Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected graph Laplacan L as follows. If (S) j > (n what follows we wll leave out the parentheses) there s a drected edge j. Thus the th row of S dentfes the edges comng nto vertex and ther weghts. Ths set of vertces are collectvely the neghbors of, and s denoted by N. The dagonal elements S are chosen such that each row sum equals. In partcular, f a vertex has no ncomng edges, we choose S =. For the purposes of ths work, we defne the Laplacan by where I s the dentty matrx. L I S, It turns out, perhaps somewhat confusngly, that dependng on the applcaton one s nterested n, sometmes S and L are the natural objects of study, and at other tmes t may be better to look at ther transpose S T and L T. As an example of the frst, consder a very smple consensus model, where the components x R n are ndvdual approvals (or lack thereof) of some tem or dea. Suppose that one s opnon changes wth a rate gven by some lnear functon of the perceved dfferences of opnon wth one s neghbors. Ths naturally leads to the model ẋ = glx, (.) where g s some arbtrary real parameter. What s mportant here, s that every opnon depends on ncomng opnons: ẋ k = g j N k w kj (x k x j ), where the w kj are non-negatve weghts wth row-sum one. One sees that n effect, the rate of change depends on the dfference of one s opnon wth a weghted average of ncomng opnons. Thus L has row-sum zero and S = I L s row-stochastc. The same thngs holds n the study of flocks and more detals for that model can be found n [2]. Farborz Maseeh Dept. of Math. and Stat., Portland State Unv., Portland, OR, USA
On the other hand, consder next a random walk on the same graph wth n vertces. Gven the probabltes p() that a walker s at vertex, the probabltes p (j) that the walker s at vertex j n the next tme step. Snce the probabltes must always be non-negatve and sum to, the random walker s a map from the (n )-dmensonal smplex to tself. Let us suppose, as before, that map s lnear and represented by a n n matrx A. So suppose p = (,, ) T, then p, the frst column of A, s a column of non-negatve probabltes summng to one. In essence, at very vertex, the outgong probabltes must sum to one, and A s column-stochastc. Thus, n ths case, t s natural to consder A S T as the lnear map determnng the random walk. It s crucal to realze that accordng to ths defnton the random walker moves n the drecton opposte to the edge arrows defned above. The am of ths note s to characterze the null spaces of both L and L T, and to descrbe ther relaton to the dynamcal processes just outlned. We buld on work done n [] and [2] where the role of the null space n the study of flockng was descrbed. In secton 2 we summarze the results of these papers. The man results of ths work concern the random walk and ts relaton to the null space of L T. These follow n Secton 3. In Secton 4 we llustrate our results wth two smple examples. In [] matrces M of the form 2 Pror Results M = D DS were consdered, where D s a non-negatve dagonal matrx and S s row stochastc. The specal case L arses by choosng D = I. We wll assume that from here on. The followng defntons (wth the excepton of Defnton 2.3) and results are taken from [] and [2]. Defnton 2. Gven any real N N matrx M = D DS, we denote by G(V, S) the drected graph wth vertces, n and an edge j whenever S j. For each vertex, set N := {j j }. We wrte j f there exsts a drected path n G S from vertex j to vertex. Furthermore, for any vertex j, we defne R(j) to be the set contanng j and all vertces such that j. We refer to R(j) as the reachable set of vertex j. Defnton 2.2 A set R of vertces n a graph wll be called a reach f t s a maxmal reachable set; n other words, R s a reach f R = R() for some and there s no j such that R() R(j) (properly). Snce our graphs all have fnte vertex sets, such maxmal sets exst and are unquely determned by the graph. For each reach R of a graph, we defne the exclusve part of R to be the set H = R \ j R. Lkewse, we defne the common part of R to be the set C = R \H. Note that, by defnton, the parwse empty ntersecton of two exclusve sets s empty: H H j = f j. The common sets can, however, ntersect. Note further that each reach R contans at least one vertex r such that ts reachable set R(r) equals the entre reach. Such a vertex s called a root of R. By defnton, any root must be contaned n the exclusve part of ts reach. Defnton 2.3 Gven a dgraph G. Then each reach R contans a set of roots P. The set P s called the root set and s contaned n H. 2
Theorem 2.4 Suppose M = D DS, where D s a nonnegatve N N dagonal matrx and S s stochastc. Suppose G S has k reaches, denoted R through R k, where we denote the exclusve and common parts of each R by H, C respectvely. Then the null space of M has a bass γ, γ 2,... γ k n R n whose elements satsfy: () γ (v) = for v H ; () γ (v) (, ) for v C ; () γ (v) = for v R ; (v) γ = n. Notce that Theorem 2.5 does not completely determne the bass of the null space. The values of the bass vectors on the common depend on the weghts of the relevant edges. Ths wll be llustrated wth an example n Secton 4. Theorem 2.5 The egenvalue of a Laplacan matrx of the form D DS wth k reaches has algebrac and geometrc multplcty k. Theorem 2.6 Any nonzero egenvalue of a Laplacan matrx of the form D DS, where D s nonnegatve dagonal and S s stochastc, has (strctly) postve real part. The consequences of ths for the smple consensus model mentoned earler are easy to descrbe. Let be the subspace of R n spanned by all the (generalzed) egenspaces of L other than ts null space and denote by {γ } n =k+ a bass of. Then the vectors {γ } n = form a bass for R n. The ntal condton x() can be wrtten n terms of ths bass as x() = n α γ. = Set g <. By Theorem 2.6, the non-zero egenvalues of gl all have negatve real part. Thus the soluton of the dfferental equaton (.) s: x(t) = k α γ + R(t), = where R(t) s as functon that decreases to exponentally fast as tme t tends to nfnty (though t may have large transents). Ths means that, for large t, the opnon vector x s entrely determned by the null space of L. The concluson for the more complcated flockng models s very smlar. We refer the nterested reader to [2]. 3 Random Walks on Drected Graphs Denote the transpose of S by A: A = S T. Everythng else s as defned n Secton 2. In partcular, edges j and drected paths j have drectons determned by S. Defnton 3. Let G be the dgraph wth (weghted) adjacency row stochastc matrx S. T S s the random walk on the dgraph G gven by the transton probabltes: prob(j ) = A j = S j. 3
Recall that the edges have the opposte drecton,.e. j s an edge f A j =prob(j ) >. We wll abbrevate T S wth T snce no confuson s lkely. Defnton 3.2 A probablty vector or a (dscrete) measure s a vector n R n such that for all, p() and p() =. The support, supp(p), of the measure p, s the set of vertces on whch p takes a postve value. To conform to the formal defnton of probablty measure, we note that p( ) =, and for any vertex set W V (G), p(w ) = W p(). Defnton 3.3 Let T be a random walk. The push forward T p of the measure p s gven by (T p)() = j prob(j )p(j) = j A j p(j), The pull back T p of the measure p s gven by (T p)() = j prob( j)p(j) = A j p(j), Therefore f the probabltes at tme step are gven by the vector p, then at tme step they are gven by the vector T p = Ap. Note that ths corresponds to left multplcaton of S by p. It s easy to drectly verfy that Ap s a probablty vector f p s a probablty vector. Frst of all, f p s a probablty vector, then (Ap) () = A j p(j), j and then, of course, (Ap) () = So that Ap satsfes Defnton 3.2. A j p(j) = j j ( ) A j p(j) =. Defnton 3.4 T has a forward nvarant probablty measure p f Ap = p. K V (G) s a forward nvarant set under T, f p K c = mples Ap K c =, where K c s the complement of K n V (G). We now use the notaton of the prevous secton. Lemma 3.5 Gven a random walk random walk T, every exclusve set H and ts root set P are forward nvarant sets under T. Proof: Suppose C s the common part of the reach correspondng to H. A walker landng n H can only leave the exclusve part f the graph G has an edge n the opposte drecton (Defnton 3.). Ths contradcts Defntons 2.2 and 2.3. Smlarly, a walker can only leave P f the graph G has edge from j V (G)\P nto P. But ths would mean that j s a root of R, whch s a contradcton. 4
Theorem 3.6 Let G be a graph wth Laplacan L = I S wth k reaches as n Theorem 2.4. Gven j V (G) and m {, k}. The probablty that a random walker under T startng at j V ends up n P m equals γ m (j). Proof: Let q(j) be the probablty that a random walker startng at j V reaches P m for some fxed m. Then q : V [, ] s well-defned and s constant n tme. Snce, by Lemma 3.5, P m s forward nvarant, q(j) s also equal to the probablty that the walker startng at j ends up and stays n P m. The probablty q(j) concerns the future (under T ) of the walker on j. Therefore t s equal to the approprately weghted average of q() of j s successors under T. Thus q(j) = prob(j )q() = j A j q(). From ths we conclude: Ths proves that q s n the null space of M: q = A T q q = Sq Lq = (I S)q = q(j) = α γ (j) However, agan by Lemma 3.5, f j s a vertex n P m, then q(j) =, and f j s n any P j wth j m, then q(j) =. Thus α m = and α l = f l m. Lemma 3.7 Let G be a dgraph wth reach R consstng of an exclusve part H, contanng the root set P, and a common part C. Under the random walk T on G, there s a unque nvarant measure p whose support equals P. Proof: Consder a reach R wth ts root set P and denote the vertex set R\P by Y and the vertex set V \R by Z. Snce drected paths n G cannot leave the reach R, we have A P Z = A Y Z =. Smlarly, drected paths n G cannot go from Z to P, nor from Y to P (see Lemma 3.5). Therefore, upon permutng vertces, the matrx A equals A P P A P Y A = A Y Y. A ZY A ZZ The matrx A P P s the transpose of the matrx S P P. Clearly P has at least one root v. Lemma 3. n [] shows that the egenvalue n S P P has algebrac and geometrc multplcty one. Its transpose A P P has the same characterstc polynomal, ts egenvalue also has algebrac multplcty (and therefore geometrc multplcty ). Smlarly, the proof of Theorem 2.7 n [] establshes that the spectral radus of S Y Y s strctly less than, and the same holds for A Y Y. We solve for p n Ap = p, where p = (a P, b Y, c Z ) T. Ths gves A P P a P + A P Y b Y = a P, A Y Y b Y = b Y and A Y Z b Y + A ZZ c Z = c Z. The mddle equaton can only be satsfed f b Y =. Ths shows that the support of p s n P. 5
Now we assume that there s a vertex k P such that p(k) =. Then = p(k) = j A kj p(j). Snce k s a root, t has outgong edges. So there must be j such that A kj >. Snce p s a probablty measure, ts components are non-negatve. Therefore p(j) must be zero. By nducton, one proves that for all vertces such that k, p() =. Snce k s a root, that mples that p = on the entre reach R, whch s absurd. Therefore p(k) >. Theorem 3.8 Let G be a graph wth Laplacan L = I S wth k reaches as n Theorem 2.4. Then the egenvalue of L T has algebrac and geometrc multplcty k. Furthermore, the null space of L T has a bass γ, γ 2,... γ k n R n whose elements satsfy: () For all {, k} and all v {, n}: γ (v) ; () v P γ (v) = ; () γ (v) = for v P. Proof: Denote the unon k = R \P by Z. Permutng rows and columns, we can wrte the matrx A n block dagonal form: A P P A P Z A P2 P 2 A P2 Z A =... A Pk P k A Pk Z A ZZ Each of the dagonal blocks, except A ZZ, s rooted and so has egenvalue wth algebrac and geometrc multplcty. As before, A ZZ has spectral radus less than. The characterstc polynomal of A s the product of the characterstc polynomals of the dagonal blocks, and the result follows. Notce that, just lke Theorem 2.5, Theorem 3.8 does not completely determne the null space. The examples n the next secton wll show that ndeed random walks on a gven dgraph but wth dfferent weghts on the edges can have non-trvally dfferent nvarant measures. 4 The Kernel of L Versus the Kernel of L T We saw that the components γ (j) of the vectors γ equal the probablty that a random walker startng at j reaches H. In contrast, the null space Ē of L T = I A s spanned by vectors { γ } k = whch gve the dfferent nvarant measures wth support n R assocated wth the random walk T. We wll now show that the role of these nvarant measures γ n the random walk s smlar to that of the vectors γ n the consensus model dscussed at the end of Secton 2. Let be the subspace of R n spanned by all the (generalzed) egenspaces of L T other than ts null space and denote by { γ } n =k+ a bass of. Then the vectors { γ } n = form a bass for R n. Clearly, any probablty vector p () can be wrtten unquely as n p () = α γ. = 6
Lemma 4. Each of the bass vectors of { γ } n =k+ of has the property that the sum of ts components equals zero. Proof: Frst, assume that v s an egenvector of L T wth egenvalue λ. We obtan A j v(j) = λ v() = ( λ) v() =,,j because A s column stochastc. Thus v() =. Next, suppose the lemma s false for some generalzed egenvector w of L T. Then, by Theorem 3.8, the assocated egenvalue must be dfferent from. Thus we have a stuaton where there s a vector v wth v() = and (A λi)w = v. Takng the sum of the components on both sdes yelds the result. Theorem 4.2 Let G be a graph wth Laplacan L = I S wth k reaches as n Theorem 2.4. Let p () = n = α γ be the ntal probablty dstrbuton. Assume S no egenvalues wth modulus, except λ =. Then the probablty measure at tme step l, p (l), s gven by: p (l) = A l p () = k α γ + A l δ, = for some δ, and A l δ tends to zero exponentally fast as l tends to zero. Proof: Lemma 4. says that j γ (j) = for > k. Therefore n j= k = α γ (j) =. Furthermore, all components are non-negatve. Ths shows that k = α γ s a probablty measure. By Gershgorn s theorem, egenvalues of A = S T have modulus at most. The egenvalue has algebrac multplcty k and the correspondng egenspace s spanned by { γ } k =. All other egenvalues must have modulus strctly less than. Thus, settng δ = n =k+ α γ, we have A l δ < Kλ l, for some K > and some λ (, ). Remark: Even though the convergence s exponentally fast, very large transents may occur for ntermedate values of l, especally for graphs that have many vertces and are hghly asymmetrc. We gve two smple examples of the contrast between the null spaces of L and L T. As the frst example suppose that V = {, 2} and ( ) ( ) x x x x S = or L =, and A s agan the transpose of S. We take x [, ]. The egenvalues of S (and A) are and x. Thus when x =, there s an egenvalue -,( volatng ) the condton on the egenvalues of Theorem4.2. The kernel of L = I S s the vector. Indeed graph G(V, S) has one reach and both vertces are (of course) n t. 7
( ) There s an nvarant measure and t s gven by (2 x) whch spans the null space of x L T = I S T. Note that ths also works for x = when the random walker hops back and forth from to 2 to, etcetera. In that case the probablty s smply /2 that the walker s at vertex and /2 that he s at vertex 2. The next example s modfed from [2]. Its vertex set s V = {, 2, 3, 4, 5} and ts (drected) Laplacan s gven by L = x x /2 /2, where x [, ]. The reaches are gven by R = {, 2, 3, 4} and R 2 = {3, 4, 5} wth exclusve parts H = {, 2} and H 2 = {5}. Its root sets are P = {} and P 2 = {5}. Snce ths graph has two reaches, the egenspace correspondng to the zero egenvalue s two-dmensonal. It s easy to verfy that t s spanned by γ = (2 x) 2 x 2 x 2( x) x and γ 2 = (2 x) x 2 x To fnd the nvarant measures we look for the kernel of the transpose of the Laplacan. Ths tme the kernel s spanned by the measures γ = and γ 2 = Snce each of the two reaches has a unque root, the nvarant measures are unquely determned. As an example, a walker startng at vertex 3 has probablty 2( x) to end up at vertex and probablty 2 x x 2( x) to end up at vertex 3. The assocated nvarant measure s γ 2 x 2 x + x γ 2 x 2. References [] J. S. Caughman, J. J. P. Veerman, Kernels of Drected Graph Laplacans, Electronc Journal of Combnatorcs 3, No, R39, 26. [2] J. J. P. Veerman, G. Lafferrere, John S. Caughman, A. Wllams, Flocks and Formatons, J. Stat.Phys. 2, Vol 5-6, 9-936, 25. 8