index is between zero point four and zero point six.

Because coal, oil, and gas consumption in these thirty-eight countries are long-term memory time series, analyzing only the Hurst index of the primary data cannot discover new features in the original data. With the assistance of the Hurst of degree, new characteristics of fossil-fuel consumption can be obtained, and fuel-consumption differences exist between different countries and within the same country.

Figure nine plots the Hurst index of the degree distribution for coal, oil, and natural-gas consumption across the thirty-eight countries, using the visibility-graph representation of each time series. The labeled points show substantial differences both between countries and among fuels within the same country—for example, Iran and Venezuela appear in the gas panel at distinct positions, while Pakistan is also separately identified.

This matters because the authors use the Hurst of degree to reveal characteristics that the Hurst index of the original long-memory data alone may not expose. The Hurst index was used to analyze fossil-fuel consumption time-series data for thirty-eight countries.

All fossil-fuel consumption data in these countries exhibited long-term memory characteristics, indicating a significant relationship between fossil-fuel consumption and economic development. The central result is that fossil-fuel consumption shows long-term memory, while visibility graphs expose differences between countries and fuels.

That combination could inform future energy-demand assessment and policy.