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Antigravity Agent
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---
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id: wiki-2026-0508-exponential-growth
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title: Exponential Growth
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [exponential, compound growth, doubling time, k-factor, viral coefficient, Moore's law]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [math, growth, exponential, viral, compound, scaling, doubling-time]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Math / Python
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applicable_to: [Growth, Modeling, Finance, Tech Forecast]
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---
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# Exponential Growth
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## 매 한 줄
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> **"매 N(t) = N₀ · e^(rt) — 매 rate ∝ size"**. 매 doubling time = ln(2)/r. 매 famous: Moore's law, COVID, viral, compound interest, ML scaling. 매 modern: 매 sigmoid (logistic) 의 의 의 cap.
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## 매 핵심
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### 매 form
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- **Continuous**: N(t) = N₀ · e^(rt).
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- **Discrete**: N_t = N₀ · (1+r)^t.
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- **Doubling time**: t₂ = ln(2)/r ≈ 0.693/r.
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- **Rule of 72**: 매 % rate 의 의 의 72 의 divide.
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### 매 응용
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1. **Population**: 매 unconstrained.
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2. **Compound interest**.
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3. **Moore's law**: 매 doubling 18-24mo.
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4. **Viral spread**: 매 R0 > 1.
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5. **Startup growth**: 매 viral coefficient k.
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6. **ML scaling laws** (Hoffmann, Kaplan).
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7. **AGI timeline** (controversial).
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### 매 cap (logistic)
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- 매 real world 의 의 logistic 의 settle (carrying capacity).
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- 매 dN/dt = rN(1 - N/K).
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### 매 sub-exponential alternatives
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- Linear: y = a + bt.
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- Polynomial: y = a + bt^n.
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- Logistic: 매 S-curve.
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- Power-law: y = at^b.
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## 💻 패턴
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### Doubling time
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```python
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import math
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def doubling_time(growth_rate_per_period):
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return math.log(2) / growth_rate_per_period
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# 매 5% per year
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print(doubling_time(0.05)) # 매 ~13.86 years
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# 매 Rule of 72: 72/5 = 14.4 (close)
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```
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### Viral coefficient (k-factor)
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```python
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def k_factor(invites_per_user, conversion_rate):
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return invites_per_user * conversion_rate
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def viral_growth(initial, k, cycles):
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"""매 k > 1 → exponential."""
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return [initial * (k ** c) for c in range(cycles)]
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```
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### Compound interest
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```python
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def compound(principal, rate, periods, n_compoundings_per_period=12):
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return principal * (1 + rate / n_compoundings_per_period) ** (n_compoundings_per_period * periods)
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def continuous_compound(principal, rate, time):
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return principal * math.exp(rate * time)
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```
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### Logistic (real-world cap)
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```python
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import numpy as np
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from scipy.integrate import odeint
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def logistic_growth(N, t, r, K):
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return r * N * (1 - N / K)
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t = np.linspace(0, 50, 500)
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N = odeint(logistic_growth, 10, t, args=(0.3, 1000))
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```
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### Detect exponential vs not
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```python
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def is_exponential(timeseries):
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"""매 log(y) 의 linear 의 fit?"""
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log_y = np.log(np.maximum(timeseries, 1e-9))
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t = np.arange(len(log_y))
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r2 = np.corrcoef(t, log_y)[0, 1] ** 2
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return r2 > 0.95
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```
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### Fit growth rate
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```python
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from scipy.optimize import curve_fit
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def exp_func(t, N0, r):
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return N0 * np.exp(r * t)
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def fit_exp(t, y):
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popt, _ = curve_fit(exp_func, t, y, p0=[y[0], 0.1])
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return {'N0': popt[0], 'r': popt[1], 'doubling_time': math.log(2) / popt[1]}
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```
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### Moore's law forecast
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```python
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def moores_forecast(transistor_count_now, year_now, year_target):
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years = year_target - year_now
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return transistor_count_now * 2 ** (years / 1.5) # 매 2x per 1.5y
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```
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### COVID-style
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```python
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def epidemic_R(cases_today, cases_5days_ago, gen_time_days=5):
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"""매 매 5d 의 doubling 매 매 R."""
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growth_rate = math.log(cases_today / cases_5days_ago) / 5
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return math.exp(growth_rate * gen_time_days)
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```
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### Cohort retention (counter-exponential)
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```python
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def retention_curve(d1=0.4, decay_rate=0.05):
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"""매 retention 의 typically 매 power-law / exponential decay."""
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return [d1 * math.exp(-decay_rate * d) for d in range(0, 365)]
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```
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### Detect inflection (saturation)
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```python
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def detect_saturation(series, window=10):
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"""매 derivative 의 decrease 의 detect."""
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deltas = np.diff(series)
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recent_delta = np.mean(deltas[-window:])
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earlier_delta = np.mean(deltas[-2*window:-window])
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return recent_delta < earlier_delta * 0.7 # 매 30% slowdown
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```
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### LLM scaling law (Chinchilla)
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```python
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def chinchilla_optimal(N_params, D_tokens):
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"""매 optimal: D ≈ 20 * N (Hoffmann 2022)."""
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optimal_D = 20 * N_params
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if D_tokens < optimal_D * 0.5: return 'undertrained'
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if D_tokens > optimal_D * 2: return 'overtrained'
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return 'near_optimal'
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```
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### Viral campaign forecast
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```python
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def viral_campaign(seed_users, k, cycles, cycle_days):
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users = [seed_users]
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for _ in range(cycles):
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users.append(users[-1] * (1 + k))
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return {'final_users': users[-1], 'days': cycles * cycle_days, 'series': users}
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```
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### Linear-log plot helper
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```python
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import matplotlib.pyplot as plt
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def plot_growth(t, y):
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fig, ax = plt.subplots(1, 2, figsize=(10, 4))
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ax[0].plot(t, y); ax[0].set_title('Linear')
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ax[1].semilogy(t, y); ax[1].set_title('Log-y (exp = straight)')
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plt.show()
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| Population | Logistic (capped) |
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| Tech transistor | Moore's law (exp) |
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| Startup | Viral k + retention |
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| Disease | R + gen time |
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| Investment | Compound |
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| Hype curve | Logistic + decay |
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**기본값**: 매 short-horizon 의 exponential model + 매 long-horizon 의 logistic + 매 detect saturation 의 monitor.
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## 🔗 Graph
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- 부모: [[Math]]
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- 변형: [[Power-Law]]
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- 응용: [[Epidemiological-Modeling]] · [[Moores-Law]]
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- Adjacent: [[Scaling-Laws]] · [[Singularity]] · [[Doubling-Time]]
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## 🤖 LLM 활용
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**언제**: 매 growth model. 매 forecast. 매 viral / scaling.
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**언제 X**: 매 saturation evident.
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## ❌ 안티패턴
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- **Extrapolate forever**: 매 cap 의 ignore.
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- **Linear intuition for exp**: 매 trillion vs million 의 underestimate.
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- **No log-y plot**: 매 detect 의 fail.
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- **Cherry-pick window**: 매 trend manipulate.
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## 🧪 검증 / 중복
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- Verified (math textbook, Hoffmann 2022 Chinchilla, COVID literature).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-04-20 | Auto-reinforced |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — exp + 매 doubling / viral / logistic / scaling code |
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