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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

Complex Networks

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

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Complex Networks

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

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Let G = (V, E) be an unweighted complex network with n nodes. Its adjacency matrix is A = (aij) ∈ Rn×n: aij = 1 if (i, j) ∈ E

• therwise

An edge (i, j) ∈ E s.t. (j, i) ∈ E is called recip- rocal. The quantities (Ar)ii, (Ar)ij count closed (resp.,

• pen) walks of length r.

Degree and Eigenvector

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

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The degree centrality di = eT

i A1 = n

• j=1

aij. It leads to d = A1. The degree centrality is “too local”.

Degree and Eigenvector

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

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The degree centrality di = eT

i A1 = n

• j=1

aij. It leads to d = A1. The degree centrality is “too local”. Bonacich introduced the eigenvector centrality: xi ∝

n

• j=1

aijxj. It leads to Ax = λx and, if A is irreducible, then x is the Perron vector of A and λ = ρ(A).

Katz Centrality

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

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Let f(x) = ∞

r=0 crxr, then, within the radius of convergence:

f(A) =

∞

• r=0

crAr

Looking closely at (f(A))ij:

• it tells us how many walks (up to infinite length) originate at node i and end

at node j

• if cr ≥ 0 and cr → 0 as r → ∞, longer walks are given less importance.

Katz centrality: k = 1 + ∞

• r=1

αrAr

• 1 = (I − αA)−11,

where α ∈ (0, 1/ρ(A)). = ⇒ solve a sparse linear system.

Nonbacktracking walks

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

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A walk is said to be backtracking if it contains at least one sequence of nodes of the form i ℓ i, nonbacktracking (NBTW ) otherwise.

Why?

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

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7 1 2 3 4 5 6

Why?

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

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NBTW-based centrality

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

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Let pr(A) ∈ Rn×n be such that (pr(A))ij = |{NBTW s of length r from node i to node j}|.

It is the nonbacktracking analogue of the matrix power Ar.

We define a NBTW-based centrality measure as: b = 1 + ∞

• r=1

trpr(A)

• 1 = φ(A, t)1,

where t > 0 is chosen so that φ(A, t) =

r trpr(A) converges.

Questions:

• is this feasible?
• restrictions on t?

Generating function

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

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✬ ✫ ✩ ✪ Theorem: Let A be the adj. matrix of a digraph, D = diag(diag(A2)), and S = A ◦ AT . Then, p0(A) = I, p1(A) = A, p2(A) = A2 − D and for all r ≥ 3 pr(A) = Apr−1(A) + (I − D) pr−2(A) − (A − S) pr−3(A).

• Undirected case: “Zeta 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).

Generating function

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

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✬ ✫ ✩ ✪ Theorem Let A, S, and D be defined as before. Moreover, let φ(A, t) = ∞

r=0 trpr(A) and

M(t) = I − At + (D − I)t2 + (A − S)t3. Then M(t)φ(A, t) = (1 − t2)I

Generating function

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

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✬ ✫ ✩ ✪ Theorem Let A, S, and D be defined as before. Moreover, let φ(A, t) = ∞

r=0 trpr(A) and

M(t) = I − At + (D − I)t2 + (A − S)t3. Then M(t)φ(A, t) = (1 − t2)I The NBTW-based centrality measure is: b = (1 − t2)(I − At + (D − I)t2 + (A − S)t3)−11, where t > 0 is chosen so that φ(A, t) =

r trpr(A) converges.

= ⇒ same cost as Katz.

Radius of convergence

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

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The power series φ(A, t) = ∞

r=0 trpr(A) converges if 0 <

t < 1/ρ(C) where ρ(C) is the spectral radius of the matrix C =   A (I − D) (S − A) I I   . Theorem

Limiting behavior

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

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Let t ∈ (0, ρ(C)−1). Then the NBTW centrality vector b(t) returns the same ranking as that returned by

• dout = A1 as t → 0+
• the first n components of x: Cx = ρ(C)x, if the rank of

(I − ρ(C)−1C) is 3n − 1 and t → (1/ρ(C))−. Theorem

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).

Special nodes

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

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1 2 3 4 6 5

• reciprocal leaf: connected through a reciprocated link to

another node, no other connections;

• dangling node: no outgoing links;
• source node: no ingoing links.

Pruning - an example

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

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Pruning - an example

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

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Pruning - an example

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

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Pruning - an example

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

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Pruning - an example

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

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Pruning - an example

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

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Pruning - an example

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

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Numerical examples

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

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Katz vs. NBTW

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

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The network Pajek/GlossGT represents connections between words from the graph/digraph glossary.

• largest weakly connected component contains 60 nodes;
• ρ(A) = ρ(C) = 1;
• we use α = t = 0.9.

Katz vs. NBTW

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

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Katz vs. NBTW

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

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Katz vs. NBTW– BA

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

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Pruning nodes

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

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NAME n m n% m% Pajek/California 9,664 16,150 2% 3% Gleich/wb-cs-stanford 9,914 35,555 57% 80% SNAP/soc-Epinions 75,888 508,837 36% 85% SNAP/wiki-talk 2,394,385 5,021,410 4% 29% SNAP/cit-Patents 3,774,768 16,518,947 0% 0%

• n: the number of nodes;
• m: the number of edges;
• n%: percentage of nodes that are retained in the network

when nodes are successively eliminated from each network;

• m%: percentage of edges that are retained in the network

when nodes are successively eliminated from each network.

Summing up

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

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Summing up

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

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We worked on directed, static networks and considered the non-backtracking version of “standard” walks. We did:

• define a NBTW-based centrality measure
• demonstrated computational feasibility
• characterized convergence
• generalized eigenvector centrality of Martin et al.
• showed that pruning certain nodes adds to efficiency
• In the paper: showed that NBTW Katz dampen localization

effects What are we doing now? We are considering other types of networks and functions.

Bibliography

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

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Thank you!

francesca.arrigo@strath.ac.uk, peter.grindrod@maths.ox.ac.uk, d.j.higham@strath.ac.uk, vnofer@essex.ac.uk [1] F. A., P. Grindrod, D. J. Higham, V. Noferini, “Nonbacktracking Walk Centrality for Directed Networks”, J. of Complex Networks, 00, pp. 1–25 (2017) - doi: 10.1093/comnet/cnx025. [2] P. Grindrod, D. J. Higham, V. Noferini, The deformed graph Laplacian and its applications to network centrality analysis. Preprint, submitted (2017). [3] T. Martin, X. Zhang, M. E. J. Newman, Localization and centrality in networks. Phys.

• Rev. E90 052808 (2014).

[4] M. Benzi and C. Klymko, On the limiting behavior of parameter-dependent network centrality measures. SIAM J. Matrix Anal. Appl. 36, 686–706 (2015). [5] E. Estrada and D. J. Higham, Network properties revealed through matrix functions. SIAM Review 52, 696–671 (2010). [6] J. Bowen and C. Reutenauer, “Zeta functions of restrictions of the shift transformation”, Global Analysis: Proc. Symp. Pure Mathematics of the AMS (1968).