<p>We study intermittent rumor resurgence on social networks using a slow-driven, threshold-based spreading model with degree-dependent adoption. High-degree “influencers” require stronger social reinforcement to participate, but exert amplified impact once activated. Driving is applied only during quiescent periods, yielding avalanche-like bursts of spreading activity. Using activity-threshold segmentation, we measure avalanche size, duration, and peak intensity across network sizes. Simulations show clear finite-size effects: the relative peak intensity decreases with system size, consistent with the declining relative coverage of the largest hub. When the per-capita drive is normalized, avalanche sizes display heavy-tailed distributions compatible with power-law behavior over intermediate scales with finite-size cutoffs. Within this explicitly driven setting, the nontrivial results concern the statistics and mechanisms of resurgence events: the finite-size scaling of peak activation, the broad but truncated avalanche-size distributions, and the disproportionate contribution of influencer-triggered avalanches to the far tail. The model should therefore be interpreted as a framework for quantifying bursty rumor reactivation under quiescence-triggered reinjection, rather than as an autonomous explanation of long-term rumor persistence in the absence of external input.</p>