Scaling Graph Cliques with Vector Projections

Exponential replication of 4-node cliques using vector space projections

Mathematical Foundation

K₄ → K₄ⁿ via Vector Projection

To scale a 4-node clique (K₄) exponentially, we can use vector projections in ℝ^d space:

Base Clique Representation:

Let V = {v₁, v₂, v₃, v₄} be position vectors of the original clique nodes.

Each vector vᵢ ∈ ℝ^d where d ≥ 3 for non-coplanar placement.

vᵢ^(k) = vᵢ + k ⋅ δ ⋅ û

Where û is a unit direction vector and δ is a scaling factor for the k-th replication.

Exponential Scaling: For n replications, we create K₄ⁿ by projecting the base clique along orthogonal vectors:

V_ijk = v + i⋅a + j⋅b + k⋅c where i,j,k ∈ {0,1,...,n-1}

This creates n³ total cliques from the original K₄.

Interactive Visualization

Replication Factor (n): 2

Spacing (δ): 5

Visualization Notes: The base K₄ clique is shown in blue. Replicated cliques are color-coded based on their replication level. Edges within cliques are solid, while inter-clique connections are dashed.

Vector Projection Equations

P_u(v) = (v ⋅ û) û

For orthogonal projection along basis vectors:

v′ = v + Σ_i=1^k αᵢ P_uᵢ(v)

Kronecker Product Approach:

For adjacency matrix A of K₄, the scaled graph adjacency is:

A_scaled = A ⊗ I_n + I₄ ⊗ B

Where B is the adjacency matrix of the replication pattern (e.g., path or grid).

Base K₄ Adjacency Matrix:

[0 1 1 1]
[1 0 1 1]
[1 1 0 1]
[1 1 1 0]
                    

Complexity Analysis

Scaling Properties:

Original K₄: 4 nodes, 6 edges

After n replications: 4n nodes, 6n + 4(n-1) edges (for linear replication)

Exponential scaling (d dimensions): 4nᵈ nodes

Computational Efficiency: Vector projection scaling is O(nᵈ) in nodes but maintains the clique property in each replication. The Kronecker product approach allows efficient computation of the scaled adjacency matrix.

Clique Preservation: Each K₄ remains fully connected after projection

Graph Clique Scaling with Vector Projections | Mathematical Network Theory