Extending the Adapted PageRank Algorithm Centrality to Multiplex Networks with Data Using the PageRank TwoLayer Approach
Abstract
:1. Introduction
1.1. Literature Review
1.2. Main Contribution
1.3. Structure of the Paper
2. Methodology
2.1. The Adapted PageRank Algorithm (APA) Model
 It is nonnegative.
 It is stochastic by columns.
 The highest eigenvalue of P is $\lambda =1$.
Algorithm 1: (Adapted PageRank algorithm (APA)). Let $G=(V,E)$ be a primary graph representing a network with n nodes. 

2.2. The Biplex Approach for Classic PageRank
 ${l}_{1}$, physical layer, it is the network G.
 ${l}_{2}$, teleportation layer, it is an alltoall network, with weights given by the personalized vector.
2.3. Constructing the APABI Centrality by Applying the TwoLayer Approach
Algorithm 2: (Adapted PageRank algorithm biplex (APABI)). Let $\mathcal{M}=(\mathcal{N},\mathcal{E},\mathcal{S})$, with layers $\mathcal{S}=({l}_{1},{l}_{2})$ and adjacency matrices ${A}_{1},{A}_{2}$ be a biplex network with n nodes. 

2.4. A Note about the Computational Cost
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
APA  Adapted PageRank algorithm 
APABI  Adapted PageRank algorithm biplex 
References
 Newman, M. Networks: An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
 Bollobas, B. Modern Graph Theory; Springer: Berlin, Germany, 1998. [Google Scholar]
 Caluset, A.; Shalizi, C.; Newman, M. Powerlaw distributions in empirical data. SIAM Rev. 2009, 51, 661–703. [Google Scholar] [CrossRef]
 Porter, M. Smallworld network. Scholarpedia 2012, 7, 1739. [Google Scholar] [CrossRef]
 Boccaleti, S.; Latora, V.; Moreno, Y.; Hwang, D. Complex networks: structure and dynamics. Phys. Rep. 2006, 424, 175–308. [Google Scholar] [CrossRef]
 Fortunato, S. Community detection in graphs. Phys. Rep. 2010, 486, 75–174. [Google Scholar] [CrossRef] [Green Version]
 De Domenico, M.; Granell, C.; Porter, M.; Arenas, A. The physics of spreading processes in multilayer networks. Nat. Phys. 2016, 12, 901–906. [Google Scholar] [CrossRef] [Green Version]
 De Domenico, M.; SolèRibalta, A.; Cozzo, E.; Kivelä, M.; Moreno, Y.; Porter, M.; Gómez, S.; Arenas, A. Mathematical formulation of multilayer networks. Phys. Rev. 2013, 3, 041022. [Google Scholar] [CrossRef]
 Kivela, M.; Arenas, A.; Barthelemy, M.; Gleeson, J.; Moreno, Y.; Porter, M. Multilayer networks. J. Complex Netw. 2014, 2, 203–271. [Google Scholar] [CrossRef] [Green Version]
 Cellai, D.; Bianconi, G. Multiplex networks with heterogeneous activities of the nodes. Phys. Rev. 2016, 93, 032302. [Google Scholar] [CrossRef]
 De Domenico, M.; SolèRibalta, A.; Gómez, S.; Arenas, A. Navigability of interconnected networks under random failures. Proc. Natl. Acad. Sci. USA 2014, 111, 8351–8356. [Google Scholar] [CrossRef] [Green Version]
 Padgett, J.; Ansell, C. Robust Action and the Rise of the Medici. Am. J. Sociol. 2016, 98, 1259–1319. [Google Scholar] [CrossRef]
 Cardillo, A.; GómezGardeñes, A.; Zanin, M.; Romance, M.; Papo, D.; del Pozo, F.; Boccaletti, S. Emergence of network features from multiplexity. SIAM Rev. 2013, 3, 1–122. [Google Scholar] [CrossRef] [PubMed]
 De Domenico, M.; Lancichinetti, A.; Arenas, A.; Rosvall, M. Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems. Phys. Rev. X 2015, 5, 011027. [Google Scholar] [CrossRef]
 Battiston, S.; Caldarelli, G.; May, R.; Roukny, T.; Stiglitz, J. The price of complexity in financial networks. Proc. Natl. Acad. Sci. USA 2016, 113, 10031–10036. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Bentley, B.; Branicky, R.; Barnes, C.; Chew, Y.; Yemini, E.; Bullmore, E.; Vértes, P. The Multilayer Connectome of Caenorhabditis elegans. PLOS Comput. Biol. 2016, 12, e1005283. [Google Scholar] [CrossRef]
 Sola, L.; Romance, M.; Criado, R.; Flores, J.; Garcia del Amo, A.; Boccaletti, S. Eigenvector centrality of nodes in multiplex networks. Chaos 2013, 23, 033131. [Google Scholar] [CrossRef] [Green Version]
 Iacovacci, J.; Rahmede, C.; Arenas, A.; Bianconi, G. Functional Multiplex PageRank. arXiv, 2016; arXiv:1608.06328v2. [Google Scholar] [CrossRef]
 Bonacich, P. Power and centrality: A family of measures. Am. J. Sociol. 1987, 92, 1170–1182. [Google Scholar] [CrossRef]
 Bonacich, P. Simultaneous group and individual centrality. Soc. Netw. 1991, 13, 155–168. [Google Scholar] [CrossRef]
 Meiss, M.; Menczer, F.; Fortunato, S.; Flammini, A.; Vespignani, A. Ranking web sites with real user traffic. In Proceedings of the 2008 International Conference on Web Search and Data Mining (WSDM ’08), Palo Alto, CA, USA, 11–12 February 2008; pp. 65–76. [Google Scholar]
 Ghoshal, G.; Barabàsi, A.L. Ranking stability and superstable nodes in complex networks. Nat. Commun. 2011, 2, 394. [Google Scholar] [CrossRef] [Green Version]
 Cristelli, M.; Gabrielli, A.; Tacchella, A.; Caldarelli, G.; Pietronero, L. Measuring the intangibles: A metrics for the economic complexity of countries and products. PLoS ONE 2013, 8, e70726. [Google Scholar] [CrossRef]
 Crucitti, P.; Latora, V.; Porta, S. Centrality measures in spatial networks of urban streets. Phys. Rev. E 2006, 73, 036125. [Google Scholar] [CrossRef] [PubMed]
 Agryzkov, T.; Oliver, J.; Tortosa, L.; Vicent, J. An algorithm for ranking the nodes of an urban network based on the concept of PageRank vector. Appl. Math. Comput. 2012, 219, 2186–2193. [Google Scholar] [CrossRef]
 Berkhin, P. A survey on PageRank computing. Internet Math. 2005, 2, 73–120. [Google Scholar] [CrossRef]
 Bianconi, G. Multilayer Networks. Structure and Functions; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
 Halu, A.; Mondragón, R.; Panzarasa, P.; Bianconi, G. Multiplex PageRank. PLoS ONE 2013, 8, e78293. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 SoléRibalta, A.; De Domenico, M.; Gómez, S.; Arenas, A. Centrality Rankings in Multiplex Networks. In Proceedings of the 2014 ACM Conference on Web Science, Bloomington, IN, USA, 23–26 June 2014; ACM: New York, NY, USA, 2014; pp. 149–155. [Google Scholar] [CrossRef]
 Bobadilla, J.; Ortega, F.; Hernando, A.; Gutiérrez, A. Recommender systems survey. Knowl.Based Syst. 2013, 46, 109–132. [Google Scholar] [CrossRef]
 Stai, E.; Kafetzoglou, S.; Tsiropoulou, E.E.; Papavassiliou, S. A Holistic Approach for Personalization, Relevance Feedback and Recommendation in Enriched Multimedia Content. Multimedia Tools Appl. 2018, 77, 283–326. [Google Scholar] [CrossRef]
 Rabieekenari, L.; Sayrafian, K.; Baras, J. Autonomous relocation strategies for cells on wheels in environments with prohibited areas. In Proceedings of the 2017 IEEE International Conference on Communications (ICC), Paris, France, 21–25 May 2017; pp. 1–6. [Google Scholar] [CrossRef]
 Tsiropoulou, E.; Koukas, K.; Papavassiliou, S. A Sociophysical and MobilityAware Coalition Formation Mechanism in Public Safety Networks. EAI Endorsed Trans. Future Internet 2018, 4. [Google Scholar] [CrossRef]
 Pedroche, F.; Romance, M.; Criado, R. A biplex approach to PageRank centrality: From classic to multiplex networks. Chaos 2016, 26, 065301. [Google Scholar] [CrossRef]
 Agryzkov, T.; Tortosa, L.; Vicent, J.; Wilson, R. A centrality measure for urban networks based on the eigenvector centrality concept. Environ. Plan. B 2017, 291, 14–29. [Google Scholar] [CrossRef]
 Page, L.; Brin, S.; Motwani, R.; Winogrand, T. The Pagerank Citation Ranking: Bringing Order to the Web; Technical Report 199966; Stanford InfoLab: Stanford, CA, USA, 1999; Volume 66. [Google Scholar]
 Pedroche, F. Métodos de cálculo del vector PageRank. Bol. Soc. Esp. Mat. Apl. 2007, 39, 7–30. [Google Scholar]
 Agryzkov, T.; Pedroche, F.; Tortosa, L.; Vicent, J. Combining the TwoLayers PageRank Approach with the APA Centrality in Networks with Data. Int. J. GeoInform. 2018, 7. [Google Scholar] [CrossRef]
 Datta, B. Numerical Linear Algebra and Applications; Brooks/Cole Publishing Company: Pacific Grove, CA, USA, 1995. [Google Scholar]
Node  Social Networks Links  Messages  Game Links  Games 

1  $\left\{2,5,7,9,16,17,19,20\right\}$  15  $\left\{2,4,5,6,9,12,13,14,18,19\right\}$  33 
2  $\left\{1,5,7,9,20\right\}$  9  $\left\{1,4,8,10,13,18,19\right\}$  26 
3  $\left\{7,9,11,13,14,15,17\right\}$  12  $\left\{4,5,6,12,14,15,17,20\right\}$  18 
4  $\left\{5,9,11,14,15,16,18,20\right\}$  19  $\left\{1,2,3,5,6,7,8,9,10,11,12,13,14,16,17,18,20\right\}$  32 
5  $\left\{1,2,4,6,7,11,12,14,18,20\right\}$  28  $\left\{1,3,4,6,7,8,11,14,17,19,20\right\}$  20 
6  $\left\{5,7,10,20\right\}$  7  $\left\{1,3,4,5,8,11,12,19\right\}$  12 
7  $\left\{1,2,3,5,6,8,9,18,20\right\}$  20  $\left\{4,5,10,11,12,13,15,16,17,18,19,20\right\}$  32 
8  $\left\{7,10,12,17,18\right\}$  7  $\left\{2,4,5,6,9,12,13,14,18,19\right\}$  6 
9  $\left\{1,2,3,4,7,10,15,18,20\right\}$  16  $\left\{1,4,8,10,13,16,18,19\right\}$  18 
10  $\left\{6,8,9,11,13,14,15,16,18,20\right\}$  21  $\left\{2,4,7,9,13,15,18,19,20\right\}$  25 
11  $\left\{3,4,,5,10,13,18,19,20\right\}$  14  $\left\{4,5,6,7,12,14,15,17,20\right\}$  24 
12  $\left\{5,8,14,17,20\right\}$  8  $\left\{1,3,4,6,7,8,11,14,19,20\right\}$  18 
13  $\left\{3,10,11,15,19,20\right\}$  11  $\left\{1,2,4,7,8,9,10,16,17,19,20\right\}$  6 
14  $\left\{3,4,,5,10,12,16,18,19\right\}$  13  $\left\{1,3,4,5,8,11,12,19\right\}$  26 
15  $\left\{3,4,9,10,13,17,20\right\}$  11  $\left\{3,7,10,11,16,17,20\right\}$  38 
16  $\left\{1,4,10,14,17,18,19\right\}$  14  $\left\{4,7,9,13,15,18,19,20\right\}$  6 
17  $\left\{1,3,8,12,15,16,20\right\}$  12  $\left\{3,4,5,7,11,13,15,18,19\right\}$  12 
18  $\left\{4,5,7,8,9,10,11,14,16,19,20\right\}$  35  $\left\{1,2,4,7,8,9,10,16,17,19,20\right\}$  30 
19  $\left\{1,11,13,14,16,18,20\right\}$  15  $\left\{1,2,5,6,7,8,9,10,12,13,14,16,17,18,20\right\}$  8 
20  $\left\{1,2,4,5,6,7,9,10,11,12,13,15,17,18,19\right\}$  27  $\left\{3,4,5,7,10,11,12,13,15,16,18,19\right\}$  25 
Node  APA Layer ${\mathit{l}}_{1}$  APA Layer ${\mathit{l}}_{2}$  APABI  

Centrality  Ranking  Centrality  Ranking  Centrality  Ranking  
1  0.05025  7  0.06394  3  0.05581  7 
2  0.03110  17  0.04635  13  0.03777  16 
3  0.04063  13  0.04330  14  0.04193  13 
4  0.04891  9  0.08152  1  0.06517  3 
5  0.07731  3  0.05134  9  0.06440  5 
6  0.02494  20  0.03356  18  0.02862  20 
7  0.06157  5  0.06867  2  0.06477  4 
8  0.02791  19  0.03133  19  0.03071  19 
9  0.05481  6  0.03965  15  0.04836  11 
10  0.06530  4  0.05433  7  0.05902  6 
11  0.04936  8  0.05341  8  0.05013  9 
12  0.02852  18  0.04671  12  0.03727  17 
13  0.03626  16  0.03407  17  0.03684  18 
14  0.04776  10  0.05047  10  0.04820  12 
15  0.03902  15  0.06387  4  0.05085  8 
16  0.04550  12  0.02807  20  0.03781  15 
17  0.04020  14  0.03830  16  0.04033  14 
18  0.09182  2  0.06305  5  0.07590  2 
19  0.04700  11  0.04686  11  0.04838  10 
20  0.09186  1  0.06111  6  0.07775  1 
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Agryzkov, T.; Curado, M.; Pedroche, F.; Tortosa, L.; Vicent, J.F. Extending the Adapted PageRank Algorithm Centrality to Multiplex Networks with Data Using the PageRank TwoLayer Approach. Symmetry 2019, 11, 284. https://doi.org/10.3390/sym11020284
Agryzkov T, Curado M, Pedroche F, Tortosa L, Vicent JF. Extending the Adapted PageRank Algorithm Centrality to Multiplex Networks with Data Using the PageRank TwoLayer Approach. Symmetry. 2019; 11(2):284. https://doi.org/10.3390/sym11020284
Chicago/Turabian StyleAgryzkov, Taras, Manuel Curado, Francisco Pedroche, Leandro Tortosa, and José F. Vicent. 2019. "Extending the Adapted PageRank Algorithm Centrality to Multiplex Networks with Data Using the PageRank TwoLayer Approach" Symmetry 11, no. 2: 284. https://doi.org/10.3390/sym11020284