Lightweight block cipher design has largely focused on incremental optimization of established paradigms such as substitution--permutation networks, Feistel structures, and ARX constructions, where security derives from the algebraic complexity of individual components. We propose a different approach based on \emph{expander-graph interaction networks}, where diffusion and security arise from sparse structural connectivity rather than component sophistication. We present \textbf{ExpanderGraph-128 (EGC128)}, a 128-bit block cipher constructed as a 20-round balanced Feistel network. Each round applies a 64-bit nonlinear transformation governed by a 3-regular expander graph whose vertices execute identical 4-input Boolean functions on local neighborhoods. Security analysis combines MILP-based differential bounds, proven optimal through 10 rounds via SCIP, establishing 147.3-bit differential security and conservatively extrapolating to 413 bits for the full cipher. Linear analysis provides MILP bounds of $\geq 2^{145}$, while related-key evaluation shows no free rounds for any nonzero key difference. Additional tests confirm rapid algebraic degree growth and the absence of invariant affine subspaces. Implementation results demonstrate practical efficiency. FPGA synthesis on Xilinx Artix-7 achieves 261~Mbps at 100~MHz using only 380 LUTs, while ARM Cortex-M4F software requires 25.8~KB Flash and 1.66~ms per encryption. These results show that expander-graph-driven diffusion provides a promising design methodology for lightweight cryptography.