Large scale computation of the trace of a matrix function Giuseppe - PowerPoint PPT Presentation

Large scale computation of the trace of a matrix function Giuseppe Rodriguez Department of Mathematics and Computer Science, University of Cagliari, Italy International Workshop on Applied Mathematics & Quantum Information Cagliari, November 3–4, 2016 G. Rodriguez Large scale computation of the trace of a matrix function

Graphs Let us consider a graph G = ( V , E ), where: V is a set of nodes that we denote by { 1 , . . . , n } ; E is a set of edges (or arcs) between nodes. If the arcs are oriented we say that the graph is directed (digraph), undirected otherwise. A graph is (strongly) connected if for any pairs of nodes there is an (oriented) path connecting them. To each unweighted graph corresponds an adjacency matrix � 1 , if nodes i and j are connected, A ij = 0 , if nodes i and j are not connected. If the graph is undirected the adjacency matrix is symmetric. G. Rodriguez Large scale computation of the trace of a matrix function

Complex networks A complex network is a graph with particular properties, which is used to model interaction between entities in various fields, e.g., computer science, sociology, economics, genetics, epidemiology, etc. Though there is often little randomness in the phenomena being studied, complex networks are generally seen as random graphs with some additional features: strong clustering, small world effect, scale-free structure (power law degree distribution). G. Rodriguez Large scale computation of the trace of a matrix function

Clustering index The friend of your friend is likely to be your friend. Global clustering is measured by the average of the clustering of the nodes � n clust( G ) = 1 C ( i ) n i =1 where C ( i ) is the probability that two nodes connected to the same node are connected themselves ( A + A T ) 3 C ( i ) = t i ii = 2[deg tot( i ) (deg tot( i ) − 2[ A 2 ] ii )] , T i i.e., ratio between existing and possible triangles. G. Rodriguez Large scale computation of the trace of a matrix function

Small world effect Milgram’s six degree of separation. In a small world network the mean distance among nodes is small : � ℓ ( G ) := 1 dist( i , j ) � log n , N ( i , j ) ∈C where C = { ( i , j ) : dist( i , j ) < ∞} . Example from [Watts and Strogatz, 1998]: G. Rodriguez Large scale computation of the trace of a matrix function

Power law degree distribution The rich gets richer. In a scale-free network the distributions of the degrees of its nodes follows a power law: F ( k ) ∼ ck − p , with p > 0. This implies the presence of hubs, i.e., super-connected nodes. Loglog scale 4 900 10 800 700 3 10 600 500 2 10 400 300 1 10 200 100 0 0 10 0 5 10 15 20 25 0 1 2 10 10 10 d min = 0 d max = 21 p = −2.27 − R = 0.98 G. Rodriguez Large scale computation of the trace of a matrix function

Ranking nodes: the degree Various indices (or metrics) have been introduced to characterize the importance of a node in terms of connection with the rest of the network. The simplest is the degree. If the network is undirected: � n � n deg( i ) = A ij = A ji ; j =1 j =1 j � = i j � = i if the network is directed: n n � � deg out ( i ) = A ij and deg in ( i ) = A ji . j =1 j =1 j � = i j � = i G. Rodriguez Large scale computation of the trace of a matrix function

Ranking nodes: f -centrality indices Various indices (or metrics) have been introduced to characterize the importance of a node in terms of connectivity. A class of indices has been defined starting from matrix functions ∞ � c m A m , f ( A ) = c m ≥ 0 . m =0 The key point is that [ A m ] ij gives the number of walks of length m starting from the node i and ending at node j k m − 1 i k 1 j · · · Then, [ f ( A )] ij is a weighted average of all the walks connecting i to j , and describes the ease of travelling between them. G. Rodriguez Large scale computation of the trace of a matrix function

f -centrality and f -communicability Two common choices for the coefficients in f ( A ) = � c m A m are f ( A ) = e A , c m = 1 / m ! ⇒ f ( A ) = ( I − γ A ) − 1 . c m = γ m ⇒ [ f ( A )] ii = e T i f ( A ) e i is the f -centrality of node i , [ f ( A )] ij = e T i f ( A ) e j is the f -communicability between i and j . [Estrada, Rodr´ ıguez-Vel´ azquez 2005, 2006], [Benzi, Boito 2010], [Estrada, Higham 2010], [Estrada, Hatano, Benzi 2012], . . . If A is not large and symmetric, we can easily compute these functions (e.g., spectral factorization, Pad´ e approx.), but the computation is not trivial for large scale problems. G. Rodriguez Large scale computation of the trace of a matrix function

Other indices The function f ( A ) plays a role in the definition of other indices for the nodes of undirected or directed networks: Katz centrality , communicability betweenness , average comm. , total subgraph comm. , starting/ending convenience [Katz 1953], [Estrada, Higham, Hatano 2009], [Estrada, Hatano, Benzi 2012], [Benzi, Klymko 2013], [Fenu, Martin, Reichel, R 2013]. These indices depend upon the row/column sums of f ( A ), i.e., n n � � [ f ( A )] ij = e T [ f ( A )] ij = e T f ( A ) e j , i f ( A ) e , j =1 i =1 which give information about walks joining node i to other nodes and walks joining other nodes to node j , respectively. G. Rodriguez Large scale computation of the trace of a matrix function

Ranking problem To extract information from f ( A ) the following quadratic forms may be computed e T i f ( A ) e j → [ f ( A )] ij , e T i f ( A ) e → row-sum, with e i = (0 , . . . , 1 , 0 , . . . 0) T and e = (1 , . . . , 1) T . It is often of interest to identify a subset of nodes for which a chosen index is largest (or smallest). Examples: find m nodes s.t. max i e T i f ( A ) e i ( f -subgraph centrality); find m nodes s.t. max i e T i f ( A ) e ( f -starting convenience). [Fenu, Martin, Reichel, R, SISC 2013], [Baglama, Fenu, Reichel, R, LAA 2013], G. Rodriguez Large scale computation of the trace of a matrix function

Approximation by Gauss quadrature The actual value of u T f ( A ) v ( A = A T , u , v = e i , e j , e , . . . ) can be obtained by Gauss quadrature [Golub, Meurant 1993. . . 2010]. This approach was proposed for subgraph centralities and communicabilities in [Benzi, Boito 2010]. Indeed, under suitable assumptions on f , starting from � n � u T f ( A ) u = f ( λ i ) ω 2 i = f ( t ) d ω ( t ) , i =1 the connection between the symmetric Lanczos process and orthogonal polynomials leads to bound the quadratic form by sequences of Gauss and Gauss–Radau quadrature formulas G k − 1 f < G k f < u T f ( A ) u < R k +1 f < R k f . G. Rodriguez Large scale computation of the trace of a matrix function

Complex networks and the trace operator The Estrada index of a graph, defined as � n � n trace(exp( A )) = [exp( A )] ii = exp( λ i ) , i =1 i =1 provides a global characterization of the graph; see [Estrada, Rodr´ ıguez-Vel´ azquez 2005], [Estrada, Hatano, Benzi 2012]. The clustering index of a network depends upon the number T of existing triangles, which can be obtained by computing T = 1 6 trace( A 3 ) . G. Rodriguez Large scale computation of the trace of a matrix function

Large scale computation of trace( f ( A )) Our approach is the following � � n trace( E T trace( f ( A )) = j f ( A ) E j ) . j =1 where E j = [ e k ( j − 1)+1 , . . . , e min { jk , m } ] , � m + k − 1 � for j = 1 , 2 , . . . , � n = . k Then, it is essential to approximate at a prescribed accuracy trace( W T f ( A ) W ) , for W ∈ R m × k , efficiently and without actually computing f ( A ). G. Rodriguez Large scale computation of the trace of a matrix function

The global Lanczos method For W 1 , W 2 ∈ R n × k , we define the inner product and induced norm � W 1 � F := � W 1 , W 1 � 1 / 2 . � W 1 , W 2 � := trace( W T 1 W 2 ) , For A = A T it holds A [ V 1 , . . . , V ℓ ] = [ V 1 , . . . , V ℓ ] � T ℓ + β ℓ +1 V ℓ +1 E T ℓ , where V j ∈ R n × k , � V i , V j � = δ ij , � T ℓ = T ℓ ⊗ I k , and   O α 1 β 2   β 2 α 2 β 3     T ℓ = β 3 · · .     · · β ℓ O β ℓ α ℓ [Jbilou, Messaoudi, Sadok 1999] [Elbouyahyaoui, Messaoudi, Sadok 2009] G. Rodriguez Large scale computation of the trace of a matrix function

Global Gauss quadrature rules In [Bellalij, Reichel, R, Sadok, ApNuM 2015] we considered the expression I f := trace( W T f ( A ) W ) V 1 = W / � W � F = � W � 2 F trace( V T A = U Λ U T 1 f ( A ) V 1 ) , = � W � 2 F trace( � 1 f (Λ) � � V T V 1 = U T V 1 V 1 ) For any i = 1 , 2 , . . . , k we can write � m � i � 1 f (Λ) � f ( λ j ) µ 2 e T V T V 1 e i = j = f ( λ ) d µ i ( λ ) , j =1 from which, letting µ ( λ ) := � k i =1 µ i ( λ ), it follows � � k � I f = � W � 2 f ( λ ) d µ i ( λ ) = � W � 2 f ( λ ) d µ ( λ ) F F i =1 G. Rodriguez Large scale computation of the trace of a matrix function

Global Gauss quadrature rules The global Lanczos method gives V j = p j − 1 ( A ) V 1 , so that � δ ij = � V j , V i � = � p j − 1 ( A ) V 1 , p i − 1 ( A ) V 1 � = p j − 1 ( λ ) p i − 1 ( λ ) d µ ( λ ) implies that G ℓ f = � W � 2 F e T 1 f ( T ℓ ) e 1 is a Gauss quadrature rule which is exact for polynomials of degree 2 ℓ − 1. Quick proof: p j ( λ ) OPs recursion coefficients → T ℓ = QDQ T . G. Rodriguez Large scale computation of the trace of a matrix function