Abstract
In this paper, we investigate network evolution dynamics using the Euler-Lagrange equation. We use the Euler-Lagrange equation to develop a variational principle based on the von Neumann entropy for time-varying network structure. Commencing from recent work to approximate the von Neumann entropy using simple degree statistics, the changes in entropy between different time epochs are determined by correlations in the degree difference in the edge connections. Our Euler-Lagrange equation minimises the change in entropy and allows to develop a dynamic model to predict the changes of node degree with time. We first explore the effect of network dynamics on three widely studied complex network models, namely (a) Erdős-Rényi random graphs, (b) Watts-Strogatz small-world networks, and (c) Barabási-Albert scale-free networks. Our model effectively captures the structural transitions in the dynamic network models. We also apply our model to a time sequence of networks representing the evolution of stock prices on the New York Stock Exchange (NYSE). Here we use the model to differentiate between periods of stable and unstable stock price trading and to detect periods of anomalous network evolution. Our experiments show that the presented model not only provides an accurate simulation of the degree statistics in time-varying networks but also captures the topological variations taking place when the structure of a network changes violently.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Wolstenholme, R.J., Walden, A.T.: An efficient approach to graphical modeling of time series. IEEE Trans. Signal Process. 63, 3266–3276 (2015). ISSN 1053-587X
Ye, C., Torsello, A., Wilson, R.C., Hancock, E.R.: Thermodynamics of time evolving networks. In: Liu, C.-L., Luo, B., Kropatsch, W.G., Cheng, J. (eds.) GbRPR 2015. LNCS, vol. 9069, pp. 315–324. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18224-7_31
Ernesto, E., Naomichi, H.: Communicability in complex networks. Phys. Rev. E 77, 036111 (2008)
Han, L., Wilson, R.C., Hancock, E.R.: Generative graph prototypes from information theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(10), 2013–2027 (2015)
Lacasa, L., Luque, B., Ballesteros, F., Luque, J., Nuno, J.C.: From time series to complex networks: the visibility graph. Proc. Nat. Acad. Sci. 105(13), 4972–4975 (2008)
Loukas, A., Simonetto, A., Leus, G.: Distributed autoregressive moving average graph filters. IEEE Signal Process. Lett. 22(11), 1931–1935 (2015)
Ye, C., Wilson, R.C., Hancock, E.R.: Correlation network evolution using mean reversion autoregression. In: Robles-Kelly, A., Loog, M., Biggio, B., Escolano, F., Wilson, R. (eds.) S+SSPR 2016. LNCS, vol. 10029, pp. 163–173. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49055-7_15
Silva, F.N., Comin, C.H., Peron, T.K.D., Rodrigues, F.A., Ye, C., Wilson, R.C., Hancock, E., Costa, L.F.: Modular dynamics of financial market networks, pp. 1–13 (2015)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Barabási, A.L., Albert, R., Jeong, H.: Mean-field theory for scale free random networks. Phys. A 272, 173–187 (1999)
Wang, J., Wilson, R.C., Hancock, E.R.: Minimising entropy changes in dynamic network evolution. In: Foggia, P., Liu, C.-L., Vento, M. (eds.) GbRPR 2017. LNCS, vol. 10310, pp. 255–265. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58961-9_23
Watts, D., Strogatz, S.: Collective dynamics of ‘small world’ networks. Nature 393, 440–442 (1998)
Wang, J., Wilson, R., Hancock, E.R.: Network entropy analysis using the Maxwell-Boltzmann partition function. In: The 23rd International Conference on Pattern Recognition (ICPR), pp. 1–6 (2016)
Wang, J., Wilson, R.C., Hancock, E.R.: Spin statistics, partition functions and network entropy. J. Complex Netw. 5, 1–25 (2017). https://doi.org/10.1093/comnet/cnx017
Han, L., Escolano, F., Hancock, E.R., Wilson, R.C.: Graph characterizations from von Neumann entropy. Pattern Recognit. Lett. 33, 1958–1967 (2012)
Passerini, F., Severini, S.: The von Neumann entropy of networks. Int. J. Agent Technol. Syst. 1, 58–67 (2008)
Yahoo! Finance. http://finance.yahoo.com
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Wang, J., Wilson, R.C., Hancock, E.R. (2018). Euler-Lagrange Network Dynamics. In: Pelillo, M., Hancock, E. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2017. Lecture Notes in Computer Science(), vol 10746. Springer, Cham. https://doi.org/10.1007/978-3-319-78199-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-78199-0_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78198-3
Online ISBN: 978-3-319-78199-0
eBook Packages: Computer ScienceComputer Science (R0)