weights $a_{ij}(k)$ and the graph sequence $G(k)$.

For each $k$, the weight matrix $A(k)$ is a doubly stochastic matrix[^4] with positive diagonal . Additionally, there exists a constant $\eta> 0$ such that if $a_{ij}(k)>0$, then $a_{ij}(k) \geq \eta$.\[weights\]

The double stochasticity assumption on the weight matrix guarantees that the average of the node values remains the same at each iteration (cf. the proof of Lemma \[vl\] below). The second part of this assumption states that each node gives significant weight to its values and to the values of its neighbors at each time $k$.

Our next assumption ensures that the graph sequence $G(k)$ is sufficiently connected for the nodes to repeatedly influence each other’s values.

There exists an integer $B \geq 1$ such that the directed graph $$\Big(N, \E(A(kB)) \bigcup
\E(A(kB+1))\bigcup \cdots \bigcup \E(A((k+1)B-1))\Big)$$ is strongly connected for all nonnegative integers $k$. \[connectivity\]

Any algorithm of the form given in Eq. (\[noquant\]) with the sequence of weights $a_{ij}(k)$ satisfying Assumptions \[weights\] and \[connectivity\] solves the averaging problem. This is formalized in the following .

\[uqc\] Let Assumptions \[weights\] and \[connectivity\] hold. Let $\{x(k)\}$ be generated by the algorithm (\[noquant\]). Then, for all $i$, we have $$\lim_{k \rightarrow \infty} x_i(k) = 