Recent years have seen an increase in the use of complex networks, in the demand for new effective tools for the analysis of complex systems and in requirement of new studies of time series. Numerous scholars have made significant contributions to this field. In 2006, Zhang and Small first implemented the complex networks for time series analysis by defining links among nodes based on temporal correlation measures . They then explored an extensive set of topological statistics for networks constructed from two pseudo-periodic time series with distinct dynamics . Xu, Zhang and Small embedded the time series in an appropriate phase space and considered each phase space point as a node with which to construct the network . Their work is the primary motivation for much of the current work on mapping time series to the complex networks . Lacasa et al. proposed the visibility graph method in which every node corresponds to series data if visibility exists between the corresponding data . They also presented the horizontal visibility algorithm, which is a geometrically simpler and analytically solvable version of their former algorithm . The horizontal visibility algorithm has been used to characterize and distinguish between correlated stochastic, uncorrelated and chaotic processes . Donner et al. constructed a recurrence network based on the concept of recurrences in phase space, which is a complex network that links different points in time if the considered states are close neighbors in phase space . Later, they used the recurrence network to identify the dynamic transitions in marine paleoclimate records . Gao et al. proposed another type of recurrence network and used it to analyze the bubbly oil-in-water flows . Gao et al. also constructed a directed weighted complex network (DWCN) from a time series and found that time series with different dynamics exhibit distinct topological properties. They used the DWCN to characterize the chaotic dynamic behavior of gas-liquid slug flow  and further proposed a multivariate recurrence network [11–14]. McCullough et al. have recently proposed the ordinal pattern partition networks that are formed from time series by symbolizing the data into ordinal patterns . The above mentioned methods have been used in different fields including medicine [16–18], astronomy , financial analysis  and geophysics .
Construction of a complex network from a time series faces two major challenges: (1) how to define nodes and (2) how to link nodes. In most of the approaches mentioned above, except for the pseudo-periodic time series and the ordinal pattern partition networks, a node is always defined as a value of the time series or a vector in the phase space reconstructed from the time series. However, a measure is required that can quantify the degree of correlation between two time series segments in order to reveal the dynamic states on different time scales in a realistic system. Kantz has presented the cross correlation interval measure that characterizes the degree of correlation between two finite sets . In this method, each set is a phase space reconstructed from a time series segment by embedding. The cross correlation interval has also been used in a separate study to construct a meta recurrence plot (MRP) for an efficient analysis of the characteristics of a time series . In this paper, the phase space reconstructed from each time series segment is considered as a node. The cross correlation interval is used to measure the degree of correlation between two nodes. An appropriate threshold is selected to determine whether two nodes are linked together or not. A complex network is then constructed based on this analysis.
Section 2 describes the proposed approach in detail. A one-dimensional time series generated from the Lorenz system is applied to select the parameters and to verify the efficiency of the approach. In Section 3, the authors construct and analyze the complex networks from time series generated from sea clutter data sets. The conclusion is presented in Section 4.
2 Constructing complex networks from time series
A time series of length N is divided into segments without overlapping and each segment is w = 2n in length. The dynamic states of the time series on different time scales can be observed by selecting different values for n.
According to Takens’ theorem , an appropriate embedding dimension m based on mutual information (MI) and embedding delay τ is selected using the approach of global false nearest neighbor (GFNN) to reconstruct the phase space for each time series segment. If represents a time series segment, then the vector in the phase subspace is (1)
Each phase space can be viewed as a node of the complex network. The number of nodes in the complex network is Nw, which is the floor-rounded N/w.
To estimate the degree of correlation between two phase spaces, the cross correlation interval previously introduced in  is used and is defined as: (2)
where I and J are the sequence numbers of the phase spaces and are also the numbers of nodes in the complex network. For the case I = J , a definition similar to Eq. (2) is used, where j < i and the normalization factor replaces w2. Θ(·) is the Heaviside function and || · || is the norm, in this work the maximum norm is selected. ε is the predetermined threshold. Based on the cross correlation interval, the distance DI,J between two phase spaces can be computed as: (3)
According to the nearest neighbor approach for the selection of the recurrence point , the values in D(ε) are sorted in ascending order. If the value fell in the top d percent, the corresponding node I and node J are considered linked.
2.2 Parameters selection
Three parameters, the length of each time series segment w (or the time scale n) and the thresholds d and ε, are important for constructing complex networks.
To explain how to select the parameters, x component of the chaotic Lorenz system (Eq. (4)) is selected to construct a complex network of length 16384. (4)
The phase space of each segment is reconstructed by assigning the values m = 3 and τ = 9. The phase space is taken as the node in the networks. The nodes should describe all types of phase space trajectories. The above mentioned Lorenz system has two attractors. Therefore, the phase space trajectories should cover one or both of the attractors. When the length of segment w is small, the phase space trajectories cannot cover both attractors (Figure 1(a) and 1(b)). When w is too large, all trajectories cover both of the attractors (Figure 1(e)). It is found that when w = 28, some phase space trajectories cover single attractor (Figure 1(c)) and the others cover both attractors (Figure 1(d)). Therefore, the time scale n = 8 is appropriate to divide the time series into segments to construct the network for the Lorenz system.
After evaluating the cross correlation intervals, the matrix Dw×w(ε) is obtained. The complex network can be constructed by sorting the values in Dw×w(ε). ε and d are two important parameters that influence the structure of the complex network. Using the software program “Pajek”, complex networks at different thresholds ε and d are plotted. Several of these complex networks are shown in Figure 2.
In Figure 2, at the same threshold ε, the complex network structures show very small change after a certain value of threshold d. Especially, when ε = 12, the complex networks at different thresholdsd (8%, 10% and 12%) have almost the same structures (Figure 2(b)). Therefore, d = 10% is selected.
Figure 2 also shows that the thresholds ε has more impact on the structures of the complex networks than the threshold d. When ε is small, a few isolated nodes exist in the complex network. Therefore, the connectivity of the network is low (Figure 2(a)). When ε is large, though two phase spaces are uncorrelated in some cases, the nodes representing these phase spaces are still linked (Figure 2(c)). Therefore, if the selected ε is too small, the connections between the nodes cannot be characterized completely. However, if the selected ε is too large, the complex networks will contain too many unnecessary links. Therefore, ε should be selected from an appropriate interval.
In order to determine an appropriate interval, two complex network properties, the clustering coefficient and the efficiency, at different ε values are calculated. The clustering coefficient C is a measure introduced by Watts and Strogatz  and can be defined as follows: (5)
where cI is the local clustering coefficient of node I and is defined as: (6)
The efficiency E of the complex network is defined as: (7)
where dIJ is the geodesic distance from node I to node J. Since any node pair belonging to disconnected components of the complex network yields a contribution equal to zero in the summation function in Eq. (7), this term can indicate the traffic capacity of a complex network and avoids the divergence of the average shortest path length (particularly when a node is disconnected from the others and dIJ = +∞) . The results are shown in Figure 3.
In Figure 3(a), the clustering coefficient are below 0.6 when the threshold ε is located in [1, 8]. This is because the nodes which should be connected are not linked at small thresholds. The clustering coefficients are above 0.6 and stable when the threshold ε is between 9 and 14. For threshold ε greater than 14, the clustering coefficients become unstable. Especially when ε = 19, the clustering coefficient decreases rapidly and then maintains the low value. It is shown that the redundancy connections make the local connectivity worse. Figure 3(b) shows that the efficiencies are lager and stable in [9, 12] and [16, 22]. In order to ensure better connectivity and transitivity of the complex network, ε should be selected from an interval in which both the clustering coefficient and the efficiency are higher. The above analysis shows that the desired interval is [9, 12].
2.3 Analysis of community structures
It is observed that the complex network in Figure 2(b) exhibits “community structures”, i.e., groups of nodes that have a high density of edges within them and have a lower density of edges between groups. Therefore, the relationship between the community structures and the dynamic states of the complex system is explored. A community is a subgroup whose nodes are tightly connected. Therefore, a fast algorithm proposed by Newman  can be used to detect these subgroups. To quantify a particular division of a network, a measure called modularity Q, is defined as follows . Let epq be the fraction of edges in the complex network that connect nodes in community p to those in community q. Let ap = ∑q epq, which represents the fraction of edges that connect to nodes in community p. Then (8)
The number of communities is determined by the maximum modularity Q. The numbers of communities yielded by at different thresholds ε is shown in Figure 4.
Figure 4 shows that the number of communities is always large for ε less than 9 due to the existence of the isolated nodes in the complex networks. The number of isolated nodes decreases with increase in ε and the number of communities becomes less accordingly. When ε is located in the interval [9, 12], the number of communities is 5 or 6. The minimum number of communities is 5 for threshold 12. When ε located in the feasible interval, as the number of communities decreases, the structures of the communities becomes clearer. Therefore, ε = 12 is selected to construct the complex network. In Figure 5, different colors are used to mark the nodes in different communities.
In order to explore the differences between the different community structures, five nodes are randomly selected from each community structure and the 2D projections of their corresponding phase spaces are plotted in Figure 6.
Figure 6 shows that the space trajectories represented by nodes in different communities are significantly different. It implies that there are 5 different states in the Lorenz system when the scale is 8. There are two attractors in the Lorenz system. It is observed from Figure 5 that the nodes of community 2, 3 and 4 are closely connected. The reason is that all of the trajectories of the nodes in community 2, 3 and 4 move between the two attractors. However, the suspended time on different attractors is different that separates them into 3 communities. Figure 6(c) shows that the suspended time is almost equal on the two attractors of the trajectory of node 53 in community 3. The trajectory of node 17 in community 2 stays longer on the upper attractor than the under one (Figure 6(b)).In contrast, the trajectory of node 29 in community 4 stays longer on the under attractor than the upper one (Figure 6(d)). However, the trajectories in community 1 and 5 are mainly within one of attractors (Figure 6(a) and Figure 6(e)). The states are two extreme situations. Therefore, the two communities are at the two extreme ends of the complex network and there is no direct link between them (Figure 5). It is found by observing the specific values of the projections that the trajectory of node 16 in community 1 stays on the upper attractor. Therefore, the community 1 and the community 2 are connected loosely. Correspondingly, the trajectory of node 62 in community 5 stays on the under attractor and the community 4 and the community 5 have loose connection.
In order to demonstrate the influence of the proposed our approach on the structures of the networks, the complex networks on different time scales are constructed in Figure 7 with parameters d = 10% and ε = 12.
Meanwhile, a time series of white Gaussian noise (10dBW) is mapped to the complex networks. The parameters for constructing the networks are same as the ones of Lorenz system: N = 16384, n = 8, m = 3, τ = 9 and d = 10%. The complex networks are plotted in Figure 8 at the thresholds ε = 1, 5, 10.
Comparing with the complex networks of the Lorenz system, Figure 8 shows that the structures of the complex networks of the noise are almost unchanged with changing threshold ε and there is no community structure in the networks. The reason is that the phase spaces of the noise time series segments have similar states and have no dynamic state. These results indicate that the different dynamic states of the complex systems can be described using the community structures. The dynamic states of the Lorenz system are already well known but the dynamic states of the realistic systems are unknown and complicated. Therefore, the proposed approach presents a new way to analyze the time series generated from realistic complex systems. The following section demonstrates the application of proposed approach for the analysis of sea clutter data.
3 Analysis of community structures of complex network constructed from sea clutter time series
Sea clutter data was collected during a measurement campaign at Osborne Head Gunnery Range (OHGR) in November 1993 with the Master University IPIX X-band radar . The radar was located on a cliff facing the Atlantic Ocean at a height of 100 feet above the mean sea level. Co and Cross-pol measurements were recorded and therefore, horizontal–horizontal (HH), vertical–vertical (VV), horizontal–vertical (HV) and vertical–horizontal (VH) polarizations are all available. Each dataset contains 14 spatial range bins of time series and the values in the time series are amplitudes. The length of time series of each range bin is 131072. In this work, the HH polarization data from the 6th range bin in dataset #17 is analyzed. The length of each time series segment is selected to be w = 29, 210, 211, 212, which is believed to be long enough to show the dynamic states of sea clutter system. Therefore, the numbers of time series segments Nw are 256, 128, 64 and 32, which are also the numbers of nodes in the complex networks. In another study  that analyzed the same sea clutter datasets as those in this paper, the embedding dimension was estimated to be 5 based on mutual information approach and the embedding delay was selected to be 11 using the global false nearest neighbor approach. Similar results are obtained in this study. According to the approach described in Section 2, the complex networks for the 6th range bin are constructed as shown in Figure 9.
From Figure 9, the complex network for w = 211 shows clearer community structure than other three complex networks. Therefore, the community structures of this complex network are analyzed and five communities are found as shown in Figure 10. The 2D phase space trajectory of each community is plotted in Figure 11.
From Figure 11(a) to Figure 11(e), the phase space trajectory changes from the contractive state to the open state. Therefore, the states of the phase spaces in the different communities are different, which demonstrates that the complex network constructed by the proposed approach is capable of uncovering the different dynamic states of sea clutter. Based on this work, the dynamics characteristics of sea clutter can be explored more deeply.
The experiment results show that for an unknown complex system, the dynamic states can be uncovered by studying the communities of the complex networks constructed by the proposed approach on the feasible time scale.
A new approach to map sea clutter time series data into a complex network is proposed in this paper. First, one-dimensional time series are divided into segments without overlapping. The phase space reconstructed from each time series segment is defined as a node of the complex network and the cross correlation interval is used to measure the degree of correlation between two nodes. Then, the clustering coefficients and the efficiencies of the complex networks under different thresholds are computed. After that an interval in which both the clustering coefficients and the efficiencies are large is selected in order to ensure that the complex networks have good connectivity and transitivity. Lastly, complex networks are constructed at the threshold values within the selected interval that can better describe the dynamic states of the time series. One-dimensional time series generated from the Lorenz system are also mapped into a complex network using the proposed approach and the community structures of the complex network are analyzed. The results have demonstrated that the reconstructed phase spaces represented by the nodes in different communities described the different dynamic states of the time series. Furthermore, the proposed approach is used to construct and analyze the complex networks of sea clutter data. The results have shown clear community structures, each corresponding to a different dynamic state. Meanwhile, the complex networks on all scales constructed from the white Gaussian noise time series have no clear community structures. It is consistent with the expected results because the white Gaussian time series has no clear dynamic state. Thus, all results have validated that the proposed approach is capable of uncovering the dynamical states of complex systems.
In this paper, the authors only focus on mapping the time series into complex networks using the proposed approach. In the further works, the dynamic characteristics of the realistic complex systems will be studied in detail by analyzing the dynamic states of the complex networks mapped by the proposed approach.
This work is supported by the Natural Science Foundation of China (41176076), the High Technology Research and Development Program of China (2014AA093400) and The National Key Research and Development Plan (2016YFC0301400).
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Published Online: 2017-04-30