Abstract
A cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. An indepth understanding of which cycles are important and what role they play on network structure and dynamics, however, is still lacking. In this paper, we define the cycle number matrix, a matrix enclosing the information about cycles in a network, and the cycle ratio, an index that quantifies node importance. Experiments on real networks suggest that cycle ratio contains rich information in addition to wellknown benchmark indices. For example, node rankings by cycle ratio are largely different from rankings by degree, Hindex, and coreness, which are very similar indices. Numerical experiments on identifying vital nodes for network connectivity and synchronization and maximizing the early reach of spreading show that the cycle ratio performs overall better than other benchmarks. Finally, we highlight a significant difference between the distribution of shorter cycles in real and model networks. We believe our indepth analyses on cycle structure may yield insights, metrics, models, and algorithms for network science.
Introduction
The last two decades have witnessed extensive development in network science (NS)^{1}, with research focuses being shifted from discovering macroscopic properties^{2,3,4} to uncovering the functional roles played by microscopic structures, or even individual nodes and links^{5,6,7}. Scientists have pieced an increasingly clear picture about the functions of specific structures in disparate dynamical processes, such as the roles of different motifs in biological and communication networks^{5}, how information and behaviors propagate along a contacting chain^{8}, and how a local star structure selfsustains an epidemic spreading process^{9,10}.
Besides extensively studied chain and star structures, cycle is another ubiquitously observed structure^{11}, which plays significant roles in both structural organization and functional implementation. A cycle, also called loop in literature, can be simply defined as a closed path with the same starting and ending node. Recent studies have uncovered the topological properties of cycles, including the distribution of cycles of different sizes in real and artificial networks^{12,13,14,15,16}, the effect of degree correlations on the loops of scalefree networks^{17}, as well as the significant roles of the cycles in network functions related to storage^{18}, synchronizability^{19}, and controllability^{20}. Cycles are also used as a tool to measure the extent of network being close to tree networks, and thus a significant difference between model networks and real networks is found, that is, the former can’t accurately reproduce the cycle structure in the latter^{21}. In addition, the organization of cycles can be utilized to characterize individual nodes and links. For example, a measure called clustering coefficient (also called local clustering coefficient)^{2} is based on counting the number of associated triangles (triangle is the cycle with smallest size), which recently considered the associated cycles with larger sizes^{11,22,23}, and was extended to the higher order cases^{22} and the weighted cases^{24,25}. The edge multiplicity measures the number of triangles passing through an edge^{26}. The effect of the addition of a noneobserved link on the local organization of cycles can be used to estimate the likelihood of the existence of this link^{27}, and the probability a selfavoid random walker returns to the target node through a cycle (cycles with different lengths are assigned to different weights) can be used to quantify the importance of the target node^{28}.
Considering a simple network where direction and weight of a link are ignored and selfloops are not allowed, then a cycle is the simplest structure providing redundant paths to all involved node pairs. That is to say, if two nodes belong to a cycle, there are at least two independent paths connecting them. Such redundancy also brings complicated feedbacks in interacting dynamics. Therefore, the indepth understanding of cycle structure may provide insights and methods on how to maintain the network connectivity under attacks^{29}, how to regulate interacting dynamics toward predesigned states^{30} and how to maximize the early reach of spreading in short time^{31}.
In this paper, according to the cyclebased statistics, we propose a matrix (named cycle number matrix) to represent cycle information of network, and an index (named cycle ratio) to quantify the importance of individual nodes. This index is essentially different from wellknown indices and methods^{7}, producing a much different ranking of nodes comparing with degree^{3}, Hindex^{32}, and coreness^{10}. Extensive experiments on real networks in identifying the most vulnerable nodes under intentional attacks^{33,34}, the most efficient nodes in pinning control^{35,36,37} and the most influential nodes in the early stage of epidemic spreading^{31,38} show that cycle ratio performs overall better than other benchmarks including degree, Hindex, and coreness. Finally, we highlight a significant difference between the distribution of shorter cycles in real and model networks.
Results
Definition of cycle ratio
Considering a simple network \(G(V,E)\), where V and E are the sets of nodes and links, respectively. The size of a cycle equals the number of links it contains. The cycles containing node i with the smallest size are defined as node i’s associated shortest cycles (also called i’s shortest cycles for simplicity) and the corresponding size is called node i’s girth^{19}. Denote by \(S_i\) the set of the shortest cycles associated with node i, and \({{{{{S}}}}} = \cup _{{{{{{i}}}}} \in V}S_{i}\) the set of all shortest cycles of G, we define the socalled cycle number matrix \(C = \left[ {c_{ij}} \right]_{N \times N}\) to characterize the cycle structure of G, where N=V is the number of nodes in G, and \(c_{ij}\) is the number of cycles in S that pass through both nodes i and j if \(i \ne j\). If \(i = j\), \(c_{ii}\) is the number of cycles in S that contain node i. Obviously, \(C\) is a symmetric matrix. Based on the cycle number matrix, we propose an index, named cycle ratio, to measure a node’s importance as
According to the above definition, if a node i doesn’t belong to any cycle in S, its cycle ratio is reasonably set to be zero. When \(c_{ii} > 0\), all items in the summation are well defined since \(c_{jj} > 0\) if \(c_{ij} > 0\). The ratio estimates the importance of node i subject to its participation to other nodes’ shortest cycles in S. Note that, in our definition, only shortest cycles associated with each node are considered since cycles with larger sizes are usually less relevant to the network functions (we have also tested on longer cycles, see details in Discussion) and to account for all cycles is infeasible for most networks due to the tremendous computational complexity^{27} (Supplementary Fig. 1 in Supplementary Note 1 shows the number of cycles with different lengths, indicating an exponential growth). Figure 1a presents an example network, and Fig. 1b shows the corresponding cycle number matrix. The process to calculate the cycle ratio of an example node (i.e., node 1) is also shown in Fig. 1b. In Eq. 1, each term represents the degree to which node i (i=1 for this example) participates in \(j\)’s associated shortest cycles (\(j = 1,2,3,4\,{{{{{{{\mathrm{and}}}}}}}}\,5\) for this example) in which denominator is the number of shortest cycles of node \(j\), and the numerator is the number of cycles associated with both node i and node \(j\). For example, the second term in the example equation in Fig. 1b, 3/4, means that three of the four shortest cycles of node 2 ({2, 3, 1}, {2, 4, 1}, {2, 1, 5}, {2, 4, 3}) contain node 1. In a word, \(r_1\) represents the degree to which node 1 participates in associated shortest cycles of other nodes. The cycle ratios of all nodes are presented in Fig. 1c. Three wellknown node centralities, degree^{3}, Hindex^{32}, and coreness^{10} (see precise definitions of these indices in Methods), are used as benchmarks for comparison. Their values for this example network are also presented in Fig. 1c.
Data
We test the performance of cycle ratio in identifying vital nodes subject to three wellstudied dynamical processes, node percolation^{33,34}, synchronization^{30}, and epidemic spreading^{38}. The first one considers nodes’ ability to maintain the network connectivity, the second one accounts for nodes’ capacity to regulate interacting dynamics toward a certain predesigned state, and the last one concentrates on infected nodes’ reach in the early stage of an epidemic outbreak. The experiments are carried out on six real networks from disparate fields, including the neural network of C. elegans (C. elegans)^{39}, the email communication network of the University at Rovira i Virgili in Spain (Email)^{40}, the collaboration network of jazz musicians (Jazz)^{41}, the collaboration network of scientists working on NS^{42}, the US air transportation network (USAir)^{32}, and the proteinprotein interaction network of yeast (Yeast)^{43}. Their basic topological features are summarized in Table 1.
Correlation analysis
Before penetrating into each index’s ability to identify vital nodes, we first see whether cycle ratio contains rich information in addition to the three benchmarks. We apply the Kendall’s Tau (\(\tau\))^{44,45} to measure the correlation between pairs of indices (see the definition of \(\tau\) in Methods). Given two indices X and Y, if \(\tau (X,Y)\) is close to 1, it indicates that X and Y are highly correlated and less differential to each other. Figure 2 shows the average correlation matrix between all index pairs for the six networks (the correlation matrix for each network is shown in Supplementary Fig. 2 in Supplementary Note 2), one can clearly observe that the correlations between degree, Hindex, and coreness are markedly high than the correlations between cycle ratio and the other three, the average of \(\tau\) over the six networks is 0.89 for the former and 0.61 for the latter. That is to say, the resulted node rankings produced by degree, Hindex, and coreness are very similar to each other. Therefore, although the performance of Hindex or coreness in some specific tasks is better than degree^{10,32}, the node rankings produced by Hindex and coreness contain less information in addition to the one produced by degree, and vice versa. In contrast, as suggested by the lower correlations, the node rankings produced by cycle ratio have rich information in addition to these produced by degree, Hindex, and coreness. This is a very important yet easytobeignored marker about the potential value of the proposed index since the lower correlations between the proposed index and known indices indicate a higher possibility that the proposed index will provide insights beyond known indices. Besides, Supplementary Note 3 shows the distributions of the four indices for the six real networks under consideration. One can observe that the distinguishability of cycle ratio with most fractions is good while the distinguishability of coreness is poor.
We are interested in comparing the difference between cycle ratio and local clustering coefficient which is the simplest index based on the neighborhood cycles. The local clustering coefficient of a node in network is the fraction of triangles that actually exist over all possible triangles in its neighborhood. In despite of the conceptual overlap, cycle ratio is largely different from local clustering coefficient in three aspects: (i) the considered shortest cycles (i.e., cycles in \(S\)) are not necessarily to be triangles; (ii) cycle ratio is not a local index since even node \(i\) and node j are distant in a network, the value of \(c_{ij}\) can be nonzero; (iii) cycle ratio is not a ratio but the sum of ratios, and thus its value can be greater than 1. Supplementary Note 4 compares the difference between cycle ratio and clustering coefficient in detail and shows that the correlations between clustering coefficient and the other four indices are the lowest. Although clustering coefficient can reflect the local connection, it cannot reflect the importance of a node. Notice that, due to the sparsity and hierarchical organization of many real networks, the local clustering coefficient is usually negatively correlated with degree (typically, local clustering coefficient scales as \(k^{  1}\))^{46,47}, and thus not a good index for influential nodes. Similarly, Supplementary Fig. 12 in Supplementary Note 5 shows that the correlations between eigenvector centrality^{48} and the other four indices are low.
Figure 3 presents visualized Yeast network corresponding to the resulted rankings by the four indices. Very intuitively, the vital nodes selected by degree, Hindex, and coreness are densely connected with each other and clustered in a certain region, in consistent to the socalled richclub phenomenon^{49,50}. As a contrast, the vital nodes selected by cycle ratio are scattered in the whole network with sparser connections among them. This is a significant advantage of cycle ratio if one would like to find out a set of vital nodes, because if the selected vital nodes tend to be clustered to each other, their influential areas will be highly overlapped and thus their collective influences are probably weaker^{10,51,52}. Therefore, we believe the indepth analyses of cycle ratio may uncover insights that cannot be directly obtained by other benchmark centralities.
Percolation
To evaluate the importance of nodes in maintaining the network connectivity, we study the node percolation dynamics^{33,34}. Given a network, we remove one node at each time step and calculate the size of the largest component of the remaining network until the remaining network is empty. The metric called Robustness^{53} is used to measure the performance, defined as
where the relative size \(g(n)\) is the number of nodes in the largest component divided by N after removing n nodes. The normalization factor 1/N ensures that the values of R of networks with different sizes can be compared. For each index, we compute once to get a fixed ranking of nodes. The node with largest index value is removed preferentially. Obviously, a smaller R means a quicker collapse and thus a better performance. Figure 4a shows the collapsing processes in the six real networks, resulted from the node removal by cycle ratio and the other three indices. For the majority of the considered networks, cycle ratio leads to much faster collapse than other indices. Figure 4b exhibits the Robustness R, from which one can see that the cycle ratio is overall the best index in identifying the most vital nodes in maintaining the network connectivity. In addition, Supplementary Figs. 10 and 13 in Supplementary Note 4 and Supplementary Note 5 respectively show the results of clustering coefficient and eigenvector centrality, respectively, and the same conclusion can be obtained.
Pinning control
We next evaluate the importance of nodes by measuring the effect caused by pinning these nodes in a synchronizing process^{35,36}. Considering a general case where a simple connected network \(G\left( {V,E} \right)\) is consisted of N linearly and diffusively coupled nodes, with an interacting dynamics as
where the vector \(x_i \in {{{{{{{\mathbf{R}}}}}}}}^n\) is the state of node i, the function \(f( \cdot )\) describes the selfdynamics of a node, the positive constant \(\sigma\) denotes the coupling strength, U_{i} is the controller applied at node i, and the inner coupling matrix \({{\Gamma }}:{{{{{{{\mathbf{R}}}}}}}}^n \to {{{{{{{\mathbf{R}}}}}}}}^n\) is positive semidefinite. The Laplacian matrix \(L = [l_{ij}]_{N \times N}\) of G is defined as follows. If \((i,j) \in E\), then \(l_{ij} =  1\); if \((i,j)\, \notin\, E\) and \(i\, \ne\, j\), then \(l_{ij} = 0\); if \(i = j\), then \(l_{ii} =  \mathop {\sum}\nolimits_{j \ne 1} {l_{ij}}\). The goal of pinning control is to drive the system from any initial state to the target state in finite time by pinning some selected nodes. Analogous to the node percolation, all nodes are ranked in the descending order by a given index. Then, we successively pin nodes one by one according to the ranking and quantify the synchronizability of the pinned networks, which can be measured by the reciprocal of the smallest nonzero eigenvalue of the principal submatrix^{54,55} (a smaller value corresponds to a higher synchronizability), namely \(1/\mu _1(L_{  Q})\), where Q is the number of pinned nodes, \(L_{  Q}\) is the principal submatrix, obtained by deleting the Q rows and columns corresponding to the Q pinned nodes from the original Laplacian matrix L, and \(\mu _1(L_{  Q})\) is the smallest nonzero eigenvalue of \(L_{  Q}\). Inspired by the metric Robustness, we propose a similar metric named pinning efficiency to characterize the performance of an index subject to pinning control, as
where \(Q_{{{\mathrm{max}}}}\) is the maximum number of pinned nodes under simulation. Here we set \(Q_{{{\mathrm{max}}}} = 0.3N\), and we have checked that the choices of \(Q_{{{\mathrm{max}}}}\) will not affect the conclusion. Figure 5a shows how \(1/\mu _1(L_{  Q})\) decays with increasing number of pinned nodes. Obviously, a faster decay corresponds to a better performance. Figure 5b compares the pinning efficiency of the four indices. Similar to the result of the node percolation, cycle ratio is overall the best index in identifying the most efficient nodes in pinning control. In addition, Supplementary Tables 1 and 2 in Supplementary Note 4 and Supplementary Note 5 respectively show the results of clustering coefficient and eigenvector centrality and the same conclusion can be obtained.
Epidemic spreading
Lastly, we consider the spreading dynamics. Since in viral marketing and online information transmission, people are more interested in maximizing the reach in short time, and in epidemiological control, the most critical issue is the spreading range and control measures in the early stage of outbreak (e.g., see the discussion of the efficacy of early control measures for COVID19^{56,57}, we concentrate on the fast influencers that play the dominant role in the early stage^{31}. To quantify the influence of a set of selected nodes, we simulate the standard susceptibleinfectedrecovered (SIR) spreading dynamics^{38}, where at each time step, each susceptible node will be infected by an infected neighbor with probability \(\beta\), and each infected node will be recovered with probability \(\gamma\). Initially, the top0.1N nodes selected by each index are set to be infected and others are susceptible. The indices are ranked by cumulative infected nodes at a certain time step t, the more the better. We consider the case at \(\beta = \beta _c\) and \(\gamma = 1\), where
is the spreading threshold^{9,38} when \(\gamma = 1\). Here \(\langle k \rangle\) and \(\langle k^2 \rangle\) are the average degree and the average squared degree, respectively. Figure 6 reports the rankings of the four indices at time steps \(t = 1\), \(t = 2\), \(t = 4\) and \(t = 8\), where the values are averaged over 2000 independent runs. The bestperformed index is ranked No. 1, the runner up is ranked No. 2, …, and the worst one is ranked No. 4. Among the 24 matches (i.e., 6 networks and 4 time steps), cycle ratio gets ranked No. 1 for 23 times and No. 2 for 1 time, it dramatically outperforms other indices. In addition, Supplementary Figs. 11 and 14 in Supplementary Note 4 and Supplementary Note 5 respectively show the results of clustering coefficient and eigenvector centrality and the same conclusion can be obtained. The results for more \(\left( {\beta ,t} \right)\) parameter sets are presented in Supplementary Note 6. In fact, the spreading capacity of cycle ratio is superior to coreness for both singlesource and multiplesource cases, including fast spreading (considering the performance at the early stage) and complete spreading (see Supplementary Note 7).
In addition to real networks, we have also analyzed two types of synthetic networks, the Erdős–Rényi (ER) networks^{58} and Barabási–Albert (BA) networks^{3}. The overall performance of cycle ratio is just in the middle of the four indices. The reason for the notsogood performance may be that the random networks are less localized (as indicated by the very small clustering coefficient) with lengths of shortest cycles (i.e., cycles in S) being relatively longer than real networks with similar sizes and densities (see Supplementary Note 8 and Supplementary Note 9), and thus effects of cycles on dynamical processes are weaker^{6,59}.
Discussion
To represent cycle information of a network, this paper defines a matrix, called cycle number matrix, with which an index, called cycle ratio, can be calculated to quantify the importance of an individual node by simply measuring to which extent it is involved in other nodes’ associated shortest cycles. The basic idea underlying such an index is that if cycles are important in maintaining connectivity and interacting dynamics, then a node involved in many cycles should be vital. Experiments on real networks show that cycle ratio outperforms the other indices in identifying vital nodes that are critical in maintaining the network connectivity, efficient in pinning control and influential in epidemic spreading. In node percolation, it should be noted that the performance will be affected by dynamics itself in the way of greedy removal, so the removal order here is fixed as the result of the first calculation. Our finding thus has potential applicability in practice. For the node percolation, the topranked nodes should be firstly protected to maintain the network connectivity if there is a risk of functional loss of nodes. Reversely, if one would like to initiate an intentional attack, the topranked nodes are considered to be the primary targets. Such scenario is relevant to power grids^{60}, air transportation networks, financial networks^{61}, Internet, and so on^{62}. Note that, when we consider an attack to an airport in the modern society, it does not mean we need to physically destroy it but disturb its information systems and signal systems. The critical nodes in pinning control can be pinned to efficiently approach the consensus of multiple agents^{63} and to ensure the coordination of unmanned aerial vehicles^{64} and mobile sensor networks^{65}. Lastly, we proved cycle ratio is an efficient index for finding the susceptible individuals that need to be vaccinated in the early stage of epidemic spreading^{26,31}.
It’s worth noting that the performance of cycle ratio is not necessarily better if longer cycles are considered. This is because when the longer cycles are counted, the difference in local cycle structure might be depressed. That is to say, the sets of associated cycles of many nodes will become more similar (i.e., with larger overlap), which may eventually lead to the decrease of the discriminability and thus the accuracy of the cycle ratio (see Supplementary Note 10).
An obvious insufficiency of cycle ratio is that it cannot be applied for trees or treelike networks. Even for normal networks, a fraction of nodes may be not associated with any cycles. These nodes’ influences may be different but they are all assigned the same cycle ratio zero. One straightforward way to solve this issue is to combine cycle ratio with some other indices, for example, a mixed index could be \(r^ \ast = r_i + \varepsilon k_i\) with \(\varepsilon\) being a tunable parameter, hence all nodes with zero cycle ratio can be ranked by their degrees. Since cycle ratio and degree will produce markedly different rankings, a subtly designed combination of cycle ratio and degree has the potential to generate much better results than the single index. Similar improvement could also be achieved by combining cycle ratio with Hindex or coreness. In contrast, the expected improvement by combining degree, Hindex and coreness is lower since they are already very similar to each other. We leave this detailed problem for future study.
In addition, the method used to characterize the cycle structure can be extended to deal with hypernetworks^{66}, where a hyperedge represents the interaction between multiple nodes. Treating hyperedges as the cycles in the set S and denoting \(\Omega\) the incidence matrix, whose element \(\Omega _{ie}\) indicates whether node i belongs to hyperedge e (\(\Omega _{ie} = 1\) indicates the belongness and \(\Omega _{ie} = 0\) otherwise), then we can obtain a matrix similar to the cycle number matrix by multiplying the incidence matrix by its transposed matrix, say \(\Omega \Omega ^T\), where the diagonal element [ΩΩ^{T}]_{ii} represents the number of hyperedges involving node i and [ΩΩ^{T}]_{ij} indicates the number of hyperedges that involving both node i and node j. Therefore, we can quantify a node’s importance in a hypernetwork by its participation to other nodes’ hyperedges.
We end this paper by presenting two open issues. Firstly, analogous to cycle ratio, one may also design cyclebased indices to quantify the likelihood of the existence of any unobserved link, which can find applications in solving the link prediction problem. Secondly, the good performance of cycle ratio, as well as the lower correlations between cycle ratio and other benchmark centralities, encourages the indepth studies on cycle structure. In terms of global statistics, the model networks have lower average clustering coefficient and lower proximity to tree networks than real networks^{21}; in terms of the distribution of shorter cycles, as shown in Supplementary Note 9, none of degreepreserved null model^{67}, Watts–Strogatz model^{2} and Barabasi–Albert model^{3} can well reproduce the cyclebased statistics of real networks, indicating that the understanding about how cycles are formed may deepen our knowledge on the mechanisms underlying network organization. In addition to the shortest cycles, higherorder cycles also play important roles in network structure and functions^{68,69}. Thus we expect to find more insights from spectral analysis of the cycle number matrix and analyzing longer and higherorder cycles in the future with the help of methodologies from algebraic topology^{69,70} and sufficient computational resource, and extend the findings and scope of applications reported in this paper.
Methods
Degree, Hindex and Coreness
Degree of a node is the number of its immediate neighbors. Hindex of a node i is the maximum integer h such that there are at least h neighbors of node i with degrees no less than h. Coreness is obtained by the kcore decomposition^{10}. The kcore decomposition process starts by removing all nodes with degree \(k = 1\). This may cause new nodes with degree \(k \le 1\) to appear. These are also removed and the process stops when all remaining nodes are of degree \(k\, > \,1\). The removed nodes and their associated links form the 1shell, and the nodes in the 1shell are assigned a coreness value 1. This pruning process is repeated to extract the twoshell, that is, in each step the nodes with degree \(k \le 2\) are removed. Nodes in the twoshell are assigned a coreness value 2. The process is continued until all higherlayer shells have been identified and all nodes have been removed. In the literature, coreness is also referred to as kshell index^{10}.
Kendall’s Tau
We consider any two indices associated with all N nodes, \(X = (x_1,x_2, \ldots ,x_N)\) and \(Y = (y_1,y_2, \ldots ,y_N)\), as well as the N twotuples \(( {x_1,y_1} ),( {x_2,y_2} ), \ldots ,(x_N,y_N)\). Any pair \(( {x_i,y_i} )\) and \(( {x_j,y_j} )\) are concordant if the ranks for both elements agree, namely if both \(x_i \, > \,x_j\) and \(y_i > y_j\) or if both \(x_i < x_j\) and \(y_i < y_j\). They are discordant if \(x_i > x_j\) and \(y_i < y_j\) or if \(x_i < x_j\) and \(y_i > y_j\). Here \(n_ +\) and \(n_ \) are used to represent the number of concordant and discordant pairs, respectively. In addition, \(t_X\) is the number of the pairs in which \(x_i = x_j\) and \(y_i \ne y_j\), and \(t_Y\) is the number of the pairs in which \(x_i \, \ne \,x_j\) and \(y_i = y_j\). Notice that if \(x_i = x_j\) and \(y_i = y_j\), the pair is not added to either \(t_X\) or \(t_Y\). Comparing all \(N(N  1)/2\) pairs of twotuples, the Kendall’s Tau is defined as^{44}
If X and Y are independent, \(\tau\) should be close to zero, and thus the extent to which τ exceeds zero indicates the strength of correlation. The above definition of Kendall’s Tau^{44} is an improved version of the original definition^{45}, specifically designed to deal with the case with many equivalent elements.
Data availability
The networks data that support the findings of this study are available through the corresponding references^{32,39,40,41,42,43} or at the following github repository: https://github.com/ftl129/CycleRatio.
Code availability
The custom code that supports the findings of this study is available at the following github repository: https://github.com/ftl129/CycleRatio.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11622538, 61673150, 61433014, 11975071), and the Zhejiang Provincial Natural Science Foundation of China (No. LR16A050001). L.L. and T.Z. acknowledges the Science Strength Promotion Programme of UESTC.
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T.F., L.L., and D.S. conceived the idea and designed the experiments and T.Z. provided a complement to the design. T.F. and L.L. performed the research. All authors analyzed the data. T.F., L.L., and T.Z. wrote the paper. D.S. edited this paper. All authors discussed the results and reviewed the paper.
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Fan, T., Lü, L., Shi, D. et al. Characterizing cycle structure in complex networks. Commun Phys 4, 272 (2021). https://doi.org/10.1038/s42005021007813
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DOI: https://doi.org/10.1038/s42005021007813
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