SOMA

Lesson 1The Graph Laplacian — Diffusion's Discrete Twin

You know $\nabla^2 f$. On a regular grid, it computes the sum of differences from neighbors. The graph Laplacian is exactly this object, freed from the grid.

Given a graph $G = (V, E)$, define:

• $A$ — the adjacency matrix: $A_{ij} = 1$ if $(i,j) \in E$
• $D$ — the degree matrix: $D_{ii} = \deg(i)$, zero off-diagonal
• $L = D - A$ — the combinatorial Laplacian

The quadratic form tells you everything: $x^T L x = \sum_{(i,j) \in E} (x_i - x_j)^2$. This measures total variation across edges. Smooth signals (constant on components) have zero energy. Spiky signals (large differences across edges) have high energy.

Play with it

Click any node to deposit heat. Watch it diffuse along edges via $dp/dt = -\alpha L p$. The sidebar shows the Laplacian matrix and the current state vector in real time.

Notice what happens: total heat is conserved under pure diffusion (no decay). The constant eigenmode $\lambda_0 = 0$ is never damped. Heat spreads but never disappears. This changes in Lesson 2.

Lesson 2The SOMA Equation — Diffusion, Decay, Stability

SOMA's Medium evolves by three forces:

The decay term $-\gamma p$ breaks conservation. Without persistent sources, all traces fade to zero — this is biologically necessary. Pheromone trails that lasted forever would make the system rigid. The source term $S(t)$ represents agents depositing traces as they work.

Equilibrium

At steady state, $dp/dt = 0$:

The equilibrium exists and is unique because $\alpha L + \gamma I$ has all eigenvalues $\alpha \lambda_i + \gamma > 0$. The system is globally attracting.

Numerical stability

SOMA uses explicit Euler: $p(t+\Delta t) = p(t) + \Delta t \cdot f(p)$. The Jacobian is $J = -(\alpha L + \gamma I)$ with eigenvalues $-(\alpha \lambda_i + \gamma)$. Explicit Euler is stable when $|1 + \Delta t \cdot \mu| < 1$ for all eigenvalues $\mu$ of $J$.

By Gershgorin, $\lambda_{\max}(L) \leq 2 d_{\max}$. So:

Play with it

Click to deposit. Adjust $\alpha$, $\gamma$, and $\Delta t$. The "destabilize" button cranks $\Delta t$ past the stability bound — watch what happens. Toggle the stability clamp to see SOMA's fix.

Lesson 3Stigmergic Agents — Coordination Without Communication

Termites build cathedrals without architects. The mechanism: indirect coordination through environmental modification. An agent modifies the shared medium; other agents respond to the modification, not to each other.

In SOMA, each agent computes the pheromone gradient at its current position:

Movement follows an $\varepsilon$-greedy policy: with probability $\varepsilon$, move randomly (explore). Otherwise, select a neighbor proportional to $\max(0, \partial p(v, u))$ (exploit).

The baseline: coupon collector

Expected time for a random walk to find $k$ bugs among $n$ nodes with $m$ agents:

Play with it

Run Random Walk (no pheromone, no gradient) and Stigmergic (agents deposit pheromone on discovery, follow gradients) side by side. The counter shows steps to find all bugs. Run multiple trials to see the distribution.

Lesson 4Resolution & Homeostasis — The Anti-Inflammatory

Here's a problem: stigmergic agents pile on. Once a bug is found, its pheromone attracts more agents. They arrive, confirm the finding, deposit more pheromone, attracting even more agents. In immunology, this is a cytokine storm — an inflammatory cascade that damages the host.

SOMA's fix: resolution traces. After discovery, agents deposit an anti-inflammatory signal:

Agents sense $p_{\text{eff}}$, not raw $p$. Resolved areas become invisible even if they still have pheromone. Agents move on to unexplored territory.

Play with it

Toggle resolution on and off. Without it, agents cluster at the first discovery. With it, they spread out and find everything.

Lesson 5From Scalars to Sheaves — and the Road Ahead

Everything so far used a constant sheaf: each node carries a scalar (pheromone level), and the restriction maps are the identity. The full SOMA framework assigns each node a typed stalk:

The sheaf Laplacian $L_{\mathcal{F}}$ generalizes the graph Laplacian. Where $L$ compares scalar values across edges, $L_{\mathcal{F}}$ compares stalks through restriction maps and measures disagreement via the coboundary operator. The spectral gap of $L_{\mathcal{F}}$ governs convergence speed — Hansen & Ghrist (2021) proved exponential convergence to global sections.

Think of it this way: the constant sheaf says "every node speaks the same language." The full sheaf says "every node has its own dialect, and the restriction maps are the translation dictionaries." Sheaf cohomology $H^1(K, \mathcal{F})$ detects obstructions — local agreements that can't be made globally consistent. When $H^1 \neq 0$, the system has structural inconsistencies. This is your debuggability tool.

The Roadmap

What follows is the mathematical territory ahead. Each item becomes a future interactive lesson.