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Non-backtracking Walk Centrality for Directed Networks F. Arrigo, P. Grindrod, D. J. Higham, and V. Noferini Networks: from Matrix Functions to Quantum Physics Oxford, August 9th, 2017 1

Complex Networks Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 2 Figure from: http://www.npr.org/2016/04/16/474396452/how-math-determines-the-game-of-thrones-protagonist

Complex Networks Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Let G = ( V, E ) be an unweighted complex network Generating function Radius of convergence with n nodes. Its adjacency matrix is A = ( a ij ) ∈ Numerical examples R n×n : � 1 Summing up if ( i, j ) ∈ E a ij = 0 otherwise An edge ( i, j ) ∈ E s.t. ( j, i ) ∈ E is called recip- rocal. The quantities ( A r ) ii , ( A r ) ij count closed (resp., open) walks of length r . 2 Figure from: http://www.npr.org/2016/04/16/474396452/how-math-determines-the-game-of-thrones-protagonist

Degree and Eigenvector Complex Networks The degree centrality Degree and Eigenvector Katz Centrality Nonbacktracking walks � n NBTW-based centrality d i = e T i A 1 = a ij . Generating function Radius of convergence j =1 Numerical examples Summing up It leads to d = A 1 . The degree centrality is “too local”. 3

Degree and Eigenvector Complex Networks The degree centrality Degree and Eigenvector Katz Centrality Nonbacktracking walks � n NBTW-based centrality d i = e T i A 1 = a ij . Generating function Radius of convergence j =1 Numerical examples Summing up It leads to d = A 1 . The degree centrality is “too local”. Bonacich introduced the eigenvector centrality: n � x i ∝ a ij x j . j =1 It leads to A x = λ x and, if A is irreducible , then x is the Perron vector of A and λ = ρ ( A ) . 3

Katz Centrality Let f ( x ) = � ∞ r =0 c r x r , then, within the radius of convergence: Complex Networks Degree and Eigenvector Katz Centrality ∞ Nonbacktracking walks � NBTW-based centrality c r A r f ( A ) = Generating function Radius of convergence r =0 Numerical examples Looking closely at ( f ( A )) ij : Summing up ◦ it tells us how many walks (up to infinite length) originate at node i and end at node j ◦ if c r ≥ 0 and c r → 0 as r → ∞ , longer walks are given less importance. Katz centrality: � ∞ � � 1 = ( I − αA ) − 1 1 , α r A r k = 1 + r =1 where α ∈ (0 , 1 /ρ ( A )). = ⇒ solve a sparse linear system. 4

Nonbacktracking walks Complex Networks A walk is said to be backtracking if it contains at least one Degree and Eigenvector sequence of nodes of the form Katz Centrality Nonbacktracking walks NBTW-based centrality i ℓ i, Generating function Radius of convergence Numerical examples nonbacktracking (NBTW ) otherwise. Summing up 5

Why? Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality 3 2 Generating function Radius of convergence Numerical examples Summing up 4 7 1 5 6 6

Why? Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 7

NBTW-based centrality Let p r ( A ) ∈ R n×n be such that Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks ( p r ( A )) ij = |{ NBTW s of length r from node i to node j }| . NBTW-based centrality Generating function Radius of convergence It is the nonbacktracking analogue of the matrix power A r . Numerical examples Summing up We define a NBTW-based centrality measure as: � ∞ � � t r p r ( A ) b = 1 + 1 = φ ( A, t ) 1 , r =1 where t > 0 is chosen so that φ ( A, t ) = � r t r p r ( A ) converges. Questions: ◦ is this feasible? ◦ restrictions on t ? 8

Generating function ✬ ✩ Complex Networks Theorem: Let A be the adj. matrix of a digraph, D = Degree and Eigenvector diag(diag( A 2 )), and S = A ◦ A T . Then, Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function p 0 ( A ) = I, Radius of convergence p 1 ( A ) = A, Numerical examples p 2 ( A ) = A 2 − D Summing up and for all r ≥ 3 p r ( A ) = Ap r− 1 ( A ) + ( I − D ) p r− 2 ( A ) − ( A − S ) p r− 3 ( A ) . ✫ ✪ ◦ Undirected case: “Z eta functions of finite graphs and coverings” , Stark & Terras, Advances in Mathematics (1996). ◦ Directed case: “Zeta functions of restrictions of the shift transformation” , Bowen et al., Global Analysis: Proc. Symp. Pure Mathematics of the AMS (1968). 9

Generating function ✬ ✩ Complex Networks Theorem Let A , S , and D be defined as before. Moreover, let Degree and Eigenvector φ ( A, t ) = � ∞ r =0 t r p r ( A ) and Katz Centrality Nonbacktracking walks NBTW-based centrality M ( t ) = I − At + ( D − I ) t 2 + ( A − S ) t 3 . Generating function Radius of convergence Numerical examples Then Summing up M ( t ) φ ( A, t ) = (1 − t 2 ) I ✫ ✪ 10

Generating function ✬ ✩ Complex Networks Theorem Let A , S , and D be defined as before. Moreover, let Degree and Eigenvector φ ( A, t ) = � ∞ r =0 t r p r ( A ) and Katz Centrality Nonbacktracking walks NBTW-based centrality M ( t ) = I − At + ( D − I ) t 2 + ( A − S ) t 3 . Generating function Radius of convergence Numerical examples Then Summing up M ( t ) φ ( A, t ) = (1 − t 2 ) I ✫ ✪ The NBTW-based centrality measure is: b = (1 − t 2 )( I − At + ( D − I ) t 2 + ( A − S ) t 3 ) − 1 1 , where t > 0 is chosen so that φ ( A, t ) = � r t r p r ( A ) converges. = ⇒ same cost as Katz. 10

Radius of convergence Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Theorem Generating function Radius of convergence The power series φ ( A, t ) = � ∞ Numerical examples r =0 t r p r ( A ) converges if 0 < Summing up t < 1 /ρ ( C ) where ρ ( C ) is the spectral radius of the matrix   ( I − D ) ( S − A ) A   . C = I 0 0 0 I 0 11

Limiting behavior Complex Networks Theorem Degree and Eigenvector Katz Centrality Nonbacktracking walks Let t ∈ (0 , ρ ( C ) − 1 ). Then the NBTW centrality vector b ( t ) NBTW-based centrality Generating function returns the same ranking as that returned by Radius of convergence d out = A 1 as t → 0 + Numerical examples ◦ Summing up ◦ the first n components of x : C x = ρ ( C ) x , if the rank of ( I − ρ ( C ) − 1 C ) is 3 n − 1 and t → (1 /ρ ( C )) − . This theorem generalizes: ◦ Benzi and Klymko , “On the Limiting Behavior of Parameter-Dependent Network Centrality Measures” , SIMAX (2015). Grindrod et al., “The deformed graph Laplacian and its applications to ◦ network centrality analysis” , submitted (2017). Martin, Zhang, and Newman, “Localization and centrality in networks” , Phys. ◦ Rev. E (2014). 12

Special nodes Complex Networks 1 Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence 2 3 Numerical examples Summing up 4 5 6 ◦ reciprocal leaf : connected through a reciprocated link to another node, no other connections; ◦ dangling node : no outgoing links; ◦ source node : no ingoing links. 13

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 14

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 15

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 16

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 17

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 18

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 19

Pruning - an example Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up 20

Complex Networks Degree and Eigenvector Katz Centrality Nonbacktracking walks NBTW-based centrality Generating function Radius of convergence Numerical examples Summing up Numerical examples 21