=Paper=
{{Paper
|id=Vol-1941/MIDAS2017_paper7
|storemode=property
|title=Profiling Smart Contracts Interactions Tensor Decomposition and Graph Mining
|pdfUrl=https://ceur-ws.org/Vol-1941/MIDAS2017_paper7.pdf
|volume=Vol-1941
|authors=Jérémy Charlier,Sofiane Lagraa,Radu State,Jérôme François
|dblpUrl=https://dblp.org/rec/conf/pkdd/CharlierLSF17
}}
==Profiling Smart Contracts Interactions Tensor Decomposition and Graph Mining==
https://ceur-ws.org/Vol-1941/MIDAS2017_paper7.pdf
Profiling Smart Contracts Interactions Tensor
Decomposition and Graph Mining
Jérémy Charlier1 , Sofiane Lagraa2 , Radu State1 , and Jérôme François1,2
1
SnT, University of Luxembourg, Luxembourg L-1855, Luxembourg,
2
Inria Nancy-Grand Est, 54600 Villers-les-Nancy, France
{jeremy.charlier, radu.state}@uni.lu,
{sofiane.lagraa, jerome.francois}@inria.fr
Abstract. Smart contracts, computer protocols designed for autonomous
execution on predefined conditions, arise from the evolution of the Bit-
coin’s crypto-currency. They provide higher transaction security and al-
low economy of scale through the automated process. Smart contracts
provides inherent benefits for financial institutions such as investment
banking, retail banking, and insurance. This technology is widely used
within Ethereum, an open source block-chain platform, from which the
data has been extracted to conduct the experiments.
In this work, we propose an multi-dimensional approach to find and pre-
dict smart contracts interactions only based on their crypto-currency ex-
changes. This approach relies on tensor modeling combined with stochas-
tic processes. It underlines actual exchanges between smart contracts and
targets the predictions of future interactions among the community. The
tensor analysis is also challenged with the latest graph algorithms to
assess its strengths and weaknesses in comparison to a more standard
approach.
Keywords: Tensor, Stochastic Process, Graph mining, Smart Contract
1 Introduction
Since the appearance of Bitcoin’s crypto-currency in 2009 by Nakamoto, different
counterparts have tried to extend Bitcoin’s design beyond the currency. The most
visible is the Ethereum platform, an open source platform relying on blockchain
technology and using Ether as a crypto-currency. They implemented a protocol
relying on smart contracts for automatic exchanges without a central gatekeeper.
Originally, smart contracts have been first mentioned by Nick Szabo in 1998. He
exposed the idea as ”a computerized transaction protocol that executes the terms
of a contract [...] to satisfy common contractual conditions, minimize exceptions
[and] the need for trusted intermediaries”. Smart contracts appear as a technol-
ogy for higher transparency without the requirement of trusted intermediaries
for currency exchanges. A smart contract is a set of transactions. A transaction
has the following fields: Hash, Block height, Sender, Receiver, Amount repre-
senting the hash of the transaction, the timestamps in terms of block height
numbers, the contract issuer, the contract recipient and the scalar value to be
transferred from the sender to the receiver, respectively. Due to the vast number
of inter-connected contracts over time, the modeling of the interactions is non
trivial and requires an efficient model to predict transactions over time. Predict-
ing and analyzing new contracts has important practical applications in financial
instruments where recommendation would be based on likely future contracts
interactions.
The main contribution of the paper resides in the modeling of smart contracts
interactions using a tensor approach. In addition, a novel technique consisting in
the combination of stochastic processes and tensors allows to reproduce existing
interactions and to predict future interactions over time. The tensor approach is
then challenged with state-of-the-art graph algorithms to assess its performance.
To describe our methodology, section 1 presents the latest research concerning
the smart contracts and the tensors. Section 2 introduces the concepts related to
the chosen tensor decomposition and the stochastic model used for interactions
simulations. In section 3, a short description of the graph theory is suggested
and section 4 demonstrates the experiments performed on the tensor approach
and challenged with the graph analytics.
2 Related Work
Since its first definition by Nick Szabo, the security and the programming pat-
terns of the automated protocols have been highlighted. More specifically, in [1],
the authors investigate the security on smart contracts. They underline the pos-
sibility of manipulation of smart contracts within the Ethereum platform with
the goal of gaining profit. They also provide a symbolic execution tool to find
potential bugs within the running Ethereum smart contracts. In [2], the authors
quantify smart contracts use according to their application domain as well as the
common coding patterns in Ethereum. However, even with security issues raised
around automated protocols, smart contracts still offer lower transactional and
running costs. In [3], the authors explain how entities can make use of smart
contracts to gain in competitiveness and push further innovation. In [4], the
authors propose a new way of programming smart contracts with a logic-based
programming. It introduces new coding behavior with the goal of further effi-
ciency. Last but not least, the use of smart contracts and its extension towards
various domains also raise the question of their legal status. In [5], the authors
address the new challenges and the evolution of the law following the use of the
smart contracts in the Legal Tech companies.
However, analysis of the smart contracts using tensor based approach has
not yet been deeply studied among the scientific community. Different decompo-
sitions have been applied on various subjects as the survey written by Kolda and
Bader in [6] exposed it. Our work concentrates on the CANDECOMP/PARAFAC
(CP) decomposition which has been presented simultaneously by Harshman et
al. in [7] and [8]. It has been applied a lot in neuroscience due to the image pro-
cessing and its uniqueness property. In [9], Andersen and Rayens used the CP
decomposition for functional magnetic resonance images (fMRI) data analysis in
three and four dimensions. Sen and Parhi also applied in [10] CP decomposition
for fMRI processing with a novel algorithm for the extraction of the common task
signals and spatial maps from a fMRI group as rank-1 tensors. Other authors
extended the use of the CP decomposition in other domains than neuroscience:
Acar et al. in [11] used for data mining, Shen et al. for Inferring network topolo-
gies in [12] and Dinç, Ertekin and Bker applied it to the quantitative resolution
of related drug substances.
In this paper, we propose a tensor-based approach based on the CP decom-
position on crypto-currency domain by combining the tensor decomposition to
stochastic process. This new approach allows to identify and predict smart con-
tracts interactions.
To the best of our knowledge, this is the first work of the tensor-based ap-
proach combined with the stochastic variance gamma model for analyzing and
modeling smart contracts.
3 Tensor Analysis
In this section, we present the tensor theoretical background and the tensor de-
composition used for the experiments. Finally, we describe the stochastic process
used for the predictions of the interactions.
3.1 Tensor Description
Notation Mathematical notations and formulation follow the notations pro-
posed by Kolda and Bader in [6]. The notations have been reused by different
articles related to tensor decomposition.
Scalars are identified by lowercase letters, a. Vectors and matrices are denoted
by boldface lowercase letters and boldface capital letters, respectively a and A.
The transpose of the matrix A ∈ RI×J is denoted by AT .
Tensor Definition Let define a N -th multidimensional array denoted by X
such as X ∈ RI1 ×I2 ×I3 ×...×IN . X is called a N -way tensor. Tensors are denoted
by Euler script letters.
Tensor Operations The square root of the sum of all tensor entries squared
determines the tensor norm.
v
u I1 I2 IN
uX X X
||X|| = t ... x2j1 ,j2 ,...,jN (1)
j1 =1 j2 =1 jN =1
The vector outer product denoted by ◦ between u∈ RI and v∈ RJ results in a
matrix W∈ RI×J .
u1 u1 v1 u1 v2 ... u1 vJ
u ◦ v = uT v = ... v1 ... vJ = ... .. .
. ... .. (2)
uI uI v1 uI v2 ... uI vJ
A N -way tensor X ∈ RI1 ×I2 ×I3 ×...×IN is a rank one tensor if it can be written
such as
X = a(1) ◦ a(2) ◦ · · · ◦ a(N ) (3)
The Khatri-Rao product between two matrices A∈ RI×K and B∈ RJ×K ,
denoted by A B, results in a matrix C of size RIJ×K . It is the column-wise
Kronecker product.
C=A B = [a1 ⊗ b1 a2 ⊗ b2 ··· aK ⊗ bK ] (4)
3.2 Tensor Decomposition
For the analysis of the interactions between smart contracts, the CP decomposi-
tion is used. It has been introduced by Harshman in [7] and Carroll and Chang in
[8]. It allows a tensor X ∈ RI1 ×I2 ×...×IN to be written as the sum of component
of rank-one tensors.
XR
X= a(1) (2) (N )
r ◦ ar ◦ · · · ◦ ar (5)
r=1
The minimization equation minX̂ ||X − X̂|| with X̂ the approximate tensor
constructed with matrices ai randomly initialized and X the original tensor is
solved with the Alternating Least Squares (ALS) method as presented by Harsh-
PR (1)
man in [7] and Carroll and Chang in [8]. The matrices A = r=1 ar ∈ RI×R ,
PR (2) R (3)
B = r=1 ar ∈ RJ×R and C = r=1 ar ∈ RI×R are successively updated
P
according to the non-negative scheme presented by Lee and Seung in [13].
[X(1) (C B)]ir
air ← air [A(C B)T (C B)]ir
[X(2) (C A)]jr
bjr ← bjr (6)
[B(C A)T (C A)]jr
[X(3) (C A)]kr
ckr ← ckr
[C(B A)T (B A)]kr
The non-negative algorithm plays a key role in the convergence of the his-
torical calibration of the parameters of the stochastic model.
3.3 Stochastic Model For Interactions Predictions
Interactions between smart contracts can be very discontinuous among time.
Some interactions appear for short period of time before vanishing, other inter-
actions are present in all time periods while some contracts never interacts. As a
consequence, a deterministic model cannot be used but a stochastic model with
jumps should be preferred.
The Variance-Gamma (VG) model has been introduced by Madan, Carr and
Chang in [14]. This stochastic model is a pure jump process. One of its main
application is in quantitative finance as illustrated by Hull in [15]. The VG model
is defined with three parameters denoted θ, σ and ν ∈ R. θ represents the drift
in the Brownian motion among time. A Brownian motion, denoted by W , is
a continuous time process representing the random motion of a small particle
immersed in a fluid having the same density as the particle. σ is the volatility
and ν is the variance rate of the gamma time change. The drift is represented
by ωt, the gamma process by h and the process to be simulated by S.
1 2
ω = ν ln(1 − θν − 0.5σ ν)
√
h = θg + σ gz with g ∼ Γ ( νt , ν), n ∼ N (0, 1) (7)
St = S0 exp(rt + ωt + h)
Based on the empirical observations, no VG drift free rate could have been
identified. As a result, the parameter r is set to 0.
4 Graph Mining
In this section, we present briefly the graph analytics used as a comparison of
the tensor-VG model in the experiments.
4.1 Graph Analytics
Centrality score measures the communication importance of a vertex. In this
paper, degree (baseline) centrality and betweenness (flow-based) centrality have
been applied. A degree centrality represents the number of vertex neighbors and
gives a local view of the graph around each node. In a directed graph, each
vertex has an indegree and an outdegree. Let G = (V, E) ∀Tx ∈ V . In smart
contracts, we can distinguish between the senders and receivers. Indegree of
vertex Tx , denoted by deg − (Tx ), is the number of edges which are coming into
the vertex Tx . Outdegree of vertex Tx , denoted by deg + (Tx ), is the number of
edges which are going out from the vertex Tx . The centrality of a sender, is based
on the amount of Ether, Ethereum crypto-currency. We measure this using the
betweenness centrality measure[16], which measures how much a given vertex
lies in the weighted shortest paths of other vertices. Let δsr = δrs denote the
number of shortest paths from Ts ∈ V to Tr ∈ V where by convention δTs Ts = 1.
Let δTs Tr (Tx ) denote the number of shortest paths from Ts to Tr that some
Tx ∈ V lies on:
X δTs Tr (Tx )
betweenness = (8)
δTs Tr
Ts 6=Tx 6=Tr
A bi-clique measure allows to discover common interaction between two sets
of objects in a graph. Often, bipartite graphs can be used to represent relation-
ships across pairs of heterogeneous data [17]. A bipartite graph is partitioned
into two sets of vertices which are non-empty disjoint partitions. In bipartite
graph, there is no edge within the same partition. In smart contracts, an in-
terpretation of common senders to a set of receivers member’s relationships is
performed by an enumeration of maximum bicliques.
Maximum biclique, the largest biclique, is used as an approach to group the
senders and receivers members into groups to identify most common senders of
a set of receivers. In the experiments, the algorithm developed in [18] has been
applied.
4.2 Link Predictions
Link predictions methods are used to identify new connections in graphs. To
predict newer links among two entities at a later point in time, the path based
link predictions approach is applied using the SimRank algorithm presented in
[19]. It assumes that two vertices are similar if they share the common connected
vertices. Numerically, this is specified by defining a score. The set of all neighbors
of vertex Tx is denoted by Γ (Tx ).
score(Tx , Tx ) = 1 P
P
a∈Γ (Tx ) b∈Γ (Ty ) score(a, b) with γ = [0, 1] (9)
score(Tx , Ty ) = γ
|Γ (Tx )|.|Γ (Ty )|
Finally, the SimRank corresponds to the link predictions in a bi-clique.
5 Experiments and Discussions
This section presents the experiments and the results of the tensor and the graph
based approaches on smart contract data.
5.1 Dataset Characteristics
Smart contracts data have been collected from the Ethereum platform from 7
August 2015 to 2 March 2016. In this period of time, two millions of transac-
tions have been realized between 241,385 sender accounts and 359,798 receiver
accounts. A sender account is defined as an account sending Ether, the Ethereum
crypto-currency, and a receiver account as an account receiving Ether. However,
60% of the sender contracts only send one Ether payment from August to March
and 70% of the receiver contracts only receive once. We concentrate on contracts
having the most activities, and thus we keep only the 1% most active contracts
over time. In 1% of contracts, the average of contracts by sender is 46.91 and by
receiver is 26.06. In our experiments, 459 smart contracts senders are considered
as well as 813 smart contracts receivers.
5.2 Tensor Decomposition And Stochastic Simulation For
Interactions Predictions
Tensor Decomposition Applied to Smart Contracts A three-way topo-
logical tensor is built using the data from Ethereum. A value of zero means no
Ether were exchanged between a sender account and a receiver account at a
given time, and a value of one corresponds to the opposite. The first dimension
of the tensor, denoted by I, characterizes the list of the sender accounts, the
second dimension, J, the list of the receiver accounts and the third dimension,
K, the time. Ether exchanges have been gathered in fifty-two time intervals. The
size of the resulting tensor is 459×813×52.
vectors associated to time dimension
Re1 Re2 Re3 · · ·
Se1 1 1 1 ···
Se2 0 0 0 ···
Re1 Re2 Re3 · · · 1 ···
= + ... + vectors associated
Se1 1 1 0 ···
..
. ... to receiver accounts
Se2 1 1 0 ···
Re1 Re2 Re3 · · · 0 ···
.. ...
Se1 1 0 0 ··· .
Sender
Se2 0 0 0 ···
e
Tim
Se3 1 1 1 ···
.. .. .. .. ...
. . . .
Receiver vectors associated to sender accounts
Fig. 1: Description of the tensor dimensions and interactions between sender and
receiver contracts
Evaluation Of Interactions Probabilities Using VG Model The VG
model is calibrated historically with the Maximum Likelihood Estimator (MLE)
as presented in [20] based on the results of the tensor decomposition as described
in Fig. 1. The first twenty-six time events are used for calibration and the sim-
ulations are done on the next twenty-six time events for a comparison between
the true data and the simulated data.
The future probabilities of Ether exchanges at maturity T are computed with
a digital function denoted by C based on the simulations of the process S.
N
1 X
CT = 1 (n) (10)
N n=1 ST ≥K
At maturity T , if the simulated series is equal or higher than K, then the value
of C is equal to one, otherwise the value of C is equal to 0. In our simulation, we
define K = 0.99 due to the numerical error from the tensor decomposition and
the calibration of the stochastic model. The number of Monte-Carlo simulations,
denoted by N , is equal to one million.
Simulation of Interaction Probabilities Using CP and VG We define
a threshold of 60% as the threshold identifying an interaction or not. At the
end of the time horizon of the simulation, if an interaction probability between
one sender and one receiver is below the threshold, it is more likely that no
interaction happen. On the opposite, if an interaction probability is higher than
the threshold, it is more likely that an interaction happen.
Regarding Fig. 2a, interaction probabilities decrease over time for a given
sender and a given receiver. At the end of the simulation, the interaction prob-
ability is around 55%. Given the threshold definition, it is more likely no inter-
actions happen. The results are cross-validated with the true data that confirm
no interaction happened.
Interaction Probabilities between [...]7d7be3b5 and [...]db757a10 Interaction Probabilities between [...]e7e74d09 and [...]aedb0efb
in function of Time in function of Time
1.0 1.0
0.25 0.25
0.8 0.8 0.20 0.20
Interaction Probabilities
Interaction Probabilities
0.15 0.15
Interaction Values
Interaction Values
0.6 0.6
0.10 0.10
0.4 0.4 0.05 0.05
0.00 0.00
0.2 0.2
0.05 0.05
0.0 0.0 0.10 0.10
5 10 15 20 25 5 10 15 20 25
Time Steps Time Steps
(a) Interaction Probabilities between (b) Interaction Probabilities between
...7d7be3b5 and ...db757a10. ...e7e74d09 and ...aedb0efb.
Fig. 2: Simulation of Interactions Predictions For 26 Time Events.
Similarly, for another sender and receiver extracted from the CP decomposi-
tion in Fig. 2b, the simulation indicates that the interaction probability is 0. It
is also confirmed by the true data where no interaction have been realized.
5.3 Constructing a smart contract graph
We define the smart contract graph to be constructed from each transaction as
a directed graph representing semantically relevant relationships between two
vertex-entities which represent senders and receivers performing a transaction
in a smart contract. Each vertex represents a sender s and receiver r and each
edge s, r indicates that the sender s sends amount of coins to r, where s 6= r.
The data size is reduced by the representation of intermediate vertex-entities,
especially for transactions with the same senders and receivers. This is a labeled
directed graph G = (V, E, β):
– V is the set of senders and receivers;
– E is a set of edges in G. Assuming Ts and Tr , sender s ∈ V and receiver
r ∈ V , there is an edge (Ts , Tr ) ∈ E if and only if it exists a transaction
between Ts and Tr .
– β is a function that assigns for each edge (Ts , Tr ) the average amount of
transactions lTs ,Tr .
The smart contract graph represents the behavior of transactions between
each vertex-entities.
100000 0.016
Degree
Betweenness centrality
0.014
Frequency (log scale)
10000 Indegree
Outdegree 0.012
1000 0.01
0.008
100 0.006
0.004
10
0.002
1 0
1 10 100 1000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000
Number of links (log scale) Degree
(a) Frequency distribution for the (b) Betweeness centrality according
number of links that users have. to users degree.
Number of vertices (log scale)
10000
Modularity class
Biclique
1000
100
10
1
1 10 100 1000 10000 100000
Number of group memberships (log scale)
(c) Frequency distributions for
group membership and group size.
Fig. 3: Main Graph Mining Results
Graph Mining Results According to Fig. 3c, the frequency of group sizes
follows a highly skewed distribution. There are few and very large groups using
modularity class as well as many very small bi-clique groups. 68.54% of bi-clique
groups have only three members. In modularity groups, the frequency distribu-
tion for group membership is even more skewed. 10.78% of modularity groups
have more than 4672 members which represents 52.95% of vertices. According to
Fig. 3a, a highly skewed distribution is also observed in frequency of users’ link
counts. Fig. 3b highlights the most important betweenness centrality vertices.
More than 51% of vertices have a degree lower than 10. The vertex degree can be
higher but less important, which is equivalent to lower betweenness centrality.
The betweenness centrality shows the importance of vertices or intermediate ver-
tices during transactions. From these experiments, bi-clique groups appear more
adequate to analyze smart contracts interactions by obtaining small groups lead-
ing to easier analysis.
5.4 Interactions Probabilities And Link Predictions Comparison
Four types of contracts have been tested. First, contracts having stable inter-
actions over time or contracts having no exchange over time have been chosen.
Then, two contracts, defined as type three, have been randomly selected. Type
three contracts have an interaction at the initial time step and no interaction at
the final time step. Finally, two contracts type four, have been selected for which
an interaction have been realized at the final time step but not at the beginning
of the experiment.
Based on the first seven lines of the Table 1, the calibrated VG model is
able to reproduce permanent interactions over time as well as the absence of
interactions. In addition, for contracts of type three, interactions probabilities
at the final time step are between 55% and 61%. It means there are very high
probabilities for having no interaction. For contracts of type four, interactions
probabilities at final time step are between 81% and 99.6%. As a result, the in-
teractions probabilities determined by the VG model appear strongly correlated
to the actual interactions values among the data set.
The graph probabilities could reproduce satisfyingly the absence of interac-
tions. However, two false positives have been found with the graph approach in
the illustrated sample for which no interactions have been found. One of them is
linked to contracts of type four for which the interactions are the most difficult
to predict. The other false positive is linked to permanent interactions which
is easier to model in theory. For interactions probabilities of 29%, one is a true
positive and another one is a false positive. Finally, for interactions probabilities
higher than 30%, only true positives have been found.
To better compare interactions probabilities between the tensor approach
and the graph approach, a curve displaying the Area Under the Curve (AUC)
for Receiver Operating Characteristic (ROC) curve is shown on Fig 4.
For an interaction threshold of 20%, all interactions probabilities higher than
20% are considered as true interactions. As a result, the Fig. 4 highlights slightly
better results for the graph approach for lower thresholds of interaction prob-
abilities. However, for thresholds higher than 61% the tensor approach gives
significantly better results. Overall, the tensor approach combined with the VG
model appears to lead to better interactions probabilities.
Value at Value at Interaction Prob.
Sender Receiver
∆T = 0 ∆T = 26 Tensor Graph
[...]35398226 [...]d5ea2e63 1 1 1.0000 0.0000
[...]35398226 [...]a1a8170c 1 1 1.0000 0.7020
[...]01825cb5 [...]8c102d88 1 1 1.0000 0.8040
[...]b3121069 [...]9f0b1b4e 1 1 1.0000 0.4865
[...]b3121069 [...]804af9db 1 1 1.0000 0.2945
[...]2f0acb76 [...]3df1b5a3 0 0 0.0000 0.2957
[...]e7e74d09 [...]aedb0efb 0 0 0.0000 0.0000
[...]7d7be3b5 [...]db757a10 1 0 0.5587 0.0000
[...]bcda34d4 [...]c6b005d0 1 0 0.6093 0.0000
[...]35398226 [...]fdd31c8a 0 1 0.8114 0.8079
[...]49a601da [...]41f79adc 0 1 0.9963 0.0000
Table 1: Comparison between graph and tensor based estimated interaction prob-
ability.
Evolution of AUC of ROC Curves
For Different Probability Level Defining An Interaction
1.0
0.8
0.6
AUC
0.4
0.2
Graph AUC
Tensor AUC
0.0
0.0 0.2 0.4 0.6 0.8
Interaction Probabilities
Fig. 4: Comparison between AUC of ROC curves for graph and tensor approaches
for varying thresholds of interactions probabilities
6 Conclusion And Future Work
We proposed a tensor-based approach combined with the stochastic variance-
gamma model for analyzing and modeling smart contracts operating over the
Ethereum blockchain platform. The approach has been challenged with graph
analysis to assess its weakness and its strengths. The tensor approach is less
efficient for detecting communities in the data, but it is more appropriate for
the modeling part since it can capture the interaction probabilities. The two
techniques complement well each other since the weakness of one technique is
the strength of the other. In addition, the accuracy of the stochastic predictions
allows to identify which contract to select for speculative investment.
Our twofold future plans consist on proposing graph based link prediction
on streaming smart contract and an automated process for a calibration of the
stochastic parameters with lower dependency to the initial guess in the tensor
approach.
Acknowledgments
This work was partially funded by HuMa, a project funded by Bpifrance and
Region Lorraine under the FUI 19 framework. It is also supported by the High
Security Lab hosted at Inria Nancy Grand Est (http://www.lhs.loria.fr).
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