[论文解读] Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities

Anton V. Proskurnikov, Ming Cao|arXiv (Cornell University)|Mar 20, 2017

Opinion Dynamics and Social Influence参考文献 31被引用 8

一句话总结

本文证明了涉及随机矩阵的一类离散时间递推不等式有界解收敛于一致共识，推广了签名网络中模一致共识的概念。关键贡献在于证明了‘一致共识二分性’性质：在较弱的连通性假设下，该不等式 $ x(k+1) \leq W(k)x(k) $ 的任意有界解均收敛于同一极限，统一并拓展了签名网络动力学与分布式优化中的现有结果。

ABSTRACT

Recently the dynamics of signed networks, where the ties among the agents can be both positive (attractive) or negative (repulsive) have attracted substantial attention of the research community. Examples of such networks are models of opinion dynamics over signed graphs, recently introduced by Altafini (2012,2013) and extended to discrete-time case by Meng et al. (2014). It has been shown that under mild connectivity assumptions these protocols provide the convergence of opinions in absolute value, whereas their signs may differ. This "modulus consensus" may correspond to the polarization of the opinions (or bipartite consensus, including the usual consensus as a special case), or their convergence to zero. In this paper, we demonstrate that the phenomenon of modulus consensus in the discrete-time Altafini model is a manifestation of a more general and profound fact, regarding the solutions of a special recurrent inequality. Although such a recurrent inequality does not provide the uniqueness of a solution, it can be shown that, under some natural assumptions, each of its bounded solutions has a limit and, moreover, converges to consensus. A similar property has previously been established for special continuous-time differential inequalities (Proskurnikov, Cao, 2016). Besides analysis of signed networks, we link the consensus properties of recurrent inequalities to the convergence analysis of distributed optimization algorithms and the problems of Schur stability of substochastic matrices.

研究动机与目标

• 将签名网络中模一致共识的概念推广至更广泛的离散时间递推不等式类别。
• 为这些不等式的有界解建立一致共识二分性性质，证明在较弱连通性假设下收敛于同一极限。
• 统一并拓展现有关于签名网络中一致共识及分布式优化算法的研究结果。
• 探索递推不等式框架与矩阵理论之间的联系，特别是亚随机矩阵的舒尔稳定性。
• 展示该框架在求解凸可行性与优化问题的分布式算法中的适用性。

提出的方法

• 分析离散时间递推不等式 $ x(k+1) \leq W(k)x(k) $，其中 $ W(k) $ 为随机矩阵序列。
• 应用闭凸集上投影理论，推导分布式优化协议的收敛性质。
• 利用投影性质 $ \|x - \xi_0\|^2 \geq \|P_{\Xi_i}(x) - \xi_0\|^2 + d_i(x)^2 $ 限制到解集的距离。
• 在假设1和假设2下，证明状态向量 $ \xi_i(k) $ 收敛于 $ \Xi = \bigcap \Xi_i $ 中的共同极限。
• 证明误差项 $ e_i(k) = \xi_i(k+1) - \eta_i(k) $ 与到集合的距离 $ d_i(\xi_i(k)) $ 渐近趋于零。
• 将连续时间一致共识二分性结果（Proskurnikov & Cao, 2016）扩展至离散时间情形，证明其具有类似的收敛行为。

实验结果

研究问题

• RQ1 在何种条件下，离散时间递推不等式 $ x(k+1) \leq W(k)x(k) $ 的有界解会收敛于一致共识？
• RQ2 连续时间微分不等式中的一致共识二分性性质如何推广至离散时间情形？
• RQ3 签名网络中的模一致共识与这类递推不等式解之间有何联系？
• RQ4 该不等式框架如何应用于分布式优化与线性方程求解算法？
• RQ5 该框架对亚随机矩阵的舒尔稳定性有何影响？

主要发现

• 任意满足较弱连通性假设的随机矩阵序列 $ W(k) $ 所构成的离散时间递推不等式 $ x(k+1) \leq W(k)x(k) $ 的有界解，均收敛于同一极限。
• 一致共识二分性成立：所有有界解均收敛于一致共识，而无界解则不收敛。
• 该框架解释并推广了离散时间Altafini模型中的模一致共识，其中意见在绝对值上收敛，但符号可能不同。
• 在假设1和假设2下，证明了分布式优化协议（10）–（12）收敛于凸集 $ \Xi_1 \cap \cdots \cap \Xi_n $ 中的共同点。
• 误差项 $ e_i(k) = \xi_i(k+1) - \eta_i(k) $ 与到可行集的距离 $ d_i(\xi_i(k)) $ 均收敛于零，确保了约束一致共识。
• 结果可推广至矩阵理论，揭示了对亚随机矩阵舒尔稳定性及迭代算法收敛性的影响。