Spectral topology analysis for component graphs.
Measure the health of the space between your microservices, modules, or any connected system. Compute Laplacian eigenvalues, Fiedler value (connectivity), Cheeger constant (bottleneck detection), and component importance.
[dependencies]
cathedral-probe = "0.1"use cathedral_probe::CathedralProbe;
let mut probe = CathedralProbe::new(vec!["web", "api", "db", "cache"]);
probe.connect("web", "api", 1.0);
probe.connect("api", "db", 1.0);
probe.connect("api", "cache", 0.5);
println!("Fiedler value: {:.3}", probe.fiedler_value());
println!("Is healthy: {}", probe.is_healthy(0.1));
println!("Cheeger upper bound: {:.3}", probe.cheeger_upper_bound());This crate treats your component graph as a mathematical object (a weighted undirected graph) and computes spectral properties of its Laplacian matrix. The "Laplacian" is a matrix that encodes how strongly each component is connected to its neighbors. Its eigenvalues reveal the structure of the space between components.
| Metric | What It Tells You |
|---|---|
| Fiedler value | Second-smallest eigenvalue. Higher = better connected. Zero = disconnected. |
| Cheeger bounds | Upper and lower bounds on edge expansion from Cheeger's inequality. |
| Fragility index | 1 / Fiedler value. Higher = more fragile. Infinity = disconnected. |
| Component importance | How much removing each component hurts connectivity. |
| Bottleneck edges | Edges whose removal most reduces connectivity. |
| Spectrum | All eigenvalues — the full "fingerprint" of your topology. |
- Microservice monitoring — Is the space between your services healthy or fragmenting?
- Dependency analysis — Which packages in your dependency graph are critical?
- Team communication — Map who talks to whom, find silos and bottlenecks.
- Network topology — Detect when a network is approaching disconnection.
- Data pipeline health — Are your ETL stages well-connected or fragile?
// Create a graph with named components
let mut probe = CathedralProbe::new(vec!["auth", "api", "db", "queue"]);
// Add weighted edges (weight = connection strength)
probe.connect("auth", "api", 1.0);
probe.connect("api", "db", 1.0);
// Spectral analysis
let eigenvalues = probe.spectrum(); // All Laplacian eigenvalues
let fiedler = probe.fiedler_value(); // Connectivity measure
let cheeger_ub = probe.cheeger_upper_bound(); // h(G) ≤ √(2·λ₂)
let cheeger_lb = probe.cheeger_lower_bound(); // λ₂/2 ≤ h(G)
let fragile = probe.fragility_index(); // 1/fiedler (infinity if disconnected)
// Health check
let healthy = probe.is_healthy(0.1); // Quick boolean: Fiedler >= threshold?
// Component analysis
let importance = probe.component_importance(); // HashMap<String, f64>
let bottlenecks = probe.bottlenecks(); // Vec<(String, String, f64)>
// Graph properties
let connected = probe.is_connected(); // Fully connected?
let components = probe.connected_components(); // Number of isolated groups
let total = probe.total_weight(); // Sum of edge weights
let avg_deg = probe.average_degree(); // Average weighted degree- Build the graph Laplacian (degree matrix minus adjacency matrix)
- Compute eigenvalues via QR iteration with Wilkinson shifts
- The sorted eigenvalues form the "spectrum" — a fingerprint of the topology
- The second-smallest eigenvalue (Fiedler value) measures algebraic connectivity
- Component importance is computed by removing each node and measuring Fiedler drop
Zero dependencies. Requires std (uses HashMap).
The spectral methods used here are well-established in the literature:
- Fiedler, M. (1973). Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 23(2), 298-305. — Introduced the Fiedler vector and algebraic connectivity.
- Chung, F. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, No. 92, AMS. — Comprehensive reference for normalized and combinatorial Laplacians.
- Mohar, B. (1989). Isoperimetric numbers of graphs. Journal of Combinatorial Theory, Series B, 47(3), 274-291. — Cheeger constants for graphs.
- Alon, N. & Milman, V. (1985). λ₁, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38(1), 73-88. — The Alon-Milman bound relating spectral gap to expansion.
MIT