The Internet of Things (IoT) has laid the foundation for smart cities by allowing devices to work in synergy through focused automation. Nonetheless, most procedures available today are missing a holistic model that allows for clear visualization of how users are interacting with, how services are being effectively rendered, and how these procedures are scalable. To remedy this oversight, our generalized IoT graph model traces the flow of and advances an IoT action from inception, in direct response to a user’s petition, to conclusion at a children’s leaf node. Our model centrally manages user interaction through a Directory, or a digital page, for requests, which travel via 3 interconnected graph-layers (Top, Middle, and Bottom) whose distinct functions and adapts to secure, scalable, personalized networks in smart city applications. The model engenders novel field of research, which we have named IoT Algebra, leading to formal services analysis, composition, and fine-tuning by researchers. We review its real-world application and precedent.