# Live Graph Lab: Towards Open, Dynamic and Real Transaction Graphs with NFT

Zhen Zhang, Bingqiao Luo, Shengliang Lu, Bingsheng He
National University of Singapore
zhen@nus.edu.sg,luo.bingqiao@u.nus.edu,lusl<EMAIL_ADDRESS>

###### Abstract

Numerous studies have been conducted to investigate the properties of large-
scale temporal graphs. Despite the ubiquity of these graphs in real-world
scenarios, it’s usually impractical for us to obtain the whole real-time
graphs due to privacy concerns and technical limitations. In this paper, we
introduce the concept of Live Graph Lab for temporal graphs, which enables
open, dynamic and real transaction graphs from blockchains. Among them, Non-
fungible tokens (NFTs) have become one of the most prominent parts of
blockchain over the past several years. With more than $40 billion market
capitalization, this decentralized ecosystem produces massive, anonymous and
real transaction activities, which naturally forms a complicated transaction
network. However, there is limited understanding about the characteristics of
this emerging NFT ecosystem from a temporal graph analysis perspective. To
mitigate this gap, we instantiate a live graph with NFT transaction network
and investigate its dynamics to provide new observations and insights.
Specifically, through downloading and parsing the NFT transaction activities,
we obtain a temporal graph with more than 4.5 million nodes and 124 million
edges. Then, a series of measurements are presented to understand the
properties of the NFT ecosystem. Through comparisons with social, citation,
and web networks, our analyses give intriguing findings and point out
potential directions for future exploration. Finally, we also study machine
learning models in this live graph to enrich the current datasets and provide
new opportunities for the graph community. The source codes and dataset are
available at https://livegraphlab.github.io.

##  1 Introduction

Temporal graphs provide an accurate representation of real-world systems,
including social networks, transaction networks and the Web [5, 38, 39, 41],
etc. By investigating temporal graphs, we can gain insights into the temporal
dynamics and understand how these systems evolve and function [86, 6].
Notably, a growing number of graph mining algorithms [36, 25, 85] and graph
systems [69, 35, 13] have been developed. However, with this continuously
growing trend, several severe issues emerge, which limit the further
development of graph community. In the current literature, studies are usually
conducted on a set of outdated and incomplete graphs. The majority of the
aforementioned graphs are either not easily available or their graph
structures are incomplete since they cannot record all the interactions in
graph. Moreover, even through all the interactions are recorded, they might
not be shareable in a public and timely evolving manner, such as social
networks in companies like Meta and Tencent. Thus, for meaningful temporal
graph analysis and benchmarks, we need open graph datasets that evolve
dynamically and are easily accessible in a timely manner.

To bridge this gap, we propose the concept of Live Graph Lab, which provides
live graphs according to blockchain transactions. Specifically, we offer a set
of tools for downloading, parsing, cleaning, and analyzing blockchain
transactions to empower the analyses of transaction graphs. It not only
alleviates the researchers’ burden of accessing massive raw transaction data,
but also brings a considerable of opportunities to conduct experiments in the
real-world scenario for temporal graph studies. Our introduced live graphs
have several unique characters like open availability, dynamic evolution and
real transactions due to the inherent characteristics of the decentralized
blockchain. Therefore, it is of great importance to investigate the properties
of these live graphs to provide new insights for graph algorithms and systems.

Today, as blockchain technology becomes more widespread, the token economy is
gradually emerging. Non-fungible tokens (NFTs) have seen tremendous growth
with its market capitalization reaching over $40 billion. Notably, one of the
digital work named “Everydays: The First 5000 Days”
111https://en.wikipedia.org/wiki/Everydays:_the_First_5000_Days by artist
Beeple was sold for $69 million, which makes NFTs become the center of
attention. This phenomenon also leads to an increasing number of enthusiasts
participating in this emerging concept. At the same times, it generates
massive, anonymous and real transaction activities, which naturally forms a
complex NFT transaction network. Although traditional networks like social
networks and citation networks have been extensively studied, they are often
outdated due to their lack of constantly updating. To enrich the current
datasets and overcome their limitations, we instantiate a live graph with NFT
transaction network by synchronizing a full Ethereum node, thus it
continuously keeps up with the latest Ethereum block and includes all the
transaction data. To investigate the characteristics of this live NFT
transaction network, we present a temporal graph extracted from a specific
time period spanning from 2017 to 2022, which comprises over 4.5 million nodes
and 124 million edges. Then, comprehensive analyses are performed, and the
results demonstrate that our presented live graph exhibits a variety of
characteristics, offering exciting opportunities for the graph community. To
summarize, the main contributions and findings are as follows:

  * •

We introduce the concept of live graph lab, which focuses on open, dynamic and
real graphs.

  * •

We instantiate a live graph with NFT transaction network and provide a
systemic analysis, which demonstrates interesting properties like fast-
growing, highly-active, ect.

  * •

Graph machine learning models are investigated in the live graph, and the
experimental results indicate that live graphs pose new challenges and
opportunities for the graph community.

##  2 Related Datasets

Table 1: Comparisons among different types of graph datasets. Categories | Datasets | Open | Timely Evolving | Complete Structure | Timestamp
---|---|---|---|---|---
Social Network | ego-Twitter [45] | ✓ | ✗ | ✗ | ✗
Citation Network | DBLP [70] | ✓ | ✗ | ✗ | ✓
The Web | web-Google [44] | ✓ | ✗ | ✗ | ✗
Blockchain | Live Graph Lab | ✓ | ✓ | ✓ | ✓

Graph has gained significant attention from both academic and industrial
communities. A wide range of benchmark datasets have been proposed to
facilitate the research in graph community. Among them, SNAP [43] and Network
Data Repository [60] provide diverse types of graphs including social
networks, web networks, etc. AMiner [70] offers comprehensive citation
networks extracted from DBLP, ACM and Microsoft Academic Graph. Chartalist
[63] presents a set of blockchain datasets to enable machine learning model
development. Although these datasets are publicly available, they are not
constantly updated in a nearly real-time manner. Take social network as an
example, the ego-Twitter [45] in SNAP project was released more than 10 years
ago. Thus, the graph properties presented in these benchmarks may no longer be
suitable in the current context. Meanwhile, their graph structures are usually
incomplete due to privacy policy constraints or technical limitations.
However, these characters are important for various downstream applications.
For instance, if the graph is incomplete or their characteristics have changed
significantly, the learning outcomes could be ineffective and even misleading
in the graph learning tasks. To enrich the current datasets and overcome their
limitations, we propose the concept of Live Graph Lab, which supports various
experiments for temporal graph algorithms. We provide a detailed comparison of
these datasets in Table 1. Specifically, the proposed live graph lab has the
following properties: (1) it is open and publicly available; (2) it is
constantly evolving in a nearly real-time manner; (3) it is complete (i.e, all
interactions are fully recorded); (4) it has realistic timestamp. Moreover,
previous blockchain datasets mainly focus on fungible token transactions.
However, Non-Fungible Tokens (NFTs), which are a vital component of the
Ethereum, have been overlooked by existing works. Our work covers this gap by
delving into this emerging NFT ecosystem.

##  3 Dataset Details

Table 2: Statistics of the dataset. Descriptions | Statistics
---|---
Start date (mm-dd-yyyy, UTC) | 07-12-2017 13:49
End date (mm-dd-yyyy, UTC) | 08-01-2022 06:50
Number of NFT collections | 97,667
Number of NFT tokens | 77,991,885
Number of account addresses | 4,531,020
Number of transactions | 124,660,813

In this paper, we instantiate a live graph with NFT transaction network in the
Ethereum blockchain. We first provide an overview of the blockchain
background. Then, we give a comprehensive explanation of the graph
construction process. Note that, our methodologies are applicable to other
bolckchains like Solana and Polygon, etc.

###  3.1 Background

Blockchain and Ethereum. Blockchain, a distributed ledger technology, has
attracted continuous attention recent years and is made up of securely linked
blocks with cryptography techniques [53], where each block contains
information of the previous block (e.g., cryptographic hash). Ethereum is a
decentralized, programmable blockchain, which means users can construct
various decentralized applications on the blockchain. Ether (ETH) is the
native cryptocurrency of Ethereum, and every transaction incurred in the
Ethereum needs a specific fee paid in ETH.

Smart Contract and Non-Fungible Token. Smart contract is an important feature
in the Ethereum blockchain [88], which is a computer program that runs on the
Ethereum to automatically execute or control relevant events and actions
according to its logic. Smart contracts have largely reduced the requirements
for trusted intermediaries, fraud losses and arbitration costs, etc. Non-
Fungible Token (NFT) is one of the most successful applications on Ethereum.
NFTs are tokens that can be used to represent the ownership of any unique
asset such as image, video and audio, etc. Different from fungible items where
one dollar is exchangeable for another one dollar, NFTs are not
interchangeable with each other, since they all have unique properties and are
not divisible.

###  3.2 Raw Data

The statistic information of our dataset is summarized in Table 2. To access
the transaction information in Ethereum, Geth, a golang implementation of
Ethereum client software, is launched to facilitate the synchronization of the
ledger on Ethereum mainnet. When the client synchronizes to the latest block,
we extract all the blocks before Aug 1st, 2022 (i.e., from block #0 to block
#15,255,104). Then, we parse all the transaction data and log data via toolkit
Ethereum ETL222https://ethereum-etl.readthedocs.io/en/latest/. Thanks to the
well-defined standards in NFT communities, it is convenient for us to extract
the information we need. Specifically, according to the standard of EIP-721,
every NFT smart contract must implement the standard interfaces including
transfer, approval, ownerof and balanceOf, etc. For those smart contract that
do not strictly follow the EIP-721 standard, we remove those smart contracts
and its relevant transactions from our data. For the remaining data, we filter
out all the transfer events triggered by its smart contracts to extract the
NFT transactions. This is because the ownership change of any NFT will emit a
transfer event identified by the event topic Keccack256 hash 0xddf…3ef. In the
NFT transfer event, it contains four key contents including the Keccack256
hash, the sender’s address, the receiver’s address, and the transferred NFT
token ID. The time when the transaction occurs can be retrieved from the block
that the event was found. Once we have obtained all of these key information,
we can know when and which NFTs are transferred among different wallet
addresses and the prices they are sold.

By parsing the transfer events, we find out that there are 97,667 NFT
collections with 77,991,885 NFT tokens, where each collection contains
different number of NFT tokens (e.g., varying from thousands to millions).
Meanwhile, 124,660,813 transactions are extracted, and among them 4,531,020
users (i.e., we regard each wallet address as a user) participate in the
transactions. It is worthy noting that there are over 100,000 NFT collections
according to Etherscan333https://etherscan.io/tokens-nft. However, as we have
mentioned, some of them are not standard NFT tokens (i.e., they do not
strictly follow the EIP-721 standard), and we remove them from our data. Thus,
our method almost extracts all the NFT collections.

###  3.3 Graph Construction

We investigate the structure and dynamics of all these transactions by
constructing a directed temporal graph
$\mathcal{G}=(\mathcal{N},\mathcal{E},\mathcal{T})$, where $\mathcal{N}$ and
$\mathcal{E}$ denote the node set and edge set, respectively. That’s to say,
we only count once for those addresses that have repetitive interactions.
$\mathcal{T}$ is a set of timestamps when the interactions happen. We use
$t(e)$ to denote the timestamp when edge $e$ is formed, and $t(u)$ represents
when the node $u$ is added into the graph. Thus, $a_{t}(u)=t-t(u)$ reflects
node $u$’s age at time $t$. For any given time $t$, graph $\mathcal{G}_{t}$
consists of all the nodes as well as edges until time $t$. Note that, since we
have accurate timestamps for the arrival of each node and edge, our
investigation of graph dynamics is at a much finer granularity compared with
the majority of existing studies [5, 41].

##  4 Observations and Analyses

To have a better understanding of the instantiated live graph, we start the
analyses from the following perspectives: (1) Structural properties, which
investigate how its nodes and edges change as time goes on; (2) Dynamic
behaviors, which are graph specific properties such as how hub-nodes and bi-
directional edges are formed. More comprehensive analyses are given in the
Appendix C and D.

###  4.1 Structural Properties

(a) Nodes

(b) Edges

(c) Edge percentages

(d) Degree distribution

Figure 1: Evolution of nodes and edges.

To fully understand how active is the NFT transaction network, we measure the
evolution of nodes and edges over the time. Specifically, we set the time
granularity to a yearly yardstick. Since our data started from the July of
2017 and ended at the August of 2022, the statistical information for 2017 and
2022 only has a half year’s data. Figure 1(a) and 1(b) show the annual growth
of nodes and edges in a log-scale. As observed, there is a rapid increase in
the number of nodes and edges, which become 28K and 165K times larger within
the six years. It means the NFT transaction network is highly active and
growing at a fast speed.

To further figure out what leads to the growth, we analyse the newly added
nodes in each pair of consecutive years. New nodes could join the network
through different ways (i.e., mint, buy or airdrop NFTs). Among them, mint is
the most common and easy way to obtain NFTs, where only a small number of gas
fee is needed to pay. Figure 1(a) presents the trend of newly added mint nodes
and non-mint nodes. We notice that the number of mint nodes added into the
network is approximately the same magnitude of the total nodes at that time.
Therefore, mint NFTs dominates the growth of nodes in the network at present,
meanwhile the number of non-mint nodes are also increasing at a high velocity.
These two phenomena result in the expansion of nodes.

Figure 1(b) also shows the trend of added bidirectional edges and self-edges.
We can see that the number of bidirectional edges only account for a small
volume. Unlike social networks where the edges are highly mutual, the NFT
market is an anonymous ecosystem and the probability of mutually interacting
with each other is relatively low. These bidirectional edges might happen in
the scenarios of swap or wash trading activities [73]. We also notice that
there exist a very few self-loops in the network, which means these addresses
transact with themselves. This abnormal phenomenon might be caused by a
mistake input for the received address or these transactions are executed for
testing. Furthermore, to characterize how active are these addresses, we
compute the percentage of edges in which new nodes have links from or to old
nodes, and the percentage of edges which only contain links between new nodes.
Here, we refer to new nodes as the nodes created in the current year, and old
nodes indicate the nodes that have existed in the previous years. Figure 1(c)
demonstrates that more than 50% percent of newly added edges are constituted
by the connections between new nodes and old nodes, except for the year of
2021. This indicates that most of the addresses remain active as the NFT
ecosystem become mature. It is also very interesting to note that the newly
created addresses are highly active in 2021, and at the same time, the whole
NFT market capitalization reaches over $40 billion USD dollars in this year.

For completeness, we also show the degree distribution of the NFT transaction
network in Figure 1(d) with log-log scale. As expected, it follows a power-law
distribution. We observe that some nodes have the degree of 1, which indicates
they did not conduct any other transactions after the NFTs were minted or
transferred. There also exist several high degree hubs that are extremely
active and interacts with more than thousands of different addresses. Among
them, a special Null address (i.e.,
0x000…000)4440x0000000000000000000000000000000000000000 has the highest
degree, since every NFT mint activity will create a link from the Null address
to the mint address.

###  4.2 Dynamic Behaviors

(a) Hub nodes in year 2018

(b) Hub nodes in year 2019

(c) Mutual edges in days

(d) Ratio of mutual edges

Figure 2: Correlation between hub nodes, new connections and mutual edges.

Evolution of Hub Nodes. The node degree shows a heavily long-tailed
distribution (i.e., Figure 1(d)), and the assortativity raises year by year
(i.e., Figure 4(a) in Appendix C). Both these two measurements are highly
relevant to the evolving of hub nodes in the transaction network. Figure 2(a)
and 2(b) illustrate the correlations between node degrees and its number of
new connections in two consecutive years. We use blue color to present the
node degree distribution in the previous year, and then red color indicates
the number of connections from new nodes in the current year. As expected, we
can find that if a node had high degree in the previous year, it would have
high probability to get more new node connections in the current year.
Moreover, this observation is also validated by measuring their Pearson
Correlation Coefficients. We evaluate the correlation coefficients between
node degrees and its number of new node connections (in year 2018-2019 and
2019-2020), which are 0.2266 and 0.5476, respectively. The results reveal that
these two factors are positively relevant. Thus, we can draw the conclusion
that the NFT transaction network follows preferential attachment growth model
[54, 32], i.e., “the rich gets richer”.

Next, we analyze the distribution of the hub nodes. Specifically, we note that
half of the top-100 largest hub nodes are smart contract addresses. This is
understandable, since smart contracts play an important role in providing
various services (e.g., NFT fractionalization and staking) to other users in
the network. Thus, they tend to have high degrees once they are frequently
used by others. The transactions with such smart contracts result in the
increase of assortativity. Furthermore, there also exist some hub nodes that
are not smart contracts. For instance, we observe that address
0x8a0…7005550x8a01fa5a77311bbcf29e293d8ecb48707cfdb700 is the fifth largest
hub nodes with the degree of 69,458. After checking its transactions manually,
we find that all its transactions are about HyperNFT tokens, and it holds more
than 180,000 such kind of tokens. Since it is strange, we then check its
transactions in details and discover that this address creates a transaction
every a few minutes, where the ID of the transacted NFT token is in a
continuous and increase order. According to these clues, it is very certain
that this address is a bot, which makes frequent transactions to pretend it’s
very prosperous to attract more users to join in. The hub nodes are
responsible for the spreading of information in the network, thus it is
necessary to investigate the properties of hub nodes.

Mutual Interaction Edges. In the previous study, we find that the reciprocity
of the transaction network is relatively low, which is around 0.1 in the end
(i.e., Figure 4(c) in Appendix C). This is quite different from the property
in the social networks, where the reciprocity is very high and the value is
above 0.7 [39]. One possible reason is that people are prone to mutually
following and interacting with their friends in the social networks, since it
does not cost anything. However, it becomes different in the NFT transaction
network, because we do not know who we are actually interacting with and every
action costs in the blockchain. To uncover the reciprocity’s characteristics,
we focus on the mutual interaction edges. Specifically, we are interested in
if two nodes are mutually interacting with each other, i.e., reciprocal edges
$\left\langle u,v\right\rangle_{t}$ and $\left\langle
v,u\right\rangle_{\hat{t}}$, what their maximum time interval distribution
$|t-\hat{t}|$ looks like, i.e., the maximum delay in days of the reciprocity.
Figure 2(c) shows that mutual interaction edges forming within one day account
for the highest proportion, and the maximum time interval can reach to more
than 1,000 days. According to Figure 2(d), we can observe that about 15% of
the reciprocal edges are formed almost simultaneously among those bi-
directional edges, and more than 70% of them are formed within 90 days. From
these observations, we can conclude that the NFT transaction network can not
be regarded as an undirected network due to its low reciprocity value, and the
simultaneously formed bi-directional edges can be a good indicator to judge
the abnormal activities. For instance, it may be caused by the token transfers
among the accounts controlled by a same person.

To further investigate this phenomenon, we want to know how many address pairs
are suspicious in these bi-directional links. We first conduct some statistics
for all NFT transaction addresses, which involves the following two factors:
the number of transactions (including from transactions and to transactions)
and the number of distinctly interacted addresses. Then, if one address has
very limited transactions and it only interacts with a specific address, it is
highly possible that these two addresses are of same person’s wallets. This is
because this address is actually not active, but it responses promptly in the
bi-directional links. Similarly, another situation is although one address has
a lot of transactions, it interacts with one specific address frequently and
the specific address accounts for a large ratio of the total transactions.
They may also belong to same person’s wallets. Based on these two
observations, we define the following two rules to identify the suspicious
address pairs: 1) one of them makes less than 5 transactions or 2) the
transaction ratios between them is larger than 0.8. If they satisfy at least
one of the rules, we then believe these two addresses are likely to be same
person’s wallets. According to these rules, we discover that 33.72% of those
simultaneously formed bi-directional links have high probability of being
suspicious. This observation provides a new perspective for detecting anomaly
transactions.

##  5 Downstream Applications

The above analyses provide a general overview of this live graph’s properties.
Our results demonstrate that the transaction network is highly active and
evolving at a fast speed. These properties put forward new challenges for
various downstream tasks. Next, we investigate three widely studied tasks.

###  5.1 Temporal Link Prediction

Link prediction lies at the core of graph analysis and mining. The link
prediction’s goal is to predict whether a pair of node would form a link,
which has been extensively used in diverse real-world scenarios like
recommendation [79, 7] and knowledge graph completion [55], just to name a
few. In this section, we focus on temporal link prediction, i.e., we try to
forecast future interactions based on the historical transactions.
Specifically, we investigate the snapshot-based representation for temporal
link prediction via graph neural networks [36, 25]. Since each edge $e$ has a
timestamp $\tau_{e}$, we utilize a sequence of graph to depict the temporal
transaction network $G=\\{\mathcal{G}_{t}\\}_{t=1}^{T}$, where each snapshot
is a static graph
$\mathcal{G}_{t}=(\mathcal{N}_{t},\mathcal{E}_{t},\mathcal{T}_{t})$ with
$\mathcal{E}_{t}=\\{e\in\mathcal{E}|\tau_{e}\leq t\\}$. Through modeling the
sequential information of different graph snapshots, we could leverage the
information available up to time $t$ to forecast possible edges at time $t+1$.
Although various approaches have been proposed for temporal link prediction,
it is unclear whether they can achieve satisfied performance in this highly
active and large-scale temporal transaction network.

Graph Neural Networks. GNNs have gained great success in numerous learning
tasks on graphs including node classification [36, 25], graph classification
[84, 83] and link prediction [82, 75]. The objective of GNN is to acquire
effective node representations through iteratively aggregating messages from
its local neighborhood. Specifically, the $l$-th layer of a GNN model can be
formulated as follows:

$\mathbf{h}_{v}^{l}=\textsc{Agg}^{l}\left(\left\\{\textsc{Msg}^{l}(\mathbf{h}_{u}^{l-1},\mathbf{h}_{v}^{l-1})|u\in\mathcal{N}(v)\right\\},\mathbf{h}_{v}^{l-1}\right)$

where $\mathbf{h}_{v}^{l}$ is the node representation for node $v$ at layer
$l$ and $\mathcal{N}(v)$ represents node $v$’s neighborhood.
$\textsc{Msg}(\cdot)$ indicates the message-passing function, which propagates
information from its neighbors. $\textsc{Agg}(\cdot)$ is the aggregation
function, which updates its representation with node’s neighborhood
representations. For a $L$ layers GNN, it aggregates information from the
$L$-hop neighborhood. Different GNN architectures can have different message-
passing and aggregation functions. The original GNN model is designed for
static graphs, thus it cannot capture the underlying temporal information
within the graph. To incorporate the evolving property, existing works utilize
RNNs [28, 14] to aggregate information from different graph snapshots. Hence,
the dynamic GNN models have much more parameters compared with the vanilla
GNNs, which is more difficult to scale to large graphs due to the back
propagation through time constraints.

Models. We compare a number of recent state-of-the-art dynamic GNN models. (1)
Dyngraph2vec [21] captures the temporal transitions with a deep architecture
with dense and recurrent layers. (2) TGCN [87] integrates GCN with GRU to
learn complex spatial and temporal dependencies. (3) EvolveGCN [56] use a RNN
to adapt the graph convolutional network parameters. (4) GCRN [62] generalize
TGCN with either GRU or LSTM. Moreover, it uses ChebNet [15] to encode the
graph structure information, and separate GNNs are applied to compute
different gates of RNNs. (5) DynGEM [22] utilizes deep autoencoder to perform
graph embedding, then local and global constraints are employed to keep node
representations being stable over time. (6) Roland [81] is an efficient
learning framework designed for temporal GNNs, which updates its parameters
through a combination of incremental training and meta-learning strategies
[19, 20].

Settings. For dataset, we remove all the transactions associated with the Null
address, which results in 3.13 million nodes and 23.13 million edges in the
directed graph. We set node feature as 1 for all the nodes and utilize area
under the curve (AUC) as well as mean reciprocal rank (MRR) as evaluation
metrics. For every node $u$ connected by a positive edge $(u,v)$ at time $t$,
we randomly select 100 negative edges originating from node $u$. Subsequently,
we determine the rank of the score for edge $(u,v)$ among all the sampled
negative edges. AUC characterizes the probability of ranking positive node
more highly than negative nodes. MRR denotes the average of reciprocal ranks
computed across all nodes. Following the settings in Roland [81], we utilize
two different train-test splits: fixed-split and live-update. Among them, the
fixed-split setting assesses the models using all the edges from the last
$20\%$ graph snapshots. Although it is widely used in the existing works, the
fixed-split might produce misleading results based on edges merely from the
last few graph snapshots, since the graph structure could constantly evolve in
the real world scenario. To eliminate this bias, we also utilize live-update
split to test the models, which evaluates their performance over all the
available graph snapshots. $10\%$ of edges are used to determine the early-
stop condition in this setting.

Table 3: Temporal link prediction performance in fixed split and live-update
settings. We repeat experiments with three different seeds to report the mean
as well as standard deviation of AUC and MRR. We also present the results
under different time snapshot granularities, e.g., days, weeks and months. OOM
means out-of-memory.

Models | Fixed Split
---|---
Snapshot Days | Snapshot Weeks | Snapshot Months
AUC | MRR | AUC | MRR | AUC | MRR
Dyngraph2vec | OOM | OOM | OOM | OOM | OOM | OOM
TGCN | 53.64$\pm$1.60 | 14.24$\pm$2.60 | 61.55$\pm$6.24 | 36.16$\pm$6.60 | 74.87$\pm$4.99 | 45.97$\pm$4.46
EvolveGCN | OOM | OOM | OOM | OOM | OOM | OOM
GCRN-GRU | 95.86$\pm$0.03 | 71.48$\pm$0.49 | 93.14$\pm$0.18 | 68.44$\pm$0.05 | 86.74$\pm$0.80 | 58.23$\pm$1.23
GCRN-LSTM | 94.12$\pm$0.92 | 68.51$\pm$2.36 | 92.90$\pm$0.43 | 67.66$\pm$0.31 | 86.44$\pm$0.92 | 58.71$\pm$0.84
DynGEM | OOM | OOM | OOM | OOM | OOM | OOM
Roland-MA | 95.93$\pm$0.15 | 66.34$\pm$0.23 | 93.53$\pm$0.13 | 65.06$\pm$0.43 | 86.23$\pm$0.85 | 54.93$\pm$1.42
Roland-MLP | 65.46$\pm$6.10 | 43.76$\pm$5.94 | 73.34$\pm$7.87 | 42.04$\pm$16.8 | 85.88$\pm$2.22 | 57.58$\pm$4.12
Roland-GRU | 73.33$\pm$11.5 | 49.45$\pm$8.91 | 91.48$\pm$1.67 | 66.28$\pm$1.86 | 86.59$\pm$0.88 | 59.37$\pm$1.54
| Live Update
Dyngraph2vec | OOM | OOM | OOM | OOM | OOM | OOM
TGCN | 58.22$\pm$5.76 | 17.77$\pm$10.7 | 59.94$\pm$8.44 | 22.67$\pm$19.3 | 75.01$\pm$2.66 | 43.16$\pm$0.75
EvolveGCN | OOM | OOM | OOM | OOM | OOM | OOM
GCRN-GRU | 80.95$\pm$1.92 | 39.13$\pm$0.39 | 85.34$\pm$0.26 | 46.08$\pm$1.43 | 81.40$\pm$0.34 | 43.68$\pm$0.39
GCRN-LSTM | 79.12$\pm$2.14 | 37.83$\pm$1.16 | 84.73$\pm$0.34 | 42.89$\pm$3.34 | 81.24$\pm$0.80 | 41.47$\pm$3.00
DynGEM | OOM | OOM | OOM | OOM | OOM | OOM
Roland-MA | 90.47$\pm$0.66 | 49.79$\pm$0.95 | 88.74$\pm$0.37 | 50.77$\pm$1.13 | 83.93$\pm$0.95 | 47.11$\pm$1.16
Roland-MLP | 56.32$\pm$8.06 | 22.16$\pm$15.4 | 70.88$\pm$10.5 | 40.91$\pm$10.8 | 79.38$\pm$4.47 | 46.51$\pm$2.61
Roland-GRU | 60.04$\pm$5.08 | 27.29$\pm$6.26 | 75.93$\pm$19.2 | 48.66$\pm$13.7 | 84.36$\pm$0.46 | 50.75$\pm$0.83

Results. We present the link prediction results in Table 3. Specifically, we
use three different time granularities (i.e, days, weeks and months) to
construct the graph snapshot sequences, which results in 1,657 graph
snapshots, 253 graph snapshots and 60 graph snapshots, respectively. All the
models’ performance is evaluated under two different settings, i.e., fixed-
split and live-update. We notice that as the time granularity becomes coarser,
the models’ performance drops. This is because the NFT transaction network is
highly active, which is about 68 million transaction volume per day on average
in 2021. Thus, it will be more difficult to predict all the possible links in
a long time period. Meanwhile, we can observe very high AUC scores in a daily
time granularity, which indicates the models have a better capability of
distinguishing the positive and the negative edges in this short time
scenarios. Also, the performance in live-update setting is a little lower than
the fixed-split setting. This is caused by the pattern shifts in different
graph snapshots, which implies the patterns indeed evolve over the time.
Therefore, we can conclude that it is necessary to model the NFT transaction
network in a finer time granularity, and at the meantime this will result in
longer graph snapshot sequence, which puts forward more challenges to model
the temporal information.

###  5.2 Temporal Node Classification

Node classification plays a crucial role in understanding the attributes,
behaviors, and relationships of nodes in temporal graphs, offering valuable
insights in diverse applications. By predicting the label of nodes at a
particular time $t$, we gain a temporal perspective that deepens our
understanding of how nodes evolve over time and their dynamic characteristics.
Specifically, we focus on categorizing nodes according to their transaction
behaviors. They can be generally classified into five distinct classes: daily
traders, weekly traders, monthly traders, yearly traders and the remaining
traders. We first filter out nodes that only have one transaction. Then, each
node’ maximum transaction interval is calculated. If the maximum interval is
within one day, we call it daily trader. Likewise, if the maximum interval is
within one week and larger than one day, we call it weekly trader, and so
forth. This process results in a large-scale directed graph with about 1.80
million nodes and 21.83 million edges. Following the setting of EvoloveGCN
[56], we use the first 80% graph snapshots as training set, the following 10%
graph snapshots as validation set and the last 10% graph snapshots as test
set. Similar to temporal link prediction task, we use the same set of GNN
models and add a multi-class classification layer. Furthermore, nodes’ degrees
are encoded as features. The number of transactions between two nodes and
their latest interaction timestamp are transformed as edge features. Two
commonly used metrics (i.e., accuracy and recall) are employed to assess the
model’s performance.

Table 4: Node classification performance in fixed split setting. We repeat
experiments with three different seeds to report the mean as well as standard
deviation of Accuracy and Recall.

Models | Snapshot Days | Snapshot Weeks | Snapshot Months
---|---|---|---
Accuracy | Recall | Accuracy | Recall | Accuracy | Recall
Dyngraph2vec | OOM | OOM | OOM | OOM | OOM | OOM
TGCN | 18.48$\pm$2.66 | 31.15$\pm$3.16 | 43.97$\pm$4.57 | 32.99$\pm$2.34 | 47.45$\pm$3.49 | 32.53$\pm$2.95
EvolveGCN | OOM | OOM | OOM | OOM | OOM | OOM
GCRN-GRU | 41.06$\pm$3.30 | 34.75$\pm$2.93 | 46.78$\pm$0.72 | 34.79$\pm$0.42 | 47.42$\pm$2.16 | 28.97$\pm$3.22
GCRN-LSTM | 46.14$\pm$3.29 | 35.19$\pm$3.61 | 48.04$\pm$2.37 | 31.58$\pm$1.75 | 49.32$\pm$2.01 | 35.39$\pm$1.49
DynGEM | OOM | OOM | OOM | OOM | OOM | OOM
Roland-MA | 51.02$\pm$2.01 | 28.77$\pm$3.23 | 50.39$\pm$0.45 | 26.33$\pm$3.95 | 47.96$\pm$2.69 | 22.33$\pm$3.07
Roland-MLP | 48.46$\pm$3.18 | 30.62$\pm$3.94 | 47.59$\pm$3.39 | 31.67$\pm$3.62 | 45.74$\pm$4.75 | 35.04$\pm$3.47
Roland-GRU | 49.88$\pm$2.15 | 33.38$\pm$3.91 | 46.63$\pm$3.07 | 33.85$\pm$1.48 | 50.04$\pm$0.37 | 32.17$\pm$2.72

Results. The node classification results are illustrated in Table 4 and key
observations are as follows. Three granularities, i.e., days, weeks and
months, are utilized to generate the temporal graph snapshot sequences. In
accordance with the temporal link prediction task, we can draw similar
conclusions. Roland could consistently outperform other baselines, and its
variant Roland-MA (moving-average) stands out due to its parameter-free design
and remarkable capability to capture evolving patterns. On the other hand,
TGCN fails to achieve satisfactory performance because of the gradient
vanishing problem in longer sequences. GCRN-GRU and Roland-GRU address this
issue by utilizing separate GNNs or leveraging previous and current snapshots.
Furthermore, the scalability of several baselines is limited and they
experience out-of-memory (OOM) issues. Among the three time granularities, the
month granularity is generally the most difficult scenarios. This is because
with a month granularity, the time period between snapshots is longer compared
with the day and week granularities. As a result, it becomes more challenging
to accurately predict the temporal patterns and changes in node behaviors. The
longer time gap between snapshots increases the complexity of modeling the
evolving dynamics of the network and introduces more uncertainty, leading to
decreased performance in terms of classification accuracy and recall.
Therefore, we can conclude that given the highly active NFT transaction
network, it is essential for the model to use a finer time granularity.
Nonetheless, this choice results in longer graph snapshot sequences,
presenting additional difficulties in accurately capturing the temporal
information.

###  5.3 Continuous Subgraph Matching

Figure 3: Five most common wash trading patterns.

Continuous Subgraph Matching (CSM) plays a vital role in numerous real-time
graph applications. The goal of CSM is to identify and report the occurrences
of a given query graph $Q$ within a temporal graph stream $G$. It can be
utilized in various scenarios. For example, when setting the query graph $Q$
as wash trading patterns in e-commerce, CSM could identify the anomaly
transaction patterns in the graph via exactly matching [58]. Similarly,
representing query graph as rumor patterns in social network could help to
detect and prevent the spread of rumors [74]. In this subsection, we set the
query graphs as the frequent wash trading patterns in the NFT transactions.
According to the definition in [73, 47], wash trading is an activity where the
seller is on the both sides of the trade. The goal of wash trading is to
influence the price or create the illusion that the item is very popular,
which produces artificial activities in the marketplace. Wash trading has been
prohibited by many countries. However, due to the anonymous nature of the
blockchain, wash trading has become a severe issue in the NFT
market666https://cryptopotato.com/over-33-of-nft-volume-is-wash-trading-
bitscrunch-ceo-interview/, which accounts for a large volume in the whole NFT
transactions. For instance, the CryptoPunk’s $9,998$-th NFT was traded between
two wallets for 124,457 ETH (about $534 million USD), in which the buyer paid
to the seller, then the seller transferred the money back to the buyer. Thus,
it is important to detect the wash trading transactions to uncover the anomaly
behaviors and reduce the risks in the market. We resort to CSM to identify the
wash trading transactions. Five most common wash trading patterns are shown in
Figure 3. As can be seen, all of them contain at least one cycle. Here is a
toy example that exemplifies Pattern 1: Address A (0x744…282) initiated the
sale of Azuki token 1,215 to Address B (0xd39…263). Subsequently, Address B
sold the token to Address C (0xeaa…c0f), and eventually, Address C sold it
back to Address A. These transactions happened within half an hour, and
interestingly, the token’s price surged from 8.98 ETH to 11.99 ETH. Such
activity can raise suspicions of wash trading. We will use them as query
graphs in the experiments to simulate the wash trading detection procedure.
More details are given in Appendix E.

Table 5: Comparison of different frameworks on query time. Query Patterns | Model Query Time (ms)
---|---
Queries | Counts | SymBi | Graphflow | TurboFlux | RapidFlow
p1 | 19,338 | 1.22$\times$104 | 1.11$\times$104 | 1.11$\times$104 | 5.93$\times$102
p2 | 2,243,232 | 1.19$\times$104 | 1.44$\times$104 | 1.51$\times$104 | 6.13$\times$102
p3 | 3,012,738 | 1.11$\times$104 | 1.13$\times$104 | 1.08$\times$104 | 5.83$\times$102
p4 | 9,472,960 | 1.24$\times$104 | 1.43$\times$104 | 6.06$\times$104 | 6.45$\times$102
p5 | 3,154,355,868 | 1.34$\times$105 | 4.24$\times$105 | 7.72$\times$104 | 5.84$\times$102

Results. Table 5 shows the key results of continuous subgraph matching. As we
can see, RapidFlow [68] is the most efficient algorithm, which demonstrates
about 18-724x speedups compared with the remaining frameworks. This is because
the query reduction technique can significantly expedite the query procedure
through an optimized matching order. We also observe that no algorithm can
dominate others in different query patterns except RapiadFlow. Among them,
SymBi [50] and Graphflow [33] perform quite worst on pattern 5, which need 10x
more time to produce the results. SJ-Tree [13] and IEDyn [30, 31] cannot
output the results within the time limit. The reason is that SJ-Tree
encounters memory exhaustion in the majority of cases, and IEDyn’s index
update bears too much overhead because of the maintenance for constant-delay
enumeration. Furthermore, the number of matched subgraphs increase from
pattern 1 to pattern 5, however the query time does not have too much
difference in almost all the frameworks. We can conclude that these four
frameworks are applicable to large-scale graphs with dense structures. They
can effectively serve as a bridge to generate ground truth for continuous
subgraph matching, providing valuable support for the training of deep graph
learning models.

##  6 Conclusion

In this paper, we propose the concept of Live Graph Lab, which includes
blockchain based temporal graphs that are openly accessible, fully recorded,
and dynamically evolving over time. Specifically, we instantiate a live graph
using the NFT transaction network and investigate its dynamic properties in a
temporal graph analysis perspective. Our findings reveal both similar and
distinct characteristics when compared to traditional networks like social
networks, citation networks, and the Web. Through comprehensive experiments on
the live graph, we uncover numerous insightful discoveries. The proposed live
graph overcomes the limitations of existing datasets and enhances the
diversity in graph research. We believe that the live graph lab will become an
indispensable resource for the graph community and open up new opportunities.
There are also various other potential use cases, such as identifying whether
an account holds a specific type of tokens or predicting the range of tokens
that the account possesses, etc.

##  7 Border Impact and Limitation

The Live Graph Lab can facilitate researchers from graph community by offering
comprehensive blockchain based graphs via an easily accessible manner.
Insights from this research can be directly applied to improve the design,
security, and user experience of NFT platforms, leading to sustainable growth
of the ecosystem. Meanwhile, as the live graph constantly records all the NFT
transactions in the Ethereum blockchain, the possibility of encountering
malicious activities could become a concern, such as bot transactions or wash
trading, etc. It might also cause potential negative societal impacts. Since
the dataset consists of complete transactions associated with the wallet
addresses, it could enable the tracking of each wallet’s behaviors, habits,
and financial activities. This kind of tracking could be exploited for
targeted advertising, manipulation, or surveillance. The dataset could also
make it possible for malicious actors to analyze the transaction patterns and
manipulate the NFT market, which could lead to unfair practices, price
manipulation, and market instability.

## Acknowledgments and Disclosure of Funding

This research is supported by the National Research Foundation, Singapore
under its Industry Alignment Fund – Pre-positioning (IAF-PP) Funding
Initiative. Any opinions, findings and conclusions or recommendations
expressed in this material are those of the author(s) and do not reflect the
views of National Research Foundation, Singapore. The authors would like to
thank reviewers for their helpful comments. The authors would also like to
thank Zhengtao Jiang for the development of website.

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##  Appendix A Detailed Background

In this section, we explain the terminologies related to Ethereum blockchain,
transactions and NFTs, etc.

###  A.1 Blockchain and Ethereum

Blockchain, a distributed ledger technology, has drawn continuous attention
recent years. Blockchains are made up of securely linked blocks with
cryptography techniques [53], where each block contains information of the
previous block (e.g., cryptographic hash). Then, consensus algorithms or
protocols are applied to validate transactions and keep them being consistent.
By this way, the blockchain transactions are immutable, traceable and publicly
available. Ethereum is a decentralized, programmable blockchain, which means
users can construct various decentralized applications on the blockchain.
Ether (ETH) is the native cryptocurrency of Ethereum, and every transaction
incurred in the Ethereum needs a specific fee paid in ETH. According to the
market capitalization, ETH is the second-largest cryptocurrency behind
Bitcoin.

###  A.2 Account and Transaction

Ethereum account is the key to access and explore the Ethereum ecosystem.
Every Ethereum account is associated with a unique address, akin to an email
address for an inbox. The address can be used to receive or send funds to the
corresponding account. Accounts in the Ethereum can be classified into two
types: (1) Externally-owned account (EOA) and (2) Contract account. All the
accounts are denoted as a 64 character hex string.

Transactions are messages sent from one account to another account. One of the
simplest transaction is transferring ETH from one account to another, which
will change the state of the Ethereum Virtual Machine (EVM) and need to be
broadcast to the whole Ethereum network. Each transaction requires amount of
fees to pay for the computation. The key information in a transaction includes
the receive address, the sender address, value, data, gas limit, and the max
fee per gas, etc. There are three categories of transactions within the
Ethereum: 1) regular transactions, which indicates the transactions between
two externally-owned accounts; 2) contract deployment transactions, which are
special transactions without receive addresses; 3) execution of a contract,
whose receive address is the smart contract address. When the transaction is
submitted, its life cycle can be simplified into the following three steps: 1)
an externally-owned account sends a transaction and generates a transaction
hash; 2) the transaction is broadcast across the network; 3) a validator
verifies the transaction and includes it into a block. Once the transaction is
successfully executed, it can never be altered.

###  A.3 Smart Contract and Non-Fungible Token

Smart contract is an important feature in the Ethereum blockchain [88], which
is a computer program that runs on the Ethereum to automatically execute or
control relevant events and actions according to its logic. Smart contracts
have largely reduced the requirements for trusted intermediaries, fraud losses
and arbitration costs, etc. As we have mentioned before, smart contracts also
belong to a type of Ethereum account, and it can interact with user accounts.
Moreover, anyone can program a smart contract and deploy it to Ethereum, as
long as the code is complied successfully and can be executed by the EVM.
Smart contracts are the fundamental building blocks for various applications
like decentralized finance (DeFi) and game finance (GameFi).

Non-Fungible Token (NFT) is one of the most successful applications on
Ethereum. NFTs are tokens that can be used to represent the ownership of any
unique asset such as image, video and audio. Different from fungible items
where one dollar is exchangeable for another one dollar, NFTs are not
interchangeable with each other, since they all have unique properties and are
not divisible. Smart contracts manage the ownership and the transferability of
NFTs. Specifically, when NFTs are minted or transferred, it triggers the code
stored in the smart contract, and then the relevant actions are executed. Each
NFT token will have an owner after mint, and this information can be easily
verified in Ethereum. The NFTs can be bought and sold on any NFT market like
OpenSea, LooksRare or X2Y2. More recently, the NFT and DeFi have been combined
to form a number of interesting applications including NFT-backed loans,
fractional ownership and certificates of authenticity.

##  Appendix B Related Work

In this section, we present the related work on graph analysis.

Analyses of Social Networks, Citation Networks and the Internet. There exist
lots of prior works focusing on analysing social networks and citation
networks like Flickr, Yahoo! 360 and LiveJournal [5, 38, 39, 41]. These
studies investigate the graph properties including density, degree
distribution and clustering coefficient, etc. Among them, [17] observed that
the node degrees followed a power-law distribution in most real-world
networks. [9] studied the graphs from the perspective of connectivity, where
large strongly connected components (i.e., SCC) widely existed in the graphs.
[23] classified social network’s links into strong ties and weak ties, with
strong ties indicating tighter clustering. [77, 37] explained the social
networks’ small-world phenomenon. [41] showed that the citation graphs
demonstrated denser densities and decreasing diameters as time goes by. Then,
a forest-fire graph generation model was proposed to simulate these phenomena.
[64] proposed a jellyfish model to describe the topology of the Internet,
which abstracted the structure in a human understandable way.

To summarize, the major findings of existing works are as follows: 1) power
law degree distribution, where some nodes exhibit significantly large degrees;
2) preferential attachment growth model, where the likelihood of a new node
establishing a connection with an existing node is directly tied to the degree
of the existing node; 3) density of the graphs follows a rapid decline, and
then becomes steady. For surveys of graph analysis, interested readers can
refer to [27, 76]. Although the graph analyses are extensively studied in
those networks, it is not clear whether these findings are still valid in this
emerging NFT transaction network.

Graph Analyses of Cryptocurrency Transaction Networks. Due to the
decentralized nature of blockchain, this makes it possible for everyone to
access all the transaction information. Several recent works have studied the
properties of Bitcoin and other cryptocurrency transaction networks. For
instance, [26, 59, 48] studied the user behaviors in Bitcoin transaction
network. [66] classified and visualized the information extracted from the
Bitcoin network. [80] and [1] forecast the price of BTC via modeling the local
topological structure of the Bitcoin graph. Apart from Bitcoin, other
cryptocurrencies like Zcash [34], EOS [29] and Monero [51] had also been
conducted similar analyses. [8] analyzed the transaction linkability in Zcash,
which revealed the underlying privacy concerns. [16, 2] discussed the key
factors that impact the scalability of various blockchain systems. [24]
identified arbitrage behaviors among multiple cryptocurrency exchange markets
(e.g., Kraken, Coinbase and Gemini) through weighted cycle detection. However,
the majority of these works are performed on analyzing static graphs, whereas
graphs are usually evolving over time in the real-world scenarios. Moreover,
the aforementioned studies that analyze Bitcoin and other cryptocurrencies
only involve transactions related to value or token transfers. Different from
Bitcoin, recent popular blockchains like Ethereum and Solana support deploying
smart contracts to provide diverse services, where human controlled accounts
and program controlled agents coexist in the network, making the transaction
network even more complicated. It is of great interest to us to investigate
this type of transaction network.

Analyses of the Ethereum Blockchain. Given the possibility to access
comprehensive information within the blockchain transaction network, some
recent efforts follow the pioneer studies on social networks, citation
networks and the Internet [39, 41, 64] to analyze the static Ethereum
transaction network. Particularly, [40] measured four interaction networks to
give new insights on the Ethereum graph properties. [18, 11] characterized
major activities including money transfer and contract creation on Ethereum
via graph analysis. [46] learned from Ethereum graph to perform price anomaly
prediction. [86, 6] investigated the evolutionary dynamics of Ethereum
activities through the lens of temporal graphs.

Instead of studying all the transactions in the Ethereum, [65] only considered
transactions relevant to ERC20 tokens which were fungible tokens circulated in
the Ethereum, and the results showed that they presented obvious social
signals in the trading network. [12] performed a systematic analyses on the
whole ERC20 token activities. [72] studied massive individual token networks
from a graph analysis perspective, and found that they were largely dominated
by a single hub and spoke pattern. Since tokens could be heavily influenced by
various events like mint, burnt, transfer or staking, these aforementioned
works only provide a general intuition about the structure of token
distributions, the flow and spread of assets on the blockchain.

Similar graph analysis approaches have also been applied to Non-fungible
tokens (NFTs), where NFT tokens are unique and not comparable with each other.
For example, [78] used the Louvain algorithm [71] to extract the community
based structures from the networks, which characterized the relationships
between buyers and sellers. [52] showed that the NFT’s sale history and visual
appearance were two good indicators to predict its price. [10] analyzed
several popular NFT projects, and concluded that the structure of NFT networks
was qualitatively similar to social networks. [73] quantified suspicious wash
trading behaviors in NFT market via closed cycle detection in the network.
Although NFTs play a crucial role in the Ethereum ecosystem, none of the
aforementioned studies explore the temporal properties of the NFT transaction
network from the temporal graph point of view.

##  Appendix C Basic Structure Properties

The following four properties are discussed in this section:

Assortativity characterizes the tendencies of nodes getting attached to
similar nodes through a specific metric. Following [4], we calculate the
degree assortativity as follows:

$\alpha=\frac{\frac{1}{|\mathcal{E}|}\sum_{(i,j)\in\mathcal{E}}k_{i}k_{j}-[\frac{1}{|\mathcal{E}|}\sum_{(i,j)\in\mathcal{E}}\frac{1}{2}(k_{i}+k_{j})]^{2}}{\frac{1}{|\mathcal{E}|}\sum_{(i,j)\in\mathcal{E}}\frac{1}{2}[k_{i}^{2}+k_{j}^{2}]-[\frac{1}{|\mathcal{E}|}\sum_{(i,j)\in\mathcal{E}}\frac{1}{2}(k_{i}+k_{j})]^{2}}$
(1)

where $k_{i}$ and $k_{j}$ represent the degrees at the ends of edge
$(i,j)\in\mathcal{E}$. $|\mathcal{E}|$ denotes the number of edges. The
assortativity $\alpha$ lies in the range of [-1, 1], where a positive $\alpha$
indicates that high degree nodes have high probabilities of linking to other
nodes with high degrees on average. In contrast, when $\alpha$ is negative, it
is a disassortative network and the high-degree nodes are more likely to link
to low-degree nodes. More specifically, when $\alpha=0$, we say the network is
neutral, and neither the tendencies of linking to high-degree nor low-degree
nodes are observed. Figure 4(a) shows that the assortativity of NFT
transaction network is negative in the recent six years, and it increases year
by year, gradually approaching to zero. This indicates that the emerging
transaction network is evolving from disassortative to assortative, which
means there are more and more hub nodes available for themselves to connect to
each other to increase the assortativity. A detailed analysis is given in
Section 4.2.

Density calculates the ratio of existing edges over the number of possible
edges [49]. The density $d$ is 1 for a complete network, and it can be larger
than 1 when self-loops or multi-edges are taken into consideration. We compute
the density of a directed network as follows:

$d=\frac{|\mathcal{E}|}{|\mathcal{V}|(|\mathcal{V}|-1)}$ (2)

where $|\mathcal{E}|$ indicates the number of edges and $|\mathcal{V}|$
represents node numbers. As we can see in Figure 4(b), the density drops
rapidly, which indicates the network becomes sparser over time. The decrease
of density is mainly caused by the increment of edges are less than the node
increment. It also means that the network utilization is quite low and the
interactions between different nodes are very limited (i.e., one node only
interacts with a few other nodes). This is understandable, since creating a
account (i.e., node) is free, but making transactions (i.e., creating edges)
cost gas fees. This property is quite different from citation networks [41],
where density becomes denser over the time.

(a) Assortativity

(b) Density

(c) Reciprocity

(d) Clustering Coefficient

Figure 4: Evolution of network global properties with different time
granularity.

Reciprocity in a directed network is determined by the proportion of
bidirectional edges to the number of total edges [67, 3]. Formally, the
reciprocity $r$ is calculated as:

$r=\frac{\sum{\mathbb{I}((i,j)\in\mathcal{E}\bigwedge(j,i)\in\mathcal{E}))}}{|\mathcal{E}|}$
(3)

where $|\mathcal{E}|$ indicates the number of edges. $\mathbb{I}(\cdot)$ is an
indicator function and it returns 1 when node $i$ and $j$ have bidirectional
edge, otherwise it returns 0. The trend of reciprocity is demonstrated in
Figure 4(c). The relation between reciprocity and time is not monotonous. In
general, it increases at the first, and then decreases in the following
several years. This may be due to the emerging of NFT swapping, which allows
users to swap their NFTs with each other. However, as time goes on, the NFT
market becomes more and more mature, and users are prone to trading instead of
swapping their NFTs to gain more profits.

Average Clustering Coefficient evaluates to what extend the nodes in a network
tend to tightly cluster together [61], which is computed by averaging local
clustering coefficient across all nodes. Node $v_{i}$’s local clustering
coefficient $c_{i}$ is the percentage of edges among its neighborhood divided
by all the possible edges between its neighborhood. We formulate the local
clustering coefficient for directed network as:

$c_{i}=\frac{|\\{e_{jk}:v_{j},v_{k}\in\mathcal{N}_{i},e_{jk}\in\mathcal{E}\\}|}{|\mathcal{N}_{i}|(|\mathcal{N}_{i}|-1)}$
(4)

where $\mathcal{N}_{i}=\\{v_{j}:e_{ij}\in\mathcal{E}\bigvee
e_{ji}\in\mathcal{E}\\}$ is the neighborhood of node $i$. Edge $e_{jk}$
indicates the link between node $j$ and node $k$. Thus, the equation for the
average clustering coefficient is as follows:

$c=\frac{1}{|\mathcal{V}|}\sum_{i=1}^{|\mathcal{V}|}c_{i}$ (5)

where $|\mathcal{V}|$ denotes the number of nodes. In Figure 4(d), we present
the clustering coefficient for the NFT transaction network across six years.
We can observe that the average clustering coefficient is growing stably,
which indicates the network is forming an increasing number of tightly
connected clusters or communities. One possible explanation is that those
accounts in a tightly connected clusters are actually controlled by the same
person, and they utilize multiple accounts to conduct money laundering and
wash trading [73], etc.

Furthermore, Figure 4 also illustrates the trends of different properties
under different time granularities. In particular, the networks constructed
with different time granularities could highlight the anomalies during its
evolving procedure. Here, the anomaly means a property value that is much more
larger or smaller than the value in its neighborhood time periods. For
instance, the reciprocity in Figure 4(c) indicates that its value in the
middle of 2018 is twice larger than that of the other time period. Since this
abnormal value is observed in a half-year granularity, we can dive into a
finer time scale, which is 3-month and monthly time scopes. The results are
illustrated in Figure 5. As we can see in Figure 5(a), the reciprocity values
are significantly different between the first and the second half of year
2018. Meanwhile, we further investigate the data with a monthly basis, which
makes it possible for us to locate the specific months. Figure 5(b)
demonstrates that the anomaly is lasting from January to August, and reaches
its peak at February. These finer granularity analyses demonstrate that
monthly data may be more helpful in the scenario of anomaly detection.

(a) 3-months granularity of year 2018

(b) Monthly granularity of year 2018

Figure 5: Finer time granularity analysis on reciprocity.

##  Appendix D Dynamic Behavior Analyses

(a) Number of active accounts

(b) Number of average transactions

Figure 6: Accounts’ active periods.

###  D.1 Active Period of Nodes

Based on our aforementioned analyses, the NFT transaction network is highly
active with new nodes continuously adding in. We now proceed to investigate
the nodes’ active periods. Here, the node’s active period is defined as the
time interval between its first transaction and its last transaction. In this
case, we discard those nodes with only one transaction in the dataset. Figure
6 presents the statistical information for node’s active duration in the scale
of days. Note that, we only show the active periods from 1 day to 40 days in
Figure 6(a). According to our data, 23.43% of the nodes have the active period
of 1 day, and up to 47.25% nodes’ active periods are 1 month. This is
consistent with our previous observations regarding the highly increasing
number of edges. It’s worth noting that only one account has the active period
of nearly 5 years. After checking the data, we find out that this node is
associated with the Null address (i.e., 0x000…000). It is not a surprise,
since all NFT mint activities would build connections with the Null address.
This also indicates that there exist new NFT tokens being continuously minted,
which reveals the fast growing speed of NFT market.

Then, we explore the impact of a node’s active period on the number of
transactions it engages in. One natural hypothesis is that the longer the
node’s active period, the more transactions it will create, which is
consistent with the user behaviours in social networks. Figure 6(b) shows the
number of average transactions with different active periods. As can be seen,
the active period and its average transactions are positively correlated. The
number of transactions increase exponentially as the active periods become
longer. We also notice two spikes with active periods of 8 days and 39 days.
Among them, address 0x610…6627770x6109DD117AA5486605FC85e040ab00163a75c662
causes the spikes of average transactions in active periods of 8 days, which
is a Ethereum Name Service (ENS) migration contract to migrate second-level
names from the old registry and registrar to the new ones. It creates more
than 685,000 transactions within 8 days. To summarize, even through there are
some addresses with rather limited active periods, we can generally conclude
that most of the addresses are quite active with lots of transactions.

###  D.2 Holding Tokens and Collections

(a) Distributions of tokens and collections

(b) Number of tokens and collections

Figure 7: Distributions of tokens and collections.

To reveal the characteristics of the NFT economy, we analyse the distributions
of holding tokens and collections for different accounts. Formally, we trace
all the received tokens and sent tokens for each account. Through this way, we
can know the owner of each token and the quantity of tokens held by an
account. Figure 7(a) shows that both the accounts’ holding tokens and
collections are power law distributions. Specifically, 42.10% of the accounts
hold only one token, and 58.50% of the accounts hold tokens from one NFT
collection. Moreover, there are about 81.70% accounts holding no more than 10
tokens, and 91.40% of accounts hold tokens from no more than 10 collections.
In the middle of year 2022, the largest “holder” is the Null address (i.e.,
0x000…000), which involves 809,125 tokens from 8,126 collections. Note that
the Null address is a special account, and sending tokens to Null address
means destroying the tokens whereas receiving tokens from Null address
indicates minting tokens. There are several reasons that people destroy their
tokens, including reducing the supply to increase a collection’s value or
rectifying error information in the tokens. Then, we look at the next valid
holding address. The second largest holder is associated with address
0x000…7a2 8880x0008d343091ef8bd3efa730f6aae5a26a285c7a2, which holds 381,570
tokens from 1,454 collections. Since it holds so many tokens, we are
interested in uncovering its identity. Therefore, we try to uncover all the
relevant activities associated with the address 0x000…7a2. First of all, we
search the address in the Ethereum Name Service and find that the address is
bound with the name
stronghands.eth999https://app.ens.domains/address/0x0008d343091ef8bd3efa730f6aae5a26a285c7a2.
Then, we search the keyword “stronghands” with Google, and the results show
that it is a blockchain community101010https://www.stronghands.io/, which
supports issuing tokens, trading NFTs and bridge integrations, etc.

Table 6 and Table 7 list the top-10 largest holders in the year of 2020 and
2021, respectively. As can be seen, most top-10 holders in 2020 continue to be
top-10 holders in 2021 as well. It is interesting to note that the relative
positions are almost same, except that address 0xd9a…6a5 and 0x721…ace swap
their positions in 2021, and the Null address becomes the second largest
“holder”. Among them, address 0xff1…2c8 drops out of the top-10 list in 2021,
and address 0xe05…9d5 enters the top-10 list in 2021, which are annotated as
bold. The little difference in top-10 list for year 2020 and 2021 indicates
that it is very difficult to be one of the top holders for new users, since it
costs a lot of money to mint or buy NFTs. Moreover, when looking at the
numbers in Table 6 and Table 7, we can find that the top holders’ tokens and
collections are rapidly growing. One exception is the address 0x09c…16b, which
holds 166K tokens from the same collection (i.e., DozerDoll, a game dirven
NFT). On average, they hold 10K more tokens in 2021 compared with the year
2020. This is because there are more and more tokens as well as collections.
Figure 7(b) also verifies this observation. Take the year of 2021 as an
example, it increases 17,428 collections and 14 million tokens. Thus, the
whole NFT ecosystem is still in its bull market.

Table 6: Top-10 accounts’ holding tokens and collections in 2020. Account Address | Tokens | Collections
---|---|---
0x0008d343091ef8bd3efa730f6aae5a26a285c7a2 | 363,962 | 36
0x26cdee4269273e1ea5dfac6b5791df2656897738 | 343,413 | 14
0xe4a8dfca175cdca4ae370f5b7aaff24bd1c9c8ef | 308,916 | 13
0xf7ee6c2f811b52c72efd167a1bb3f4adaa1e0f89 | 216,477 | 39
0x09c1e4c1adad99436b5c22a395174a1320ee716b | 166,321 | 1
0xf33bd4edc6dcd7240966f20401014ad0018d065b | 161,368 | 19
0xd9ab699e5e196139b8a1c8f70ead01b2137fc6a5 | 152,788 | 16
0x721931508df2764fd4f70c53da646cb8aed16ace | 149,115 | 49
0x0000000000000000000000000000000000000000 | 143,849 | 683
0xff18298382948028f9d93c4e32be1382204022c8 | 140,025 | 22
Table 7: Top-10 accounts’ holding tokens and collections in 2021. Account Address | Tokens | Collections
---|---|---
0x0008d343091ef8bd3efa730f6aae5a26a285c7a2 | 378,893 | 198
0x0000000000000000000000000000000000000000 | 365,565 | 1,914
0x26cdee4269273e1ea5dfac6b5791df2656897738 | 343,413 | 14
0xe4a8dfca175cdca4ae370f5b7aaff24bd1c9c8ef | 308,880 | 13
0xf7ee6c2f811b52c72efd167a1bb3f4adaa1e0f89 | 217,172 | 83
0x09c1e4c1adad99436b5c22a395174a1320ee716b | 166,321 | 1
0xe052113bd7d7700d623414a0a4585bcae754e9d5 | 163,499 | 1,467
0xf33bd4edc6dcd7240966f20401014ad0018d065b | 161,413 | 20
0x721931508df2764fd4f70c53da646cb8aed16ace | 159,464 | 204
0xd9ab699e5e196139b8a1c8f70ead01b2137fc6a5 | 152,788 | 16

###  D.3 Evolution of Diameters

We study the diameter of the NFT transaction network in this section, which
reflects the communication efficiency among different nodes. Generally, the
network’s diameter is defined as the largest shortest path among all pairs of
nodes in the network. As pointed out by [41, 42], this metric is very
sensitive to the noise in the network. For instance, a single long path would
result in a large diameter. Thus, we resort to the effective diameter used in
[41], which is defined as the 90-th percentile of the shortest path length
among all pairs of nodes. Figure 8 presents the diameters as it evolves over
time by years. The blue line shows the diameter calculated with the whole
transaction network, which includes the special Null address. In contrast, the
red line indicates the diameter computed without the Null address, which means
all the NFT minting and destroying activities are removed from the network. We
can observe that these two lines show totally different trends, i.e., one is
increasing and the other is decreasing. The final diameter is about 3.0 in
2022 when including the Null address, and it is only half of the value when
removing the Null address. It is surprised to see that the Null address has
such large influence on this property. With in-depth analysis, we find that
there are about 3.5 million addresses connecting with the Null address, which
accounts for 77.3% of the total addresses. Thus, the Null address is a huge
hub node, and provides a shortcut for nodes to reach each other. As a result,
the diameter is smaller in this situation, and every non-mint action would
increase its value when the network expands.

To eliminate the effect of the Null address, we focus on the analysis of
diameter without Null address, which reveals the transaction behavior patterns
without mint. Similar to citation networks that the diameter is shrinking
observed by Leskovec et al [41], our results show that the diameter also
shrinks in NFT transaction network, i.e., decreasing from 5.94 to 4.72. Then,
we explore the graph structure to see whether we can explain the diameter
shrinking phenomenon. Specifically, we observe that about 42.28% of the nodes
have the degree of 1 in the final network. Those nodes with degree 1 are the
key factor that leads to the increase of diameter. Thus, we remove these
degree 1 nodes, and calculate the effective diameter of the remaining part.
The value is 4.20, which shrinks again. After that, we repeat this process
again, and find that it only has 2.26% nodes with degree 1 in the remaining
part. Similarly, we delete those degree 1 nodes, and compute its effective
diameter again, which is 4.28 and stays the same as before. This suggests that
there is a giant component with extremely high connectivity, and nodes with
degree 1 is only a thin layer at the outside of the well-connected giant
component. This observation brings new challenges for some downstream tasks.

Figure 8: Effective diameter by years.

##  Appendix E More Analyses on Continuous Subgraph Matching

Frameworks. We evaluate the performance of six recent CSM frameworks in this
section. (1) SJ-Tree [13] proposes a lazy search algorithm, and the search
strategy is determined by a vertex-to-vertex basis, which depends on the
likelihood of a matching in the vertex neighborhood. (2) IEDyn [30, 31]
randomly selects a node from query graph, then it products matching order by
conducting DFS on the query graph. (3) SymBi [50] employs a dynamic candidate
space as an auxiliary data structure for filtering. (4) Graphflow [33] first
generates matching order offline, and then retrieves it in online processing.
(5) TurboFlux [35] employs a concise representation of the intermediate
results, and a novel edge transition model is proposed to identify the update
operations that may affect the current solutions. (6) RapidFlow [68] performs
batch subgraph matching via designing a query reduction technique, then dual
matching is utilized to leverage the duality of the graph in the matching
procedure.

Settings. Similar to the link prediction task, we also remove all the
transactions related to the Null address, which gets rid of the impact of the
extreme large degree node. In previous studies [69, 35, 13], the initial graph
is constructed using the first 90% of edges, while the rest 10% of edges are
employed as insertion streams. In our scenario, we know the exact time of each
transaction, thus we use NFT transactions from year 2017 to the end of 2021 as
the initial graph, and then the transactions in the year of 2022 are regarded
as the insertion streams. Since the original nodes and edges do not have fine-
grained labels, each node is assigned a label randomly selected from a pool of
30 labels. We do not assign labels for edges following the previous works [69,
35]. For evaluation metrics, we report the query time, which denotes the time
taken by the online matching procedure to execute. We discard the graph update
time, since it’s same for all the frameworks. To complete the experiments
under an affordable time, a one-hour time limit is imposed for query
processing (i.e., $3.6\times 10^{6}$ ms). Additionally, we also calculate the
number of matched subgraphs for each query graph.

(a) Query time on pattern 1

(b) Query time on pattern 3

Figure 9: Query time with deleted hub nodes.

Results. As we have discussed in Section 4.2, there exist lots of hub nodes in
the NFT transaction network. Therefore, we are interested in whether updating
the index of hub nodes will have a great impact on the query time.
Specifically, except Graphflow, all the remaining frameworks belong to the
index-based incremental computation. In this context, the query time consists
of the index time, which signifies the time spent on updating the index, and
the enumeration time that enumerates the matched results. Therefore, we first
sort the node degrees in the descending order, and the results show that the
top-50 largest hub nodes have the largest degree with 433,114 and the smallest
degree with 7,623. Then, we delete the top-50 hub nodes sequentially and
report the query time in Figure 9 for pattern 1 and pattern 3. Note that, we
remove the randomly assigned node labels in this situation, which counts the
query time associated with the hub nodes. As can be seen, the query time
almost remains the same with acceptable fluctuations across all the frameworks
except for RapidFlow. This is because RapidFlow is extremely efficient (i.e,
only taking several hundred milliseconds), thus a slight fluctuation will be
very obvious in the figure. We can conclude that these four frameworks are
applicable to graphs with dense structures. They can effectively serve as a
bridge to generate ground truth for continuous subgraph matching, providing
valuable support for the training of deep graph learning models.

##  Appendix F Temporal Link Prediction Results on Sampled Subset

To fully evaluate the models’ performance, we also sample a subset of data
spanning from January 1st 2020 to August 31st 2020. This subset contains
88,112 nodes and 203,221 edges. We provide the temporal link prediction
results under live update setting in Table 8. As can be seen from the results,
we can draw similar conclusions. Notably, we also observe that with the
increase in time granularity, the performance of the models exhibits a
corresponding decline within this sub-sampled dataset.

Table 8: Temporal link prediction results with live-update settings in sampled subset. We repeat experiments with 3 random seeds to report the mean as well as standard deviation of AUC and MRR. We also present the results under different time snapshot granularities, e.g., days, weeks and months. Models | Snapshot Days | Snapshot Weeks | Snapshot Months
---|---|---|---
AUC | MRR | AUC | MRR | AUC | MRR
Dyngraph2vec | 77.39$\pm$2.04 | 40.29$\pm$5.75 | 75.33$\pm$7.15 | 36.84$\pm$4.97 | 72.03$\pm$2.73 | 34.83$\pm$4.20
TGCN | 95.49$\pm$0.78 | 62.40$\pm$7.89 | 91.09$\pm$0.46 | 41.69$\pm$7.82 | 86.13$\pm$1.33 | 43.67$\pm$6.23
EvolveGCN | 84.34$\pm$2.81 | 40.91$\pm$7.96 | 79.13$\pm$0.92 | 40.72$\pm$3.82 | 78.40$\pm$3.33 | 42.86$\pm$3.71
GCRN-GRU | 91.41$\pm$1.24 | 33.65$\pm$9.48 | 90.78$\pm$2.43 | 42.88$\pm$0.48 | 82.64$\pm$0.22 | 38.99$\pm$0.18
GCRN-LSTM | 93.97$\pm$1.66 | 41.63$\pm$3.22 | 90.16$\pm$1.83 | 39.20$\pm$7.19 | 83.38$\pm$0.71 | 37.22$\pm$2.63
DynGEM | 95.67$\pm$0.74 | 49.25$\pm$6.01 | 93.34$\pm$0.98 | 45.41$\pm$5.29 | 90.75$\pm$1.29 | 41.38$\pm$0.61
Roland-MA | 97.55$\pm$0.59 | 37.18$\pm$6.76 | 94.59$\pm$0.50 | 44.26$\pm$1.26 | 87.69$\pm$2.17 | 37.98$\pm$4.61
Roland-MLP | 96.38$\pm$1.07 | 60.44$\pm$10.0 | 89.25$\pm$0.89 | 52.32$\pm$3.20 | 87.89$\pm$0.96 | 42.02$\pm$3.37
Roland-GRU | 96.52$\pm$0.08 | 69.34$\pm$2.68 | 92.83$\pm$0.82 | 52.74$\pm$2.60 | 91.01$\pm$0.77 | 41.59$\pm$2.02

(a) TEA plot with day intervals

(b) TET plot with sampled 2 million edges

Figure 10: TEA and TET plots.

##  Appendix G TEA and TET Plots

Following the definitions in [57], we present Temporal Edge Appearance (TEA)
and Temporal Edge Traffic (TET) plots in Figure 10. Specifically, a TEA plot
visualizes the proportion of recurring edges compared to newly joined edges at
each timestamp within a temporal graph. In Figure 10(a), the gray bar
represents the count of edges that are previously observed, while the red bar
signifies the quantity of new edges generated at each subsequent time step.
The TEA plot illustrates that our dataset exhibits a substantial proportion of
newly formed edges at each time step, which means the transaction graph is
highly active. In contrast, a TET plot represents the recurrent pattern of
edges across different time intervals. Edges are colored to indicate whether
they appear solely in the training set (green), solely in the test set (red),
or in both sets (orange). As it is time consuming to plot all the 124 million
edges in the figure, we randomly sample 2 million edges and show the results
in Figure 10(b). We have tried to use all the data, but the plotting process
was unable to complete within 12 hours. The sampled 2 million edges subset
showcases a subtle recurrence pattern, which is consistent with our analyses
in previous sections.


Pascal Giorgi
Fabien Laguillaumie LIRMM UM5506 - CNRS, Université de Montpellier Master ALGO
Cryptography is the discipline that allows securing of the exchange of
information. In this internship, we will focus on a certain branch of this
discipline, secure computation in a network. The main goal of this internship,
illustrated in this report, is to adapt a roster of protocols intended to do
linear algebra. We want to adapt them to do algebra for matrices with
polynomial coefficients. We then wish to make a complete analysis of the
different complexities of these protocols.

# Distributed and secure linear algebra

Hugo Mirault