$G$ Method and Finite-Time Consensus

We give an extension of the $G$ method, with results, the extension and results being partly suggested by the finite Markov chains and specially by the finite-time consensus problem for the DeGroot model and that for the DeGroot model on distributed systems. For the (homogeneous and nonhomogeneous) DeGroot model, using the $G$ method, a result for reaching a partial or total consensus in a finite time is given. Further, we consider a special submodel/case of the DeGroot model, with examples and comments – a subset/subgroup property is discovered. For the DeGroot model on distributed systems, using the $G$ method too, we have a result for reaching a partial or total (distributed) consensus in a finite time similar to that for the DeGroot model for reaching a partial or total consensus in a finite time. Then we show that for any connected graph having $2^m$ vertices, $m\ge 1,$ and a spanning subgraph isomorphic to the $m$-cube graph, distributed averaging is performed in $m$ steps – this result can be extended – research work – for any graph with $n_1{n}_2\mathrm{.}\mathrm{.}\mathrm{.}{n}_t$ vertices under certain conditions, where $t,$ $n_1,n_2,\mathrm{.}\mathrm{.}\mathrm{.},n_t\ge 2,$ and, in this case, distributed averaging is performed in $t$ steps.