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9. Stable Network Dynamics in a Tokenized Financial Ecosystem

Published onMay 14, 2020
9. Stable Network Dynamics in a Tokenized Financial Ecosystem


A central concern with any economic system is its stability and resilience. Use of distributed data resources and distributed encryption methods helps with many security problems. However, as the wild price history of Bitcoin shows, use of distributed function and secure methods does little to guarantee stability. A further concern is centralization and monopoly. During the last decade a handful of companies have contributed to an alarmingly centralized world wide web. Google and Facebook directly influence over 70% of internet traffic [1].

Moreover, dominance on the internet affects the entire economy. For instance, in digital advertising, almost 60% of spending went to Google and Facebook in 2016. Amazon accounted for 43% of e-commerce sales in the United States that same year. Such digital monopolies also have important implications for journalism, politics, and society [2]. Such centralization, associated data scandals and other side effects have led to suggestions that breaking up these large network monopolies may ultimately be the only solution [3].

But how can one tell that a company is too large? When the network is becoming unstable? At what point should regulators interfere and how? We need answers to those and similar questions, using a framework that is largely independent of the system under scrutiny. At the heart of all these networks is the ubiquitous law of proportional growth, stating that a firm’s growth rate is proportional to its size and more specifically, we consider a system of agents, whereby the rate at which any given agent establishes connections to other agents is proportional to its already existing number of connections and its intrinsic fitness. This representation is known to apply to of a large class of interacting systems [4], and carries a risk of centralization [5], in the sense that removal of a few dominating agents would collapse the entire system.

Dynamics of a Digital Economy

Most of today’s network economies include distributed communication networks (e.g., the Internet), but transactions occur in centralized and conventional enterprise management software, and humans are part of the accounting and audit systems. The purest examples of network economies today have both communication and transaction execution on the distributed network, including standard accounting, and audit functions. Efficiency, transparency, cost, and security are pushing all economic ecosystems toward this more distributed, all-digital network technology, leaving only proprietary, private, or legal functions on the periphery.

Such fully distributed digital systems are often called Distributed Ledger or Blockchain Technologies (DLT, BT). This category includes systems like Estonia’s long-standing government infrastructure, China’s new “smart city” infrastructure, Singapore’s UBIN trade and logistics infrastructure, national digital currencies being deployed by China and Singapore, along with grass-roots systems like Bitcoin. These systems are proliferating throughout the globe and giving birth to a new types of economic ecosystems. These new ecosystems present new types of challenges for policy makers and regulators, due to their potential economic and social impact that could fundamentally alter traditional financial and social structures.

Perhaps the archetypical example of such a new economy is a system called Ethereum that was launched in July 2015 [6], and which today has 1.4 million users and $25 Billion dollar capitalization. The Etherium network includes a public ledger that keeps records of all Ethereum related transactions. The Ethereum ledger is able to store not only ownership, similar to Bitcoin, but also execution code, in the form of “Smart Contracts”. This has led to the creation of a large number of new types of “tokens”, which are tradable securities, offered by a wide variety of service providers, with all transactions carried out by their corresponding Smart Contracts and all publicly accessible, although encrypted, on the Ethereum ledger. As a result, the Ethereum network ecosystem constitutes a unique example of a large-scale, highly varied financial ecosystem, whose entire transaction history activity is publicly available from its inception.

In this Chapter we will analyze the dynamical properties of the Ethereum financial ecosystem, with an eye toward understanding its stability and resilience. We show that the Ethereum financial ecosystem, despite being hugely heterogeneous in terms of users, services, tradable certificates (called tokens), and accounts (called wallets) it still satisfies the key stability and resilience properties that characterize older financial networks. We will make use of all Ethereum network transactions between February 2016 and June 2018, resulting in 88, 985, 493 token trades performed by 17, 611, 649 unique wallets.

To accomplish this task we will observe γ, the power of the degree-distribution, e.g., the exponent that characterizes the distribution of transaction connections between actors within the Ethereum network. We will show how this meta-parameter is able to describe the dynamics and consolidation process of the network through time, despite exponential growth of the network and churn of services, investments, and users. In particular, we demonstrate that the dynamics of the Ethereum economy, as captured by γ, can be modeled as a damped harmonic oscillator, enabling the prediction of an economic network’s future dynamics.

The ability to predict future dynamics of the Ethereum network (or other digital economies) can provide policy makers with a useful tool for designing regulations and mechanisms to control instabilities within the network. Finally, drawing on further work by Lera, Pentland, and Sornette [7], we will describe the intervention required to control monopolies, and argue that this same method can be applied to other financial networks as a better method for anti-trust regulators to prevent monopolies.

2. Background and Related Work

The Ethereum network is perhaps the first to support execution of “Smart Contracts” [6]. Smart Contracts are computer programs, formalizing digital agreements, automatically enforce agreed-upon conditions thereby allowing the execution of a contractual agreements while enforcing their correctness. The development of “digital law” for regulation of such contracts is discussed in the last Chapter of this book and at

There has been a surge in recent years in the attempt to model social dynamics via statistical physics tools [8]. Frisch [9], who started this trend, has suggested to use a damped oscillator model to the economy post wars or disasters, with the assumption that there is an equilibrium state that has been perturbed. Network science has added a new dimension to our understanding of these dynamics by modeling the transaction graph connecting individual actors. This type of analysis has been usefully applied to social networks [10], computer communication networks [11], biological systems [12], transportation [13], emergency detection [14] and financial trading systems [15].

Because the first objective in this paper is to explore the dynamics of the diverse Ethereum network over time, we begin by observing weekly rolling window snapshots of the network’s transactional data. This data is shown in Figure 1, where it is obvious that there is exponential growth within a highly unpredictable ecosystem. This erratic behavior, across multiple properties, might suggest that the Ethereum network is unstable.

Figure 1. Number of weekly active wallets and transactions volume.

However, traditional metrics such as number of transactions or number of wallets are only a surface representation of the economic network. We are instead interested in its dynamics and predicting its stability. To accomplish this goal, we first explore its networks’ degree distribution, and verify that the Ethereum network has the same network properties as other real-world networks. We therefore construct a directed graph, consisting of all transactions during the examined two and one-half year period.

The resulting graph consists of 6,890,237 vertices and 17,392,610 edges. Out-going edges depict transactions in which wallet u sold a token to other wallets, and in-coming edges to u are formed as result of transactions in which u bought any token from others. Out-degree of vertex u represents the number of unique wallets buying tokens from u and its in-degree depicts the number of unique wallets selling tokens to it.

Surprisingly, despite the great heterogeneity of tokens and exponential growth, the degree distribution very accurately follows the standard power-law pattern of other financial networks, such as [10,15]. As shown in Figure 2, the Ethereum network has a few very connected nodes (wallets) with exponentially fewer and fewer connections present in more and more nodes (wallets).

Figure 2. Analysis of Ethereum network dynamics for a 2-year period from February 2016 to June 2018. The networks nodes are wallets and edges are formed by buy-sell transactions. Outgoing degree of a node reflects the number of unique wallets receiving tradable securities (tokens) from that node, and vice-versa for incoming degree. Both outgoing and incoming degrees have a power-law distribution, similarly to what was demonstrated in analysis of mobile phone, citation data and many other real-world networks [10].

In order to apply network theory to modeling Ethereum network’s dynamics over time, we also verify that temporal snapshots of this network adhere to a power-law model. We therefore form and analyze weekly transactions graphs, each of which is based on one week of all Ethereum transactions. Similar to the overall power-law fit shown in Figure 2, each of the weekly graphs also follow power-law patterns, and their goodness-of-fit, measured by their R2, is lower bounded by 0.8 and converges to 1.0 (a perfect fit).

The Damped Oscillator Model

We can also calculate the γ exponents separately for both in and out degree distributions for each of these weekly graphs respectively The exponents of both γtin\gamma_{t}^{\text{in}} and γtout\gamma_{t}^{\text{out}} are shown in Fig. 8, and it is clear that the time evolution of both the in-degree exponent and the out-degree exponent can be modeled as an harmonic damped oscillator.

Figure 3. Ethereum transaction network temporal development, demonstrating the underlying consolidation process the network undergoes. Evolution of the incoming degree distribution, γtin\gamma_{t}^{\text{in}}​, is shown in the upper panel and out-degree distribution γtout\gamma_{t}^{\text{out}} is displayed in the lower panel. Both converge toward stable states following a harmonic damped oscillator model.

The damped oscillator model has a constant stable state, and is governed by five parameters: (i) λ, the exponential decay, (ii) ω\omega, the angular frequency, (iii) γ , the stable state to which the system converges, (iv) A, the maxim amplitude of the oscillation and (v) ϕ, the phase of the oscillation.

As illustrated in Figure 3, our analysis demonstrates the underlying temporal consolidation process of the Ethereum ecosystem during the 30 months, reaching an equilibrium with respect to the essential network characteristics, γtout\gamma_{t}^{\text{out}} and γtin\gamma_{t}^{\text{in}}. Though unpredictable in many surface aspects, such as exchange rates, number of active wallets and activity volume, when observed as a dynamic process it is clear that the Ethereum network undergoes a steady consolidation process, reaching an equilibrium which it has maintained to this day.

We conclude that the Ethereum network conforms to the statistics of other social networks [10,11,12,13]. There is no a priori theoretical justification as to why a highly heterogeneous amalgamation of unrelated tokens, each with a different source and functionality, will result in a cohesive, single network behaving according to the well-established principles of other human networks unless all the elements of the transaction network share a common dynamics originating from the character of the underlying market. Our results therefore support the idea that the Ethereum network has condensed into a single community.


To better understand the character of the Ethereum networks’ oscillating dynamics, it is useful to understand the meaning of γ, the exponent of the power-law degree distribution. The degree distribution slope (in the log-log scale) intuitively describes the ratio between the number of sparsely connected wallets as the network’s edge and the number of connections of the network’s largest hub. For instance, a small γ means the ratio between number of sparsely connected wallets and the size of the largest hub is small.

We note that the Etherium network dynamics can be roughly divided into two phases, with a transition occurring around April 2017, when exponential growth of the number of “edge” wallets began due to a rush of new human users. During the first phase with relatively small numbers of participants, the numbers of buyers and sellers in the network and the sizes of the associated hubs were all quite comparable, however both γin and γout show large, anti-phase oscillations, signifying an “overshoot” of the system beyond its equilibrium state. This is the hallmark of a potentially dangerous, under-damped oscillator. This overshoot in γin and γout may represent a “herd” behavior of many individuals/wallets entering the community, making a small number of buying transactions.

During the second phase however, the number of sellers and buyers started undergoing exponential growth. During this period the largest selling hub became excessively large, accompanied with a substantially lower number of active sellers in the network correlating with its rather low γout and corresponding to the much more damped oscillations in that period. We further note that γin continued presenting oscillatory behavior in the second phase, with γin values higher than 2, as the ratio between buyers in the network and the size of the largest buying hub remained high.


Once the damped oscillator modeling parameters of the Etherium networks’ dynamics are established, the resulting analytic model can also be used for predictive purposes. We can, for instance, predict future values of γ based on fitting to a damped oscillator during the first eighteen months of data and then extending that fit to predict the second year.

The prediction for γin is shown in Figure 4 (upper panel) and shows that damped oscillations during the last year of data can be accurately predicted by the oscillator model trained during the first phase of network growth. Figure 4 (bottom panel), shows that future γout dynamics are predicted fairly well, but not as well predicted as for γin.

Figure 4. Prediction of γ dynamics, for both in-degree (top panel) and out-degree (bottom panel). Training performed until estimation date ed = June 25th, 2017, denoted by the vertical line. The test data contains γin and γout values from the last year of data, starting from the vertical line up until June 2018.

One explanation for the greater precision of γin predictions is that the vertical line marks the beginning of the period of exponential growth in number of wallets. This characteristic of γout is described in [16], where it is claimed that networks with γ<2\gamma < 2 are anomalous among naturally-occuring scale-free networks, since their largest hub grows faster than the network’s size, N .

Regulation of Financial Networks

The analysis so far shows that the Ethereum network converged to stable dynamics once the level of transaction activity became large. Can we use this example to better understand what sorts of regulatory policies will enforce this sort of stability and prevent monopolies? The answer is yes, this network perspective gives us a new way to understand how to make better predictions of impending financial problems and what sort of financial regulation will be most effective at preventing these problems.

For instance, Lera, Pentland, and Sornette [7] use a model similar to that used in the Ethereum analysis shown here, and develop a way to map a system of interacting agents into a phase-space in which the ‘distance’ from a regime of unhealthy centralization can be measured. This allows anticipation of the emergence of overly dominant agents ex ante and construct methods for early intervention. This distance they develop is quite similar to the oscillation damping factor for γin and γout that we have shown for the Ethereum network.

They show that in a sufficiently active economic system, seemingly overly fit traders may temporarily emerge essentially out of a streak of luck, and then vanish as quickly as they have appeared. In other words, the dynamics of the fitnesses themselves change quickly enough such that no systematic instability can grow explosively. However, in a more slow paced economic system, such a self-correction does not necessarily happen.

This change in system dynamics can be seen between the first phase of growth in the Ethereum network, where there were relatively few transactions and correspondingly large oscillations in the degree distribution, and the second phase where the system became much more active and the dangerous oscillations subsided. In such situations, an external regulator (e.g. the state) may wish to interfere and assert system stability. However, ad-hoc punishment of individual, particularly fit agents does not only seem ‘unfair’, but is in the end useless, because the emergence of WTA-situations is a consequence of the distribution of fitnesses as a whole.

In fact the dynamics of growth gives rise to two distinct asymptotic regimes: the fit-get richer (FGR) regime and the winner-takes-all (WTA) regime. In the later, the system is largely dependent and controlled by just a few agents. In statistical quantum mechanics, this regime corresponds to a Bose-Einstein condensate [5]. In socio-economic context, the dominant agents have been termed dragon-kings [17].

The quantitative difference between the two regimes can be well understood in terms of how the network degree distributions in γin and γout change over time. The FGR regime corresponds to a power-law degree distribution, with agents’ influences distributed over a wide range of values. In the WTA regime, the distribution exhibits a truncated power-law, with one or more agents that have degree larger than the point of truncation and dominate the system disproportionally.

What this result implies is that the common counter-measure of stopping the most dominant agents from growing further does not solve the problem sustainably. Instead, it is the weakest agents that need to be supported, to generate an overall more balanced fitness landscape and thus a healthier, more stable system.

Currently, governments address centralization issues with progressive taxes, anti-trust laws and similar legislation. However, this analysis of network dynamics reveals that this naive approach addresses only the symptom of disproportionally dominant firms rather than the underlying cause, which is that the financial network structure is fundamentally imbalanced. Instead of punishing the most competitive firms, one should foster more balanced competition by improving the relative fitness of underperforming agents.

The important implication for practical purposes is that controlling the system by taming the few largest or fittest agents is not effective. Dampening or rewiring of dominant nodes might delay but not prevent a WTA dominance, akin to similar observations in the context of explosive percolation [18]. Instead, the fitness landscape as a whole must be considered to effectively prevent the formation of winner-takes all situations.


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