Ranking in evolving complex networks

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. Many popular ranking algorithms (such as 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. At the same time, 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 network traffic, prediction of future links, and identification of significant nodes

Generalized quantum PageRank algorithm with arbitrary phase rotations

CAM/FEDER Project [S2018/TCS-4342]; Spanish MINECO/FEDER Project [PGC2018-099169- B-I00FIS2018]; MCIN; European Union NextGenerationEU [PRTR-C17.I1]; Ministry of Economic Affairs Quantum ENIA project; U.S. Army Research Office [W911NF-14-1-0103]; QUITEMAD grant; Universidad Complutense de Madrid-Banco Santander [CT58/21-CT59/21]The quantization of the PageRank algorithm is a promising tool for a future quantum internet. Here we present a modification of the quantum PageRank, introducing arbitrary phase rotations (APR) in the underlying Szegedy's quantum walk. We define three different APR schemes with only one phase as a degree of freedom. We have analyzed the behavior of these algorithms in a small generic graph, observing that a decrease of the phase reduces the standard deviation of the instantaneous PageRank, so the nodes of the network can be distinguished better. However, the algorithm takes more time to converge, so the phase cannot be decreased arbitrarily. With these results we choose a concrete value for the phase to later apply the algorithm to complex scale-free graphs. In these networks, the original quantum PageRank is able to break the degeneracy of the residual nodes and detect secondary hubs that the classical algorithm suppresses. Nevertheless, not all of the detected secondary hubs are real according to the PageRank's definition. Some APR schemes can overcome this problem, restoring the degeneration of the residual nodes and highlighting the truly secondary hubs of the networks. Finally, we have studied the stability of the new algorithms. The original quantum algorithm was known to be more stable than the classical. We have found that one of our algorithms, whose PageRank distribution resembles the classical one, has a stability similar to the original quantum algorithm.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEMinisterio de Economia y Competitividad (MINECO)/FEDERMinisterio de Ciencia e Innovación (MICINN)/AEIMinisterio de Economía y Competitividad (MINECO)Comunidad de Madrid/FEDERU.S. Army Research Office W911NF-14-1-0103Universidad Complutense de Madrid/Banco de Santanderpu

Graph Learning and Its Applications: A Holistic Survey

Graph learning is a prevalent domain that endeavors to learn the intricate relationships among nodes and the topological structure of graphs. These relationships endow graphs with uniqueness compared to conventional tabular data, as nodes rely on non-Euclidean space and encompass rich information to exploit. Over the years, graph learning has transcended from graph theory to graph data mining. With the advent of representation learning, it has attained remarkable performance in diverse scenarios, including text, image, chemistry, and biology. Owing to its extensive application prospects, graph learning attracts copious attention from the academic community. Despite numerous works proposed to tackle different problems in graph learning, there is a demand to survey previous valuable works. While some researchers have perceived this phenomenon and accomplished impressive surveys on graph learning, they failed to connect related objectives, methods, and applications in a more coherent way. As a result, they did not encompass current ample scenarios and challenging problems due to the rapid expansion of graph learning. Different from previous surveys on graph learning, we provide a holistic review that analyzes current works from the perspective of graph structure, and discusses the latest applications, trends, and challenges in graph learning. Specifically, we commence by proposing a taxonomy from the perspective of the composition of graph data and then summarize the methods employed in graph learning. We then provide a detailed elucidation of mainstream applications. Finally, based on the current trend of techniques, we propose future directions.Comment: 20 pages, 7 figures, 3 table

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples