Networked SEIR models describe epidemic spread within and between interacting subpopulations through contact-supported nonlinear transmission. Standard polynomial liftings based on complete ordered Kronecker tensors yield linear higher-dimensional representations, but their dimensions grow rapidly because they retain interactions absent from the transmission graph. This paper develops a graph-induced tensor lifting whose observables are selected from the effective transmission support. An exact edge-based quadratic representation separates linear compartmental transitions from nonlinear infection terms. A homogeneous hierarchy is then constructed recursively. The quadratic transmission field generates the next degree. The linear compartmental field saturates the resulting dictionary within that degree. The first edge-closure dynamics are linear up to an explicit cubic truncation residual, and higher-order truncations contain only next-degree terms. The first lifted dimension scales with the numbers of subpopulations and effective transmission channels. At fixed order, graph-induced dictionaries grow linearly with network size under uniformly bounded local connectivity, whereas complete polynomial liftings retain order-dependent polynomial growth. Uniform first edge-closure residual bounds depend on the transmission rate and the maximum weighted incoming transmission intensity. Numerical illustrations compare equal intensity per active channel with equal total incoming intensity. They confirm that dictionary dimensions depend only on graph support, whereas residual trajectories also reflect weight accumulation, weight distribution, and nonlinear propagation. These results provide a structured basis for reduced modeling and subsequent model-specific analysis and control.
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