Geometry-Driven Localization in Self-Similar Photonic Graphs with Fractional Dispersion and Temporal Memory

Fractional operators, fractal geometry, and localization frequently appear together in descriptions of wave transport, although they correspond to distinct physical mechanisms. Here, we introduce a graph-wave framework that allows their respective roles to be examined separately. Hierarchical geometry is represented by a finite Sierpiński-gasket network and compared with ordered and topology-randomized controls. Spatial fractional dynamics are defined spectrally through a power of the graph Laplacian, while temporal nonlocality is introduced independently through a Caputo derivative. We find that self-similar connectivity reshapes the Laplacian eigenmode structure, producing a population of scale-dependent confined modes alongside extended states, together with slow spreading and enhanced recurrence. In contrast, changing the spatial fractional order remaps the spectrum without modifying the eigenvectors and therefore does not, by itself, produce modal localization. The temporal fractional order controls memory and relaxation while leaving the underlying spatial mode structure unchanged. These results separate the roles of the three ingredients: geometry governs the spatial structure in which modal confinement can emerge, spatial fractionality remaps the eigenvalue spectrum and the associated phase accumulation while preserving the eigenvectors, and temporal fractionality modifies the memory-dependent evolution and persistence of the excited modes. The framework therefore provides a direct way to distinguish these mechanisms and to analyze wave transport in hierarchical photonic networks.