We investigate the sandpile model on complex networks by developing a branching-process framework that explicitly incorporates dissipation during avalanche propagation. Unlike classical branching descriptions, which assume conservative transport and locally treelike independence, the present approach introduces grain-loss effects directly into the offspring distribution, yielding generalized generating functions for dissipative avalanche dynamics. In the dissipative regime, avalanche-size distributions acquire exponential cutoffs while preserving topology-dependent scaling behavior. Numerical simulations confirm the theoretical predictions on sparse random networks and reveal systematic deviations in highly structured topologies. In particular, using Holme-Kim clustered scale-free networks, we show that the avalanche exponent is relatively insensitive to weak-to-moderate clustering but decreases substantially in the strongly clustered regime, especially as the triangle-closing probability approaches unity. This decrease is accompanied by an enhanced probability of large cascades, showing that sufficiently abundant short cycles generate strong correlations that challenge the classical independent-branch approximation. Surprisingly, trees also exhibit substantial deviations from a power law because low edge density and the abundance of leaves constrain avalanche propagation. These results show that dissipation, clustering, and sparse connectivity fundamentally reshape the avalanche size distribution of the sandpile model on networks and establish quantitative limits for branching-process descriptions of avalanche dynamics.

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• Complex networks
• Sandpile models
• Self-organized criticality