我们通过构建一个显式纳入雪崩传播过程中耗散效应的分支过程框架，研究复杂网络上的沙堆模型。与假设保守输运及局部树状独立性的经典分支描述不同，本方法将颗粒损失效应直接引入后代分布，从而导出适用于耗散型雪崩动力学的广义生成函数。在耗散机制主导下，雪崩尺寸分布呈现指数截断，但仍保留依赖于网络拓扑的标度行为。数值模拟验证了该理论对稀疏随机网络的预测，并揭示其在高度结构化拓扑中存在系统性偏差。特别地，利用Holme-Kim聚类无标度网络，我们发现随着聚类系数增大，雪崩指数持续减小，大规模级联事件的概率增强，表明短环路引发强关联，使经典的独立分支近似失效。出人意料的是，树状网络亦显著偏离幂律分布，原因在于边密度低且叶节点丰富，从而限制了雪崩传播。这些结果表明，耗散、聚类及稀疏连通性从根本上重塑了网络上沙堆模型的雪崩尺寸分布，并为雪崩动力学的分支过程描述确立了定量适用边界。

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 tree-like 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, by using Holme-Kim clustered scale-free networks, we show that increasing clustering continuously lowers the avalanche exponent and enhances the probability of large cascades, demonstrating that short cycles generate strong correlations that invalidate the classical independent-branch approx imation. Surprisingly, trees also exhibit substantial deviations from 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 avalanche size distribution of the sandpile model on networks and establish quantitative limits for branching-process descriptions of avalanche dynamics.