We propose a local transformation on bicolored graphs, which we call local homophily, inspired by adaptive networks and based on majority dynamics and homophily. In this transformation, a vertex updates its color to match the majority of its neighbors, while neighbors of the same color become connected and neighbors of the opposite color become disconnected. We show how to simulate Boolean circuits using local homophily and establish that determining whether a given pair of vertices becomes connected under iterative applications of local homophily is $\mathbf{P}$-complete under logspace reductions.

翻译：我们提出一种双色图上的局部变换方法，称之为局部同质性，其灵感源于自适应网络，并基于多数动力学与同质性原理。在该变换中，顶点更新自身颜色以匹配其邻居中的多数颜色，同时具有相同颜色的邻居之间建立连接，而颜色相反的邻居之间则断开连接。我们展示了如何利用局部同质性模拟布尔电路，并证明在logspace归约下，判断给定顶点对是否能在局部同质性的迭代应用中建立连接是一个$\mathbf{P}$-完全问题。