Falls among older adults represent a critical public health challenge, with approximately 37.3 million fall-related incidents reported globally each year. Early and accurate prediction of falls is essential to enable timely, proactive interventions and to reduce associated injuries and fatalities. This work introduces a graph-based machine learning framework that leverages data from the cStick, a smart assistive Internet of Medical Things (IoMT) device. Bipartite graphs are constructed to model static correlations between multivariate sensor inputs— including heart rate variability (HRV), pressure, distance, SpO2, blood sugar levels, and accelerometer readings— and fall outcomes encoded as no fall, predicted fall, or definite fall. SHAP (SHapley Additive exPlanations) values are further integrated to enhance model interpretability and to identify the most influential sensor features through feature-only graph projections. Kernel Density Estimation (KDE) plots and pairplots are used to visualize feature distributions across fall categories. The proposed framework demonstrates that Pressure and Distance exhibit the strongest correlations with fall decisions (1.000 and −0.946, respectively), providing actionable insights for risk stratification. The integration of graph-based analysis with SHAP interpretability improves both predictive accuracy and transparency, facilitating proactive interventions and enhancing the safety, autonomy, and well-being of elderly individuals in real-world assistive care settings.

The efficient allocation of finite resources to a dynamic patron base represents a core challenge in modern library management. Traditional heuristic approaches often lack the formal rigor needed for verifiable optimization and proactive planning. This paper introduces a novel formal framework grounded in automata theory to model library operations, patron behavior, and resource allocation strategies. We define a Library Resource Automaton (LRA), a deterministic finite automaton whose states represent distinct configurations of resource availability, whose input alphabet encapsulates patron interactions, and whose transition function formally encodes allocation policies. By interpreting sequences of patron actions as strings in a formal language, the LRA provides a computationally tractable and analytically powerful model for simulating library states, predicting bottlenecks, and synthesizing optimal allocation strategies. We elaborate on the theoretical foundations of the model, present a detailed multi-layer automata architecture for handling complex, multi-resource scenarios, and discuss algorithms for state space analysis and policy optimization. Furthermore, we explore the integration of temporal logic for specifying and verifying critical system properties such as fairness and liveness. This work establishes a rigorous bridge between theoretical computer science and library information science, offering a new paradigm for building predictable, efficient, and patron-centric library management systems.