For Mathematicians

Open Problems in Stigmergic Optimization

A Formula That Creates Emergence

f(e, α) = w_e / (1 + p_e × α)

Where:

• w_e = base weight of edge e
• p_e = pheromone level on edge e
• α = agent sensitivity parameter

This is the STAN formula. It determines how agents choose paths through a graph.

From this simple formula, complex behavior emerges:

• Power-law distributions (0.4% of edges carry 94.9% of pheromone)
• Self-organizing highways
• Adaptive optimization
• Collective intelligence

The mathematics of this emergence is not fully understood.

The Optimization Problem

Given a weighted graph G = (V, E) with edge weights w: E → ℝ⁺, and a target node t, find the minimum cost path from any start node s.

Classical solution: Dijkstra, A*, etc. Time complexity O((V + E) log V).

Stigmergic solution: Deploy agents that:

1. Traverse edges with probability inversely proportional to effective cost
2. Deposit pheromone p on traversed edges when reaching target
3. Pheromone decays: p(t+1) = τ × p(t) where τ < 1

Question: Does this converge? To what? How fast?

Open Problems

1. Convergence

Conjecture: For finite graphs with connected components containing the target, STAN converges to a stationary pheromone distribution.

What’s known:

• Classical ACO has convergence proofs for specific variants (Dorigo & Blum, 2005)
• STAN differs in the cost formula and agent heterogeneity

What’s open:

• General convergence conditions
• Rate of convergence
• Basin of attraction characterization

2. Optimality

Question: Under what conditions does the stationary distribution encode the optimal path?

Intuition: High pheromone → Low cost → More traversals → More pheromone. But this could lock into local optima.

Related: This resembles reinforcement learning’s exploration-exploitation tradeoff. Can we borrow bounds from RL theory?

3. Complexity Classification

Question: What is the complexity of the following decision problem?

Given graph G, pheromone dynamics (deposit, decay), and threshold k, is there a stationary pheromone configuration where minimum path cost ≤ k?

Conjecture: This is in P for certain graph classes and NP-hard in general.

4. Power-Law Emergence

Observation: In simulations, pheromone follows a power-law distribution.

Question: Why?

Hypothesis: The dynamics create a preferential attachment process (rich-get-richer), which provably generates power laws.

Challenge: Formalize this. Derive the exponent.

5. Phase Transitions

Observation: There appears to be a phase transition when trails crystallize into highways.

Question: Is this a genuine phase transition in the statistical mechanics sense?

If yes:

• What is the order parameter?
• What universality class?
• What are the critical exponents?

The Mathematical Objects

The Pheromone Landscape

A function p: E → ℝ⁺ assigning pheromone levels to edges.

Dynamics:

• Deposit: When agent traverses path π, for each e ∈ π: p(e) ← p(e) + Δ
• Decay: For all e: p(e) ← τ × p(e)

Fixed points: Stationary distributions where expected deposit equals decay.

The Effective Cost Graph

A transformed graph where edge weights are:

w'(e) = w(e) / (1 + p(e) × α)

This is agent-dependent (α varies by agent type).

Key insight: Different agents see different graphs. Scouts (α = 0.3) see nearly the original graph. Harvesters (α = 0.9) see heavily modified weights.

The Agent Population

A distribution over agent types with different α values.

Question: What population distribution optimizes colony performance?

Connection: This resembles portfolio optimization—balancing exploration (scouts) and exploitation (harvesters).

Theorems to Prove

Theorem 1 (Convergence)

For any finite connected graph G and pheromone dynamics with τ ∈ (0,1) and bounded deposit Δ, the expected pheromone distribution converges to a unique fixed point.

Theorem 2 (Optimality Bound)

The expected path cost under stationary distribution is within factor f(G, τ, Δ) of optimal.

Theorem 3 (Power-Law Distribution)

Under conditions [X], the stationary pheromone distribution is power-law with exponent γ = [Y].

Theorem 4 (Complexity)

The stigmergic optimization decision problem is in [complexity class] for graphs with [property].

What We Provide

Data

• Complete pheromone snapshots over time
• Empirical distribution of pheromone levels
• Agent traversal traces
• Graph structure (35 entity types, 17 relation types)

Computational Resources

• TypeDB Cloud access for queries
• Agent simulation environment
• GPU credits for large-scale experiments

Collaboration

• Access to CS and Physics collaborators
• Mentorship from algorithm designers
• Co-authorship opportunities

Hackathon Challenges for Mathematicians

Challenge: Prove Convergence

Prove or disprove convergence of STAN for general graphs.

Approach:

• Model as Markov chain on pheromone configurations
• Show existence of stationary distribution
• Prove convergence from any initial state

Prize bonus: $1,000

Challenge: Derive Power-Law Exponent

Prove power-law emergence and derive the exponent.

Approach:

• Model pheromone dynamics as preferential attachment
• Apply theory of Yule-Simon processes
• Validate against empirical data

Prize bonus: $500

Challenge: Complexity Analysis

Classify the computational complexity of stigmergic optimization.

Approach:

• Define the decision problem precisely
• Attempt reduction from known hard problems
• Or find polynomial algorithm

Prize bonus: $2,000 for definitive result

Challenge: Optimal Population Distribution

What mix of agent types (α values) optimizes colony performance?

Approach:

• Formulate as optimization problem
• Analyze fixed points for different distributions
• Validate with simulation

Your Heroes Worked on Related Problems

Paul Erdős studied random graphs—our pheromone landscape is a dynamic random graph.

László Lovász worked on random walks—our agents are random walkers with pheromone-biased transitions.

Avi Wigderson proved hardness results—stigmergic optimization might have similar structure.

Noga Alon combined probabilistic and algebraic methods—both are relevant here.

Publication Opportunities

The Beauty

There’s something mathematically beautiful here.

A simple formula creates complexity. Local rules generate global structure. Random agents produce deterministic patterns.

This is emergence captured in equations.

The proofs are waiting to be written.

Register Your Team

[REGISTER NOW]

Include at least one non-math team member (we recommend CS or Physics).

The best proofs come from unexpected connections.

“The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.”

— Henri Poincaré

You’ve proved theorems about graphs, Markov chains, and complexity.

Now prove theorems about emergence.

[JOIN THE HACKATHON]