Abstract We study opinion diffusion on social graphs where agents hold binary opinions and where social pressure leads them to conform to the opinion manifested by the majority of their neighbors. We provide bounds relating the number of agents that suffice to spread an opinion to all other agents with the number of required propagation steps. Bounds are established constructively, via polynomial time algorithms that identify the agents that must act as seeds. In particular, we show that, on any given social graph G = ( N , E ) , it is possible to efficiently identify a set formed by half of the agents that can lead to consensus in min ⁡ { ⌊ | N | / 2 ⌋ , even ( G ) + 1 } propagation steps, where even ( G ) is the number of agents with an even number of neighbors in G. The result marks the boundary of tractability, since we show that the existence of sets of seeds consisting of less than half of the agents depends on certain features of the underlying graphs, which are NP-hard to identify. In fact, other NP-hardness results emerge from our analysis. In particular, by closing a problem left open in the literature, we show that it is intractable to decide whether further stable configurations exist in addition to the “consensus” ones (where all agents hold the same opinion). Eventually, for all these problems related to reasoning about opinion diffusion, we show that islands of tractability can be identified by focusing on classes of tree-like social graphs.

中文翻译：

---

论多数动力学下意见扩散推理的复杂性

摘要 我们研究了社交图上的意见扩散，其中代理持有二元意见，社会压力导致他们符合大多数邻居所表达的意见。我们提供了与足以将意见传播给所有其他代理的代理数量相关的界限以及所需的传播步骤数量。边界是通过多项式时间算法建设性地建立的，这些算法识别必须充当种子的代理。特别是，我们表明，在任何给定的社交图 G = ( N , E ) 上，可以有效地识别由一半代理形成的集合，这些代理可以在 min ⁡ { ⌊ | 达成共识。否 | / 2 ⌋ , even ( G ) + 1 } 传播步骤，其中 even ( G ) 是在 G 中具有偶数个邻居的代理数量。结果标志着易处理性的边界，因为我们表明由少于一半的代理组成的种子集的存在取决于底层图的某些特征，这些特征是 NP 难以识别的。事实上，我们的分析中还出现了其他 NP 硬度结果。特别是，通过解决文献中未解决的问题，我们表明除了“共识”配置（所有代理持有相同意见）之外，确定是否存在进一步的稳定配置是难以处理的。最终，对于与意见扩散推理相关的所有这些问题，我们表明可以通过关注树状社交图的类别来识别易处理岛。特别是，通过解决文献中未解决的问题，我们表明除了“共识”配置（所有代理持有相同意见）之外，确定是否存在进一步的稳定配置是难以处理的。最终，对于与意见扩散推理相关的所有这些问题，我们表明可以通过关注树状社交图的类别来识别易处理岛。特别是，通过解决文献中未解决的问题，我们表明除了“共识”配置（所有代理持有相同意见）之外，确定是否存在进一步的稳定配置是难以处理的。最终，对于与意见扩散推理相关的所有这些问题，我们表明可以通过关注树状社交图的类别来识别易处理岛。