Editing Openai/693b7cce-38f4-800d-92f7-5e56a467bfa7 (section)

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For linearized system (ignoring mutation noise & nonlinearity), the continuous dynamics are stable if the real parts of eigenvalues of the effective update matrix are negative. A sufficient practical condition is:

max⁡i(λi+∑jW~ij)>0,\max_i \big( \lambda_i + \sum_j \widetilde W_{ij} \big) > 0,imax(λi+j∑Wij)>0,
and choose Δt\Delta tΔt small enough for explicit Euler:

Δt<1max⁡i(λi+∑jW~ij)\Delta t < \frac{1}{\max_i(\lambda_i + \sum_j \widetilde W_{ij})}Δt<maxi(λi+∑jWij)1
In practice, using normalized W~\widetilde WW keeps ∑jW~ij≤1\sum_j \widetilde W_{ij}\le 1∑jWij≤1, so Δt<1/(1+max⁡λi)\Delta t < 1/(1+\max\lambda_i)Δt<1/(1+maxλi) is safe. For stiff systems consider RK4 or implicit integrators.

===== For nodes with extremely high degree, compute influence via sparse sampling of neighbors each tick (subsample k neighbors proportional to ωik\omega_{ik}ωik), or precompute aggregated group influences (community-level summarization). =====