2nd/10_Wiki/Topics/AI_and_ML/Exponential-Growth.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-exponential-growth	Exponential Growth	10_Wiki/Topics	verified	self	exponential compound growth doubling time k-factor viral coefficient Moore's law	none	A	0.95	applied	math growth exponential viral compound scaling doubling-time		2026-05-10	pending	language applicable_to Math / Python Growth Modeling Finance Tech Forecast

Exponential Growth

매 한 줄

"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.

매 핵심

매 form

• Continuous: N(t) = N₀ · e^(rt).
• Discrete: N_t = N₀ · (1+r)^t.
• Doubling time: t₂ = ln(2)/r ≈ 0.693/r.
• Rule of 72: 매 % rate 의 의 의 72 의 divide.

매 응용

1. Population: 매 unconstrained.
2. Compound interest.
3. Moore's law: 매 doubling 18-24mo.
4. Viral spread: 매 R0 > 1.
5. Startup growth: 매 viral coefficient k.
6. ML scaling laws (Hoffmann, Kaplan).
7. AGI timeline (controversial).

매 cap (logistic)

• 매 real world 의 의 logistic 의 settle (carrying capacity).
• 매 dN/dt = rN(1 - N/K).

매 sub-exponential alternatives

• Linear: y = a + bt.
• Polynomial: y = a + bt^n.
• Logistic: 매 S-curve.
• Power-law: y = at^b.

💻 패턴

Doubling time

import math
def doubling_time(growth_rate_per_period):
    return math.log(2) / growth_rate_per_period

# 매 5% per year
print(doubling_time(0.05))  # 매 ~13.86 years
# 매 Rule of 72: 72/5 = 14.4 (close)

Viral coefficient (k-factor)

def k_factor(invites_per_user, conversion_rate):
    return invites_per_user * conversion_rate

def viral_growth(initial, k, cycles):
    """매 k > 1 → exponential."""
    return [initial * (k ** c) for c in range(cycles)]

Compound interest

def compound(principal, rate, periods, n_compoundings_per_period=12):
    return principal * (1 + rate / n_compoundings_per_period) ** (n_compoundings_per_period * periods)

def continuous_compound(principal, rate, time):
    return principal * math.exp(rate * time)

Logistic (real-world cap)

import numpy as np
from scipy.integrate import odeint

def logistic_growth(N, t, r, K):
    return r * N * (1 - N / K)

t = np.linspace(0, 50, 500)
N = odeint(logistic_growth, 10, t, args=(0.3, 1000))

Detect exponential vs not

def is_exponential(timeseries):
    """매 log(y) 의 linear 의 fit?"""
    log_y = np.log(np.maximum(timeseries, 1e-9))
    t = np.arange(len(log_y))
    r2 = np.corrcoef(t, log_y)[0, 1] ** 2
    return r2 > 0.95

Fit growth rate

from scipy.optimize import curve_fit

def exp_func(t, N0, r):
    return N0 * np.exp(r * t)

def fit_exp(t, y):
    popt, _ = curve_fit(exp_func, t, y, p0=[y[0], 0.1])
    return {'N0': popt[0], 'r': popt[1], 'doubling_time': math.log(2) / popt[1]}

Moore's law forecast

def moores_forecast(transistor_count_now, year_now, year_target):
    years = year_target - year_now
    return transistor_count_now * 2 ** (years / 1.5)  # 매 2x per 1.5y

COVID-style

def epidemic_R(cases_today, cases_5days_ago, gen_time_days=5):
    """매 매 5d 의 doubling 매 매 R."""
    growth_rate = math.log(cases_today / cases_5days_ago) / 5
    return math.exp(growth_rate * gen_time_days)

Cohort retention (counter-exponential)

def retention_curve(d1=0.4, decay_rate=0.05):
    """매 retention 의 typically 매 power-law / exponential decay."""
    return [d1 * math.exp(-decay_rate * d) for d in range(0, 365)]

Detect inflection (saturation)

def detect_saturation(series, window=10):
    """매 derivative 의 decrease 의 detect."""
    deltas = np.diff(series)
    recent_delta = np.mean(deltas[-window:])
    earlier_delta = np.mean(deltas[-2*window:-window])
    return recent_delta < earlier_delta * 0.7  # 매 30% slowdown

LLM scaling law (Chinchilla)

def chinchilla_optimal(N_params, D_tokens):
    """매 optimal: D ≈ 20 * N (Hoffmann 2022)."""
    optimal_D = 20 * N_params
    if D_tokens < optimal_D * 0.5: return 'undertrained'
    if D_tokens > optimal_D * 2: return 'overtrained'
    return 'near_optimal'

Viral campaign forecast

def viral_campaign(seed_users, k, cycles, cycle_days):
    users = [seed_users]
    for _ in range(cycles):
        users.append(users[-1] * (1 + k))
    return {'final_users': users[-1], 'days': cycles * cycle_days, 'series': users}

Linear-log plot helper

import matplotlib.pyplot as plt
def plot_growth(t, y):
    fig, ax = plt.subplots(1, 2, figsize=(10, 4))
    ax[0].plot(t, y); ax[0].set_title('Linear')
    ax[1].semilogy(t, y); ax[1].set_title('Log-y (exp = straight)')
    plt.show()

매 결정 기준

상황	Approach
Population	Logistic (capped)
Tech transistor	Moore's law (exp)
Startup	Viral k + retention
Disease	R + gen time
Investment	Compound
Hype curve	Logistic + decay

기본값: 매 short-horizon 의 exponential model + 매 long-horizon 의 logistic + 매 detect saturation 의 monitor.

🔗 Graph

• 부모: Math
• 변형: Power-Law
• 응용: Epidemiological-Modeling · Moores-Law
• Adjacent: Scaling-Laws · Singularity · Doubling-Time

🤖 LLM 활용

언제: 매 growth model. 매 forecast. 매 viral / scaling. 언제 X: 매 saturation evident.

❌ 안티패턴

• Extrapolate forever: 매 cap 의 ignore.
• Linear intuition for exp: 매 trillion vs million 의 underestimate.
• No log-y plot: 매 detect 의 fail.
• Cherry-pick window: 매 trend manipulate.

🧪 검증 / 중복

• Verified (math textbook, Hoffmann 2022 Chinchilla, COVID literature).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-04-20	Auto-reinforced
2026-05-08	Phase 1
2026-05-10	Manual cleanup — exp + 매 doubling / viral / logistic / scaling code