Localized eigenvectors of the non-backtracking matrix

Kawamoto, Tatsuro

doi:10.1088/1742-5468/2016/02/023404

Cited by 24 publications

(26 citation statements)

References 33 publications

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“…Recently, Pastor-Satorras and Castellano [85] performed an extensive analytic and numerical investigation of the mathematical properties of the H-index, finding analytically that the H-index is expected to be strongly correlated with degree in uncorrelated networks. The same strong correlation is found also in real networks, and Pastor-Satorras and Castellano [85] point out that the H-index is a poor indicator of node spreading influence as compared to the non-backtracking centrality [102]. While the comparison of different metrics with respect to their ability to identify influential spreaders is not one of the main focus of the present review, we remind the interested reader to [48,85,87] for recent reports on this important problem.…”

Section: Coreness Centrality and Its Relation With Degree And H-indexsupporting

confidence: 69%

Ranking in evolving complex networks

et al. 2017

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Complex networks have emerged as a simple yet powerful framework to represent and analyze a wide range of complex systems. The problem of ranking the nodes and the edges in complex networks is critical for a broad range of real-world problems because it affects how we access online information and products, how success and talent are evaluated in human activities, and how scarce resources are allocated by companies and policymakers, among others. This calls for a deep understanding of how existing ranking algorithms perform, and which are their possible biases that may impair their effectiveness. Well-established ranking algorithms (such as the popular Google's PageRank) are static in nature and, as a consequence, they exhibit important shortcomings when applied to real networks that rapidly evolve in time. The recent advances in the understanding and modeling of evolving networks have enabled the development of a wide and diverse range of ranking algorithms that take the temporal dimension into account. The aim of this review is to survey the existing ranking algorithms, both static and time-aware, and their applications to evolving networks. We emphasize both the impact of network evolution on well-established static algorithms and the benefits from including the temporal dimension for tasks such as prediction of real network traffic, prediction of future links, and identification of highly-significant nodes.Comment: 54 pages, 16 figure

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Section: Coreness Centrality and Its Relation With Degree And H-indexsupporting

confidence: 69%

Ranking in evolving complex networks

et al. 2017

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“…For the purpose of ranking nodes with an eigenvector-based centrality, localization can sometimes be problematic, because the centrality concentrates onto a (potentially very small) subset of the nodes, and this makes it difficult to reliably rank nodes outside of that subset. This has prompted the introduction and investigation of new centrality measures that, for example, do not exhibit localization (or at least exhibit less severe localization) due to the presence of nodes with large degree [68,86], although localization can still arise due to other network structures [98]. We also remark that the "non-backtracking centrality" (also called "Hashimoto centrality") introduced in Ref.…”

Section: Mgp: Pagerank Centralitymentioning

confidence: 99%

Eigenvector-Based Centrality Measures for Temporal Networks

Taylor¹,

Myers²,

Clauset³

et al. 2017

Multiscale Model. Simul.

173

198

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Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network during different time windows, and we couple centrality matrices for the layers into a supra-centrality matrix of size NT × NT whose dominant eigenvector gives the centrality of each node i at each time t. We refer to this eigenvector and its components as a joint centrality, as it reflects the importances of both the node i and the time layer t. We also introduce the concepts of marginal and conditional centralities, which facilitate the study of centrality trajectories over time. We find that the strength of coupling between layers is important for determining multiscale properties of centrality, such as localization phenomena and the time scale of centrality changes. In the strong-coupling regime, we derive expressions for time-averaged centralities, which are given by the zeroth-order terms of a singular perturbation expansion. We also study first-order terms to obtain first-order-mover scores, which concisely describe the magnitude of nodes’ centrality changes over time. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among top-billed actors during the Golden Age of Hollywood, and citations of decisions from the United States Supreme Court.

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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%

“…We will use BTW and NBTW as shorthand for backtracking walk and nonbacktracking walk, respectively. We note that NBTWs have typically been studied on undirected networks [2,17,20,[24][25][26]32], but the definition continues to make sense in the directed case. We will denote by p r (A) the matrix whose (i, j)th entry counts the number of NBTWs of length r from node i to node j.…”

Section: Background and Problem Settingmentioning

confidence: 99%

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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Localized eigenvectors of the non-backtracking matrix

Cited by 24 publications

References 33 publications

Ranking in evolving complex networks

Ranking in evolving complex networks

Eigenvector-Based Centrality Measures for Temporal Networks

Non-backtracking walk centrality for directed networks

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