  Distributed consensus has appeared as one of the most important and primary
problems in the context of distributed computation and it has received renewed
interest in the field of sensor networks (due to recent advances in wireless
communications), where solving fastest distributed consensus averaging problem
over networks with different topologies is one of the primary problems in this
issue. Here in this work analytical solution for the problem of fastest
distributed consensus averaging algorithm over Chain of Rhombus networks is
provided, where the solution procedure consists of stratification of associated
connectivity graph of the network and semidefinite programming, particularly
solving the slackness conditions, where the optimal weights are obtained by
inductive comparing of the characteristic polynomials initiated by slackness
conditions. Also characteristic polynomial together with its roots
corresponding to eigenvalues of weight matrix including SLEM of network is
determined inductively. Moreover to see the importance of rhombus graphs it is
indicated that convergence rate of path network increases by replacing a single
node by a rhombus sub graph within the path network.
