## 4 System Architecture and Graph Modeling  
To robustly model autonomous vehicle (AV) navigation in urban environments characterized by GNSS signal degradation, perceptual occlusion, and dynamic risk, we introduce a mathematically grounded, multi-layered architecture that integrates graph theory, spatiotemporal risk forecasting, and reinforcement learning-based decision-making. Let the urban environment be represented as a time-varying directed graph \(G_{t}=(V,E_{t})\), where  
* \(V=\{v_{1},v_{2},\ldots,v_{n}\}\) denotes the set of spatial clusters (e.g., crime blocks or Police Service Areas);
* \(E_{t}\subseteq V\times V\) denotes directed edges representing navigable road segments at time \(t\). Each edge \(e_{ij}\in E_{t}\) is assigned a composite weight \(w_{ij}(t)\) defined as  
\[w_{ij}(t)=\alpha\cdot\tau_{ij}(t)+\beta\cdot\rho_{j}(t), \tag{1}\]  
where \(\tau_{ij}(t)\) is the estimated travel time from \(v_{i}\) to \(v_{j}\), and \(\rho_{j}(t)\) is the predicted crime risk at node \(v_{j}\). The scalar coefficients \(\alpha\) and \(\beta\) balance routing efficiency and risk aversion. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Authors (Year)** & **Problem** & **Solution and Approach** \\ \hline [PERSON] (2008); [PERSON] (2010) [3, 4] & Static crime hotspot identification & GIS-based hotspot analysis using kernel density estimation (KDE) in tools like CrimeStat and ArcGIS; enabled spatial targeting but lacked temporal adaptability. \\ \hline [PERSON] et al. (2015) [1] & Predicting repeat and near-repeat crimes & Self-exciting point process model to capture temporal clustering and event-triggered risk spikes for short-term neighborhood-level forecasting. \\ \hline [PERSON] et al. (2017) [13] & Low granularity and accuracy in crime forecasts & Deep learning model using residual convolutional layers to model crime intensity as a spatiotemporal continuous function; enhanced prediction accuracy. \\ \hline [PERSON] et al. (2018) [5] & Limited spatiotemporal dependency modeling in urban systems & Developed Spatiotemporal Graph Convolutional Networks (ST-GCN) for dynamic forecasting over urban networks; later extended to crime prediction. \\ \hline [PERSON] et al. (2019); [PERSON] et al. (2020) [14, 15] & Static patrol planning without real-world constraints & Q-learning and Deep Q-Network (DQN) for urban police scheduling and patrol robots; foundational use of RL but lacked sensor or partial observability modeling. \\ \hline [PERSON] et al. (2023) [16] & Dynamic police patrol scheduling under incident uncertainty & Multi-agent deep RL (APPO) for dynamic bi-objective patrol dispatch; enables continuous reoptimization and reactive patrol adjustment. \\ \hline [PERSON] et al. (2024) [17] & Coordinating autonomous patrol vehicles under noisy, constrained conditions & Multi-agent PPO with Reinforced Inter-Agent Learning (RIAL); adaptive fault-tolerant patrolling with AV congestion, failure handling, and cooperative learning. \\ \hline [PERSON] et al. (2025) [18] & Poor modeling of cross-type crime dependencies & Multi-type Relation-Aware Graph Neural Network (MRAGNN) combining spatiotemporal and type-level features; outperforms prior GNNs on multi-label forecasting. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of key studies in crime analysis using deep learning and AV policy optimization. To estimate the dynamic risk \(\rho_{j}(t)\), we employ a Spatiotemporal Graph Convolutional Network (ST-GCN) defined as  
\[\rho_{j}(t)=f_{\text{ST-GCN}}(X_{t},A_{t}), \tag{2}\]  
where \(X_{t}\in\mathbb{R}^{n\times d}\) is the feature matrix comprising \(d\)-dimensional vectors \(x_{i}(t)\) for each node \(v_{i}\) at time \(t\), and \(A_{t}\) is the adjacency matrix of \(G_{t}\). Each \(x_{i}(t)\) encodes temporal variables (e.g., hour of day, weekday indicator), static spatial descriptors (e.g., population density), and historical incident statistics (e.g., prior offense frequency, severity scores).