Entropic Geometry and Information Dynamics in Green Cryptocurrency Markets

Abstract

Cryptocurrencies play a central role in modern financial markets; however, geopolitical tensions and environmental concerns raise critical questions about their stability and informational efficiency. This study distinguishes between green cryptocurrencies (GCs), based on low-energy validation mechanisms, and dirty cryptocurrencies (DCs), which rely on energy-intensive protocols, to examine their behaviour under geopolitical stress. The objective of this paper is to assess how information dynamics, market resilience, and efficiency differ between GCs and DCs during periods of heightened geopolitical uncertainty, with particular focus on the Russia–Ukraine war. Using daily data from 28 April 2019 to 5 October 2023, we employ advanced information-theoretic measures, including mutual information, the rolling local nearest-neighbour entropy estimator (RLNNEE), and approximate entropy. The results show that DCs exhibit stronger information dominance than GCs, with this gap widening during the conflict. In contrast, GCs display lower but more stable mutual information, indicating greater informational resilience. Approximate entropy further reveals a decline in market complexity during the war period. Overall, the findings highlight the relevance of entropy-based tools for evaluating stability and risk in cryptocurrency markets facing geopolitical shocks.

1. Introduction

Since Bitcoin’s emergence in 2009, cryptocurrencies have established themselves as a global asset class, reshaping financial market architecture through decentralisation, technological innovation, and highly volatile valuation dynamics (Nakamoto 2008; Giudici et al. 2020). Their rapid diffusion and pronounced price fluctuations highlight an investment universe that is particularly sensitive to uncertainty, information flows, and exogenous shocks. In this context, the key issue is no longer whether crypto assets can contribute to diversification, but rather under what conditions these markets become fragile, interdependent, and less informative, especially during periods of heightened macro-financial stress.

A significant turning point occurred on 24 February 2022, when Russia invaded Ukraine, triggering a global geopolitical shock and reactivating multiple risk transmission channels related to energy markets, inflationary pressures, expectations, and financial stability (Caldara and Iacoviello 2022; IMF 2022; BIS 2025). Beyond traditional equity and commodity markets, this episode also affected digital assets. Several studies show that war-related attention and geopolitical risk significantly influence cryptocurrency prices, volatility, and dependency structures (Khalfaoui et al. 2023; Patel et al. 2023). However, an important limitation persists while the effects on returns and volatility are now relatively well documented, the informational dimension, namely, how information flows, concentrates, or fragments across crypto assets during geopolitical conflicts, remains insufficiently explored. This gap is particularly relevant given the growing role of cryptocurrencies in debates on sustainable finance. So-called dirty cryptocurrencies (DCs), which rely on energy-intensive validation mechanisms such as Proof-of-Work, raise substantial environmental concerns related to energy consumption, emissions, and negative externalities (De Vries 2020; Sedlmeir et al. 2020). In contrast, alternative mechanisms, notably Proof-of-Stake, are frequently presented as environmentally friendlier solutions that can support more sustainable crypto-ecosystems (Saleh 2021). Consequently, a growing strand of the literature distinguishes between green cryptocurrencies (GCs) and dirty cryptocurrencies DCs and examines their financial properties, diversification potential, and behaviour under stress (Ren and Lucey 2022a, 2022b; Ali et al. 2024). Nevertheless, empirical evidence remains mixed and strongly dependent on the analysed periods, particularly around the Russia–Ukraine war, during which notable reconfigurations of connectivity and spillover patterns between (GCs) and (DCs) have been observed (Patel et al. 2023).

At this stage, the literature exhibits two significant limitations. First, most comparative analyses of GCs and DCs rely on returns, volatility measures, or standard connectivity models, which may fail to capture nonlinear dependencies and shifts in information regimes. Second, although information-theoretic tools have been applied to cryptocurrency markets, their use remains fragmented for systematically characterising informational stability and disruptions associated with major geopolitical shocks (Lahmiri and Bekiros 2020; Assaf et al. 2022). However, during crisis periods, the critical question is not only which assets gain or lose value, but also how information is reorganised, how predictability evolves, and how dependency structures across assets are reshaped. To address these limitations, this study adopts an information-theoretic perspective rooted in Shannon’s entropy framework (Shannon 1948) and aligned with the Adaptive Market Hypothesis (Lo 2004), which posits that market efficiency is neither static nor universal but evolves in response to changing economic, institutional, and geopolitical conditions. In this context, entropic approaches are particularly well-suited, as they allow quantifying uncertainty and complexity, detecting nonlinear relationships, and identifying regime shifts that may remain invisible in volatility-based analyses (Lahmiri and Bekiros 2020). Rather than relying solely on a return–volatility framework, we propose an information-oriented approach that is especially relevant for cryptocurrency markets given their nonstationarity and sensitivity to external shocks.

More specifically, we examine the informational dynamics of GCs and DCs before and during the Russia–Ukraine war using three complementary measures: (i) mutual information (MI) to capture nonlinear interdependence and information sharing, (ii) approximate entropy (ApEn) to assess signal regularity and predictability, and (iii) the rolling local nearest neighbour entropy estimator (RLNNEE), designed to detect local and time-varying information regimes. The originality of this approach lies in the joint use of these tools to analyse both the GCs vs DCs distinction and the informational regime shifts induced by a major geopolitical shock.

Accordingly, this study addresses the following research questions:

• (Q1) Did the Russia–Ukraine war strengthens or weaken information sharing between cryptocurrencies, and did these effects differ between GCs and DCs?
• (Q2) Did signal regularity and predictability, as measured by approximate entropy, change during the conflict in a manner consistent with the Adaptive Market Hypothesis?
• (Q3) Does RLNNEE uncover local regimes of informational instability that global measures fail to detect, particularly during periods of extreme stress?

This study makes three main contributions. First, it provides a novel interpretation of the Russia–Ukraine war through the lens of GCs versus DCs, focusing on nonlinear dependence, predictability, and informational stability. Second, it demonstrates the relevance of a rolling local entropy estimator in identifying regime shifts that are imperfectly captured by traditional approaches. Third, by linking information theory with the Adaptive Market Hypothesis, it contributes to the broader debate on the adaptive efficiency of cryptocurrency markets under geopolitical stress, with implications for risk management, diversification, and sustainable finance (Shannon 1948; Lo 2004; Patel et al. 2023; Ali et al. 2024).

The remainder of the paper is organised as follows. Section 2 reviews the literature on GCs and DCs, geopolitical risk, and informational approaches. Section 3 describes the data and methodology. Section 4 presents and discusses the empirical results. Section 5 concludes.

2. Literature Review

2.1. Geopolitical Risk and Cryptocurrency Markets

A growing body of research documents that geopolitical shocks amplify uncertainty, alter asset dependence structures, and weaken diversification benefits. In the cryptocurrency domain, empirical evidence increasingly suggests that geopolitical tension reshapes market dynamics beyond price levels, affecting volatility patterns and cross-market linkages. For example, heightened geopolitical uncertainty has been associated with sharper volatility regimes in cryptocurrencies (Bouri et al. 2020), while the Russia–Ukraine war has been shown to disrupt financial markets globally, with spillovers extending to digital assets (Boungou and Yatié 2022). Recent studies further indicate that investor attention to war-related news and geopolitical risk can influence cryptocurrency pricing and dependence patterns (Khalfaoui et al. 2023; Patel et al. 2023). More recent evidence highlights the growing role of cryptocurrencies in geopolitical conflicts, documenting how digital assets interact with international tensions, sanctions, and geopolitical uncertainty (Tiwari et al. 2024). More recently, Boungou and Yatié (2024) provide direct evidence that geopolitical risks and global uncertainty measures play a central role in shaping crypto-asset volatility and market behaviour.

Despite these advances, most contributions in this stream remain centred on returns, volatility, and broad connectedness outcomes. Consequently, the informational dimension of crypto markets under geopolitical stress, how information is transmitted, concentrated, or destabilised across assets, has received comparatively less systematic attention. This limitation is especially relevant in crisis settings, where the structure of dependence and predictability may change rapidly and nonlinearly, calling for tools that can go beyond classical volatility-centred assessments.Recent evidence also shows that geopolitical risk and market-implied volatility significantly shape the connectedness of green financial assets, including green cryptocurrencies, reinforcing the sensitivity of digital markets to global geopolitical shocks (Bajra et al. 2025).

2.2. GCs and DCs and Energy Concerns

In parallel with geopolitical concerns, the environmental footprint of crypto-assets has become a major topic in finance. Proof-of-Work cryptocurrencies have been widely criticised for high energy intensity and related externalities. In contrast, Proof-of-Stake and other low-energy mechanisms have been proposed as more sustainable alternatives. This debate has encouraged researchers to distinguish between GCs and DCs and to assess whether GCs exhibit different risk–return profiles, spillovers, or hedging properties. Recent spillover analyses explicitly distinguish between green and non-green cryptocurrencies, revealing asymmetric volatility transmission mechanisms with oil and clean energy markets (Zhou and Wang 2024). In this context, studies examine interactions between cryptocurrencies and energy markets, as well as the role of environmental considerations in shaping connectedness and portfolio outcomes (Ren and Lucey 2022a; Pham et al. 2022; Zhou and Wang 2024; Yan and Yuan 2025).

Beyond market dynamics, recent work has also approached green cryptocurrencies from a strategic and sustainability-oriented perspective, highlighting their implications for business strategies and stewardship-based frameworks (Arora et al. 2025).

Other contributions extend the sustainability lens to broader asset classes and GCs finance instruments. In this vein, recent studies analyse the connectedness between green bonds, cryptocurrencies, and energy-related assets across different frequencies and crisis periods, highlighting diversification and hedging channels within sustainable finance ecosystems (Zeng et al. 2025; Zhong et al. 2023). For instance, research has explored the interplay between green bonds, equities, commodities, and cryptocurrencies, often emphasising time-varying dependence patterns and environmental awareness channels (Hassan et al. 2022; Umar et al. 2023; Kamal and Bouri 2025; Lee et al. 2023). Patel et al. (2024) further show that environmental concerns can influence connectedness between GCs, energy-related crypto-assets, and green financial instruments.

However, this literature remains fragmented on two fronts. First, evidence on whether GCs are systematically more resilient than DCs ones is mixed and often sensitive to sample periods and methodologies. Second, relatively few studies explicitly connect the GCs vsDCs distinction to geopolitical conflict settings, particularly through the lens of informational resilience (information sharing stability, predictability, and changes in complexity during war-related stress).

2.3. Information Transmission and Entropy-Based Approaches

Financial markets, and cryptocurrency markets in particular, display nonlinear dynamics, abrupt regime changes, and complex dependence structures that are poorly captured by standard correlation- or volatility-based models. Information-theoretic approaches offer an attractive alternative by quantifying uncertainty, complexity, and nonlinear dependence in a model-free way. Early work demonstrates the relevance of entropy and information measures for capturing complex market behaviour and dependence beyond linear correlation (Dionisio et al. 2004). More recent contributions explicitly rely on information-theoretic quantifiers to characterise entropy, complexity, and nonlinear dynamics in cryptocurrency time series (Suriano et al. 2025). Subsequent surveys and theoretical contributions further formalise the role of entropy as a unifying framework for asset valuation, market efficiency, and information dynamics in complex financial systems, including cryptocurrencies (Chortane and Naoui 2022).

In cryptofinance, entropy-based tools and mutual information have increasingly been used to detect structural changes, crisis-driven shifts in dependence, and variations in predictability (Lahmiri and Bekiros 2020; Assaf et al. 2022; Wang et al. 2022). Recent evidence further highlights the time-varying nature of efficiency and predictability in cryptocurrency markets, emphasising forward-looking dynamics that evolve across market regimes (Vukovic et al. 2025). These methods are particularly valuable during periods of turmoil, when traditional econometric frameworks may fail to capture local and time-varying information flows. Recent contributions further extend the literature by exploring nonlinear connectedness, quantile-based dependence, and time–frequency approaches (Patel et al. 2024; Haq 2023; Haq et al. 2025; Touhami et al. 2025; Shao et al. 2025). However, much of this work remains focused on dependence strength rather than on informational stability and complexity as market states evolve, including recent evidence on time-varying connectedness between green financial assets and cryptocurrencies based on TVP-VAR and wavelet-based risk measures (Sebai and Jaber 2025).

A key limitation persists, although information-theoretic tools are increasingly applied to cryptocurrencies, their use remains unsystematic when comparing GCs vs. DCs under geopolitical stress. Moreover, capturing the short-lived instability windows of local informational regimes that are critical during crises remains underdeveloped in many empirical setups, even when time-varying models are employed (e.g., TVP-VAR frameworks).Recent entropy-based evidence in cryptocurrency markets further challenges traditional mean–variance frameworks and highlights the relevance of alternative entropy measures for capturing informational stability and crisis dynamics (Chortane and Naoui 2025).

2.4. Research Gap and Positioning of the Study

Taken together, existing research shows growing interest in the intersection of (i) geopolitical risk, (ii) sustainability considerations in crypto markets, and (iii) advanced dependence modelling. Nevertheless, three aggregated gaps remain. First, most GCs vs-DCs comparisons rely on returns, volatility, and standard connectedness frameworks and therefore may not fully capture nonlinear information transmission, informational resilience, and complexity changes (Bouri et al. 2020; Ren and Lucey 2022b; Zhou and Wang 2024; Yan and Yuan 2025; Patel et al. 2024). Second, although several studies investigate the effects of geopolitical risk on cryptocurrencies, the war context is typically evaluated through volatility and spillovers, leaving the information flow and predictability dimensions comparatively underexplored (Boungou and Yatié 2022; Khalfaoui et al. 2023; Patel et al. 2023). Third, entropy-based approaches and mutual information have demonstrated strong potential to detect nonlinear dependence and crisis-driven shifts. However, they are rarely mobilised to jointly evaluate GCs vs. DCs behaviour under a major geopolitical shock, and to identify local/time-varying informational regimes (Dionisio et al. 2004; Lahmiri and Bekiros 2020; Assaf et al. 2022; Wang et al. 2022).

To address these gaps, this study proposes an information-oriented framework combining mutual information, approximate entropy, and the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE) to analyse daily data before and during the Russia–Ukraine war. By doing so, we provide a systematic comparison of GCs and DCs in terms of information sharing, predictability, and regime instability, offering new evidence on informational resilience and adaptive efficiency during extreme geopolitical stress.

Table 1. Summary of related literature and research.

Dimension	What the Literature Mainly Does	Aggregated Limitations/Gaps	What This Study Adds
Geopolitical risk and crypto markets	Examines the effects of geopolitical tensions and wars on cryptocurrency returns, volatility, and market instability.	Focuses mainly on price dynamics and volatility; limited attention to information transmission and informational stability during armed conflicts.	Shifts the analysis toward information dynamics, comparing informational behaviour before and during the Russia–Ukraine war.
Sustainability perspective (GCs vs. DCs cryptocurrencies)	Classifies cryptocurrencies by energy intensity (PoW vs. PoS) and examines spillovers, diversification, and links to energy and GCs finance markets.	Evidence is fragmented and mixed; geopolitical shocks are rarely integrated into GCs vs. DCs comparisons; informational resilience remains unexplored.	Provides a unified framework linking GCs/DCs classification, geopolitical stress, and informational resilience.
Main methodological approaches	Relies on volatility models, spillover indices, connectedness measures, and parametric econometric frameworks.	Predominantly volatility-centred and often unable to capture nonlinear dependence and information regime changes.	Introduces information-theoretic tools to capture nonlinear dependence and shifts in complexity.
Information transmission and dependence	Uses correlation-based or time-varying connectedness measures; limited use of information theory.	Nonlinear information sharing and changes in predictability are insufficiently modelled, especially during crisis periods.	Employs mutual information to assess nonlinear information sharing across cryptocurrencies.
Market complexity and predictability	Assesses efficiency mainly through returns and volatility persistence.	Ignores signal regularity, predictability, and complexity changes under extreme uncertainty.	Uses approximate entropy to evaluate predictability and market complexity dynamics.
Local and time-varying informational regimes	Captures global or average dynamics using rolling or time-varying models.	Local instability regimes and short-lived informational disruptions remain largely undetected.	Applies RLNNEE to identify local and time-varying informational regimes during geopolitical stress.
Theoretical interpretation	Implicitly assumes static market efficiency.	Limited integration of the Adaptive Market Hypothesis (AMH) in geopolitical and sustainability contexts.	Interprets AMH results, highlighting adaptive efficiency under geopolitical shocks.

Note: This table provides an aggregated synthesis of the main gaps identified in the literature reviewed in Section 2 and positions the contribution of the present study.

summarises the aggregated gaps and positions this paper’s contribution.

3. Data and Methods

3.1. Data

This study analyses daily price data for eight major cryptocurrencies, classifying them as either GCs or DCs based on their energy intensity and consensus mechanisms. Following the energy-based classification proposed by Ren and Lucey (2022a), GCs rely on low-energy validation protocols. In contrast, DCs cryptocurrencies use energy-intensive mechanisms that result in substantial electricity consumption. The GCs group includes Ripple (XRP), Polygon (MATIC), Stellar (XLM), and Cardano (ADA). In contrast, the DCs group comprises Bitcoin (BTC), Ethereum (ETH), Bitcoin Cash (BCH), and Ethereum Classic (ETC). These cryptocurrencies are selected based on their market capitalisation to ensure sufficient liquidity, market representativeness, and data reliability across the entire sample period.

The sample period spans from 28 April 2019 to 5 October 2023. It is divided into two sub-periods: a pre-war period (28 April 2019 to 23 February 2022) and a war period (24 February 2022 to 5 October 2023), corresponding to the outbreak of the Russia–Ukraine conflict. Although the conflict extends beyond 2023, the analysis deliberately ends in October 2023 to focus on a sufficiently long and economically meaningful post-shock regime following the initial geopolitical disruption. From an economic perspective, extending the sample beyond this date would introduce a different macro-financial regime characterised by monetary policy inflexions, regulatory developments, and renewed sources of geopolitical uncertainty, which could obscure the identification of information dynamics specifically attributable to the Russia–Ukraine war. The selected period, therefore, allows us to isolate informational stabilisation processes and adaptive dynamics associated with the geopolitical shock, while ensuring consistency and comparability with recent empirical studies on cryptocurrency markets and financial crises.

Daily continuously compounded returns are computed as:

$$\mathrm{r}\mathrm{i},\mathrm{t}= \mathrm{l}\mathrm{n}\left(\mathrm{p}\mathrm{i},\mathrm{t}\right)-\mathrm{l}\mathrm{n}\left(\mathrm{p}\mathrm{i},\mathrm{t}-1\right).$$

where signifies return for a particular asset i at period t, denotes the log of price level for an asset $i$ at time $t$, and the log of price level at time. We selected the GCs and DCs based on market capitalisation, i.e., the currencies with the highest capitalisations. Data are collected from https://investing.com (accessed on 23 December 2025) and https://coinmarketcap.com (accessed on 23 December 2025).

3.2. Methods

This study uses a new framework to understand digital asset markets. We track market behaviour and informational structures during geopolitical stress, such as the Russia–Ukraine conflict. We combine three tools to achieve this. These include Mutual Information (MI), Approximate Entropy (ApEn), and a new indicator called the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE). These techniques work together to analyse interconnected parts of the cryptocurrency market. Mutual Information shows how assets depend on one another and detects contagion. Approximate Entropy measures the complexity of prices and their predictability. RLNNEE gives a detailed, time-varying view of local instability. These tools provide a comprehensive assessment of market stability, resilience, and the spread of information. This analysis is instrumental in unstable environments. The analysis sorts of cryptocurrencies into two broad categories. This relies on their consensus mechanisms and environmental impact. The first group lists non-GCs like Bitcoin (BTC), Ethereum (ETH), Bitcoin Cash (BCH), and Ethereum Classic (ETC). The second group lists GCs like Cardano (ADA), Polygon (MATIC), Stellar (XLM), and Ripple (XRP).

3.2.1. Why Mutual Information as a Measure of Dependence?

Mutual information (Shannon 1948) is a fundamental measure of how much information two random variables X and Y share. Specifically, it is given by:

$$MI(X,Y)=H\left(X\right)+H\left(Y\right)-H(X,Y)$$

(1)

where $H\left(X\right)$ and $H\left(Y\right)$ the marginal entropies, and $H(X,Y)$ the joint entropy. Unlike traditional correlation coefficients, MI can identify both linear and nonlinear dependencies, making it a powerful tool for studying complex relationships among financial assets.

Marginal entropy is invoked to measure the uncertainty corresponding to a single random variable.

$$H(X)=-\sum _{x\in X}P\left(X\right)logP\left(X\right)$$

(2)

Alternatively, H(X) denotes the Entropy of the random variable.

P(X) is the Probability of the event X is in the set of possible values of $X$,

The summation over $x\in \mathrm{X}$ accounts for all possible states of the random variable $X$, thereby aggregating uncertainty across its entire support.

The logarithm is typically taken in base 2 for measuring entropy in bits.

Joint entropy quantifies the total uncertainty associated with two random variables $X$ and $Y$.

$$H(X,Y)=- \sum _{x\in X}\sum _{y\in Y}P(X,Y)log P(X,Y)$$

(3)

$P (X ,Y)$ is the Joint probability of simultaneous occurrence of events $X$ and $Y$.

Here, P(X,Y) denotes the joint probability of the simultaneous occurrence of events X and Y, where X ∈ $𝒳$ and Y ∈ $𝒴$ denote all possible states of the random variables X and Y, respectively. This entropy measure captures the overall uncertainty when the two variables are considered jointly (Cover and Thomas 2006).

The conditional entropy measures the uncertainty of X given Y.

$$H(X/Y) = - \sum _{x\in X}\sum _{y\in Y}P(X,Y) logP(X/Y)$$

(4)

where

P(X/Y) is the conditional probability of x donnée, and

H(X/Y) quantifies how much uncertainty remains about x when y is known.

3.2.2. Approximate Entropy

Approximate Entropy (ApEn) measures the complexity of a time series by quantifying its regularity and predictability. The concept was initially introduced by Pincus (1991) and subsequently refined by Pincus and Goldberger (1994). ApEn evaluates the likelihood that patterns of observations that are similar over a given length remain similar when the sequence length increases. Formally, it computes a logarithmic probability by examining sequences of length m that match within a tolerance r and then assessing whether these sequences continue to match when extended to length m + 1. The parameter m controls the embedding dimension, while r defines the similarity threshold. In financial markets, lower ApEn values indicate more regular, predictable price dynamics, whereas higher values reflect greater randomness, instability, and informational inefficiency. ApEn has been widely used to distinguish between regular and irregular behaviour in complex systems (Richman and Moorman 2000). Its application to financial markets has been examined by several studies, including Andrieş and Căpraru (2014) and Bhaduri (2014). In the context of cryptocurrencies, ApEn serves as a valuable indicator of informational resilience, as assets with lower entropy exhibit more stable informational structures, even during periods of crisis. Technical details and parameter choices are provided in Appendix A.1.

3.2.3. Rolling Local Nearest Neighbour Entropy Estimator

We introduce the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE). It tracks the local, time-varying nature of informational instability. The model extends the classical k-nearest-neighbour entropy estimator of Kozachenko and Leonenko (1987). We use a rolling-window mechanism. This fits non-stationary environments marked by sudden regime changes and short-term turbulence.

Let $\left(X_t{)}_{t=1}^n\right.$ be a strictly stationary stochastic process defined on a metric space $\left(\mathcal{X},d\right)$. For a fixed window size $W\ge 2k$, define

$${\mathcal{W}}_{\mathit{t}}=\{{\mathit{X}}_{\mathit{t}-\mathit{W}+\mathbf{1}},\dots ,{\mathit{X}}_{\mathit{t}}\},\mathit{t}=\mathit{W},\dots ,\mathit{n}.$$

(5)

For each $x_i\in {\mathcal{W}}_t$, denote by $r_{\left(k\right)}\left(x_i\right)$ the distance from $x_i$ to its $k$-th nearest neighbour within the window. The local density estimate follows the Kozachenko–Leonenko approach:

$${\widehat{\mathit{f}}}_{\mathit{k}}\left({\mathit{x}}_{\mathit{i}}\right)={\displaystyle \frac{\mathit{k}}{\mathit{W}\hspace{0.17em}{\mathit{V}}_{\mathit{d}}\left({\mathit{r}}_{\left(\mathit{k}\right)}\left({\mathit{x}}_{\mathit{i}}\right)\right)}},$$

(6)

where $V_d\left(r\right)$ is the volume of a $d$-dimensional Euclidean ball of radius $r$:

$${\mathit{V}}_{\mathit{d}}\left(\mathit{r}\right)={\displaystyle \frac{{\mathit{\pi }}^{\frac{\mathit{d}}{\mathbf{2}}}}{\mathsf{\Gamma }\left(\frac{\mathit{d}}{\mathbf{2}}+\mathbf{1}\right)}}{\mathit{r}}^{\mathit{d}},$$

(7)

The local entropy at $x_i$ is:

$${\widehat{\mathit{H}}}_{\mathit{i}}^{\left(\mathit{k}\right)}=-\mathbf{l}\mathbf{o}\mathbf{g} ({\widehat{\mathit{f}}}_{\mathit{k}}({\mathit{x}}_{\mathit{i}}))=\mathbf{l}\mathbf{o}\mathbf{g} {\mathit{V}}_{\mathit{d}}({\mathit{r}}_{\left(\mathit{k}\right)}({\mathit{x}}_{\mathit{i}}))+\mathbf{l}\mathbf{o}\mathbf{g} \mathit{W}-\mathbf{l}\mathbf{o}\mathbf{g} \mathit{k}.$$

(8)

The rolling entropy at time $t$ is the window average:

$${\widehat{\mathit{H}}}_{\mathit{t}}^{\left(\mathit{k},\mathit{W}\right)}={\displaystyle \frac{\mathbf{1}}{\mathit{W}}}\sum _{\mathit{i}=\mathit{t}-\mathit{W}+\mathbf{1}}^{\mathit{t}}{\widehat{\mathit{H}}}_{\mathit{i}}^{\left(\mathit{k}\right)},$$

(9)

The sequence $\left\{{\widehat{H}}_t^{\left(k,W\right)}{\}}_{t=W}^n\right.$ constitutes the RLNNEE process. The theoretical foundations of the Rolling Local Nearest Neighbour Entropy Estimator are presented in Appendix A.2, where Equations (A2)–(A17) formalise the estimator, and the corresponding computational procedure is summarized in Algorithm A1.

From an economic perspective, the entropy measures employed in this study characterise the production, dissemination, and assimilation of information within cryptocurrency markets. High entropy indicates a more disorderly and unpredictable information environment, typically linked to increased uncertainty, varied investor expectations, and market inefficiencies. Conversely, lower entropy indicates a more stable and predictable information structure, suggesting the market’s greater ability to integrate available information.

Mutual information quantifies the degree of information dependence between cryptocurrencies, regardless of whether their relationships are linear or nonlinear. High mutual information indicates a high degree of information sharing and increased interconnection between assets. A lower value, meanwhile, suggests relative informational autonomy and is often interpreted as a sign of diversification and resilience. Finally, the rolling local nearest neighbour entropy estimator (RLNNEE) captures these dynamics locally and temporally, providing an in-depth analysis of how information regimes evolve.

4. Result and Discussion

4.1. Preliminary Analysis

Figure 1 shows a gap between the prices and returns of GCs and polluting cryptocurrencies. The GCs group includes XRP, MATIC, XLM, and ADA. The DCs group contains BTC, ETH, BCH, and ETC. This contrast is sharpest around the outbreak of the Russian Ukrainian war. A red dotted line marks that event. Polluting coins rose in price during the crisis. Bitcoin and Ethereum showed wild swings in their returns. GCs cryptocurrencies followed a moderate path. Their returns stayed stable. This suggests they react less to market chaos. These trends confirm that crypto markets worsen vulnerabilities during systemic shocks (Urquhart 2016). The data also shows distinct behaviours for GCs versus polluting assets.

Table 2. Descriptive Statistics.

	BTC	ETH	BCH	ETC	XRP	MATIC	XLM	ADA
Panel A: Full sample
Mean	0.001	0.001	0.000	0.001	0.000	0.003	0.000	0.001
Std. Dev	0.037	0.047	0.054	0.058	0.057	0.078	0.054	0.053
Max	0.172	0.231	0.421	0.352	0.549	0.498	0.559	0.279
Min	−0.465	−0.551	−0.561	−0.506	−0.551	−0.716	−0.410	−0.504
Skewness	−1.290	−1.307	−0.513	0.087	0.457	0.085	1.069	−0.232
Kurtosis	18.464	15.486	15.701	10.756	20.217	14.049	18.100	7.673
Jarque–Bera	23,542.74	16,708.022	16,770.616	7840.543	27,741.327	13,373.540	22,501.344	4004.77
Panel B: Pre-war period
Mean	0.002	0.003	0.000	0.002	0.001	0.006	0.001	0.002
Std. Dev	0.040	0.051	0.059	0.061	0.063	0.090	0.060	0.059
Max	0.172	0.231	0.421	0.352	0.445	0.498	0.559	0.279
Min	−0.465	−0.551	−0.561	−0.506	−0.551	−0.716	−0.410	−0.504
Skewness	−1.454	−1.494	−0.815	−0.140	−0.114	0.011	0.756	−0.296
Kurtosis	18.880	15.948	15.851	11.562	15.017	11.679	14.277	7.181
Jarque–Bera	15,776.60	11,382.89	10,979.37	5784.95	9754.357	5899.179	8913.157	2246.57
Panel C: During the war
Mean	−0.001	−0.001	0.000	−0.001	0.000	−0.002	−0.001	−0.002
Std. Dev	0.029	0.037	0.044	0.050	0.044	0.051	0.041	0.041
Max	0.136	0.166	0.305	0.281	0.549	0.325	0.476	0.214
Min	−0.174	−0.192	−0.177	−0.187	−0.217	−0.290	−0.178	−0.204
Skewness	−0.490	−0.438	0.806	0.763	3.138	0.159	2.416	−0.078
Kurtosis	5.730	4.671	7.368	5.604	42.029	7.042	31.353	4.359
Jarque–Bera	836.365	559.455	1406.884	834.879	44,564.27	1229.439	24,838.69	471.488

Notes: The table in question presents statistical data on cryptocurrency returns, examined in two distinct temporal contexts—namely, prior to and during the Russia–Ukraine conflict. The values exhibited herein encompass several statistical measures: Mean, Max (representing the maximum value), Min (denoting the minimum value), and S.D. (which signifies standard deviation) pertinent to each distribution. Furthermore, moments of Skewness and Kurtosis are also represented. The assessment of normality is conducted utilising the Jarque–Bera test as proposed in 1980. The timeframe of this investigative study spans from 28 April 2019, through to 5 October 2023, with returns computed using continuous compounding.

displays descriptive statistics for cryptocurrency returns. It lists GCs, such as XRP, MATIC, XLM, and ADA, alongside DCs, such as BTC, ETH, BCH, ETC. The analysis covers the entire sample and the periods before and during the Russian Ukrainian War. Average returns remain low. Standard deviations show different risk levels across assets. MATIC and BCH display higher volatility than others. GCs mostly showed positive skewness before the conflict. However, some polluting cryptocurrencies showed negative skewness. Ethereum Classic (ETC) is a specific example. Distributions showed high skewness and kurtosis during the war. These traits confirm that cryptocurrency returns have fat tails. They also follow non-Gaussian patterns.

Figure 2 displays a three-dimensional heat map. It tracks correlations between eight GCs and eight (DCs), The friendly coins include XRP, MATIC, XLM, and ADA. (DCs) are BTC, ETH, BCH, ETC. Coefficients range from −1 to 1. This number shows how assets move together. The map reveals higher correlations among some (DCs), However, GCs mostly have moderate correlations. These results indicate distinct dependency structures between the two groups.

4.2. Mutual Information Analysis

Table 3. Mutual information values.

	BTC	ETH	BCH	ETC	XRP	MATIC	XLM	ADA
Panel A: Full sample
BTC	NA	0.623	0.517	0.410	0.408	0.305	0.370	0.427
ETH	0.623	NA	0.545	0.532	0.494	0.364	0.451	0.532
BCH	0.517	0.545	NA	0.559	0.441	0.287	0.445	0.447
ETC	0.410	0.532	0.559	NA	0.417	0.311	0.408	0.439
XRP	0.408	0.494	0.441	0.417	NA	0.300	0.563	0.474
MATIC	0.305	0.364	0.287	0.311	0.300	NA	0.296	0.391
XLM	0.370	0.451	0.445	0.408	0.563	0.296	NA	0.503
ADA	0.427	0.532	0.447	0.439	0.474	0.391	0.503	NA
Panel B: Pre-war period
BTC	NA	0.562	0.542	0.378	0.403	0.240	0.360	0.382
ETH	0.562	NA	0.586	0.504	0.515	0.292	0.451	0.512
BCH	0.542	0.586	NA	0.594	0.522	0.267	0.475	0.481
ETC	0.378	0.504	0.594	NA	0.447	0.269	0.430	0.402
XRP	0.403	0.515	0.522	0.447	NA	0.270	0.572	0.491
MATIC	0.240	0.292	0.267	0.269	0.270	NA	0.257	0.320
XLM	0.360	0.451	0.475	0.430	0.572	0.257	NA	0.523
ADA	0.382	0.512	0.481	0.402	0.491	0.320	0.523	NA
Panel C: During the war
BTC	NA	0.834	0.552	0.574	0.504	0.593	0.485	0.605
ETH	0.834	NA	0.545	0.683	0.529	0.641	0.536	0.646
BCH	0.552	0.545	NA	0.598	0.399	0.415	0.490	0.485
ETC	0.574	0.683	0.598	NA	0.468	0.521	0.475	0.619
XRP	0.504	0.529	0.399	0.468	NA	0.495	0.632	0.527
MATIC	0.593	0.641	0.415	0.521	0.495	NA	0.474	0.670
XLM	0.485	0.536	0.490	0.475	0.632	0.474	NA	0.534
ADA	0.605	0.646	0.485	0.619	0.527	0.670	0.534	NA

Note: The table presents mutual information estimations for the whole period, including the War Russia Ukraine and War Russia Ukraine periods. Mutual information estimates are significantly influenced by the Russia–Ukraine war. During this period, estimates indicate heightened volatility and uncertainty across sectors. Data from this period is essential for assessing the geopolitical impact on economic indicators. Values represent the mutual information transferred between each pair of cryptocurrencies. When MI = 0, the two variables are independent; a value of 1 indicates maximum mutual information. Mutual information was estimated using entropy-based metrics and time-series analysis. Initially, the entropy of each cryptocurrency’s price data was calculated to evaluate its intrinsic unpredictability. Subsequently, the joint entropy between pairs of cryptocurrencies was assessed to ascertain the extent of shared information. This allowed for mutual information computation.

displays mutual information values between GCs and polluting cryptocurrencies. The data covers the full sample. The table also tracks the periods before and during the Russia–Ukraine war. These values range from 0 to 1. They measure the intensity of non-linear dependencies between return series. Polluting cryptocurrencies shows higher mutual information on average across the whole timeline. This pattern is strongest for pairs involving Bitcoin and Ethereum. The mutual information between BTC and ETH is 0.623 across the entire sample. Mutual information levels were moderate before the conflict. Several GCs cryptocurrencies showed lower numbers. This points to weaker dependency structures. The values of all assets rose during the war. Polluting cryptocurrencies saw the most significant increases. The BTC and ETH values climbed to 0.834 in this period. Their interdependence grew stronger. GCs cryptocurrencies show lower levels by comparison. The results reveal distinct dependency patterns for GCs and polluting cryptocurrencies. Informational structures change along with market regimes.

4.3. Changes in Mutual Information Between War and Prewar

Table 4. Changes in mutual information (prewar and during the war period).

	BTC	ETH	BCH	ETC	XRP	MATIC	XLM	ADA
BTC		1	1	1	1	1	1	1
ETH			1	1	1	1	1	1
BCH				0	0	1	1	0
ETC					1	1	1	1
XRP						1	1	1
MATIC							1	1
XLM								1
ADA

Note: This table illustrates changes in mutual information sharing between GCs and DCs compared to the pre-war period. A value of 1 implies an increase in mutual information, while 0 indicates a decrease. Mutual information quantifies the extent to which one variable carries information about another. Greater mutual information in cryptocurrencies indicates a stronger correlation between the price fluctuations of beneficial and harmful digital currencies. This implies a potential rise in market correlation or the presence of external factors that impact both cryptocurrency categories. A decline in mutual information between GCs and DCs indicates reduced price correlation, suggesting that investors increasingly differentiate between GCs sand DCs. Consequently, market dynamics may change, leading to a preference for environmentally friendly cryptocurrencies as awareness of sustainability concerns grows.

compares mutual information in cryptocurrency pairs before and during the war between Russia and Ukraine. A value of 1 shows an increase in mutual information during the war. A value of 0 means the level decreased. The results show a significant rise in mutual information for several ‘polluting’ cryptocurrencies. This trend primarily affects Bitcoin (BTC), Ethereum (ETH), and Bitcoin Cash (BCH). Informational dependencies grew stronger during the conflict. GCs cryptocurrencies showed different results. Their changes in mutual information were mixed and limited. This points to distinct information patterns during geopolitical shocks.

4.4. Estimating Mutual Information over Time Using the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE)

Table 5. Estimates of mutual information using RLNNEE.

	BTC	ETH	BCH	ETC	XRP	MATIC	XLM	ADA
Panel A: Full sample
BTC	NA	0.692	0.573	0.446	0.437	0.307	0.404	0.457
ETH	0.692	NA	0.591	0.577	0.520	0.385	0.506	0.586
BCH	0.573	0.591	NA	0.602	0.462	0.311	0.507	0.482
ETC	0.446	0.577	0.602	NA	0.449	0.312	0.440	0.478
XRP	0.437	0.520	0.462	0.449	NA	0.313	0.639	0.478
MATIC	0.307	0.385	0.311	0.312	0.313	NA	0.306	0.403
XLM	0.404	0.506	0.507	0.440	0.639	0.306	NA	0.548
ADA	0.457	0.586	0.482	0.478	0.478	0.403	0.548	NA
Panel B: Pre-war period
BTC	NA	0.610	0.585	0.424	0.403	0.208	0.390	0.400
ETH	0.610	NA	0.643	0.552	0.535	0.296	0.473	0.591
BCH	0.585	0.643	NA	0.651	0.522	0.263	0.516	0.517
ETC	0.424	0.552	0.651	NA	0.472	0.240	0.451	0.438
XRP	0.403	0.535	0.522	0.472	NA	0.241	0.648	0.476
MATIC	0.208	0.296	0.263	0.240	0.241	NA	0.255	0.319
XLM	0.390	0.473	0.516	0.451	0.648	0.255	NA	0.541
ADA	0.400	0.591	0.517	0.438	0.476	0.319	0.541	NA
Panel C: During the war
BTC	NA	0.872	0.517	0.554	0.498	0.522	0.447	0.621
ETH	0.872	NA	0.530	0.694	0.476	0.642	0.485	0.603
BCH	0.517	0.530	NA	0.544	0.355	0.369	0.507	0.429
ETC	0.554	0.694	0.544	NA	0.420	0.534	0.440	0.628
XRP	0.498	0.476	0.355	0.420	NA	0.466	0.643	0.468
MATIC	0.522	0.642	0.369	0.534	0.466	NA	0.421	0.652
XLM	0.447	0.485	0.507	0.440	0.643	0.421	NA	0.517
ADA	0.621	0.603	0.429	0.628	0.468	0.652	0.517	NA

Note: The table shows the mutual information estimates obtained via the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE) for three different periods: the entire sample, the pre-war period (28 April 2019, to 23 February 2022), and the war period (24 February 2022, to 5 October 2023).

displays mutual information estimates from the RLNNEE. It covers the full sample. It also breaks down the periods before and during the Russia–Ukraine war. Pairs with polluting cryptocurrencies showed high mutual information across the whole timeline. Bitcoin (BTC) and Ethereum (ETH) stood out, suggesting strong interdependence between the two. These ties were moderate but constant before the fighting started. Mutual information values rose sharply during the war. This occurred across several polluting pairs, with the BTC–ETH pair showing the most significant increase. GCs behaved differently. They maintained lower, steadier levels of mutual information both before and during the conflict. Their dependency structures appear less concentrated. Figure 3a,b adds to these findings. They map how entropy changed over time according to the RLNNEE estimates. GCs stayed on stable entropy paths during the geopolitical shock. Polluting coins showed much sharper swings during the war. These outcomes highlight the contrasting patterns of GCs and DCs. They react differently to geopolitical shocks. The intensity and stability of their dependencies vary over time.

4.5. Approximate Entropy Analysis

Approximate entropy analysis (ApEn) measures the complexity of cryptocurrency returns. It also assesses predictability across the studied periods.

Table 6. Approximate entropy analysis.

	Full Sample	Pre-War	During-War
BTC	1.650	1.563	1.312
ETH	1.653	1.583	1.287
BCH	1.599	1.488	1.348
ETC	1.556	1.467	1.284
XRP	1.499	1.395	1.367
MATIC	1.593	1.517	1.353
XLM	1.620	1.534	1.342
ADA	1.614	1.523	1.334

Note: This table shows the approximate entropy of selected cryptocurrencies for three periods: the entire sample (28 April 2019, to 5 October 2023), the pre-war period (28 April 2019, to 23 February 2022), and the war period (24 February 2022, to 5 October 2023). The approximate entropy quantifies the regularity and unpredictability of time series data, essential for understanding market dynamics. A lower entropy value indicates that price patterns are more predictable, while a higher value suggests high complexity and unpredictability.

displays the average ApEn values for GCs and DC_S. The data covers the full sample. It includes the time before the war and the conflict between Russia and Ukraine. The results reveal a drop in approximate entropies during the war. This trend reflects a shift in the information patterns of cryptocurrency markets. The change is most substantial for polluting cryptocurrencies like Bitcoin (BTC) and Ethereum (ETH). Their ApEn values signal unstable dynamics. They also show higher sensitivity to outside shocks. GCs like MATIC and ADA behave differently. They show lower and stable levels of approximate entropy across all periods. This stability suggests a regular information structure. It also indicates they adapt better to geopolitical disruptions. Figure 4a,b confirms these observations. GCs showed lower dispersion during the war period. They also had a more stable median entropy. DC_S showed increased variability.

Figure 5 adds to this analysis. It shows that the approximate entropy for BTC and ETH became increasingly unstable around the time the conflict started. GCs acted differently. They kept a steady pattern.

Figure 6 summarises the data by comparing normalised volatility and approximate normalised entropy. The chart separates GCs from DCs. GCs feature moderate volatility and relative predictability. This combination is favourable. DCs differ. They display high volatility and unstable informational complexity.

4.6. Discussion

Beyond the descriptive interpretation of the results, this study’s findings contribute to the growing literature on information efficiency and market resilience in cryptocurrency markets. The observed increase in mutual information among DCs during the Russia–Ukraine war is consistent with previous evidence showing that systemic shocks tend to strengthen dependencies among assets sharing similar technological and structural characteristics (Bariviera and Merediz-Solà 2021; Vidal-Tomás et al. 2021). In contrast, the relatively stable and lower levels of information sharing among GCs suggest a more fragmented and adaptive informational structure, which may limit the propagation of systemic risk during periods of extreme uncertainty.

From a theoretical perspective, these results align closely with the Adaptive Market Hypothesis (Lo 2004), which posits that market efficiency evolves in response to changing economic and geopolitical environments. Rather than converging toward a single efficiency regime, cryptocurrency markets display heterogeneous adaptive behaviours. DCs appear to become more synchronised and informationally constrained under geopolitical stress, while GCs exhibit greater informational flexibility and resilience. This heterogeneity challenges the notion of uniform efficiency across digital assets and supports the view that market dynamics depend on underlying technological and environmental characteristics.

The entropy-based evidence further reinforces this interpretation. The decline and stabilisation of approximate entropy among GCs during the war period indicate a transition toward more regular and predictable information patterns, whereas the heightened entropy volatility observed for polluting cryptocurrencies reflects persistent informational instability. This finding is consistent with prior studies suggesting that entropy measures capture crisis-induced regime changes that are often overlooked by volatility-based approaches (Bariviera 2017; Lahmiri and Bekiros 2020). Notably, the use of the Rolling Local Nearest Neighbour Entropy Estimator (RLNNEE) reveals local and time-varying informational regimes, highlighting that crisis dynamics are neither uniform nor static across assets.

From an economic and financial standpoint, these results have direct implications for portfolio diversification, risk management, and sustainable investment strategies. GCs appear to offer more stable informational properties during geopolitical turmoil, making them potentially valuable defensive assets in diversified portfolios. Conversely, the strong informational coupling among DCs increases their exposure to contagion and systemic risk during crises. These findings suggest that environmental characteristics are not merely ethical attributes but play a substantive role in shaping informational resilience and adaptive market behaviour.

Overall, the results underscore the importance of integrating environmental considerations and information-theoretic tools when assessing cryptocurrency markets under geopolitical stress. By linking sustainability, information dynamics, and adaptive efficiency, this study provides a broader framework for understanding how digital assets respond to extreme uncertainty. It offers insights relevant to investors, policymakers, and researchers concerned with the stability of emerging financial systems.

5. Conclusions and Implications

This study examines how cryptocurrency markets handle information and maintain resilience. We separate GCs, such as XRP, MATIC, XLM, and ADA, from DCs, such as BTC, ETH, BCH, ETC. The context is the geopolitical shock from the war between Russia and Ukraine. Traditional methods focus on volatility or linear correlations. We used complementary entropic tools instead. These included mutual information, approximate entropy, and the new dynamic estimator RLNNEE. These tools measure complexity and adaptability. They also track how information spreads during a crisis. The results show that geopolitical instability affects coins differently based on their structure. DCs showed stronger informational dependencies. Their prices changed more in sync during the war. This reflects a higher risk of system-wide failure. High integration into global markets explains this sensitivity. High liquidity plays a role, too. Their energy-intensive consensus mechanisms encourage copycat behaviour, leading to rapid shock transmission. GCs kept more fragmented information structures. Their entropy levels remained stable. This shows they adapt better and resist outside shocks. Investors might treat these assets differently during a crisis. Assets with high mutual information and unstable entropy attract synchronised speculation. This increases volatility and spreads adverse effects. GCs look like defensive assets. They are relatively predictable and react less to systemic shocks. This supports their role in diversification strategies for long-term strength.

The implications for sustainable finance matter. Environmental sustainability is more than a non-financial metric. It directly changes the informational properties and stability of digital assets. GCs offer options for institutional investors. These investors can incorporate ESG factors while maintaining strong portfolios. Regulators should treat crypto-assets differently based on these findings. They should look at energy footprints and behaviour during stress, not just market cap or liquidity. Protocol developers and issuers should focus on efficient consensus mechanisms to improve project credibility. This makes them more attractive over time. These results challenge the Efficient Market Hypothesis. They show that the efficiency and complexity of cryptocurrencies vary across market regimes. Structural characteristics matter too. The findings align better with the Adaptive Markets Hypothesis. This theory states that investor behaviour and market structure evolve. Economic and geopolitical conditions drive these changes.

The study has limitations. We analysed a small sample of major cryptocurrencies. The period covered a specific geopolitical crisis. Future research could look at other digital asset classes. It could study different stress periods. Researchers might combine entropic measures with detailed behavioural and ESG indicators. The proposed approach proves that dynamic entropy tools are useful even with these limits. The RLNNEE helps analyse market resilience in uncertain environments.