Localization and centrality in networks

Martin, Travis; Zhang, Xiao; Newman, M. E. J.

doi:10.1103/physreve.90.052808

Cited by 269 publications

(369 citation statements)

References 24 publications

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“…These walks include at least one back-and-forth flip between a pair of nodes. One concrete justification for the use of nonbacktracking walks is that localization effects can sometimes be avoided [17,24,31]. In the context of community detection, a nonbacktracking version of spectral clustering was proposed in [20].…”

Section: Motivationmentioning

confidence: 99%

“…Moreover, quantities defined using the concept of nonbacktracking walks have been proved to be useful in the framework of undirected networks when tackling decycling problems, as well as dismantling problems and the problem of optimal percolation (see, e.g., [25,26] and references therein). A nonbacktracking analogue of eigenvector centrality was developed in [24] for undirected networks, and a Katz version was proposed in [14] and studied from a matrix polynomial perspective. However, none of those references handle directed edges.…”

Section: Motivationmentioning

confidence: 99%

“…Section 6 considers the limiting values of the downweighting parameter. In particular, by allowing the downweighting parameter to approach its upper bound, we arrive at a generalization to directed networks of the nonbacktracking eigenvector centrality measure proposed in [24]. Section 7 gives a rigorous analysis of the new measure on a synthetic network.…”

Section: Motivationmentioning

confidence: 99%

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Non-backtracking walk centrality for directed networks

Arrigo

Grindrod

Higham

et al. 2017

Journal of Complex Networks

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The theory of zeta functions provides an expression for the generating function of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is related to the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter. We find that three important additional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation parameter. Second, a natural approach for determining a suitable parameter range is invariant under the removal of certain types of nodes, we can gain computational efficiencies through reducing the dimension of the resulting eigenvalue problem. Third, the dimension of the linear system defining the centrality measures may be reduced in the same manner. We show that the new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where nonreciprocal edges are augmented with negative weights. We also give a multilayer interpretation, where negatively weighted walks between layers compensate for backtracking walks on the only non-empty layer. Studying the limit as the attenuation parameter approaches its upper bound also allows us to propose a generalization of eigenvector-based nonbacktracking centrality measure to this directed network setting. In this context, we find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. Finally, we give results for real networks.

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Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

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Non-backtracking walk centrality for directed networks

Arrigo

Grindrod

Higham

et al. 2017

Journal of Complex Networks

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“…Such localisation poses a problem for conventional eigenvector centrality because most of the weight of the centrality concentrates on a small number of nodes, thus necessitating the random-hop term in the Google matrix [33]. On the other hand, the relatively homogeneous degree distribution in Erdös-Rényi random networks is expected to favour a delocalised phase of the walker, as observed in [7].…”

Section: Localisation-delocalisation Transitionmentioning

confidence: 99%

Comparing classical and quantum PageRanks

Loke

Tang

Rodriguez

et al. 2016

Quantum Inf Process

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Following recent developments in quantum PageRanking, we present a comparative analysis of discrete-time and continuous-time quantum-walk-based PageRank algorithms. For the discrete-time case, we introduce an alternative PageRank measure based on the maximum probabilities achieved by the walker on the nodes. We demonstrate that the required time of evolution does not scale significantly with increasing network size. We affirm that all three quantum PageRank measures considered here distinguish clearly between outerplanar hierarchical, scale-free, and Erdös-Rényi network types. Relative to classical PageRank and to different extents, the quantum measures better highlight secondary hubs and resolve ranking degeneracy among peripheral nodes for the networks we studied in this paper.

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“…In fact, both quantities have been already used to characterize the eigenfunctions of the adjacency matrices of random network models (see some examples in Refs. [31,36,39,42,48,[53][54][55][56][57][66][67][68][69]). …”

Section: B Entropic Eigenfunction Localization Lengthmentioning

confidence: 99%

Universality in the spectral and eigenfunction properties of random networks

et al. 2015

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By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

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Localization and centrality in networks

Cited by 269 publications

References 24 publications

Non-backtracking walk centrality for directed networks

Non-backtracking walk centrality for directed networks

Comparing classical and quantum PageRanks

Universality in the spectral and eigenfunction properties of random networks

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