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Egocentraities and the Ethereum Transaction Network

CS Edwards

University of Hawaii at Manoa
cedward2@hawaii.edu

Draft version 1: 5/14/2022

© The Author 2022


Keywords: Blockchain ᐧ Ethereum ᐧ Network Data Analysis ᐧ Egocentralities ᐧ  Graph Algorithms  ᐧ  Complex Networks

Introduction:
With a market cap of over $230B (May 2022), Ethereum is the world’s largest public blockchain with smart contract capability. This exploratory paper discusses foundational node-based centrality metrics and performs network analysis on a sample of the Ethereum Network. Centrality metrics are applied to the sample network to identify influential contract addresses and their owner organizations within the Ethereum ecosystem.

Background:
Blockchain is a distributed ledger of transactions whose state is shared among all members of the network. Each time a new transaction takes place they are added to a block. The block is then validated using a consensus protocol and then added to the most recent valid block, forming a chain. Through the consensus protocol, blockchain allows for a “trustless” system (Peyrott, 2017), where participants do not need to trust each other or a third party to validate transactions. The most prominent blockchain implementation is Bitcoin. Bitcoin is a peer-to-peer transaction system that only supports the buying and selling of the cryptocurrency with the same name.

Ethereum is a Turing-complete blockchain that provides the ability to not only facilitate transactions of cryptocurrency, but can also, more notably, execute smart contracts. As such, Etherum serves as a platform for digital money and decentralized applications. Ethereum can perform any computation as part of a transaction (Ethereum).

Ether is the native cryptocurrency of the Ethereum blockchain. In order to execute a transaction, Ether must be spent. Executing computations on the blockchain is computationally expensive, thus in order to limit computations there is a cost associated with them. The amount of Ether associated with a particular transaction is called gas. If a transaction runs out of gas before it is completed, it is ended. This is designed to prevent any code from running on an infinite loop on the network (Peyrott, 2017).

There are two types of account addresses on the Ethereum network. The first is called an Externally Owned Account (EOA), this account is controlled by a (human) user. The second is a contract account. A "smart contract" is simply a program that runs on the Ethereum blockchain”(Ethereum). Contracts  “ are autonomous computer programs that, once started, execute automatically and mandatorily according to the program logic defined beforehand” (Chen,Ting,et al., 2018). Within the Ethereum network there are three groups of transactions: user-to-user transactions, which is a transaction between EOAs, user-to- smart contract/smart contract-to-user, a transaction between an EOA and contract address, and finally smart contract-to-smart contract, a transaction between contract addresses.

 

The structure of the blockchain network naturally lends itself to graph analysis (Cuneyt et al. 2017). In a blockchain network, addresses serve as nodes and transactions serve as the edges. Both static (Tikhomirov et al., 2018) and temporal analysis (Zhao et al.,2021) of the Ethereum blockchain, use graph theoretic measures to provide a foundational understanding of Ethereum community structure (Said et al., 2021) and network activity (Anoaica and Hugo, 2018; Serena et al.,2022). Applications of centrality metrics on the Ethereum network (Said et al., 2021; Sui et al., 2019; Lee et al., 2020) reveal influential organizations and contract owners within the ecosystem.

Data and Methods:
The data for this project is a sample of the Ethereum Transaction Network. Data was extracted from the crypto_ethereum.transactions dataset under the Google Cloud bigquery-public-data. The data represented in this sample is sourced from ten consecutive blocks: #14375070 to #14375079, inclusive. All transaction data occurred on March 13, 2022, an arbitrary date chosen for this sample. Sample date macro market data for Ethereum can be found in Table 1.

From the queried data, we construct an application layer content graph to reflect the transfer of assets on the Ethereum network, during the specified time frame.Addresses accounts are represented by graph nodes and transactions between the accounts as edges. In the construction of the content graph parallel edges were summed and serve as edge weights. Within the scope of this study edge weight intentionally represents the frequency of transactions between nodes. In other studies of blockchain and Ethereum, transaction value is used as the edge weight. The content graph is a directed, weighted,address graph. To construct the graph the queried data was converted from a CSV to a graphml file to allow for analysis in Gephi and R Studio.

In this paper the transaction graph is referred to as ‘the sample’ , ‘the sample network’ or the like. The following graph metrics and measures were used to analyze the sample:

All metric algorithms, with one exception, are defined and implemented by the igraph library package version 1.3.0. The centiserve library package version 1.0.0  was used for the implementation of the Katz centrality algorithm as igraph does not provide this function. All mathematical analysis was performed in R Studio with visualizations in Gephi.

After conducting centrality analysis, the site EthScan was used to examine network addresses and associate the addresses with organizations.

Network Structure Summary:

Degree distribution provides some insight into the structure of an overall network (Barabasi, 2016) . The Ethereum Network follows a power-law degree distribution (Guo et al., 2019; Liang et al., 2018; Wang et al.,2021), making it a scale-free network. In the sample data for this paper, we also observe an in-degree (Figure 1) and out-degree (Figure 2) degree distribution that fits the power law.

In directed networks we distinguish between incoming degree, kiin, representing the number of links that point to node i, and outgoing degree, kiout, representing the number of links that point from node i to other nodes (Barabasi 2016).

In the case of the Ethereum Network, kin  represents incoming transactions to the address represented by the node. An incoming transaction could be a transfer of cryptocurrency (ie. ETH) to an EOA address, or a call to smart contract at the address. Kout represents outgoing transactions. On the network an outgoing transaction could be the transfer of assets from an EOA address, or the result/output of a smart contract execution.

The sample is disassortative, reflected by the negative assortativity metric. This also matched the observed behavior of the Ethereum network at large (Liang et al.,2018;Zhao et al. 2021). Assortativty is the measure of the tendency for nodes to attach to like-nodes. For example in an associative network, a high degree node would likely share a link with another high degree node. In a disassortative network, nodes attach to dissimilar nodes. The disassortative nature of the Ethereum network reflects the tendency of high degree addresses (popular smart contracts) to share a link (transaction) with either marginally used contracts or user addresses. For example a smart contract belonging to a crypto exchange is likely to have many transactions with user addresses, instead of transactions with another exchange.

The sample network is supercritical as expected by <k> > 1. In the supercritical regime we observe a giant component co-existing with smaller clusters of nodes. This can be observed visually in Image 1 . This aligns with Barabasi’s (2016) observation that real-networks are supercritical. Sample network structure summary data in Table 2.

Ego Centralities:

Degree Centrality:
Degree centrality is a simple yet fundamental centrality. The degree of a node is the count of edges attached to it. Degree centrality values all edges equally. A high degree centrality score indicates that a node has a larger than average number of connections for that graph. Using an adjacency matrix degree, centrality is calculated using the equation:

(1)


Where degree,
xi, is the sum of all edges ij.

As noted previously, as a directed graph the Ethereum transaction network has both in-degree and out-degree measurements.To calculate in-degree using the degree centrality equation e( i→j), and out-degree e(j→i). High-degree nodes in blockchain networks are generally the most central based on betweenness, closeness, and PageRank metrics(Lee, X.T. et al., 2020). In Table 3  the top five Ethereum addresses from the sample are ranked in descending order based on their in-degree. The in-degree is understood to be the indicator of importance(Newman, 2010)  in most applications of centrality, as indegree represents the count of references to a specific node.

Eigenvector Centrality:
Eigenvector centrality, also called eigencentrality,  builds on degree centrality by ranking the importance of a node based on their neighbor count. However, in eigenvector centrality not all node neighbors are considered equally important. Eigenvector extends  degree centrality and awards a number of points proportional to the centrality score of their neighbors:

 (2)

Where K-1 is the constant of proportionality and xj is the centrality score of the node neighbor of i.

In eigenvector centrality a node can achieve a high score by being connected to a few high scoring neighbors, or being connected to numerous moderately scoring neighbors (Newman, 2010). With this in mind, it is possible for a node to achieve a high score in degree centrality based on their neighbor count, but a low score in eigenvector centrality if many of their neighbors are low scoring nodes.

In Anoaica and Levard’s (2018) study of internal activity on the Ethereum Network, degree centrality and eigen centrality were used to identify major actors or activity monopolies within the network. The study illustrates a varied ranking of node importance when comparing centrality metrics:

Among the 21 unique addresses identified as most central, none of them belong to the 20 most central addresses in terms of eigenvector centrality. Because the eigenvector centrality awards higher score to nodes connected to other nodes showing a high connectivity, it can be concluded that the most central nodes, from this perspective, are individuals that interact often with major actors, rather than the latter interacting with themselves. (p.6)

Considering the disassortative nature of the Ethereum network as observed in the sample and in other studies, it makes sense that high degree nodes have low eigenvector centrality since  high degree nodes are expected to be connected to low degree nodes.

Calculating the eigenvector centrality is an iterative process, where at t0 each node is assigned a score of 1. At the next time step, t1, each node score is updated to be the sum of the scores of their neighbors. Note at t1, the scores of the nodes are simply their degree. The process continues recursively where at each time step the node score is updated based on the sum of the scores of their neighbors until the values stabilize. These normalized values proved the eigenvector centrality score for each node.

The process described above is used to calculate the eigen centrality in an undirected, strongly connected network. In a  weakly connected, directed network, such as the sample Ethereum transaction network, the eigenvector centrality encounters some complications. First, in the sample network the adjacency matrix is asymmetric, therefore the network has two sets of eigenvectors; a left eigenvector and a right eigenvector(Newman 2010). Eigenvector centrality in a directed graph is defined as:

 (3)

It is worth noting that there are varying interpretations of which eigenvector is appropriate for a directed network. It is generally agreed upon that the in-degree eigencentrality confers the importance of a node. However, in some texts and documentation the eigenvector capturing the in-degree is the right leading eigenvector (i.e. Newman), while in other texts (i.e. Networkx and Anoaica & Levard, 2018) the left eigenvector is denoted as capturing the in-degree eigencentrality. The discrepancy in approach could be rooted in the interpretation of directed links in the adjacency matrix where one text interprets Aij as i as the source and j as the destination (i→j) and another interprets Aij as j as the source and i  as the destination (j→i). For successful interpretation and analysis either can be used as long as there is consistency in the directed interpretation.

In addition to the left vs right eigenvector, there is another issue to consider when using eigenvector centrality on a directed graph. It is possible for a node to have an eigenvector centrality of zero if their inbound neighbors have a centrality score of zero. Newman uses Figure 3 as a reference and describes the scenario as follows:

Figure 3.         

Node A in this network has only outgoing edges and hence will have eigenvector centrality zero. Node B has outgoing edges one ingoing edge, but the ingoing edge originates at A hence node B will also have a centrality of zero.

In the Newman figure above, following the iterative process of the eigenvector calculation, the zero centralities of A and B will cascade throughout the nodes shown; eventually causing all of (except one, see Figure 3a) the nodes to have a centrality of zero. This is not very useful.

Figure 3a.

Annotated figure 3a. Depending on the centrality of node x, node C might be the only node in the figure with a non-zero centrality.

Katz Centrality and PageRank are variants of eigenvector centrality that address the aforementioned issue.

In Table 4  the Ethereum addresses from the sample are ranked in descending order based on their undirected eigencentrality. While Ethereum is a directed graph, the directed eigencentrality does not provide useful information; for the sample transaction network sample the directed eigencentrality is zero which illustrates the aforementioned issue. The undirected eigencentrality provides a non-zero metric that can be used to determine centrality.

Katz Centrality:

The Katz centrality approach mitigates the ‘power leak’ (Beveridge, 2020) or zero centrality, issue that arises from using eigenvector centrality on a directed network. With Katz centrality each node in the network receives a small amount of centrality ‘for free’(Newman, 2010), while attenuating the influence of indirect neighbors (Katz,1953) . In the equation below the eigencentrality equation is modified to include positive constants alpha and beta:

(4)

Beta is the ‘free’ constant positive value given to each node. Katz centrality uses β  to solve the zero centrality issue. As Newman(2010) explains:

By  adding the second term, we ensure that even nodes with zero in-degree still get centrality β, and once they have none-zero centrality they can pass it along to the other nodes they point to. This means that any node that is pointed to by many other nodes will have a high centrality.(p.164)


The purpose of the alpha constant is to discount the impact of the centralities of increasingly distant neighbors of
i; by a factor  𝛼>0 (Avella-Medina et al., 2020). Avella-Medina et al.(2020) summarize the impact of 𝛼 on the measure:

If α is close to zero, the relative weight given to neighbors further away decreases fast, and [xi ] is driven mainly by the one-hop neighbors just like degree centrality…In contrast, if α is close to 1/λ1 is almost a scaled version of [eigenvector centrality]. (p.3)


The suggested value of the alpha constant in Katz centrality is 0 < α < 1/λ1 (Avella-Medina et al., 2020; Newman 2010). Although Katz centrality provides a solution to the issues of eigen centrality and is said to be a more useful centrality metric on directed graphs (Newman 2010), some of the articles referenced in this paper use eigenvector centrality coupled with degree centrality in their analysis of the Ethereum network, with no mention of Katz centrality explicitly.

In Table 4 the Ethereum addresses from the sample are ranked in descending order based on their Katz centrality. For the sample, the Katz centrality alpha and beta parameters are α = 0.1, and β= 1.0; these are the default parameter values set by Jalili, M. (2017) in the design of the Katz centrality function. Alpha can be adjusted to analyze the attenuation impact on the centrality measure; holding the beta parameter constant at  β= 1.0,  Zhan et al. (2017) test the impact of alpha parameters on their network dataset within the range of  0 < α < 0.148.

PageRank:
PageRank is a centrality measure that modifies the Katz centrality approach. In Katz centrality, the node centrality is passed along to each out-degree neighbor which causes an artificial inflation of the neighbor node’s centrality. This is a motivating issue solved by PageRank. In the equation below, the Katz centrality is modified to divide the neighbors centrality by their out-degree:

(5)

As Newman(2010)  states:

…the added ingredient of dividing by the out-degree of pages ensures that a page that simply points to an enormous number of others does not pass much centrality to any of them. (p.166)

The modification ensures that network hubs do not have a disproportionate influence on the centrality metric of the nodes that it points to.

In Table 4  the Ethereum addresses from the sample are ranked in descending order based on their PageRank. Note,the five addresses from the table are ranked equivalently by Katz centrality and PageRank, however beyond the selected addresses the centrality measures diverge in their ranking.  

Kleinberg Centrality:
Thus far, the centralities that handle directed networks have emphasized in-degree as a determinant of node importance. In Kleinberg centrality, the out-degree of a node is also considered to be a measure of importance particularly in the case of identifying network hubs. A hub is described as a node that points to authorities. An authority is described as a node that ‘contains useful information on a topic of interest’ (Newman 2010). In other words, an authority is referenced by hubs and other important nodes. Klienberg defines hub centrality as:

where alpha is a constant. And authority centrality as:

where beta is a constant. In this centrality measure, nodes that are not referenced by any other node have an authority score of zero. The hub and authority scores for each of the selected addresses of the Ethereum sample are listed in Table 5 .  Newman (2010) notes that Kleinberg centrality as a measure is not widely used in practice (p.170). Sui et al. (2019) use PageRank, instead of Klienberg centrality, to identify hubs and authorities in their analysis of the Ethereum transaction network.

Activity on Ethereum Network:

According to Anoaica and Levard’s (2018) analysis of the internal activity of the Ethereum network, the majority of network transactions (64.6%) are user-to-user transactions between EOAs. The remaining transactions are transactions interacting with smart contracts. Ethereum data analysis by Said et al.(2021) found six smart contract addresses were responsible for approximately 30% of transaction activity. Across the literature, high transaction addresses, identified by centrality metrics, are associated with cryptocurrency exchanges, token distribution, mining and services (i.e. name service registrars, notaries, etc.) (Anoaica and Hugo, 2018; Said et al., 2021).

This section discusses selected prominent contracts on the Ethereum network as observed by external studies and the sample transaction network.

ENS-Registrar: Ethereum Name Service Registrar[1]
According to ENS Documentation, the purpose of an ENS registrar is to map human readable names to machine readable identifiers such as Ethereum addresses and other crypto-addresses. The ENS registrar operates similarly to the Domain Name System (DNS) used to resolve urls to IP addresses. The contract associated with the ENS-Registrar is used to process naming requests and assign ownership to requesting addresses.


YoCoin: Cryptocurrency[2]
YoCoin (SYM: YOCO)  is the native token of the yoco.finance crypto project built on the Binance Smart Chain. The contract associated with YOCO is used to create and allocate the tokens. According to YoCoin promotional material, the token is a ‘safe’ ,’long-term’, crypto investment that differentiates itself from other tokens with low trading fees and automatic rewards for users. YoCoin market data is displayed in
Table 6.


Bittrex_2: Cryptocurrency Exchange[3]
Bittrex_2 is a contract owned by the Bittrex Cryptocurrency Exchange. According to
CoinRanking, Bittrex is currently ranked 33rd among other crypto exchanges, with 327 listed currencies in 688 markets. As of this writing, Bittrex has 0.06% market share in the crypto exchange space, and a 24hour trading volume of $115.36M.

Acronis_Contract: Acronis Notary[4]

According to their documentation, Acronis is a blockchain based cyber notary cloud, offering e-signing and verification for their customers. Use cases for the Acronis Notary service include property records, court documents and long term archives for legal or tax audits. The notarization certificate ensures that data has not been altered, outside of an authorized update,  since the file was stored on the network.

Said et al.(2021) also identified Poloniex_1 and Kraken_5 as prominent smart contracts in their study. Both Polinex and Kraken are crypto exchanges.

The following organizations are associated with the ‘most central’ contract addresses in the sample network: Wyvern Exchange Protocol, Uniswap Protocol, Binance, USD Coin and Bitfinex.

The Wyvern protocol underpins the OpenSea digital asset exchange, where users can purchase and trade NFTs (non fungible tokens). The Uniswap Protocol is a decentralized crypto trading protocol that allows users to swap, earn and build tokens. Binance is a prominent crypto currency exchange. Both Tether (owned by  Bitfinex) and USD Coin are stable coins pegged against the US Dollar. Data summary available in Table 7.

The activity highlighted by centrality metrics on the sample network are similar to the results produced in aforementioned studies (Anoaica and Hugo, 2018, Said et al., 2021), where the most significant (transaction intensive) addresses in the blocks were associated with digital asset exchanges and token distribution contracts.

Conclusion and Further Analysis:
This study has explored the foundational degree-based network centrality metrics and applied them to a sample of the Ethereum Transaction Network. The analysis demonstrates the usefulness of centrality metrics in identifying key players in the Ethereum space. Further analysis could include a granular exploration of the internal (smart contract to smart contract) activity and a deep dive into the executed code contained in those contracts. Additionally, the impact of the alpha parameter on the Katz centrality metric could provide insight on how to fine tune the measure for optimal information.

Appendix: Tables, Figures and Images

Tables:

Table 1.

ETH Market Summary: March 13, 2022

Market Cap

301.89 B

Open:

2573.49

Close

2518.94

Delta

-2.11%

Volume

8.63 M

Transactions

1.042 M

Table 2.

Sample Network Structure Summary

<k>

1.763911

Mean Dist

1.40685

Degree Assortativity

-0.12515649

Component Count

457

Max Component Size

4064

Table 3.

Sample Network Top-K Addresses

ID

Address

A

0x7f268357a8c2552623316e2562d90e642bb538e5

B

0x68b3465833fb72a70ecdf485e0e4c7bd8665fc45

C

0xdac17f958d2ee523a2206206994597c13d831ec7

D

0x28c6c06298d514db089934071355e5743bf21d60

E

0xa0b86991c6218b36c1d19d4a2e9eb0ce3606eb48

Table 4.

Address Rankings by Centrality

Rank

In-Degree

Eigenvector

Katz

PageRank

1

A

B

A

A

2

B

A

B

B

3

C

D

D

D

4

D

E

E

E

5

E

C

C

C

Table 5.

Kleinberg Centrality

ID

Authority

Hub

A

0

0

B

0

0

C

0.968

0

D

0

0.009

E

0.071

0

Table 6.

YoCoin: Market Data

Fully Diluted Market Cap ($USD):

536927.67

Total Transactions:

9343

Total Transfers

56494

Holder Addresses

5396

Total Supply (YOCO)

94,730,320.15

Data as of 5/9/2022

Source: BSC Scan

token/0xdd17629d05e068a9d118ee35d11101d4140d0586

Table 7.

Sample Network Top-K Addresses and Associations

ID

Address

Association

A

0x7f268357a8c2552623316e2562d90e642bb538e5

Wyvern Exchange

B

0x68b3465833fb72a70ecdf485e0e4c7bd8665fc45

Uniswap

C

0xdac17f958d2ee523a2206206994597c13d831ec7

Tether via Bitfinex

D

0x28c6c06298d514db089934071355e5743bf21d60

Binance

E

0xa0b86991c6218b36c1d19d4a2e9eb0ce3606eb48

USD Coin

Figures:

Figure 1:Sample Ethereum In- Degree

Figure 2: Sample Ethereum Out Degree

Images:

Image 1: Sample Ethereum Network

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[1] Contract identified in Said et al.(2021)

[2] Contract identified in Said et al.(2021)

[3] Contract identified in Said et al.(2021)

[4] Contract identified in Said et al.(2021)