The $K$-core of a graph is the unique maximum subgraph within which each vertex connects to at least $K$ other vertices. The $K$-core optimal attack problem asks to construct a minimum-sized set of vertices whose removal results in the complete collapse of the $K$-core. In this paper, we construct a hierarchical cycle-tree packing model which converts a long-range correlated $K$-core pruning process into static patterns and analyze this model through the replica-symmetric (RS) cavity method of statistical physics. The cycle-tree guided attack (CTGA) message-passing algorithm exhibits superior performance on random regular and Erdos-Renyi graphs. It provides new upper bounds on the minimal cardinality of the $K$-core attack set. The model of this work may be extended to construct optimal initial conditions for other irreversible dynamical processes.

翻译：图的$K$-核是满足每个顶点至少与$K$个其他顶点相连的唯一最大子图。$K$-核最优攻击问题要求构造一个最小规模的顶点集合，其移除会导致$K$-核完全崩溃。本文构建了一个分层环-树包装模型，将长程相关的$K$-核剪枝过程转化为静态模式，并通过统计物理的副本对称（RS）空穴方法分析该模型。基于环-树指导的攻击（CTGA）消息传递算法在随机正则图和Erdos-Renyi图上展现出优越性能，为$K$-核攻击集的最小基数给出了新的上界。本工作提出的模型可推广至其他不可逆动力学过程的最优初始条件构造。