On the Hierarchical Ergodicity Coefﬁcient

—In this paper, we deal with the fundamental concepts and properties of ergodicity coefﬁcients in a hierarchical sense by making use of partition. Moreover, we establish a hierarchial Hajnal’s inequality improving some previous results.


I. INTRODUCTION
T HROUGHOUT the paper all matrices are of fixed size n × n.A stochastic matrix P is a square matrix with non-negative entries and unit row sums.For an n-dimensional column vector x, we say it is stochastic if x has non-negative entries and x T 1 = 1, where 1 is the n-dimensional column vector with all entries equal to 1. Ergodicity coefficient [5], [15] is a continuous scalar function μ(•) on the set of stochastic matrices P (P is regarded as a point in R n 2 ) that satisfies 0 ≤ μ(P ) ≤ 1.An ergodicity coefficient is called proper if μ(P ) = 0 ⇐⇒ P = 1x T , for some stochastic vector x (i.e., all rows of P are identical).
Ergodicity coefficient is used to measure the convergence rate of infinite products of stochastic matrices especially in the context of inhomogeneous Markov chains [17], [18].Roughly speaking, a Markov chain is said to be ergodic if the associated matrix products converge to a stochastic matrix whose rows are all identical, that is, a rank-one matrix.For relevant backgrounds on ergodicity and some classical stochastic matrices such as Markov matrices and scrambling matrices, we refer to [7], [8], [16] and references therein.
For a stochastic matrix P , denote p ij the entry of P on the ith row and jth column.Some common ergodicity coefficients are defined as follows: τ is called the Markov-Dobrushin ergodicity coefficient [4], [9].It is easy to check [16] that they are all proper, and furthermore, the following relation holds for every stochastic matrix P α(P ) ≤ τ (P ) ≤ β(P ).
Other variants of ergodicity coefficient can be found in e.g.
Y. Shang is with the Institute for Cyber Security, University of Texas, San Let • be a vector norm in R n .Define the Hajnal diameter of a stochastic matrix P with row vectors P 1 , P 2 , • • • , P n as Clearly, Δ(P ) = 0 if and only if all rows of P are identical.As a useful tool in studying the ergodicity of Markov chains, the well known Hajnal's inequality (see the comment below), in its general form, is established as follows.
Theorem 1. [20] For any two stochastic matrices P and Q, Note that when we take the L 1 norm • 1 , The resulting inequality is sharper than Hajnal's in [5].Historically, the classical form of the inequality was due to Markov [9], and was rediscovered by others after [5] was published.Some authors also call it Hajnal's inequality (see [14], [16] for historical development).
In this paper, motivated by applications in search of a cluster consensus between a set of agents [6], [19], we aim to refine ergodicity coefficients as well as Hajnal's inequality by means of partition (in a hierarchical nature).A partition and S s ∩ S t = ∅ for s = t.As we will see below the partition S allows us to extend some known results on ergodicity elegantly.
We should mention that, in the literature, the application of partitions to ergodicity coefficients has been made in some papers (c.f.[10], [11], [12]).However, the coefficients studied here have totally different definitions and the results are different (see the discussion below).

II. HIERARCHICAL ERGODICITY COEFFICIENT
By virtue of the partition S, in this section we extend the concepts of ergodicity coefficient mentioned above to those in the hierarchical sense.
For a given partition , is a sort of ergodicity coefficient defined for stochastic matrices P .We say μ S is hierarchial proper if T over all i ∈ S s , and x s is a stochastic vector.
A hierarchical proper ergodicity coefficient is equal to zero if the rows corresponding to each subset S s for s = 1, • • • , K of the stochastic matrix are identical.As such, the rank of the stochastic matrix equals K. Clearly, we reproduce the concept of proper ergodicity coefficient when K = 1.In what follows, we instantiate the above general μ S in some special cases.Specifically, the hierarchical counterparts for ergodicity coefficients τ, α and β can be formulated as It is direct to check that all these ergodicity coefficients are hierarchical proper.The definition of τ S first appears in [11] geared towards the study of uniform weak S-ergodicity of Markov chains.A sub-multiplicative property τ S (P Q) ≤ τ S (P )τ S (Q) was established.In [10], an analogous generalized Markov-Dobrushin ergodicity coefficient was introduced as Note that τS measures the inter-group differences rather than the intra-group ones.Under a decomposability condition, for any non-unit eigenvalue λ of P , Pǎun [10] showed that |λ| ≤ τS (P ).
A stochastic matrix P is said to be doubly stochastic if its transpose P T is also stochastic.We have the following result for hierarchial ergodicity coefficient of doubly stochastic matrices.Proposition 1.
For a given partition S = {S 1 , S 2 , • • • , S K }, if P is a doubly stochastic matrix and μ S a hierarchical proper ergodicity coefficient, then μ S (P ) = μ S (P T ) = 0 ⇐⇒ P = P T .
To see how Proposition 1 works, a simple non-trivial example could be n = 3, S = {{1, 3}, {2}} and A useful relation between the above mentioned hierarchical ergodicity coefficients τ S , α S and β S are stated below.Analogous and further relations in non-hierarchial sense can be found in e.g.[8], [13], [16].

Proposition 2.
For a given partition

III. EXTENDED HAJNAL'S INEQUALITY
In this section, we extend the Hajnal inequality to the hierarchical case.
To this end, we first introduce a hierarchical Hajnal diameter [6].Given a partition S = {S 1 , S 2 , • • • , S K }, and a stochastic matrix P , which has row vectors P 1 , P 2 , • • • , P n , the hierarchial Hajnal diameter is defined as Note that Δ S (P ) = 0 if and only if Given a partition S, a stochastic matrix P is said to be hierarchical balanced if, for i ∈ S s , is independent of i.In other words, γ st only depends on the partition indices s and t.This property is dubbed "inter-cluster common influence" in [6].

Theorem 2. For a given partition
In particular, if Q = I (i.e., the identity matrix), then Δ S (P ) ≤ τ S (P ).
Fix i ∈ S t and j ∈ S t for some 1 ≤ t ≤ K.We have Define a set Y as follows In other words, y is an index vector with the sth element belonging to S s , for s = 1, • • • , K. By the arbitrariness of i, j and t, we finally arrive at Δ S (R) ≤ τ S (P )Δ S (Q) as desired.

kk 2 kmin 1≤s≤K k min i∈Ss p ik = 1 −
then the following relation holds α S (P ) ≤ τ S (P ) ≤ β S (P ).Proof.We start with the first inequality.By definition of ergodicity coefficient α S , we have α S (P ) = p mr − p lr ≥ 0 for some indices l, m and r.Hence τ S (P |p mk − p lk |.We now partition [n] into two subsets: P m , consisting of indices k such that p mk ≥ p lk , and P l , consisting of indices k such that p mk < p lk .Since P is a stochastic matrix which has unit row sums, we obtain|p mk − p lk | = k∈Pm (p mk − p lk ) + k∈P l (p lk − p mk ) = 2 k∈Pm (p mk − p lk ).World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:1, 2013 183 International Scholarly and Scientific Research & Innovation 7(1) 2013 ISNI:0000000091950263 Open Science Index, Mathematical and Computational Sciences Vol:7, No:1, 2013 waset.org/Publication/9996769Recall that for any k ∈ P m , we have p mk − p lk ≥ 0. Thus, we obtain τ S (P ) ≥ 1 |p mk − p lk | = k∈Pm (p mk − p lk ) = p mr − p lr + k∈Pm k =r (p mk − p lk ) ≥ α S (P ), as desired.As for the second inequality, for every s = 1, • • • , K, we can choose two indices i s and j s such that min i,j∈Ss k min{p ik , p jk } = k min{p isk , p jsk }.Hence, we obtain 1 − τ S (P ) = min 1≤s≤K k min{p isk , p jsk } ≥ β s (P ).
For every y ∈ Y , we define a convex combination ofQ 1 , • • • , Q n asSince R i and R j are in the convex hull of {C y : y ∈ Y }, we obtainR i − R j ≤ max y,y ∈Y C y − C y ≤ max y,y ∈Y K s=1 γ st − k∈Ss min{p ik , p jk } • Q ys − Q y s ≤ τ S (P )Δ S (Q).