TLDR
 This paper presents the first systematic study of the interactions occurring in a number of NFT ecosystems.
 They illustrate how to retrieve transaction data available on the blockchain and structure it as a graphbased model.
 The structure of NFT networks is qualitatively very similar to the one measured for interactions in social networks.
Core Research Question
What kind of graph pattern will the interactions occurring across a number of NFT networks generate?
Citation
S. CasaleBrunet, P. Ribeca, P. Ribeca, M. Mattavelli “Networks of Ethereum NonFungible Tokens: A graphbased analysis of the ERC721 ecosystem”, 2021, IEEE
Background
 NonFungible token (NFT): A unique and noninterchangeable unit of data stored on a blockchain, a form of digital ledger.
 Ethereum: A decentralized, opensource blockchain with smart contract functionality.
 ETH (Ether): The native cryptocurrency of the Ethereum blockchain.
 WETH (Wrapped ETH): An ERC20 compatible version of ether. 1 ETH = 1 WETH.
 ERC20 (ERC20): A token standard on Ethereum whose tokens are fungible.
 ERC721 (ERC721): A token standard on Ethereum whose tokens are nonfungible.
 Externally owned account (EoA): An account controlled by a private key.
 Contract Account (CA): An account controlled by contract code.
 CryptoPunks: A nonfungible token collection on the Ethereum blockchain by the Larva Labs studio.
 CryptoKitties: A blockchain game on Ethereum developed by Dapper Labs that allows players to purchase, collect, breed and sell virtual cats.
 HashMasks: Virtual artworks created by a globallydistributed team of 70 artists managed by Suum Cuique Labs.
 Meebits: 20,000 unique 3D voxel characters, created by a custom generative algorithm.
 Bored Ape Yacht Clubs: A collection of 10,000 unique Bored Ape NFTs.
 Moon Cats: An Ethereumbased NFT Collectible launched back in 2017, rescued this year by catloving NFT Holders.
 CryptoVoxels: A virtual world and metaverse, powered by the Ethereum blockchain.
 Decentraland: An opensource 3D virtual world platform.
 ArtBlocks: Ethereumbased platform providing unique, programmable, and ondemand generative NFT art to collectors worldwide.
 Unspent Transaction Output (UTXO): The amount of digital currency someone has left remaining after executing a cryptocurrency transaction such as bitcoin.
 CDF: A function that gives the probability that a random variable is less than or equal to the independent variable of the function.
 KolmogorovSmirnov goodnessoffit test: Compares your data with a known distribution and lets you know if they have the same distribution.
 Power Laws: Distributions of the form , in which the dependent variable, the probability for a node to have degree 𝑘, varies as an inverse power of the independent variable, the degree 𝑥. In other works, P(x) decreases monotonically, but significantly slower than the exponential decay of normal distributions.
 Assortativity: The graph is said to be strongly assortiative, weakly assortative, neutral, weakly disassortative, and strongly disassortative, if the assortativity coefficient falls into the ranges [0.6, 1], [0.2, 0.6], [0.2, 0.2], [0.6, 0.2], and [1, 0.6], respectively.
Summary
 The paper presents the state of the art in the analysis of graphs related to web networks, social networks, and Ethereum ERC20 (fungible) tokens.
 It describes which NFT collections have been analyzed for this work and how the data has been collated; crucially, it defines the structure of an NFT transaction graph.
 It illustrates the topological and clustering analysis of such transaction graphs.
 It presents methodologies for estimating the value of a collection and how to identify flows between major wallets and the most successful investors.
Method

The paper selected 8 different NFT projects of significance and collected data from their ecosystems such as the number of wallets, transactions, and volume.

The authors summarize measurements for each transaction graph and compare them with social network graphs.

They initially structured their data into a multidirected weighted graph, with nodes being wallets and edges being transactions.

They evaluated and discussed the properties such as distributions of degree, density, component clustering coefficients, and assortativity.

In and outdegree of nodes are considered as fundamental properties of a directed graph because they indicate what the ratio between purchases (i.e., accumulation) and sales for the various wallets that make up the network are.

They analyze the frequency distribution of degrees in transaction graphs to get insight into user behavior when trading a particular collection of NFTs.

The contemporary cumulative distribution function (CCDF) is applied for nodes, and the level of homophily of the graph is measured by the assortativity coefficient.

The number of daily transactions together with the daily volume traded during hype cycles is observed.

They identify for a given collection which wallets have had the largest amount of tokens over time since the minting date and follow the flow of sales and purchases.
Results
 Transaction graph analysis.

The following table summarizes measurements for each transaction graph (in terms of the number of nodes and edges, diameter, and mean distance) when each NFT project is considered separately.

To get a visual idea of the structure, they make graphs where the size of the nodes are directly proportional to their input degree (i.e., the larger a node, the more tokens it has accumulated from a given collection) and the edges are colored according to the project the transferred token belongs to.


Real, separated wallet communities do exist according to each project, but also those wallets trading on more than one project are frequent.

They also analyze social network graphs that show the estimated powerlaw coefficient, diameter, and mean distance of web, social networks, and ERC20 token networks.

Interestingly, the diameter (i.e. the length of the longest path, measured as the number of edges) and the mean distance (i.e. the average number of edges between any two nodes in the network) are very similar to those identified from the analysis of social network graphs.

They analyze the frequency distribution of in and out degrees of nodes in transaction graphs.
 Many realworld graphs for social media and the Internet show highly skewed, heavytailed degree distributions, generally indicating that a significant portion of the information about node interactions needs to be extrapolated from the analysis of their trails and demonstrating the existence of (several) high degree hubs.
 Ethereum ERC20 token networks hubs are exchanges, in social networks they are influencer nodes, and in web networks, they are popular and highranked websites.
 Several kinds of networks have been confirmed to follow power laws in their degree distribution.

The following table shows the complementary cumulative distribution function (CCDF) of in and outdegree, and the cumulative distribution function (CDF) of the indegree to outdegree ratio for each measured NFT network.

The following table shows the estimated powerlaw coefficient for each NFT network along with the KolmogorovSmirnov goodnessoffit metric.

Figures show that all the powerlaw coefficients for the separate networks are similar, but the one for the combined network is different due to the different proportion of hubs in the separate and combined network.

Studies on the distribution of in and outdegree in web networks have consistently helped with the identification of better methods to find relevant information on the web. NFT transaction networks have a large number of active nodes with a low in to outdegree ratio, (i.e., nodes selling NFTs), and a much more limited number of nodes with high ratios(i.e., nodes accumulating NFTs). We can then draw a natural parallel between web influencers and accumulators of NFTs.

Network graph connectivity and clustering statistics for each NFT collection in terms of reciprocity, transitivity, and assortativity.
 The reciprocity coefficient measures the proportion of mutual connections on a directed graph. From the table, it’s possible to see that reciprocity levels are close to zero, indicating in fact that buy/sell trades are only made in one direction, which is also the case when considering the global graph. In other words, exchanges between wallets are still uncommon.
 The transitivity coefficient (also known as the clustering coefficient) measures the probability for adjacent nodes of a network to be connected. From the graphs, they all tend towards 0, indicating that collaboration between wallets is very rare.
 Assortativity coefficient measures the level of homophily of the graph (i.e., how nodes are connected with respect to a given property) and its value range is [1, 1]. The table illustrates the assortativity for NFT transaction graphs. It shows that most of them are neutral associative, with the exception of Acclimated Moon Cats that are strongly assortative, and ArtBlocks and CryptoPunks that are weakly assortative.

The following figure shows the CCDF of both Pagerank and coreness values for the NFT transaction network graphs considered.

The distributions of Pagerank and coreness show that the presence of nodes is more important than others and that high coreness nodes are not always influencers, although it is highly likely that influencer nodes are high coreness nodes.

The following figures show the widespread presence across a number of NFT networks of both investors accumulating NFTs and individuals who make large profits.

Looking at the example of the Bored Ape Yacht Club NFTs, which have the highest exchange probability, they are also characterized by a substantial increase in the value of the collection in a very short time frame.

Average wallet values show the widespread presence across a number of NFT networks of both investors accumulating NFTs and individuals who make large profits.

As for typical successful projects, both the number of unique wallets holding at least one token and the overall value of the network grows with time.

The number of daily transactions and the daily volume traded help us understand how interesting a collection is.

The earlier a purchase is made (i.e., during the mint or just after), the greater the probability of making a profit by reselling a token.

A large proportion of the tokens in each collection were bought at minting time and have not been sold since. This indicates how much NFT technology is in its infancy. NFTs are seen as a storeofvalue investment, both to be resold only at some future time.

NFTs are typically accumulated in the same wallet (e.g., to be displayed in online galleries and associated with a social media profile) because NFTs are assets generally made for display and have very high entry prices.

HashMask and Meebits have a very similar trend, where the first 3 wallets have consolidated their positioning during the time; on the other hand, for the Bored Ape Yacht Club, the top wallet has, after an initial dormant phase, put all its tokens on the market, making the value of the whole collection increase as a consequence.
Discussion and Key Takeaways
 Distribution: It was shown that analysis graphs follow a powerlaw in their nodes degree distribution.
 Similarity: Topological values such as graph diameter and mean distance are similar to state of art results for graphs derived for web networks and social networks.
Implications and Followups
 This paper presents a systematic methodology for the analysis of NFT communities, analysis of such communities over time, and identification of hub nodes (i.e., wallets) and supernodes (i.e., wallets across different NFT collections) that influence the market.
 Further research may refine our understanding of wallets holding many NFTs. In this work, they were able to identify them, but further investigation is needed to identify how their strategies (in terms of accumulation and subsequent selling) on one or more collections affect the markets and the overall value of the different NFTs.
 The analysis might be refined by considering the total costs of each transaction (e.g., split payments when the transaction goes through a smart contract and Ethereum gas fees).
Applicability
 By filtering information publicly available on the blockchain, and structuring it in a graph model, one can perform systematic analysis and define metrics that help establish how the wallets participating in this ecosystem interact.
 This study can help shed quantitative light on a market that might otherwise be prone to hype and misleading information.