Directed Weighted Complex Network for Characterizing Gas-Liquid Slug Flow
Recently, we have proposed a framework for inferring a directed weighted complex network from a time series [1 ]. We here introduce the analytical framework as follows: we start from construction of the Directed weighted complex network (DWCN).
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Directed Weighted Complex Network for Characterizing Gas-Liquid Slug Flow
8.1 Methodology Recently, we have proposed a framework for inferring a directed weighted complex network from a time series [1]. We here introduce the analytical framework as follows: we start from construction of the Directed weighted complex network (DWCN). Our first step is phase space reconstruction. Given a time series z(it) (i = 1, 2…, M), where t is the sampling interval and M is the sample size, we construct a sequence of phase-space vectors according to the standard delay-coordinate embedding method [2–4]: *
Xk ¼ fxk ð1Þ; xk ð2Þ; . . .:; xk ðmÞg ¼ fzðktÞ; zðkt þ sÞ; ; zðkt þ ðm 1ÞsÞg
ð5:1Þ
where s is the delay time, m is the embedding dimension, k = 1, 2,…, N, and N ¼ M ðm 1Þs=t is the total number of vector points in the reconstructed phase space. To construct a network, we then regard each vector point as a node and use the phase-space distance to determine the edges. Given two vector points *
*
Xi and Xj ði [ jÞ, the phase-space distance is defined to be dij ¼
m X Xi ðnÞ Xj ðnÞ
ð5:2Þ
n¼1 *
for all where Xi ðnÞ ¼ zði þ ðn 1ÞsÞ is the nth element of Xi . This generates, nodes (vector points) in the network, a distance matrix D ¼ dij where i [ j. By choosing a critical threshold value rc , we obtain the connections of the network: an edge connecting node i and j ði [ jÞ exists if dij rc ; while there is no edge between i and j if dij [ rc . We regard the time ði jÞ t as the weight of an edge that connected nodes i and j and the edge direction is from node i to j. Finally, we obtain the weight matrix W¼ wij , where wij ¼ 0 means node i and j are not connected, otherwise, wij 6¼ 0 implies an edge from node i to j exists and the edge
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_8, The Author(s) 2014
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8 Directed Weighted Complex Network for Characterizing Gas-Liquid Slug Flow
weight is wij ¼ ði jÞ t. The topology of the reconstructed DWCN is determined entirely by W. A key issue in extracting DWCN from time series is then the choice of the critical threshold rc . In this chapter, we exploit normalized maximum size of subgraph to determine the critical threshold. Take the Tent map 2x; if x\1=2 ð5:3Þ f ðxÞ ¼ 2ð1 xÞ; if x 1=2 as an example, we here theoretically demonstrate how to properly select the critical threshold. The locations of the periodic orbits of the tent map can be obtained explicitly. At each iteration, the map has two line segments of slope 2 and -2 in the unit square of the plane xnþ1 versus xn . The pth-iterated map has 2p line segments in the unit square. Fixed points of the pth-iterated map, which contain all periodic orbits of period p, are located at cross points of these 2p line segments with the line xnþ1 ¼ xn . Thus we have the following set of points that belong to different periodic orbits of period p: 2j=ð2P þ 1Þ; if
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