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Kernels of directed graph Laplacians

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Kernels of directed graph Laplacians
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  Kernels of Directed Graph Laplacians J.S. Caughman and J.J.P. Veerman Department of Mathematics and StatisticsPortland State UniversityPO Box 751, Portland, OR 97207.caughman@pdx.edu, veerman@pdx.edu Submitted: Oct 28, 2005; Accepted: Mar 14, 2006; Published: Apr 11, 2006Mathematics Subject Classification: 05C50 Abstract. Let G denote a directed graph with adjacency matrix Q and in-degree matrix D . We consider the Kirchhoff matrix  L = D − Q , sometimesreferred to as the directed Laplacian  . A classical result of Kirchhoff asserts thatwhen G is undirected, the multiplicity of the eigenvalue 0 equals the numberof connected components of  G . This fact has a meaningful generalization todirected graphs, as was observed by Chebotarev and Agaev in 2005. Sincethis result has many important applications in the sciences, we offer an inde-pendent and self-contained proof of their theorem, showing in this paper thatthe algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace – namely, thenumber of reaches of the directed graph. We also extend their results by deriv-ing a natural basis for the corresponding eigenspace. The results are proved inthe general context of stochastic matrices, and apply equally well to directedgraphs with non-negative edge weights. Keywords: Kirchhoff matrix. Eigenvalues of Laplacians. Graphs. Stochastic matrix. 1 Definitions Let G denote a directed graph with vertex set V  = { 1 , 2 ,...,N  } and edge set E  ⊆ V  × V  . To each edge uv ∈ E  , we allow a positive weight ω uv to be assigned. The adjacency matrix  Q is the N  × N  matrix whose rows and columns are indexed by the vertices, andwhere the ij -entry is ω  ji if  ji ∈ E  and zero otherwise. The in-degree matrix  D is the N  × N  diagonal matrix whose ii -entry is the sum of the entries of the i th row of  Q . Thematrix L = D − Q is sometimes referred to as the Kirchhoff matrix  , and sometimes asthe directed graph Laplacian  of  G .A variation on this matrix can be defined as follows. Let D + denote the pseudo-inverseof  D . In other words, let D + be the diagonal matrix whose ii -entry is D − 1 ii if  D ii  = 0 the electronic journal of combinatorics 13 (2006), #R39 1  and whose ii -entry is zero if  D ii = 0. Then the matrix L = D + ( D − Q ) has nonnegativediagonal entries, nonpositive off-diagonal entries, all entries between -1 and 1 (inclusive)and all row sums equal to zero. Furthermore, the matrix S  = I  −L is stochastic.We shall see (in Section 4) that both L and L can be written in the form D − DS  where D is an appropriately chosen nonnegative diagonal matrix and S  is stochastic. Wetherefore turn our attention to the properties of these matrices for the statement of ourmain results.We show that for any such matrix M  = D − DS  , the geometric and algebraic multi-plicities of the eigenvalue zero are equal, and we find a basis for this eigenspace (the kernel  of  M  ). Furthermore, the dimension of this kernel and the form of these eigenvectors canbe described in graph theoretic terms as follows.We associate with the matrix M  a directed graph G , and write j  i if there existsa directed path from vertex j to vertex i . For any vertex j , we define the reachable set  R (  j ) to be the set containing j and all vertices i such that j  i . A maximal reachableset will be called a reach  . We prove that the algebraic and geometric multiplicity of 0 asan eigenvalue for M  equals the number of reaches of  G .We also describe a basis for the kernel of  M  as follows. Let R 1 ,... R k denote the reachesof  G . For each reach R i , we define the exclusive part  of  R i to be the set H  i = R i \∪  j  = i R  j .Likewise, we define the common part  of  R i to be the set C  i = R i \ H  i . Then for eachreach R i there exists a vector v i in the kernel of  M  whose entries satisfy: (i) ( v i )  j = 1 forall j ∈ H  i ; (ii) 0 < ( v i )  j < 1 for all j ∈ C  i ; (iii) ( v i )  j = 0 for all j ∈ R i . Taken together,these vectors v 1 , v 2 ,..., v k form a basis for the kernel of  M  and sum to the all 1’s vector 1 .Due to the recent appearance of Agaev and Chebotarev’s notable paper [1], we wouldlike to clarify the connections to their results. In that paper, the matrices studied havethe form M  = α ( I  − S  ) where α is positive and S  stochastic. A simple check verifiesthat this is precisely the set of matrices of the form D − DS  , where D is nonnegativediagonal. The number of reaches corresponds, in that paper, with the in-forest dimension  .And where that paper concentrates on the location of the Laplacian eigenvalues in thecomplex plane, we instead have derived the form of the associated eigenvectors. 2 Stochastic matrices A matrix is said to be (row) stochastic if the entries are nonnegative and the row sumsall equal 1. Our first result is a special case of Gerˇsgorin’s theorem [3, p.344]. 2.1 Lemma. Suppose  S  is stochastic. Then each eigenvalue  λ satisfies  | λ |≤ 1 . 2.2 Definition. Given any real N  × N  matrix M  , we denote by G M  the directedgraph with vertices 1 ,...,N  and an edge j → i whenever M  ij  = 0 . For each vertex i , set  N  i := {  j |  j → i } . We write j  i if there exists a directed path in G M  from vertex j tovertex i . Furthermore, for any vertex j , we define R (  j ) to be the set containing j and allvertices i such that j  i . We refer to R (  j ) as the reachable set  of vertex j . Finally, wesay a matrix M  is rooted  if there exists a vertex r in G M  such that R ( r ) contains everyvertex of  G M  . We refer to such a vertex r as a root  . the electronic journal of combinatorics 13 (2006), #R39 2  2.3 Lemma. Suppose  S  is stochastic and rooted. Then the eigenspace  E  1 associatedwith the eigenvalue 1 is spanned by the all-ones vector  1 . Proof. Conjugating S  by an appropriate permutation matrix if necessary, we may assumethat vertex 1 is a root. Since S  is stochastic, S  1 = 1 so 1 ∈ E  1 . By way of contradiction,suppose dim( E  1 ) > 1 and choose linearly independent vectors x,y ∈ E  1 . Suppose | x i | ismaximized at i = n . Comparing the n -entry on each side of the equation x = Sx , we seethat | x n | ≤   j ∈N  n S  nj | x  j | ≤ | x n |   j ∈N  n S  nj = | x n | . Therefore, equality holds throughout, and | x  j | = | x n | for all j ∈ N  n . In fact, since   j ∈N  n S  nj x  j = x n , it follows that x  j = x n for all j ∈ N  n . Since S  is rooted at vertex1, a simple induction now shows that x 1 = x n . So | x i | is maximized at i = 1. The sameargument applies to any vector in E  1 and so | y i | is maximized at i = 1.Since y 1  = 0 we can define a vector z such that z i := x i − x 1 y 1 y i for each i . Thisvector z , as a linear combination of  x and y , must belong to E  1 . It follows that | z i | is alsomaximized at i = 1. But z 1 = 0 by definition, so z i = 0 for all i . It follows that x and y are not linearly independent, a contradiction.  2.4 Lemma. Suppose  S  is stochastic  N  × N  and vertex 1 is a root. Further assume  N  1 is empty. Let P  denote the principal submatrix obtained by deleting the first row andcolumn of  S  . Then the spectral radius of  P  is strictly less than 1. Proof. Since N  1 is empty, S  is block lower-triangular with P  as a diagonal block. Sothe spectral radius of  P  cannot exceed that of  S  . Therefore, by Lemma 2.1, the spectralradius of  P  is at most 1. By way of contradiction, suppose the spectral radius of  P  is equalto 1. Then by the Perron-Frobenius theorem (see [3, p. 508]), we would have Px = x forsome nonzero vector x .Define a vector v with v 1 = 0 and v i = x i − 1 for i ∈ { 2 ,...,N  } . We find that Sv =  1 0 ··· 0 S  21 ... S  N  1 P   0 x  =  0 x  = v. So v ∈ E  1 . But v 1 = 0, so Lemma 2.3 implies x = 0. This contradiction completes theproof.  2.5 Corollary. Suppose  S  is stochastic and N  × N  . Assume the vertices of  G S  canbe partitioned into nonempty sets  A , B such that for every  b ∈ B , there exists  a ∈ A with a  b in G S  . Then the spectral radius of the principal submatrix  S  BB obtained by deleting from S  the rows and columns of  A is strictly less than 1. Proof. Define the matrixˆ S  byˆ S  =  1 0u S  BB  , the electronic journal of combinatorics 13 (2006), #R39 3  where u is chosen so thatˆ S  is stochastic. We claim thatˆ S  is rooted (at 1). To see this,pick any b ∈ B . We must show 1  b in G ˆ S  . By hypothesis there exists a ∈ A with a  b in G S  . Let a = x 0 → x 1 →···→ x n = b be a directed path in G S  from a to b . Let i be maximal such that x i ∈ A . Then the x i +1 ,x i entry of  S  is nonzero, so the x i +1 row of  S  BB has row sum strictly less than 1.Therefore, the x i +1 entry of the first column of ˆ S  is nonzero. So 1 → x i +1 in G ˆ S  andtherefore 1  b in G ˆ S  as desired. Soˆ S  is rooted, and the previous lemma gives the result.  2.6 Definition. A set R of vertices in a graph will be called a reach  if it is a maximalreachable set; in other words, R is a reach  if  R = R ( i ) for some i and there is no j suchthat R ( i ) ⊂ R (  j ) (properly). Since our graphs all have finite vertex sets, such maximalsets exist and are uniquely determined by the graph. For each reach R i of a graph, wedefine the exclusive part  of  R i to be the set H  i = R i \∪  j  = i R  j . Likewise, we define the common part  of  R i to be the set C  i = R i \ H  i . 2.7 Theorem. Suppose  S  is stochastic  N  × N  and let R denote a reach of  G S  withexclusive part H  and common part C  . Then there exists an eigenvector  v ∈ E  1 whose entries satisfy (i) v i = 1 for all  i ∈ H  ,(ii) 0 < v i < 1 for all  i ∈ C  ,(iii) 0 for all  i ∈ R . Proof. Let Y  denote the set of vertices not in R . Permuting rows and columns of  S  if necessary, we may write S  as S  =  S  HH  S  HC  S  HY  S  CH  S  CC  S  CY  S  YH  S  YC  S  YY   =  S  HH  0 0 S  CH  S  CC  S  CY  0 0 S  YY   Since S  HH  is a rooted stochastic matrix, it has eigenvalue 1 with geometric multiplicity1. The associated eigenvector is 1 H  .Observe that S  CC  has spectral radius < 1 by Corollary 2.5. Further, notice that S  ( 1 H  , 0 C  , 0 Y  ) T  = ( 1 H  ,S  CH  1 H  , 0 Y  ) . T  Using this, we find that solving the equation S  ( 1 H  , x , 0 C  ) T  = ( 1 H  , x , 0 C  ) T  for x amounts to solving  1 H  S  CH  1 H  + S  CC  x0 Y   =  1 H  x0 Y   . the electronic journal of combinatorics 13 (2006), #R39 4  Solving the above, however, is equivalent to solving ( I  − S  CC  ) x = S  CH  1 H  . Since thespectral radius of  S  CC  is strictly less than 1, the eigenvalues of  I  − S  CC  cannot be 0. So I  − S  CC  is invertible. It follows that x = ( I  − S  CC  ) − 1 S  CH  1 H  is the desired solution.Conditions (i) and (iii) are clearly satisfied by ( 1 H  , x , 0 Y  ) , T  so it remains only to verify(ii). To see that the entries of  x are positive, note that ( I  − S  CC  ) − 1 =  ∞ i =0 S  iCC  , so theentries of  x are nonnegative and strictly less than 1. But every vertex in C  has a pathfrom the root, where the eigenvector has value 1. So since each entry in the eigenvectorfor S  must equal the average of the entries corresponding to its neighbors in G S  , all entriesin C  must be positive.  3 Matrices of the form D − DS  We now consider matrices of the form D − DS  where D is a nonnegative diagonalmatrix and S  is stochastic. We will determine the algebraic multiplicity of the zeroeigenvalue. We begin with the rooted case. 3.1 Lemma. Suppose  M  = D − DS  , where  D is a nonnegative diagonal matrix and S  is stochastic. Suppose  M  is rooted. Then the eigenvalue  0 has algebraic multiplicity  1 . Proof. Let M  = D − DS  be given as stated. First we claim that, without loss of generality, S  ii = 1 whenever D ii = 0. To see this, suppose D ii = 0 for some i . If  S  ii  = 1,let S   be the stochastic matrix obtained by replacing the i th row of  S  by the i th row of the identity matrix I  , and let M   = D − DS   . Observe that M  = M   , and this proves ourclaim. So we henceforth assume that S  ii = 1 whenever D ii = 0 . (1)Next we claim that, given (1), ker( M  ) must be identical with ker( I  − S  ). To see this, notethat if ( I  − S  ) v = 0 then clearly Mv = D ( I  − S  ) v = 0. Conversely, suppose Mv = 0.Then D ( I  − S  ) v = 0 so the vector w = ( I  − S  ) v is in the kernel of  D . If  w has a nonzeroentry w i then D ii = 0. Recall this implies S  ii = 1 and the i th row of  I  − S  is zero. But w = ( I  − S  ) v , so w i must be zero. This contradiction implies w must have no nonzeroentries, and therefore ( I  − S  ) v = 0. So M  and I  − S  have identical nullspaces as desired.By Lemma 2.3, S  1 = 1 , so M  1 = 0. Therefore the geometric multiplicity, and hencethe algebraic multiplicity, of the eigenvalue 0 must be at least 1. By way of contradiction,suppose the algebraic multiplicity is greater than 1. Then there must be a nonzero vector x and an integer d ≥ 2 such that M  d − 1 x  = 0 and M  d x = 0 . Now, since ker M  = ker( I  − S  ), Lemma 2.3 and the above equation imply that M  d − 1 x must be a multiple of the vector 1 . Scaling M  d − 1 x appropriately, we find there exists avector v such that Mv = − 1 . the electronic journal of combinatorics 13 (2006), #R39 5
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