A consensus based distributed algorithm to compute the spectral radius of a network is proposed. The spectral radius of the graph is the largest eigenvalue of the adjacency matrix, and is a useful characterization of the network graph. Conventionally, centralized methods are used to compute the spectral radius, which involves eigenvalue decomposition of the adjacency matrix of the underlying graph. Our distributed algorithm uses a simple update rule to reach consensus on the spectral radius, using only local communications. We consider time-varying graphs to model packet loss and imperfect transmissions, and provide the convergence characteristics of our algorithm, for both static and time-varying graphs. We prove that the convergence error is a function of principal eigenvector of adjacency matrix of the graph and reduces as $\mathcal {O}(1/t)$, where $t$ is the number of iterations. The algorithm works for any connected graph structure. Simulation results supporting the theory are also presented.

中文翻译：

---

基于共识的分布式谱半径估计

提出了一种基于共识的分布式算法来计算网络的谱半径。图的谱半径是邻接矩阵的最大特征值，是网络图的一个有用表征。传统上，集中方法用于计算谱半径，这涉及底层图的邻接矩阵的特征值分解。我们的分布式算法使用简单的更新规则来就频谱半径达成共识，仅使用本地通信。我们考虑时变图来模拟数据包丢失和不完美传输，并为静态和时变图提供我们算法的收敛特性。我们证明收敛误差是图的邻接矩阵的主特征向量的函数，并减少为$\mathcal {O}(1/t)$， 在哪里 $t$是迭代次数。该算法适用于任何连通图结构。还提供了支持该理论的仿真结果。