Network centrality: an introduction

Centrality is a key property of complex networks that influences the behavior of dynamical processes, like synchronization and epidemic spreading, and can bring important information about the organization of complex systems, like our brain and society. There are many metrics to quantify the node centrality in networks. Here, we review the main centrality measures and discuss their main features and limitations. The influence of network centrality on epidemic spreading and synchronization is also pointed out in this chapter. Moreover, we present the application of centrality measures to understand the function of complex systems, including biological and cortical networks. Finally, we discuss some perspectives and challenges to generalize centrality measures for multilayer and temporal networks.Comment: Book Chapter in "From nonlinear dynamics to complex systems: A Mathematical modeling approach" by Springe

Centrality metrics and localization in core-periphery networks

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it has been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to a large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the Stochastic Block Model, or its degree corrected version, with a strong core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES}, showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.Comment: 15 pages, 8 figure

Centralities in complex networks

In network science complex systems are represented as a mathematical graphs consisting of a set of nodes representing the components and a set of edges representing their interactions. The framework of networks has led to significant advances in the understanding of the structure, formation and function of complex systems. Social and biological processes such as the dynamics of epidemics, the diffusion of information in social media, the interactions between species in ecosystems or the communication between neurons in our brains are all actively studied using dynamical models on complex networks. In all of these systems, the patterns of connections at the individual level play a fundamental role on the global dynamics and finding the most important nodes allows one to better understand and predict their behaviors. An important research effort in network science has therefore been dedicated to the development of methods allowing to find the most important nodes in networks. In this short entry, we describe network centrality measures based on the notions of network traversal they rely on. This entry aims at being an introduction to this extremely vast topic, with many contributions from several fields, and is by no means an exhaustive review of all the literature about network centralities.Comment: 10 pages, 3 figures. Entry for the volume "Statistical and Nonlinear Physics" of the Encyclopedia of Complexity and Systems Science, Chakraborty, Bulbul (Ed.), Springer, 2021 Updated versio

On the exponential generating function for non-backtracking walks

We derive an explicit formula for the exponential generating function associated with non-backtracking walks around a graph. We study both undirected and directed graphs. Our results allow us to derive computable expressions for non-backtracking versions of network centrality measures based on the matrix exponential. We find that eliminating backtracking walks in this context does not significantly increase the computational expense. We show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network. Insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions. Rigorous analysis on the star graph illustrates the effect of non-backtracking and shows that localization can be eliminated when we restrict to non-backtracking walks. We also investigate the localization issue on synthetic networks

On Spectral Graph Embedding: A Non-Backtracking Perspective and Graph Approximation

Graph embedding has been proven to be efficient and effective in facilitating graph analysis. In this paper, we present a novel spectral framework called NOn-Backtracking Embedding (NOBE), which offers a new perspective that organizes graph data at a deep level by tracking the flow traversing on the edges with backtracking prohibited. Further, by analyzing the non-backtracking process, a technique called graph approximation is devised, which provides a channel to transform the spectral decomposition on an edge-to-edge matrix to that on a node-to-node matrix. Theoretical guarantees are provided by bounding the difference between the corresponding eigenvalues of the original graph and its graph approximation. Extensive experiments conducted on various real-world networks demonstrate the efficacy of our methods on both macroscopic and microscopic levels, including clustering and structural hole spanner detection.Comment: SDM 2018 (Full version including all proofs

Effects of Backtracking on PageRank

In this paper, we consider three variations on standard PageRank: Non-backtracking PageRank, $\mu$-PageRank, and $\infty$-PageRank, all of which alter the standard formula by adjusting the likelihood of backtracking in the algorithm's random walk. We show that in the case of regular and bipartite biregular graphs, standard PageRank and its variants are equivalent. We also compare each centrality measure and investigate their clustering capabilities

Non-backtracking Walk Centrality for Directed Networks

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