On the Limits of Stochastic Push-Pull over Time-Varying Digraphs

Stochastic Push-Pull achieves linear speedup with respect to the network size $n$ on static strongly connected digraphs, but whether this speedup persists on time-varying networks remains unclear. In this work, we construct a quadratic decentralized problem on period-two networks showing that stochastic Push-Pull may fail to attain linear speedup over time-varying digraphs. For $p$-periodic networks, we identify an effective node count $n_{\mathrm{eff}}\le n$ that governs the leading stochastic term. Specifically, for smooth nonconvex objectives, we establish convergence upper bounds and matching lower bounds for stochastic Push-Pull with constant stepsizes, both exhibiting the leading-order dependence $Θ(1/\sqrt{n_{\mathrm{eff}} T})$. When $p=1$, the identity $n_{\mathrm{eff}}=n$ recovers the usual linear speedup for static digraphs. However, periodic networks can have $n_{\mathrm{eff}}=o(n)$, demonstrating the loss of linear speedup under temporal variation. Numerical experiments illustrate effective-count scaling and the separation between static and periodic networks, consistent with our theoretical findings.