Online change point detection in dynamic graphs requires comparing graphs as they arrive, in time linear in the number of edges, without parametric assumptions. Recent spectral methods address scale via the Kernel Polynomial Method (KPM): SCPD computes Chebyshev moments of the normalized Laplacian, discretizes them into a density-of-states histogram, and scores the histogram with SVD plus cosine similarity. We introduce SWORD, which computes the same moments and instead compares their mean across two adjacent time windows by their $L_1$ distance. On three real-world benchmarks (MIT Reality, AskUbuntu, Enron), this lifts mean $F_1$ from SCPD's $0.27$ to $0.79$, with SCPD failing to detect any change on Enron. A controlled cascading ablation attributes the gap to two design choices: the two-window mean structure (dominant on MIT) and the $L_1$ metric on those mean vectors (dominant on Enron). A bin-width sweep rules out histogram discretization -- SCPD's most visible architectural choice -- as the driver. SWORD inherits SCPD's KPM core, so per-graph cost is $O(KRm)$ with no eigendecomposition, scaling to $86{,}000$-node networks. With per-dataset tuning it matches the offline TIRE autoencoder on mean $F_1$ and attains the highest precision among online methods ($0.91$, only $2$ false positives across the three benchmarks).

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