Exploratory graph analysis of the network data of the Ethereum blockchain

Related works

Small-world models were proposed by Watts et al.⁴ These creators were interested in the way that numerous networks in the genuine work show significant levels of clustering, yet with the little distance between most vertices. They recommended rather starting with a network graph with a cross-section structure, and afterward arbitrarily 'reworking' a little the level of the edges. Assuming we have many N vertices that are instigated on an intermittent style, each of the vertex joins its neighbors to their respective side.⁴

Barabási et al. proposed a preferential attachment model for modeling networks.⁵ The more associated a vertex is, the more probable it is to get new connections. A vertex with a further extent has a more grounded capacity to get joins added to the network.⁶

Exponential random graph models (ERGMs) have relations to generalized linear models. However, the appropriate specification and fitting of exponential random graph models can be more unobtrusive than that of a standard generalized linear model. Furthermore, a significant part of the standard inferential foundation accessible for generalized linear models, based upon asymptotic approximations to fitting chi-square distributions, is not in the consideration for exponential random graph models.⁷

Stochastic block models (SBM) are a class of models in the statistical data analysis of the network or graph data, and they can be utilized to find or comprehend the structure and the latent of a network graph for clustering. This model will in general deliver a graph containing networks, subsets described by being associated with edge densities.⁷

Methodology

Dataset

The dataset of transaction records of Ethereum was taken from the BigQuery public dataset (Ethereum in BigQuery).⁸ The attributes from_address and to_address will be used as nodes (vertices) and the relationship between them will be the edges of the nodes in this study. The attributes and their descriptions are shown in Table 1. In this exploratory study, only arbitrarily selected 1000 contiguous rows of transactions in July 2019 are considered.¹⁵

Software for analysis

The analysis is performed in R version 4.0.2, utilising the igraph R package for network analysis. The source code used can be found at Zenodo.¹⁶

Network graph

A graph $G=\left(V,E\right)$ is a set of $E$ of links (edges), a set $V$ of nodes (vertices), and where components of $E$ are unordered pairs $\left\{u,v\right\}$ of distinct vertices $u,v\in V$ . The number of vertices $N_v=\left|V\right|$and the number of edges$N_e=\left|V\right|$ are also called the order and size of the graph $G$, respectively.⁷

Visualizing the network

To visualize a large network having over 1000 vertex and edge attributes, few methods exist. Kamada, et al. proposed a technique to draw general graphs.⁹ This technique tends to be broadly utilized in the network structures that are managed by the system. The mathematical displacement between the node in the drawing can be identified by graph-theoretic displacement. The spring algorithm proposed by Kamada, et al. has good properties such as symmetric drawings, edge crossing with a relatively small number of edges, and isomorphic graphs with almost congruent drawings.⁹

The DrL method provides two-dimensional visualizations of exceptionally huge theoretical chart structures.¹⁰ The graph is drawn employing a force-directed algorithm based on recreated strengthening. This clustering is utilized to create a coarsened graph (less vertex) which is at that point redrawn. This handle is rehashed until an adequately small graph is produced.¹⁰

Charactering network cohesion

One way to deal with characterizing the network cohesion of a specific network graph is through the identification of the subgraph of interest. The standard case is that of finished subgraphs and henceforth are subgroups of vertices that are completely strong, as in all vertices inside the subgroup are associated by edges. Large groups of subgraphs are considered rare in practice because they need the required network graph itself to be dense.⁷

$\mathrm{den}\left(H\right)=\frac{\left|E_H\right|}{\left|V_H\right|\left(\left|V_H\right|-1\right)∕2}$ is the general formula of the density of a subgraph $H=\left(V_H,E_H\right)$, that the recurrence of acknowledged edges is comparative with possible edges, in an undirected graph G which has no self-loops and various edges. In the situation that if $G$ is a directed graph, $\left|V_H\right|\left(\left|V_H\right|-1\right)$ will be replaced with the denominator of the equation above. The value of zero and one in $\mathrm{den}\left(H\right)$ is to justify how near the subgraph $H$ is to be becoming a clique in the network.⁷

In graph clustering, the relative frequency also can be one of the methods to defining a cluster. The standard use of the term clustering coefficient normally means the quantity${\mathrm{cl}}_T\left(G\right)=\frac{3{\tau }_{\Delta }\left(G\right)}{{\tau }_3\left(G\right)}$ , where ${\tau }_{\Delta }\left(G\right)$ is the number of triangles in the graph $G$, and ${\tau }_3\left(G\right)$ is the number of connected triples. The value of ${\mathrm{cl}}_T\left(G\right)$ is the transitivity of the graph and is also referred to as the fraction of transitive triples A graph $G$ is fully connected if each node is reachable from all other nodes, and that a graph’s connected component is a maximally connected subgraph.⁷

Hierarchical clustering

Various methods have been proposed for clustering, varying in essentially by the way they assess the nature of the proposed clustering and quality of the enhancement brought about by the algorithms.¹¹ These strategies adopt a greedy strategy to looking through the states of all potential partitions C, through altering progressive applicant partitions iteratively.

One of the most popular measures is modularity.¹⁰ Let $C=\left\{C_1,\dots ,C_k\right\}$ as a given candidate partition and define $f_{\mathit{ij}}=f_{\mathit{ij}}\left(C\right)$ to be the fraction of edges in the original network that connect vertices in $C_1$ with vertices in $C_j$. The modularity of $C$ is the value $mod\left(C\right)={\sum }_{k=1}^k{\left[f_{\mathit{kk}}\left(C\right)-f_{\mathit{kk}}^{\ast }\right]}^2$, where $f_{\mathit{kk}}^{\ast }$ is the expected value of $f_{\mathit{kk}}$ under some model of random edge assignment.¹⁰

Classical random graph modelling

Graph modelling refers to a collection $\left\{{\mathrm{\mathbb{P}}}_{\theta }\left(G\right),G\in g:\theta \in \omega \right\}$, where $g$ is a collection of possible graphs, ${\mathrm{\mathbb{P}}}_{\theta }$ is a probability distribution on $g$, and $\theta$ is a vector of parameters, ranging over possible values in $\omega$. A collection $g$ and a uniform probability $\mathrm{\mathbb{P}}\left(·\right)$ over $g$ is usually used to refer to a random graph model.⁷ Erdos et al. established a series of the seminal paper on the classical theory of random graph models.⁸ Erdos et al.’s model specifies a collection of $g{N}_v,N_e$ of all graph $G=\left(V,E\right)$ with $\left|V\right|=N_v$ and $\left|E\right|=N_e$, and assigns probability $\mathrm{\mathbb{P}}\left(G\right)={\left(\begin{array}{c}N\\ N_e\end{array}\right)}^{-1}$ to each $G\in {g\mathrm{N}}_v,N_e$, where $N=\left(\begin{array}{c}{N}_v\\ 2\end{array}\right)$ is the total number of distinct vertex pairs.⁸

Network block models

In a random graph, G, let $G=\left(V,E\right)$. The graph can be categorized into one of Q classes. Each element of the block model for a particular graph can be specified as each element of the adjacency matrix Y, labelled as q and r of vertices i and j, respectively, with a probability of Bernoulli random variable. Snijders, et al. introduced a parametric limit of the form.¹³

Underlying data

Zenodo: Extracted Ethereum transactions for July 2019 from the Ethereum in BigQuery public dataset. https://doi.org/10.5281/zenodo.5263166.¹⁵

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Extended data

Analysis code available from: https://github.com/tzenvun/exploratory-network-eth.

Archived analysis code as at time of publication: https://doi.org/10.5281/zenodo.5336085.¹⁶

License: GNU General Public License v3.0.