Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 1 / 31ana

Transcription

1 Topology identification of complex networks from noisy time series using ROC curve analysis Juan CHEN and Jun-an LU School of Mathematics and Statistics Wuhan University Oct , 2010 CCCN at Soochow University Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 1 / 31ana

2 Outline 1 Background 2 Noise Bridges Dynamics and Network Topology 2.1 A General Approach to Bridge Dynamics and Network Topology 2.2 Steps to Infer Network Topology from Dynamics 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis ROC Curves Construction Threshold Selection in ROC Curves Coupling Strength Comparison Noise Strength Comparison 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 2 / 31ana

3 1 Background 2 Noise Bridges Dynamics and Network Topology 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 3 / 31ana

4 Real-Life Networks Complex networks widely exist in natural and man-made systems Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 3 / 31ana

5 Real-Life Networks Complex networks widely exist in natural and man-made systems The study of complex networks pervades through almost all scientific disciplines Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 3 / 31ana

6 Real-Life Networks Complex networks widely exist in natural and man-made systems The study of complex networks pervades through almost all scientific disciplines Current focus: how the topology influence networks dynamics behavior Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 3 / 31ana

7 Structure vs Dynamics From topology to dynamics: stability of networks Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 4 / 31ana

8 Structure vs Dynamics From topology to dynamics: stability of networks From dynamics to topology: inferring network structures Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 4 / 31ana

9 Structure vs Dynamics From topology to dynamics: stability of networks From dynamics to topology: inferring network structures Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 4 / 31ana

10 Adaptive Control The exact network topological structures are sometimes unknown or uncertain, to identify their topological structures becomes a key problem Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 5 / 31ana

11 Adaptive Control The exact network topological structures are sometimes unknown or uncertain, to identify their topological structures becomes a key problem Can we estimate a network s topology by using its nodes dynamical behaviors? Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 5 / 31ana

12 Adaptive Control The exact network topological structures are sometimes unknown or uncertain, to identify their topological structures becomes a key problem Can we estimate a network s topology by using its nodes dynamical behaviors? Adaptive control: linear independence condition or persistent exciting condition Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 5 / 31ana

13 1 Background 2 Noise Bridges Dynamics and Network Topology 2.1 A General Approach to Bridge Dynamics and Network Topology 2.2 Steps to Infer Network Topology from Dynamics 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 6 / 31ana

14 2.1 A General Approach Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 6 / 31ana

15 2.2 Steps to Infer Network Topology Construct a dynamical correlation matrix C: C ij =< [x i (t) x(t)] [x j (t) x(t)] >, where x(t) = (1/N) N i=1 x i(t), and x i is some component with noise of the ith oscillators; Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 7 / 31ana

16 2.2 Steps to Infer Network Topology Construct a dynamical correlation matrix C: C ij =< [x i (t) x(t)] [x j (t) x(t)] >, where x(t) = (1/N) N i=1 x i(t), and x i is some component with noise of the ith oscillators; Compute ˆL = [σ 2 /(2c)]C + ; Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 7 / 31ana

17 2.2 Steps to Infer Network Topology Construct a dynamical correlation matrix C: C ij =< [x i (t) x(t)] [x j (t) x(t)] >, where x(t) = (1/N) N i=1 x i(t), and x i is some component with noise of the ith oscillators; Compute ˆL = [σ 2 /(2c)]C + ; Set l is twice the total number of links, calculate l through l = (Sσ 2 + S 2 σ 4 + 8cNSσ 2 )/4c and keep its integral part, where S = N i=1 1/C ii; Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 7 / 31ana

18 2.2 Steps to Infer Network Topology Rank all non-diagonal elements of the matrix ˆL in an ascending order, and set the threshold as the value of the lth one in the ascending-ordered matrix elements; Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 8 / 31ana

19 2.2 Steps to Infer Network Topology Rank all non-diagonal elements of the matrix ˆL in an ascending order, and set the threshold as the value of the lth one in the ascending-ordered matrix elements; Set all elements in ˆL with values above the threshold to be zero and others to be 1, the latter corresponding to existent links; Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 8 / 31ana

20 2.2 Steps to Infer Network Topology Rank all non-diagonal elements of the matrix ˆL in an ascending order, and set the threshold as the value of the lth one in the ascending-ordered matrix elements; Set all elements in ˆL with values above the threshold to be zero and others to be 1, the latter corresponding to existent links; The obtained new matrix is the Laplacian matrix L. Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 8 / 31ana

21 1 Background 2 Noise Bridges Dynamics and Network Topology 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 9 / 31ana

22 Consensus Dynamics Consider consensus dynamics: N ẋ i = c P ij (x j x i ) + η i, (1) j=1 where P = (P ij ) N N is the adjacency matrix, the matrix L = P K is to be identified. Network topology is chosen as friendship network of karate club. Topology identification of complex networks from noisy time series CCCN using 2010 ROC curve 9 / 31ana

23 Example 1 Set the coupling strength c = 0.1 and the noise strength σ 2 = 2. The distribution of non-diagonal elements of [σ 2 /(2c)]C + is shown as follows SREL= and SRNL= 1. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 10 curve / 31ana

24 Example 2 Set the coupling strength c = 1 and the noise strength σ 2 = 2. The distribution of non-diagonal elements of [σ 2 /(2c)]C + is shown as follows SREL= and SRNL= Topology identification of complex networks from noisy time series CCCNusing 2010ROC 11 curve / 31ana

25 Example 3 Set the coupling strength c = 0.1 and the noise strength σ 2 = The distribution of non-diagonal elements of [σ 2 /(2c)]C + is shown as follows SREL= and SRNL= Topology identification of complex networks from noisy time series CCCNusing 2010ROC 12 curve / 31ana

26 Reason for Failures Threshold is determined based on the hypothesis that L = [σ 2 /(2c)]C + accurately holds. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 13 curve / 31ana

27 Reason for Failures Threshold is determined based on the hypothesis that L = [σ 2 /(2c)]C + accurately holds. The Laplacian matrix L produced by the dynamical correlation matrix may not be completely consistent with the true topological structure in real case. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 13 curve / 31ana

28 Reason for Failures Threshold is determined based on the hypothesis that L = [σ 2 /(2c)]C + accurately holds. The Laplacian matrix L produced by the dynamical correlation matrix may not be completely consistent with the true topological structure in real case. The dynamical correlation matrix C cannot contain all the information about the true topology of a network, with improper coupling strength or noise strength. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 13 curve / 31ana

29 1 Background 2 Noise Bridges Dynamics and Network Topology 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis ROC Curves Construction Threshold Selection in ROC Curves Coupling Strength Comparison Noise Strength Comparison 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCNusing 2010ROC 14 curve / 31ana

30 4.1 ROC Curves Construction Receiver Operating Characteristic (ROC) curves analysis First used during World War II for the analysis of radar signals contaminated by noise Topology identification of complex networks from noisy time series CCCNusing 2010ROC 14 curve / 31ana

31 4.1 ROC Curves Construction Receiver Operating Characteristic (ROC) curves analysis First used during World War II for the analysis of radar signals contaminated by noise Then in medicine to determine a cutoff value for a clinical test Topology identification of complex networks from noisy time series CCCNusing 2010ROC 14 curve / 31ana

32 4.1 ROC Curves Construction Receiver Operating Characteristic (ROC) curves analysis First used during World War II for the analysis of radar signals contaminated by noise Then in medicine to determine a cutoff value for a clinical test to find a cutoff value that discriminate existent links from non-existent links Topology identification of complex networks from noisy time series CCCNusing 2010ROC 14 curve / 31ana

33 4.1 ROC Curves Construction Receiver Operating Characteristic (ROC) curves analysis First used during World War II for the analysis of radar signals contaminated by noise Then in medicine to determine a cutoff value for a clinical test to find a cutoff value that discriminate existent links from non-existent links to evaluate the impact of the coupling strength and noise strength on the network topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 14 curve / 31ana

34 4.1 ROC Curves Construction All the possible links into two groups Existent link positive; Non-existent link negative. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 15 curve / 31ana

35 4.1 ROC Curves Construction All the possible links into two groups Existent link positive; Non-existent link negative. The success rate of existent links (SREL): the ratio of the existent links found by the identification to the total number of existent links in a network true positive rate sensitivity. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 15 curve / 31ana

36 4.1 ROC Curves Construction All the possible links into two groups Existent link positive; Non-existent link negative. The success rate of existent links (SREL): the ratio of the existent links found by the identification to the total number of existent links in a network true positive rate sensitivity. The success rate of non-existent links (SRNL): the ratio of the non-existent links found by the identification to the total number of non-existent links true negative rate specificity. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 15 curve / 31ana

37 4.1 ROC Curves Construction Table: Schematic outcomes Actual value Identification Existent Non-existent Total Positive a (True positive) b (False positive) a + b Negative c (False negative) d (True negative) c + d Total a + c b + d a + b + c + d Sensitivity=True Positive Rate=SREL= a/(a + c) Specificity=True Negative Rate=SRNL= d/(b + d) False Positive Rate= b/(b + d) = 1 Specificity Topology identification of complex networks from noisy time series CCCNusing 2010ROC 16 curve / 31ana

38 4.1 ROC Curves Construction ROC curve: the true positive rate (sensitivity) versus the false positive rate (1-specificity) for all possible values of the cut-off points. Start at (0, 0): a threshold at which all are classified as non-existent links; end at (1, 1): a threshold at which all are classified as existent links. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 17 curve / 31ana

39 4.1 ROC Curves Construction ROC curve: the true positive rate (sensitivity) versus the false positive rate (1-specificity) for all possible values of the cut-off points. Start at (0, 0): a threshold at which all are classified as non-existent links; end at (1, 1): a threshold at which all are classified as existent links. Pass through (0, 1): 100% sensitivity(srel) and 100% specificity (SRNL) Topology identification of complex networks from noisy time series CCCNusing 2010ROC 17 curve / 31ana

40 4.1 ROC Curves Construction ROC curve: the true positive rate (sensitivity) versus the false positive rate (1-specificity) for all possible values of the cut-off points. Start at (0, 0): a threshold at which all are classified as non-existent links; end at (1, 1): a threshold at which all are classified as existent links. Pass through (0, 1): 100% sensitivity(srel) and 100% specificity (SRNL) The closer the ROC curve is to the upper left corner, the higher the overall accuracy of the identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 17 curve / 31ana

41 4.1 ROC Curves Construction ROC curve: the true positive rate (sensitivity) versus the false positive rate (1-specificity) for all possible values of the cut-off points. Start at (0, 0): a threshold at which all are classified as non-existent links; end at (1, 1): a threshold at which all are classified as existent links. Pass through (0, 1): 100% sensitivity(srel) and 100% specificity (SRNL) The closer the ROC curve is to the upper left corner, the higher the overall accuracy of the identification. Area under the ROC curve (AUC): 1 represents perfect discrimination; values close to 1.0 indicates high identification effectiveness. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 17 curve / 31ana

42 4.2 Threshold Selection in ROC Curves ROC curve passes the upper left corner (0, 1) with AUC = 1: the optimal corresponding threshold can discriminate perfectly the existent links from the non-existent links Topology identification of complex networks from noisy time series CCCNusing 2010ROC 18 curve / 31ana

43 4.2 Threshold Selection in ROC Curves Cut-off Sensivity Specificity False positive rate values (SREL) (SRNL) (1 SRNL) AUC= 1 Topology identification of complex networks from noisy time series CCCNusing 2010ROC 19 curve / 31ana

44 4.3 Coupling Strength Comparison AUC can be used as a measure of which coupling strength is better overall Topology identification of complex networks from noisy time series CCCNusing 2010ROC 20 curve / 31ana

45 4.3 Coupling Strength Comparison AUC can be used as a measure of which coupling strength is better overall The curve with larger AUC represents the coupling strength that will perform better Fix the noise strength σ 2 = 2 and change the coupling strength Topology identification of complex networks from noisy time series CCCNusing 2010ROC 20 curve / 31ana

46 4.3 Coupling Strength Comparison ROC curves with different coupling strength. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 21 curve / 31ana

47 4.3 Coupling Strength Comparison AUC first increases, then trends toward the constant 1.0, and finally decreases as the coupling strength increases. Too small or too large coupling strength leads to a failure in topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 22 curve / 31ana

48 4.3 Coupling Strength Comparison AUC first increases, then trends toward the constant 1.0, and finally decreases as the coupling strength increases. Too small or too large coupling strength leads to a failure in topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 22 curve / 31ana

49 Why? Small coupling strength: dynamical correlation matrix cannot contain all the information about the network topology, since the dynamical relation between every two nodes is very weak. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 23 curve / 31ana

50 Why? Small coupling strength: dynamical correlation matrix cannot contain all the information about the network topology, since the dynamical relation between every two nodes is very weak. Strong coupling: nodes in the network approximatively synchronize. That is to say, x i (t) ˆx(t) = 1 N N i=1 x i(t), t, 1 i N. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 23 curve / 31ana

51 Why? dynamical correlation C ij =< [x i (t) ˆx(t)] [x j (t) ˆx(t)] > between node i and node j will trends toward zero, C singular! Topology identification of complex networks from noisy time series CCCNusing 2010ROC 24 curve / 31ana

52 4.4 Noise Strength Comparison Noise: a bridge between dynamical correlation and topology; cannot be neglected in practice Topology identification of complex networks from noisy time series CCCNusing 2010ROC 25 curve / 31ana

53 4.4 Noise Strength Comparison Noise: a bridge between dynamical correlation and topology; cannot be neglected in practice Use AUC to measure the impact of the noise strength on the topology identification Topology identification of complex networks from noisy time series CCCNusing 2010ROC 25 curve / 31ana

54 4.4 Noise Strength Comparison Topology identification of complex networks from noisy time series CCCNusing 2010ROC 26 curve / 31ana

55 4.4 Noise Strength Comparison For strong coupling, AUC remains unchanged except the noise with extremely low strength; Topology identification of complex networks from noisy time series CCCNusing 2010ROC 26 curve / 31ana

56 4.4 Noise Strength Comparison For strong coupling, AUC remains unchanged except the noise with extremely low strength; For weak coupling, AUC always goes up and down with the change in the noise strength Topology identification of complex networks from noisy time series CCCNusing 2010ROC 26 curve / 31ana

57 4.4 Noise Strength Comparison The accuracy and efficiency of topology identification crucially depends on the coupling strength. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 27 curve / 31ana

58 4.4 Noise Strength Comparison The accuracy and efficiency of topology identification crucially depends on the coupling strength. Noise is an important existence to topology identification, however, decreasing or increasing the noise strength cannot clearly improve the effectiveness of topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 27 curve / 31ana

59 1 Background 2 Noise Bridges Dynamics and Network Topology 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCNusing 2010ROC 28 curve / 31ana

60 Summary Provide concrete examples for showing possible failure in topology identification from noisy time series; Use ROC curves to determine the optimal threshold to discriminate existent links from non-existent links; Use AUC to evaluate the impact of coupling strength and noise strength on the effectiveness of topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 28 curve / 31ana

61 Summary Provide concrete examples for showing possible failure in topology identification from noisy time series; Use ROC curves to determine the optimal threshold to discriminate existent links from non-existent links; Use AUC to evaluate the impact of coupling strength and noise strength on the effectiveness of topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 28 curve / 31ana

62 Summary Provide concrete examples for showing possible failure in topology identification from noisy time series; Use ROC curves to determine the optimal threshold to discriminate existent links from non-existent links; Use AUC to evaluate the impact of coupling strength and noise strength on the effectiveness of topology identification. Topology identification of complex networks from noisy time series CCCNusing 2010ROC 28 curve / 31ana

63 1 Background 2 Noise Bridges Dynamics and Network Topology 3 Examples Showing Failure of Topology Identification 4 Topology Identification by ROC Analysis 5 Summary 6 Related Work Topology identification of complex networks from noisy time series CCCNusing 2010ROC 29 curve / 31ana

64 Related work J. Ren, W.-X. Wang, B. Li and Y.-C. Lai, Noise briges dynamical correlation and topogy in complex oscillator networks, Physical Review Letters 104, (2010). J. Zhou, and J. Lu, Topology identification of weighted complex dynamical networks, Physica A 386, (2007). L. Chen, J. A. Lu and C. K. Tse, Synchronization: An Obstacle to Identification of Network Topology, IEEE Trans. Circuits Syst.-II 56, (2009). H. Liu, J. A. Lu, J. H. Lü and D. J. Hill, Structure Identification of Uncertain General Complex Dynamical Networks with Time Delay, Automatica-regular papers 45, (2009). Topology identification of complex networks from noisy time series CCCNusing 2010ROC 29 curve / 31ana

65 Related work J. Zhao, Q. Li, J. Lu and Z. P. Jiang, Topology Identification of complex dynamical networks, Chaos 20, (2010). W. Lin, and H. F. Ma, Failure of parameter identification based on adaptive synchronization techniques, Physical Review E 75, (2007). W. Yu, G. Chen, J. Cao, J. Lü, and U. Parlize, Parameter identification of dynamical systems from time series, Physical Review E 75, (2007). Topology identification of complex networks from noisy time series CCCNusing 2010ROC 30 curve / 31ana

66 Thank you! Author: CHEN Juan Address: School of Mathematics & Statistics Wuhan University Wuhan, , China Topology identification of complex networks from noisy time series CCCNusing 2010ROC 31 curve / 31ana