Dissipative Avalanche Regimes Driven by Memory-Biased Random Walks on Networks

We investigate a network model in which a single random walker combines local diffusion with preferential resetting to previously visited nodes. Each arrival deposits one unit of stress on the target node, and threshold crossings trigger sandpile-like relaxation cascades. The fixed per-neighbor transfer rule produces a brittle transition on Watts--Strogatz networks: below the stress-balance condition $\alpha k \simeq T$ cascades remain short, whereas mildly supercritical transfer values generate runaway-capped events at large system sizes. A subtractive dissipative rule -- in which a toppling node loses $T$ units and redistributes only $\beta T$ across its neighbors -- stabilizes broad, finite cascades over a significantly wider parameter range. For $\beta = 0.995$ and $0.998$, the dissipative model remains non-runaway through $N = 4096$ and favors power-law tails by AIC model selection; however, system-scale event fractions decrease with $N$, a branching-ratio proxy remains below unity, and bootstrap Kolmogorov--Smirnov tests reject a pure power law. Shuffled-order controls that preserve node-visit frequencies while randomizing the temporal sequence of arrivals yield nearly identical avalanche macrostatistics for $\beta < 1$ across memory strengths $q = 0$--$0.6$, demonstrating that dissipation and redistribution rules dominate over temporal memory ordering in the regime we can reliably characterize. On Barab\'{a}si--Albert networks, fixed per-neighbor transfer is strongly hub-sensitive, while degree-normalized transfer suppresses runaways but yields distributions better described by exponentials. The central conclusion is therefore regime-based: memory-biased driving localizes stress injection and shapes visitation hotspots, but broad cascade behavior is governed primarily by stress balance, dissipation strength, and network topology.