From Hubs to Holography: How Graph Regularity Unlocks Entanglement

Description: We investigate the origin of boundary entanglement in random tensor networks defined on emergent causal graphs. Building on a previous model that exhibits perfect linear scaling of classical minimal cuts—a hallmark of de Sitter holography—we embed quantum tensors and compute the ratio R = S_q(1)/C_{\text{min}}(1) for a boundary interval consisting of the first spacelike edge. Uncapped graphs (containing high-degree "hubs") yield R \approx 0.05 at bond dimension \chi=4. Imposing a hard cap on node degree (max degree 6) increases the ratio to 0.138—a 2.6× increase (Welch's t-test, t(57.2)=18.3, p < 10^{-15}). Replacing random tensors on degree‑4 nodes with AME(4,4) perfect tensors yields only a marginal increase to 0.144 (p = 0.22), not statistically significant. To explain this dramatic effect, we perform a classical hub analysis on uncapped graphs of 2000 nodes. Every boundary interval of length L \geq 2 has at least one hub (degree \geq 8) in its causal past, demonstrating that hubs act as global entanglement sinks. Capping removes all hubs, allowing entanglement to flow freely to the boundary. We then systematically study the dependence of R on bond dimension \chi using graphs of size 12–15, obtaining 15–17 successful contractions per \chi = 2,3,4,5,6,8. The mean ratio decreases from 0.45 \pm 0.21 at \chi=2 to 0.30 \pm 0.07 at \chi=8. Control experiments reveal that compressed contraction (max_bond=64) underestimates R by \sim 18\% at \chi=4 compared to exact contraction, and that running \chi=5 on larger graphs (size 15) yields a higher ratio (0.41 \pm 0.09) than on size‑12 graphs—though with a low success rate (6/20) that may bias the result. Together, these controls indicate that the apparent decrease is dominated by finite‑size effects and compression artifacts, rather than a genuine physical trend. The findings serve as a cautionary tale for small‑graph studies and underscore the need for larger‑scale simulations. Together, these results establish that graph regularity is the dominant factor enabling holographic entanglement. Perfect tensors play only a secondary role, becoming relevant only when the graph is already regular. The hub hypothesis provides a mechanistic explanation: high‑degree nodes trap information, and their removal frees it. Bond dimension scaling reveals the subtlety of finite‑size effects, highlighting the path toward larger‑scale investigations. We discuss implications for tensor network models of holography and outline a program for fu ture work.