Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions. Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned. In this paper we develop an original harmonic-analysis framework for directed networks centered on the \emph{adjacency} operator. We propose a \emph{Biorthogonal Graph Fourier Transform} (BGFT) built from left/right eigenvectors, formulate directed ``frequency'' and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departure-from-normality functional. We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs. A simulation protocol compares undirected versus directed cycles (asymmetry without non-normality) and a perturbed directed cycle (genuine non-normality), demonstrating that BGFT yields coherent reconstruction and filtering across asymmetric regimes.

翻译：经典的谱图理论与图信号处理依赖于一个对称性原理：无向图产生对称（自伴）的邻接/拉普拉斯算子，从而得到正交特征基和能量守恒的傅里叶展开。现实世界网络通常是有向且非对称的，这会产生非自伴且经常非正规的算子，导致正交性失效，谱坐标可能呈现病态条件。本文针对有向网络，围绕\emph{邻接}算子发展了一套原创的调和分析框架。我们提出了一种基于左/右特征向量的\emph{双正交图傅里叶变换}（BGFT），在非厄米特设定下构建了有向“频率”与滤波理论，并通过条件数与偏离正规性泛函量化了非对称性与非正规性对稳定性的影响。我们在可对角化条件下证明了精确的合成/分析恒等式，为BGFT带限信号建立了采样与重构的理论保证，并推导了扰动/稳定性界，从而解释了为何在非正规有向图上，基于正交GFT的朴素假设会失效。通过仿真实验，我们比较了无向环与有向环（仅非对称但正规）以及扰动有向环（真正的非正规性），结果表明BGFT能在各种非对称体系中实现一致的重构与滤波效果。