Convexity Evolution: Nonlinear Growth Model of Ethereum Ecology

Vitalik 2025-10-27 17:01 95527人围观 ETH

The market is accustomed to using linear thinking to predict Ethereum, but the most critical growth often occurs at critical point mutations. When investors use the traditional DCF model to value Ethereum, they assume an implicit premise: ecological growt

The market is accustomed to predicting Ethereum using linear thinking, but the most critical growth often occurs at critical point mutations.

When investors use the traditional DCF model to value Ethereum, they assume an implicit premise: ecological growth is uniform and predictable. However, historical data shows that every jump in Ethereum is not the result of linear accumulation, but a non-linear pattern of "long-term silence-sudden burst".

This is not an accident, but a mathematical manifestation of the convexity of the system. Understanding this nonlinear characteristic may be more important than focusing on the number of daily active addresses.
1. Linear vs. Convexity: Two completely different growth logics

Most financial assets follow a linear logic. In A-share margin trading, if you borrow twice as much money and pay twice as much risk, you will get twice as much potential income.

This is the world we are familiar with: input is directly proportional to output, and risk is directly proportional to return.

But convex assets are different. Its core characteristic is that when tail events occur, returns grow much faster than earnings decline. Described in mathematical language, this is a function with a positive second-order derivative - the curve is convex upward and the marginal benefit is increasing.

The growth curve of the Ethereum ecosystem naturally has this characteristic.

It is not a linear system with a steady growth of 20% every year, but a step-like system of "long-term sideways - short-term surge". This pattern appears repeatedly at three levels: network effects, technical architecture, and developer ecosystem.

Understanding this explains why Ethereum always "looks like it's dying" and then suddenly explodes.
2. Non-linear growth in three dimensions
1. Network Effects’ “Ketchup Moment”"

In the ketchup bottle, nothing comes out after a few squeezes for a long time, but most of it comes out after a few squeezes.

This accurately describes the explosion pattern of Ethereum’s network effects.

Case 1: 2015-2017 ICO outbreak

The mainnet was launched in July 2015, and the ecosystem was almost blank in the first two years.

After the DAO incident in 2016, the market once thought that Ethereum was "dead""

2017 suddenly exploded: ICO financing soared from $95M in 2016 to $5.6B in 2017 (a 58-fold increase)

This is not the result of linear accumulation. The silence in the first two years is not a failure, but a process of "squeezing the ketchup bottle" - the underlying tools mature, the developer community is formed, and smart contract standards are established. Once the tipping point is crossed, the release is explosive.

Case 2: 2018-2020 DeFi Summer

In the 2018-2019 bear market, Ethereum TVL has been hovering below $1B for a long time

In June 2020, TVL exceeded $1B

In November 2021, TVL reached $180B (180x growth in 18 months)

Verify again: Nonlinear growth is not a constant climb, but a phase change process of "balance-disequilibrium-reorganization".

Data support:

Number of DeFi protocols: about 50 at the end of 2019, more than 500 at the end of 2021 (10 times growth)

But the TVL growth is 180 times - this is a typical power law distribution

Composability effect: The number of trading strategies combined by Uniswap + Aave + MakerDAO has increased exponentially

Non-linearity is quite counter-intuitive. We are accustomed to reaping the benefits of hard work. But the essence of the network effect is: the first 90% of the efforts only produce 10% of the results, and the last 10% of the efforts produce 90% of the results.

If investors use a linear model for valuation, they will give up in the first half of the "ketchup bottle" period and chase higher in the second half.
2. Layer2’s “10-meter step effect”"

If a person jumps down from a 1-meter-high step 10 times, nothing will happen. But I jumped down from the 10-meter-high steps and died immediately. This is the negative manifestation of nonlinearity—risk does not accumulate linearly, but suddenly explodes at a threshold.

But Ethereum's Layer 2 strategy does the opposite: dispersing "10 meters of pressure" into "10 times 1 meter."

In DeFi Summer 2021, mainnet gas fees soared to $200+. If we follow linear logic, the solution is to "increase TPS by 10 times." But this will bring centralization risks - this is a "10-meter step" type of systemic vulnerability.

Ethereum has chosen the modular path:

Migrate the execution layer to Rollup (Arbitrum, Optimism, Base)

Data availability layer remains on mainnet (further optimized via EIP-4844)

The settlement layer is still guaranteed by the mainnet consensus

This is not a simple expansion, but an engineering implementation of convexity.

Data validation:

In 2023, the total transaction volume of Layer 2 will exceed the main network by 10 times.

But security is inherited by the mainnet – risk does not increase linearly

User costs are reduced by 100 times, while the degree of system decentralization is increased (more verification nodes participate)

This is a counter-intuitive result: by "distributing pressure", the system achieves super-linear growth in throughput while maintaining sub-linear risk in security.

Expressed by formula:

Traditional expansion: performance = f(risk), linear relationship

Modular architecture: performance = f(risk^0.5), risk grows slower than performance

This is the mathematical expression of convexity - the yield curve is convex upward and the risk curve is concave downward.
3. Power law growth of developer ecosystem

The growth in the number of Ethereum developers also shows non-linear characteristics.

Key figures (Electric Capital report):

In 2015, there were about 50 full-time developers

In the bear market of 2018, the number of full-time developers increased to 500 (a 10-fold increase)

In 2023, there will be more than 5,000 full-time developers (another 10-fold increase)

But what matters more is not the quantity, but the non-linear growth of marginal product.

The exponential effect of composability:

Value of 1 agreement = 1

10 protocols can be combined to create 10² = 100 applications

100 protocols can be combined to create 100² = 10,000 applications

This is a variation of Metcalfe's Law: the value of a network is proportional to the square of the number of nodes.

Actual case:

After Uniswap V2 went online, it spawned aggregators (1inch, Matcha), derivatives (Perp, dYdX), and revenue optimizers (Yearn).

These applications have become new "Lego bricks", assembled by next-generation protocols

Form a positive feedback loop of "protocol→application→protocol"

The first 1,000 developers contribute 10% of the value, and the next 10,000 developers contribute 90% of the value.

This is not a difference in developer capabilities, but the leverage effect of composability - latecomers stand on the shoulders of giants, and their marginal output increases exponentially.

Comparative data:

Bitcoin: The number of developers increased by 2 times, and the protocol functionality increased by about 1.5 times (sublinear)

Ethereum: The number of developers has increased by 2 times, and the number of ecological applications has increased by 5-10 times (super linear)

This is the manifestation of convexity at the technical level: input growth is slower than output growth, and system efficiency increases rather than decreases.
3. Why does the market underestimate non-linear growth?

The limitation of traditional valuation models is that they assume the future is a linear extension of the past.

The DCF model requires the input of a "stable growth rate", but Ethereum's growth rate itself is variable - close to zero before the critical point and close to infinity after the critical point.

Cognitive Misunderstanding 1: Using daily activity data to predict long-term value

The number of daily active addresses in Ethereum has been stagnant for a long time in 2018-2019, but innovations at the protocol layer (such as AMM and lending protocols) are brewing. If you only look at the daily activity data, you will come to the conclusion that "growth has stagnated."

But the characteristic of nonlinear systems is that changes in key variables occur at the invisible bottom. The number of developers, the amount of code submissions, and the establishment of protocol standards—these are the real indicators of the squeeze of the "ketchup bottle."

Cognitive Misunderstanding 2: Ignoring the Unpredictability of Critical Points

The moment when a phase change occurs is difficult to predict. Water is still liquid at 99°C and boils suddenly at 100°C.

The same goes for Ethereum:

Before the ICO explosion in 2017, no one predicted the magnitude of the growth

Before DeFi Summer 2020, the TVL curve looked dull

Before the outbreak of Layer 2 in 2023, the market is still questioning "Why not directly increase the main network TPS""

The common feature of these critical points is that linear indicators tend to be weakest on the eve of a sudden change. This is the counterintuitive thing about convex systems—the greatest opportunities are hidden in the most boring moments.
4. Risks and Contrarian Views

Rational investment requires facing up to the negative possibilities of non-linear growth.

Risk 1: Non-linearity may also be downward

"The 10-meter step metaphor also applies to risk. If there is a fundamental flaw in the system (such as a flaw in the consensus mechanism), the collapse may also be non-linear.

In 2022, the Ronin cross-chain bridge was attacked, resulting in a single loss of $625M.

If a similar vulnerability occurs on the mainnet, the impact may be exponential

Risk 2: The "reflexivity" of composability"

Protocols can be combined to bring not only positive feedback, but also negative feedback. On "Black Thursday" in March 2020, MakerDAO's liquidation mechanism failed, resulting in a chain reaction of $4M in bad debts.

Composability means that bugs in a protocol will be transmitted to the entire ecosystem in a non-linear manner.

Risk three: Unpredictability of critical points

The flip side of non-linear growth is that we can’t tell where we are in the ketchup bottle.

If "most of it" has been squeezed out, future growth may return to linearity or even attenuate.

If you are still in the "first few times" stage, the waiting period may be several years.

Contrary view:

Some investors believe that non-linear growth is an ex post facto narrative rather than a law that can be verified ex ante. Each outbreak has unique triggers (ICO boom, liquidity mining, Layer 2 technology maturity), which cannot be easily copied.

This doubt has its validity. But the key is not to predict "when" it will explode, but to identify "whether" it has a convex structure:

Whether the system has enough "pressure reserve" (developers, capital, technology accumulation)

Is there any "potential energy" that can be released (unmet needs, unresolved bottlenecks)

Is there a positive feedback mechanism (network effect, composability, community consensus)

If these three conditions are true, even if the timing cannot be predicted, it can be confirmed that the "ketchup bottle" will be squeezed out sooner or later.
5. Conclusion: Embrace nonlinearity rather than predict it

"The AI revolution is a disruptive innovation, and there will not be all winners. "The same applies to Ethereum.

The essence of technological change is not linear optimization, but a spiral of "balance-imbalance-reorganization". Every time Ethereum seems to be stagnant, it is the brewing period for the next phase change.

Investment inspiration:

Traditional valuation models fail in the face of nonlinear systems, not because the model is wrong, but because the assumptions are wrong. If growth itself is a sudden change, then the input term "stable growth rate" is a false proposition.

A more reasonable framework is:

Don't predict "when" it will grow, but evaluate "whether" it has a convex structure

Don't pursue "accurate valuation", but focus on "odds asymmetry""

Do not expect "even-speed returns", but accept "long-term silence + short-term surges""

Ethereum is like a life system that explodes at critical points rather than moving forward at a constant speed. Only by understanding this can you persevere in the first half of the "ketchup bottle" and reap the rewards in the second half.

The market always underestimates nonlinear systems because human intuition is inherently linear. And this is precisely where excess returns come from.