The Hierarchical Product of Graphs

Transcription

1 The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 March 22, 2007

2 Outline 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

3 Introduction 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

4 Introduction Motivation Complex networks: randomness, heterogeneity, modularity M.E.J. Newman. The structure and function of complex networks. SIAM Rev. 45 (2003) Hierarchical networks: degree distribution, modularity S. Jung, S. Kim, B. Kahng. Geometric fractal growth model for scale-free networks. Phys. Rev. E 65 (2002) E. Ravasz, A.-L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E 67 (2003) E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-L. Barabási, Hierarchical organization of modularity in metabolic networks, Science 297 (2002)

5 Introduction Our work Deterministic graphs Algebraic methods

6 Introduction Our work Deterministic graphs Algebraic methods Far from real networks but a beautiful mathematical object!!!

7 Introduction Previous work N. Biggs. Algebraic Graph Theory. Cambridge UP, Cambridge, D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs. Theory and Applications, Academic Press, New York, M.A. Fiol, M. Mitjana. The local spectra of regular line graphs. Discrete Math., submitted. C. D. Godsil. Algebraic Combinatorics. Chapman and Hall, New York, A.J. Schwenk, Computing the characteristic polynomial of a graph, Lect. Notes Math. 406 (1974) J. R. Silvester, Determinants of block matrices, Maths Gazette 84 (2000)

8 The hierarchical product 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

9 The hierarchical product Definition and basic properties Definition For i = 1,... N, G i graph rooted at 0 H = G N G 2 G 1 vertices x N... x 3 x 2 x 1, x i V i if x j y j in G j then x N... x j+1 x j x N... x j+1 y j Example The hierarchical products K 2 K 3 and K 3 K 2

10 The hierarchical product Definition and basic properties G N G 2 G 1 is a spanning subgraph of G N G 2 G 1 Example The hierarchical product P 4 P 3 P 2

11 The hierarchical product Definition and basic properties G N = G N G is the hierachical N-power of G Example The hierarchical powers K 2 2, K 4 2 and K 5 2

12 The hierarchical product Definition and basic properties Example The hierarchical power C 3 4

13 The hierarchical product Definition and basic properties Order and size n i = V i and m i = E i H = G N G 2 G 1 n H = n N n 3 n 2 n 1 N m H = m k n k+1... n N k=1 Properties of Associativity. G 3 G 2 G 1 = G 3 (G 2 G 1 ) = (G 3 G 2 ) G 1 Right-distributivity. (G 3 G 2 ) G 1 = (G 3 G 1 ) (G 2 G 1 ) Left-semi-distributivity. G 3 (G 2 G 1 ) = (G 3 G 2 ) n 3 G 1, where n 3 G 1 = K n3 G 1 is n 3 copies of G 1 G K 1 = K 1 G = G (G, ) is a monoid

14 The hierarchical product Vertex hierarchy Degrees If δ i = δ Gi (0), then N δ G (0) = i=1 δ i x = x N x N 1... x k , x k 0 k 1 δ H (x) = δ i + δ Gk (x k ) i=1 If G is δ-regular, the degrees of the vertices of G N follow an exponential distribution, P(k) = γ k, for some constant γ For k = 1,..., N 1, G N contains (n 1)n N k vertices with degree kδ and n vertices with degree Nδ

15 The hierarchical product Vertex hierarchy Example T m = K2 m has 2m k vertices of degree k = 1,..., m 1, and two vertices of degree m

16 The hierarchical product Vertex hierarchy Modularity H = G N G 2 G 1, z an appropriate string H zx k... x 1 = H[{zx k... x 1 x i V i, 1 i k}] H x N... x k z = H[{x N... x k z x i V i, k i N}] Lemma H zx k... x 1 = G k G 1, for any fixed z H x N... x k 0 = G N G k H x N... x k z = (n N n k )K 1, for any fixed z 0 H = H 0 Lemma (G N G 2 G 1 ) = N k=1 (G k G k 1 G 1 ) (K2 N) = N 1 k=0 K 2 k K2 N {{0, 10}} = K N 1 2 K N 1 2

17 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

18 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

19 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

20 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

21 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

22 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

23 The hierarchical product Metric parameters Eccentricity, radius and diameter H = G N G 2 G 1 ε i = ecc Gi (0), r GN = r N and D GN = D N ρ i shortest path routing of G i, i = 1,..., N Proposition {ρ i } i=1...n induce a shortest path routing ρ in H The eccentricity, radius and diameter of H are N N 1 ecc H (0) = ε i, r H = r N + ε i, D H = D N + 2 i=1 i=1 N 1 i=1 ε i

24 The hierarchical product Metric parameters Proof.

25 The hierarchical product Metric parameters Proof.

26 The hierarchical product Metric parameters Mean distance G graph of order n 1 Mean distance. d G = dist G (v, w) n(n 1) v w V Local mean distance. dg 0 = 1 dist G (0, v) n v V Proposition { d 00 H = G 2 G 1 H = d d 2 0 d H = 1 [ n 1 (n1 1)d 1 + n 1 (n 2 1)(d 2 + 2d1 0)]

27 The hierarchical product Metric parameters Mean distance G graph of order n 1 Mean distance. d G = dist G (v, w) n(n 1) v w V Local mean distance. dg 0 = 1 dist G (0, v) n v V Proposition { d 00 H = G 2 G 1 H = d d 2 0 d H = 1 [ n 1 (n1 1)d 1 + n 1 (n 2 1)(d 2 + 2d1 0)] Proof. Just compute!

28 The hierarchical product Metric parameters Corollary H = G N, d = d G, d 0 = d 0 G ecc N (0) = Nε, d 0 N = Nd 0 r N = r + (N 1)ε, D N = D + 2(N 1)ε ( ) (N 1)n N +1 1 n N 1 n 1 d 0 d N = d + 2 Asymptotically, d N N d + 2d 0 ( N n n 1 ) d + 2Nd 0 n Example G = K 2 ecc = r = D = 1, d 0 = 1/2 and d = 1 The metric parameters of T m = K2 m are ecc m (0) = m, d 0 m = m/2 r m = m, D m = 2m 1 d m = m2m 2 m 1 1 m 1

29 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

30 Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar

31 Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar Lemma H = G 2 G 1 A H = A 2 D 1 + I 2 A 1 = D1 A 2 + A 1 I 2 where D 1 = diag(1, 0,... 0)

32 Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar Lemma H = G 2 G 1 A H = A 2 D 1 + I 2 A 1 = D1 A 2 + A 1 I 2 where D 1 = diag(1, 0,... 0) Example H = G K n, G of order N A H = D 1 A G + A Kn I N = A G I N I N I N 0 I N... I N I N 0

33 Theorem (Silvester, 2000) R commutative subring of F n n, the set of all n n matrices over a field F (or a commutative ring), and M R m m. Then, det F M = det F (det R M) Corollary ( (Silvester, ) 2000) A B M = where A, B, C, D commute with each other. C D Then, det M = det(ad BC)

34 Spectral properties of G K m 2 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

35 Spectral properties of G K m 2 G K 2 Example The Petersen graph, hierarchically multiplied by K 2

36 Spectral properties of G K m 2 G K 2 G graph of order n, A adjacency matrix of G and φ G characteristic polynomial of G The adjacency matrix of H = G K 2 is ( ) A In A H = I n 0 The characteristic polynomial of H( is xin A I φ H (x) = det(xi 2n A H ) = det n I n xi n = det((x 2 1)I n xa) = x n φ G (x 1 x ) ) =

37 Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x ) Proposition H = G K 2 and spg = {λ m 0 0 < λ m 1 1 <... < λ m d sph = {λ m 0 00 < λm 1 01 <... < λm d 0d < λm 0 10 < λm 1 11 <... < λm d 1d } where λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 d

38 Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x ) Proposition H = G K 2 and spg = {λ m 0 0 < λ m 1 1 <... < λ m d sph = {λ m 0 00 < λm 1 01 <... < λm d 0d < λm 0 10 < λm 1 11 <... < λm d 1d } where λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 Proof. λ sph φ H (λ) = λ n φ G (λ 1 λ ) = 0 λ 1 λ spg λ i spg λ 2 λ i λ 1 = 0 d

39 Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x )

40 Spectral properties of G K m 2 H m = G K m 2 H m = H m 1 K 2, m 1. The adjacency matrix of H m is ( ) Am 1 I A m = m 1 I m 1 0 where I m denotes the identity matrix of size n2 m (the same as A m ) H 0 = G, A 0 = A the adjacency matrix of G Example

41 Spectral properties of G K m 2 Let {p i, q i } i 0 be the family of polynomials satisfying the recurrence equations p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 with initial conditions p 0 = x and q 0 = 1 Proposition For every m 0, the characteristic polynomial of H m = G K2 m ( ) φ m (x) = q m (x) n pm (x) φ 0 q m (x) is Lemma If p and q are arbitrary polynomials, then ( pin qa qi ) det n = det((p 2 q 2 )I qi n pi n pqa) n

42 Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma

43 Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x.

44 Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x. m 1. By induction on i, we prove that φ m = det(p i I m i q i A m i ) i = 0 : φ m = det(xi m A m ) = det(p 0 I m q 0 A m ) i 1 i : φ m = det(p i 1 I m i+1 q i 1 A m i+1 ) = = det((p 2 i 1 q2 i 1 )I m i p i 1 q i 1 A m i ) = = det(p i I m i q i A m i )

45 Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x. m 1. By induction on i, we prove that φ m = det(p i I m i q i A m i ) i = 0 : φ m = det(xi m A m ) = det(p 0 I m q 0 A m ) i 1 i : φ m = det(p i 1 I m i+1 q i 1 A m i+1 ) = = det((p 2 i 1 q2 i 1 )I m i p i 1 q i 1 A m i ) = = det(p i I m i q i A m i ) The case i = m gives ( φ m (x)( = det(p m (x)i)) 0 q m (x)a 0 ) = ( ) = det q m (x) pm(x) q I m(x) 0 A 0 = q m (x) n φ pm(x) 0 q m(x)

46 The spectrum of the binary hypertree T m = K m 2 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

47 The spectrum of the binary hypertree T m = K m 2 T m = K m 2 Corollary φ Tm (x) = p m (x) φ T m (x) = q m (x) p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 p 0 = x, q 0 = 1

48 The spectrum of the binary hypertree T m = K m 2 T m = K m 2 Corollary φ Tm (x) = p m (x) φ T m (x) = q m (x) p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 p 0 = x, q 0 = 1 Proof. ( ) G = K 1 φ 0 (x) = x φ Tm (x) = q m (x) n φ pm(x) 0 q m(x) = p m (x) Tm = T m 0 = m 1 i=0 T i φ T m (x) = m 1 i=0 p i(x) = q m (x)

49 The spectrum of the binary hypertree T m = K m 2 Proposition T m, m 1, has distinct eigenvalues λ m 0 < λm 1 < < λm n 1, with n = 2 m, satisfying the following recurrence relation: λ m λm 1 k + n +k = 2 λ m n k 1 = λm k for m > 1 and k = n 2, n 2 + 1,..., n 1 (λ m 1 k )

50 The spectrum of the binary hypertree T m = K m 2 Proposition T m, m 1, has distinct eigenvalues λ m 0 < λm 1 < < λm n 1, with n = 2 m, satisfying the following recurrence relation: λ m λm 1 k + n +k = 2 λ m n k 1 = λm k for m > 1 and k = n 2, n 2 + 1,..., n 1 Proof. (λ m 1 k ) λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 T m bipartite its spectrum is symmetric with respect to 0 spt 0 = {0 1 } the multiplicity of every λ m 1 is 1

51 The spectrum of the binary hypertree T m = K m 2 Properties of spt m λ i spg λ 2 λ i λ 1 = 0 f 0 (x) = x x f 1 (x) = x + x m = 0 spt 0 = {0} m = 1 λ 0 = f 0 (0) = 1, λ 1 = f 1 (0) = 1 m = 2 λ 0 = f 0 ( 1) = f 0 (f 0 (0)) = λ 1 = f 0 (1) = f 0 (f 1 (0)) = λ 2 = f 1 ( 1) = f 1 (f 0 (0)) = λ 3 = f 1 (1) = f 1 (f 1 (0)) = m fixed, i = i m 1... i 1 i 0 Z m 2 λ i = (f im 1 f i1 f i0 )(0)

52 The spectrum of the binary hypertree T m = K m 2 The distinct eigenvalues of the hypertree T m for 0 m 6.

53 The spectrum of the binary hypertree T m = K m 2 Proposition The asymptotic behaviors of the spectral radius ρ k = max 0 i n 1 { λ i } = λ , the second largest eigenvalue θ k = λ , and the minimum positive eigenvalue σ k = min 0 i n 1 { λ i } = λ of the hypertree T m are: ρ k 2k, θ k 2k, σ k 1/ 2k

54 The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k.

55 The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1

56 The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1 ρ k and θ k verify the recurrence λ k+1 = f 1 (λ k ) = 1 2 (λ k + λ 2 k + 4)

57 The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1 ρ k and θ k verify the recurrence λ k+1 = f 1 (λ k ) = 1 2 (λ k + λ 2 k + 4) Assuming λ k αk β α(k + 1) β αkβ + α 2 k 2β α 2 (k + 1) β [(k + 1) β k β ] 1 2(k + 1) (k + 1) [(k + 1) 2 k 2 ] = 1 (k + 1) k 1 2

58 The spectrum of a generic two-term product G 2 G 1 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

59 The spectrum of a generic two-term product G 2 G 1 Theorem Let G 1 and G 2 be two graphs on n i vertices, with adjacency matrix A i and characteristic polynomial φ i (x), i = 1, 2. Consider the graph G1 = G 1 0, with adjacency matrix A 1 and characteristic polynomial φ 1. Then the characteristic polynomial φ H (x) of the hierarchical product H = G 2 G 1 is: ( ) φ H (x) = φ 1(x) n φ1 2 (x) φ 2 φ 1 (x)

60 The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ).

61 The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2

62 The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2 The characteristic polynomial( of H is ) xi2 A φ H (x) = det(xi A H ) = det 2 B B (xi 1 A 1) I 2

63 The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2 The characteristic polynomial( of H is ) xi2 A φ H (x) = det(xi A H ) = det 2 B B (xi 1 A 1) I 2 Computing the determinant in R n 2 n 2 : φ H (x) = det([xi 2 A 2 ]φ 1 (x)i 2 + ( φ 1 (x)i 2 [ xi 2 φ 1 (x)) ]) = = det(φ 1 (x)i 2 φ 1 (x)a 2) = det φ 1 (x) φ1 (x) φ (x)i 2 A 2 = ( ) 1 = φ 1 (x)n 2 φ φ1 (x) 2 φ 1 (x)

64 The spectrum of a generic two-term product G 2 G 1 Corollary ( ) φ G 1 walk-regular φ H (x) = 1 (x) n2 ( ) n1 φ 1 (x) φ 2 φ 1 (x) Proof. φ 1 (x) = 1 n 1 φ 1 (x) n 1 Corollary Ggraph of order n 2 = N and characteristic polynomial φ G the characteristic polynomial of H = G K( n is ) (x + 1)(x n + 1) φ H (x) = (x + 1) N(n 2) (x n + 2) N φ G (x n + 2) Proof. K n is walk-regular, φ Kn = (x n + 1)(x + 1) n 1 and φ K n = (x + 1) n 1 + (n 1)(x n + 1)(x + 1) n 2

65 Generalization of the hierarchical product 1 Introduction 2 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters 3 Spectral properties of G K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 4 Generalization of the hierarchical product 5 Conclusions

66 Generalization of the hierarchical product Definition of the generalized hierarchical product G i = (V i, E i ), U i V i, i = 1, 2,..., N 1 H = G N G N 1 (U N 1 ) G 1 (U 1 ) is the graph: vertices V N V 2 V 1 if x j y j in G j and u i U i, i = 1, 2,..., j 1 then x N... x j+1 x j u j 1... u 1 x N... x j+1 y j u j 1... u 1 Example For every i, U i = V i G N G N 1 (U N 1 ) G 1 (U 1 ) = G N G N 1 G 1 For every i, U i = {0} G N G N 1 (U N 1 ) G 1 (U 1 ) = G N G N 1 G 1

67 Generalization of the hierarchical product Example Two views of a generalized hierarchical product K 3 3 with U 1 = U 2 = {0, 1}.

68 Conclusions Summary 1 Definition of the hierarchical product of graphs 2 Spectral properties 3 The particular case of T m 4 Definition of the generalized hierarchical product Further work 5 T m, spt m and spt m are structures with nice properties m 6 Properties of the generalized hierarchical product

69 Conclusions Thank you!!!