  Solving fastest mixing Markov chain problem (i.e. finding transition
probabilities on the edges to minimize the second largest eigenvalue modulus of
the transition probability matrix) over networks with different topologies is
one of the primary areas of research in the context of computer science and one
of the well known networks in this issue is K-partite network. Here in this
work we present analytical solution for the problem of fastest mixing Markov
chain by means of stratification and semidefinite programming, for four
particular types of K-partite networks, namely Symmetric K-PPDR, Semi Symmetric
K-PPDR, Cycle K-PPDR and Semi Cycle K-PPDR networks. Our method in this paper
is based on convexity of fastest mixing Markov chain problem, and inductive
comparing of the characteristic polynomials initiated by slackness conditions
in order to find the optimal transition probabilities. The presented results
shows that a Symmetric K-PPDR network and its equivalent Semi Symmetric K-PPDR
network have the same SLEM despite the fact that Semi symmetric K-PPDR network
has less edges than its equivalent symmetric K-PPDR network and at the same
time symmetric K-PPDR network has better mixing rate per step than its
equivalent semi symmetric K-PPDR network at first few iterations. The same
results are true for Cycle K-PPDR and Semi Cycle K-PPDR networks. Also the
obtained optimal transition probabilities have been compared with the
transition probabilities obtained from Metropolis-Hasting method by comparing
mixing time improvements numerically.
