Distributed Averaging in Opinion Dynamics

We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node $u$ has an initial value $\xi_u(0)$. In the first process, the NodeModel, at each time step $t\ge 0$, a random node $u$ and a random sample of $k$ of its neighbours $v_1,v_2,\cdots,v_k$ are selected. Then, $u$ updates its current value $\xi_u(t)$ to $\xi_u(t+1) = \alpha \xi_u(t) + \frac{(1-\alpha)}{k} \sum_{i=1}^k \xi_{v_i}(t)$, where $\alpha \in (0,1)$ and $k\ge 1$ are parameters of the process. In the second process, the EdgeModel, at each step a random pair of adjacent nodes $(u,v)$ is selected, and then node $u$ updates its value equivalently to the NodeModel with $k=1$ and $v$ as the selected neighbour. For both processes, the values of all nodes converge to $F$, a random variable depending on the random choices made in each step. For the NodeModel and regular graphs, and for the EdgeModel and arbitrary graphs, the expectation of $F$ is the average of the initial values $\frac{1}{n}\sum_{u\in V} \xi_u(0)$. For the NodeModel and non-regular graphs, the expectation of $F$ is the degree-weighted average of the initial values. Our results are two-fold. We consider the concentration of $F$ and show tight bounds on the variance of $F$ for regular graphs. We show that, when the initial values do not depend on the number of nodes, then the variance is negligible, hence the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time $T_\varepsilon$ required to make all node values `$\varepsilon$-close' to each other. Our bounds are asymptotically tight under assumptions on the distribution of the initial values.