# Core-based Trend Detection in Blockchain Networks [Scalable Data Science]

Jason Zhu  University of ManitobaWinnipegCanada<EMAIL_ADDRESS>,
Arijit Khan  Aalborg UniversityAalborgDenmark<EMAIL_ADDRESS>and  Cuneyt
Gurcan Akcora  University of ManitobaWinnipegCanada<EMAIL_ADDRESS>
###### Abstract.

Blockchains are now significantly easing trade finance, with billions of
dollars worth of assets being transacted daily. However, analyzing these
networks remains challenging due to the large size and complexity of the data.
We introduce a scalable approach called InnerCore for identifying key actors
in blockchain-based networks and providing a sentiment indicator for the
networks using data depth-based core decomposition and centered-motif
discovery. InnerCore is a computationally efficient, unsupervised approach
suitable for analyzing large temporal graphs. We demonstrate its effectiveness
through case studies on the recent collapse of LunaTerra and the Proof-of-
Stake (PoS) switch of Ethereum, using external ground truth collected by a
leading blockchain analysis company. Our experiments show that InnerCore can
match the qualified analysis accurately without human involvement, automating
blockchain analysis and its trend detection in a scalable manner.

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##  1\. Introduction

Blockchain technology has been gaining increasing popularity over the past few
years, revolutionizing the way we store and transfer digital assets (Nakamoto,
2008; Wood, 2014). Public blockchain networks are also completely open, where
anonymous addresses use transactions to move cryptocurrencies and trade/invest
in assets. While the technology offers numerous benefits, it also poses
significant challenges, particularly in the area of cybersecurity. Blockchains
enable electronic crimes in a variety of ways (Wu et al., 2022), ranging from
demands for ransomware (Huang et al., 2018) to transactions in darknet markets
(Jiang et al., 2021).

One of the biggest challenges in securing blockchain networks is detecting and
preventing e-crimes. E-crimes detection requires scalable analysis of large-
scale blockchain graphs in real-time, where results are both qualified and
manageable by human analysts. To address this challenge, researchers have
developed tools and algorithms for analyzing blockchain networks (Victor et
al., 2021; Akcora et al., 2018; Abay et al., 2019; Su et al., 2022; Khan and
Akcora, 2022).

Unfortunately, analyzing blockchain networks is an arduous task, given their
large size and the anonymity of actors involved. It is crucial to devise
scalable and effective methods that can analyze blockchain networks in real-
time, to avoid future losses. The failure to conduct timely analysis of
blockchain networks has already resulted in a staggering loss of over billions
of dollars to blockchain users, as evident by the recent collapse of LunaTerra
(Faux and Shen, 2022).

In this article, we introduce a new approach to e-crimes and trends detection.
Our approach, InnerCore, involves identifying influential addresses with data
depth-based core decomposition and further filtering out the role of addresses
by using a centered-motif approach. InnerCore analysis can reduce large graphs
having more than 400K nodes and 1M edges to an induced subgraph of less than
300 nodes and 90K edges, while still being able to detect the most influential
nodes. InnerCore is unsupervised and highly scalable, yielding only
$\sim$4-second running times on daily Ethereum graphs with $\sim$500K nodes
and $>$1M edges. We apply InnerCore to two recent influential events in the
blockchain world: the collapse of LunaTerra in May 2022 and the Proof-of-Stake
(PoS) switch of Ethereum in September 2022. Empirical study demonstrates that
our proposed approach effectively detects changes within the network without
the need for human intervention.

Additionally, our method excels at efficiently identifying the most
influential addresses within the network with a high level of precision,
highlighting its utility in identifying key actors in blockchain-based
networks.

Our key novelties are summarized below.

  * •

InnerCore: We propose InnerCore, a data depth-based core discovery method that
can identify the most influential investors and traders in blockchain-based
asset networks (§3.1).

  * •

Explainable behavior: We develop two metrics, InnerCore expansion and decay
(§3.2), that can provide a sentiment indicator for the networks and explain
investor mood (§3.3).

  * •

Unsupervised address discovery: Through conducting node ranking with a
centered-motif approach in temporal asset networks, we demonstrate that
InnerCore tracking can detect e-crime behavior and warn the network about
possible long-term instability, without the need for supervised address
discovery (§3.4).

  * •

Scalability: Due to their computational efficiency and ability to utilize only
a small portion of graph nodes and edges to analyze overall behavior, the
InnerCore discovery and expansion/decay calculations are suitable for
application on large temporal graphs including Ethereum transaction and
stablecoin networks (§4).

We discuss preliminaries and our problem in §2. Additional related works are
specified in the Appendix.

##  2\. Background and Problem Formulation

We discuss preliminaries on blockchain, smart contracts, and stablecoins
(§2.1, §2.2), followed by one key technique AlphaCore decomposition (§2.3). We
introduce our problem in §2.4.

###  2.1. Blockchain and Smart Contracts

A blockchain is an immutable public ledger that records transactions in
discrete data structures called blocks. The earliest blockchains are
cryptocurrencies such as Bitcoin and Litecoin where a transaction is a
transfer of coins.

The Ethereum project (Wood, 2014) was created in July 2015 to provide smart
contract functionality on a blockchain. Smart contracts are Turing complete
software codes that are replicated across a blockchain network. The contracts
ensure deterministic code execution that can be verified publicly.

Smart contracts have implemented mechanisms to trade digital assets, known as
tokens (Victor and Lüders, 2019). Similar to cryptocurrencies, a token is
transferred publicly between accounts (addresses), and may have an associated
value in fiat currency which is arbitrated by token demand and supply in the
real world.

###  2.2. Stablecoins

A stablecoin is a smart contract-based asset whose price is protected against
volatility by i) collateralizing the stablecoin with one or more offline real-
life assets (e.g., USD, gold), ii) using a dual coin, or by iii) employing
algorithmic trading mechanisms (Moin et al., 2019).

In the pegged asset mechanism, an increase in the price is countered by
creating more stablecoins (i.e., coin minting) and selling them to investors
at the pegged price. The dual coin mechanism operates by having a management
coin, referred to as the dual coin, to oversee a stablecoin. The investors of
the dual coin participate in decision making through voting and receive
benefits from the stablecoin’s transactions. In the event that the
stablecoin’s price rises, some of the dual coin will be sold to purchase and
decrease the supply of the stablecoin. Conflicting demand and supply dynamics
of the two coins are assumed to stabilize the stablecoin’s price. However,
investors may lose faith in the stablecoin to such a degree that they might
also not buy the dual coin, however cheap it becomes. Stablecoins that are
based on algorithmic trading do not require collateral for stability. They
achieve stability through the utilization of a blockchain-based algorithm that
adjusts the supply of tokens automatically in response to changes in demand.

###  2.3. Data Depth-Based Core Decomposition

Core decomposition (Malliaros et al., 2020) is a central technique used in
network science to determine the significance of nodes and to find community
structures in a wide range of applications such as biology (Luo et al., 2009),
social networks (Al-garadi et al., 2017), and visualization (Zhang and
Parthasarathy, 2012). One of the best-known representatives of core
decomposition algorithms, graph-$k$-core (Seidman, 1983; Batagelj and
Zaversnik, 2011), finds the maximal subgraph where each node has at least $k$
neighbors in that subgraph. Although the graph-$k$-core algorithm demonstrates
high utility for analysis of graph structural properties, it does not account
for important graph information such as the direction of edges, edge weights,
and node features.

To address these limitations, multiple modifications of graph-$k$-core have
been proposed to tackle task-specific graphs, e.g., graph-$k$-core in weighted
and directed graphs, generalized $k$-core (Al-garadi et al., 2017; Zhou et
al., 2021; Liao et al., 2022; Batagelj and Zaveršnik, 2002; Garas et al.,
2012; Giatsidis et al., 2011). Different from them, AlphaCore (Victor et al.,
2021) is a recent core decomposition algorithm that combines multiple node
properties using the statistical methodology of data depth (Mosler, 2012). The
key idea of data depth is to offer a center-outward ordering of all
observations by assigning a numeric score in $(0,1]$ to each data point with
respect to its position within a cloud of a multivariate probability
distribution. Using such a data depth function designed for directed and
weighted graphs, AlphaCore maps a node with multiple features to a single
numeric score, while preserving its relative importance with respect to other
nodes.

As the AlphaCore decomposition unfolds, the data depth values are repeatedly
updated through the calculation of node property functions and the application
of data depth to the resulting values.

Consider a directed and weighted multigraph, $G(\mathcal{V},\mathcal{E},w)$,
where $\mathcal{V}$ represents the set of nodes and $\mathcal{E}$ is a
multiset of edges. The weight of each edge is designated by the weight
function $w:\mathcal{E}\rightarrow\mathbb{R}^{+}$. In accordance with the
generalized core definitions introduced in (Batagelj and Zaveršnik, 2002), a
node property function can assign a real value to each node $v\in\mathcal{V}$,
based on edge properties such as weight. A node $v$ can be represented by its
feature vector $\textbf{x}\in\mathbb{R}^{d}$, where $d$ features have been
computed for the node $v$.

###### Definition 0 (Mahalanobis depth to the origin (MhDO)).

Let $\textbf{x}\in\mathbb{R}^{d}$ be an observed data point, then Mahalanobis
(MhD) depth of x in respect to a $d$-variate probability distribution $F$ with
mean vector $\mu_{F}\in\mathbb{R}^{d}$ and covariance matrix
$\Sigma_{F}\in\mathbb{R}^{d\times d}$ is given by

(1)
$MhDO_{F}(\textbf{x})=\bigl{(}1+\textbf{x}^{\top}\Sigma^{-1}_{F}\textbf{x}\bigr{)}^{-1},$

$\Sigma_{F}$ is the covariance matrix of $F$. The Mahalanobis data depth to
origin (MhDO) measures the degree of ”outlyingness” of point x (in this
context, the node property column vector) in relation to origin $\mathbf{0}$.

The core value $\alpha$ of a node is established using a data depth threshold
$\epsilon\in[0,1]$ that is applied to remove neighboring high depth nodes
iteratively. Nodes with high property values, such as large edge weights,
generally have a low depth, while nodes with low property values often have a
high depth, such as most blockchain nodes that trade small amounts of tokens.
However, node property values are not the only factor that determines depth;
the community structure around the node also plays a role. Nodes are
considered to be in the $\alpha=(1-\epsilon)$-core if their depth, relative to
themselves, is no more than $\epsilon$.

Why Data Depth? Data depth provides a more precise identification of crucial
nodes compared to state-of-the-art core decomposition algorithms and acts as a
combination of centrality measure and core decomposition (Victor et al.,
2021). Unlike traditional decomposition algorithms, a depth-based
decomposition does not require the specification of multiple feature weighting
parameters to perform effectively on a particular task. We provide a running
example in the Appendix.

###  2.4. Problem Definition

Given a snapshot of weighted, directed, multi-graphs over successive
timestamps, where $G_{t}(\mathcal{V}_{t},\mathcal{E}_{t},w_{t})$ denotes the
graph at timestamp $t$, $\mathcal{V}_{t}$ its set of nodes (i.e.,
investors/traders), and $\mathcal{E}_{t}$ its multiset of edges (i.e.,
transactions) representing the transfer of underlying asset, we aim to (i)
locate the most important node set $S_{t}\subseteq\mathcal{V}_{t}$ at time $t$
such that the behavior of nodes in $S_{t}$ can be used to characterize the
future success of the underlying asset at $t^{\prime}>t$, and (ii) categorize
investors’ behavior in terms of the future health and success of the
underlying asset.

To resolve above problems, we identify nodes in the innermost AlphaCore, as
well as characterize three-node motifs in these innermost cores from our
transaction networks. In our experiments, we demonstrate that nodes in the
innermost AlphaCore are more useful, compared to other notions of important
nodes, in characterizing and predicting the future success of the blockchain
assets.

##  3\. Methodology

In keeping with the daily routine of daily life, blockchain networks are
frequently examined on a daily basis (Casale-Brunet et al., 2021; Chen et al.,
2020). We divide a blockchain network into daily intervals, using a reference
time zone to create a set of snapshot graphs. In a snapshot graph of a
blockchain network, a node represents a trader/investor, whereas an edge
denotes a financial transaction. Next, we define InnerCore, InnerCore expand,
and InnerCore decay on the snapshot graphs.

We use in-degree, out-degree, in-strength, and out-strength as node properties
(defined in Table 1) to compute the AlphaCore decomposition of a snapshot
graph, as these node features can be defined easily for a weighted, directed,
multi-graph. Core decomposition helps us eliminate unimportant edges and nodes
(e.g., addresses trading small amounts). Using the results of the core
decomposition, we then identify an InnerCore of nodes, which helps us pinpoint
the most influential nodes.

Table 1. Example node property functions. Function | Definition
---|---
$N(v)$ |  neighbors of $v$
$N_{out}(v)$ |  neighbors reachable with outgoing edges from $v$
$N_{in}(v)$ |  neighbors reachable with incoming edges to $v$
$deg(v)$ |  edges to/from $v$ (Degree)
$deg_{out}(v)$ |  outgoing edges from $v$ (Out-Degree)
$deg_{in}(v)$ |  incoming edges to $v$ (In-Degree)
$S(v)$ | sum of edge weights incident to a node (Strength)
$S_{out}(v)$ | sum of outgoing edge weights (Out-Strength)
$S_{in}(v)$ | sum of incoming edge weights (In-Strength)

###  3.1. InnerCore of a Graph

The data depth of a node $v\in V$ is defined as the degree of ”outlyingness”
of the node property function in relation to the origin $\mathbf{0}$. We
define the InnerCore of $G$ as the set of nodes $\mathcal{V}^{inner}$ whose
data depth, relative to themselves, is less than an $\epsilon$ value. We set
$\epsilon$ to a small value, and iteratively recompute the depth of each node
as we remove nodes whose data depth is greater than $\epsilon$ in each
iteration. This process continues until no more nodes can be removed. The
resulting set of nodes is the InnerCore of the graph. The InnerCore
computation is illustrated in Algorithm 1.

Input: Directed, weighted, multigraph $G(V,E,w)$,

Set of node property functions $p_{1},...,p_{n}\in P$,

Data depth threshold $\epsilon$

Output: innerCore $V^{inner}$

// Compute feature matrix

1 $F=[f_{1},...,f_{n}]=\forall p_{i}\in P:f_{i}=p_{i}(v,G),\forall v\in V$;

$\Sigma_{F}^{-1}$ = cov$(F)^{-1}$;

// compute only once

// Compute initial depth values

2 $z=[z_{1},...,z_{n}]=\forall v_{i}\in
V:z_{i}=[1+(F_{i,*})^{\prime}\Sigma^{-1}_{F}(F_{i,*})]^{-1}$;

3 do

4 foreach  _$z_{i}\geq\epsilon$_ do

5 $\mathcal{V}=\mathcal{V}\setminus\\{v_{i}\\}$;

6

// recompute node properties

7 $F=\forall p_{i}\in P:p_{i}(v,G),\forall v\in V$;

// recompute depth

8 $z_{i}=[1+(F_{i,*})^{\prime}\Sigma^{-1}_{F}(F_{i,*})]^{-1},\forall v_{i}\in
V$;

9

while  _$\exists z_{i}:(z_{i}\geq\epsilon)\wedge(v_{i}\in V)$ // one iteration
_

10;

return _$\mathcal{V}$ // as innerCore $V^{inner}$ _

Algorithm 1 InnerCore Discovery

Algorithm 1 computes a feature matrix based on each node property function in
line 1. For instance, this could include a node’s neighborhood size as listed
in Table 1. The feature matrix $F$ is used to compute the inverse covariance
matrix $\Sigma_{F}$ in line 1, which will be utilized for future data depth
calculations. The initial depth of each node is determined using the
Mahalanobis depth with respect to the origin at line 1. Nodes with depth
greater than or equal to input $\epsilon$ are removed from the node set
$\mathcal{V}$ at line 1. Once one batch of node removals has been performed,
the feature matrix and depth values are re-evaluated in lines 1–1. If any
remaining nodes still have a depth greater than or equal to $\epsilon$, the
next batch is initiated at the same $\epsilon$ level. When there are no nodes
left with depth larger than $\epsilon$, the algorithm is considered complete,
and the remaining nodes in $\mathcal{V}$ are returned as the InnerCore.

Scalability. Computing the InnerCore requires performing Cholesky
decomposition on the covariance matrix at line 1 once, which has time
complexity $O(d^{3})$ for $d$ features. Node features need to be recomputed at
each iteration of the while loop with a cost of $O(|\mathcal{V}|\times deg)$,
where $deg$ is the average degree in the graph. There are at most
$|\mathcal{V}|$ iterations (number of nodes). In the worst case, the total
time complexity is $O(d^{3}+|\mathcal{V}|\times deg\times|\mathcal{V}|)$.
However, since the neighborhood of a node can be sparse, the value of $deg$ is
small. Moreover, since multiple nodes are removed in batches, the number of
iterations is much smaller than $|\mathcal{V}|$. For example, in a network
with approximately 480,000 nodes and 1 million edges (§4), only 4 iterations
on average are needed for an $\epsilon$ of 0.1.

InnerCore vs. Alphacore. InnerCore discovery of a graph $G$ does not require a
complete decomposition of all graph cores by varying $\epsilon$, as it is done
in AlphaCore (Victor et al., 2021). Instead, we set an $\epsilon$ value (e.g.,
$\epsilon=0.1$) just once, and then use the value to iteratively prune nodes
until all remaining nodes, relative to themselves, satisfy a data depth less
than $\epsilon$. The InnerCore approach is also different from graph-$k$-core
decomposition (Batagelj and Zaversnik, 2011), where the outer cores are
computed first before the higher $k$-core can be determined. As a result,
InnerCore discovery is quite scalable and can be applied to very large graphs.
Our experiments in §4 reveal that InnerCore discovery has a running time that
is only one-tenth of that required for AlphaCore decomposition.

###  3.2. InnerCore Expansion and Decay

By analyzing how a temporal graph expands and shrinks in relation to the entry
and exit of nodes on a daily basis, we can gain valuable insights into market
sentiment. Specifically, we investigate how the InnerCore of a network expands
and decays on a given day compared to previous days. We define two measures to
quantify the activity of influential nodes in the network: expansion and
decay. Expansion measures the ratio of influential nodes on day $t$ that were
also influential in the preceding $i$ days, while decay quantifies the ratio
of influential nodes from the previous $i$ days that are not present in the
influential nodes of day $t$. We define the influential nodes of a graph as
its InnerCore nodes (i.e., $\mathcal{V}_{t}^{inner}$).

To this end, we first discover $\mathcal{V}_{t}^{inner}$ as the set of nodes
in the InnerCore of the snapshot graph at timestamp $t$, and define
$\mathcal{V}_{\cup(t-i)}^{inner}=\bigcup_{i}\mathcal{V}_{t-i}^{inner}$ as the
union set of nodes in the InnerCore of snapshot graphs from timestamps
$\\{t-1,t-2,\ldots,t-i\\}$ for $i\geq 1$. Next, we define the expansion and
decay measures at timestamp $t$ as follows:

###### Definition 0 (Expansion).

$\mathbb{E}_{t}=\left|\mathcal{V}^{inner}_{t}\setminus\mathcal{V}^{inner}_{\cup(t-i)}\right|/\left|\mathcal{V}^{inner}_{\cup(t-i)}\right|.$

The expansion values have a range $[0,\infty)$, where a value greater than 1
indicates a growth in the size of InnerCore.

###### Definition 0 (Decay).

$\mathbb{D}_{t}=\left|\mathcal{V}^{inner}_{\cup(t-i)}\setminus\mathcal{V}^{inner}_{t}\right|/\left|\mathcal{V}^{inner}_{\cup(t-i)}\right|.$

The decay values have a range $[0,1]$, where a value of 0 indicates that all
InnerCore members from $t$ are present at $t+1$ as well.

###### Example 1 (Expansion and Decay).

Suppose we have a temporal graph that produces two daily snapshot graphs at
days $t$ and $t+1$. On day $t$, the InnerCore is composed of five nodes:
$\mathcal{V}^{inner}_{t}=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$. On day $t+1$,
the InnerCore has expanded to include eight nodes:
$\mathcal{V}^{inner}_{t+1}=\\{v_{3},v_{4},v_{5},v_{6},v_{7},v_{8},v_{9},v_{10}\\}$.

If we set $i=1$, we can calculate the expansion and decay measures for day
$t+1$ based on the previous day. In this case, the union of the InnerCores is
$\mathcal{V}^{inner}_{\cup(t-i)}=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$.
Therefore, we have:

The expansion measure $\mathbb{E}_{t+1}$ is equal to
$\frac{\left|\\{v_{6},v_{7},v_{8},v_{9},v_{10}\\}\right|}{\left|\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}\right|}$,
which yields a value of 1. The decay measure $\mathbb{D}_{t+1}$ is equal to
$\frac{\left|\\{v_{1},v_{2}\\}\right|}{\left|\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}\right|}$,
which yields a value of 0.4.

###  3.3. Behavioral Patterns in Temporal Networks

Temporal networks, such as blockchain networks, are constantly evolving and
may undergo significant changes in user sentiment and node activity due to
technological updates and cataclysmic events in as little as a few days.

By utilizing expansion and decay, we have identified four behavioral patterns
that capture user sentiment and node activity. These patterns serve as the
foundation for network analysis in our experiments detailed in §4. Figure 1
illustrates the expansion and decay values for each pattern. To gain a better
understanding of these patterns, particularly when examining the temporal
graph of a financial network such as the Ethereum transaction network, it is
helpful to consider the network’s underlying transaction semantics.

  * •

The Despair pattern is characterized by a reduction in expansion and an
increase in decay, implying that previously influential nodes are leaving the
network, while the InnerCore is shrinking due to a decrease in the number of
new influential nodes.

  * •

The Uncertainty pattern is distinguished by an increase in both expansion and
decay. This is primarily due to the influx of many new investors into the
network who do not remain active for a significant period of time.

  * •

The Hope pattern is characterized by a reduction in decay and an increase in
expansion, indicating the presence of many newcomers to the network who remain
active within the network.

  * •

The Faith pattern is identified by a decrease in both decay and expansion,
which initially suggests a state of confusion. On the positive side, nodes,
such as investors, may have faith in the network’s ability to withstand a
catastrophic event, as demonstrated in the LunaTerra case in our experimental
results. On the negative side, it may indicate a sense of hopelessness as
investors may hold onto their assets without engaging in transactions or
exiting the system altogether.

Figure 1. In a temporal graph (e.g., transaction network), changes in decay
and expansion reflect varying levels of hope, despair, uncertainty, and faith
in the asset being represented.

###  3.4. Motif Analysis in Innercore

Our rationale behind using motif analysis in conjunction with InnerCore is to
accurately discover larger and potentially influential players in the daily
network. The structure of a motif defines a behavior of interest and its
existence in a network indicates the presence of such behavior.

Figure 2. Five three-node motifs exhibiting buy and sell behaviors. Nodes
labeled C denote the center where a center with an in-degree = 2 indicates buy
behavior and an out-degree = 2 indicates sell behavior. Out of the 16
connected three-node motifs (see Figure 1B in (Milo et al., 2002)), only the
five given above (motifs 1, 4, 5, 6, and 11) contain a center node.

Motif analysis has been a popular tool to identify subgraph patterns and the
addresses involved in them (Lee et al., 2020; Bailey et al., 2009; Zhang and
Parthasarathy, 2012; Paranjape et al., 2017; Milo et al., 2002). We have
decided to use three-node motifs since they can be identified more quickly
than higher-order motifs, while still capturing the direct buying or selling
behavior between addresses. Our decision is consistent with previous research
on temporal motifs (Paranjape et al., 2017).

Scalability. The fastest triangular motif discovery algorithm has time
complexity $O(|\mathcal{V}^{inner}|^{\omega})$, where $\omega<2.376$ is the
fast matrix product exponent (Latapy, 2008; Coppersmith and Winograd, 1987).
The number of nodes in the InnerCore is denoted by $|\mathcal{V}^{inner}|$. We
demonstrate in §4 that triangular motif discovery on InnerCores has low time
costs because of the relatively small size of daily networks’ InnerCores.

We define the center of each 3-node motif as a node that either receives
incoming edges from the two other nodes (buy behavior) or delivers outgoing
edges to two other nodes (sell behavior). This definition ensures that motif
centers exhibit buy or sell behavior, and they do not act as intermediary
nodes between the other two nodes in a motif.

Out of the 16 connected three-node motifs (see Figure 1B in (Milo et al.,
2002)), only five of them contain a center node (Figure 2). We identify all
instances of these five motifs and their centers from our daily networks’
InnerCores. Finally, we utilize the well-known TF-IDF measure from information
retrieval (Salton and Buckley, 1988) to rank the discovered center nodes.

TF-IDF is a statistical measure to reflect the relevance of a word in a
collection of documents. In our setting, we treat each discovered center
address as a word and daily instances of each motif as a collection of
documents to propose a novel node relevance score for temporal graphs: NF-IAF.

Formally, let $M={m_{1},m_{4},m_{5},m_{6},m_{11}}$ be the set of five motifs
of interest, and let $T={t_{1},t_{2},\dots,t_{n}}$ be the set of $n$ days
under consideration. For each $m_{i}\in M$ and $t_{j}\in T$, let
${c(v,m_{i},t_{j})}$ denote the number of occurrences of node
$v\in\mathcal{V}^{inner}$ in all instances of motif $m_{i}$ on day $t_{j}$.
For all $v\in\mathcal{V}^{inner}$, $m_{i}\in M$, and $t_{j}\in T$, we define
the node frequency (NF) and inverse-appearance frequency (IAF) as follows:

###### Definition 0 (Node Frequency).

We define the node frequency of node $v$ for motif $m_{i}$ on day $t_{j}$ as

$NF(v,m_{i},t_{j})=\frac{c(v,m_{i},t_{j})}{\sum\limits_{v\in\mathcal{V}_{j}^{inner}}{c(v,m_{i},t_{j})}}.$

The NF measures how frequently a particular node occurs in a specific motif on
a specific day relative to the total number occurrences of all nodes in that
motif on that day.

###### Definition 0 (Inverse Appearance Frequency).

We define the inverse appearance frequency of node $v$ for motif $m_{i}$ as

$IAF(v,m_{i})=\log\frac{|T|}{df(v,m_{i})}$

where $|T|$ is the total number of days in the dataset, and $df(v,m_{i})$ is
defined as the number of days $t_{j}\in T$ where $c(v,m_{i},t_{j})>0$.

The IAF measures the importance of a node by how frequently it appears across
all days for a motif. If a node appears in many days for a motif, its IAF will
be low, indicating that it is not very informative. On the other hand, if a
node appears in only a few days for a motif, its IAF will be high, indicating
that it is a rare and potentially important node.

###### Definition 0 (NF-IAF Score).

The NF-IAF score of node $v$ for motif $m_{i}$ on day $t_{j}$ is given as

$NF{\text{-}}IAF(v,m_{i},t_{t})=NF(v,m_{i},t_{j})\times IAF(v,m_{i}).$

A greater NF-IAF score of a center node on a particular day indicates greater
relevance between that node and the behavior associated with the motif type.
Therefore, a node corresponding to a motif center on a particular day with a
high NF-IAF score has an increased likelihood that it has more influence on
the network on that day, while a lower NF-IAF score indicates the opposite.

###### Example 2.

Table 2 shows the number of occurrences of three nodes over three days for
motifs $m_{4}$ and $m_{5}$. For example, to compute
${NF}(v_{1},m_{4},t_{1})=\frac{5}{5+15+0}=0.25$, we divide the number of times
$v_{1}$ appears in instances of $m_{4}$ on day $t_{1}$, by the total number of
occurrences of all nodes in instances of $m_{4}$ on day $t_{1}$. Similarly, we
compute $IAF(v_{1},m_{4},t_{1})=\frac{3}{3}=1$ as $v_{1}$ appears in all three
days for $m_{4}$. Thus, $NF{\text{-}}IAF(v_{1},m_{4},t_{1})=0.25\times 1$. The
resulting NF-IAF score for each node, motif, and day combination is given in
the right panel of Table 2.

Table 2. Occurrences and NF-IAF scores of nodes $v_{1},v_{2}$, and $v_{3}$ across three days $t_{1},t_{2}$, and $t_{3}$ in instances of motifs $m_{4}$ and $m_{5}$. $v_{3}$ does not appear for motif $m_{4}$ on any day, whereas $v_{1}$ does not appear on days $t_{1}$ and $t_{2}$ for motif $m_{5}$. | Occurrence | NF-IAF Score
---|---|---
| $m_{4}$ | $m_{5}$ | $m_{4}$ | $m_{5}$
node | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{1}$ | $t_{2}$ | $t_{3}$
$v_{1}$ | 5 | 4 | 3 | 0 | 0 | 15 | 0.25 | 0.25 | 0.25 | 0 | 0 | 0.92
$v_{2}$ | 15 | 12 | 9 | 4 | 7 | 9 | 0.75 | 0.75 | 0.75 | 0.16 | 0.23 | 0.26
$v_{3}$ | 0 | 0 | 0 | 21 | 23 | 35 | 0 | 0 | 0 | 0.84 | 0.74 | 0.71

##  4\. Experimental Study

In this section we first describe two large temporal blockchain graphs that we
use to answer our research questions (§2.4). Next, we analyze the scalability
of InnerCore discovery and centered-motif analysis on these graphs. Upon
demonstrating our scalability results, we illustrate how our methods provide
predictive insights into anomalies stemming from external events and identify
the addresses that played a significant role in such events. Our code and
datasets are available at https://github.com/JZ-FSDev/InnerCore.

###  4.1. Environment Setup

####  4.1.1. Datasets

Our experiments investigate the Ethereum transaction network and Ethereum
stablecoin networks during three events: the LunaTerra collapse, Ethereum PoS
(proof-of-stake) phase 1, and Ethereum PoS phase 2.

Ethereum Stablecoin Networks. We retrieve transaction data for the top five
stablecoins based on market capitalization (USDT, USDC, DAI, UST, PAX) and
WLUNA from the Chartalist repository (Shamsi et al., 2022). The data pertains
only to transactions conducted on the Ethereum blockchain; each transaction in
the data set corresponds to a transfer of the asset indicated by the contract
address. However, the UST collapse event that we are studying involved another
blockchain called Terra with its own network, and the cryptocurrency called
Luna, acting as a parallel to ether on Ethereum. Terra also issued a
stablecoin named TerraUSD, which offered high-interest rates to lenders.
Additionally, Terra’s owners created another stablecoin called UST on the
Ethereum blockchain, which was pegged to the value of USD$1. Furthermore, a
Wrapped LUNA (WLUNA) token was established to trade Luna tokens on Ethereum.

In May 2022, the Terra blockchain and its cryptocurrency Luna collapsed, owing
to TerraUSD loans that could not be repaid. A Luna coin that was valued at
$USD116 in April plummeted to a fraction of a penny during the collapse. This
resulted in a loss of confidence in both WLUNA and UST on Ethereum. On May
9th, 2022, UST lost its $USD1 peg and fell as low as 35 cents. The dataset
covers the period from April 1st, 2022, to November 1st, 2022, spanning about
one month before the crash to six months after the crash. In addition to the
transactions, we also use the address labels dataset from (Shamsi et al.,
2022) where labels of 296 addresses from 149 centralized and decentralized
Ethereum exchange addresses are listed publicly.

Ethereum Transaction Network. We collected ether transactions from the
Ethereum blockchain for the period between August 23rd and September 29th,
2022. On an average day during this period, there were 480,000 addresses, with
approximately 1 million edges connecting them. Ether is a type of
cryptocurrency, similar to bitcoin, and its value can be converted to various
fiat currencies such as USD and JPY. The nodes on the graph could represent
investors who anticipate future price increases for ether, or traders who
engage in buying and selling goods and services. Ethereum changed its block
creation process during this time, moving from the costly Proof-of-Work method
to the more efficient Proof-of-Stake algorithm in two phases on September 9th
and 15th, 2022.

####  4.1.2. Parameters

In the context of InnerCore expansion and decay, a greater $i$ (i.e., the
history parameter from §3.2) produces an averaging effect, coupled with the
tendency to depress expansion rates and inflate decay rates. Setting a
specific $i$ value depends on the application; in general, higher $i$ reduces
fluctuations between expansion and decay rates of each day. In our
applications, we use $i$ = 1 to improve the accentuation of expansion and
decay in the InnerCore to better depict the shift in market sentiment during
the days of significant events in our case studies.

In AlphaCore decomposition, depth values range between $(0,1]$; nodes with
high property values (e.g., many transactions, higher transacted amounts) tend
to have low depth, while nodes with low property values tend to have high
depth (Victor et al., 2021). With data depth threshold $\epsilon=1$, all nodes
will be returned as InnerCore members; while for $\epsilon=0$, the empty set
will be returned. Setting an appropriate $\epsilon$ depends on the desired
size of the InnerCore returned specific to an application. In our
applications, we set $\epsilon=0.1$ to ensure that the average number of nodes
in each daily InnerCore is above 150.

###  4.2. Scalability Analysis

System Specifications. The machine used for experiments is an Intel Core
i7-8700K CPU @ 3.70GHz processor, 32.0GB RAM, Windows10 OS, and GeForce
GTX1070 GPU. A combination of Python and R was used for coding.

Figure 3. Comparison between the running times of AlphaCore with the starting
$\epsilon=1.0$ and stepsize $s=0.1$ and InnerCore with $\epsilon=0.1$ on daily
Ethereum transaction networks to return the InnerCore of depth ¡ 0.1. An
average of approximately 480,000 nodes (addresses) and 1 million edges
(transactions) exist in each network. The average computation time for
InnerCore is 4.06 seconds (max 8.1s), which is approximately 0.10485 times the
average computation time of AlphaCore.

InnerCore Discovery. Since we are interested in directly finding the
InnerCore, compared to AlphaCore decomposition (Victor et al., 2021), our
InnerCore discovery method (§3.1) does not associate different $\epsilon$
values to intermediate cores generated in an iterative stepwise fashion.
Instead, a fixed threshold $\epsilon$, or upper bound for depth, is set and
all nodes with a depth greater than $\epsilon$ are pruned repetitively until
all remaining nodes relative to each other in the resulting network have a
depth ¡ $\epsilon$. Effectively, this allows InnerCore discovery to run
approximately 1/stepsize times faster than AlphaCore decomposition, since the
computations of all intermediate cores are skipped. As depicted in Figure 3,
InnerCore discovery has a running time of only one-tenth of that for AlphaCore
decomposition. Furthermore, the average running time for InnerCore discovery
is only 4.06 seconds on graphs with approximately 480,000 nodes and 1 million
edges, which demonstrates the scalability of our approach.

Three-Node Motifs Counting. Instead of conducting motif analysis on all nodes,
our approach utilizes the InnerCore. By focusing on this core subset of nodes,
we are able to reduce the number of nodes in a daily network consisting of
approximately 480,000 nodes and 1 million edges to an induced subgraph of
roughly 300 nodes and 90,000 edges (counting multi-edges), resulting in a more
manageable and efficient approach. Although motif counting on each snapshot
graph takes $>$ 1 day to complete, motif counting inside InnerCore
significantly improves the processing speed, requiring only $<$ 1 sec to
complete, which illustrates our scalability.

###  4.3. Case 1: The Collapse of LunaTerra

Stablecoins are meant to be a safe house as they are generally pegged to and
maintain a 1:1 ratio with a fiat currency, resisting the volatility associated
with other popular cryptocurrencies. Commonly, investors keep blockchain
assets not needed for immediate use in a transaction as a stablecoin,
analogous to people keeping extra money in a bank. For this reason, the
LunaTerra collapse was a historic event in the decentralized financial space
as it questioned investors’ trust in cryptocurrencies; if even stablecoins are
susceptible to collapse, then is any cryptocurrency truly safe?

First, we analyze this event from the perspective of investors’ market
sentiment in the stablecoin network. Specifically, we examine behavioral
patterns in the expansion and decay measures of the temporal stablecoin
network of the days surrounding the collapse. From Figure 4, we observe that
four days after the collapse unfolded, on May 13, 2022, there is a substantial
increase in decay and a decrease in expansion: a prime indicator of the
despair behavioral pattern (§3.3). We can infer from this signal that a large
majority of regular investors stopped trading by this time, either from the
conversion or sale of any assets stored as UST out of the stablecoin
ecosystem, or simply due to uncertainty and inaction in response to the
collapse. Following this cue, for approximately two weeks afterward, we see a
consistent behavioral pattern of faith characterized by low expansion and low
decay. During this period, few new investors entered or left the stablecoin
network. There was still faith in the remaining investors that perhaps a large
stablecoin such as UST could rebound and restore its peg with USD and thus,
they refrained from engaging in any transactions. On the other hand, decay and
expansion values also indicate a sign of hopelessness as bulk of investors
already exited the network since the first signal of despair. We understand
from this behavioral analysis that there is a delayed reaction from investors
when a significant unannounced event occurs due to indecision, and there is a
general trend of inactivity in the following period.

Figure 4. LunaTerra decay and expansion measures. On May 9 (shown with the vertical blue line), UST loses its $1 peg and falls to as low as 35 cents. Figure 5. Sizes of stablecoin network cores during LunaTerra collapse. Innercores accentuate the collapse better than nodes found in the highest cores via graph-$k$-core decomposition.  Table 3. Numbers of center addresses in motifs identified by our method (§3.4) that are known exchanges. The numbers represent total counts per motif across all days. | # Unique Addresses | # Exchange Addresses
---|---|---
Motif 1 | 1221 | 15
Motif 4 | 1762 | 15
Motif 5 | 1447 | 17
Motif 6 | 1513 | 4
Motif 11 | 939 | 11
Table 4. NF-IAF score-based percentile ranks of InnerCore motif centers matching highlighted addresses by Nansen.ai that played key roles on May 8, 2022 in LunaTerra collapse.  Address/Motif Center | $C_{1}$ | $C_{4}$ | $C_{5a}$ | $C_{5b}$ | $C_{6}$ | $C_{11}$
---|---|---|---|---|---|---
Celsius | - | 81.2 | - | 78.9 | - | -
hs0327.eth | 88.2 | 67.0 | 96.2 | 69.7 | 81.6 | -
Smart LP: 0x413 | - | 68.4 | - | - | 95.1 | -
Token Millionaire 1 | 84.6 | 89.7 | - | 86.2 | 74.2 | 88.9
Token Millionaire 2 | 69.7 | 99.5 | - | 98.7 | 98.9 | 37.6
masknft.eth | 90.9 | 90.7 | - | 82.1 | 93.3 | 92.2
Heavy Dex Trader | 71.3 | 96.2 | - | - | 81.2 | -
Oapital | 91.6 | 78.9 | 60.5 | 58.5 | 71.8 | 92.5
Hodlnaut | 39.9 | 98.9 | - | 90.6 | 99.4 | -

Compared to traditional graph-$k$-core decomposition where the target is the
highest $k$-core, AlphaCore with a fixed core target of $\epsilon$ = 0.1
(i.e., $\alpha$ = 1 - 0.1 = 0.9) better accentuates the LunaTerra collapse
when applied to temporal networks and comparing the sizes of the target cores.
From Figure 5, it is evident that the size of the highest graph-$k$-core
fluctuates randomly across each day and does not respond to changes in the
health of the network. This can be attributed to the fact that graph-$k$-core
decomposition only considers the degree of nodes irrespective of the edge
weights. Conversely, AlphaCore considers all the provided node features (§3
and Table 1) into consideration when computing each node’s depth, which
includes edge weights in addition to node degree. Therefore, if a node is
incident to a single edge of extremely high edge weight, the highest
graph-$k$-core will exclude this node; whereas AlphaCore will, depending on
the target $\epsilon$, maintain it in its InnerCore as the high edge weight
offsets the low node degree.

We next focus on the unknown addresses that occurred most frequently as motif
centers in InnerCores (defined in §3.4) on days immediately before the Luna
Terra crash, since they could have influenced the initial phase of the crash.

Generally, a large amount of tokens transferred from one address to another is
easily detectable due to the sheer volume. However, if an investor tries to
confiscate detection, the investor could produce multiple transactions with
smaller volumes. Additionally, often in a transaction where one token is
exchanged for another, a series of multiple transfers can arise for a single
conversion transaction due to interactions with
exchanges.111https://etherscan.io/tx/0xa3663b813b2c13a88daeeb5b48b32b7024fc07cbf250f2c2a9318ec1950c9da9
Therefore, an investor is more likely to create transfers to various receiving
addresses and receive transfers from various sending addresses, making the
investor a prime candidate as a 3-node motif center.

To capture the transactions of investors converting between different
stablecoins, we have included four stablecoins in our network along with UST.
Specifically, before the LunaTerra collapse, it is reasonable to assume that
investors responsible for the collapse would prepare for the anticipated
negative consequences by exiting the UST network and entering another reliable
stablecoin.

In a conversion transaction, participants sell their current assets to acquire
different assets. Therefore, we anticipate that the investors responsible for
the LunaTerra collapse will be highly relevant motif centers exhibiting both
selling and buying behaviors.

Ground Truth. Nansen (https://www.nansen.ai/) is a prominent blockchain
analytics platform that frequently publishes comprehensive analyses of
blockchain events, which are followed with great interest by the industry. Six
forensics experts from Nansen.ai conducted a thorough analysis of the
LunaTerra collapse in May 2022 and identified 11 important addresses that
played central roles in the collapse (Barthere et al., 2022). We aim to
compare the addresses of interest detected by our InnerCore analysis using the
centered-motif approach with those identified by Nansen.ai (Table 4) as the
primary candidates for triggering the collapse.

Exchanges are an intermediary hub to facilitate transfers between investors.
The addresses of exchanges are well-known for this reason, making them not
very interesting in our context. In contrast, addresses that are not exchanges
are mostly owned by investors and thus, the existence of such addresses and
their edges in a network is a direct consequence of an investor’s activity in
the network. From Table 3, we observe that motif centers identified from
InnerCores have a high ratio of non-exchange addresses to exchange addresses
($\approx$99%). This shows the effectiveness of our method to identify
potentially meaningful addresses in a network different from high-traffic
exchange addresses.

In particular, we capture 9 of 11 externally owned addresses (EoAs) in Table 4
identified by Nansen.ai that occurred as center addresses for our motif types
(Figure 2) on days immediately leading up to the LunaTerra collapse. We notice
that the NF-IAF score-based percentile ranks of these addresses are higher
compared to that of other center addresses for the same motif type on the same
day, indicating that these addresses were important investors contributing to
the buy or sell behavior associated with the motif on the day. We surmise the
possibility that certain EoAs found by our InnerCore method, coupled with
centered-motif analysis, could have been responsible for the initial phase of
the collapse.

Recall that in Figure 2, we defined motif centers $C_{1}$, $C_{5a}$, and
$C_{11}$ as exhibiting sell behavior; while motif centers $C_{4}$, $C_{5b}$,
and $C_{6}$ as exhibiting buy behavior. It is evident from Table 4 that every
motif center on May 8, 2022 has at least one corresponding investor with an
NF-IAF score-based percentile rank above 91. This suggests that addresses with
greater NF-IAF percentiles exhibit a higher buy or sell behavior associated
with the particular motif type on the day before the collapse. Specifically,
we identify the two investors, masknft.eth and Oapital, as the most likely
candidates for influencing the initial phase of the crash, since they are the
two addresses with greater NF-IAF percentiles (above 90) occurring
consistently across at least two motif types exhibiting sell behavior.

K-Core vs. InnerCore. We compare the addresses identified by InnerCore \+
centered-motif analysis with those in the highest graph-$k$-core on the May 8
(i.e., one day before the crash) stablecoin temporal network. We find that
graph-$k$-core cannot identify any of the 11 addresses indicated by Nansen.ai
as prime candidates for triggering the initial phase of the LunaTerra
collapse. In comparison, InnerCore \+ centered-motif analysis captures
potentially anomalous buy and sell behaviors by identifying 9 of the 11
addresses.

###  4.4. Case 2: Ethereum’s Switch to Proof-of-Stake

Ethereum’s transition from Proof-of-Work (PoW) to Proof-of-Stake (PoS) came
with many benefits including enhanced security for users and lower energy
consumption. Together, these positives incentivized new investors to
participate in the Ethereum network due to increased trust in the blockchain
and lower barriers to entry. The transition occurred in two phases; the first
phase was a preparatory hard forking of the blockchain into a PoS structure
and the second phase was a finalization of the upgrade. A pattern of hope was
expected as the upgrade was highly anticipated due to the positives,
transparency, and consistent updates regarding the official dates of the
upgrade. From Figure 6, we indeed verify this behavioral pattern of hope
characterized by inflated expansion values, coupled with relatively stable
decay values, on three separate occasions. The first occurrence of hope is
observed approximately a week before the first phase of the upgrade took
place. It was around this time, the end of August 2022, that official news
regarding the concrete dates of when the upgrade would be expected to take
place was released to the public. We observe a surge of new hopeful investors
participating in the Ethereum network and a significant dip in existing
investors leaving the network in anticipation of the upgrade. The other two
instances of hope are seen during the immediate days surrounding and between
each of the phases of the upgrade. These occurrences provide insight into the
market sentiment during the upgrade as positive and the overall transition of
Ethereum to PoS as being well-received by investors.

Figure 6. Ethereum decay and expansion measures. The move of Ethereum to
Proof-of-Stake mining took place in two stages, indicated by 2 vertical blue
lines (Sep 6 and 15, 2022).

##  5\. Conclusions

In this article, we introduced InnerCore, which utilizes data depth-based core
discovery to identify the most influential nodes in temporal blockchain
networks. Furthermore, we proposed two metrics, InnerCore expansion and decay,
that provide a sentiment indicator for the networks and explain investor mood.
These metrics help in understanding the behavior of the network and predicting
its long-term stability. The scalability and computational efficiency of
InnerCore discovery make it well-suited for analyzing large temporal graphs,
including those found in Ethereum transaction and stablecoin networks. Our
experiments, which included comparing our findings with those of six forensic
experts from the prominent blockchain data analytics company Nansen.ai, show
that InnerCore can efficiently extract useful information from large networks.
As a future work, we plan to use InnerCore to explore network robustness
against Decentralized Finance (DeFi) attacks.

##  6\. Appendix

###  6.1. An Example of AlphaCore

To better illustrate the differences between the traditional graph-$k$-core
and AlphaCore decomposition methods, we showcase an example in Figure 7. We
are interested in the inner core that captures only the most important
investors of a financial network. In the case of graph-$k$-core, the innermost
core is the 3-core, whereas the inner core of AlphaCore would be the core of
$\alpha$ ¿ 0.75. Note that the 3-core consists of investors that trade
frequently with themselves, but their trade volumes with themselves are not
that significant compared to other transactions which exist in the network. In
certain analyses of financial networks such as anomalous address detection,
being able to filter out these negligible transactions and their participating
investors, while still capturing more meaningful ones, significantly improves
the accuracy and scalability of subsequent computations on the decomposed
network core. On the other hand, the AlphaCore of $\alpha$ ¿ 0.75 is able to
capture both the investors that participate in the largest transactions which
occur in the example network, while filtering the negligible transactions and
their participating investors. We point out that the main limitation with
graph-$k$-core is that it only considers node degrees, whereas AlphaCore is
flexible and can consider any combination of node features as outlined in
Table 1, without requiring to specify any feature weighting parameters to
perform effectively on a particular task. Therefore, in networks where edge
weights fall under a broad range and they are meaningful distinguishing
factors, we recommend AlphaCore over the traditional graph-$k$-core
decomposition.

###  6.2. Additional Related Work

In recent years, several studies focused on analyzing different aspects of the
blockchain networks (Chen et al., 2019; Akcora et al., 2020; Kalodner et al.,
2017; Guidi and Michienzi, 2020), particularly in the Ethereum network.
Researchers working on natural language processing and sentiment analysis
using tweets, news articles, cryptocurrency prices and charts, Google Trends
about blockchains (Vo et al., 2019; Kraaijeveld and D. Smedt, 2020) could find
supporting evidences based on blockchain data analysis. Oliveira et al.
(Oliveira et al., 2022) performed an analysis of the effects of external
events on the Ethereum platform, highlighting short-term changes in the
behavior of accounts and transactions on the network. Aspembitova et al.
(Aspembitova et al., 2021) used temporal complex network analysis to determine
the properties of users in the Bitcoin and Ethereum markets and developed a
methodology to derive behavioral types of users.

Other studies focused on specific aspects of the Ethereum network. For
instance, Casale et al. (Casale-Brunet et al., 2021) analyzed the networks of
Ethereum Non-Fungible Tokens using a graph-based approach, while Silva et al.
(Silva, 2020) characterized relationships between primary miners in Ethereum
using on-chain transactions. Meanwhile, Victor et al. (Victor and Lüders,
2019) measured Ethereum-based ERC20 token networks and Kiffer et al. (Kiffer
et al., 2018) examined how contracts in Ethereum are created and how users
interact with them.

Zhao et al. (Zhao et al., 2021) investigated the evolutionary nature of
Ethereum interaction networks from a temporal graphs perspective, detecting
anomalies based on temporal changes in global network properties and
forecasting the survival of network communities using relevant graph features
and machine learning models. Li et al. (Li et al., 2021) analyzed the
magnitude of illicit activities in the Ethereum ecosystem using proprietary
labeling data and machine learning techniques to identify additional malicious
addresses. Kilic et al. (Kılıç et al., 2022) predicted whether given addresses
are blacklisted or not in the Ethereum network using a transaction graph and
local and global features.

Our approach for analyzing the effects of external events on a blockchain
platform is similar to the one used by Anoaica et al. (Anoaica and Levard,
2018). The authors examined the temporal variation of transaction features in
the Ethereum network and observed an increase in activity following the
announcement of the Ethereum Alliance creation. Gaviao et al. (Gaviao
Mascarenhas et al., 2020) also studied the evolution of users and transactions
over time, showing the centralization tendency of the transaction network.
Kapengut et al.(Kapengut and Mizrach, 2022) study the Ethereum blockchain
around the BeaconChain phase of the PoS transition (September 15, 2022), but
the authors focus on the power efficiency and miners’ rewards around the
transition.

Finally, Khan (Khan, 2022) conducted a survey of datasets, methods, and future
work related to graph analysis of the Ethereum blockchain data, while Ramezan
Poursafaei’s PhD thesis (Ramezan Poursafaei, 2022) presented results on
temporal anomaly detection in blockchain networks.

Figure 7. A running example to compare between the graph-$k$-core and
AlphaCore decomposition methods. The Coreness of nodes according to
graph-$k$-core decomposition is shown with different node colors, whereas
AlphaCore is run with in-strength and out-strength as node features with a
step size of 0.25. Different AlphaCores are shown using dotted boundaries.

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$\displaystyle h(t)=C\>t^{1-\alpha/2}\quad\text{with
$C=\frac{\Gamma\left((d-\alpha)/2\right)}{2^{\alpha/2}(2-\alpha)\Gamma\left(1+d/2\right)}$}.$

  2. (ii)

(Ornstein-Uhlenbeck kernel) If $f(x)=\exp(-|x|^{\alpha})$ with $\alpha>0$,
then

$\displaystyle h(t)\asymp\begin{dcases}\sqrt{t},&d=1,\\\ \log(t),&d=2,\\\
1,&d\geq 3,\end{dcases}\quad\text{as $t\to\infty$,}\quad\text{and}\quad
h(t)\asymp t\quad\text{as $t\downarrow 0$,}$ (A.1)

and in particular, when $\alpha=2$,

$\displaystyle h(t)=\begin{dcases}\sqrt{2t+1}-1&\text{if $d=1$,}\\\
4^{-1}\log(1+2t)&\text{if $d=2$,}\\\
\left[(d-2)d\right]^{-1}\left(1-(1+2t)^{1-d/2}\right)&\text{if $d\geq
3$.}\end{dcases}$ (A.2)

  3. (iii)

(Brownian case: $W(t,x)=W(t)$) If $f(x)\equiv 1$, then $h(t)=t$ for all $t\geq
0$.

  4. (iv)

(One-dimensional space-time white case) If $d=1$ and $f(x)=\delta(x)$, then
$h(t)=\sqrt{t/\pi}$.

  5. (v)

(Bessel potential; see [Gra14, Section 1.2.2], aka. Matérn correlation family;
see [GG06]) Suppose that

$\displaystyle
f_{\nu}(x)=\int_{\mathbb{R}^{d}}\frac{e^{-ix\cdot\xi}}{(1+|\xi|^{2})^{\nu/2}}\mathrm{d}\xi,\quad\nu>0.$
(A.3)

Then,

$\displaystyle h(t)\asymp\begin{dcases}\sqrt{t},&d=1,\\\ \log(t),&d=2,\\\
1,&d\geq 3,\end{dcases}\qquad\text{as $t\to\infty$,}\quad\text{and}\quad
h(t)\asymp t\quad\text{as $t\downarrow 0$.}$ (A.4)

(Note that the large time asymptotics is the same as the Ornstein-Uhlenbeck
kernel because $f_{\nu}(x)$ behaves like $e^{-\frac{|x|}{2}}$ for $|x|\geq 2$;
see [Gra14, Proposition 1.2.5]).

  6. (vi)

(Bessel potential as the spectral measure) Suppose that
$f_{\nu}(x)=\left(1+|x|^{2}\right)^{-\nu/2}$ with $\nu>0$. Then

$\displaystyle h(t)\asymp\begin{dcases}t^{1-\frac{\nu\wedge d}{2}},&\nu\wedge
d<2,\\\ \log(t),&\nu\wedge d=2,\\\ 1,&\nu\wedge d>2,\end{dcases}\quad\text{as
$t\to\infty$},\quad\text{and}\quad h(t)\asymp t\quad\text{as $t\downarrow
0$.}$ (A.5)

###### Proof.

Parts (i), (iii), (iv) and (A.2) are from Examples 1.2–1.5 in Appendix of
[CK19]. Moreover, the asymptotics at $0$ in (A.1), (A.4) and (A.5) can be
formulated by evaluating the following limit by L’Hôpital’s rule,

$\displaystyle\lim_{t\downarrow 0}h(t)/t=2\lim_{t\downarrow 0}\langle
p_{2t},f\rangle=2f(0),$

for all $f$ that is locally continuously bounded and has growth of order
$o(e^{\epsilon x^{2}})$ for all $\epsilon>0$ at infinity.

Eq. (A.1) in part (ii):  Note that the correlation function is bounded by $1$.
Thus,

$\displaystyle
k(t)=\int_{\mathbb{R}^{d}}\mathrm{d}zp_{t}(z)f(z)\leq\int_{\mathbb{R}^{d}}\mathrm{d}zp_{t}(z)=1.$

On the other hand, using the polar coordinates, we can write

$\displaystyle
k(t)=Ct^{-\frac{d}{2}}\int_{\mathbb{R}}\mathrm{d}x|x|^{d-1}\exp\left(-\frac{x^{2}}{2t}-|x|^{\alpha}\right)\leq
Ct^{-\frac{d}{2}}\int_{\mathbb{R}}\mathrm{d}x|x|^{d-1}\exp\left(-|x|^{\alpha}\right)\leq
Ct^{-\frac{d}{2}}.$

Therefore, for $t\geq 1$

$\displaystyle
h(t)=\int_{0}^{2t}\mathrm{d}s\,k(s)\leq\int_{0}^{1}\mathrm{d}s+C\int_{1}^{2t}\mathrm{d}s\,s^{-d/2}\leq\begin{dcases}C\left(1+\sqrt{t}\right),&d=1,\\\
C\left(1+\log t\right),&d=2,\\\ C\left(1+t^{1-d/2}\right),&d\geq
3.\end{dcases}$

For the lower bound, using the polar coordinates again, we can deduce that for
$t\geq 1$,

$\displaystyle h(t)\geq
C\int_{1}^{2t}\mathrm{d}s\,s^{-\frac{d}{2}}\int_{\mathbb{R}}\mathrm{d}x\,|x|^{d-1}\exp\left(-\frac{x^{2}}{2}-|x|^{\alpha}\right)\geq
C\int_{1}^{2t}\mathrm{d}s\,s^{-d/2}.$

This completes the proof of the asymptotics of $h(\cdot)$ at infinity.

Part (v):  From the proof of [CE22a, Proposition 5.12], we have that

$\displaystyle k(2t)=h^{\prime}(t)\asymp
U\left(\frac{d}{2},\frac{2+d-\nu}{2},t\right),\qquad\mathrm{as}\ t\to\infty,$

where $U$ denotes the confluent hypergeometric function; see [Olv+10, Formula
13.4.4 on page 326]. Using asymptotic behavior of $U(a,b,t)$ as $t\to\infty$
(see Formula 13.2.6 on page 322, ibid.), one obtains the large time
asymptotics in (A.4).

Part (vi):  From the proof of [CE22a, Proposition 5.13] and by the asymptotic
analysis for $U$ as in the previous case, we deduce that

$\displaystyle
k(2t)\asymp\,t^{-\frac{d}{2}}U\left(\frac{d}{2},\frac{2+d-\nu}{2},\frac{1}{4t}\right)\asymp\,t^{-\frac{\nu\wedge
d}{2}},\qquad\mathrm{as}\ t\to\infty,$

from which one obtains the large time asymptotics of $h(t)$ in (A.5). This
completes the whole proof of Proposition A.1. ∎

###  A.2 One-dimensional stochastic wave equations

######  Proposition A.2.

Let $h(t)$ be defined as in (5.15). The following statements hold:

  1. (i)

(Riesz Kernel) If $f(x)=|x|^{-\alpha}$ with $\alpha\in(0,1)$, then for all
$t>0$,

$\displaystyle h(t)=Ct^{3-\alpha}\quad\text{with
$C=\frac{2^{1-\alpha}}{(1-\alpha)(2-\alpha)(3-\alpha)}$.}$

  2. (ii)

(Ornstein-Uhlenbeck kernel) If $f(x)=\exp(-|x|^{\alpha})$ with $\alpha>0$,
then,

$\displaystyle h(t)\asymp\,t^{2}\quad\text{as $t\to\infty$ and}\quad
h(t)\asymp\,t^{3}\quad\text{as $t\downarrow 0$}.$ (A.6)

  3. (iii)

(Brownian case: $W(t,x)=W(t)$) If $f(x)\equiv 1$, then $h(t)=t^{3}/3$ for all
$t\geq 0$.

  4. (iv)

(Space-time white case) If $f(x)=\delta(x)$, then $h(t)=t^{2}/2$ for all
$t\geq 0$.

  5. (v)

(Bessel potential) Suppose that $f_{\nu}(\cdot)$ is the Bessel potential given
in (A.3) with $d=1$ and the parameter $\nu>0$. Then

$\displaystyle h(t)\asymp t^{2}\quad\text{as
$t\to\infty$,}\quad\text{and}\quad h(t)\asymp t^{3}\quad\text{as $t\downarrow
0$.}$ (A.7)

  6. (vi)

(Bessel potential as the spectral measure) Suppose that
$f_{\nu}(x)=\left(1+x^{2}\right)^{-\nu/2}$ with $\nu>0$ and $x\in\mathbb{R}$.
Then,

$\displaystyle h(t)\asymp\begin{dcases}t^{3-\nu},&\nu<1,\\\
t^{2}\log(t),&\nu=1,\\\ t^{2},&\nu>1,\end{dcases}\qquad\text{as
$t\to\infty$,}\quad\text{and}\quad h(t)\asymp t^{3}\quad\text{as $t\downarrow
0$.}$ (A.8)

###### Proof.

Parts (i), (iii) and (iv) are straightforward. Similar as in Proposition A.1,
the asymptotics for $t\downarrow 0$ in (A.6)–(A.8) are obtained because

$\displaystyle\frac{c_{1}t^{3}}{3}=c_{1}\int_{0}^{t}\mathrm{d}s\int_{\mathbb{R}^{2}}\mathrm{d}y\mathrm{d}y^{\prime}\,G(s,y)G(s,y^{\prime})\leq\int_{0}^{t}\mathrm{d}s\int_{\mathbb{R}^{2}}\mathrm{d}y\mathrm{d}y^{\prime}\,G(s,y)G(s,y^{\prime})f(y-y^{\prime})\leq\frac{c_{2}t^{3}}{3},$

for all $t\in[0,\epsilon)$ and $f(\cdot)$ such that $0<c_{1}\leq f(x)\leq
c_{2}$ for all $x\in[-2\epsilon,2\epsilon]$ with some $\epsilon>0$.

For part (ii), we first note that

$\displaystyle h(t)\leq
C\int_{0}^{t}\int_{-s}^{s}\mathrm{d}y\int_{\mathbb{R}}\mathrm{d}y^{\prime}f(y-y^{\prime})\leq
Ct^{2}.$

For the low bounds, assume that$t>1$. Then

$\displaystyle h(t)\geq$ $\displaystyle
C\int_{1}^{t}\mathrm{d}s\int_{[-s,s]^{2}}\mathrm{d}y\mathrm{d}y^{\prime}f(y-y^{\prime}){{\rm
1\mkern-1.5mu}\\!{\rm I}}_{[-1,1]}(y-y^{\prime})\geq
C2^{\frac{3}{2}}e^{-2^{\alpha}}\int_{1}^{t}\mathrm{d}s\,(s-1)\geq C(t-1)^{2}.$

This complete the proof of the large time asymptotics in (A.6).

As for part (v), by using the Fourier transform of the wave kernel, we see
that

$\displaystyle h(t)=$
$\displaystyle(2\pi)^{-1}\int_{0}^{t}\mathrm{d}s\int_{\mathbb{R}}\mathrm{d}\xi\frac{\sin^{2}(s\xi)}{\xi^{2}(1+\xi^{2})^{\nu/2}}$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}\mathrm{d}s\left(\int_{0}^{\frac{\pi}{2s}}\mathrm{d}\xi\frac{(s\xi)^{2}}{\xi^{2}(1+\xi^{2})^{\nu/2}}+\int_{\frac{\pi}{2s}}^{\infty}\mathrm{d}\xi\frac{1}{\xi^{2}(1+\xi^{2})^{\nu/2}}\right)$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}\mathrm{d}s\left(\int_{0}^{\frac{\pi}{2s}}s^{2}\mathrm{d}\xi+\int_{\frac{\pi}{2s}}^{\infty}\frac{1}{\xi^{2}}\mathrm{d}\xi\right)\leq
C\int_{0}^{t}\mathrm{d}s(s+s)\leq Ct^{2}.$

On the other hand, it is clear that

$\displaystyle h(t)\geq$ $\displaystyle
C\int_{0}^{t}\mathrm{d}s\int_{0}^{\frac{\pi}{2s}}\mathrm{d}\xi\frac{\sin^{2}(s\xi)}{\xi^{2}(1+\xi^{2})^{\nu/2}}\geq
C\int_{1}^{t}\mathrm{d}s\int_{0}^{\frac{\pi}{2s}}\mathrm{d}\xi\frac{(s\xi)^{2}}{\xi^{2}(1+\xi^{2})^{\nu/2}}$
$\displaystyle\geq$ $\displaystyle
C\int_{1}^{t}\mathrm{d}s\int_{0}^{\frac{\pi}{2s}}\mathrm{d}\xi\frac{s^{2}}{(1+(\pi/2)^{2})^{\nu/2}}\geq
C(t-1)^{2},$

for all $t\geq 1$. This proves the large time asymptotics in (A.7).

As for part (vi), by changing of variables $y-y^{\prime}=z$ and
$y+y^{\prime}=z^{\prime}$, we have that
$\mathrm{d}y\mathrm{d}y^{\prime}=2^{-1}\mathrm{d}z\mathrm{d}z^{\prime}$. Thus,
for $t>2$,

$\displaystyle
h(t)\leq\frac{1}{8}\int_{0}^{t}\mathrm{d}s\int_{-2s}^{2s}\int_{-2s}^{2s}\frac{\mathrm{d}z\mathrm{d}z^{\prime}}{(1+z^{2})^{\nu/2}}\leq
C\left(1+\int_{0}^{t}\mathrm{d}s\,s\int_{1}^{2s}\mathrm{d}z\,z^{-\nu}\right)$

and

$\displaystyle
h(t)\geq\frac{1}{8}\int_{2}^{t}\mathrm{d}s\int_{-s/2}^{s/2}\int_{-s/2}^{s/2}\frac{\mathrm{d}z\mathrm{d}z^{\prime}}{(1+z^{2})^{\nu/2}}\geq\frac{1}{8}\int_{2}^{t}\mathrm{d}s\,s\int_{1}^{s/2}\frac{\mathrm{d}z}{(1+z^{2})^{\nu/2}}\geq\frac{1}{2^{3+\nu/2}}\int_{2}^{t}\mathrm{d}s\,s\int_{1}^{s/2}\mathrm{d}zz^{-\nu}.$

By considering three cases $\nu\in(0,1)$, $\nu=1$, and $\nu>1$ in the above
two inequalities, we obtain the large time asymptotics in (A.8). This
completes the proof of Proposition A.2. ∎

##  Appendix B Supplementary proof for Proposition 6.2

###### Proof of Proposition 6.2.

All formulas in the special cases (i)–(iii), except (6.4) and (6.5), can be
deduced by plugging in the specific $h(t)$ given in Appendix A.1. For (6.4),
by [CE22a, Example 5.8], we find that

$\displaystyle\mathcal{J}_{0}(t,x)\leq
c_{1}t^{-\ell/2}\quad\text{and}\quad\mathcal{J}_{1}(t,x)\leq
c_{2}t^{1-\ell-\beta/2}.$

Then, plugging these inequalities into (6.3), one gets (6.4) immediately.

The proof of (6.5) also relies on the estimates for $\mathcal{J}_{0}(t,x)$ and
$\mathcal{J}_{1}(t,x)$. By using the equivalence of norms in Euclidean spaces,
there exists $c_{2}\geq c_{1}>0$ such that

$\displaystyle
c_{1}\sum_{i=1}^{d}|x_{i}|\leq|x|=\left(\sum_{i=1}^{d}x_{i}^{2}\right)^{1/2}\leq
c_{2}\sum_{i=1}^{d}|x_{i}|.$

It follows that

$\displaystyle
e^{\ell|x|}\leq\exp\left(c\ell\sum_{i=1}^{d}|x_{i}|\right),\quad\text{for all
$x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$,}$

where $c=c_{2}$ if $\ell\geq 0$ and otherwise $c=c_{1}$. As a result, by using
[CD15b, Proposition A.11], we can deduce that

$\displaystyle\mathcal{J}_{0}(t,x)\leq\prod_{i=1}^{d}\int_{\mathbb{R}}\mathrm{d}y_{i}e^{c\ell|y_{i}|}\frac{1}{\sqrt{2\pi
t}}e^{-\frac{(x_{i}-y_{i})^{2}}{2t}}\leq 2e^{c^{2}d\ell^{2}t+c\ell|x|}.$ (B.1)

We still need to estimate $\mathcal{J}_{1}(t,x)$. Using (6.4), by [CE22a,
Example 5.8], we have

$\displaystyle\int_{\mathbb{R}^{d}}\mathrm{d}yp_{t}(x-y)|y|^{-\beta}\leq
c_{3}t^{-\beta/2},$ (B.2)

with some universal constant $c_{3}>0$. Hence, plugging (B.1) and (B.2) into
(1.11), it follows that

$\displaystyle\mathcal{J}_{1}(t,x)\leq
2c_{3}\int_{0}^{t}\mathrm{d}s(t-s)^{-\beta/2}\int_{\mathbb{R}^{d}}\mathrm{d}y\>p_{t-s}(x-y)e^{2c^{2}d\ell^{2}s+2c\ell|y|}.$

Finally, following the same argument as in the proof of (B.1), one can show
that

$\displaystyle\mathcal{J}_{1}(t,x)\leq$ $\displaystyle
2c_{3}\int_{0}^{t}\mathrm{d}s(t-s)^{-\beta/2}\exp\left(2c^{2}d\ell^{2}s+4c^{4}d\ell^{2}(t-s)+2c^{2}\ell|x|\right)$
$\displaystyle\leq$ $\displaystyle
t^{1-\beta/2}\exp\left(C_{1}\ell^{2}t+C_{2}\ell|x|\right).$

This completes the proof of (6.5). ∎

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