Measure for degree heterogeneity in complex networks and its application to recurrence network analysis
aa r X i v : . [ phy s i c s . s o c - ph ] N ov Measure for degree heterogeneity in complex networks and its application torecurrence network analysis
Rinku Jacob ∗ and K. P. Harikrishnan † Department of Physics, The Cochin College, Cochin-682 002, India
R. Misra ‡ Inter University Centre for Astronomy and Astrophysics, Pune-411 007, India
G. Ambika § Indian Institute of Science Education and Research, Pune-411 008, India
We propose a novel measure of degree heterogeneity, for unweighted and undirected complexnetworks, which requires only the degree distribution of the network for its computation. Weshow that the proposed measure can be applied to all types of network topology with ease andincreases with the diversity of node degrees in the network. The measure is applied to computethe heterogeneity of synthetic (both random and scale free) and real world networks with its valuenormalized in the interval [0 , N tends to infinity. We numerically study the variation of heterogeneity for random graphs (asa function of p and N ) and for scale free networks with γ and N as variables. Finally, as aspecific application, we show that the proposed measure can be used to compare the heterogeneityof recurrence networks constructed from the time series of several low dimensional chaotic attractors,thereby providing a single index to compare the structural complexity of chaotic attractors. Keywords: complex networks, heterogeneity measure, recurrence network analysis
I. INTRODUCTION
A network is an abstract entity consisting of a certain number of nodes connected by links or edges. The numberof nodes that can be reached from a reference node ı in one step is called its degree denoted by k i . If equal numberof nodes can be reached in one step from all the nodes, the network is said to be regular or homogeneous. A regularlattice where nodes are associated with fixed locations in space and each node connected to equal number of nearestneighbours, is an example of a regular network. However, in the general context of complex networks, it is definedin an abstract space with a set of nodes N = 1 , , ....N and a set of links denoted by K = k , k , k ....k N − . As thespectrum of k values of the nodes increases, the network becomes more and more irregular and complex. Over thelast two decades, the study of such complex networks has developed into a major field of inter-disciplinary researchspanning across mathematics, physics, biology and social sciences [1–3].Many real world structures [4] and interactions [2, 5] can be modeled using the underlying principles of complexnetworks and analysed using the associated network measures [6]. In such contexts, the corresponding complexnetwork can be weighted [7] or unweighted and directed [8] or undirected depending on the system or interaction itrepresents. In this paper, we restrict ourselves to unweighted and undirected networks and the possible extensionsfor weighted and directed networks are discussed in the end. The topology or structure of a complex network isdetermined by the manner in which the nodes are connected in the network. For example, in the case of the classicalrandom graphs (RG) of Erd˝os and R´enyi (E-R) [9], two nodes are connected with a constant and random probability p . In contrast, many real world networks are found to have a tree structure with the network being a combination ofsmall number of hubs on to which large number of individual nodes are connected [10]. An important measure thatdistinguishes between different topologies of complex networks is the degree distribution P ( k ) that determines howmany nodes in the network have a given degree k . For the RGs, P ( k ) is a Poisson distribution around the averagedegree < k > [6] while many real world networks follow a fat-tailed power law distribution given by P ( k ) ∝ k − γ , with ∗ Electronic address: [email protected] † Electronic address: kp˙[email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] the value of γ typically between 1 and 3 [11]. Such networks are called scale free (SF) [12, 13] due to the inherentscale invariance of the distribution.Though topology is an important aspect of a complex network, that alone is not sufficient to characterize andcompare the interactions that are so vast and diverse. A number of other statistical measures have been developed forthis purpose, each of them being useful in different contexts. Two such commonly used quantifiers are the clusteringcoefficient (CC) and the characteristic path length (CPL). There are also characteristic properties of local structureused to compare the complexity of networks in particular cases, such as, the hierarchy or community structure [3]in social networks and motifs [14] and super family profiles [5] in genetic and neuronal networks. However, a singleindex that can quantify the diversity of connections between nodes in networks even with different topologies, is theheterogeneity measure [15]. It is also indicative, in many cases, of how stable and robust [16] a network is with respectto perturbations from various external parameters. An important example is the technological network of NorthAmerican power grid [4]. Recent studies have also revealed the significance of the heterogeneity measure in variousother contexts, such as, epidemic spreading [17], traffic dynamics in networks [18] and network synchronization [19].The network heterogeneity has been defined in various ways in the literature which we will discuss in detail in thenext section where, we will also present the motivations and need for a new measure. While all the existing measuresare based on the degree correlations k i and k j of nodes ı and in the network, the measure proposed in this paperuses only the degree distribution P ( k ) to compute the heterogeneity of the network. However, we show that thisnew measure varies directly with the k spectrum, or the spectrum of k values in the network, and hence gives a truerepresentation of the diversity of node degrees present in the network. In other words, it serves as a single index toquantify the node diversity in the network.In this work, we also include a class of networks not considered so far in the context of heterogeneity measurein any of the previous works. These are complex networks constructed from the time series of chaotic dynamicalsystems, called recurrence networks [20]. They have a wide range of practical applications [21, 22] and the measuresfrom these networks are used to characterize strange attractors in state space, typical of chaotic dynamical systems,as discussed in § V. The diversity of node degrees in the RNs was actually one of the motivations for us to searchfor a heterogeneity measure that could be used to compare the structural complexities of different chaotic attractorsthrough the construction of RNs.Our paper is organized as follows: In the next section, we discuss briefly all the previous measures of heterogeneityand give reasons why we have to look for a new measure. The measure that we propose is based on the idea of whatwe consider as a completely heterogeneous network of N nodes, that is illustrated in § III. The proposed measureof heterogeneity is presented in § IV while § V and § VI are devoted to computation of this new measure for varioussynthetic as well as real world networks. Our conclusions are summarised in § VII.
II. EXISTING MEASURES OF HETEROGENEITY
If we carefully analyze the heterogeneity measures proposed in the literature, it becomes clear that two differentaspects of a complex network can be quantified through a heterogeneity measure. They are the diversity in node degreesand the diversity in the structure of the network. For example, the initial attempts to measure the heterogeneity tryto capture the diversity in the node degrees of the network and were mainly motivated by the random graph theory.The first person to propose a measure of heterogeneity was Snijders [23] in the context of social networks and it wasmodified by Bell [24] as the variance of node degrees:
V AR = 1 N N X i ( k i − < k > ) (1)where < k > represents the average degree in the network. Though this is still one of the popular measures ofheterogeneity, its applicability is mainly limited to RGs where one can effectively define an average k . Anothermeasure was proposed by Albertson [25] as: A = X i,j | k i − k j | (2)which is a sum of the local differences in the node degrees in the network. This index also is not completely adequatein quantifying correctly the heterogeneity of networks with different topologies. Apart from the above two measuresdefined in the context of social networks, another measure [26] has recently been proposed to quantify the degreeheterogeneity. It uses a measure of inequality of a distribution, called the Gini coefficient [27], which is widely used ineconomics to describe the inequality of wealth. Here a heterogeneity curve is generated using the ratio of cumulativepercentage of the total degree of nodes to the cumulative percentage of the number of nodes. The heterogeneityindex is then measured as the degree inequality in a network. Though the authors compute heterogeneity of severalstandard exponential and power law networks, the measure turns out to be very complicated and works mainly fornetworks of large size with N → ∞ . In short, none of these measures, though useful in particular contexts, trulyreflects heterogeneity as represented by the diversity of node degrees in a network. A comparative study of the aboveheterogeneity measures has been done by Badham [28].The second aspect of heterogeneity discussed in the literature is the topological or structural heterogeneity possiblein a complex network which is especially important in real world networks. An example for this is the measureproposed by Estrada [15] recently, given by ρ = X i,j ( 1 √ k i − √ k j ) (3)which can also be normalised to get a measure ρ n within the unit interval [0 ,
1] as: ρ n = ρN − p ( N −
1) (4)If we analyse this measure closely, we find that it is basically different with respect to the earlier measures. Thereason is that the measure proposed by Estrada is based on the Randic index [29] given by R − / = X i,j ( k i k j ) − / (5)Now, the Randic index was originally proposed [30] as a topological index under the name branching index tomeasure the branching of Carbon atom skeletons of saturated Hydrocarbons. This index is so designed to getextremum value for the “star” structure which is the most heterogeneous branching structure and is bounded byvalues given by √ N − ≤ R − / ≤ N/ I ij , where I ij = 0 if k i = k j and I ij → k i = 1 and k j → ∞ .It is obvious that a measure based on this definition will be maximum for a “star network” of N nodes compared toall other networks since there are ( N −
1) connections with I ij having maximum value.The above discussion makes it clear that Estrada’s measure elegantly captures the structural aspect of heterogeneityassociated with a complex network. This is also evident in the results given by the author. Out of all possible branchingstructures, the heterogeneity is maximum for the star structure. While the star network has ρ n = 1, the values fornetworks with other topologies are much less with a typical SF network having ρ n ∼ .
1. This measure is importantin the context of real world networks with different topology and structure and can be used to classify such networksas shown by Estrada.Our focus here is the heterogeneity associated with the diversity in node degrees (analogous to the earlier attemptsof heterogeneity) to propose a measure applicable to networks of all topologies. An important difference is that weuse the frequencies of the node degrees, rather than k i directly, to define this measure. We show that, as the spectrumof k values in the network increases, the value of the measure also increases correspondingly. We call the measureproposed here degree heterogeneity in order to distinguish it from the measure in [15]. Also, the two measures capturecomplimentary features of heterogeneity in a complex network. A network having high heterogeneity in one measuremay not be so in the other measure and vice versa. For example, the star network is nearly homogeneous in ourdefinition of heterogeneity, as shown below. It is also possible to correlate the robustness or stability of a networkwith the measure proposed here, with the SF networks having comparatively high value of heterogeneity. On theother hand, the star network is most vulnerable since disruption of just one node can destroy the entire network. Todefine the new measure, we require a network with a limiting value of heterogeneity to play a role similar to that ofstar network in the earlier measure. This network is presented in the next section.Finally, the heterogeneity measure that we define below can be shown to have direct correspondence with theentropy measure of a complex network [31], characterized by the standard Shannon’s measure of information S . Inparticular, this measure can be so adjusted to get the value zero for completely homogeneous networks and the value FIG. 1: A comparison of the completely heterogeneous networks (see text) with N = 4, 5, 6 and 7. In each case, all the possible k - values from 1 to ( N −
1) are present in the network as shown. One degree (one k value) has to be shared by two nodes sincethe N th node will have the degree of any one of other nodes. It is empirically shown that this degree of N th node, denoted by k ∗ , is automatically fixed (if the network has all possible degrees from 1 to ( N − N/ N is even and ( N − / N is odd. For example, for N = 4 and 5, k ∗ = 2 and for N = 6 and 7, k ∗ = 3 and so on. S → , FIG. 2: Change in the degree distribution for a typical complex network as it is transformed from complete homogeneity tocomplete heterogeneity, for N = 10. static measure characterizing the structure and diversity of connections between the nodes and not directly concernedwith the information transfer. That is why traditionally the two measures have been treated seperately, though thevalues of both for the extreme cases can be made identical.Moreover, the measure that we propose below has the following advantages:i) Only the degree distribution of the network is required to compute the heterogeneity in contrast to all the previousmeasures proposed so far.ii) The specific condition that we apply for the completely heterogeneous network provides analytical values forheterogeneity in terms of network size.iii) Based on the proposed measure, we are able to give a structural characterization index for a chaotic attractorthrough the construction of a complex network called recurrence network. III. COMPLETELY HETEROGENEOUS COMPLEX NETWORK
Here we present what we consider as the logical limit of a completely heterogeneous network of N nodes. Thereader may find that this is an ideal case. Nevertheless, it helps to put the concept of heterogeneity of a complexnetwork in a proper perspective. Consider an unweighted and undirected complex network of N nodes, with all thenodes connected to the network having a degree of at least one. If all the nodes have the same degree k , the networkis completely homogeneous with the degree distribution P ( k ) being a δf unction peaked at k .Let us now consider the other extreme where no two nodes have the same degree. The maximum possible degreefor a node is ( N − N th node will have to take a degree equal to that of any one of the other nodes having degree from 1 to ( N − N th node under the given condition, we start with taking small number of nodes asshown in Fig. 1, where we show 4 different cases of N ranging from 4 to 7. In each case, the N th node is representedas a pentagon shape with its degree denoted as k ∗ . It is clear that if all the node degrees are to be different, there isonly one possible value of k ∗ for the N th node, which is N if N is even and ( N − if N is odd.We now give a simple argument that this result is true in general for any N . The degree of node 1 is 1 which meansthat it is connected only to the node with degree ( N − N th node. Node 2is connected only to two nodes with degree ( N −
1) and ( N −
2) and hence it is also not connected to node N . Byinduction, one can easily show that the r th node is connected only to nodes with degree ( N − N − N − r ).Suppose N is even. When r = N , this node is connected to nodes with degree from ( N −
1) to N . To avoid self loop,this node should be connected to node N . Thus all nodes with higher degree from N to ( N −
1) are connected tonode N whose degree becomes N . By a similar argument, one can show that the degree of N th node is ( N − if N isodd.Let us now consider the degree distribution P ( k ) of this completely heterogeneous network. All the nodes have FIG. 3: A snapshot of different types of complex networks in the increasing order of their heterogeneity( H m ), taking N = 50in all cases. From top to bottom, the nature of the network varies from completely homogeneous, star, RG, SF and finally tocompletely heterogeneous network. The degree distribution and k spectrum are also shown for each case to indicate that H m is a measure of the diversity in the node degrees. different k values and only two nodes share the same k value, k ∗ . One can easily show that: P ( k ) = P = 1 N , ( k = k ∗ ) P ( k ) = 2 N , ( k = k ∗ )Our definition of heterogeneity is derived in such a way that this network has maximum heterogeneity, which is donein the next section. IV. A NEW MEASURE OF DEGREE HETEROGENEITY
It is very well accepted that a network of N nodes with all nodes having equal degree k is a completely homogeneousnetwork with P ( k ) being a δf unction centered at k . The value of k can be anything in the range 2 ≤ k ≤ ( N −
1) andall these networks have heterogeneity measure zero, for any N . In principle, the heterogeneity of a network shouldmeasure the diversity in the node degrees with respect to a completely homogeneous network of same number ofnodes. All the measures defined so far in the literature directly use the k values present in the network for computingthe heterogeneity measure. Here we argue that a much better candidate to define such a measure is P ( k ) rather than k . Since P ( k ) is a probability distribution, as the spectrum of k values increase, the value of P ( k ) gets shared betweenmore and more nodes with the condition P k P ( k ) = 1. In other words, this variation in P ( k ) reflects the diversity ofnode degrees and hence the heterogeneity of the network. A typical variation of P ( k ) as the network changes fromcomplete homogeneity to complete heterogeneity is shown in Fig. 2. Note that for RGs, this variation in P ( k ) is withrespect to P ( < k > ), with < k > being the average degree, while for SF networks, it is with respect to P ( k min ).To get the heterogeneity measure, we first define a heterogeneity index h for a network of N nodes as the varianceof P ( k ) with respect to the peak value corresponding to the completely homogeneous case: h = 1 N k max X k min (1 − P ( k )) , P ( k ) = 0 (7)The condition implies that the summation is only over k values for which P ( k ) = 0. For a completely homogeneousnetwork, P ( k ) is non zero only for one value of k , say k ∗ , and P ( k ∗ ) = 1, making h = 0, for all N .We now consider the other extreme of completely heterogeneous case. From the results in the previous section forthe completely heterogeneous case, we have h het = 1 N ( N − X k =1 (1 − P ( k )) (8)Putting the values of P ( k ) and simplifying, we get h het = 1 − N + N + 2 N (9)This is the maximum possible heterogeneity measure for a network of N nodes. For large N , as a first approximation,we have h het ≈ r − N (10)For finite N , its value is < N → ∞ , h het →
1. To define the heterogeneity measure ( H m ) for a network,we normalize the heterogeneity index of the network with respect to the completely heterogeneous network of samenumber of nodes to get the value in the unit interval [0 , H m = hh het (11)If N is sufficiently large, say N > h het ∼ H m ≈ h .We note the following features regarding H m :i) It is defined here for unweighted and undirected complex networks and represents a unique measure applicable toany network independent of the topology or degree distribution and increases with the diversity in the node degrees.ii) However, certain topologies have inherent limitations in diversity. For example, H m for a star network is veryclose to zero and hence the star network is nearly homogeneous in our definition. This is because, in the star topology,the degree of only one node is different from the rest of the nodes.iii) Since we use the counts of the node degrees rather than directly k i to find H m , we cannot express the measurein terms of the elements of the Laplacian matrix, as some authors have done.iv) For two networks of the same size N independent of the topology, the measure we propose has a direct corre-spondence with the degree diversity in the network. To show this explicitly, we present the k spectrum , the spectrum FIG. 4: Degree distribution of E-R networks (RGs) for four different p values with N fixed at 2000. The k spectrum is shownbelow the degree distribution. The value of H m and < k > are also indicated in each case. Note that H m varies directly withthe degree diversity or the spectrum of k values in the network. of k values in the network in the form of a discrete line spectrum. In Fig. 3, we compare some standard networks in theincreasing order of their H m , taking N = 50. For each network, we show the degree distribution (as histogram), the k spectrum and the value of H m . Note that, of different topologies, the SF network is the most heterogeneous. Herethe star network has a reasonably high value of H m since N is only 50. We also show the completely heterogeneousnetwork with H m = 1, for comparison.v) The heterogeneity index h is defined as a measure normalized with respect to the size of the network N . Forlarge N , since h ∼ H m , the measure H m can also be used to compare the heterogeneities of two networks even if N is different. This is especially important for real world networks where N varies from one network to another,as discussed in § VI. However, a network with larger N generally tends to have lower H m since, to keep the sameheterogeneity, the range of non zero k values should also increase correspondingly. In other words, a network with100 nodes attains complete heterogeneity if the k values range from 1 to 99 whereas, to attain complete heterogeneityfor a network of 1000 nodes, the k values should range from 1 to 999.The above result also implies that for any network that is evolving or growing, for example the SF network wherethe nodes are added with preferential attachment [33], the value of H m generally keeps on decreasing with increasing N . In the next section, we numerically study the variation of H m with different network parameters for varioussynthetic networks. V. DEGREE HETEROGENEITY OF SYNTHETIC NETWORKS
In this section, we analyze 3 different classes of complex networks, namely, the RGs of Erdos-Renyi, the SF networksand the networks derived from the time series of chaotic dynamical systems, called recurrence networks (RNs) whosedetails are discussed in § V.C.
FIG. 5: Variation of H m with p for RGs for a fixed value of N , as shown. We expect the profile for a higher N value to bewithin that of a lower N as H m decreases with N for any fixed p . A. Classical random graphs
For RGs, the degree distribution is Poissonian centered around an average degree < k > ≡ pN where p is theprobability that two nodes in the network is connected. In Fig. 4, we show the degree distribution and the k spectrumfor RGs of 4 different p values with N fixed at 2000. The values of H m for all these networks are also shown. Themain result here is that the value of H m increases correspondingly with the range of k values for a fixed N .We next consider how H m depends on p and N , the two basic parameters of the RG. The effect of changing p for a fixed N as well as changing N for a fixed p are to shift the average k value of the nodes in the RG. Since thedegree distribution is approximately Gaussian for large N , the spectrum of k values depends directly on the varianceof the Gaussian profile. As p increases from zero for a fixed N , the spectrum of k values and hence H m increasecorrespondingly. Due to the obvious symmetry of the network with respect to the transformation p → (1 − p ), as p increases beyond 0 . H m starts decreasing. Thus the maximum value of H m is obtained for p = 0 . N . On the other hand, by increasing N for any fixed p , one expects the Gaussian profile of the degree distributionto become sharper, thus decreasing H m . These results are compiled in Fig. 5 for three values of N . Note that theminimum p value that can be used for N = 500 is 0 .
004 and this decreases as N increases. In the figure, we show theresults starting from p = 0 .
1. Higher values of N would involve very large computer memory requirements for large p . However, we have checked the variation of H m with N for smaller p values, say 0 .
005 and 0 .
01, for N up to 50000 FIG. 6: Degree distributions (inset) and the distributions in log scale along with the k spectrum for synthetic SF networks withfour different values of γ and N fixed at 2000. In all cases, the values of H m and the minimum degree k min of the network arealso shown. As the spectrum of k values increases, H m increases correspondingly for a fixed N . Note that there appears to bea second peak with a gap in the distribution for logk >
4. This is size dependent effect due to the presence of many k valueshaving P ( k ) very close to zero. It is also evident from the k spectrum shown below each distribution. For example, in the case( d ) where the k spectrum is almost continuous without a visible gap, the scaling becomes more evident. and have found that the decrease is approximately exponential. B. Scale free networks
For SF networks, the degree distribution obeys a power law P ( k ) ∝ k − γ . To construct the SF network synthetically,we use the basic scheme proposed by Barabasi et al. [34]. In this scheme, we start with a small number of initial nodesdenoted as m . As a new node is connected, a fixed number of edges, say m , is added to the network. This numberrepresents the minimum number of node degree, k min , in the network. The new edges emerging at node creation aredistributed according to the preferential attachment mechanism. The two parameters, m and k min , determines thevalue of γ as the network evolves. We have constructed SF networks of different γ by changing both m and k min .In Fig. 6, we show the degree distribution and the corresponding k spectrum for SF networks with four different γ and k min , with N fixed at 2000. We find that the k spectrum and hence the value of H m depend directly on k min ascan be seen from the figure. In other words, for a SF network of fixed N , H m increases as the value of k min increases.The variation is approximately linear for k min in the range 1 to 10. More interesting is the variation of H m with N for a fixed k min . In Fig. 7, we show the variation of H m as N increases from 1000 to 10000 for two different SFnetworks with k min = 5 and 10. This variation is also shown in the inset in a log scale in the same figure indicatingthat H m varies as H m ∝ N − ρ , where the value of ρ is found to be 0 . k min = 5 and 0 . k min = 10 forthe given range of N values. However, we do not claim that this variation is, in general, a power law since we haveonly tested a limited range of N values. This needs to be explicitly tested with other alternatives with a wider rangeof N values.1 FIG. 7: Variation of H m with N for synthetic SF networks with two different values of k min . The same variation is shown inthe inset in log scale indicating a clear power law in both cases. C. Recurrence networks
Recently, a new class of complex networks has been proposed for the characterization of the structural propertiesof chaotic attractors, called the recurrence networks (RNs)[20, 35]. They are constructed from the time series of anyone variable of a chaotic attractor. From the single scalar time series, the underlying attractor is first constructedin an embedding space of dimension M using the time delay embedding [36] method. Any value of M equal to orgreater than the dimension of the attractor can be used for the construction of the attractor. The topological andthe structural properties of this attractor can be studied by mapping the information inherent in the attractor to acomplex network and analyzing the network using various network measures.To construct the network, an important property of the trajectory of any dynamical system is made use of, namely,the recurrence [37]. By this property, the trajectory tends to revisit any infinitesimal region of the state space of adynamical system covered by the attractor over a certain interval of time. To convert the attractor to a complexnetwork, one considers all the points on the embedded attractor as nodes and two nodes ı and are considered to beconnected if the distance d ij between the corresponding points on the attractor in the embedded space is less than orequal to a recurrence threshold ǫ . Selection of this parameter is crucial in getting the optimum network that representsthe characteristic properties of the attractor. The resulting complex network is the RN which, by construction, is anunweighted and undirected network. The adjacency matrix A of the RN is a binary symmetric matrix with elements A ij = 1 (if nodes ı and are connected) and 0 (otherwise). More details regarding the construction of the RN canbe found elsewhere [38, 39]. Here we follow the general framework recently proposed by us [39] to construct the RNfrom time series.For generating the time series, we use the equations and the parameter values given in [36] for all chaotic systems.For continuous systems, we have used the sampling rate 0 .
05 for generating the time series. The time delay used for2
FIG. 8: Top panel shows the variation of H m with N for RNs constructed from Lorenz (filled triangle) and Henon (filled circle)attractor time series. Bottom panel shows the variation of H m with M for fixed N for Lorenz (filled triangle) and Duffing(filled circle) attractors. In both graphs, the error bar comes from the standard deviation of values for H m computed from timeseries with ten different initial conditions. embedding is the first minimum of the autocorrelation. We first study how the value of H m varies with the numberof nodes N for RNs. In Fig. 8 (top panel), we show the results for the Lorenz attractor and the Henon attractor. Itis evident that for large value of N , H m converges to a finite value. We have checked and verified that this is true forother low dimensional chaotic attractors as well. It is found that once the basic structure of the attractor is formed,the value of H m remains independent for further increase in N . In other words, the range of k values increases with N to keep the value of H m approximately constant. This result also follows from the statistical invariance of the degreedistribution of the RN as has already been shown [39].Next, we consider the variation of H m with embedding dimension M . This is also shown in Fig. 8 (bottom panel) fortwo standard chaotic attractors. It is clear that the value of H m converges for M ≥ <
3. We have already shown [39] that the degree distribution ofthe RN from any chaotic attractor converges beyond the actual dimension of the system. Thus, H m turns out to bea unique measure for any chaotic attractor independent of both M and N .3From the construction of RNs, the range of connection between two nodes is limited by the recurrence threshold ǫ .Hence the degree of a node in the RN and the probability density around the corresponding point over the attractorare directly related. For example, for the RN from a random time series, every node has degree close to the averagevalue < k > since the probability density over the attractor is approximately the same. One can show that thedegree distribution of the RN from a random time series is Gaussian for large N . Thus, the k spectrum of the RN isindicative of the range in the probability density variations over the attractor, which in turn, is characteristic of thestructural complexity of the attractor.We have already shown that the measure H m proposed here is indicative of the diversity in the k spectrum.Moreover, it is found to have a specific value for a given attractor independent of M and N . It is well known thatthe statistical measures derived from the RNs characterize the structural properties of the corresponding chaoticattractor. In particular, since every point on the attractor is converted to a node in the RN and the local variation inthe node degree is a manifestation of the variation in the local probability density over the attractor, the measure H m can serve as a single index to quantify the structural complexity of a chaotic attractor through RN construction. InTable I, we compare the values of H m for RNs constructed from several standard chaotic attractors. In all cases, thesaturated value of H m converged upto M = 5 is shown. In each case, ten different RNs are constructed changing theinitial conditions and the average is shown with standard deviation as the error bar. The results indicate that thatamong the continuous systems compared, the Lorenz attractor is structurally the most complex while in the case of2D discrete systems, Lozi attractor is found to be the most diverse in terms of the probability density variations.Finally, it will also be interesting to see how the value of H m is affected by adding noise to the chaotic time series. Totest this, we generate data adding different percentages of noise to Lorenz data. When the value of H m is computed,it is found that the value reduces systematically with the increase in the noise percentage and approaches the valueof noise for a noise level > H m for random time series with N = 2000 and M = 3 is found to be 0 . ± . H m for a RG with p = 0 . < k > as that of the RN from random time series and a typical SF network with γ = 2 .
124 with N = 2000 in bothcases. The average of ten different simulations is taken. We find that H m = 0 . ± .
012 for the RG which is exactlysame as the RN from random time series and H m = 0 . ± .
06 for the SF network.
System
Lorenz R¨ossler Duffing Ueda Henon Lozi Cat Map H m . ± .
056 0 . ± .
042 0 . ± .
058 0 . ± .
038 0 . ± .
044 0 . ± .
072 0 . ± . H m for several standard chaotic attractors. VI. REAL WORLD NETWORKS AND POSSIBLE EXTENSION TO WEIGHTED NETWORKS
So far, we have been discussing the degree heterogeneity measure of synthetic networks of different topologies.In this section, we consider some unweighted and undirected complex networks from the real world and see whatinformation regarding the degree heterogeneity of such networks can be deduced using the proposed measure. Weuse data on networks from a cross section of fields, such as, biological, technological and social networks. In Fig. 9,we show the degree distribution and k spectrum of two such networks. In Table II, we compile the details of thesenetworks and the values of H m computed by us for each.Since we have to restrict to the case of unweighted and undirected networks, we could use only a small subset fromthe very large variety of real world networks that are mostly weighted or directed. To extend the measure to directednetworks, one has to consider the in-degree and out-degree distributions and find the heterogeneity separately. Inorder to generalise the measure to weighted networks, the distribution of the weight or strength of the nodes in thenetwork [7, 8], rather than the simple degree distribution is to be considered and define the measure accordingly. Forexample, for unweighted and undirected networks, all the links are equivalent and hence the degree of ı th node k i isjust the sum of the links connected to node ı . On the other hand, for weighted networks, each link is associated witha weight factor w ij and hence the degree k i should be generalised to the sum of the weights of all the links attachedto node ı : s i = X j w ij (12)Thus the degree distribution needs to be generalised to the strength distribution P ( s ), which is the probability that agiven node has a strength equal to s [44, 45]. The equation for heterogeneity for weighted networks can be modified4 FIG. 9: Degree distribution and the k spectrum of the protein interaction network are shown in the top panel. In the bottompanel, the same for the network of Western Power Grid. System Reference
N H m accordingly. However, it should be noted that the weight factors are assigned based on different criteria dependingon the specific system or interaction the network tries to model. Modifying the measure by incorporating the specificaspects of interaction, the measure itself becomes network specific. The measure that we propose here is independentof such details and is representative of only the diversity of node degrees in a network, determined completely by thesimple degree distribution. VII. CONCLUSION
Complex networks and the network based quantifiers have become useful tools for the analysis of many real worldphenomena. Physical, biological and social interactions are increasingly being modeled and characterized throughthe language of complex network. An important measure for the characterization of any complex network is itsheterogeneity measured in terms of the diversity of connection reflected through its node degrees. Here we introducea measure to quantify this diversity which is applicable to networks of different topologies. This measure is minimum(equal to zero) for a completely homogeneous network with all k i ≡ k . To get the upper bound for the measure, weconsider the logical limit of heterogeneity possible in a network of N nodes where nodes of degree varying from 1 to( N −
1) are present, whose heterogeneity is normalised as 1. While considering this network of limiting heterogeneity,we also prove that the degree that is repeated (or shared by two nodes) is N/ N is even and ( N − if N is odd.Also, the measure that we propose here uniquely quantifies the diversity in the node degrees in the network which ischaracteristic of the type and range of interactions the network represents. The diversity also depends on the topologyof the resulting network. For example, for RGs, the diversity is limited since most degrees are centered around theaverage value < k > while the SF networks are comparatively more diverse due to the presence of hubs. The proposedmeasure can quantify this diversity in the node degrees irrespective of the topology of the network.By applying the proposed measure, we compute the heterogeneity of various unweighted and undirected networks,synthetic as well as real world. We study numerically how the heterogeneity varies for RG with respect to the twoparameters p and N , while for SF networks the variation of heterogeneity with respect to γ as well as N are analysed.To illustrate the practical relevance of the measure, we analyse the RNs constructed from the time series of chaoticdynamical systems and highlight its utility as a quantifier to compare the structural complexities of different chaoticattractors. As has already shown by us [39], the nonlinear character and chaotic dynamics underlying the timeseries can be distinguished from the RNs through the usual characteristic measures of complex networks like CC orCPL. However the subtle differences between the degree distributions of RNs from different chaotic systems are notclearly evident from their CC or CPL. We find these can be quantified uniquely using the proposed measure of degreeheterogeneity. Data accessibility : We use the open source software GEPHI for the construction of allnetworks. All the codes used for computing the heterogeneity and other network measures are available at https://sites.google.com/site/kphk11/home.
Author contributions : The initial idea for the paper started from the discussions of RJ and KPH. The codes forcomputations of network measures were developed in association with RM. The interpretation of the results and the6expert guidance for the manuscript were finalised after a series of discussions with GA. All the authors gave the finalapproval of the manuscript.
Competing interests : The authors have no competing interests.
Funding : RJ, KPH and RM acknowledge the financial support from the Science and Engineering Research Board(SERB), Govt. of India in the form of a Research Project No. SR/S2/HEP-27/2012.
Acknowledgements : The authors thank one of the anonymous Referees for a thorough review of the manuscriptand especially for pointing out the correspondence of our heterogeneity measure with the entropy measure of complexnetworks. RJ and KPH acknowledge the computing facilities in IUCAA, Pune. [1]
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