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

Move ordering and communities in complex networks describing the game of go

We analyze the game of go from the point of view of complex networks. We construct three different directed networks of increasing complexity, defining nodes as local patterns on plaquettes of increasing sizes, and links as actual successions of these patterns in databases of real games. We discuss the peculiarities of these networks compared to other types of networks. We explore the ranking vectors and community structure of the networks and show that this approach enables to extract groups of moves with common strategic properties. We also investigate different networks built from games with players of different levels or from different phases of the game. We discuss how the study of the community structure of these networks may help to improve the computer simulations of the game. More generally, we believe such studies may help to improve the understanding of human decision process.Comment: 14 pages, 21 figure

Google matrix analysis of directed networks

In past ten years, modern societies developed enormous communication and social networks. Their classification and information retrieval processing become a formidable task for the society. Due to the rapid growth of World Wide Web, social and communication networks, new mathematical methods have been invented to characterize the properties of these networks on a more detailed and precise level. Various search engines are essentially using such methods. It is highly important to develop new tools to classify and rank enormous amount of network information in a way adapted to internal network structures and characteristics. This review describes the Google matrix analysis of directed complex networks demonstrating its efficiency on various examples including World Wide Web, Wikipedia, software architecture, world trade, social and citation networks, brain neural networks, DNA sequences and Ulam networks. The analytical and numerical matrix methods used in this analysis originate from the fields of Markov chains, quantum chaos and Random Matrix theory.Comment: 56 pages, 58 figures. Missed link added in network example of Fig3

A new look into the evolution of clusters literature. A bibliometric exercise

In the contemporary globalising knowledge-based economies, local clusters have become crucial elements of regional development, assuming a significant role in both academic and political fields. Although there is an intuitive awareness about the raising importance of the theoretical debate on clusters, there is a substantial lack of empirical support of its precise magnitude and evolution. Moreover, the majority of literature surveys on clusters are exclusively qualitative-based. Aiming at filling this gap, the main purpose of this paper is to provide a quantitative survey of the cluster literature, using bibliometric techniques based on articles. Based on a throughout analysis of all abstracts of articles on clusters published in journals indexed on the Econlit and EBSCO databases, covering the period 1962-2007, our research show that besides their importance in academic fields, the role of clusters has also been widely acknowledged in political spheres. In parallel with the increasing interest in the ‘local’, there has been, as well, an emergent body of literature on global networks and clusters. Moreover, on the basis of the recent boom on clusters literature stands the emergent themes of ‘local networks and social approaches’ and ‘knowledge-based theories’. Literature associated to ‘regional and national innovation systems’ and to ‘institutional approaches’ (local enrooted cultures, governance and customs) has been object of a particular dynamism since the 1990s. Despite the evidence of a clear positive correlation between journals ‘quality’ and formal related research, the evolution of the literature on clusters continues to be mostly appreciative led.Clusters, Industrial Clusters, Industrial Location; Bibliometrics

The game of go as a complex network

We study the game of go from a complex network perspective. We construct a directed network using a suitable definition of tactical moves including local patterns, and study this network for different datasets of professional tournaments and amateur games. The move distribution follows Zipf's law and the network is scale free, with statistical peculiarities different from other real directed networks, such as e. g. the World Wide Web. These specificities reflect in the outcome of ranking algorithms applied to it. The fine study of the eigenvalues and eigenvectors of matrices used by the ranking algorithms singles out certain strategic situations. Our results should pave the way to a better modelization of board games and other types of human strategic scheming.Comment: 6 pages, 9 figures, final versio

Quantum Google in a Complex Network

We investigate the behavior of the recently proposed quantum Google algorithm, or quantum PageRank, in large complex networks. Applying the quantum algorithm to a part of the real World Wide Web, we find that the algorithm is able to univocally reveal the underlying scale-free topology of the network and to clearly identify and order the most relevant nodes (hubs) of the graph according to their importance in the network structure. Moreover, our results show that the quantum PageRank algorithm generically leads to changes in the hierarchy of nodes. In addition, as compared to its classical counterpart, the quantum algorithm is capable to clearly highlight the structure of secondary hubs of the network, and to partially resolve the degeneracy in importance of the low lying part of the list of rankings, which represents a typical shortcoming of the classical PageRank algorithm. Complementary to this study, our analysis shows that the algorithm is able to clearly distinguish scale-free networks from other widespread and important classes of complex networks, such as Erd\H{o}s-R\'enyi networks and hierarchical graphs. We show that the ranking capabilities of the quantum PageRank algorithm are related to an increased stability with respect to a variation of the damping parameter $\alpha$ that appears in the Google algorithm, and to a more clearly pronounced power-law behavior in the distribution of importance among the nodes, as compared to the classical algorithm. Finally, we study to which extent the increased sensitivity of the quantum algorithm persists under coordinated attacks of the most important nodes in scale-free and Erd\H{o}s-R\'enyi random graphs

Graph Convolutional Neural Networks for Web-Scale Recommender Systems

Recent advancements in deep neural networks for graph-structured data have led to state-of-the-art performance on recommender system benchmarks. However, making these methods practical and scalable to web-scale recommendation tasks with billions of items and hundreds of millions of users remains a challenge. Here we describe a large-scale deep recommendation engine that we developed and deployed at Pinterest. We develop a data-efficient Graph Convolutional Network (GCN) algorithm PinSage, which combines efficient random walks and graph convolutions to generate embeddings of nodes (i.e., items) that incorporate both graph structure as well as node feature information. Compared to prior GCN approaches, we develop a novel method based on highly efficient random walks to structure the convolutions and design a novel training strategy that relies on harder-and-harder training examples to improve robustness and convergence of the model. We also develop an efficient MapReduce model inference algorithm to generate embeddings using a trained model. We deploy PinSage at Pinterest and train it on 7.5 billion examples on a graph with 3 billion nodes representing pins and boards, and 18 billion edges. According to offline metrics, user studies and A/B tests, PinSage generates higher-quality recommendations than comparable deep learning and graph-based alternatives. To our knowledge, this is the largest application of deep graph embeddings to date and paves the way for a new generation of web-scale recommender systems based on graph convolutional architectures.Comment: KDD 201