c 2009 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J OPTIM Vol 20, No 2, pp c 2009 Society for Idustrial ad Applied Mathematics FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES STEPHEN BOYD, PERSI DIACONIS, PABLO PARRILO, AND LIN XIAO Abstract We show how to exploit symmetries of a graph to efficietly compute the fastest mixig Markov chai o the graph (ie, fid the trasitio probabilities o the edges to miimize the secod-largest eigevalue modulus of the trasitio probability matrix) Exploitig symmetry ca lead to sigificat reductio i both the umber of variables ad the size of matrices i the correspodig semidefiite program, ad thus eable umerical solutio of large-scale istaces that are otherwise computatioally ifeasible We obtai aalytic or semiaalytic results for particular classes of graphs, such as edge-trasitive ad distace-trasitive graphs We describe two geeral approaches for symmetry exploitatio, based o orbit theory ad block-diagoalizatio, respectively, ad establish a formal coectio betwee them Key words Markov chais, fast mixig, eigevalue optimizatio, semidefiite programmig, graph automorphism, group represetatio AMS subject classificatios 05C25, 20C30, 60J22, 65F15, 90C22, 90C51 DOI / Itroductio I the fastest mixig Markov chai problem [3], we choose the trasitio probabilities o the edges of a graph to miimize the secod-largest eigevalue modulus of the trasitio probability matrix As show i [3], this ca be formulated as a covex optimizatio problem, i particular a semidefiite program Thus it ca be solved, up to ay give precisio, i polyomial time by iterior-poit methods I this paper, we show how to exploit symmetries of a graph to make the computatio much more efficiet 11 The fastest mixig Markov chai problem We cosider a udirected graph G =(V, E) with vertex set V = {1,,} ad edge set E, ad assume that G is coected We defie a discrete-time Markov chai by associatig with each edge {i, j} Ea trasitio probability P ij (P ii deotes the holdig probability at vertex i) We assume the trasitio betwee two vertices coected by a edge is symmetric, ie, P ij = P ji Thus the trasitio probability matrix, P R,satisfies P = P T, P 0, P1 = 1, where the superscript T deotes the traspose of a matrix, the iequality P 0 meas elemetwise, ad 1 deotes the vector of all oes Sice P is symmetric ad stochastic, the uiform distributio (1/)1 T is statioary I additio, the eigevalues of P are real, ad o more tha oe i modulus We Received by the editors April 24, 2007; accepted for publicatio (i revised form) March 31, 2009; published electroically Jue 17, Iformatio Systems Laboratory, Departmet of Electrical Egieerig, Staford Uiversity, Staford, CA (boyd@stafordedu) Departmet of Statistics ad Departmet of Mathematics, Staford Uiversity, Staford, CA Departmet of Electrical Egieerig ad Computer Sciece, Massachusetts Istitute of Techology, Cambridge, MA (parrilo@mitedu) Microsoft Research, 1 Microsoft Way, Redmod, WA (lixiao@microsoftcom) 792

2 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 793 list them i oicreasig order as 1=λ 1 (P ) λ 2 (P ) λ (P ) 1 We deote by μ(p ) the secod-largest eigevalue modulus (SLEM) of P, ie, μ(p )= max i=2,, λ i(p ) =max{λ 2 (P ), λ (P )} This quatity is widely used to boud the asymptotic covergece rate of the Markov chai to its statioary distributio, i the total variatio distace or chi-squared distace (see, eg, [16, 19]) I geeral the smaller μ(p ) is, the faster the Markov chai coverges For more backgroud o Markov chais, eigevalues, ad rapid mixig, see, eg, the text [8] The fastest mixig Markov chai (FMMC) problem [3] is to fid the optimal P that miimizes μ(p ) This ca be posed as the followig optimizatio problem: (11) miimize μ(p ) subject to P = P T, P 0, P1 = 1, P ij =0, i j ad {i, j} / E It turs out that this is a covex optimizatio problem [3] This ca be see, eg, by expressig the objective fuctio as μ(p )= P (1/)11 T 2,where 2 deotes the spectral orm of a matrix Moreover, it ca be trasformed ito a semidefiite program (SDP): (12) miimize subject to s si P (1/)11 T si, P = P T, P 0, P1 = 1, P ij =0, i j ad {i, j} / E Here I deotes the idetity matrix, ad the variables are the matrix P ad the scalar s Thesymbol deotes matrix iequality, ie, X Y meas Y X is positive semidefiite We should ote that there are other importat measures of rapid mixig, eg, the log-sobolev costat [33, 55, 17], ad other methods to speed up mixig, eg, liftig [11] We focus o the approach of miimizig the SLEM o a fixed graph topology I additio to its direct coectio to Markov chai Mote Carlo simulatio, the FMMC problem has foud may practical applicatios i fast load balacig of parallel computig systems (ofte with highly symmetric cofiguratios, as discussed i this paper) [56, 59], ad i average cosesus ad gossip algorithms i sesor etworks [58, 7] There has bee some follow-up work o the FMMC problem Boyd et al [6] proved aalytically that o a -path the fastest mixig chai ca be obtaied by assigig the same trasitio probability half at the 1 edges ad two loops at the two eds Roch [51] used stadard mixig-time aalysis techiques (variatioal characterizatios, coductace, caoical paths) to boud the fastest mixig time Gade ad Overto [23] have cosidered the fastest mixig problem for a oreversible Markov chai Here, the problem is ocovex ad much remais to be doe Fially, closedform solutios of fastest mixig problems have recetly bee applied i statistics to give a geeralizatio of the usual spectral aalysis of time series for more geeral discrete data; see [53]

3 794 S BOYD, P DIACONIS, P PARRILO, AND L XIAO 12 Exploitig problem structure Whe solvig the SDP (12) by iteriorpoit methods, i each iteratio, we eed to compute the first ad secod derivatives of the logarithmic barrier fuctios (or potetial fuctios) for the matrix iequalities, ad assemble ad solve a liear system of equatios (the Newto system) Let be the umber of vertices ad let m be the umber of edges i the graph (m is the umber of variables i the optimizatio problem) The Newto system is a set of m liear equatios with m ukows Without exploitig ay structure, the umber of flops per iteratio i a typical barrier method is o the order max{m 3,m 2 2,m 3 }, where the first two terms come from computig ad assemblig the Newto system, ad the third term amouts to solvig it (see, eg, [2, sectio 1183]) (Other variats of iterior-poit methods have similar orders of flop cout) Exploitig problem structure ca sigificatly improve solutio efficiecy As for may other problems defied o a graph, sparsity is the most obvious structure to cosider here I fact, may curret SDP solvers already exploit sparsity However, it is a well-kow fact that exploitig sparsity aloe i iterior-poit methods for SDP has limited effectiveess The sparsity of P, ad the sparsity plus rak-oe structure of P (1/)11 T, ca be exploited to greatly reduce the complexity of assemblig the Newto system, but typically the Newto system itself is dese The computatioal cost per iteratio is still at the order O(m 3 ), domiated by solvig the dese liear system (see aalysis i [58]) We ca also solve the FMMC problem i the form (11) by subgradiet-type (firstorder) methods The subgradiets of μ(p ) ca be obtaied by computig the SLEM of the matrix P ad the associated eigevectors This ca be doe very efficietly by iterative methods, specifically the Laczos method, for large sparse symmetric matrices (see, eg, [30, 52]) Compared with iterior-poit methods, subgradiet-type methods ca solve much larger problems but oly to a moderate accuracy; they also do t have polyomial-time worst-case complexity I [3], we used a simple subgradiet method to solve the FMMC problem o graphs with up to a few hudred thousad edges More sophisticated first-order methods, for solvig large-scale eigevalue optimizatio ad SDPs, have bee developed i, eg, [10, 35, 38, 45, 46] A successive partial liear programmig method was developed i [47] I this paper, we focus o the FMMC problem o graphs with large symmetry groups, ad we show how to exploit symmetries of the graph to make the computatio more efficiet A result by Erdős ad Réyi [21] states that with high probability (asymptotically with probability oe), the symmetry group of a suitably defied radom graph is trivial; ie, it cotais oly the idetity elemet Nevertheless, may of the graphs of theoretical ad practical iterest, particularly i egieerig applicatios, have very iterestig, ad sometimes very large, symmetry groups Symmetry reductio techiques have bee explored i several differet cotexts, eg, dyamical systems ad bifurcatio theory [31], polyomial system solvig [25, 57], umerical solutio of partial differetial equatios [22], ad Lie symmetry aalysis i geometric mechaics [40] I the cotext of optimizatio, a class of SDPs with symmetry has bee defied i [36], where the authors study the ivariace properties of the search directios of primal-dual iterior-poit methods I additio, symmetry has bee exploited to prue the eumeratio tree i brach-ad-cut algorithms for iteger programmig [39] ad to reduce matrix size i a spectral radius optimizatio problem [34] Closely related to our approach i this paper, the recet work [14] cosidered geeral SDPs that are ivariat uder the actio of a permutatio group ad developed a techique based o matrix -represetatio to reduce problem size This

4 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 795 techique has bee applied to simplify computatios i SDP relaxatios for graph colorig ad maximal clique problems [20] ad to stregthe SDP bouds for some codig problems [37] 13 Cotets The paper is orgaized as follows I sectio 2, we first explai some basic backgroud o graph automorphisms ad symmetry groups We show that the FMMC problem always attais its optimum i the fixed-poit subset of the feasible set uder the automorphism group This allows us to cosider oly a umber of distict trasitio probabilities that equals the umber of orbits of the edges I sectio 3, we give closed-form solutios for the FMMC problem o some special classes of graphs, amely, edge-trasitive graphs ad distace-trasitive graphs Alog the way we also discuss FMMC o graphs formed by takig Cartesia products of simple graphs I sectio 4, we first review the orbit theory for reversible Markov chais developed i [4], which gives sufficiet coditios o costructig a orbit chai that cotais all distict eigevalues of the origial chai This orbit chai is usually o loger symmetric but always reversible We the solve the fastest reversible Markov chai problem o the orbit graph, from which we immediately obtai a optimal solutio to the origial FMMC problem I sectio 5, we focus o the approach developed i [26], which block-diagoalizes the liear matrix iequalities i the FMMC problem by costructig a symmetryadapted basis The resultig blocks usually have much smaller sizes, ad repeated blocks ca be discarded i computatio We establish a formal coectio betwee this approach ad the orbit theory, ad demostrate their coectio o several examples More examples ca be foud i [5] I sectio 6, we coclude the paper by poitig out some possible future work 2 Symmetry aalysis I this sectio we explai the basic cocepts that are essetial i exploitig graph symmetry, ad we derive our result o reducig the umber of optimizatio variables 21 Graph automorphisms ad classificatio The study of graphs that possess particular kids of symmetry properties has a log history (see, eg, [1, 9]) The basic object of study is the automorphism group of a graph, ad differet classes ca be defied depedig o the specific form i which the group acts o the vertices ad edges A automorphism of a graph G = (V, E) isapermutatioσ of V such that {i, j} Eif ad oly if {σ(i),σ(j)} E The (full) automorphism group of the graph, deoted by Aut(G), is the set of all such permutatios, with the group operatio beig compositio For a vertex i V, the set of all images σ(i), as σ varies through a subgroup G Aut(G), is called the orbit of i uder the actio of G Distict orbits form equivalet classes ad they partitio the set V The actio is trasitive if there is oly oe sigle orbit i V AgraphG = (V, E) issaidtobevertex-trasitive if Aut(G) acts trasitively o V The actio of a permutatio σ o V iduces a actio o E with the rule σ({i, j}) ={σ(i),σ(j)} A graph G is edge-trasitive if Aut(G) acts trasitively o E Graphs ca be edge-trasitive without beig vertex-trasitive, ad vice versa; simple examples are show i Figure 21 A graph is called 1-arc-trasitive if, give ay four vertices u, v, x, y such that {u, v}, {x, y} E, there exists a automorphism σ Aut(G) such that σ(u) =x ad σ(v) =y Notice that, as opposed to edge-trasitivity, here the orderig of the

5 796 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Fig 21 The graph o the left side is edge-trasitive but ot vertex-trasitive The oe o the right side is vertex-trasitive but ot edge-trasitive Distace-trasitive 1-arc-trasitive Edge-trasitive Vertex-trasitive Fig 22 Classes of symmetric graphs ad their iclusio relatioship vertices is importat, eve for udirected graphs I fact, a 1-arc-trasitive graph must be both vertex-trasitive ad edge-trasitive, ad the reverse may ot be true The 1-arc-trasitive graphs are called symmetric graphs i [1], but the moder use exteds this term to all graphs that are simultaeously edge- ad vertex-trasitive Fially, let δ(u, v) deote the distace betwee two vertices u, v V A graph is called distace-trasitive if, for ay four vertices u, v, x, y with δ(u, v) = δ(x, y), there is a automorphism σ Aut(G) such that σ(u) =x ad σ(v) =y The cotaimet relatioship amog the four classes of graphs described above is illustrated i Figure 22 Explicit couterexamples are kow for each of the oiclusios It is geerally believed that distace-trasitive graphs have bee completely classified This work has bee doe by classifyig the distace-regular graphs It would take us too far afield to give a complete discussio See the survey i [18, sectio 7] The cocept of graph automorphism ca be aturally exteded to weighted graphs by requirig that the permutatio also preserve the weights o edges (see, eg, [4]) This extesio allows us to exploit symmetry i more geeral reversible Markov chais, where the trasitio probability matrix is ot ecessarily symmetric 22 FMMC with symmetry costraits A permutatio σ Aut(G) ca be represeted by a permutatio matrix Q, whereq ij =1ifi = σ(j) adq ij =0 otherwise The permutatio σ iduces a actio o the trasitio probability matrix by σ(p )=QP Q T We deote the feasible set of the FMMC problem (11) by C, ie, C = {P R P 0, P1 = 1, P= P T,P ij =0for{i, j} / E} This set is ivariat uder the actio of graph automorphism To see this, let h = σ(i)

6 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 797 ad k = σ(j) The we have (σ(p )) hk =(QP Q T ) hk = l (QP ) hl Q kl =(QP ) hj = l Q hl P lj = P ij Sice σ is a graph automorphism, we have {h, k} E if ad oly if {i, j} E,sothe sparsity patter of the probability trasitio matrix is preserved It is straightforward to verify that the coditios P 0, P 1 = 1, adp = P T are also preserved uder this actio Let F deote the fixed-poit subset of C uder the actio of Aut(G); ie, (21) F = {P C σ(p )=P, σ Aut(G)} We have the followig theorem Theorem 21 The FMMC problem always has a optimal solutio i F Similar results have appeared i, eg, [14, 26] Here we iclude the proof for completeess Proof Let μ deote the optimal value of the FMMC problem (11), ie, μ = if{μ(p ) P C} Sice the objective fuctio μ is cotiuous ad the feasible set C is compact, there is at least oe optimal trasitio matrix P such that μ(p )=μ Let P deote the average over the orbit of P uder Aut(G): P = 1 Aut(G) σ Aut(G) σ(p ) This matrix is feasible because each σ(p ) is feasible ad the feasible set is covex By costructio, it is also ivariat uder the actios of Aut(G) Moreover, usig the covexity of μ, wehaveμ(p ) μ(p ) It follows that P F ad μ(p )=μ As a result of Theorem 21, we ca replace the costrait P Cby P Fi the FMMC problem ad get the same optimal value I the fixed-poit subset F, the trasitio probabilities o the edges withi a orbit must be the same So we have the followig corollaries Corollary 22 The umber of distict edge trasitio probabilities we eed to cosider i the FMMC problem is at most equal to the umber of orbits of E uder Aut(G) Corollary 23 If G is edge-trasitive, the all the edge trasitio probabilities ca be assiged the same value Note that the holdig probabilities at the vertices ca always be elimiated usig P ii =1 j P ij (of course we also eed to add the costrait j P ij 1; see sectio 23) So it suffices to cosider oly the edge trasitio probabilities 23 Formulatio with reduced umber of variables With the results from the previous sectio, we ca give a explicit parametrizatio of the FMMC problem with a reduced umber of variables Recall that the adjacecy matrix of a graph with vertices is a matrix A whose etries are give by A ij =1if{i, j} E ad A ij =0otherwise Letν i be the valecy (degree) of vertex i TheLaplacia matrix of the graph is give by L = Diag(ν 1,ν 2,,ν ) A, where Diag(ν) deotes a diagoal matrix with the vector ν o its diagoal A extesive accout of the Laplacia matrix ad its use i algebraic graph theory are provided i, eg, [12, 28, 42]

7 798 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Suppose that there are N orbits of edges uder the actio of Aut(G) For each orbit, we defie a orbit graph G k =(V, E k ), where E k is the set of edges i the kth orbit Note that the orbit graphs are discoected if the origial graph is ot edgetrasitive Let L k be the Laplacia matrix of G k The diagoal etries (L k ) ii equal the valecy of ode i i G k (which is zero if vertex i is discoected with all other vertices i G k ) By Corollary 22, we ca assig the same trasitio probability o all the edges i the kth orbit Deote this trasitio probability by p k ad let p =(p 1,,p N ) The the trasitio probability matrix ca be writte as (22) N P (p) =I p k L k k=1 This parametrizatio of the trasitio probability matrix automatically satisfies the costraits P = P T, P 1 = 1, adp ij =0for{i, j} / E The etrywise oegativity costrait P 0 ow traslates ito p k 0, k =1,,N, N (L k ) ii p k 1, i =1,,, k=1 where the secod set of costraits comes from the oegativity of the diagoal etries of P It ca be verified that the parametrizatio (22), together with the above iequality costraits, is the precise characterizatio of the fixed-poit subset F Therefore we ca explicitly write the FMMC problem restricted to the fixed-poit subset as (23) ( ) N miimize μ I p k L k k=1 subject to p k 0, k =1,,N, N (L k ) ii p k 1, i =1,, k=1 3 Some aalytic results For some special classes of graphs, the FMMC problem ca be cosiderably simplified ad ofte solved by exploitig symmetry oly I this sectio, we give some aalytic results for the FMMC problem o edge-trasitive graphs, Cartesia product of simple graphs, ad distace-trasitive graphs The optimal solutio is ofte expressed i terms of the eigevalues of the Laplacia matrix of the graph It is iterestig to otice that eve for such highly structured classes of graphs, either the maximum-degree or the Metropolis Hastigs heuristics discussed i [3] gives the optimal solutio Throughout, we use α to deote the optimal trasitio probability o all the edges, ad μ to deote the optimal SLEM

8 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES FMMC o edge-trasitive graphs Theorem 31 Suppose the graph G is edge-trasitive, ad let α be the trasitio probability assiged o all the edges The the optimal solutio of the FMMC problem is (31) (32) { 1 α =mi, ν max μ =max { 1 λ 1(L) ν max, } 2, λ 1 (L)+λ 1 (L) λ 1 (L) λ 1 (L) λ 1 (L)+λ 1 (L) where ν max =max i V ν i is the maximum valecy of the vertices i the graph, ad L is the Laplacia matrix defied i sectio 23 Proof By defiitio of a edge-trasitive graph, there is a sigle orbit of edges uder the actios of its automorphism group Therefore we ca assig the same trasitio probability α o all the edges i the graph (Corollary 23), ad the parametrizatio (22) becomes P = I αl Sowehave λ i (P )=1 αλ +1 i (L), i =1,,, ad the SLEM μ(p )=max{λ 2 (P ), λ (P )} =max{1 αλ 1 (L), αλ 1 (L) 1} To miimize μ(p ), we let 1 αλ 1 (L) =αλ 1 (L) 1adgetα =2/(λ 1 (L) + λ 1 (L)) But the oegativity costrait P 0 requires that the trasitio probability must also satisfy 0 <α 1/ν max Combiig these two coditios gives the optimal solutio (31) ad (32) 311 Example: Cycle graphs The cycle (or rig) graph C is a coected graph with 3 vertices, where each vertex has exactly two eighbors Its Laplacia matrix is L = which has eigevalues 2 2cos(2kπ/), k =1,, The two extreme eigevalues are λ 1 (L) =2 2cos 2 /2 π, λ 1 (L) =2 2cos 2π, where /2 deotes the largest iteger that is o larger tha /2, which is /2 for eve or ( 1)/2 for odd By Theorem 31, the optimal solutio to the FMMC problem is (33) α = 2 cos 2π 1 cos 2 /2 π,, }, (34) μ = cos 2π 2 cos 2π cos 2 /2 π cos 2 /2 π

9 800 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Whe, the trasitio probability α 1/2 adtheslemμ 1 2π 2 / 2 32 Cartesia product of graphs May graphs we cosider ca be costructed by takig Cartesia product of simpler graphs The Cartesia product of two graphs G 1 =(V 1, E 1 )adg 2 =(V 2, E 2 ) is a graph with vertex set V 1 V 2,where two vertices (u 1,u 2 )ad(v 1,v 2 ) are coected by a edge if ad oly if u 1 = v 1 ad {u 2,v 2 } E 2,oru 2 = v 2 ad {u 1,v 1 } E 1 LetG 1 G 2 deote this Cartesia product Its Laplacia matrix is give by (35) L G1 G 2 = L G1 I V1 + I V2 L G2, where deotes the matrix Kroecker product [32] The eigevalues of L G1 G 2 are (36) λ i (L G1 )+λ j (L G2 ), i =1,, V 1, j =1,, V 2, where each eigevalue is obtaied as may times as its multiplicity (see, eg, [43]) Combiig Theorem 31 ad the above expressio for eigevalues, we ca easily obtai solutios to the FMMC problem o graphs formed by takig Cartesia products 321 Example: Mesh o a torus Mesh o a torus is the Cartesia product of two copies of C WewriteitasM = C C By (36), its Laplacia matrix has eigevalues 4 2cos 2iπ 2cos2jπ, i, j =1,, By Theorem 31, we obtai the optimal trasitio probability α 1 = 3 2cos 2 /2 π cos 2π ad the smallest SLEM 2 /2 π μ 1 2cos +cos 2π = 3 2cos 2 /2 π cos 2π Whe, the trasitio probability α 1/4 adtheslemμ 1 π 2 / Example: Hypercubes The d-dimesioal hypercube, deoted Q d, has 2 d vertices, each labeled with a biary word with legth d Two vertices are coected by a edge if their words differ i exactly oe compoet This graph is isomorphic to the Cartesia product of d copies of K 2, the complete graph with two vertices The Laplacia of K 2 is [ L K2 = whose two eigevalues are 0 ad 2 The oe-dimesioal hypercube Q 1 is just K 2 Higher-dimesioal hypercubes are defied recursively: Q k+1 = Q k K 2, k =1, 2, Usig (35) ad (36) recursively, the Laplacia of Q d has eigevalues 2k, k =0, 1,,d, each with multiplicity ( d k ) The FMMC is achieved for α = 1 d +1, μ = d 1 d +1 This solutio has also bee obtaied, for example, i [43] ],

10 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES FMMC o distace-trasitive graphs Distace-trasitive graphs have bee studied extesively i the literature (see, eg, [9]) I particular, they are both edge- ad vertex-trasitive I previous examples, the cycles ad the hypercubes are actually distace-trasitive graphs I a distace-trasitive graph, all vertices have the same valecy, which we deote by ν The Laplacia matrix is L = νi A, witha beig the adjacecy matrix Therefore λ i (L) =ν λ +1 i (A), i =1,, We ca substitute the above equatio i (31) ad (32) to obtai the optimal solutio i terms of λ 2 (A) adλ (A) For distace-trasitive graphs, it is more coveiet to use the itersectio matrix, which has all the distict eigevalues of the adjacecy matrix Let d be the diameter of the graph For a oegative iteger k d, chooseay two vertices u ad v such that their distace satisfies δ(u, v) =k Leta k, b k,adc k be the umber of vertices that are adjacet to u ad whose distace from v are k, k +1,adk 1, respectively That is, a k = {w V δ(u, w) =1,δ(w, v) =k}, b k = {w V δ(u, w) =1,δ(w, v) =k +1}, c k = {w V δ(u, w) =1,δ(w, v) =k 1} For distace-trasitive graphs, these umbers are idepedet of the particular pair of vertices u ad v chose Clearly, we have a 0 =0,b 0 = ν, adc 1 =1 Theitersectio matrix B is the followig tridiagoal (d +1) (d +1)matrix: B = a 0 b 0 c 1 a 1 b 1 c 2 a 2 bd 1 c d a d Deote the eigevalues of the itersectio matrix, arraged i decreasig order, as η 0,η 1,,η d These are precisely the (d + 1) distict eigevalues of the adjacecy matrix A (see, eg, [1]) I particular, we have λ 1 (A) =η 0 = ν, λ 2 (A) =η 1, λ (A) =η d The followig corollary is a direct cosequece of Theorem 31 Corollary 32 The optimal solutio to the FMMC problem o distacetrasitive graphs is { } 1 α =mi ν, 2 (37), 2ν (η 1 + η d ) { } μ η1 =max ν, η 1 η d (38) 2ν (η 1 + η d )

11 802 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Fig 31 The Peterse graph 331 Example: Peterse graph The Peterse graph, show i Figure 31, is a well-kow distace-trasitive graph with 10 vertices ad 15 edges The diameter of the graph is d = 2, ad the itersectio matrix is B = , with eigevalues η 0 =3,η 1 =1,adη 2 = 2 Applyig (37) ad (38), we obtai α = 2 7, μ = Example: Hammig graphs The Hammig graphs, deoted H(d, ), have vertices labeled by elemets i the Cartesia product {1,,} d, with two vertices beig adjacet if they differ i exactly oe compoet By the defiitio, it is clear that Hammig graphs are isomorphic to the Cartesia product of d copies of the complete graph K Hammig graphs are distace-trasitive, with diameter d ad valecy ν = d ( 1) Their eigevalues are give by η k = d ( 1) k for k =0,,d These ca be obtaied usig a equatio for eigevalues of adjacecy matrices, similar to (36), with the eigevalues of K beig 1ad 1 Therefore we have { α =mi μ =max 1 d ( 1), 2 (d +1) { 1 d( 1), d 1 d +1 We ote that hypercubes (see sectio 322) are special Hammig graphs with =2 333 Example: Johso graphs The Johso graph J(, q) (for1 q /2) is defied as follows: the vertices are the q-elemet subsets of {1,,}, with two vertices beig coected with a edge if ad oly if the subsets differ exactly by oe elemet It is a distace-trasitive graph, with ( ) q vertices ad 1 2 q ( q)( ) q edges It has valecy ν = q ( q) ad diameter q The eigevalues of the itersectio matrix ca be computed aalytically, ad they are η k = q ( q)+k (k 1), }, } k =0,,q Therefore, by Corollary 32, we obtai the optimal trasitio probability { } α 1 =mi q ( q), 2 q + + q q 2

12 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 803 ad the smallest SLEM μ =max { 1 q( q), 1 } 2 q + + q q 2 4 FMMC o orbit graphs For graphs with large automorphism groups, the eigevalues of the trasitio probability matrix ofte have very high multiplicities To solve the FMMC problem, it suffices to work with oly the distict eigevalues without cosideratio of their multiplicities This is exactly what the itersectio matrix does for distace-trasitive graphs I this sectio we develop similar tools for more geeral graphs, based o the orbit theory developed i [4] More specifically, we show how to costruct a orbit chai which is much smaller i size tha the origial Markov chai, but cotais all its distict eigevalues (with much fewer multiplicities) The FMMC o the origial graph ca be foud by solvig a much smaller problem o the orbit chai 41 Orbit theory Let P be a symmetric Markov chai o the graph G = (V, E), ad let H be a group of automorphisms of the graph Ofte, it is a subgroup of the full automorphism group Aut(G) The vertex set V partitios ito orbits O v = {σ(v) : σ H} For otatioal coveiece, i this sectio we use P (v, u), for v, u V, to deote etries of the trasitio probability matrix We defie the orbit chai by specifyig the trasitio probabilities betwee orbits: (41) P H (O v,o u )=P(v, O u )= P (v, u ) u O u This trasitio probability is idepedet of which v O v is chose, so it is well defied ad the lumped orbit chai is ideed Markov (see [4]) The orbit chai is i geeral o loger symmetric, but it is always reversible Let π(i), i V, be the statioary distributio of the origial Markov chai The the statioary distributio o the orbit chai is obtaied as (42) π H (O v )= π(i) i O v It ca be verified that (43) π H (O v )P H (O v,o u )=π H (O u )P H (O u,o v ), which is the detailed balace coditio to test reversibility The followig is a summary of the orbit theory developed i [4], which relate the eigevalues ad eigevectors of the orbit chai P H to those of the origial chai P Liftig If λ is a eigevalue of P H with associated eigevector f, the λ is a eigevalue of P with H-ivariat eigefuctio f(v) = f(o v ) Coversely, every H-ivariat eigefuctio appears uiquely from this costructio Projectio Let λ be a eigevalue of P with eigevector f Defie a fuctio o the orbits: f(ov )= σ H f(σ 1 (v)) The λ appears as a eigevalue of P H, with eigevector f, if either of the followig two coditios holds: (a) H has a fixed poit v ad f(v ) 0 (b) f is ozero at a vertex v i a Aut(G)-orbit which cotais a fixed poit of H

13 804 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Equipped with this orbit theory, we would like to costruct oe or multiple orbit chais that retai all the distict eigevalues of the origial Markov chai The followig theorem (Theorem 37 i [4]) gives sufficiet coditios for this to happe Theorem 41 Suppose that V = O 1 O K is a disjoit uio of the orbits uder Aut(G) LetH i be the subgroup of Aut(G) that has a fixed poit i O i Theall eigevalues of P occur amog the eigevalues of {P Hi } K i=1 Further, every eigevector of P occurs by liftig a eigevector of some P Hi Observe that if H G Aut(G), the the eigevalues of P H cotai all eigevalues of P G This allows disregardig some of the H i i Theorem 41 I particular, it is possible to costruct a sigle orbit chai that cotais all distict eigevalues of the origial chai Therefore we have the followig corollary Corollary 42 Suppose that V = O 1 O k is a disjoit uio of the orbits uder Aut(G), adh is a subgroup of Aut(G) If H has a fixed poit i every O i, the all distict eigevalues of P occur amog the eigevalues of P H Remark To fid H i the above corollary, we ca just compute the correspodig stabilizer, ie, compute the largest subgroup of Aut(G) that fixes oe poit i each orbit Note that the H promised by the corollary may be trivial i some cases; see [4, Remark 310] 42 Fastest mixig reversible Markov chai o orbit graph Sice i geeral the orbit chai is o loger symmetric, we caot directly use the covex optimizatio formulatio (11) or (12) to miimize μ(p H ) Fortuately, the detailed balace coditio (43) leads to a simple trasformatio that allows us to formulate the problem of fidig the fastest reversible Markov chai as a covex program [3] Suppose the orbit chai P H cotais all distict eigevalues of the origial chai Let π H be the statioary distributio of the orbits, ad let Π = Diag(π H ) The detailed balace coditio (43) ca be writte as ΠP H = PH T Π, which implies that the matrix Π 1/2 P H Π 1/2 is symmetric (ad, of course, has the same eigevalues as P H ) The eigevector of Π 1/2 P H Π 1/2 associated with the maximum eigevalue 1 is ( q = πh (O 1 ),, ) π H (O k ) The SLEM μ(p H ) equals the spectral orm of Π 1/2 P H Π 1/2 restricted to the orthogoal complemet of the subspace spaed by q This ca be writte as μ(p H )= (I qq T )Π 1/2 P H Π 1/2 (I qq T ) 2 = Π 1/2 P H Π 1/2 qq T 2 Itroducig a scalar variable s to boud the above spectral orm, we ca formulate the fastest mixig reversible Markov chai problem as a SDP: (44) miimize subject to s si Π 1/2 P H Π 1/2 qq T si, P H 0, P H 1 = 1, ΠP H = P T H Π, P H (O, O )=0, (O, O ) / E H The optimizatio variables are the matrix P H ad scalar s, ad problem data are give by the orbit graph ad the statioary distributio π H Note that the reversibility costrait ΠP H = PH T Π ca be dropped sice it is always satisfied by the costructio of the orbit chai; see (43) By pre- ad postmultiplyig the matrix iequality by

14 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 805 Π 1/2, we ca the write aother equivalet formulatio: (45) miimize subject to s sπ ΠP H π H πh T sπ, P H 0, P H 1 = 1, P H (O, O )=0, (O, O ) / E H To solve the fastest mixig reversible Markov chai problem o the orbit graph, we eed the followig three steps: 1 Coduct symmetry aalysis o the origial graph: idetify the automorphism graph Aut(G) ad determie the umber of orbits of edges N By Corollary 22, this is the umber of trasitio probabilities we eed to cosider 2 Fid a group of automorphisms H that satisfies the coditios i Corollary 42 Costruct its orbit chai by computig the trasitio probabilities usig (41), ad compute the statioary distributio usig (42) Note that the etries of P H are multiples of the trasitio probabilities o the origial graph 3 Solve the fastest mixig reversible Markov chai problem (44) The optimal SLEM μ(ph ) is also the optimal SLEM for the origial chai, ad the optimal trasitio probabilities o the origial chai ca be obtaied by simple scalig of the optimal orbit trasitio probabilities We have assumed a sigle orbit chai that cotais all distict eigevalues of the origial chai Sometimes it is more coveiet to use multiple orbit chais Let P Hi, i =1,,K, be the collectio of orbit chais i Theorem 41 I this case we eed to miimize max i μ(p Hi ) This ca be doe by simply addig the set of costraits i (44) for every matrix P Hi Remark The mai challege of implemetig the above procedure is the idetificatio of automorphism groups ad costructio of the orbit chais Discussios o efficiet algorithms or software that ca automate these computatioal tasks are beyod the scope of this paper We will give further remarks i our coclusios i sectio 6 43 Example: K -K We demostrate the above computatioal procedure o the graph K -K This graph cosists of two copies of the complete graph K joied by a bridge (see Figure 41(a)) We follow the three steps described i sectio 42 First, it is clear by ispectio that the full automorphism group of K -K is C 2 (S 1 S 1 ) The actios of S 1 S 1 are all possible permutatios of the two sets of 1 vertices, distict from the two ceter vertices x ad y, amog themselves The group C 2 acts o the graph by switchig the two halves The semidirect product symbol meas that the actios of S 1 S 1 ad C 2 do ot commute By symmetry aalysis i sectio 2, there are three edge orbits uder the full automorphism group: the bridgig edge betwee vertices x ad y, the edges coectig x ad y to all other vertices, ad the edges coectig all other vertices Thus it suffices to cosider just three trasitio probabilities p 0, p 1,adp 2, each labeled i Figure 41(a) o oe represetative of the three edge orbits I the secod step, we costruct the orbit chais The orbit chai of K -K uder the full automorphism group is depicted i Figure 41(b) The orbit O x icludes vertices x ad y, adtheorbito z cosists of all other 2( 1) vertices The

15 806 S BOYD, P DIACONIS, P PARRILO, AND L XIAO p 2 z p 1 x p 0 y O z ( 1)p 1 p 1 O x u v (a) The graph K -K (b) Orbit chai uder C 2 (S 1 S 1 ) z O u ( 1)p 1 p 0 ( 1)p 1 x y p 1 p 1 O v ( 2)p 2 O u p 1 p p 1 2 p p 0 1 x ( 2)p 1 ( 1)p 1 y O v (c) Orbit chai uder S 1 S 1 (d) Orbit chai uder S 2 S 1 Fig 41 The graph K -K ad its orbit chais uder differet automorphism groups Here O x,o z,o u,o v represet orbits of the vertices x, z, u, v labeled i Figure 41(a), respectively, uder the correspodig automorphism groups i each subgraph trasitio probabilities of this orbit chai are calculated from (41) ad are labeled o the directed edges i Figure 41(b) Similarly, the orbit chai uder the subgroup S 1 S 1 is depicted i Figure 41(c) While these two orbit chais are the most obvious to costruct, oe of them cotais all eigevalues of the origial chai, or does their combiatio For the oe i Figure 41(b), the full automorphism group does ot have a fixed poit i either its orbit O x or O z For the oe i Figure 41(c), the automorphism group S 1 S 1 has a fixed poit i O x (either x or y), but does ot have a fixed poit i O z (ote here that O z is the orbit of z uder the full automorphism group) To fix the problem, we cosider the orbit chai uder the group S 2 S 1, which leaves the vertices x, y, adz fixed, while permutig the remaiig 2 vertices o the left ad the 1 poits o the right, respectively The correspodig orbit chai is show i Figure 41(d) By Corollary 42, all distict eigevalues of the origial Markov chai o K -K appear as eigevalues of this orbit chai Thus there are at most five distict eigevalues i the origial chai o matter how large is To fiish the secod step, we calculate the trasitio probabilities of the orbit chai uder H = S 2 S 1 usig (41) ad label them i Figure 41(d) If we order the vertices of this orbit chai as (x, y, z, O u,o v ), the the trasitio probability matrix is P H = 1 p 0 ( 1)p 1 p 0 p 1 ( 2)p 1 0 p 0 1 p 0 ( 1)p ( 1)p 1 p p 1 ( 2)p 2 ( 2)p 2 0 p 1 0 p 2 1 p 1 p p p 1 By (42), the statioary distributio of the orbit chai is ( ) 1 π H = 2, 1 2, 1 2, 2 2, 1 2

16 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 807 As the third step, we solve the SDP (44) with the above parametrizatio Here we oly eed to solve a SDP with 4 variables (three trasitio probabilities p 0, p 1, p 2, ad the extra scalar s) ad5 5 matrices o matter how large the graph () is We will revisit this example i sectio 544, where we preset a aalytic expressio for the exact optimal SLEM ad correspodig trasitio probabilities 5 Symmetry reductio by block-diagoalizatio By defiitio of the fixed-poit subset F i (21), ay trasitio probability matrix P Fis ivariat uder the actios of Aut(G) More specifically, for ay permutatio matrix Q give by σ Aut(G), we have QP Q T = P,equivaletlyQP = PQ I this sectio we show that this property allows the costructio of a coordiate trasformatio matrix that ca block-diagoalize every P F The resultig blocks usually have much smaller sizes, ad repeated blocks ca be discarded i computatio The method we use i this sectio is based o classical group represetatio theory (see, eg, [54]) It was developed for more geeral SDPs i [26] ad has foud applicatios i sum-of-squares decompositio for miimizig polyomial fuctios [48, 49, 50] ad cotroller desig for symmetric dyamical systems [13] A closely related approach is developed i [14], which is based o a low-order represetatio of the commutat (collectio of ivariat matrices) of the matrix algebra geerated by the permutatio matrices 51 Some group represetatio theory Let G be a group A represetatio ρ of G assigs a ivertible matrix ρ(g) toeachg G i such a way that the matrix assiged to the product of two elemets i G is the product of the matrices assiged to each elemet: ρ(gh) = ρ(g)ρ(h) The matrices we work with are all ivertible ad are cosidered over the real or complex umbers We thus regard ρ as a homomorphism from g to the liear maps o a vector space V The dimesio of ρ is the dimesio of V Two represetatios are equivalet if they are related by a fixed similarity trasformatio If W is a subspace of V ivariat uder G, theρ restricted to W gives a subrepresetatio Of course the zero subspace ad the subspace W = V are trivial subrepresetatios If the represetatio ρ admits o otrivial subrepresetatio, the ρ is called irreducible We cosider first complex represetatios, as the theory is cosiderably simpler i this case For a fiite group G there are oly fiitely may iequivalet irreducible represetatios ϑ 1,,ϑ h of dimesios 1,, h, respectively The degrees i divide the group order G ad satisfy the coditio h i=1 2 i = G Every liear represetatio of G has a caoical decompositio as a direct sum of irreducible represetatios ρ = m 1 ϑ 1 m 2 ϑ 2 m h ϑ h, where m 1,,m h are the multiplicities Accordigly, the represetatio space C has a isotypic decompositio (51) C = V 1 V h, where each isotypic compoets cosists of m i ivariat subspaces (52) V i = V 1 i V mi i, each of which has dimesio i ad trasforms after the maer of ϑ i A basis of this decompositio trasformig with respect to the matrices ϑ i (g) is called symmetryadapted ad ca be computed usig the algorithm preseted i [54, sectios 26 27]

17 808 S BOYD, P DIACONIS, P PARRILO, AND L XIAO or [22, sectio 52] This basis defies a chage of coordiates by a matrix T collectig the basis as colums By Schur s lemma (see, eg, [54]), if a matrix P satisfies (53) ρ(g)p = Pρ(g) g G, the T 1 PT has block-diagoal form with oe block P i for each isotypic compoet of dimesio m i i, which further decomposes ito i equal blocks B i of dimesio m i That is, (54) T 1 PT = P 1 0, P i = B i 0 0 P h 0 B i For our applicatio of semidefiite programs, the problems are preseted i terms of real matrices, ad therefore we would like to use real coordiate trasformatios I fact a geeralizatio of the classical theory to the real case is preseted i [54, sectio 132] If all ϑ i (g) are real matrices, the irreducible represetatio is called absolutely irreducible Otherwise, for each ϑ i with complex character its complex cojugate will also appear i the caoical decompositio Sice ρ is real, both will have the same multiplicity, ad real bases of V i + V i ca be costructed So two complex cojugate irreducible represetatios form oe real irreducible represetatio of complex type There is a third case, real irreducible represetatios of quaterio type, rarely see i practical examples I this paper, we assume that the represetatio ρ is orthogoal, ie, ρ(g) T ρ(g) = ρ(g)ρ(g) T = I for all g G As a result, the trasformatio matrix T ca also be chose to be orthogoal Thus T 1 = T T (for complex matrices, it is the cojugate traspose) For symmetric matrices the block correspodig to a represetatio of complex type or quaterio type simplifies to a collectio of equal subblocks For the special case of circulat matrices, complete diagoalizatio reveals all the eigevalues [15, p 50] 52 Block-diagoalizatio of SDP costrait As i sectio 22, for every σ Aut(G) we assig a permutatio matrix Q(σ) by lettig Q ij (σ) =1ifi = σ(j) ad Q ij (σ) =0otherwiseThisisa-dimesioal represetatio of Aut(G), which is ofte called the atural represetatio As metioed i the begiig of this sectio, every matrix P i the fixed-poit subset F has the symmetry of Aut(G); ie, it satisfies the coditio (53) with ρ = Q Thus a coordiate trasformatio matrix T ca be costructed such that P ca be block-diagoalized ito the form (54) Now we cosider the FMMC problem (23), which ca be formulated as the followig SDP: (55) miimize subject to s N si I p k L k (1/)11 T si, k=1 p k 0, k =1,,N, N (L k ) ii p k 1, i =1,, k=1 Here we have expressed the trasitio probability matrix as P (p) =I N k=1 p kl k, where L k is the Laplacia matrix for the kth orbit graph, ad p k is the commo

18 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES 809 trasitio probability assiged o all edges i the kth orbit graph Sice the matrix P (p) has the symmetry of Aut(G), we ca fid a coordiate trasformatio T to blockdiagoalize the liear matrix iequalities Thus we obtai the followig equivalet problem: (56) miimize s subject to si mi B i (p) J i si mi, i =1,,h, p k 0, k =1,,N, N (L k ) ii p k 1, i =1,,, k=1 where B i (p) correspod to the small blocks B i i (54) of the trasformed matrix T T P (p)t,adj i are the correspodig diagoal blocks of T T (1/)11 T T Theumber of matrix iequalities h is the umber of iequivalet irreducible represetatios, ad the size of each matrix iequality m i is the multiplicity of the correspodig irreducible represetatio Note that we oly eed oe out of i copies of each B i i the decompositio (54) Sice m i cabemuchsmallertha (the umber of vertices i the graph), the improvemet i computatioal complexity over the SDP formulatio (55) ca be sigificat (see the flop couts discussed i sectio 12) This is especially the case whe there are high-dimesioal irreducible represetatios (ie, whe i is large; see, eg, K -K defied i sectio 43) 53 Coectio betwee block-diagoalizatio ad orbit theory With the followig theorem, we establish a iterestig coectio betwee the blockdiagoalizatio approach ad the orbit theory i sectio 4 Theorem 51 Let H be a subgroup of Aut(G), adlett be the coordiate trasformatio matrix whose colums are a symmetry-adapted basis for the atural represetatio of H Suppose a Markov chai P defied o the graph has the symmetry of H The the matrix T T (1/)11 T T has the same block-diagoal form as T T PT Moreover, there is oly oe ozero block Without loss of geerality, let this ozero block be J 1 ad the correspodig block of T T PT be B 1 These two blocks relate to the orbit chai P H by (57) (58) B 1 =Π 1/2 P H Π 1/2, J 1 = qq T, where Π=Diag(π H ), q = π H,adπ H is the statioary distributio of P H Proof First we ote that P always has a sigle eigevalue 1 with associated eigevector 1 Thus1 spas a ivariat subspace of the atural represetatio, which is obviously irreducible The correspodig irreducible represetatio is isomorphic to the trivial represetatio (which assigs the scalar 1 to every elemet i the group) Without loss of geerality, let V 1 be the isotypic compoet that cotais the vector 1 Thus V 1 is a direct product of H-fixed vectors (each correspods to a copy of the trivial represetatio), ad 1 is a liear combiatio of these vectors Let m 1 be the dimesio of V 1, which is the umber of H-fixed vectors We ca calculate m 1 by Frobeius reciprocity, or Burside s lemma ; see, eg, [54] To do so, we ote that the character χ of the atural represetatio Q(g), g H, isthe umber of fixed poits of g, ie, χ(g) =Tr Q(g) =FP(g) =#{v V: g(v) =v}

19 810 S BOYD, P DIACONIS, P PARRILO, AND L XIAO Burside s lemma says that 1 FP(g) =#orbits H g H The left-had side is the ier product of χ with the trivial represetatio It thus couts the umber of H-fixed vectors i V So m 1 equals the umber of orbits uder H Suppose that V = O 1 O m1 as a disjoit uio of H-orbits Let b i (v) = 1/ O i if v O i ad zero otherwise The b 1,,b m1 are H-fixed vectors, ad they form a orthoormal symmetry-adapted basis for V 1 (these are ot uique) Let T 1 =[b 1 b m1 ] be the first m 1 colums of T They are orthogoal to all other colums of T Sice 1 is a liear combiatio of b 1,,b m1, it is also orthogoal to other colums of T Therefore the matrix T T (1/)11 T T has all its elemets zero except for the first m 1 m 1 diagoal block, which we deote as J 1 More specifically, J 1 = qq T,where q = 1 T T 1 1 = 1 [ b T 1 1 b T m 1 1 ] T [ = 1 O 1 O1 ] T [ O m1 O1 = Om1 ] T Om1 Note that by (42) the statioary distributio of the orbit chai P H is [ O1 π H = O m1 ] T Thus we have q = π H This proves (58) Fially we cosider the relatioship betwee the two matrices B 1 = T T 1 PT 1 ad P H We prove (57) by showig It is straightforward to verify that Π 1/2 T1 T = b T 1 Π 1/2 B 1 Π 1/2 =Π 1/2 T T 1 PT 1 Π 1/2 = P H b T m 1, b i (v) = 1 if v O i, O i 0 if v/ O i, { T 1 Π 1/2 = 1 [ b 1 b ] m 1, b 1 if v Oi, i (v) = 0 if v/ O i The etry at the ith row ad jth colum of the matrix Π 1/2 T T 1 PT 1Π 1/2 is give by b T i Pb j = 1 O i P (v, u) = 1 P H (O i,o j )=P H (O i,o j ) O i u O j v O i v O i I the last equatio, we have used the fact that P H (O i,o j ) is idepedet of which v O i is chose This completes the proof

20 FASTEST MIXING MARKOV CHAIN ON GRAPHS WITH SYMMETRIES a O 2 b O 1 O 5 2a 4b Fig 51 Left: the 3 3 grid Right: its orbit chai uder D 4 From Theorem 51, we kow that B 1 cotais the eigevalues of the orbit chai uder H OtherblocksB i cotai additioal eigevalues (ot icludig those of P H ) of the orbit chais uder various subgroups of H (Note that the eigevalues of the orbit chai uder H are always cotaied i the orbit chai uder its subgroups) With this observatio, it is possible to idetify the multiplicities of eigevalues i orbit chais uder various subgroups of Aut(G) by relatig to the decompositios (51), (52), ad (54) (some prelimiary results are discussed i [4]) 54 Examples We preset several examples that use the block-diagoalizatio method ad draw coectios to the method based o orbit theory i sectio 4 Some of the examples may be difficult if oe uses the orbit theory aloe, but are icely hadled by block-diagoalizatio 541 The 3 3grid Cosider the symmetric Markov chai o a 3 3 gridg; see Figure 51(left) The automorphism group Aut(G) is isomorphic to the 8-elemet dihedral group D 4, ad correspods to flips ad 90-degree rotatios of the graph The orbits of Aut(G) actig o the vertices are ad there are two orbits of edges {1, 3, 7, 9}, {5}, {2, 4, 6, 8}, {{1, 2}, {1, 4}, {2, 3}, {3, 6}, {4, 7}, {7, 8}, {6, 9}, {8, 9}}, {{2, 5}, {4, 5}, {5, 6}, {5, 8}} So G is either vertex- or edge-trasitive By Corollary 22, we associate trasitio probabilities a ad b to the two edge orbits, respectively The trasitio probability matrix has the form P = 1 2a a 0 a a 1 2a b a 0 b a 1 2a 0 0 a a a b b 0 a b 0 b 1 4b b 0 b a 0 b 1 2a b 0 0 a a a a b 0 a 1 2a b a a 0 a 1 2a The matrix P satisfies Q(σ)P = PQ(σ) for every σ Aut(G) Usig the algorithm i [22, sectio 52], we foud a symmetry-adapted basis for the represetatio Q, which

21 812 S BOYD, P DIACONIS, P PARRILO, AND L XIAO we take as colums to form T = With this coordiate trasformatio matrix, we obtai 1 4b 0 2b 0 1 2a 2a 2b 2a 1 2a b 1 2a T T PT = 1 2a b 1 2a 2a 2a 1 2a b 1 2a 2a 2a 1 2a b The three-dimesioal block B 1 cotais the eigevalue 1, ad it is related to the orbit chai i Figure 51 (right) by (57) The correspodig ozero block of T T (1/)11 T T is J 1 = Next, we substitute the above expressios ito the SDP (56) ad solve it umerically Sice there are repeated 2 2blocks,theorigial9 9 matrix is replaced by four smaller blocks of dimesio 3, 1, 1, 2 The optimal solutios are a 0363, b 02111, μ Iterestigly, it ca be show that these optimal values are ot ratioal, but istead algebraic umbers with defiig miimal polyomials: a a a a a 16 = 0, b b b b b 256 = 0, μ μ μ μ μ = Complete k-partite graphs The complete k-partite graph, deoted K 1,, k,hask subsets of vertices with cardialities 1,, k, respectively Each vertex is coected to all the vertices i a differet subset, ad is ot coected to ay of the vertices i the same subset I this case, the trasitio probability matrix has dimesios i i ad the structure (1 j 1 jp 1j )I 1 p p 1k 1 1 k p (1 j 2 jp 2j )I 2 p 2k 1 2 k P (p) = p k1 1 k 1 p k2 1 k 2 (1, j k jp kj )I k