Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-20T06:42:07.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of a network’s asymptotic behavior via its structure involving its strongly connected components

Published online by Cambridge University Press:  01 October 2019

Jan Treur Open the ORCID record for Jan Treur [Opens in a new window]
Jan Treur*
Affiliation:
Social AI Group, Department of Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, the Netherlands
*
Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, it is addressed how network structure can be related to asymptotic network behavior. If such a relation is studied, that usually concerns only strongly connected networks and only linear functions describing the dynamics. In this paper, both conditions are generalized. A couple of general theorems is presented that relates asymptotic behavior of a network to the network’s structure characteristics. The network structure characteristics, on the one hand, concern the network’s strongly connected components and their mutual connections; this generalizes the condition of being strongly connected to a very general condition. On the other hand, the network structure characteristics considered generalize from linear functions to functions that are normalized, monotonic, and scalar-free, so that many nonlinear functions are also covered. Thus, the contributed theorems generalize the existing theorems on the relation between network structure and asymptotic network behavior addressing only specific cases such as acyclic networks, fully, and strongly connected networks, and theorems addressing only linear functions. This paper was invited as an extended (by more than 45%) version of a Complex Networks’18 conference paper. In the discussion section, the differences are explained in more detail.

Keywords

Hocine CherifiLuis RochaStanley Wassermanasymptotic network behaviorstrongly connected componentsrelating network structure and network behavior
Type
Research Article
Information
Network Science , Volume 8 , Issue S1: Complex Networks 2018 , July 2020 , pp. S82 - S109
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ashby, W. R. (1960). Design for a brain: The origin of adaptive behavior, second extended edition (1st ed., 1952). London, England: Chapman and Hall.10.1037/11592-000CrossRefGoogle Scholar
Bloem, R., Gabow, H. N., & Somenzi, F. (2006). An algorithm for strongly connected component analysis in n log n symbolic steps. Formal Methods in System Design, 28, 37–56.10.1007/s10703-006-4341-zCrossRefGoogle Scholar
Bosse, T., Duell, R., Memon, Z. A., Treur, J., & van der Wal, C. N. (2015). Agent-based modelling of emotion contagion in groups. Cognitive Computation, 7(1), 111–136.10.1007/s12559-014-9277-9CrossRefGoogle Scholar
Chen, Y. (2009). General spanning trees and reachability query evaluation. In Desai, B. C. (Ed.), Proceedings of the 2nd Canadian conference on computer science and software engineering, C3S2E’09 (pp. 243–252). New York: ACM Press.Google Scholar
Drechsler, R. (2004). Advanced formal verification. Dordrecht: Kluwer Academic Publishers.10.1007/b105236CrossRefGoogle Scholar
Fisher, M. S. (2007). Software verification and validation: An engineering and scientific approach. New York, NY: Springer Science+Business Media.Google Scholar
Fleischer, L. K., Hendrickson, B., & Pınar, A. (2000). On identifying strongly connected components in parallel. In Rolim, J. (Ed.), Parallel and distributed processing. IPDPS 2000. Lecture Notes in Computer Science (vol. 1800, pp. 505–511). Berlin Heidelberg New York: Springer-Verlag.10.1007/3-540-45591-4_68CrossRefGoogle Scholar
Gentilini, R., Piazza, C., & Policriti, A. (2003). Computing strongly connected components in a linear number of symbolic steps. In Proceedings of the SODA’03 (pp. 573–582). Philadelphia: SIAM.Google Scholar
Haghighi, R., & Namazi, H. (2015). Algorithm for identifying minimum driver nodes based on structural controllability. Mathematical Problems in Engineering, 2015, Article ID 192307, 10.1155/2015/19230710.1155/2015/192307CrossRefGoogle Scholar
Harary, F., Norman, R. Z., & Cartwright, D. (1965). Structural models: An introduction to the theory of directed graphs. New York, NY: Wiley.Google Scholar
Kalman, R. E. (1963). Mathematical description of linear dynamical systems. Journal of the Society for Industrial and Applied Mathematics Series A Control, 1, 152.10.1137/0301010CrossRefGoogle Scholar
Karlsen, M., & Moschoyiannis, S. (2018). Evolution of control with learning classifier systems. Applied Network Science, 3, 30. 10.1007/s41109-018-0088-xCrossRefGoogle ScholarPubMed
Kuich, W. (1970). On the entropy of context-free languages. Information and Control, 16, 173–200.10.1016/S0019-9958(70)90105-1CrossRefGoogle Scholar
Łacki, J. (2013). Improved deterministic algorithms for decremental reachability and strongly connected components. ACM Transactions on Algorithms, 9(3), Article 27.10.1145/2483699.2483707CrossRefGoogle Scholar
Li, G., Zhu, Z., Cong, Z., & Yang, F. (2014). Efficient decomposition of strongly connected components on GPUs. Journal of Systems Architecture, 60(1), 1–10.10.1016/j.sysarc.2013.10.014CrossRefGoogle Scholar
Lin, C.-T. (1974). Structural controllability. IEEE Transactions on Automatic Control, 19, 201–208.Google Scholar
Liu, Y. Y., Slotine, J. J., & Barabasi, A. L. (2011). Controllability of complex networks. Nature, 473, 167–173.CrossRefGoogle ScholarPubMed
Liu, Y. Y., Slotine, J. J., & Barabasi, A. L. (2012). Control centrality and hierarchical structure in complex networks. PLOS One, 7(9), e44459.10.1371/journal.pone.0044459CrossRefGoogle ScholarPubMed
Moschoyiannis, S., Elia, N., Penn, A. S., Lloyd, D. J. B., & Knight, C. (2016). A web-based tool for identifying strategic intervention points in complex systems. In Brihaye, T., Delahaye, B., Jezequel, L., Markey, N., & Srba, J. (Eds.), Cassting workshop on games for the synthesis of complex systems and 3rd international workshop on synthesis of complex parameters (Cassting’16/SynCoP’16). EPTCS 220 (pp. 39–52). Electronic Proceedings in Theoretical Computer Science http://about.eptcs.org/.Google Scholar
Port, R. F., & van Gelder, T. (1995). Mind as motion: Explorations in the dynamics of cognition. Cambridge, MA: MIT Press.Google Scholar
Schoenmaker, R., Treur, J., & Vetter, B. (2018). A temporal-causal network model for the effect of emotional charge on information sharing. Biologically Inspired Cognitive Architectures, 26, 136–144.10.1016/j.bica.2018.10.003CrossRefGoogle Scholar
Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2), 146–160.10.1137/0201010CrossRefGoogle Scholar
Treur, J. (2016a). Network-oriented modeling: Addressing the complexity of cognitive, affective and social interactions. Cham: Springer Publishers.10.1007/978-3-319-45213-5CrossRefGoogle Scholar
Treur, J. (2016b). Verification of temporal-causal network models by mathematical analysis. Vietnam Journal of Computer Science, 3, 207–221.CrossRefGoogle Scholar
Treur, J. (2017). On the applicability of network-oriented modeling based on temporal-causal networks: Why network models do not just model networks. Journal of Information and Telecommunication, 1(1), 23–40.10.1080/24751839.2017.1295653CrossRefGoogle Scholar
Treur, J. (2018a). Relating emerging network behaviour to network structure. In Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds.) Proceedings of the 7th international conference on complex networks and their applications, ComplexNetworks’18, vol. 1. Studies in computational intelligence (vol. 812, pp. 619–634). Berlin, Heildelberg, New York: Springer.Google Scholar
Treur, J. (2018b). Mathematical analysis of a network’s asymptotic behaviour based on its strongly connected components. In Proceedings of the 7th international conference on complex networks and their applications, ComplexNetworks’18, vol. 1. Studies in computational intelligence (vol. 812, pp. 663–679). Springer.CrossRefGoogle Scholar
Treur, J. (2019). The ins and outs of network-oriented modeling: From biological networks and mental networks to social networks and beyond. Transactions on Computational Collective Intelligence, 32, 120–139.Google Scholar
Watts, D. J. (2002, April 30). A simple model of global cascades on random networks. Proceedings of the National Academy of Sciences of the United States of America, 99(9), 5766–5771.10.1073/pnas.082090499CrossRefGoogle ScholarPubMed
Wijs, A., Katoen, J. P., & Bošnacki, D. (2016). Efficient GPU algorithms for parallel decomposition of graphs into strongly connected and maximal end components. Formal Methods in System Design, 48, 274–300.10.1007/s10703-016-0246-7CrossRefGoogle Scholar