On any graph, every signal on edges can be uniquely decomposed into three parts. This is the Hodge decomposition. It is the discrete analog of Helmholtz’s theorem: every vector field splits into an irrotational part, a solenoidal part, and a harmonic residual. In the continuous case, the decomposition is classical. On graphs, it is exact and computable. And it has something specific to say about trust.
Consider a directed graph where each edge carries a weight — a reputation flow. Alice attests that she trusts Bob with weight 0.8. Bob attests trust in Carol at 0.6. Carol trusts Alice at 0.7. These are edge signals: real numbers on directed edges. The Hodge decomposition says that any such signal ω on edges can be written uniquely as:
Three orthogonal components. Three entirely different stories about what the trust network is doing.
The gradient part is ωgrad = B₀¹f where f is a potential function on nodes and B₀ is the node-edge incidence matrix. A gradient signal flows “downhill” from high-potential nodes to low-potential nodes. If the entire reputation signal were gradient, there would exist a global ranking of every participant — a single number per node such that all edge weights are explained by differences in rank.
Gradient signals satisfy curl = 0. There are no inconsistencies, no cycles of disagreement. Everyone agrees on the hierarchy. This is the world PageRank assumes: a directed acyclic reputation landscape where trust flows from the authoritative to the peripheral.
In practice, reputation is never purely gradient. But the gradient component tells you how much of the signal can be explained by consensus. The larger the gradient fraction, the more the network agrees on who is trustworthy.
The curl part lives in im(B₁¹) where B₁ is the edge-triangle incidence matrix. It captures circulation — signal that flows around triangles without any net potential difference. If A trusts B trusts C trusts A more than the reverse directions, that excess is curl.
Curl reveals inconsistency. In a reputation network, inconsistency often means collusion. Three accounts endorsing each other in a ring, each boosting the other’s reputation without any external validation — that is a curl signal. Sybil clusters generate curl because their mutual endorsement creates closed loops of trust that don’t connect to the broader gradient landscape.
The curl component is the adversarial detector. When you decompose a reputation graph and the curl fraction spikes, something is circulating that shouldn’t be.
The harmonic part is the residual — what remains after gradient and curl are removed. It lives in ker(L₁) where L₁ = B₁B₁¹ + B₀¹B₀ is the Hodge Laplacian on edges. Harmonic signals are simultaneously curl-free and divergence-free. They are the topological fingerprint of the graph.
In a reputation network, harmonic components reveal community structure. Trust accumulates within communities but does not flow between them. The harmonic signal traces the boundaries — the places where the network’s topology prevents trust from propagating. Two tightly-knit clusters connected by a single bridge will have a strong harmonic component along that bridge: the signal is trapped by the structure.
The dimension of the harmonic space equals the first Betti number β₁ of the graph — the number of independent cycles. More cycles, more room for harmonic signals, more community structure that resists being explained by either global ranking or local manipulation.
The mathematical fact that curl ∘ grad = 0 has a direct interpretation: global consensus and local manipulation live in orthogonal subspaces. You can measure them independently. The Sybil signal does not contaminate the consensus signal, and vice versa. This is not an approximation — it is an algebraic identity.
This is why spectral methods work for Sybil detection. Algorithms like SybilRank and EigenTrust implicitly exploit this orthogonality. They project the trust signal onto the gradient subspace (where legitimate reputation lives) and discard the rest. The Hodge decomposition makes explicit what these algorithms do implicitly — and adds the harmonic component, which neither gradient nor curl can explain.
In a reputation protocol — say, one built on Nostr where edge weights represent signed attestations — the decomposition gives you three independent measurements from one dataset. The gradient tells you who is broadly trusted. The curl tells you where trust is being manufactured. The harmonic tells you where communities begin and end, where trust accumulates but does not cross.
For a small network the computation is direct. Build B₀ (the signed incidence matrix, nodes to edges) and B₁ (edges to triangles). The gradient projection is B₀¹(B₀B₀¹)†B₀ω. The curl projection is B₁¹(B₁B₁¹)†B₁ω. The harmonic residual is everything else. For large networks, iterative methods on the Hodge Laplacian scale to millions of edges. The mathematics is the same. Only the linear algebra changes.
Toggle the visualization above. Add a Sybil ring and watch the curl component light up. Inject a hierarchy and see the gradient fraction grow. The harmonic component — quiet, structural — traces what neither consensus nor manipulation can reach.