Superfamily phenomena and motifs of networks induced from time series

We introduce a transformation from time series to complex networks and then study the relative frequency of different subgraphs within that network. The distribution of subgraphs can be used to distinguish between and to characterize different types of continuous dynamics: periodic, chaotic, and periodic with noise. Moreover, although the general types of dynamics generate networks belonging to the same superfamily of networks, specific dynamical systems generate characteristic dynamics. When applied to discrete (map-like) data this technique distinguishes chaotic maps, hyperchaotic maps, and noise data.

[1]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[2]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[3]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[4]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[5]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[6]  Jürgen Kurths,et al.  Recurrence plots for the analysis of complex systems , 2009 .

[7]  A Vázquez,et al.  The topological relationship between the large-scale attributes and local interaction patterns of complex networks , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[8]  J. B. Gao,et al.  Recurrence Time Statistics for Chaotic Systems and Their Applications , 1999 .

[9]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[10]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[11]  O A Rosso,et al.  Distinguishing noise from chaos. , 2007, Physical review letters.

[12]  S. Shen-Orr,et al.  Superfamilies of Evolved and Designed Networks , 2004, Science.

[13]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[14]  Michael Small,et al.  Detecting temporal and spatial correlations in pseudoperiodic time series. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  R. Gilmore Topological analysis of chaotic dynamical systems , 1998 .

[16]  B. Pompe,et al.  Permutation entropy: a natural complexity measure for time series. , 2002, Physical review letters.

[17]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[18]  M Small,et al.  Complex network from pseudoperiodic time series: topology versus dynamics. , 2006, Physical review letters.

[19]  D. Ruelle,et al.  Recurrence Plots of Dynamical Systems , 1987 .

[20]  Schwartz,et al.  Method for generating long-range correlations for large systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Lucas Lacasa,et al.  From time series to complex networks: The visibility graph , 2008, Proceedings of the National Academy of Sciences.