Blockchain technology is set to transform economics and finance by enabling secure, transparent, and decentralized transactions. Some significant examples in this sense are cryptocurrencies and decentralized finance, which leverage blockchain technology to provide fast, low-cost financial services without a central authority, as well as the tokenization of finance, already forecast by Larry Fink, CEO of BlackRock. As crypto economies and blockchain applications gain global relevance, the need to measure and assess their eficiency is becoming increasingly important. While blockchain eficiency is often evaluated in terms of transactions per second or energy consumption, cryptocurrency eficiency is implicitly assessed through various indexes, such as capitalization, price trends, average transaction value, mining profitability, and others. What is lacking is an index capable of comprehensively and coherently describing the actual functioning of a crypto economic system, accounting for its key economic characteristics - such as supply mechanisms and token distribution - and the level of user participation within the specific crypto economy. In this study, we introduce a new theoretical framework based on Shannon entropy to assess the economic eficiency of a cryptocurrency through the Entropy Balance index (EB-index). Our approach integrates on-chain parameters - sourced from Coin Metrics® - by mapping them to economic quality attributes. To illustrate how our entropy-based approach works, we apply it to two distinct sets of attributes across six leading cryptocurrencies by market capitalization and use-case diversity: Bitcoin, Ethereum, Ripple, USD Coin, Dogecoin, and Cardano. For either set of attributes, the six EB-index values provide us with a comprehensive way of comparing the considered cryptocurrencies from an economic eficiency viewpoint. Our approach is fully customizable with respect to the selection of attributes as well as their weights.
Internet is transforming economics and finance, with blockchain raising as a key technology with
its secure, transparent, and immutable distributed ledger [
With the expansion of applications of cryptocurrencies and blockchains, attention has increasingly turned to assessing their properties, especially in terms of eficiency. Despite the various possible interpretations of this word – such as performance eficiency based on transactions per second [ 14, 15], energy consumption in blockchain networks [16, 17], or price trends in crypto markets [18] – its use in the realm of cryptocurrencies introduces new dimensions. This is due to the unique features associated with blockchain technology, characterized by decentralized governance, peer-to-peer interactions, and a heterogeneous structure and range of use cases for digital assets. For instance, to the best of our knowledge, no study has thoroughly explored the economic eficiency of cryptocurrencies by considering key factors such as supply distribution, user participation, and exchange activity; all of these are dimensions in which wealth concentration [19, 20], low engagement [21, 22], and asset dormancy [23, 24, 25] can undermine network economic vitality. Table 1 shows these issues with regard to various cryptocurrencies up to April 2025. Notably, Bitcoin has a high asset dormancy (up to 99.56%), while Ripple exhibits a low engagement (4.4%). Moreover, the lack of a clear conceptual framework for analyzing economic eficiency in this context underscores the need to define it through measurable links between network behavior and core economic indicators.
This study examines the concept of economic eficiency for the class of cryptocurrencies that use public distributed ledgers to record transfers, ofer accessible data, and leverage blockchain for transparency, traceability, and consistency [29, 30]. It is worth noting that similar analyses in traditional economies are hindered by limited access to raw data – controlled by centralized authorities – and inconsistencies in reporting across sources, such as variable monetary aggregates calculated by central banks [31]. Since data accessibility is ensured within blockchains, the problem to tackle is the lack of a unifying index expressing economic eficiency that enables a comprehensive comparison of cryptocurrencies instead of proceeding parameter by parameter.
We propose a theoretical framework that uses Shannon entropy [32] – a foundational concept in information theory, originally introduced by Claude Shannon in 1948 [32] – to define the Entropy Balance index (EB-index), an economic index capable of aggregating a set of parameters describing economic qualities of a cryptocurrency. Our method follows these guiding principles: () rest on a solid theoretical foundation, () remain adaptable to changes in parameters, () reward cryptocurrencies exhibiting well-balanced values for the chosen parameters, and () support parameter weighting without compromising the theoretical foundation.
Claude Shannon’s seminal paper “A Mathematical Theory of Communication” [32] introduced a rigorous, axiomatic framework for quantifying information in communication systems. Entropy has subsequently found extensive use across diverse fields – from linguistics and neurobiology to machine learning and medical diagnostics – underscoring its flexible role as an index of information exchanged, the degree of fragmentation of a set, or the balance in the distribution of resources in both natural and engineered systems [33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Its role as a measure of economic (in-)equality was already formalized by Theil [43], who proposed entropy-based and divergence-based indices as alternatives to the Gini index, highlighting their decomposability across population subgroups. There are some applications also in the context of blockchain, for example to quantify the degrees of decentralization [44, 45], to measure the stability in blockchain consensus dynamics [46], and to express portfolio diversification [ 47].
To illustrate the robustness of our approach, we apply the entropy framework by using two distinct
sets of economic qualities, Set1 and Set2, each representing structural and behavioral aspects of
cryptocurrency economies. Set1 focuses on a minimal combination of basic financial activity metrics,
while Set2 adopts a richer, more granular perspective on network dynamics and internal economic
organization. These attributes guide the selection of corresponding on-chain parameters, which we
retrieve from the Coin Metrics® platform [48], ensuring an empirical and data-driven rooting. The
entropy based on these two sets of features is then employed to evaluate six major cryptocurrencies
with high market capitalization, spanning various economic roles and use cases (payments, smart
contracts, stablecoins, and memecoins): Bitcoin (BTC) [
This study paves the way to a foundational and flexible methodology for evaluating economic eficiency in cryptoassets and ofering insights to investors, policymakers, and researchers navigating the evolving landscape of blockchain-based economies. As part of this work, we release our refined dataset (available at https://doi.org/10.5281/zenodo.15221823) for result reproducibility as well as independent analysis execution.
Roadmap. §2 surveys existing literature and related works. §3 outlines the conceptual study for an entropy-based economic framework. §4 shows an illustrative example. Finally, §5 concludes the paper and discusses future works.
Blockchain is a decentralized digital ledger technology that securely records transactions across a
network of nodes. Each block of data is linked to the previous one, forming a chain that is tamper-proof and
transparent [
The literature review reveals the absence of a universally accepted definition of eficiency in this setting. Among the various interpretations, some focus on energy consumption and expenditure [17], while others emphasize market price trends by analyzing fluctuations in cryptocurrency prices and their implications for economic stability and eficiency [18]. Still others assess eficiency based on the production objectives of the considered economy, including the presence or absence of technological components [49]. In [50], the discussion centers on technical and scale eficiency, which difer in scope. The former measures how efectively resources are allocated to maximize output, so as to ensure the optimal utilization of inputs, while the latter examines the relationship between input growth and output expansion, in order to identify whether economies of scale are being achieved [50]. Additionally, in [50], eficiency is evaluated both qualitatively – by comparing actual delivery times with planned schedules – and quantitatively – by analyzing the ratio of actual versus expected outputs relative to expenditures. In traditional economies, metrics like the Gini coeficient [ 19, 51] – which quantifies wealth distribution – and Pareto eficiency [ 52] – which describes an optimal allocation of resources where no individual’s well-being can improve without negatively afecting another – are used to identify economic ineficiencies and potential resource underutilization. Furthermore, it is interesting that resource allocation analyses often incorporate concepts such as the cost of unused capacity [53].
As for the research gap in relation to economic eficiency within blockchain and cryptocurrency ecosystems, we identify numerous studies that examine supply distribution, active user participation, and idle resources within these ecosystems.
Supply distribution impacts fairness, decentralization, and network stability [20]. Consensus mechanisms and other aspects impact cryptocurrency distribution, where wealth concentration can afect security, and exchange rates [ 54, 19]. Actually, cryptocurrencies exhibit high inequality patterns similar to traditional economies (e.g., DOGE [27] Gini coeficient – 0.82 – is similar to US one – 0.84 [ 19]).
User participation afects liquidity and network value. In [ 21], an improved version of the PageRank algorithm – a Google technology for ranking web pages based on their importance – is used to evaluate Bitcoin user participation by taking into account both the consistency and the variability of transaction patterns. Key metrics like active addresses, transaction volume, and circulation frequency strongly correlate with price trends and economic activity [55, 22, 56].
Idle assets signal ineficiencies in resource use [ 23, 24, 25]. A key metric here is “Bitcoin Days Destroyed”, which measures transaction volume while accounting for how long bitcoins have remained unused – highlighting the economic impact of previously dormant coins becoming active again [23]. Another useful metric is average dormancy, which tracks the duration that bitcoins remain inactive before reuse thus providing insights into circulation patterns, although it does not directly measure monetary velocity or account for the full money supply and price levels [24, 25].
Together, these elements highlight the need for an integrated framework to evaluate uniformly economic eficiency in blockchain systems [57].
A possible way to address the divergent interpretations of economic eficiency recalled in §2 is to define this concept in cryptocurrencies by linking transfer dynamics with basic economic parameters, thus enabling a more structured and comprehensive analysis. In particular, it is necessary to aggregate parameters in a coherent way, with an index able to satisfy the guiding principles mentioned in §1. We propose using Shannon entropy to define a theoretical framework that measures economic eficiency in cryptocurrency ecosystems through the Entropy Balance index (EB-index), aligning with established standards for reliability [58, 59, 60] and leveraging the transparency of blockchain data. Starting from a probability distribution associated with a set of events or parameters chosen to represent the qualities that determine the economic eficiency of a cryptocurrency, we employ Shannon entropy to measure how good the balance is within this set. The EB-index captures the complex internal structure of cryptoassets without relying on arbitrary single metrics. By aggregating heterogeneous dimensions – such as user activity, transactions, and supply – it reveals patterns that univariate analyses may miss.
In this section we recall the basics of the entropy measure (§3.1) and then we discuss a number of economic parameters among which to select the ones to be used in the entropy formula (§3.2).
Let 1,2,..., be a set of quality parameters, with ≤ ≤ , where and represent, respectively, the lower and upper bounds of the interval of variability for the corresponding parameter. The idea behind our economic eficiency index for a cryptocurrency is to combine these parameters in a way that rewards cryptocurrencies exhibiting a well-balanced set of parameters with good performance, which leads to a wealthy economic development of the corresponding cryptocurrency. This composition must allow for changes in the number of parameters without afecting the nature or compromising the coherence of the index.
The first step is to normalize each within the interval [
The economic eficiency index should intuitively reach its maximum value when all attain the
upper bound of the interval [
ℐ() = −log 2( ) = − =1 ∑︁log2 =
∑︀
=1
parameters in the interval [
if is degenerate log2 if is uniform By taking the logarithms to the base , we obtain a normalization of the entropy: degenerate means it is in the form 0,...,1,...,0, while uniform corresponds to = 1/ ∀ [32, 61]. (1) (2) (3) (4) (5) (7) (8) 0 ≤ ( ) ≤ 1 (6) reasonably possess [32, 62, 63]. of events: The motivation for choosing the Shannon entropy lies in the fact that it can be proven, through a theorem, that Shannon entropy is the unique function, among infinitely many possible ones, that satisfies a specific set of postulates outlining the natural properties an information measure should The distinguishing postulate is the so-called branching property, expressed as follows in the case (1,2,...,) = (1 +2,3,...,)+(1 +2) ︂( 1 that describes how entropy behaves when a p.d. is broken down into successive steps. The relation expresses the average information loss incurred when two events are grouped together and made indistinguishable; this loss is given by the entropy of the two events, weighted by the sum of their probabilities. Furthermore, it should be noted that this branching property can be generalized to a partition { | 1 ≤ ≤ }, < , of {1,2,...,} with p.d. = { 1,2,...,}: (1,2,..., ) = (1,2,..., )+∑︁() =1 where () = − ∑︀∈ ︁( )︁ log︁( )︁ is the entropy associated with and ∑︀∈ = 1 so that / is efectively a p.d. In this last case, the relation expresses the average information loss incurred when some subsets of events are grouped together and made indistinguishable; this loss is given by the sum of the entropies related to the subsets, weighted by their probabilities [62]. These equations provide the theoretical foundation underpinning entropy and our approach. 1An analytic function is a function that () is locally representable by a convergent power series, hence it can be expressed as a sum of terms based on powers of the variable, and () is diferentiable at every point in its domain.
(9) (10) (11)
Another important advantage of the Shannon entropy is that it is possible to associate a set of property. This leads to the definition of the Beliş-Guiaşu weighted entropy [64] weights or utilities with the p.d. , say = {1,2,...,}, ≥ 0 , without losing the branching ℋ( ;) = − ∑︁ · log2 =1 It is simple to check that the branching property still holds as soon as we correctly define the weight of the composition of two events 1 and 2 as a weighted sum of the utilities of the single events: Weight (1,2) = 12 =
1 1 +2 1 +
2 1 +2 2 The branching property can then be expressed in the following way (12 = 1 +2): ℋ(1,...,;1,...,) = ℋ(12,3,...,;12,3,...,)+12 ·ℋ ︂( 1 , 2 ;1,2
︂) The presence of the weights in the weighted entropy (9) causes the function to be zero when = 0, for all . Moreover, it is also reduced to zero when the useful events are impossible or the possible events are useless. The last case occurs also when is degenerate.
Among the most commonly used methods to measure complex socio-economic phenomena – such as Gross Domestic Product (GDP) or Human Poverty Index (HPI) – composite indices play a central role. They are widely used due to their adherence to established theoretical and functional requirements that ensure their reliability [58, 59]. According to [59, 58, 60], guidelines and steps to summarize a set of economic indicators and construct a composite index are the following: 1. Phenomenon definition, i.e., the theoretical framework to achieve a clear conceptualization of what is being measured and establish selection criteria for determining whether an indicator should be included.
their correlation to minimize redundancy. 2. Data modeling, i.e., the selection of relevant, timely, and accessible data sources, considering 3. Data processing, composed by () normalization, which ensures comparability between indicators, () weighting, which assigns priority to indicators based on their relevance, and () aggregation, which combines the normalized indicators into a unified framework.
As discussed in §3.1, our entropy-based approach adheres to all these phases, providing a solid foundation for evaluations. In the following, we illustrate how each phase contributes to the construction of our framework for assessing economic eficiency.
Phenomenon Definition.
The economic eficiency analysis we aim to conduct goes beyond purely economic parameters measured in some currency. It also considers additional aspects, such as the intensity and frequency of transfers, as well as user participation. To establish a solid foundation for this study on economic eficiency, we introduce in Table 2 several basic cryptocurrency parameters. We use these basic parameters to clearly define and formalize the set of key economic quality attributes that characterize these digital economic systems. In particular, we identify primitive quality (PQ) attributes, which are defined by a single parameter, and derived quality (DQ) attributes, which result from a mathematical combination of multiple PQs. These are presented in Table 3 along with their definitions, formulas, units, and range intervals. When applicable, we explicitly indicate the use of the native cryptocurrency unit (marked as "Ntv") rather than USD.
The set of unique addresses, counted only once, that are involved in sending transfers over a given period. It refers to all activities leading to a change in the ledger, excluding the null address used for issuance purposes.
The total amount of a digital asset (coin or token) available within the tokenomics of a cryptocurrency.
DQ7 - Mean Transfer per Market Cap
MTMC
MTMC = (value ·USD price)/Cap
USD
[0,+∞)
Formula value = ∑︀∈ amount()
count = | | NAfunded = |funded| NAactive = |active| Cap = ·USD price
MTS = value/count MTAactive = count/NAactive
TR = value/active
Unit Interval Ntv
[0,+∞) Count [0,+∞) Count [0,+∞) Count [0,+∞) Ntv USD USD [0,+∞) (0,+∞) (0,+∞) Ntv
[0,+∞) Count [0,+∞) Count [0,+∞) participation The proportion of active addresses relative to the total addresses holding at least 1 USD. A higher ratio suggests a more engaged user base and a healthy level of participation within the cryptocurrency ecosystem.
participation = NAactive/NAfunded
Count
[
Count
[
Count
[
MTAactive DQ4 - Active Supply Turnover Rate DQ5 - Active Supply Ratio DQ6 - Wealth Distribution
Cap MTS TR ASR WD
The total Ntv value exchanged within the system over a given period, excluding issuance account transfers, which record asset creation (e.g., Bitcoin coinbase transactions).
The total number of transfers executed within the network during a given period.
The number of addresses continuously holding at least 1 USD during a given period.
The total number of unique addresses, counted only once, that are involved in sending transfers over a given period.
The amount of coins/tokens that have been moved at least once within a given time period, excluding double counting of the same units being recycled.
The price in USD per native unit of the coin or token at the close of the day.
The total value of a cryptocurrency in USD, calculated by multiplying its current price by the total circulating supply of coins.
The mean size of a transfer, measured in Ntv. It is calculated by dividing the total value transferred by the number of transfers between distinct addresses during a given period.
The mean number of transfers per active address. It is calculated by dividing the total number of transfers by the number of active addresses during a given period. This metric provides insights into the intensity of usage per user.
The ratio between the total value transferred and the active supply over a given period. A higher TR indicates more frequent economic activity and greater liquidity.
The proportion of the current supply that is actively participating in transactions. A higher ASR signifies that a larger portion of the available cryptocurrency is being used rather than held passively. the network participants, where: is a set of balance intervals each of which represents a range of account balances; NAfunded() is the number of accounts whose balance falls within interval ; current() is the total supply in interval ; avgbal() = current()/NAfunded() is the average balance per address in interval .
The mean size of a transfer over a specific period relative to the cryptocurrency’s market capitalization.
Data Modeling. Having properly defined the quality attributes, we can now identify the on-chain parameters to be measured for each quality, based on a selection criterion that reflects their relevance to the corresponding attribute. To achieve this, we can exploit blockchain data intelligence platforms that provide several on-chain data in user-friendly formats. Table 4 compares some of these platforms with respect to the available metrics, the number of accessible assets and protocols, and the type of API access.
We rely on Coin Metrics [48] as its indicators categorization aids us in identifying both blockchain and cryptocurrency components from economic and technological perspectives. Additionally, it ofers a wide range of protocols and provides researchers with the opportunity to access premium data for free up to 2 years (upon request). In Table 5 we present the on-chain parameters from Coin Metrics that best represent the previously defined quality attributes, enabling a comprehensive study focused on economic eficiency within cryptocurrency networks [ 57, 55]. We use Coin Metrics to download this data with a daily granularity.
Data Processing. The data processing stage is essential to ensure that economic eficiency is grounded in solid theoretical principles. To meet the methodological requirements of normalization, weighting, and aggregation, we adopt the rigorous framework of Shannon entropy [32].
After downloading the relevant metrics from Coin Metrics, we align all datasets to a common time reference, using the earliest available timestamp (2010-07-18) for PQ6 as the baseline (USD Denominated Closing Price). Next, we convert into USD all metrics originally expressed in native units. This applies to the following metrics from Table 5: Xfer’d_Val (PQ1), 1-Day_Active_Supply (PQ5), Current_Supply, Val_in_Addrs_w/Bal≥X_Ntv . Particular attention should be paid to the calculation of DQ6 (Wealth Distribution), which considers balance class intervals rather than individual address balances as in the original Gini formula. These intervals range from 0.001 units of the native currency up to 100K. Furthermore, since a user may control multiple wallets, we focus on the address-level granularity provided by Coin Metrics, which allows us to distinguish between wallets and individual addresses.
We then normalize the data. Normalization ensures that each metric contributes equally, independent
of scale diferences (e.g., market cap or address count). While metrics may reflect protocol-specific
factors, the entropy-based approach remains agnostic to context, focusing on internal distributions. For
fairer comparisons, assets should ideally belong to the same category. For each metric and asset (except
PQ7), we use min-max normalization [59, 58, 60] based on the individual asset’s range, taken within the
entire lifetime of the cryptocurrency. For PQ7 (Market Capitalization), we apply a global normalization
by using the min and max values across all assets to preserve comparability. To handle DQ6 (WD),
where lower values indicate greater eficiency, we use the specific transformation 1 − _ to
reflect this inverse relationship. This adjustment ensures that all metrics behave consistently – higher
normalized values always reflect greater economic eficiency – while remaining within the [
Finally, we compute the EB-index derived from the chosen parameters for an illustrative example by using both the standard formulation (3) and its weighted variant (9). Our analysis, encompassing all data and resulting output, is conducted on a daily granularity. The corresponding results are presented in §4.
In this section, we present a case study to illustrate the application of our EB-index to evaluate economic eficiency in cryptocurrencies. Our intention is not to provide any conclusive assessment of the performance or quality of the selected cryptocurrencies, but rather to demonstrate the practical implementation of the proposed theoretical framework within a real-world financial context.
We conduct a comparative analysis of the entropies obtained by the six major cryptocurrencies cited in §1 – BTC, ETH, XRP, USDC, DOGE, and ADA – over the period from 2010 to 2025; their main financial and technical characteristics, such as capitalization, max supply, underlying blockchain, and consensus mechanism, are briefly summarized in Table 6. The choice of the aforementioned cryptocurrencies also takes into account the fact that not all cryptocurrencies have the desired parameters available on Coin Metrics.
The analysis is based on two distinct sets of derived quality attributes, constructed from selected parameters available on Coin Metrics (see §3.2): Set1 and Set2, each designed to capture diferent layers of cryptocurrency economic behavior. Set1 is composed of a minimal combination of fundamental ifnancial activity metrics – such as market capitalization – primarily reflecting broader macroeconomic sentiment. In contrast, Set2 ofers a more granular view, incorporating a richer selection of attributes that describe structural network dynamics and internal economic organization, thereby capturing slower-moving, user-driven behavioral trends. The composition of these two sets is detailed in Table 7.
The entropy variations for the chosen cryptocurrencies, computed by using the standard formula (3), are illustrated in Figure 1. The comparison across diferent attribute sets highlights how the selection of metrics can substantially afect the observed performance and behavior of cryptoassets.
In Set1, which combines user activity with financial metrics, most assets exhibit a gradual growth with peak values between 0.85 and 0.9 – particularly DOGE, ADA and ETH, which display the highest levels of entropy for the entire period. This suggests a peak in homogeneity and well-balanced development It comprises six dynamic qualities, with a focus on user engagement, transaction characteristics, and the distribution of wealth within the network.
Participation (participation) Mean Transfer Size (MTS ) Mean Transfers per Active Address (MTAactive) Active Supply Turnover Rate (TR) Active Supply Ratio (ASR)
Wealth Distribution (WD ) 1.0 0.8 y0.6 p o r t En0.4 0.2 0.0 1.0 0.8 y0.6 p o r t En0.4 0.2 0.0 btc eth xrp usdc doge ada btc eth xrp usdc doge ada 2012 (a) Entropy variations for Set1 (b) Weighted entropy variations for Set1 during that period. After 2020, ETH stabilizes and becomes the most consistent asset across the entire set, maintaining values steadily above 0.6 – a sign of structural resilience. BTC, on the other hand, undergoes a temporary recovery between 2022 and 2023, followed by another sharp decline. The dynamics of the entropy values are generally well pronounced (0.25–0.9), with a lot of sharp peaks and bottoms, highlighting that the index is capable of capturing key aspects associated with variations in the economic parameters.
Set2, built on dynamic, interaction-based features such as transaction size, turnover rate, and wealth distribution, reveals higher and less dispersed values, with only the period of time from 2023 to mid 2024 characterized by some steep declines in value.
Ethereum consistently exhibits the highest entropy, reflecting a well-balanced and resilient ecosystem, especially during market stress. As represented in Figure 2, in Set1 a drop in DQ3 ofset by rising Market Cap preserved balance, while in Set2 opposing shifts in DQ5 and DQ6 had a compensatory efect.
It is worth noting that some assets can perform diferently when using diferent sets of parameters. For example XRP consistently ranks as the least eficient asset in Set2, with values averaging near 0.6 for the majority of the period, reflecting poorer distribution and transactional engagement. On the contrary, it can be considered average compared to the others in Set1. This underlines again the importance of carefully selecting parameters based on the characteristics of the cryptocurrency one aims to highlight.
While some global events – such as market crashes, regulatory changes, or liquidity shocks – may coincide with entropy shifts, their causal impact on the distributional structure of the parameters is not addressed in this work. Our primary goal remains the introduction of a well-founded economic index, flexible in both composition and number of parameters, capable of expressing the balanced integration of economic qualities as an indicator of systemic eficiency – irrespective of exogenous causes. It remains the responsibility of the researcher to choose the set of parameters and economic qualities that best suit her/his research.
The proposed EB-index assumes equal relevance for all selected economic qualities. However, this approach has a limitation: it rewards only the good balance among qualities, regardless of their absolute value. As an example, consider two cryptocurrencies: the first one with all normalized qualities reaching the maximum value = 1,∀; the second one with = 0.5,∀. From (4) it is clear that both the two corresponding p.d.’s are uniform and correspond to the maximum value of normalized entropy, i.e., ( ) = 1 with the base of the logarithm. But the first cryptocurrency has all absolute values greater than that of the second one. To overcome this limitation we can use the weighted entropy (9) and set = , so each quality contributes proportionally to its normalized absolute value. In this way the entropy metric becomes sensitive not only to a good balance of the qualities, but also to the absolute performance of the system, thus better capturing the economic eficiency of the evaluated blockchain protocol. Note, however, that when we use the term "absolute", we are always comparing the value of the given quality to its own highest value over the historical series, within the context of the same cryptocurrency.
In Figure 1, the unweighted entropy calculation is shown alongside its weighted counterpart with = for each quality . This highlights a significant shift in the interpretation of the results, particularly for Set2, where interestingly the USDC stablecoin emerges as the most balanced and highly entropic representation. Moreover, it is worth noting that the entropy score drops significantly below 0.3, indicating that a full balance among the selected economic features has not yet been achieved within the sets.
Another possibility ofered by the weighted entropy is that of increasing the importance of some qualities to the detriment of others, by assigning = , where is the utility (upper bound) of the corresponding quality. Note that in any case we can consider the last two weighting principles in a joint manner by setting = · , without losing the formal properties satisfied by the Shannon entropy.
The EB-index adopted in this study ofers a novel perspective on assessing economic eficiency in cryptoassets. The key idea is the introduction of Shannon entropy as a global metric, capable of expressing good performance and balance among several economic parameters of a cryptocurrency. This is accomplished in a rigorous axiomatic framework, which ensures the coherent behavior of the index when changing the number of parameters and/or the weights associated with them, without compromising the theoretical foundation. In other words, our entropy-based approach ofers a way to evaluate the harmonious development of the analyzed cryptocurrency, in the sense of a global, healthy, and well-balanced improvement of all of its parameters.
By analyzing multiple sets of attributes over time, one could identify how structural and valuationdriven components influence the internal systemic behavior of these assets. Future work may explore 2012 2014 2016 2018
ETH - Set1 causal methods to better assess the link between external shocks and systemic dynamics.
The quality and completeness of the analysis are directly influenced by the availability and granularity of on-chain data. Similarly, the Gini-based Wealth Distribution (DQ6) had to be computed by using balance class intervals between 0.001 units of the native currency up to 100,000, as this range best matched the practical constraints of the available dataset.
The framework currently assumes equal relevance for all selected attributes, which may not align with their actual economic significance. In this regard, we have proposed the adoption of the weighted entropy formulation by Beliş and Guiaşu [64], to better reflect the diferent importance of each parameter by attributing a specific weight to it, without losing the axiomatic coherence of the index. Future work will explore weighted formulations based on domain-specific criteria, data-driven methods, or expert input. The proposed methodology remains flexible and adaptable to various research objectives and data analysis strategies.
The EB-index flexibly assesses structural balance by employing user-defined parameters. We do not seek an optimal metric set, as eficiency is context-dependent. Changes in the index resulting from adding or removing metrics are context-specific and not generalizable. A sensitivity analysis, which is beyond the scope of this work, is left for future investigation.
Another consideration involves the treatment of certain on-chain metrics. For example, Active Addresses Sent aggregates various activities that result in a ledger change, not exclusively economic transfers. While this metric is broadly representative of network usage, it may introduce noise in contexts where distinguishing between economic and non-economic activity is crucial.
We recall that the study datasets are available at https://doi.org/10.5281/zenodo.15221823
This research has been supported by the PRIN 2020 project NiRvAna – Noninterference and Reversibility Analysis in Private Blockchains. The scholarship of the first author at the Italian PhD Program in Blockchain and Distributed Ledger Technology is funded by PNRR – Piano Nazionale di Ripresa e Resilienza according to D.M. 351/2022. We also wish to thank the Coin Metrics team for providing access to their data and their kind support.
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