ORIGINAL ARTICLE

Analysing the topology of seismicity in the Hellenic arcusing complex networks

Eleni Daskalaki & Gerassimos A. Papadopoulos & Konstantinos Spiliotis &

Constantinos Siettos

Received: 18 January 2013 /Accepted: 9 September 2013 /Published online: 27 September 2013# Springer Science+Business Media Dordrecht 2013

Abstract Based on the theory of complex networks,we quantify for the first time correlations betweenseismic events occurring in the Hellenic arc andTrench system, which is the most seismogenic struc-ture in the European–Mediterranean region. We ex-amine how relatively strong events with their after-shock sequences trigger phase transitions of the un-derlying network topology of seismic activity fromrandom to scale-free structures. In particular, we showthat the network is characterized by a highly clusteredspatial structure giving rise to heterogeneous net-works exhibiting enhanced small-world attributes.

Keywords Seismicity . Hellenic arc . Complexnetworks . Scale-free structures . Earthquakes

1 Introduction

The investigation of seismicity patterns has attractedmuch interest over the last years, not only because suchpatterns may carry important information for the fore-casting of the earthquakes but mainly because they canfacilitate a better understanding of the earthquake prep-aration process. For background, non-clustered, seis-micity, the earthquake generation in time is consideredas a random (Poissonian) point process (e.g. Lomnitz1974). The most fundamental statistical law for clus-tered seismicity is the one describing the power-lawdecay of aftershock sequences following strongmainshocks (Omori 1894; Utsu 1962). However, theaftershock evolution often deviates from the simplepower-law decay mainly because strong aftershocksare followed by their own aftershocks (e.g. Jiang andWu 2012). To address this issue, Ogata (1998) intro-duced the model of epidemic type aftershock sequencesto describe the complex pattern of aftershock decay(Zhuang 2012). Mogi (1962) suggested that anotherpattern of clustered seismicity is the foreshock activitypreceding mainshocks with a rate increasing with theinverse of time towards the mainshock (Jones andMolnar 1979). Today, it is known that somemainshocksare preceded by foreshocks but others are not. An ex-ceptional recent example of strong foreshock signal wasrecognized to occur before the mainshock (Mw=6.3) ofApril 6, 2009 in L’Aquila, central Italy (Papadopouloset al. 2010). On the other hand, spatiotemporal seismic-ity clusters that exhibit a gradual rise and fall in seismicmoment release, lacking a mainshock–aftershock pat-tern, are termed earthquake swarms (Yamashita 1998).

J Seismol (2014) 18:37–46DOI 10.1007/s10950-013-9398-8

E. Daskalaki :G. A. PapadopoulosInstitute of Geodynamics, National Observatory of Athens,Athens 11810, Greece

E. Daskalakie-mail: edaskal@noa.gr

G. A. Papadopoulose-mail: papadop@noa.gr

K. Spiliotis : C. Siettos (*)School of Applied Mathematics and Physical Sciences,National Technical University of Athens,15780( Politechnioupoli, Zografos, Athens, Greecee-mail: ksiet@mail.ntua.gr

K. Spiliotise-mail: spiliotiskos@gmail.com

In the earthquake magnitude domain, the magni-tude–frequency (or G–R) relation (Ishimoto and Iida1939; Gutenberg and Richter 1944) reads:

logN ¼ a−bM

The above relation which describes the power-law de-crease of the number of events with the increase ofmagnitude has been found to apply in both clusteredand non-clustered types of seismicity (see review inUtsu 2002a, b, c); N is the cumulative number of eventsof magnitude equal to or larger than M and a, b areparameters determined by the data. Theoretically, b=1,which has been confirmed by global scale seismicity dataanalysis (Frohlich and Davis 1993). However, the valueof this parameter depends on the local seismotectonicconditions, such as the material heterogeneity, the degreeof symmetry in stress distribution and on the existence ornot of asperities in the fault zone (Tsapanos 1990). Suchconditions are reflected in the mode of seismic activity,e.g. background seismicity, swarms, foreshocks, after-shocks. For example, it is quite common that in a swarm,several events share almost the same maximum magni-tude in the sequence which results in high or even veryhigh values of b.

Over the last years, the complex network theoryprovides a new insight and perspective in analysingthe seismicity process (Abe and Suzuki 2004, 2007;Baiesi and Paczuski 2004, 2005; Albert and Barabasi2002; Barrat et. al. 2008). Abe and Suzuki (2004) ex-amined the seismicity in southern California and Japanand showed that long range spatiotemporal correlationsof earthquakes exhibit a power-law distribution. Apower-low function for spatiotemporal distribution ofevents of Japan and some Greek cities is reported inMusson et al. (2002). Baiesi and Paczuski (2005) pro-posed a method based on metrics of complex networksto discriminate between aftershocks and mainshocks.Baesi (2006) showed that simple motifs in the structureof network seismicity in southern California could serveas precursors of major events. Lotfi and Darooneh(2012) investigated the impact of the size of grid parti-tion of a seismic region on the network’s properties inIran. They showed that an increase in the resolutionresults in random and uncorrelated network topologies.This resembles the hypothesis that earthquake time se-ries distributions become random or deviate from ran-domness with the change of the counting time intervalselected to test the validity of the theoretical distribution

model against the experimental data (Papadopoulos1993). It is worth mentioning though that due to thestrong heterogeneity of the underlying dynamics thatpertain to the behaviour of region-dependent earthquakeevents, the generalization of the outcomes of the rela-tively new approach of Lotfi and Darooneh (2012) isstill an open question.

Since the complex network theory is a new, quitepromising tool for the investigation of seismicity pat-terns, in this paper we analyse, for the first time, theseismicity occurring along the Hellenic arc and Trench(HA-T) system, which is one of the most activeseismogenic structures in the western Eurasia. Our anal-ysis was focused in the period from 2000 to 2009. Byemploying a sliding window approach on the methods ofBaiesi and Paczuski (2005) and Abe and Suzuki (2004),we identified four major, distinct phase transitions of thestructure of the HA-T network of seismicity correspond-ing one by one to cases of increased seismicity due to theoccurrence of four strong seismic sequences, namely theones of Samos 2005, Zakynthos 2006, Cephalonia 2007and of Methoni 2008. All four mainshocks involvedwere followed by aftershock sequences characterizedby a highly clustered structure. We show that duringthese major events, the underlying seismic network ofthe HA-T undergoes a phase transition from random toscale-free structure characterized by simultaneous sharpdecrease of the power-law exponent, γ of the degreedistribution and increase of the network’s entropy.Importantly, we show that there are strong, suddenchanges of the topological characteristics. This is the firsttime that such an analysis is presented for the HA-Twhich is quite promising for further seismicity analysisin other seismogenic regions of the world.

2 Seismotectonic setting

The eastern Mediterranean region is characterized byactive lithospheric plate interactions and consequentlyby very high seismicity. One of the most active geotec-tonic structures in the region is the HA-T system situatedon a convergent plate boundary comprising the activesubduction of the African plate under the southern mar-gin of the Eurasian plate from about SSW to NNE (e.g.McKenzie 1970, 1972; Papazachos 1990; Fig. 1). Thekinematics of the system becomes more complex due to(a) the counterclockwise rotation of the African plate withrespect to the Aegean Sea area and relative to an Euler

38 J Seismol (2014) 18:37–46

rotation pole situated at a geographical point with coor-dinates 40° N 18° E (Le Pichon and Angelier 1979) and(b) the rollback of the subducting lithosphere beneath theHellenic Trench (e.g. Reilinger et al. 2006). GPS mea-surements of crustal motions indicated that the centraland southern Aegean is characterized by coherentmotion, with small internal deformation of <2 mm/year,toward the SW at 30±1 mm/year relative toEurasia (McClusky et al. 2000; Kreemer andChamot-Rooke 2004).

From seismicity data of the instrumentally record-ing period that is of about the last 100 years, it comesout that the HA-T system generates large, shallow andintermediate-depth earthquakes with magnitudes up toabout 7.5. In the long-term sense, however, historicalseismicity data combined with archaeological andgeological field observations have shown that bigearthquakes of magnitude around 8 or even largerruptured segments of HA-T in the past. Such “mega”

events occurred on AD 365 and 1303 (see review inPapadopoulos 2011), which very likely represent thelargest earthquakes that may be expected in the west-ern and eastern segments of HA-T, respectively. Thetoday’s modern seismograph network of the Instituteof Geodynamics, National Observatory of Athens(http://www.gein.noa.gr/) provides monitoring capa-bilities for the determination of earthquake events assmall as 2 in magnitude and in some instances lessthan this.

3 Data and method

Here, we investigate network properties of the HA-Tshallow seismicity (focal depth h<40 km) for the timeperiod from 2000 to 2009 inclusive. The region ofHA-T is determined by the geographical coordinatesfrom 19.5° N to 30° N in longitude and from 34.5° E

Fig. 1 Map of the eastern Mediterranean showing thegeodynamic framework and the principal regional lithosphericmotions. HA-T Hellenic arc and Trench system, NAFZ North

Anatolian Fault Zone. Inset seismicity along the HA-T from2000 to 2009, with events of magnitude M≥4.0

J Seismol (2014) 18:37–46 39

to 40° E in latitude. The seismic data were retrievedfrom the earthquake catalogue which is produced bythe Institute of Geodynamics, National Observatoryof Athens (http://www.gein.noa.gr/). The earthquakecatalogue was tested for data completeness on thebasis of the G–R diagram (Fig. 2) resulting to a totalnumber of 39,450 events. The roll-off of the linearpart of the diagram in its left-hand side indicates thatthe magnitude cutoff for data completeness for theentire period examined is Mc=2.7. This is an impor-tant task as mishandling data may subsequently leadto wrong evaluations for a series of important fea-tures, such as seismic rate changes, with implicationsfor the identification of earthquake sequences, e.g.aftershocks.

The HA-T was tessellated into square cells ofdimension 0.1° for both longitude and latituderesulting to a total number of 5,775 cells. Here, thenetwork was constructed by linking cells successivelyin time, when seismic activity is observed within thecells, i.e. cells serve as vertices or nodes of the un-derlying network, in two different ways: (a) succes-sively in time, independently from the distance of theearthquake epicentres as proposed by Abe and Suzuki(2004, 2007) and (b) depending on the spatiotemporaland magnitude characteristics of the seismic events(Baiesi and Paczuski 2004, 2005). Other scales ofresolution (e.g. 0.2°) were also tested showing noqualitative difference.

The approach of Abe and Suzuki analyses spatio-temporal properties of seismicity from the viewpointof non-extensive statistical mechanics. According to

this approach, the spatial distance and time intervalbetween two successive earthquake events are corre-lated, independently of their distance.

The second approach by Baiesi and Paczuski (2004,2005) combines two statistical laws of seismicity: theG–R (see the relation above) and the law of fractaldistribution of epicentres (Turcotte 1997; Hirata 1989).Given that two seismic events j, i occurring at cells lj, liat times instances Tj, Ti, respectively (with Tj>Ti), theexpected number of earthquake events nij, occurring inthe a rea , o f magni tude wi th in an in te rva lMi−ΔM≤Mi≤Mi+ΔM and within a time and space in-terval of Δt=Tj−Ti and Δl=lj−li, respectively, is yieldedby nij ¼ cΔt10−bMi ld fΔM , where b is the G–R con-stant, Mi the magnitude of the i event, c is constant anddf is the fractal dimension of the epicentral distribution.The correlation between two seismic events is definedas cij=1/nij. Unexpected events nij that is small nij aremore likely to be correlated. Two cells are linked (i→j)if cij≥cth (cth is the correlation threshold).

In order to analyse the evolution of the network topol-ogy, we applied two types of sliding windows: (a) withfixed number of seismic events and (b) with fixed timeintervals. For the area under study andwithin each slidingwindow, we computed several statistical properties of thenetwork including the average clustering coefficient(ACC), the inverse efficiency 1/E, the power-law expo-nent of the approximate scale-free degree distribution (γ)and the entropy of the degree distribution (Newman2003; Albert and Barabasi 2002; Krishna Mohan andRevathi 2011; Stam and Reijneveld 2007) (see also inthe “Appendix” for definitions).

For both approaches, we used a window of 100seismic events sliding forward with a step of tenevents. The following typical parameters were usedfor the approach by Baiesi and Paczuski (2004,2005): b=0.95, Δm=0.1. The correlation thresholdwas chosen as cth=10,000.

4 Results

Figure 3 illustrates the evolution of ACC (Fig. 3a), 1/E(Fig. 3b) and γ (Fig. 3c) of the network constructed withthe approach of Abe and Suzuki (2004) using a fixednumber of events sliding window for the HA-T in theperiod from 2000 to 2009. Within this period, four majorearthquake sequences occurred: that of Samos 2005,Zakynthos 2006, Cephalonia 2007 and Methoni 2008.

Fig. 2 G–R diagram for the segment of the earthquake cata-logue selected to use: cumulative number, N of earthquakesagainst local magnitude, Mc magnitude cutoff for data com-pleteness is 2.7

40 J Seismol (2014) 18:37–46

Fig. 3 Statistical network measures as calculated using the Abeand Suzuki sliding window approach for the Hellenic arc andTrench seismicity: a average clustering coefficient (ACC), breverse efficiency 1/E, c critical exponent γ and d entropy ofthe degree distribution. Earthquakes in the magnitude ranges5.0≤M<5.5, 5.5≤M<6.0 and M≥6.0 are marked with circles,squares and stars, respectively

Fig. 4 As in Fig. 3 for average clustering coefficient using awindow of 200 events and overlapping step of length 50earthquake events

Fig. 5 As in Fig. 3 using a fixed period of time as width of thesliding window: a average clustering coefficient (ACC), breverse efficiency 1/E and c critical exponent γ. Earthquakesin the magnitude ranges 5.0≤M<5.5, 5.5≤M<6.0 and M≥6.0 aremarked with circles, squares and stars, respectively

J Seismol (2014) 18:37–46 41

In the periods of normal seismic activity (i.e. in theabsence of strong mainshocks), the network resembles arandom structure, characterized by small values of theACC (around 0.01) and an inverse efficiency (corre-sponding to the average path length (APL)) around 15.Relatively strong mainshocks of magnitude M≥5.0 withtheir aftershock sequences result to phase transitions fromthe random structure as this is reflected by sudden simul-taneous changes in the topological characteristics. Inparticular, the average clustering coefficient jumps up tohigher values after the mainshock (around 0.4 for Samos,Zakynthos and Cephalonia and around 0.3 for Methoni)(Fig. 3a); both the average path length (Fig. 3b) and γ(Fig. 3c) decrease (around 4 for the average path length;around 1 for γ) while the entropy of the degree distribu-tion increases (Fig. 3d); then, all quantities graduallydecay to the values prevailing before the occurrence ofthe mainshocks and their aftershocks. It is interesting tonote that equivalent behaviour, the so-called self-organized criticality, has been observed in other studiesbased on spring-block models (Olami et.al. 1992).

This critical behaviour can be explained as fol-lows: The power exponent, γ, of the approximatepower-law distribution (see “Appendix”) decreasesindicating the creation of hubs. In fact, aftershocksfollowing a mainshock revolve around the locus ofthe mainshock, and therefore, they contribute to thelarge degree of connectivity of the vertex (cell) wherethe mainshock occurs. Strong seismic events enhancethe heterogeneity of the network, and consequently,the corresponding cells act as hubs (Abe and Suzuki2007). The existence of hubs is also confirmed by theincrease of the entropy, which is an index of thenetwork heterogeneity. Thus, the emergence of highvalues of the average clustering coefficient and lowvalues of the average path length is characteristic ofsmall world networks (Watts and Strogatz 1998;Newman 2003; Albert and Barabasi 2002).

In order to test the robustness of the analysis, weconstructed networks using different values of slidingwindow lengths and advancing steps. Within a widerange of values, the outcomes were—for all practical

Fig. 6 Illustrative snapshot of the resulting network using theAbe and Suzuki (2004) method for the earthquake sequences ofa Samos island (17–27 October 2005), b Zakynthos (4–19April 2006), c Cephalonia (25 March–10 April 2007) and d

Methoni (14–29 February 2008). Case e is for randomnetwork. Earthquakes in the magnitude ranges M≤3.0,3.0<M<5.0 and M≥5.0 are marked with circles, squaresand stars, respectively

42 J Seismol (2014) 18:37–46

purposes—equivalent. For example, Fig. 4 depicts theaverage clustering coefficients computed using a slid-ing window of 200 events and a step of 50 events.The outcomes are comparable with the ones obtainedusing a sliding window of 100 events and a step often events (Fig. 3). We also repeated the analysisusing windows of fixed time intervals. The correspond-ing statistical measures are shown in Fig. 5. Again, theoutcomes of the analysis are equivalent with the onesanalysed with the fixed number of events sliding win-dow. Our analysis identified four major, distinct phasetransitions of the structure of the HA-T network ofseismicity corresponding to phases of increased seismic-ity due to the occurrence of the four strong mainshocks(marked with vertical lines in Figs. 3, 4 and 5). Figure 6illustrates geographical snapshots of the HA-T networkstructure as computed with the use of the Abe andSuzuki (2004) method using data within the period ofoccurrence of each one of the four earthquake sequencesexamined. Figure 6a depicts the snapshot of the networkconstructed with the data retrieved from 17th to 27th ofOctober of 2005. In particular on the 17th of October of

2005, a mainshock of magnitude M=5.5 occurred closeto Samos Island. Figure 6b depicts the snapshot of thenetwork constructed for the period 4th to 19th of April2006. On the 12th of April, there was a mainshock ofM=5.2 close to Zakynthos Island. Figure 6c depicts thesnapshot of the network constructed for the period 25thMarch to 10th of April 2007. On the 25th of March, amainshock of M=5.5 occurred in an area close toCephalonia Island. Finally, Figure 6d depicts the snap-shot of the network constructed for the period 14th to29th of February 2008. On the 14th of February, therewas a large earthquake ofM=6.8 in the area of Methonioffshore SW Peloponnese.

The snapshots of the network structure show arelatively large amount of seismic events being local-ized as hubs. For comparison, Fig. 6e illustrates asnapshot of the HA-T network for the period fromJuly to September 2005 which was of normal seis-micity given that no strong events of M≥5.0 occurred.For such periods, the analysis showed that the HA-Tnetwork follows a random structure. The first phaseof the seismic activity near Samos Island was veryintriguing. It started with a strong earthquake of M=

Fig. 7 Statistical measures of resulting network using Baiesiand Paczuski (2004, 2005) method: a average clustering coef-ficient (ACC) and b reverse efficiency 1/E. Earthquakes in themagnitude range 5.0≤M<5.5, 5.5≤M<6.0 and M≥6.0 aremarked with circles, squares and stars, respectively

Fig. 8 Resulting average clustering coefficient using a the Abeand Suzuki (2004) and b the Baiesi and Paczuski (2004, 2005)methods (from Figs. 3a and 7a)

J Seismol (2014) 18:37–46 43

5.5 on 17th of October 2005 followed by a secondstrong earthquake of M=5.4 4 h later. These twoevents are marked with circles in Fig. 3. This se-quence caused an abrupt significant increase of ACCfrom 0.02 to 0.4 (Fig. 3a) and a simultaneous abruptdecrease of both 1/E (Fig. 3b) and γ (Fig. 3c). Thesecond case of increased seismicity in ZakynthosIsland during the April of 2006 was similar to thatof Samos. An initial strong event of magnitude M=5.2caused synchronous change of ACC, 1/E and γ. TheACC increased rapidly while two subsequent eventsof magnitude M=5.2 and M=5.4 strengthened its fur-ther increment; at the same time, the quantities 1/Eand γ decreased in an abrupt way (Fig. 3b, c). Thesame behaviour was observed during the seismicevent of M=5.5 which occurred in the region ofCephalonia on the 25th of March 2007. Again, thestatistical measures of the underlying networkchanged abruptly indicating a phase transition of thenetwork topology. In particular, the earthquake se-quence caused a sudden increase of both ACC(Fig. 3a) and entropy (Fig. 3d) and a simultaneousabrupt decrease of 1/E (Fig. 3b) and γ (Fig. 3c). Thesame outcomes are shown in Fig. 5. Similar resultswere observed for the earthquake sequence ofMethoni, which initiated with the mainshock of 14February 2008 (M=6.8). The outcomes of the analysisobtained by the approach of Baiesi and Paczuski(2004, 2005) were consistent with the ones describedabove for all four cases (Fig. 7). For comparison, theevolution of the ACC as derived with both methods isillustrated in Fig. 8. The high values of the ACC andthe small values of the average path length suggestthe small-word character of the HA-T earthquakenetwork: a sub-network with a high clustering formedlocally around the region of the mainshock isconnected with few links with the seismic eventsoccurring within the rest of the HA-T system.

5 Conclusions

We analysed the topological features of earthquakesequences occurring in the Hellenic arc and Trenchduring the period from 2000 to 2009. The HA-T is animportant region on the active tectonics of the EasternMediterranean region producing large earthquakeswith magnitudes up to M=8.0. Based on two differentapproaches, exploiting the complex network theory,

we constructed the underlying earthquake networkand analysed the seismicity with respect to basicstatistical network measures, namely the inverse effi-ciency (corresponding to the average path length), theaverage clustering coefficient and the power-law ex-ponent of the approximate degree distribution.

Using sliding windows of both fixed number ofevents and fixed time intervals, we showed that in theabsence of strong earthquakes (indicatively M>5.0),the topological structure of the HA-T system resem-bles a random network. Importantly, we show that themonitoring of the statistical properties of the networkcarry important information for the detection andcharacterization of the seismicity. In particular, wewere able to identify four mainshocks and their asso-ciated aftershock sequences that occurred during thisperiod, namely the cases of Samos in 2005,Zakynthos in 2006, Cephalonia in 2007 and inMethoni, 2008. The onset of these events resulted toa sharp and big simultaneous change in the statisticalproperties of the network marking its phase transitionto scale-free structures as this can be quantitativelymeasured by the value of the power-law exponent, γ,as well as the entropy of the network. At this point,we should remark that the average clustering coeffi-cient changes an order of magnitude before and justafter the main event (from ∼0.01 to ∼0.4) whichtogether with a significant decrease of the averagepath length to values around 5 suggests the strongsmall-world character of the underlying network.Within the same small time window, the number ofseismic events did not change a lot. These results arepromising regarding the characterization of the seis-micity as can be used as an alternative and/or supple-mentary to well-established traditional statisticalmethods in seismology such as the laws of Omori,Gutenberg–Richter and Mogi. To our knowledge, thisis the first time that such an analysis is performed forthe Hellenic arc region. Further research could pro-ceed towards the systematic classification of seismicactivity in swarms, foreshocks, aftershocks andmainshocks in comparison and in connection to theabove-mentioned traditional seismological laws.

Appendix

Usually, the statistical properties of networks are stud-ied in terms of the following basic quantitative

44 J Seismol (2014) 18:37–46

measures (Watts and Strogatz 1998; Albert andBarabasi 2002; Newman 2003):

(a) The average path length (APL). It is defined asthe mean value of all the shortest paths betweenany two nodes, reading:

APL ¼X

di→ j

N N−1ð Þwhere di→j is the shortest path between i and j nodes andN is the size of the network. The average path length is aglobal property of a network indicating the averagenumber of steps needed to reach any two nodes.

(b) The global efficiency. In the case where a net-work is disconnected that is at least two nodesdo not communicate, the APL is infinity. Inorder to overtake this problem, the global effi-ciency E is defined as

E ¼

X 1

di→ j

N N−1ð ÞIf two nodes are disconnected, then 1

di→ j¼ 0 , i.e. the

efficiency is zero. The inverse of global efficiency 1/Eis the harmonic mean of the shortest paths, and it issimilar to APL.

(c) The clustering coefficient ci. The clustering co-efficient ci of the node i is the number Ei oflinked triangles it forms with its neighbours,divided by the number of all possible trianglesthat i node forms. In the case of directed net-works, the ci takes the form (Fagiolo 2007):

ci ¼Aþ AT� �3

ii

2 ktot ktot−1ð Þ−2 A2� �

ii

h i

where A is the adjacency matrix of the network, ktot isthe summation of inward and outward degrees, i.e.ktot=kin+kout and the parenthesis (⋅)ii indicate the maindiagonal of the ⋅ matrix. The ACC of the wholenetwork is defined as the mean value of the clusteringcoefficients ci.

(d) The degree distribution P(k) which gives the frac-tion of nodes with exactly k edges connected to it.Characteristic examples of almost symmetric-

around the mean value of the degree distributionsare the Erdős–Rényi networks (Albert andBarabasi 2002; Barrat et al. 2008; Newman2003); scale-free networks are characterized frompower-law distributions with exponent γ (Barratet al. 2008; Newman 2003). In the case of directednetwork, the degree distribution is replaced by thedistribution of inward or outward degrees.

(e) The entropy of the degree distribution is defined as

H ¼ �X

k¼1

kmax

P kð Þlog P kð Þ

which provides a measure for the heterogeneity of thenetwork. Higher values of H indicate the existence ofheterogeneous degree distribution (Costa et. al 2007):There are few nodes acting as hubs (i.e. having bigdegree). Examples of such distributions are the powerlaws (or scale-free) with heavy tails in their degreedistribution.

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