SpectralGraph trait

[mxfactorial]

current

the SpectralGraph trait is absent from geonum

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spectral graphs traditionally rely on eigendecomposition of graph matrices (adjacency, Laplacian) with O(n³) complexity

this is impractical for large-scale graphs

with geonum's angle-based representation, we can implement the same theoretical power with O(1) complexity per operation, enabling analysis of massive graphs

add a SpectralGraph trait to src/traits/spectral_graph.rs:

#[cfg(feature = "spectral_graph")]
pub trait SpectralGraph: Sized {
    /// convert graph representation to multivector form
    fn to_multivector(&self) -> Multivector;
    
    /// compute graph spectrum directly through angle stability
    fn spectrum(&self) -> Multivector;
    
    /// perform spectral clustering via angle proximity
    fn spectral_cluster(&self, k: usize) -> Vec<Vec<usize>>;
    
    /// compute graph partitioning through angle thresholding
    fn spectral_partition(&self) -> (Vec<usize>, Vec<usize>);
    
    /// compute algebraic connectivity (Fiedler value) via angle gap
    fn algebraic_connectivity(&self) -> f64;
    
    /// compute stationary distribution of random walks via angle stability
    fn stationary_distribution(&self) -> Multivector;
    
    /// compute eigenvalue gaps for community detection
    fn eigenvalue_gaps(&self) -> Vec<f64>;
    
    /// compute diffusion distance between vertices
    fn diffusion_distance(&self, i: usize, j: usize, t: f64) -> f64;
    
    /// perform spectral embedding into k dimensions
    fn spectral_embedding(&self, k: usize) -> Multivector;
    
    /// compute graph conductance via angle flow
    fn conductance(&self, partition: &[usize]) -> f64;
    
    /// perform dimensionality reduction through spectral projection
    fn spectral_dimension_reduction(&self, dim: usize) -> Multivector;
    
    /// analyze community structure through angle patterns
    fn community_structure(&self) -> Vec<Vec<usize>>;
    
    /// compute vertex centrality through angle influence
    fn spectral_centrality(&self) -> Vec<f64>;
    
    /// perform Cheeger cut via angle separation
    fn cheeger_cut(&self) -> (Vec<usize>, Vec<usize>);
    
    /// perform heat kernel analysis
    fn heat_kernel(&self, t: f64) -> Multivector;

    /// returns the fiedler vector l2 eigenvector of the laplacian
    fn fiedler_vector(&self) -> Multivector

    /// returns the maximum spatial frequency in the graph’s structure
    /// computed without matrices using geometric derivative estimates
    /// consistent with ∇²ψ = λψ form where λ is the spectral radius
    fn spectral_radius(&self) -> f64

    /// performs the graph fourier transform of a signal over nodes
    fn graph_fourier_transform(&self, signal: Multivector) -> Multivector

    /// estimates the eigenvalue density using gaussian kernel smoothing
    fn spectral_density(&self) -> Multivector

    /// returns lower and upper bounds from cheeger inequality
    fn cheeger_bound(&self) -> (f64, f64)

    /// returns the directed laplacian for non-symmetric graphs
    /// preserves directional flow using asymmetric edge angles
    fn directed_laplacian(&self) -> Multivector

    /// applies a spectral filter given a weight vector over frequencies
    fn spectral_filter(&self, weights: &[f64]) -> Multivector

    /// returns a low rank homological signature using laplacian rank collapse
    /// captures persistent cycles as minimal energy modes in the graph geometry
    fn homological_signature(&self) -> Multivector
}

Impl for Multivector

example for Multivector:

#[cfg(feature = "spectral_graph")]
impl SpectralGraph for Multivector {
    fn spectrum(&self) -> Multivector {
        // extract graph structure (vertices and edges)
        let vertices = self.vertices();
        let edges = self.edges();
        
        // use geometric product to identify angle-stable components
        // these are equivalent to eigenvectors/eigenvalues without matrix operations
        let eigencomponents = vertices.wedge(&edges);
        
        // extract eigenvalues through length scaling factors
        let eigenvalues = eigencomponents.0.iter().map(|g| {
            Geonum {
                length: g.length,   // eigenvalue magnitude
                angle: g.angle,     // eigenvalue phase
                blade: 0,           // scalar (grade 0) for eigenvalue
            }
        }).collect();
        
        Multivector(eigenvalues)
    }
    
    fn algebraic_connectivity(&self) -> f64 {
        // compute algebraic connectivity (Fiedler value) via direct angle gap
        // traditional design: second smallest eigenvalue of Laplacian matrix
        
        // in geonum, find the smallest non-zero angle gap
        // between components of the graph spectrum
        let spectrum = self.spectrum();
        
        // sort spectrum by angle
        let mut angles: Vec<f64> = spectrum.0.iter()
            .map(|g| g.angle)
            .collect();
        angles.sort_by(|a, b| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal));
        
        // find smallest non-zero gap (equivalent to Fiedler value)
        let mut min_gap = std::f64::MAX;
        for i in 1..angles.len() {
            let gap = (angles[i] - angles[i-1]).abs();
            if gap > 1e-10 && gap < min_gap {
                min_gap = gap;
            }
        }
        
        min_gap
    }
    
    fn spectral_cluster(&self, k: usize) -> Vec<Vec<usize>> {
        // perform spectral clustering through angle proximity
        // instead of computing eigenvectors and then k-means clustering
        
        // 1. compute spectrum via angle stability
        let spectrum = self.spectrum();
        let vertices = self.vertices();
        
        // 2. extract top-k eigencomponents by sorting
        let mut eig_components: Vec<&Geonum> = spectrum.0.iter().collect();
        eig_components.sort_by(|a, b| b.length.partial_cmp(&a.length).unwrap_or(std::cmp::Ordering::Equal));
        let top_k = eig_components.iter().take(k).cloned().cloned().collect::<Vec<Geonum>>();
        
        // 3. group vertices by angle proximity to eigenmodes
        // This is equivalent to clustering in eigenspace
        let mut clusters = vec![Vec::new(); k];
        
        // for each vertex, find closest eigencomponent by angle
        for (i, vertex) in vertices.0.iter().enumerate() {
            let mut closest_idx = 0;
            let mut min_angle_dist = std::f64::MAX;
            
            for (j, eigen) in top_k.iter().enumerate() {
                let angle_dist = vertex.angle_distance(eigen);
                if angle_dist < min_angle_dist {
                    min_angle_dist = angle_dist;
                    closest_idx = j;
                }
            }
            
            clusters[closest_idx].push(i);
        }
        
        clusters
    }
    
    fn spectral_partition(&self) -> (Vec<usize>, Vec<usize>) {
        // use Fiedler vector (second eigenvector) for partitioning
        // in geonum, use second angle component
        
        // get spectrum ordered by stability (eigenvalue magnitude)
        let spectrum = self.spectrum();
        let mut components: Vec<Geonum> = spectrum.0.clone();
        components.sort_by(|a, b| a.length.partial_cmp(&b.length).unwrap_or(std::cmp::Ordering::Equal));
        
        // get second component (Fiedler vector equivalent)
        let fiedler = if components.len() >= 2 {
            components[1].clone()
        } else {
            // default value if not enough components
            Geonum {
                length: 1.0,
                angle: 0.0,
                blade: 0,
            }
        };
        
        // partition vertices based on angle alignment with Fiedler component
        let vertices = self.vertices();
        let mut partition_a = Vec::new();
        let mut partition_b = Vec::new();
        
        for (i, vertex) in vertices.0.iter().enumerate() {
            // compute angle alignment with Fiedler component
            let alignment = vertex.dot(&fiedler);
            
            if alignment >= 0.0 {
                partition_a.push(i);
            } else {
                partition_b.push(i);
            }
        }
        
        (partition_a, partition_b)
    }
    
    fn stationary_distribution(&self) -> Multivector {
        // compute stationary distribution of random walks via angle stability
        // traditional: leading eigenvector of transition matrix
        
        // extract graph structure and adjacency information
        let adjacency = self.adjacency();
        
        // find the most stable angle component under adjacency transformation
        // this is equivalent to the leading eigenvector
        
        // apply adjacency operation until convergence
        let mut state = Multivector(vec![
            Geonum {
                length: 1.0,
                angle: 0.0,
                blade: 1,
            }
        ]);
        
        // iterate a fixed number of times (in practice would use convergence check)
        for _ in 0..10 {
            // Transform state by adjacency (equivalent to matrix multiplication)
            state = state.mul(&adjacency);
            
            // Normalize state
            let norm = state.0.iter().map(|g| g.length.powi(2)).sum::<f64>().sqrt();
            state = Multivector(state.0.iter().map(|g| {
                Geonum {
                    length: g.length / norm,
                    angle: g.angle,
                    blade: g.blade,
                }
            }).collect());
        }
        
        state
    }
    
    // added implementations for other methods...
}

add doc.rs comments and unit test coverage

feature flag

add to Cargo.toml:

[features]
spectral_graph = []

module integration

add spectral_graph module to src/traits/mod.rs:

#[cfg(feature = "spectral_graph")]
pub mod spectral_graph;
#[cfg(feature = "spectral_graph")]
pub use spectral_graph::SpectralGraph;

re-export from lib.rs:

#[cfg(feature = "spectral_graph")]
pub use traits::SpectralGraph;

test implementation

add tests/spectral_graph_test.rs with the following test:

#[cfg(feature = "spectral_graph")]
mod tests {
    use geonum::*;
    use std::f64::consts::PI;

    // include O(1) performance assertions in tests:

    // // spectral embedding & structure
    // it_embeds_graph_in_low_dimensions();
    // it_discovers_latent_structure_with_low_rank_spectrum();
    // it_recovers_shape_signatures_from_cycle_basis();

    // // spectral operators & transforms
    // it_performs_graph_fourier_transform_using_geometric_angles();
    // it_computes_diffusion_distances_between_vertices();
    // it_models_diffusion_processes_with_angle_weighted_steps();
    // it_estimates_spectral_radius_without_matrix_factorization();
    // it_derives_spectral_density_from_low_rank_modes();
    // it_filters_graph_signal_in_frequency_domain();

    // // topology & homology
    // it_extracts_homological_cycles_from_laplacian_rank_collapse();
    // it_identifies_homology_through_spectral_degeneracy();
    // it_tests_duality_between_energy_and_topology();

    // // flow & directionality
    // it_constructs_directed_laplacian_preserving_flow_directions();
    // it_detects_flow_asymmetry_through_directed_blades();

    // // optimization & inference
    // it_solves_minimum_cut_problems();
    // it_bounds_cheeger_constant_using_geometric_edges();
    // it_identifies_bottlenecks_using_fiedler_geometry();

    // // signal processing
    // it_improves_graph_signals_with_directional_spectral_filter();
    // it_establishes_graph_similarity_via_fourier_signatures();

    // // centrality, community, conductance
    // it_measures_graph_conductance();
    // it_identifies_communities_in_social_network();
    // it_computes_centrality_measures();

    // // misc geometry
    // it_encodes_graph_topology_with_bivector_rotation();
    // it_visualizes_high_frequency_modes_as_topological_noise();
    // it_analyzes_path_structures_through_angles();


    #[test]
    fn its_a_graph_spectrum() {
        // create a simple graph with 4 vertices and 5 edges
        // 0 -- 1 -- 2
        // |         |
        // +----3----+
        
        // represent vertices as geometric numbers
        let vertices = Multivector(vec![
            Geonum {
                length: 1.0,
                angle: 0.0,
                blade: 1,
            },
            Geonum {
                length: 1.0,
                angle: PI/2.0,
                blade: 1,
            },
            Geonum {
                length: 1.0,
                angle: PI,
                blade: 1,
            },
            Geonum {
                length: 1.0,
                angle: 3.0*PI/2.0,
                blade: 1,
            },
        ]);
        
        // represent edges as geometric numbers
        let edges = Multivector(vec![
            // edge 0-1
            Geonum {
                length: 1.0,
                angle: PI/4.0,  // halfway between vertices 0 and 1
                blade: 2,       // bivector (grade 2) representing edge between vertices
            },
            // edge 1-2
            Geonum {
                length: 1.0,
                angle: 3.0*PI/4.0,
                blade: 2,
            },
            // edge 2-3
            Geonum {
                length: 1.0,
                angle: 5.0*PI/4.0,
                blade: 2,
            },
            // edge 3-0
            Geonum {
                length: 1.0,
                angle: 7.0*PI/4.0,
                blade: 2,
            },
            // edge 0-2 (diagonal)
            Geonum {
                length: 1.0,
                angle: PI/2.0,
                blade: 2,
            },
        ]);
        
        // create graph as combined multivector
        let mut graph_components = Vec::new();
        graph_components.extend(vertices.0.clone());
        graph_components.extend(edges.0.clone());
        let graph = Multivector(graph_components);
        
        // compute spectrum
        let spectrum = graph.spectrum();
        
        // basic test - verify we get non-empty spectrum
        assert!(!spectrum.0.is_empty());
        
        // perform spectral partition
        let (part_a, part_b) = graph.spectral_partition();
        
        // the graph should be partitioned into two approximately equal groups
        assert!(!part_a.is_empty());
        assert!(!part_b.is_empty());
        
        // find algebraic connectivity
        let conn = graph.algebraic_connectivity();
        
        // test non-zero for a connected graph
        assert!(conn > 0.0);
        
        // stationary distribution has entries for all vertices
        let stationary = graph.stationary_distribution();
        assert!(stationary.0.len() > 0);
        
        // spectral clustering - test with k=2
        let clusters = graph.spectral_cluster(2);
        assert_eq!(clusters.len(), 2);
        
        // all vertices are assigned to a cluster
        let total_vertices = clusters[0].len() + clusters[1].len();
        assert_eq!(total_vertices, vertices.0.len());
    }
    
    #[test]
    fn it_handles_extreme_dimensions() {
        // create a graph in 1000-dimensional space
        let high_dim = Dimensions::new(1000);
        
        // create vertices in high dimensions
        let vertices = high_dim.multivector(&[0, 1, 2, 3, 4]);
        
        // create edges connecting vertices
        let edges = Multivector(vec![
            // create some sample edges
            Geonum {
                length: 1.0,
                angle: PI/4.0,
                blade: 2,
            },
            Geonum {
                length: 1.0,
                angle: PI/2.0,
                blade: 2,
            },
        ]);
        
        // create the graph
        let mut graph_components = Vec::new();
        graph_components.extend(vertices.0.clone());
        graph_components.extend(edges.0.clone());
        let graph = Multivector(graph_components);
        
        // measure time to compute spectral properties
        let start = std::time::Instant::now();
        
        // compute spectrum
        let _spectrum = graph.spectrum();
        
        // compute algebraic connectivity
        let _conn = graph.algebraic_connectivity();
        
        // elapsed time
        let elapsed = start.elapsed();
        
        // completes in under 1 second despite high dimensionality
        assert!(elapsed.as_secs() < 1); // TODO: this can be a smaller assertion
    }
}

benchmark

add benchmarks to compare against traditional eigendecomposition:

#[cfg(feature = "spectral_graph")]
fn bench_spectral_operations(c: &mut Criterion) {
    let mut group = c.benchmark_group("Spectral Graph Operations");
    
    // benchmark spectral analysis in million-node graph
    group.bench_function("spectrum_million_nodes", |b| {
        b.iter(|| {
            let dims = Dimensions::new(1_000_000);
            let vertices = dims.multivector(&[1, 2, 3]);
            let graph = Multivector(vertices.0);
            
            black_box(graph.spectrum())
        })
    });
    
    // compare against traditional method (will be much slower)
    group.bench_function("traditional_spectrum_thousand_nodes", |b| {
        b.iter(|| {
            // create a 1000x1000 adjacency matrix for a graph
            let n = 1000;
            let mut adj_matrix = vec![vec![0.0; n]; n];
            
            // fill with some edge data (sparse random graph)
            for i in 0..n {
                for j in 0..i {
                    if (i + j) % 100 == 0 {  // Sparse connections
                        adj_matrix[i][j] = 1.0;
                        adj_matrix[j][i] = 1.0;
                    }
                }
            }
            
            // traditional design computes eigenvalues using 
            // an iterative algorithm like Lanczos or Arnoldi
            // since we can't do that in this benchmark, we'll simulate
            // the computational complexity with adjusted delay
            
            // simulate O(n³) complexity with matrix operations
            let mut sum = 0.0;
            for i in 0..n {
                for j in 0..n {
                    for k in 0..n {
                        if k % 100 == 0 {  // reduce work but keep complexity
                            sum += adj_matrix[i][j] * (i * j * k) as f64;
                        }
                    }
                }
            }
            
            // return a vector of simulated eigenvalues
            let eigenvalues = (0..10).map(|i| sum / (i+1) as f64).collect::<Vec<f64>>();
            black_box(eigenvalues)
        })
    });
}

advantages

this trait will provide significant advantages over traditional spectral graph theory implementations:

1. Constant-time operations: O(1) complexity regardless of graph size, compared to O(n³) for traditional methods
2. Massive scalability: enables spectral analysis of graphs with millions of nodes
3. Geometric intuition: provides clear geometric meaning to spectral properties
4. Direct implementation: avoids the complexity of numerical eigendecomposition algorithms
5. Memory efficiency: requires only O(|V|+|E|) storage instead of O(|V|²) for matrix-based designs

applications

the SpectralGraph trait will support advanced graph applications including:

• graph clustering and partitioning
• graph visualization and layout
• link prediction
• graph neural networks
• node embedding
• structural role discovery
• anomaly detection in graphs
• graph diffusion models
• dimensionality reduction for graph-structured data
• network community detection

tags

• feature
• spectral graph theory
• graph algorithms
• eigenvalues
• angle-based computation

[mxfactorial]

hours worked

requirement writing: 1.5