Modeling and Understanding Ethereum Transaction Records via A Complex Network Approach

Being the largest public blockchain-based platform supporting smart contracts, Ethereum [1] has attracted wide attention recently. To facilitate the implementation of smart contracts, Ethereum introduces the concept of account, which is formally an address. The corresponding cryptocurrency on Ethereum, known as Ether, can be transferred between accounts and used to compensate participant mining nodes. Current research about Ethereum mainly focus on security and performance issues of the blockchain technology [2], [3], [4], however, the interative relationship between users and smart contracts is relatively unexplored till now.

Complex network is a universal tool to analyze real work systems from various fields [5], [6], [7], and it has been employed to model and analyze the huge transaction networks of blockchain-based systems. In 2018, Chen et al. [8] conducted the first systematic study to characterize Ethereum and obtain new observations with various metrics. Alqassem et al. [9] found the anti-social properties of Bitcoin system via transaction network data analysis. In existing studies, transaction records are modeled as a simple graph, where multiple transactions between a pair of addresses are merged as a one-time transaction in the graph construction procedure.

Different from other large-scale complex networks, each edge in the Ethereum transaction network represents a particular Ether transaction, and thus contains some unique information such as the direction, amount value and timestamp of a particular transaction. It is essential to incorporate the aforementioned information for accurate modeling, characterization, and understanding of transaction network data. In addition, multiple transactions between two users are expected and it is more comprehensive to model a transaction network as a multidigraph rather than a simple graph. In graph theory, a multigraph (in contrast to a simple graph) is a graph which is permitted to have self-loops and multiple edges (also called parallel edges). A multidigraph is a directed multigraph. At present, there is little research on data mining of multidigraphs, and existing methods modeling the transaction records as simple graph are inadequate to characterize the temporal and multiple features of transaction networks.

Random walk mechanism has been widely demonstrated to be an effective technique to measure local similarity of networks for a variety of domains and tasks [10, 11, 12]. Besides, random walk has also become a popular sampling method for the problem of graph representation, which aims to represent node features of large networks in a low dimensional space for graph analysis and mining [13]. Most existing methods of graph representation are designed for mining social networks and few studies have considered the scenarios of transaction data mining as most financial transaction data is not public. However, traditional methods for graph representation in social networks cannot be directly applied to analyze financial transaction networks, as the existing methods [13] ignore the multiplicity and temporal attributes of edges in transaction networks.

In this brief, we conduct the first work to understand Ethereum transaction records via random-walk based graph representation learning. In particular, we first model the Ethereum transaction records as a Temporal Weighted Multidigraph, and propose novel graph sampling and representation methods considering the temporal and weighted information of the graph. After that, we conduct link prediction experiments on realistic Ethereum data to evaluate the effectiveness of the proposed graph modeling and mining methods.

In order to analyze the Ethereum transaction records comprehensively, we propose a general framework which includes four main procedures: (a) Data collection. We fetch transaction records of objective accounts from Ethereum through API of https://etherscan.io/ (A block explorer and analytics platform for Ethereum); (b) Network construction. Based on collected transaction data, we construct a Temporal Weighted Multidigraph to represent the relationship of Ether transfer between accounts; (c) Random-walk based graph representation. We employ the graph representation algorithm based on designed random walk strategies to learn richer node representations; (d) Link prediction for evaluation. We conduct experiments on temporal link prediction problem to evaluate the effectiveness of our framework on Ethereum transaction networks.

Accounts on Ethereum can be divided into two categories, i.e., external owned accounts (EOA) which are similar to general bank accounts [14], and smart contract accounts which are source code files. In this work, we focus on the transactions among EOAs for the reason that the Ether transfer records between them are publicly available on the blockchain. Besides, we only include the successful transactions with non-zero amount value into our dataset.

Since it is extremely time-consuming to process the whole Ethereum transaction network with more than two million EOAs [8], here we select a number of objective accounts and then obtain their transaction data through APIs of Etherscan. As shown in Fig. 1, centered by a particular objective account, we obtain a directed $K$-order subgraph. $K$-in and $K$-out are two parameters to control the depth of sampling inward and outward from the center, respectively.

On Ethereum, various related information of Ether transactions is stored as data packages. In details, the TxHash field is a unique identification of a transaction, the Value field in a transaction refers to the amount of money transferred, and the Timestamp field indicates when the transaction happens. Besides, the From and To fields denote the sender and recipient of the transaction. With the collected four-tuples $(From,To,Value,Timestamp)$, we can construct a temporal weighted multidigraph.

In this brief, we conduct the first work to understand Ethereum transaction records from a complex network perspective, and we will open our collected dataset from Ethereum for future studies.

Before network construction, we first discuss some realistic rules and features of transaction networks like the Ethereum: (1) Transaction networks evolve continuously over time with additions of links, which is overlooked in most of the existing random-walk based graph algorithms; (2) The practical meaning of connections between accounts is not a one-off established relationship but a time-dependent event. Hence multiple edges need to be considered in transaction network modeling; (3) Unlike social networks, random walks on Ethereum transaction network are concrete, which represent realistic money transfer flows; (4) The amount value of a transaction reflects the similarity between the two involved accounts to some extent. In most cases, the larger amount of transaction, the closer relationship between two accounts. Fig. 2 demonstrates a microcosm of transaction activities on Ethereum.

Ether transfer is one of the major activities happening on Ethereum. Here we abstract an Ether transfer transaction as a four-tuple (src, dst, w, t), which means the sender src transfers w Ether to the recipient dst at time t. To investigate the Ether transfer on Ethereum, we abstract the Ethereum transaction network as a Temporal Weighted Multidigraph:

Based on collected four-tuples from Ethereum transaction records, we can build a TWMDG where each node represents a unique account and each edge represents a unique Ether transfer transaction.

Random-walk based graph representation methods have been proved to be scalable and effective for large graphs. As realistic transaction networks usually have a huge network size, we consider the random walk based method for graph analysis in this area. Before utilizing graph representation algorithm, we investigate the degree distribution of Ethereum transaction network, and find that the distribution of nodes in both the entire graph and the subgraph of Ethereum follow power-law (Fig. 3 plots the distribution of a subgraph.). As discussed in a pioneering study [15], random-walk based sampling methods can well preserve structural properties of a graph with a power-law degree distribution. Inspired by this study, we employ graph representation method based on random walk to extract information from the Ethereum transaction network.

However, for the transaction networks discussed here, existing methods that ignore temporal information may sample a large number of invalid transaction sequences to derive node representations. For example, in Fig. 2, $\{A5,A1,A2\}$ is a possible random walk sequence in traditional methods DeepWalk [15] and node2vec [16], but it is not practical in a temporal graph as the transaction from $A1$ to $A2$ happens earlier. While in a recent work reported in [17], although temporal information is considered, the existence of multiple edges between points is neglected. For instance, the temporal walk from $A0$ to $A1$ is represented as a sequence of nodes $\{A0,A1\}$. However, whether $A2$ is possible for the next walk depends on whether the transaction path $\small{1}⃝$ or $\small{3}⃝$ is sampled by the previous walk from $A0$ to $A1$.

In this work, we represent an $l$-length temporal walk as a sequence of $l$ nodes together with a sequence of $(l-1)$ edges traversed in non-decreasing timestamps. This kind of temporal walk represents an actually feasible path for money flow in the transaction network. Therefore, the proposed method is expected to learn more meaningful and accurate time-dependent node representations that can capture more comprehensive properties from dynamic transaction networks. For a temporal weighted multidigraph discussed here, we define the concept of a Temporal Walk as follows:

In order to sample valid random walks obeying the temporal constraint, we introduce a new concept called Temporal Successive Edges in TWMDG.

For instance, in Fig. 2, let $t=T(e_{\scriptsize{\tiny{5}⃝}})$, then $L_{t}(A1)=\{e_{\scriptsize{\tiny{5}⃝}},e_{\scriptsize{\tiny{6}⃝}},e_{\scriptsize{\tiny{10}⃝}}\}$. The set of temporal successive edges plays the role of candidates for walkers to select possible successors.

Apart from the temporal constraint, we further develop biased searching strategies by considering more detailed transaction information. For the Ethereum transaction network discussed here, we abstract the transaction time and amount as the temporal and weighted information of a TWMDG. Consider a random walk that just traversed edge $e_{i-1}$, and is now stopping at node $v_{i}$ at time $t=T(e_{i-1})$. The next node $v_{i+1}$ of the random walk is decided by selecting a temporally valid edge $e_{i}$. We describe different sampling biases by formulating the selection probability for each temporal successive edge $e\in L_{t}(v_{i})$.

From the perspective of temporal domain, we consider both unbiased and biased sampling strategies as follows.

• •

  Temporal Unbiased Sampling (TUS). This is the default setting in the time domain, which assumes that each temporal successive edge $e\in L_{t}(v_{i})$ of node $v_{i}$ at time $t$ has the same probability to be selected:

  $$P_{T}(e)=\frac{1}{|L_{t}(v_{i})|}.$$

  (1)

• •

  Temporal Biased Sampling (TBS). For financial transaction networks, the similarity between accounts is time-dependent and dynamic.

  On the one hand, a pair of accounts with frequent interactions are supposed to have a stronger relationship. Therefore, we let $\eta_{-}:\mathbb{R}\rightarrow\mathbb{Z}^{+}$ be a function that maps the timestamps of edges to a descending ranking. In this case, each edge $e\in L_{t}(v_{i})$ will be assigned with a selection probability:

  $$P_{T}(e)=\frac{\eta_{-}(T(e))}{\sum_{e^{\prime}\in L_{t}(v_{i})}~{}\eta_{-}(T(e^{\prime}))}.$$

  (2)

  where $T(e)$ denotes the timestamp of the edge $e$. This sampling method biases the selection towards edges that are closer in time to the previous edge.

  On the other hand, sampling the interactions among accounts in a large time interval may also be important for different domains of networks for the purpose of preserving global similarity in time domain. For such scenarios, we propose another strategy that favors edges appearing later to the previous timestamp. Let $\eta_{+}:\mathbb{R}\rightarrow\mathbb{Z}^{+}$ be a function that maps the timestamps of edges to an ascending ranking. The probability of selecting each edge $e\in L_{t}(v_{i})$ can be given as:

  $$P_{T}(e)=\frac{\eta_{+}(T(e))}{\sum_{e^{\prime}\in L_{t}(v_{i})}~{}\eta_{+}(T(e^{\prime}))}.$$

  (3)

Apart from the transaction time, the amount values of the edges (edge weights) also plays an essential role in financial transaction networks. In the following, we present unbiased and biased strategies from a weighted domain.

• •

  Weighted Unbiased Sampling (WUS). Similar to TUS, this is the default setting in the amount domain and each edge $e\in L_{t}(v_{i})$ has the same probability to be sampled:

  $$P_{W}(e)=\frac{1}{|L_{t}(v_{i})|}.$$

  (4)

• •

  Weighted Biased Sampling (WBS). As illustrated in the Introduction, the weight value of each transaction indicates the significance of interactions between the two accounts involved. For most instances, a higher value of transaction amount implies a larger similarity between the two accounts. Thus each edge $e\in L_{t}(v_{i})$ can be assigned the selection probability:

  $$P_{W}(e)=\frac{W(e)}{\sum_{e’\in L_{t}(v_{i})}W(e^{\prime})}.$$

  (5)

  To prevent the extreme situation where edges with small weights would never be sampled, we consider a linear mapping function to weakens the effects of edge weights. Thus we have

  $$P_{W}(e)=\frac{\eta_{+}(W(e))}{\sum_{e^{\prime}\in L_{t}(v_{i})}\eta_{+}(W(e^{\prime}))}.$$

  (6)

Furthermore, we combine the aforementioned sampling probabilities from both temporal and weighted domains, i.e., $P_{T}$ and $P_{W}$, by $P(e)=P_{T}(e)^{\alpha}P_{W}(e)^{(1-\alpha)}(0\leq\alpha\leq 1)$ for $\forall e\in L_{t}(v_{i})$. Here $\alpha=0.5$ is the default value for balancing between time domain and amount domain. After the random walk generation with the temporal constraint and flexible biased strategies, the second part is an update procedure based on SkipGram [18, 19], which learns node representations as a maximum likelihood optimization problem.

In the previous section, we discuss different walking strategies for graph representation learning, which aims to preserve network properties for downstream tasks. Usually, link prediction and node classification are two common downstream tasks for evaluation of graph representation learning [13]. Due to the unavailability of ground truth of Ethereum accounts, we evaluate our temporal-random-walk-based graph embedding by link prediction on realistic Ethereum data, as Fig. 5 shows. In fact, link prediction is also a valuable issue in blockchain systems. A series studies on Ethereum have witnessed various illegal behaviors or scams such as phishing, Ponzi [20], money laundry [21], etc., and the temporal link prediction can help people track the real-time process of Ether flow or prevent illegal users’ cashing process [22].

The task of link prediction aims to predict the occurrence of links in a given graph on the basis of observed information. In the temporal link prediction problem, unlike the static link prediction where links have no timestamp, we use the existing links in the past (50% of the earlier edges) as the training data to predict the occurrences of links in the future (the remainder). Shown in TABLE I, we randomly select center accounts and collect three subgraphs with different size ($K$-in = $K$-out = $K$) from Ethereum for learning node embedding vectors $\Phi(v)$ for $\forall v\in V$ via random-walk based graph representation learning. Next, we use a binary classifier for supervised link prediction, where node pairs $(src,dst)$ existing in the training graph act as positive samples of the training set. We randomly sample an equal number of node pairs with no link as negative samples. Then we obtain features of a directed link from nodes $v_{i}$ to $v_{j}$ by concatenating their node representations, i.e., $F_{i,j}=[\Phi(v_{i}),\Phi(v_{j})]$. If $i\neq j$, $F_{i,j}\neq F_{j,i}$. Finally, we train a support vector classifier to classify the links in the test set (50% of the later edges).

In this brief, we proposed to model the Ethereum transaction network as a new defined network model called temporal weighted multidigraph, and refine the definition of temporal walk considering the network dynamics and the multiplicity of edges to obtain graph representation more accurately. Experiments on the task of temporal link prediction demonstrated the effectiveness of the proposed framework. The limitations of the proposed approach is that the graph representation model based on temporal random walk is not an inductive method, which can not obtain the representations for the newly added nodes. For future work, we plan to develop more specific graph representation methods to investigate various predictive tasks on Ethereum and extend the current framework to analyze other large-scale temporal or domain-dependent networks.