Physics Reports 689 (2017) 1–54 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Ranking in evolving complex networks Hao Liao a, Manuel Sebastian Mariani b,a, *, Matú² Medo c,d,b, *, Yi-Cheng Zhang b, Ming-Yang Zhou a a National Engineering Laboratory for Big Data System Computing Technology, Guangdong Province Key Laboratory of Popular High Performance Computers, College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060, PR China b Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland c Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, PR China d Department of Radiation Oncology, Inselspital, Bern University Hospital and University of Bern, 3010 Bern, Switzerland article info a b s t r a c t Article history: Complex networks have emerged as a simple yet powerful framework to represent and Accepted 15 May 2017 analyze a wide range of complex systems. The problem of ranking the nodes and the Available online 9 June 2017 edges in complex networks is critical for a broad range of real-world problems because Editor: I. Procaccia 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 Keywords: Complex networks policymakers, among others. This calls for a deep understanding of how existing ranking Ranking algorithms perform, and which are their possible biases that may impair their effectiveness. Centrality metrics Many popular ranking algorithms (such as Google's PageRank) are static in nature and, Temporal networks as a consequence, they exhibit important shortcomings when applied to real networks Recommendation that rapidly evolve in time. At the same time, recent advances in the understanding and Network science 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. ' 2017 Elsevier B.V. All rights reserved. Contents 1. Introduction...............................................................................................................................................................................................3 2. Ranking with static centrality metrics ....................................................................................................................................................4 2.1. Getting started: basic language of complex networks ..............................................................................................................4 2.2. Degree and other local centrality metrics ..................................................................................................................................4 2.2.1. Degree............................................................................................................................................................................4 2.2.2. H-index..........................................................................................................................................................................5 2.2.3. Other local centrality metrics ......................................................................................................................................5 2.3. Metrics based on shortest paths..................................................................................................................................................5 2.3.1. Closeness centrality......................................................................................................................................................6 2.3.2. Betweenness centrality ................................................................................................................................................6 2.4. Coreness centrality and its relation with degree and H-index .................................................................................................6 * Corresponding authors at: Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland. E-mail addresses: [email protected] (M.S. Mariani), [email protected] (M. Medo). http://dx.doi.org/10.1016/j.physrep.2017.05.001 0370-1573/' 2017 Elsevier B.V. All rights reserved. 2 H. Liao et al. / Physics Reports 689 (2017) 1–54 2.5. Eigenvector-based centrality metrics .........................................................................................................................................7 2.5.1. Eigenvector centrality ..................................................................................................................................................7 2.5.2. Katz centrality...............................................................................................................................................................8 2.5.3. Win–loss scoring systems for ranking in sport...........................................................................................................8 2.5.4. PageRank .......................................................................................................................................................................9 2.5.5. PageRank variants......................................................................................................................................................... 10 2.5.6. HITS algorithm .............................................................................................................................................................. 11 2.6. A case study: node centrality in the Zachary's karate club network ........................................................................................ 11 2.7. Static ranking algorithms in bipartite networks........................................................................................................................ 12 2.7.1. Co-HITS algorithm ........................................................................................................................................................ 13 2.7.2. Method of reflections ................................................................................................................................................... 13 2.7.3. Fitness-complexity metric ........................................................................................................................................... 13 2.8. Rating-based ranking algorithms on bipartite networks .......................................................................................................... 14 3. The impact of network evolution on static ranking algorithms ............................................................................................................ 15 3.1. The first-mover advantage in preferential attachment and its suppression............................................................................ 15 3.2. PageRank's temporal bias and its suppression........................................................................................................................... 16 3.3. Illusion of influence in social systems ........................................................................................................................................ 18 4. Time-dependent ranking algorithms ...................................................................................................................................................... 19 4.1. Striving for time-balance: Node-based time-rescaled metrics................................................................................................. 19 4.2. Metrics with explicit penalization for older edges and/or nodes ............................................................................................. 20 4.2.1. Time-weighted degree and its use in predicting future trends................................................................................. 20 4.2.2. Penalizing old edges: Effective Contagion Matrix ...................................................................................................... 21 4.2.3. Penalizing old edges: TimedPageRank........................................................................................................................ 21 4.2.4. Focusing on a temporal window: T-Rank, SARA ........................................................................................................ 22 4.2.5. PageRank with time-dependent teleportation: CiteRank.......................................................................................... 22 4.2.6. Time-dependent reputation algorithms ..................................................................................................................... 22 4.3. Model-based ranking of nodes.................................................................................................................................................... 23 5. Ranking nodes in temporal networks.....................................................................................................................................................