Dynamical Systems Induced on Networks Constructed from Time Series

Abstract

Several methods exist to construct complex networks from time series. In general, these methods claim to construct complex networks that preserve certain properties of the underlying dynamical system, and hence, they mark new ways of accessing quantitative indicators based on that dynamics. In this paper, we test this assertion by developing an algorithm to realize dynamical systems from these complex networks in such a way that trajectories of these dynamical systems produce time series that preserve certain statistical properties of the original time series (and hence, also the underlying true dynamical system). Trajectories from these networks are constructed from only the information in the network and are shown to be statistically equivalent to the original time series. In the context of this algorithm, we are able to demonstrate that the so-called adaptive k-nearest neighbour algorithm for generating networks out-performs methods based on -ball recurrence plots. For such networks, and with a suitable choice of parameter values, which we provide, the time series generated by this method function as a new kind of nonlinear surrogate generation algorithm. With this approach, we are able to test whether the simulation dynamics built from a complex network capture the underlying structure of the original system; whether the complex network is an adequate model of the dynamics.

Stability of Bounded Dynamical Networks with Symmetry

“…For recent applications using the mathematics of networks, see [15][16][17][18]. Some common results with graph theory can be found in [19][20][21][22][23].…”

“…with conditions of the following form: 5,9,13,17,21,25,29,33,36; 7,11,15,19,23,27,31; 5,9,13,17,21,25,29,33,36; 7,11,15,19,23,27,31; 10,14,22,26,34; 8,16,20,28,32. Under these conditions, we have some sort of symmetry at the steady states for the network under study.…”

“…A i = A i+1 = A i+2 , i = 1, 2, ..., 40; 10,14,22,26,34; 8,16,20,28,32. Note that because of the symmetry that appears in several parts at the steady states, every solution of a difference equation related to the B i ' s is periodic with a period of three.…”

Regenerating time series from ordinal networks

“…2 To investigate how much information is encoded in a network model of a time series, some studies have been undertaken to recover the original time series from the network, to use the network to reconstruct the phase space topology of the original system, or to generate new time series from the networks and compare these with the original. [3][4][5][6][7] In this paper, we apply this type of approach to the study of a special class of network based time series models called ordinal networks. 8 These are transition networks constructed based on the temporal succession of states defined by the ordinal symbolic dynamics of a time series.…”

Is Bach’s brain a Markov chain? Recurrence quantification to assess Markov order for short, symbolic, musical compositions

“…Specifically, McCullough et al 27 considered random walks on weighted ordinal transition networks constructed from time series derived from the Lorenz 28 and other dynam-150 ical systems. Campanharo et al 29 and Hou et al 30 performed analogous investigations of random walks on non-ordinal transition networks, but without employing RQA.…”