Simplex Geometry of Graphs Piet Van Mieghem in collaboration with - PDF document

Simplex Geometry of Graphs Piet Van Mieghem in collaboration with Karel Devriendt 1 Google matrix: fundamentals, applications and beyond (GOMAX) IHES, October 15-18, 2018 Outline Background: Electrical matrix equations Geometry of a graph 1

Adjacency matrix A " % N = 6 0 1 1 0 0 1 L = 9 $ ' 1 3 1 0 1 0 1 1 $ ' $ 1 1 0 1 0 0 ' 4 2 A N × N = $ ' 0 0 1 0 1 0 $ ' 6 5 0 1 0 1 0 1 $ ' $ ' 1 1 0 0 1 0 # & For an undirected graph: A = A T is symmetric N ∑ Number of neighbors of node i is the degree: d i = a ik k = 1 if there is a link between node i and j, then a ij = 1 else a ij = 0 Incidence matrix B N = 6 Label links ( e.g.: l 1 = (1,2), l 2 = (1,3), l 3 = (1,6), L = 9 • 1 3 l 4 =(2,3), l 5 =(2,5), l 6 =(2,6), l 7 =(3,4), l 8 =(4,5), l 9 =(5,6) ) 4 Col k for link l k = (i,j) is zero, except: • 2 source node i = 1 à b ik = 1 6 5 destination node j = -1 à b jk = -1 # & 1 1 − 1 0 0 0 0 0 0 % ( − 1 0 0 1 − 1 1 0 0 0 % ( % 0 − 1 0 − 1 0 0 1 0 0 ( B N × L = % ( 0 0 0 0 0 0 − 1 − 1 0 % ( 0 0 0 0 1 0 0 1 − 1 % ( % 0 0 1 0 0 − 1 0 0 1 ( $ ' Col sum B is zero: u T B = 0 where the all-one vector u = (1,1,…,1) 4 B specifies the directions of links 2

Laplacian matrix Q N = 6 # & L = 9 3 − 1 − 1 0 0 − 1 1 3 % ( − 1 4 − 1 0 − 1 − 1 % ( 4 2 % ( − 1 − 1 3 − 1 0 0 Q N × N = % ( 0 0 − 1 2 − 1 0 6 5 % ( 0 − 1 0 − 1 3 − 1 % ( % ( − 1 − 1 0 0 − 1 3 $ ' Since BB T is symmetric, so are = = D - T Q BB A A and Q . Although B specifies D = diag ( d d ! d ) directions, A and Q lost this info here. 1 2 N u is an eigenvector of Q Basic property: Qu = 0 Belonging to eigenvalue µ = 0 5 because 0 = u T B = B T u Qu = BB T u = 0 Network: service(s) + topology transport of items from A to B Service (function) A B software, algorithms Topology (graph) hardware, structure Service and topology own specifications • both are, generally, time-variant • service is often designed independently of the topology • often more than 1 service on a same topology • 6 3

Function of network • Usually, the function of a network is related to the transport of items over its underlying graph • In man-made infrastructures: two major types of transport o Item is a flow (e.g. electrical current, water, gas,…) o Item is a packet (e.g. IP packet, car, container, postal letter,…) • Flow equations (physical laws) determine transport (Maxwell equations (Kirchhoff & Ohm), hydrodynamics, Navier- Stokes equation (turbulent, laminar flow equations, etc.) • Protocols determine transport of packets (IP protocols and IETF standards, car traffic rules, etc.) 7 Linear dynamics on networks Linear dynamic process: “proportional to” ( ~) graph of network Examples : x i water (or gas) flow ~ pressure • displacement (in spring) ~ force • heat flow ~ temperature • v i electrical current ~ voltage • i link flow y ij x = Q . v # $ − # & ~' $& injected weighted nodal j v j nodal Laplacian potential current of the vector x j vector graph 8 P. Van Mieghem, K. Devriendt and H. Cetinay, 2017, "Pseudoinverse of the Laplacian and best spreader node in a network", Physical Review E, vol. 96, No. 3, p 032311. 4

Pseudoinverse of the Laplacian (review) The inverse of the current-voltage relation x = Qv is the voltage-current relation v = ! " x subject to # $ % = 0 and # $ ( = 0 The spectral decomposition ./- 0 ) $ ! = ∑ +,- 1 + 2 + 2 + allows us to compute the pseudoinverse ( or Moore-Penrose inverse ) ! " = ∑ +,- ./- - $ 4 5 2 + 2 + 3 Ω = #8 $ + 8# $ − 2! " , The effective resistance N x N matrix is 6 " , ! == " , ⋯ , ! .. " where the N x 1 vector 8 = ! -- An interesting graph metric is the effective graph resistance ./- 1 ? @ = A# $ 8 = Atrace ! " = A G 1 + +,- 10 P. Van Mieghem, K. Devriendt and H. Cetinay, 2017, "Pseudo-inverse of the Laplacian and best spreader node in a network", Physical Review E, vol. 96, No. 3, p 032311. r ik k i r ij j v= ! " x with voltage reference u T v = 0 Inverses: x = Qv ! " : pseudoinverse of the weighted Laplacian obeying !! " = ! " ! = $ − & ' ( ( = )) * : all-one matrix u : all-one vector Unit current injected in node i nodal potential of i x = e i – 1/N u " + , = - ,, " ≤ - ,, " for 1 ≤ 4 ≤ 5 The best spreader is node k with - .. 5

Outline Background: Electrical matrix equations Geometry of a graph Three representations of a graph Topology domain Spectral domain Geometric domain 2 N = 6 L = 9 1 3 ! = ! # = $Λ$ # 4 2 4 6 5 1 " % 3 0 1 1 0 0 1 $ &×& : orthogonal $ ' 1 0 1 0 1 1 eigenvector matrix $ ' Undirected graph on $ ' 1 1 0 1 0 0 A N × N = N nodes $ ' Λ &×& : diagonal 0 0 1 0 1 0 $ ' = simplex in Euclidean eigenvalue matrix 0 1 0 1 0 1 $ ' (N-1) -dimensional space $ 1 1 0 0 1 0 ' # & Devriendt, K. and P. Van Mieghem, 2018, "The Simplex Geometry of Graphs", Delft University of Technology, report20180717. (http://arxiv.org/abs/1807.06475). 6

Miroslav Fiedler (1926-2015) Father of “algebraic connectivity” His 1972 paper: > 3400 citations “This book comprises, in addition to auxiliary material, the research on which I have worked for over 50 years.” 14 appeared in 2011 What is a simplex? Vertex Facet Edge Point Line Segment Triangle Tetrahedron Roughly : a simplex is generalization of a triangle to N dimensions Potential : Euclidean geometry is the oldest, mathematical theory 15 T. L. Heath, The Thirteen Books of Euclid’s Elements , Vol. 1-3, Cambridge University Press, 1926 7

Spectral decomposition weighted Laplacian (1) ! = #$# % Spectral decomposition: where $ = ./01 , ) , , 3 , ⋯ , , *+) , 0 , because Q u = 0 and the eigenvector matrix Z obeys Z T Z =Z Z T = I with structure ) ⋯ 8 - ) ) - 3 ) - ) ) - 3 ) ⋯ - * ) * - ) 3 - 3 3 ) ⋯ - ) 3 - 3 3 ⋯ - * 3 8 node # = = * ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮ - ) * - 3 * ⋯ - * * ) ⋯ - ) * - 3 * 8 * frequencies (eigenvalues) 16 *+) , ' - ' - ' % ! = ∑ '() Spectral decomposition weighted Laplacian (2) Only for a positive semi-definite matrix, it holds that 0 - = ./. 0 = . / . / 0 obeys - = ! 0 ! and has rank N-1 The matrix ! = . / (row N = 0 due to # ( = 0 ) # $ % $ $ # $ % $ & ⋯ # $ % $ ( # & % & $ # & % & & ⋯ # & % & ( ! = ⋮ ⋮ ⋱ ⋮ # (*$ % (*$ $ # (*$ % (*$ & # (*$ % (*$ ( ⋯ 0 0 0 17 (*$ # 2 % 2 % 2 0 - = ∑ 23$ 8

Geometrical representation of a graph $ % & % % $ % & % ( ⋯ $ % & % * $ ( & ( % $ ( & ( ( ⋯ $ ( & ( * , = ⋮ ⋮ ⋱ ⋮ $ *0% & *0% % $ *0% & *0% ( $ *0% & *0% * ⋯ 0 0 0 The i -th column vector ! " = $ % & % " , $ ( & ( " , ⋯ , $ * & * " = 0 represents a point p i in ( N-1 )-dim space (because S has rank N-1 ) ! % Simplex ! ( ! 4 18 Simplex geometry: omit zero row, , *×* → , (*0%)×* Faces of a simplex Each connected, undirected graph on N nodes corresponds to 1 vertex specific simplex in N-1 dimensions (Fiedler) 2 V is a set of vertices of the simplex in ℝ '() , corresponding to a set of face nodes in the graph G 4 edge 1 3 A face ! " = $ ∈ ℝ '() |$ = +, " -./ℎ , " 1 ≥ 0 456 7 8 , " = 1 19 The vector , " ∈ ℝ ' is a barycentric coordinate with : , " 1 ∈ ℝ .; . ∈ < , " 1 = 0 .; . ∉ < 9

Centroids % & ! " = $ |"| is the centroid of face ( " with - " . = 1 .∈" ! 1 = 6 1 ! 7 = $ - 8 = 0 ! 2 = − 1 2 ! 2 ! 4 ! 2 ! 1 = ! 2,4 a centroid of a face is a vector centroid of simplex is origin 20 - " = - − - " |:|! " = $ - − - " = − 8 − : ! " Geometric representation of a graph % = ! " − ! $ ' ! " − ! $ = ! " % = - " ' ! " +! $ ' ! $ -2! " ' ! $ ! " % ! " − ! $ % = , "" + , $$ − 2, "$ = - " +- $ + 2. "$ /01 2 ≠ 4, 67!6 8610 % = - < ! < % ! < The matrix with off-diagonal elements - " + - $ + 2. "$ is a distance matrix ! % (if the graph G is connected) ' ! % = , <% = −. <% ! < The geometric graph representation is not unique (node relabeling changes Z ) =>< ? : 8 : 8 : ' ! $ = ∑ :;< =>< ' ! " ? : 8 : " ? : 8 : $ = ∑ :;< "$ = , "$ 21 =>< ? : 8 : 8 : ' and , = @ ' @ , = ∑ :;< 10