The Math: Network Coverage Analysis
How many users does UFOBeep need for effective coverage?
The Question
How many users does UFOBeep need in the United States for the network to be effective - meaning when someone reports a UFO sighting, there's a high probability that at least one other nearby user receives the alert and can verify?
Key Assumptions
Alert Radius
50 km
(user-adjustable, default)
Covers ~7,854 km² area
Active Fraction (α)
25%
Users actively connected and able to receive alerts at any given moment
Urban Clustering
80%
Users cluster in urban areas (top 50 metros = 52% of US population)
Target Success
95%
Desired probability that a beep reaches at least one other user
The Formula: Poisson Distribution
The probability that at least one user receives an alert follows a Poisson distribution:
What this means:
- For 80% probability: need μ = 1.61 → ~6 registered users per 50 km
- For 90% probability: need μ = 2.30 → ~9 registered users per 50 km
- For 95% probability: need μ = 3.00 → ~12 registered users per 50 km
- For 99% probability: need μ = 4.61 → ~18 registered users per 50 km
Mathematical Minimum vs Strategic Target
Model 1: Mathematical Minimum (Instantaneous)
Goal: 95% chance that at least one other active user receives the alert RIGHT NOW
| Metro Size | Area (km²) | Bubbles | Users Needed |
|---|---|---|---|
| Small city | ~8,000 | 1 | 12 |
| Medium metro | ~15,000 | 2 | 24 |
| Large metro | ~30,000 | 4 | 48 |
| Major metro | ~60,000 | 8 | 96 |
Mathematical Minimum for 50 metros: ~600-1,200 total users (12-24 per metro, depending on size)
Model 2: Strategic Target (Real-World)
Goal: Reliable multi-witness verification + uneven clustering + network effects
Why so much higher than the minimum?
- Uneven clustering: Users won't be perfectly distributed - need redundancy
- Multiple witnesses: Want 2-3+ people to see it, not just 1
- Network perception: Need density for users to feel it's “active”
- Engagement variance: α varies by time of day (10% at 3am, 40% at 8pm)
- Retention: Higher density = better experience = lower churn
Strategic Target for 50 metros: 15,000-20,000 total users (300-400 per metro)
Scaling to 50 Major Metros
The top 50 US metropolitan areas contain 52% of the US population (~170M people). Here's how different user counts translate to coverage:
| Total Users | Users/Metro | Active (25%) | Coverage Quality |
|---|---|---|---|
| 600 | 12 | 3 | 95% instant (mathematical min) |
| 1,200 | 24 | 6 | Near-certain instant |
| 5,000 | 100 | 25 | Guaranteed + multi-witness |
| 10,000 | 200 | 50 | Strong multi-witness coverage |
| 15,000-20,000 ★ | 300-400 | 75-100 | STRATEGIC TARGET |
| 30,000 | 600 | 150 | Excellent redundancy |
| 50,000 | 1,000 | 250 | National scale |
★ Why 15-20K vs 600 minimum? The mathematical minimum (600-1,200) assumes perfect distribution and single-witness verification. The strategic target accounts for real-world clustering, multi-witness needs, time-of-day variance, and network effects that drive retention and growth.
Why the Strategic Target is 25-30x Higher
1. Uneven Geographic Distribution
Users won't be uniformly spread across metros. Some will cluster in specific neighborhoods, college campuses, or enthusiast communities. Others will be isolated. The mathematical model assumes perfect uniform distribution - reality requires 10x more users to compensate for clustering.
2. Multi-Witness Verification
The minimum calculates P(≥1 user). But for credible verification, you want 2-3+ witnesses. For P(≥3 users) at 95% confidence, you need roughly 3× the users. This alone pushes the target from 600 to ~2,000.
3. Time-of-Day Variance
Active fraction (α) isn't constant. At 3am: maybe 5-10% active. At 8pm: maybe 40-50% active. The 25% is an average. To maintain 95% coverage at low-activity times, you need higher baseline density.
4. Network Effects & Retention
With only 12 users per metro, most people will NEVER experience a successful connection. They'll install, never see alerts, and uninstall. Higher density creates positive feedback: more alerts → more engagement → better retention → organic growth.
5. Growth Headroom
Users are concentrated in fewer than 50 metros initially. Maybe 10-20 cities dominate. Those cities need extra density to feel “alive” while coverage expands. The 15-20K target provides headroom for uneven early adoption.
Real-World Growth Scenarios
Sensitivity Analysis: What If We're Wrong?
Our model depends on the active fraction (α). Here's how the target changes:
| Active Fraction (α) | Scenario | Users/50km (95%) | Total (50 metros) |
|---|---|---|---|
| 10% | Low engagement | 30 | 1,500-3,000 |
| 25% | Conservative (our model) | 12 | 600-1,200 |
| 40% | High engagement (peak hours) | 7.5 | 375-750 |
| 50% | Very high engagement | 6 | 300-600 |
Key Insight: Even if α doubles (50% vs 25%), the strategic target stays 15-20K because we're optimizing for multi-witness verification and network effects, not just bare minimum coverage.
The Answer (Two Perspectives)
📐 Mathematical Minimum: ~600-1,200 Users
This is the bare minimum for 95% probability of instant connection:
- •12-24 users per metro (depending on size)
- •Assumes perfect distribution (unrealistic)
- •Only guarantees 1 witness, not multiple
- •Insufficient for network effects
🎯 Strategic Target: 15,000-20,000 Users
This is the realistic target for an effective, sustainable network:
- ✓300-400 users per major metro
- ✓75-100 active users at any moment per metro
- ✓Multiple witnesses for verification (2-5+)
- ✓Robust to uneven clustering and churn
- ✓Network feels “alive” - drives retention
- ✓Enables viral growth and word-of-mouth
Model Validation
Both the conservative (instantaneous, α=25%) and optimistic (window-based, 70% reach) models were validated using Monte Carlo simulations with 10,000 iterations per scenario.
- At 600-1,200 users: Mathematical minimum confirmed (95% instant)
- At 5,000 users: 99%+ connection probability
- At 10,000 users: Near-certain with multi-witness capability
- At 20,000 users: Guaranteed coverage with strong redundancy
The mathematical models closely match simulation results across all scenarios.
Conclusion
UFOBeep's coverage requirements depend on what you're optimizing for:
Proves the concept works, but sparse coverage and poor user experience
Network starts feeling useful, but concentrated in 10-20 cities
Network becomes self-sustaining with reliable multi-witness verification, strong retention, and viral growth potential across 50+ metros
Bottom line: While the math shows UFOBeep could theoretically work with 600 users, the realistic target of 15,000-20,000 userscreates a network that actually delivers value, retains users, and grows organically.