Secondly, we considered the network partitioning and we proposed a clustering method by reference to the

stochastic matrix using the MCL algorithm.

(II) We consider (1) with a well-known doubly

stochastic matrix which has applications in communication theory and graph theory [15].

For irreducible

stochastic matrix [M.sub.11], there exists a vector [mathematical expression not reproducible] with positive elements satisfying that [[beta].sup.T.sub.1][M.sub.11] = [[beta].sup.T.sub.1].

However, since n must also be a

stochastic matrix, the sum of its components must equal 1.

where p is the stationary probability vector of the irreducible

stochastic matrix A = [[summation].sup.N.sub.i=0] [A.sub.i] and [beta] = [[summation].sup.N.sub.i=0] [iA.sub.i]e.

Demetrius and Manke [31] propose the analysis of the

stochastic matrix in the context of network robustness.

However, there are some cases where [??] is a

stochastic matrix, even if it does not correspond to carries processes.

Let P be a

stochastic matrix defined on the countable state space G.

We refer to the index set of a

stochastic matrix as its state space.

Let S be a

stochastic matrix. If G(S) contains at least one spanning tree such that the root vertex of that spanning tree has a self-loop in G(S), then S is SIA.

[??] [there exists] 3 double

stochastic matrix P such that PXw = Xv we have another criterion for SSD relation:

Additionally, transition probability matrix P=[[p.sub.j]] has the property of a doubly

stochastic matrix since: