Community Detection in Flow Pattern Complex Network
Flow Pattern Complex Network (FPCN) [1 ], extracted from the conductance fluctuating signals, is an abstract network, in which each flow condition is represented by a single node and the edge is determined by the strength of correlation between nodes. Flo
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Community Detection in Flow Pattern Complex Network
4.1 Community Detection in Gas-Water Flow Pattern Complex Network 4.1.1 Flow Pattern Complex Network Flow Pattern Complex Network (FPCN) [1], extracted from the conductance fluctuating signals, is an abstract network, in which each flow condition is represented by a single node and the edge is determined by the strength of correlation between nodes. Flow condition refers to the flow behavior under different proportions of gas flow rate and water flow rate in the pipe. Since we configured 90 different proportions of gas flow rate and water flow rate to obtain 90 conductance fluctuating signals in the gas-water two-phase flow experiment, there are 90 different flow conditions (i.e., the number of nodes contained in FPCN is 90), in which each node corresponds to one of these 90 conductance fluctuating signals. Note that the correlation between two nodes characterizes the correlation between two corresponding conductance fluctuating signals. We now demonstrate how the strength of correlation between conductance fluctuating signals can be used to establish edges. With respect to the nonlinear characteristics of the gaswater two-phase flow, we first apply the method of Time-Delay Embedding [2] to process the conductance fluctuating signals. That is, we use C–C method [3] to calculate the delay time s from 90 conductance fluctuating signals, respectively, and choose the proper s that can maximize the FPCN modularity [4]. Then we extract six time-domain features and four frequency-domain features from each processed conductance fluctuating signal to conduct the characteristic vector (see the following Part A for details). Therefore, there are 90 characteristic vectors and each vector contains ten elements. For each pair of characteristic vectors, Ti and Tj , the correlation coefficient can be written as:
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_4, The Author(s) 2014
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Community Detection in Flow Pattern Complex Network
PM k¼1 ½Ti ðkÞ hTi i Tj ðkÞ Tj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:1Þ 2ffi ; PM PM 2 ½ T ðkÞ T T ðkÞ T h i i i j j k¼1 k¼1 M P Ti ðkÞ M, where M is the dimension of the characteristic vector and hTi i ¼ k¼1 M P Tj ðkÞ M. The elements Cij are restricted to the range 1 Cij 1, Tj ¼ k¼1
where Cij = 1, 0 and -1 correspond to perfect correlations, no correlations and perfect anti-correlations, respectively. C is a symmetric matrix and Cij describes the state of connection between node i and j. Finally, choosing a critical threshold rc (see the following Part B for details), the correlation matrix C can be turned into adjacency matrix A, the rules of which read: 1; ðCij rc Þ Aij ¼ ; ð4:2Þ 0; ðCij \rc Þ Which means there will be an edge connecting nodes i and j, if Cij rc . While there will not be an edge
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