Large-scale decentralized learning frameworks such as federated learning (FL), require both communication efficiency and strong data security, motivating the study of secure aggregation (SA). While information-theoretic SA is well understood in centralized and fully connected networks, its extension to decentralized networks with limited local connectivity remains largely unexplored. This paper introduces \emph{topological secure aggregation} (TSA), which studies one-shot, information-theoretically secure aggregation of neighboring users' inputs over arbitrary network topologies. We develop a unified linear design framework that characterizes TSA achievability through the spectral properties of the communication graph, specifically the kernel of a diagonally modulated adjacency matrix. For several representative classes of $d$-regular graphs including ring, prism and complete topologies, we establish the optimal communication and secret key rate region. In particular, to securely compute one symbol of the neighborhood sum, each user must (i) store at least one key symbol, (ii) broadcast at least one message symbol, and (iii) collectively, all users must hold at least $d$ i.i.d. key symbols. Notably, this total key requirement depends only on the \emph{neighborhood size} $d$, independent of the network size, revealing a fundamental limit of SA in decentralized networks with limited local connectivity.

翻译：大规模去中心化学习框架，如联邦学习（FL），需要同时满足通信效率与强数据安全性，这推动了对安全聚合（SA）的研究。虽然信息论安全聚合在集中式和全连接网络中已有充分理解，但其在本地连接受限的去中心化网络中的扩展仍很大程度上未被探索。本文提出了**拓扑安全聚合**（TSA），研究在任意网络拓扑上对相邻用户输入进行一次性、信息论安全聚合的问题。我们开发了一个统一的线性设计框架，通过通信图的谱特性——具体而言是对角调制邻接矩阵的核——来刻画TSA的可实现性。针对包括环形、棱柱形和完全拓扑在内的几类代表性$d$-正则图，我们建立了最优通信与密钥速率区域。特别地，为安全计算邻域和的一个符号，每个用户必须（i）存储至少一个密钥符号，（ii）广播至少一个消息符号，并且（iii）所有用户总共必须持有至少$d$个独立同分布的密钥符号。值得注意的是，这一总密钥需求仅取决于**邻域大小**$d$，而与网络规模无关，揭示了在本地连接受限的去中心化网络中安全聚合的一个基本极限。