Smart contracts, computer protocols designed for autonomous execution on prede ned conditions, arise from the evolution of the Bitcoin's crypto-currency. They provide higher transaction security and allow economy of scale through the automated process. Smart contracts provides inherent bene ts for nancial 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 nd and predict smart contracts interactions only based on their crypto-currency exchanges. This approach relies on tensor modeling combined with stochastic 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.
Since the appearance of Bitcoin's crypto-currency in 2009 by Nakamoto, di erent 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 rst 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 technology 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 elds: Hash, Block height, Sender, Receiver, Amount representing 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 e cient model to predict transactions over time. Predicting and analyzing new contracts has important practical applications in nancial 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
Since its rst de nition by Nick Szabo, the security and the programming
patterns of the automated protocols have been highlighted. More speci cally, in [
However, analysis of the smart contracts using tensor based approach has
not yet been deeply studied among the scienti c community. Di erent
decompositions have been applied on various subjects as the survey written by Kolda and
Bader in [
In this paper, we propose a tensor-based approach based on the CP decomposition on crypto-currency domain by combining the tensor decomposition to stochastic process. This new approach allows to identify and predict smart contracts interactions.
To the best of our knowledge, this is the rst work of the tensor-based approach combined with the stochastic variance gamma model for analyzing and modeling smart contracts. 3
In this section, we present the tensor theoretical background and the tensor decomposition used for the experiments. Finally, we describe the stochastic process used for the predictions of the interactions. 3.1
Notation Mathematical notations and formulation follow the notations
proposed by Kolda and Bader in [
Scalars are identi ed 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 2 RI J is denoted by AT .
Tensor De nition Let de ne a N -th multidimensional array denoted by X such as X 2 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.
vu I1 jjXjj = tu X j1=1 j2=1
I2 IN X ::: X jN =1 xj21;j2;:::;jN The vector outer product denoted by matrix W2 RI J .
between u2 RI and v2 RJ results in a u
2u13 2u1v1 u1v2 ::: u1vJ 3 v = uT v = 64 ... 75 v1 ::: vJ = 64 ... ... ::: ... 75 uI uI v1 uI v2 ::: uI vJ (1) (2) A N -way tensor X 2 RI1 I2 I3 ::: IN is a rank one tensor if it can be written such as
X = a(1) a(2) a(N)
The Khatri-Rao product between two matrices A2 RI K and B2 RJ K , denoted by A B, results in a matrix C of size RIJ K . It is the column-wise Kronecker product.
C = A
For the analysis of the interactions between smart contracts, the CP
decomposition is used. It has been introduced by Harshman in [
X =
R X a(1)
r r=1 a(2) r a(N) r
The minimization equation min X^ jjX X^ jj with X^ 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
Harshman in [
bjr > > > > >>:ckr
[X(1)(C air [A(C B)T (C
[X(2)(C bjr [B(C A)T (C
[X(3)(C ckr [C(B A)T (B
B)]ir
B)]ir A)]jr
A)]jr A)]kr
A)]kr
The non-negative algorithm plays a key role in the convergence of the historical calibration of the parameters of the stochastic model. 3.3
Interactions between smart contracts can be very discontinuous among time. Some interactions appear for short period of time before vanishing, other interactions 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 [
g >:St = S0 exp(rt + !t + h) ( t ; ); n
N (0; 1) (7) Based on the empirical observations, no VG drift free rate could have been identi ed. As a result, the parameter r is set to 0. 4
In this section, we present brie y the graph analytics used as a comparison of the tensor-VG model in the experiments. 4.1
Centrality score measures the communication importance of a vertex. In this
paper, degree (baseline) centrality and betweenness ( ow-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) 8Tx 2 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[
X Ts6=Tx6=Tr
TsTr (Tx)
TsTr (8)
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
relationships across pairs of heterogeneous data [
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 [
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
[
8>score(Tx; Tx) = 1 < >score(Tx; Ty) = :
P a2 (Tx) j (Tx)j:j (Ty)j
Pb2 (Ty) score(a; b) with = [0; 1] (9) Finally, the SimRank corresponds to the link predictions in a bi-clique. 5
This section presents the experiments and the results of the tensor and the graph based approaches on smart contract data. 5.1
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 transactions have been realized between 241,385 sender accounts and 359,798 receiver accounts. A sender account is de ned 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
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 rst 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 fty-two time intervals. The size of the resulting tensor is 459 813 52.
vectors associated to time dimension + ::: + vectors associated to receiver accounts
Re1 Re2 Re3 Se1 1 1 1
Se2 0 0 0 Se1 R1e1 R1e2 R0e3 ...1 ... =
Se2 1 1 0
Re1 Re2 Re3 0 r Se1 1 0 0 ... ... eendS SSee23 01 01 01 Time ... ... ... ... ...
Receiver
vectors associated to sender accounts
model is calibrated historically with the Maximum Likelihood Estimator (MLE)
as presented in [
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 CT = 1 X 1S(n) K
N n=1 T (10)
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 de ne 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.
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 probability is around 55%. Given the threshold de nition, it is more likely no interactions happen. The results are cross-validated with the true data that con rm no interaction happened.
Interaction Probabilities between [...]7d7be3b5 and [...]db757a10
in function of Time 1.0 Interaction Probabilities between [...]e7e74d09 and [...]aedb0efb in function of Time
Similarly, for another sender and receiver extracted from the CP decomposition in Fig. 2b, the simulation indicates that the interaction probability is 0. It is also con rmed by the true data where no interaction have been realized. We de ne 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; ): { V is the set of senders and receivers; { E is a set of edges in G. Assuming Ts and Tr, sender s 2 V and receiver r 2 V , there is an edge (Ts; Tr) 2 E if and only if it exists a transaction between Ts and Tr. { 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. 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 distribution 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 vertices during transactions. From these experiments, bi-clique groups appear more adequate to analyze smart contracts interactions by obtaining small groups leading to easier analysis. Four types of contracts have been tested. First, contracts having stable interactions over time or contracts having no exchange over time have been chosen. Then, two contracts, de ned as type three, have been randomly selected. Type three contracts have an interaction at the initial time step and no interaction at the nal time step. Finally, two contracts type four, have been selected for which an interaction have been realized at the nal time step but not at the beginning of the experiment.
Based on the rst 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 nal 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 nal time step are between 81% and 99.6%. As a result, the interactions 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 interactions. 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 di cult 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 probabilities. However, for thresholds higher than 61% the tensor approach gives signi cantly better results. Overall, the tensor approach combined with the VG model appears to lead to better interactions probabilities.
Sender
Receiver
Value at Value at Interaction Prob.
T = 0 T = 26 Tensor Graph We proposed a tensor-based approach combined with the stochastic variancegamma 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 e cient 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.
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).