Recurrence Network for Characterizing Bubbly Oil-in-Water Flows
More recently, we have used the recurrence network to characterize the flow behavior of bubbly oil-in-water flows [1 ]. We here introduce methodology and obtained results as follows: Mapping a time series into a complex network allows quantitatively chara
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Recurrence Network for Characterizing Bubbly Oil-in-Water Flows
10.1 Recurrence Network Analysis of Time Series from Dynamic System More recently, we have used the recurrence network to characterize the flow behavior of bubbly oil-in-water flows [1]. We here introduce methodology and obtained results as follows: Mapping a time series into a complex network allows quantitatively characterizing the structural characteristics of complex systems that are composed of a large numbers of entities interacting with each other in a complex manner. Now we introduce how to transform a time series into a complex network in the framework of recurrence network. When dealing with a time series x(t) (t = 1,…,M), we can use a proper m-embedding dimension and a suitable stime delay to reconstruct x(t) [2], *
X ðtÞ ¼ ðxðtÞ; xðt þ sÞ; ; xðt þ ðm 1ÞsÞÞ
ð10:1Þ
*
to obtain a phase space with N phase space vector points X ðtÞ; t ¼ 1; 2; . . .; N, where N ¼ M ðm 1Þs
ð10:2Þ
Specifically, we use the method based on distinguishing false nearest neighbors (FNNs) [3] to determine the embedding dimension m and employ the correlationintegral-based algorithm [4] to determine the delay time s. According to Refs. [5–7], *
in the framework of recurrence network, a phase space vector X ðti Þ is said to be recurrent if there is tj 6¼ ti such that * * d X ðti Þ; X tj \e ð10:3Þ where d(, ) is the distance measure in phase space
Z.-K. Gao et al., Nonlinear Analysis of Gas-Water/Oil-Water Two-Phase Flow in Complex Networks, SpringerBriefs on Multiphase Flow, DOI: 10.1007/978-3-642-38373-1_10, The Author(s) 2014
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10 Recurrence Network for Characterizing Bubbly Oil-in-Water Flows
* * * * d X ðti Þ; X tj ¼ X ðti Þ X tj
ð10:4Þ
Then we can obtain the recurrence matrix that represents the configuration of recurrences in the phase space in terms of d(, ) as follows * * ð10:5Þ Rij ¼ H e d X ðti Þ X tj where HðÞ is the Heaviside function. Consequently, we can infer an un-weighted and un-directed complex network from a given time series x(t) by interpreting the recurrence matrix as the network adjacency matrix. In particular, in order to avoid self-loops in the network, the adjacency matrix of the recurrence network can be defined as: Aij ¼ Rij dij
ð10:6Þ
where dij is the Kronecker delta introduced in [5–7]. For a recurrence network, individual phase space vector serves as a node and the existence of an edge indicates the occurrence of a recurrence, i.e., the distance measure between a pair of nodes in the phase space is smaller than the threshold value e. We using the above method generate a recurrence network from a chaotic time series from the x component of Duffing system €x þ d_x bx þ ax3 ¼ c cos wt
ð10:7Þ
where a ¼ b ¼ 1:0; d ¼ 0:2; c ¼ 0:36 and w ¼ 1:0. We show the structure of the generated network in Fig. 10.1a. It should be noted that the network structure shown in Fig. 10.1a is drawn by the software ‘‘Pajek’’, and the computational algorithm is the Kamada-Kawai sp
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