The skyrocketing market value of cryptocurrencies has prompted more investors to pour funds into cryptocurrencies to seek asset hedging. However, the anonymity of blockchain makes cryptocurrency naturally a tool of choice for criminals to commit smart contract scams. Consequently, smart contract scam detection is particularly critical for investors to avoid economic loss. Previous methods mainly leverage specific code logic of smart contracts and/or design rules based on abnormal transaction behaviors for scam detection. Although these methods gain success at detecting particular scams, they perform worse when applied to scams with highly similar codes. Besides, well-designed decision rules rely on expert knowledge and tedious data collection steps, which causes poor flexibility. To combat these challenges, we consider the problem of smart contract scam detection via mining topological features of account interaction information that dynamically evolves. We adopt interactive features extracted from dynamic interaction information of accounts and propose a framework named TTG-SCSD to utilize the features and Topological Data Analysis for smart contract scams detection. The TTG-SCSD constructs discrete dynamic interaction graphs for each contract and designs interactive features that characterize account behaviors. The features are modeled combined with a topology quantification mechanism to capture contract intentions in transactions. Experimental results on real-world transaction datasets from Ethereum show that TTG-SCSD obtains better generalizability and improves the performance of the bare versions of the comparison methods.
Abstract With the the advent era of big data, the secure computation calculates data on the premise of protecting data privacy, to realize the availability and invisibility of data. Secure multi-party computation, as one of three major technical tools of privacy computing, can still securely carry out data collaborative computation without a trusted third party. As an important branch of secure multi-party computation, the secure computing geometric problem can solve practical problems in the military, national defense, finance, life, and other fields, which has important research significance. In this paper, the graphic similarity problem is studied. Firstly, this paper proposes the adjacency matrix vector coding method of isomorphic graphics and uses the Paillier variant cryptosystem to securely solve the graphic similarity judgment under the semi-honest model. By using an elliptic curve cryptosystem and zero-knowledge proof to solve the possible malicious attacks under the semi-honest model, a graphic similarity judgment protocol under the malicious model is designed. The protocol can resist malicious attacks, has high computational efficiency, and has wide application value.
When trying to fit a deep neural network (DNN) to a $G$-invariant target function with $G$ a group, it only makes sense to constrain the DNN to be $G$-invariant as well. However, there can be many different ways to do this, thus raising the problem of ``$G$-invariant neural architecture design'': What is the optimal $G$-invariant architecture for a given problem? Before we can consider the optimization problem itself, we must understand the search space, the architectures in it, and how they relate to one another. In this paper, we take a first step towards this goal; we prove a theorem that gives a classification of all $G$-invariant single-hidden-layer or ``shallow'' neural network ($G$-SNN) architectures with ReLU activation for any finite orthogonal group $G$, and we prove a second theorem that characterizes the inclusion maps or ``network morphisms'' between the architectures that can be leveraged during neural architecture search (NAS). The proof is based on a correspondence of every $G$-SNN to a signed permutation representation of $G$ acting on the hidden neurons; the classification is equivalently given in terms of the first cohomology classes of $G$, thus admitting a topological interpretation. The $G$-SNN architectures corresponding to nontrivial cohomology classes have, to our knowledge, never been explicitly identified in the literature previously. Using a code implementation, we enumerate the $G$-SNN architectures for some example groups $G$ and visualize their structure. Finally, we prove that architectures corresponding to inequivalent cohomology classes coincide in function space only when their weight matrices are zero, and we discuss the implications of this for NAS.
In this piece, we overview Isaac Asimov's most iconic work, the Foundation series, with two primary goals: to provide quantitative insights about the novels and bridge data science with digital art. First, we rely on data science and text processing tools to describe certain properties of Asimov's career and the novels, focusing on the different worlds in Asimov's universe. Then we transform the books' texts into a network centered around Asimov's planets and their semantic context. Finally, we introduce the world of crypto art and non-fungible tokens (NFTs) by transforming the visualized network into a high-end digital piece of art minted as an NFT. Additionally, to pay tribute to Asimov's devotion to robotics and artificial intelligence, we use OpenAI's Generative Pre-trained Transformer 3 (GPT-3) to draft several paragraphs of this paper.
Yuzhou Chen, Ignacio Segovia-Domínguez, Yulia R. Gel
There recently has been a surge of interest in developing a new class of deep learning (DL) architectures that integrate an explicit time dimension as a fundamental building block of learning and representation mechanisms. In turn, many recent results show that topological descriptors of the observed data, encoding information on the shape of the dataset in a topological space at different scales, that is, persistent homology of the data, may contain important complementary information, improving both performance and robustness of DL. As convergence of these two emerging ideas, we propose to enhance DL architectures with the most salient time-conditioned topological information of the data and introduce the concept of zigzag persistence into time-aware graph convolutional networks (GCNs). Zigzag persistence provides a systematic and mathematically rigorous framework to track the most important topological features of the observed data that tend to manifest themselves over time. To integrate the extracted time-conditioned topological descriptors into DL, we develop a new topological summary, zigzag persistence image, and derive its theoretical stability guarantees. We validate the new GCNs with a time-aware zigzag topological layer (Z-GCNETs), in application to traffic forecasting and Ethereum blockchain price prediction. Our results indicate that Z-GCNET outperforms 13 state-of-the-art methods on 4 time series datasets.
Cüneyt Gürcan Akçora, Yulia R. Gel, Murat Kantarcıoğlu
Abstract Blockchain is an emerging technology that has enabled many applications, from cryptocurrencies to digital asset management and supply chains. Due to this surge of popularity, analyzing the data stored on blockchains poses a new critical challenge in data science. To assist data scientists in various analytic tasks for a blockchain, in this tutorial, we provide a systematic and comprehensive overview of the fundamental elements of blockchain network models. We discuss how we can abstract blockchain data as various types of networks and further use such associated network abstractions to reap important insights on blockchains' structure, organization, and functionality. This article is categorized under: Technologies > Data Preprocessing Application Areas > Business and Industry Fundamental Concepts of Data and Knowledge > Data Concepts Fundamental Concepts of Data and Knowledge > Knowledge Representation
P. Saengduean, Sutthipong Noisagool, Farida Chamchod
In this study, we investigate financial crashes in the cryptocurrency market including both mini and major crashes for two cryptocurrencies, Bitcoin and Ethereum, during the period that the digital market crashed in 2018. By applying techniques in topological data analysis, we are able to predict financial transitions and explore optimal values of the window size and the dimension of point cloud data to obtain good early warning signals. Our results demonstrate good early warning signals before the financial crashes and also show that the L1-norm and C1– norm of persistent landscapes peak before the crashes occur.
Cüneyt Gürcan Akçora, Yitao Li, Yulia R. Gel, Murat Kantarcıoğlu
Recent proliferation of cryptocurrencies that allow for pseudo-anonymous transactions has resulted in a spike of various e-crime activities and, particularly, cryptocurrency payments in hacking attacks demanding ransom by encrypting sensitive user data. Currently, most hackers use Bitcoin for payments, and existing ransomware detection tools depend only on a couple of heuristics and/or tedious data gathering steps. By capitalizing on the recent advances in Topological Data Analysis, we propose a novel efficient and tractable framework to automatically predict new ransomware transactions in a ransomware family, given only limited records of past transactions. Moreover, our new methodology exhibits high utility to detect emergence of new ransomware families, that is, detecting ransomware with no past records of transactions.
Abstract Partial orders and directed acyclic graphs are commonly recurring data structures that arise naturally in numerous domains and applications and are used to represent ordered relations between entities in the domains. Examples are task dependencies in a project plan, transaction order in distributed ledgers and execution sequences of tasks in computer programs, just to mention a few. We study the problem of order preserving hierarchical clustering of this kind of ordered data. That is, if we have $$a<b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>a</mml:mi><mml:mo><</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:math> in the original data and denote their respective clusters by [ a ] and [ b ], then we shall have $$[a]<[b]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mi>a</mml:mi><mml:mo>]</mml:mo><mml:mo><</mml:mo><mml:mo>[</mml:mo><mml:mi>b</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math> in the produced clustering. The clustering is similarity based and uses standard linkage functions, such as single- and complete linkage, and is an extension of classical hierarchical clustering. To achieve this, we develop a novel theory that extends classical hierarchical clustering to strictly partially ordered sets. We define the output from running classical hierarchical clustering on strictly ordered data to be partial dendrograms ; sub-trees of classical dendrograms with several connected components. We then construct an embedding of partial dendrograms over a set into the family of ultrametrics over the same set. An optimal hierarchical clustering is defined as the partial dendrogram corresponding to the ultrametric closest to the original dissimilarity measure, measured in the p -norm. Thus, the method is a combination of classical hierarchical clustering and ultrametric fitting. A reference implementation is employed for experiments on both synthetic random data and real world data from a database of machine parts. When compared to existing methods, the experiments show that our method excels both in cluster quality and order preservation.
Mohd Sabri Ismail, Saiful Izzuan Hussain, Mohd Salmi Md Noorani
This study explores persistent homology to detect early warning signals of the 2017 and 2019 major financial crashes in Bitcoin. Sliding window is used to obtain point cloud datasets from a multidimensional time series (Bitcoin, Ethereum, Litecoin and Ripple). We apply persistent homology to quantify transient loops that appear in multiscale topological spaces, which associated on each point cloud dataset and encode the quantified information in a persistence landscape. Temporal changes in persistence landscapes are measured via their L1-norms. Consequently, a new representative is attained, called L1-norms time series. The L1-norms is associated with indicators: autocorrelation function at lag 1, variance and mean power spectrum at low frequencies to detect the signals. By using Kendall's tau correlation and significance test, significant rising trend events that occur before major financial crashes in Bitcoin are defined as the signals. A threshold is determined to scan entire data and record all the significant rising trend events. Lastly, we compare L1-norms with residuals time series, which is another representative obtained from de-trending approach. Our result portrays that autocorrelation function at lag 1 and variance of the L1-norms successfully detect early warning signals before the 2017 and 2019 major financial crashes. However, variance of the L1-norms is better since it able to signal another 2018 major financial crash. For the residuals, no early warning signals are detected. Hence, persistent homology provides a better representative than de-trending approach. Overall, persistent homology is a promising method to detect early warning signals of major financial crashes in Bitcoin.
The Blockchain technology and, in particular blockchain-based cryptocurrencies, offer us information that has never been seen before in the financial world. In contrast to fiat currencies, all transactions of crypto-currencies and crypto-tokens are permanently recorded on distributed ledgers and are publicly available. This allows us to construct a transaction graph and to assess not only its organization but to glean relationships between transaction graph properties and crypto price dynamics. The goal of this paper is to facilitate our understanding on horizons and limitations of what can be learned on crypto-tokens from local topology and geometry of the Ethereum transaction network whose even global network properties remain scarcely explored. By introducing novel tools based on Topological Data Analysis and Functional Data Depth into Blockchain Data Analytics, we show that Ethereum network (one of the most popular blockchains for creating new crypto-tokens) can provide critical insights on price changes of crypto-tokens that are otherwise largely inaccessible with conventional data sources and traditional analytic methods.
Blockchain technology and, in particular, blockchain-based cryptocurrencies offer us information that has never been seen before in the financial world. In contrast to fiat currencies, all transactions of crypto-currencies and crypto-tokens are permanently recorded on distributed ledgers and are publicly available. As a result, this allows us to construct a transaction graph and to assess not only its organization but to glean relationships between transaction graph properties and crypto price dynamics. The ultimate goal of this paper is to facilitate our understanding on horizons and limitations of what can be learned on crypto-tokens from local topology and geometry of the Ethereum transaction network whose even global network properties remain scarcely explored. By introducing novel tools based on topological data analysis and functional data depth into Blockchain Data Analytics, we show that Ethereum network (one of the most popular blockchains for creating new crypto-tokens) can provide critical insights on price strikes of crypto-tokens that are otherwise largely inaccessible with conventional data sources and traditional analytic methods.
Portfolio management is essential for any investment decision. Yet, traditional methods in the literature are ill-suited for the characteristics and dynamics of cryptocurrencies. This work presents a method to build an investment portfolio consisting of more than 1500 cryptocurrencies covering 6 years of market data. It is centred around Topological Data Analysis (TDA), a recent approach to analyze data sets from the perspective of their topological structure. This publication proposes a system combining persistence landscapes to identify suitable investment opportunities in cryptocurrencies. Using a novel and comprehensive data set of cryptocurrency prices, this research shows that the proposed system enables analysts to outperform a classic method from the literature without requiring any feature engineering or domain knowledge in TDA. This work thus introduces TDA-based portfolio management of cryptocurrencies as a viable tool for the practitioner.
Nazmiye Ceren Abay, Cüneyt Gürcan Akçora, Yulia R. Gel, Murat Kantarcıoğlu · 7 authors
With emergence of blockchain technologies and the associated cryptocurrencies, such as Bitcoin, understanding network dynamics behind Blockchain graphs has become a rapidly evolving research direction. Unlike other financial networks, such as stock and currency trading, blockchain based cryptocurrencies have the entire transaction graph accessible to the public (i.e., all transactions can be downloaded and analyzed). A natural question is then to ask whether the dynamics of the transaction graph impacts the price of the underlying cryptocurrency. We show that standard graph features such as degree distribution of the transaction graph may not be sufficient to capture network dynamics and its potential impact on fluctuations of Bitcoin price. In contrast, the new graph associated topological features computed using the tools of persistent homology, are found to exhibit a high utility for predicting Bitcoin price dynamics. %explain higher order interactions among the nodes in Blockchain graphs and can be used to build much more accurate price prediction models. Using the proposed persistent homology-based techniques, we offer a new elegant, easily extendable and computationally light approach for graph representation learning on Blockchain.
Cüneyt Gürcan Akçora, Yitao Li, Yulia R. Gel, Murat Kantarcıoğlu
Proliferation of cryptocurrencies (e.g., Bitcoin) that allow pseudo-anonymous transactions, has made it easier for ransomware developers to demand ransom by encrypting sensitive user data. The recently revealed strikes of ransomware attacks have already resulted in significant economic losses and societal harm across different sectors, ranging from local governments to health care. Most modern ransomware use Bitcoin for payments. However, although Bitcoin transactions are permanently recorded and publicly available, current approaches for detecting ransomware depend only on a couple of heuristics and/or tedious information gathering steps (e.g., running ransomware to collect ransomware related Bitcoin addresses). To our knowledge, none of the previous approaches have employed advanced data analytics techniques to automatically detect ransomware related transactions and malicious Bitcoin addresses. By capitalizing on the recent advances in topological data analysis, we propose an efficient and tractable data analytics framework to automatically detect new malicious addresses in a ransomware family, given only a limited records of previous transactions. Furthermore, our proposed techniques exhibit high utility to detect the emergence of new ransomware families, that is, ransomware with no previous records of transactions. Using the existing known ransomware data sets, we show that our proposed methodology provides significant improvements in precision and recall for ransomware transaction detection, compared to existing heuristic based approaches, and can be utilized to automate ransomware detection.
István András Seres, László Gulyás, Dániel Nagy, Péter Burcsi
Bitcoin's Lightning Network (LN) is a scalability solution for Bitcoin allowing transactions to be issued with negligible fees and settled instantly at scale. In order to use LN, funds need to be locked in payment channels on the Bitcoin blockchain (Layer-1) for subsequent use in LN (Layer-2). LN is comprised of many payment channels forming a payment channel network. LN's promise is that relatively few payment channels already enable anyone to efficiently, securely and privately route payments across the whole network. In this paper, we quantify the structural properties of LN and argue that LN's current topological properties can be ameliorated in order to improve the security of LN, enabling it to reach its true potential.
Marian Gidea, Daniel Goldsmith, Yuri A. Katz, Pablo Roldan · 5 authors
We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum,\nLitecoin, and Ripple) before the digital market crash at the end of 2017 -\nbeginning 2018. We introduce a methodology that combines topological data\nanalysis with a machine learning technique -- $k$-means clustering -- in order\nto automatically recognize the emerging chaotic regime in a complex system\napproaching a critical transition. We first test our methodology on the complex\nsystem dynamics of a Lorenz-type attractor, and then we apply it to the four\nmajor cryptocurrencies. We find early warning signals for critical transitions\nin the cryptocurrency markets, even though the relevant time series exhibit a\nhighly erratic behavior.\n
Andrei Yu. Khrennikov, Sergei V. Kozyrev, W. A. Zúñiga-Galindo
Introduction The connections between the Archimedean heat equations with number theory and geometry are well known and deep. Let us mention here the connection with the Riemann zeta function which leads naturally to trace-type formulae, see e.g. [48] and the references therein, and the connection with the Atiyah–Singer index theorem, see e.g. [178] and the references therein. The study of non-Archimedean counterparts of the above-mentioned matters is quite relevant, especially taking into account that the Connes and Deninger programs to attack the Riemann hypothesis lead naturally to these matters, see e.g. [112], [121], [309] and the references therein. For instance, several types of p -adic trace formula have been studied, see e.g. [13], [96], [449] and the references therein. In this chapter we study heat traces and spectral zeta functions attached to certain p -adic Laplacians, denoted as A β , following [105]. Using an approach inspired by the work of Minakshisundaram and Pleijel, see [340]–[342], we find a formula for the trace of the semigroup e − tAβ acting on the space of square integrable functions supported on the unit ball with average zero, see Theorem 12.13. The trace of e − t A β is a p -adic oscillatory integral of Laplace–type. We do not know the exact asymptotics of this integral as t tends to infinity; however, we can obtain a good estimation for its behavior at infinity, see Theorem 12.13 (ii). Several unexpected mathematical situations occur in the p -adic setting. For instance, the spectral zeta functions are p -adic Igusa-type integrals, see Theorem 12.18. The p -adic spectral zeta functions studied here may have infinitely many poles on the boundary of their domain of holomorphy. Thus, to the best of our knowledge, the standard Ikehara Tauberian theorems cannot be applied to obtain the asymptotic behavior for the function encompassing the eigenvalues of A β less than or equal to T ≥ 0. However, we are still able to find good estimates for this function, see Theorem 12.18, Remark 12.19, and Conjecture 12.20. The proofs require several results on certain “boundary-value problems” attached to p -adic heat equations associated with operators A β , see Proposition 12.5, Theorem 12.11, and Proposition 12.12.
Marian Gidea, Daniel Goldsmith, Yuri Katz, Pablo Roldan · 5 authors
We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 - beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -- $k$-means clustering -- in order to automatically recognize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor, and then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions in the cryptocurrency markets, even though the relevant time series exhibit a highly erratic behavior.
In this issue, we present two very different papers written in two very different styles. The first is a survey of the multiple timescales method for approximating solutions to differential equations. Multiple timescale methods are common in the literature and an integral part of many graduate programs. However, like riding a bicycle, you need some practice, experience, and insight to use it properly and have meaningful results. The second is an exposition on the Mountain Pass Lemma and related mathematical ideas underlying the existence of saddle points. Despite its name, the second article is no ordinary hike through the hills. In “Profits and Pitfalls of Timescales in Asymptotics,” author Ferdinand Verhulst presents a survey of multiple timescale methods. A colleague of mine once sarcastically pointed out that a tremendous amount of insight can be gleaned from the observation that in almost all problems, parameters are either larger than one or smaller than one, leading to an asymptotic approximation in one form or another. However, one does not have to look far to find problems where it is hard to handle the resulting asymptotic series using a simple Taylor series. Multiple timescales can resolve these problems, but the challenge remains of how to know what the multiple timescales should be without having special knowledge of the problem. Verhulst does an admirable job presenting the basic ideas behind determining timescales a priori using two basic concepts: normal forms and bifurcation theory. In the former case, one transforms the problem into a simpler expression to reveal underlying timescales. In the latter case, understanding the dynamics of a system in terms of bifurcations reveals the qualitative structure of the solution and therefore the timescales. Thus, the author puts order to a body of knowledge that can often appear to students as a disjoint collection of tricks for special problems. In “Mountain Passes and Saddle Points,” author James Bisgard develops the Mountain Pass Lemma of Ambrosetti and Rabinowitz which specifies sufficient conditions for the existence of saddle points. Beginning with accessible examples of smooth functions $F: R^2 \rightarrow R$, we can think of $F$ as the height of the landscape. The central element of this manuscript is a very clear proof of the Mountain Pass Lemma, which essentially states that if there is a local minimum in a valley surrounded by a mountain range and there is a point somewhere beyond the mountain range that is lower than the local minimum, then with an additional special requirement, it can be shown that there must be a mountain pass (saddle point) somewhere. While it may seem that there should always be a mountain pass without any additional requirements, the authors present some counterexamples early in the paper to show that this is not a trivial issue. (I could not resist the urge to fire up my tablet and explore some of the sample surfaces.) The special requirement is the Palais--Smale condition, which is the seemingly peculiar condition that every sequence $x_n$ having two properties, (1) that the height above these points is bounded and (2) that the $\| \nabla F(x_n) \|$ approaches zero, must have a convergent subsequence. The author goes on to extend the Mountain Pass Lemma to domains of any finite dimension and from there to Hilbert spaces. Finally, the author uses the concepts involved in the proof to develop methods for finding saddle points. In summary, the Education section in this issue has something for everyone. The first offering focuses on methods and techniques and would be ideal for a graduate course on perturbation methods or applied mathematics. The second paper is analytic, anchored to theorems and proofs but having ample discussion. It would find a home in an undergraduate and graduate real analysis course. Both take a fresh look at classic subjects in mathematics and could be used to liven up traditional courses in most undergraduate and graduate programs.
This thesis is concerned with stochastic perturbation theory of the symmetric eigen-value problem. In particular, we provide results about the probability of interchanges in the ordering of the eigenvalues and changes in the eigenvectors of symmetric matrices subject to stochastic perturbations. In this analysis we use a novel combination of traditional Numerical Linear Algebra, Perturbation Theory and Probability Theory. The motivation for this study arises from reliability of spectral clustering of networks, when network data is subject to noise. As far as we are aware, there is nothing comparable in the literature. Further, we make conjectures from which we derive an asymptotic relation between the distributions of the largest eigenvalue and the 2-norm of random symmetric ma- trices, whose entries above the main diagonal are independent, identically distributed random variables with probability density functions being symmetric with respect to zero, including matrices from the Gaussian Orthogonal Ensemble (GOE). As far as we know, some of these conjectures are not new (possibly only as conjectures) but we are not aware of any proofs. Also, we consider networks of coupled oscillators. In their analysis we use both, knowledge of dynamical systems and spectral properties of non-negative matrices. As a result, we present an algorithm, which uncovers the \\master-slave" structure of the network. With its help, the analysis of the dynamics and the entrainment of the entire network can be reduced to considering only few of the oscillators, those whose dynamics determine the behaviour of the rest. This can be helpful in large networks exhibiting the \\master-slave" structure. Finally, we consider similarities of spectral clustering with respect to di®erent matrices which can be associated with a given network. In particular, we compare clustering of products of Path graphs with respect to two di®erent matrices: the Laplacian and the Normalised Laplacian matrices of the graph. We make the comparison by constructing a Homotopy between two eigenvalue problems and, using some Linear Algebra techniques, we show that the two matrices give similar spectral clusterings when applied to products of Path graphs.