We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.

翻译：我们研究了分布式系统中节点间可交换信息的平均共识问题。针对此类问题，我们聚焦于一类算法：每个节点维护一个状态，并以其入邻节点（即直接接收信息的节点）状态的线性组合进行迭代更新。该类平均算法可视为无外部驱动输入的离散时变线性系统，其状态矩阵为列随机矩阵。因此，算法具有全局不变性：所有状态变量之和始终保持恒定。本文揭示了上述平均算法类别的另一不变性质。该性质具有局部性，反映节点保存的某些守恒量——这些量汇聚了算法执行过程中节点从其入邻节点接收的全部值，以及该节点向其出邻节点（即直接发送信息的节点）发送的全部值。我们展示了如何利用这一新发现的不变性在平均算法执行过程中检测错误。