Directedfiltered

Directedfiltered is a term used in graph signal processing to describe a family of filtering operations applied to signals defined on directed graphs. It denotes filters that respect edge orientation, enabling direction-aware smoothing and feature extraction that reflect the flow of information in the network.

Formally, for a directed graph G=(V,E) with a weighted adjacency matrix W, a directedfiltered operator F acts on a signal x in R^|V| to produce x' = F x, where F is constructed to emphasize interactions along edge directions. Common constructions include polynomials in a row-stochastic transition matrix P = D_out^{-1} W and directional diffusion operators that combine forward and backward components with asymmetric weights. Some approaches define F as a weighted sum of neighborhood aggregations restricted to out-neighbors (forward filtering) or in-neighbors (backward), while spectral methods may use a directed Laplacian or a directed Fourier basis to implement filters in a directional frequency domain.

Construction choices reflect a balance between locality, stability, and interpretability. Directednesstivity can be achieved by imposing asymmetry in the weighting scheme, and normalization is often used to preserve scale across layers or iterations. Computationally, directedfiltered methods tend to be more involved than symmetric graph filters but remain compatible with sparse graph representations.

Applications include smoothing and denoising of sensor measurements on networks with flow, traffic forecasting, modeling influence in social or information networks, and feature extraction on knowledge or language graphs where edge directions encode precedence or entailment. Limitations include potential sensitivity to edge direction noise and the need for careful parameter tuning to avoid distortion of symmetric structures.

See also graph signal processing, directed graphs, and graph filters.