Machine learning on graphs has become a central tool in modern data science, with applications in social networks, biology, chemistry, recommendation systems, and infrastructure monitoring. Despite the success of graph embeddings and Graph Neural Networks (GNNs), many existing models are biased toward local neighborhoods and may struggle to capture global, multiscale structure. This limitation becomes particularly pronounced when different networks share similar local statistics, such as degree distributions, yet differ significantly in hierarchical organization.

Fractal geometry provides a complementary perspective by characterizing networks through scale invariance, self-similarity, and multiscale complexity. Fractal descriptors such as box-counting dimension and multiscale box-cover signatures encode global structural information that is robust to local perturbations and informative under coarsegraining. This project develops a computational pipeline for fractal-aware machine learning on graphs, integrating fractal descriptors with classical machine learning models and optional GNN baselines. Through experiments on synthetic and real networks, the study evaluates when and why fractal features enhance classification, anomaly detection, and cross-domain generalization.

Curvature is a central concept in geometry, measuring how a space deviates from flatness. In Riemannian geometry, Ricci curvature governs fundamental phenomena such as volume growth, geodesic dispersion, and diffusion processes. In recent years, extending curvature concepts to discrete structures such as graphs has emerged as a powerful approach in network science, enabling a geometric understanding of complex networks. This research proposal aims to investigate Forman–Ricci curvature, a combinatorial and computationally efficient notion of Ricci curvature defined on the edges of a graph.

Starting from discrete exterior calculus and the Hodge Laplacian on graphs, the proposed research will derive the Forman–Ricci curvature formula, study its behavior in weighted and unweighted networks, and analyze its geometric interpretation in terms of local branching, expansion, and bottlenecks. Beyond the theoretical framework, the proposed work will explore applications of Forman–Ricci curvature in network analysis, including hub and bottleneck detection, robustness assessment, community structure characterization, and diffusion dynamics. The project combines rigorous mathematical foundations with computational experiments on synthetic and real-world networks, providing a geometric perspective on modern network science.

Diffusion and spreading processes on networks play a fundamental role in modeling information flow, epidemic propagation, opinion dynamics, and transport phenomena in complex systems. Classical diffusion models are typically defined on crisp or probabilistic graphs, where edges either exist or are associated with probabilities. While such models are mathematically tractable, they often fail to capture vagueness, subjectivity, and imprecision that characterize many real-world interactions. Fuzzy networks offer an alternative modeling framework in which edges are associated with degrees of membership in the interval [0, 1], representing graded interactions such as trust, influence, or confidence. This project investigates diffusion and spreading dynamics on fuzzy networks. It develops mathematical models for fuzzy diffusion, fuzzy random walks, and fuzzy contagion processes, and compares their behavior with classical crisp and probabilistic models. Through theoretical analysis and numerical simulations, the study aims to understand how fuzziness affects convergence, equilibrium, spreading speed, and robustness of diffu- sion processes