Asynchronous Graph Trace Approximation via Randomized Sampling

The patent applies partial action by computing only a subset of the full matrix diagonalization process. Instead of calculating the complete trace through full diagonalization, the method computes a randomized trace approximation using only k randomly selected rows, where k < n (the dimension of the matrix). This partial computation achieves sufficient accuracy for many applications while dramatically reducing computational requirements from O(n³) to approximately O(k·n²).

The patent uses copying by creating a randomized approximation of the trace rather than the exact trace. The method generates a random vector x and computes the expectation E[x^T A x], which serves as an unbiased estimator of the trace. This copied/estimated value provides sufficient accuracy for many graph analytics applications without requiring the expensive exact computation.

The patent reduces computational time by performing partial diagonalization through random sampling. The method selects k random rows from the n×n adjacency matrix and computes trace approximations using only these rows, reducing the time complexity from O(n³) for full diagonalization to approximately O(k·n²) where k is typically much smaller than n. This partial action maintains sufficient accuracy for many graph analytics tasks.

The patent creates a time-efficient copy of the trace calculation through randomized estimation. By computing E[x^T A x] for random vectors x, the method produces an unbiased estimator of the trace that can be computed much faster than exact diagonalization. This copied approach trades minimal accuracy loss for significant time savings, making it feasible for large-scale graphs.

The patent simplifies the computational system by implementing only the essential components needed for randomized trace estimation. Instead of requiring full matrix diagonalization capabilities, the system only needs to: (1) generate random vectors, (2) perform matrix-vector multiplications, and (3) compute expectations. This partial implementation reduces device complexity while maintaining sufficient measurement precision for graph analytics applications.

The patent uses a simplified copying approach where the trace calculation is replaced by computing E[x^T A x] for random vectors. This copied method requires significantly simpler computational operations - essentially only random vector generation and matrix-vector multiplications - rather than complex full diagonalization algorithms, thereby reducing device complexity while preserving adequate accuracy.