You are an expert at summarizing long articles. Proceed to summarize the following text: 
the emergence of the world wide web ( www ) belongs arguably to the most relevant events of the present time . the interest in this system and in networks in general permeated through all the society , including physics . this led , at the turn of the century , to a large amount of studies of what with the time came to be known as `` network science '' . most studies of the www were performed , however , in the early 2000s @xcite and large - scale studies of the www are rather hard to find nowadays , despite the immense growth of the internet in the last 10 years . a remarkable finding of the first generation studies of the www is the emergence of scale - free degree distributions , which can be explained potentially from the view of preferential attachment , although the exponents obtained are not universal @xcite . generally , one can assume that the growth process of a complex network will be influenced by inter - node correlations and that these dependencies will be reflected in the resulting network topology . however , such correlations are not easy to detect and characterize , and have not been studied in depth . it is expected that a simple rule as preferential attachment can not completely reproduce the structures found in real - world networks , and therefore more complicated models have been developed to replicate the behavior @xcite . correlations between different properties are generally used as a proxy to study the internal structure of the network . for instance , vespagnani studied correlations between the in - degree of a node and that of a first neighbor of said node @xcite , showing a scale free property ( recently modeled by takagi @xcite ) , barabasi and albert studied the local clustering coefficient as a function of the in - degree @xcite , in order to obtain information regarding the hierarchical structuring of the network . however , real - world data about said correlations is not abundant .    in the present work we study the complete dominant core of the www by crawling @xmath0 million domains , including all domains with the largest traffic ( all domains with an alexa rank of one million or less are included ) . collapsing the data , by neglecting link multiplicities , we study the network of inter - domain hyperlinks ( not webpages ) , containing about half a billion directed edges . we find non - trivial correlations between in- and out - degree , between the in - degree and the local clustering coefficient and between the degrees of neighboring sites . in addition to evaluate averaged quantities , we study the full density plots , finding novel scaling features for the boundaries of several correlation functions . we present , in addition , a formula for the clustering coefficient of random directed graph characterized by given arbitrary in- and out - degree sequences . finally we present an analysis of the correlations between the number of in - links and the alexa rank of a domain .     and out - degree @xmath2 . right : two types of in - degree clusters , with the edges always directed towards the central site ( a).,title="fig:",scaledwidth=11.0% ]   and out - degree @xmath2 . right : two types of in - degree clusters , with the edges always directed towards the central site ( a).,title="fig:",scaledwidth=11.0% ] for directed graphs we have to distinguish between the distribution @xmath3 and @xmath4 of the in- and the out - degrees @xmath5 and @xmath6 , respectively . there are , in addition , two kinds of nearest neighbors , in - neighbors and out - neighbors . site b1 in fig . [ fig : graph - illustrations ] is a nearest in - neighbor of site a and a nearest out - neighbor of site b2 . alternatively one could call b1 an ancestor of a and a descendent of b2 @xcite . for bi - directional links , as between b3 and b4 in fig . [ fig : graph - illustrations ] , in - neighbors are also out - neighbors . the total number of in - links equals the total number of out - links , the in- and out - degree coordination numbers @xmath7 are hence identical .      in order to calculate the relevance of correlations between in- and out- degree in the structure of the network , we have developed a statistical model of the clustering coefficient for given distributions of in- and out- degree which are uncorrelated . we define with @xmath8 the excess distribution @xcite of outgoing links of a nearest in - neighbor . the normalization constant @xmath9 is just the coordination number @xmath10 , see ( [ def_z ] ) . equivalently we define via @xmath11 the degree distribution ( not excess ) of incoming links of a nearest in - neighboring site . here @xmath12 is the probability that a site has @xmath6 out - links and @xmath5 in - links ( joint distribution function ) , with the usual relations @xmath13 for the marginal distribution functions . the normalization constant @xmath14 in ( [ def_bar - q - in ] ) is given by the coordination number @xmath10 , @xmath15 for the clustering coefficient @xmath16 ( the ` hat ' symbol stands here for the clustering coefficient of a directed graph ) we now consider two in - neighbors , having respectively , with probabilities @xmath17 and @xmath18 , @xmath5 in - links and @xmath6 excess out - links ( as stubs ) . we now assume that the distributions @xmath19 and @xmath20 of the two neighbors are independent of each other . the probability , for a graph with @xmath21 nodes , that a given pair of in- and out - stubs are connected is then @xmath22 , where @xmath23 is the total number of in- or out - stubs , and hence @xmath24 transforming now into a sum @xmath25 over sites , every site @xmath26 being characterized by an in - degree @xmath27 and out - degree @xmath28 , one obtains @xmath29 which coincides with the usual expression @xcite for non - directed graphs ( apart from a factor @xmath28 instead of @xmath30 in the first factor ) , by taking @xmath31 for @xmath32 . a fully - connected network results in @xmath33 under this formula . we note that the expression ( [ eq : hat - c - sites ] ) for @xmath16 may actually violate the sum rule @xmath34 , due to the neglect of inter - site degree correlations , when applied to a real - world graph . as an example consider a network composed out of a single star , like the site c in fig . [ fig : graph - illustrations ] , but with bi - directional edges . for a un - directed ( and loopless ) star the degree sequence is @xmath35 with an intensive coordination number @xmath36 . the statistical formula ( [ eq : hat - c - sites ] ) for the clustering coefficient would , one the other hand , diverge @xmath37 in the thermodynamic limit @xmath38 . a substantial deviation of @xmath16 from the true clustering coefficient is hence a measure for the strength of inter - site degree correlations , the expression ( [ eq : hat - c - sites ] ) being valid for graphs with vanishing inter - node correlations .     for the in - degree ( main panel ) and the out - degree ( inset ) , log - log plot . the dashed lines , corresponding to power - law distributions , have slopes @xmath39 and @xmath40 , respectively for the in- and the out - degree.,scaledwidth=45.0% ] using the crawlers of the former file search engine findfiles.net @xcite we crawled , mostly in 2011 , @xmath0 million domains ( of type http://www.domain.com ) with a total of @xmath41 million subdomains ( of type http://subdomain.domain.com ) . these @xmath0 million domains have 223 million hyperlinks in between them , linking in addition to 50 million other sites . for the network analysis we neglected these 50 million external sites , as we did not crawl them separately . the network of 223 million inter - domain directed links has an average degree of 32 and @xmath42 million of the @xmath0 million domains are isolated in the sense that they have no in - links , they can not be reached from the core of the world wide web . a further one million sites have just a single hyperlink directed to them . the crawling strategy started from the set of the about 32 million subdomains referred - to in wikipedia and dmoz ( all languages ) , with further systematic additional extensions . we included , in particular , the one million domains with the largest traffic volume , in terms of the alexa rank . this data set , which we denote with ff-2011 , hence corresponds essentially to the complete relevant part of the world wide web , in terms of traffic volume . the degree distribution of hyperlinks have been observed to follow a power law @xmath43 , with an exponent close to the limiting case @xmath44 ( when the mean degree would diverge in the thermodynamic limit ) @xcite . in fig . [ fig : in - out - degree - distribution ] we present the complementary cumulative distribution functions @xcite for both the in - degree and the out - degree .    over a range of about 2.5 - 3 orders of magnitude , the data can be approximated quite nicely by power law distributions , with exponents @xmath45 and @xmath46 respectively for the in- and the out - degree . these results confirm earlier studies @xcite finding consistently @xmath47 . the absolute magnitude of the values reported for the scaling exponents vary slightly from study to study , either because of the evolution of the internet with time passing , or due to the size of the respective databases .     of domains with in - degree @xmath5 and out - degree @xmath48 . the density is shown in log scale , as well as both axes . the solid line represents the average @xmath49 . the probability density is normalized , @xmath50.,scaledwidth=45.0% ]      in fig . [ fig : kvskout ] , the density distribution of nodes having an in - degree @xmath5 and an out - degree @xmath48 , is presented , together with the average out - degree @xmath51 , for sites having an in - degree @xmath5 . in- and out - degree do not seem to be particularly correlated , on a first sight . however , the average out - degree shows two regimes with approximated power - law scaling , for @xmath52 and @xmath53 , with exponents @xmath54 and @xmath55 respectively . in the case that the joint distribution @xmath56 would factorize , @xmath57 , the mean out - degree @xmath58 would functionally follow the in - degree distribution @xmath3 , where @xmath59 is the average ( in- and out- ) degree of our internet data . however , as shown in fig . [ fig : in - out - degree - distribution ] , the marginal in - degree distribution @xmath3 , falls approximately like @xmath60 , viz substantially faster than ( [ eq : in_out ] ) would imply . in- and out - degree are hence non - trivially correlated . we will discuss the nature of the respective correlations in more detail further below when discussing the distribution of local clustering coefficients . the local clustering coefficient @xmath61 is given by the number of linked nearest neighbors of site @xmath62 , relative to the total number of possible links between the neighbors . for directed graphs there are in- and out - neighbors and various possible 3-site loops , as illustrated in fig . [ fig : graph - illustrations ] , also known as network motifs @xcite . here we examine the in - clustering coefficient . for a given site @xmath62 the in - clustering coefficient @xmath61 is given by the average number of links in between the in - neighbors of site @xmath62 . in fig . [ fig : graph - illustrations ] , the sites ( b1,a , b2 ) form an in - loop of site a , contributing to @xmath63 , while the sites ( b4,a , b3 ) contribute two in - loops . we focus on the in - clustering coefficient since the number of in - links is a measure for the importance of a site , contributing to its traffic volume . we find , for the ff-2011 network data , a mean clustering coefficient @xmath64 of @xmath65 . this is , for two reasons , a surprising high value . firstly the connection probability @xmath66 is very low , being just @xmath67 . secondly a quite large number of sites , 0.27% , has a vanishing local clustering @xmath68 , and only a small fraction , 0.3% , of domains , mostly with small degrees , have a maximal local clustering coefficient of unity . we can assess the impact of correlations on the formation of local loops by considering identical degree sequences for the in- and out- degree , as extracted from the ff-2011 network data , but considering various types of correlations between the in- and out- degree of each node . * applying eq . ( [ eq : hat - c - sites ] ) to the actual network , the obtained value amounts to @xmath69 . this value is over the maximum @xmath70 , indicating towards very strong correlations between the in- and the out - degree distributions , compare the discussion below eq . ( [ eq : hat - c - sites ] ) . * for a network having the same degree distributions @xmath3 and @xmath4 for the in- and the out - degree as the actual network , but without correlations between these degrees , viz assuming a joint probability distribution @xmath71 , the clustering coefficient obtained by eq . ( [ eq : hat - c - sites ] ) would amount to @xmath72 . * a network where the in- and out- degrees are anticorrelated ( nodes with largest in - degree are mapped to the smallest out - degree ) , would amount to an even lower @xmath73 . * for a network with a maximally correlated distribution of in- and out- degree ( nodes with the largest in - degree being mapped to the largest out - degree ) , would result again in a higher - than - unity clustering coefficient @xmath74 , when using eq . ( [ eq : hat - c - sites ] ) . we hence conclude that the in- and out - degree are quite strongly correlated positively for the world wide web .     of pairs of in - degree @xmath5 and local clustering coefficient @xmath75 . the density , and both axes are given in log scales . the solid line represents the average value @xmath76 , as a function of in - degree @xmath5 . the probability density is normalized , @xmath77.,scaledwidth=45.0% ]      in fig . [ fig : kvsc ] , the density of @xmath78 pairs is shown in a log - log scale , where @xmath5 is the in - degree and @xmath75 the local clustering coefficient . the density distribution has upper and lower cutoffs scaling approximatively like @xmath79 , with @xmath80 and @xmath81 . the lower limit has a simple explanation . the lowest non - zero local clustering coefficient is realized when just a single loop exist out of the @xmath82 possible triangles , @xmath83 when setting the number of loops @xmath84 to one . the exponent of the upper limit , @xmath85 , implies , compare ( [ eq : c_min ] ) , that the number of local loops scales like @xmath86 . we have presently no explanation for this scaling behavior .    the average value of @xmath87 , as a function of in - degree @xmath5 , follows mostly a power law for small @xmath88 , with an exponent @xmath89 . for larger @xmath90 the exponent changes toward @xmath91 for the mean local clustering coefficient . this last exponent is in agreement with previous observations found in @xcite , and are a fingerprint for a hierarchical network structure . the change in behavior at the point @xmath92 is also observable in the correlation between the in - degree and the degree of nearest neighbors , as we will show in the next sections . there is a group of nodes with very high clustering coefficients @xmath93 around the @xmath94 region ( close to where the upper limit with the @xmath95 slope intersects the abscissa ) , which somewhat falls of the line . after analyzing some of the domains involved in this region , we conclude that this group of nodes does not represent the intrinsic network structure of the www , belonging most probably to _ link farms_. these nodes are however responsible for the jumps in @xmath87 at @xmath94 .     of the in - degree @xmath5 and the in - degree @xmath96 of nearest neighbor sites . both the density distribution and the axis are given in log scale . the solid line corresponds to the average value @xmath97 . the probability distribution is normalized , such that @xmath98.,scaledwidth=45.0% ]      in fig . [ fig : kvsknn ] , the density of pairs @xmath99 is shown , where @xmath96 is the in - degree of a first neighbor , and @xmath5 the in - degree . the dots with higher densities for low @xmath5 values are relatively large groups of linked domains which share exactly the same pairs of in - degree and first - neighbor in - degree . the domains do not seem to be particularly related although we do not discard the possibility that they may belong to _ link farms _ , as they clearly stand out of the general behavior of the density distribution . when analyzing the average @xmath100 as a function of the in - degree @xmath5 , we observe a very weak increase for small @xmath5 until @xmath101 . we can fit this increase fairly good with a power law of exponent @xmath102 . this behavior would be in agreement with the one observed in the canonical barabasi - albert model @xcite , though it differs with with a 1998 www network study @xcite .    in the range from @xmath101 to @xmath103 we observe a change in the behavior of the average @xmath100 , as it starts decaying with increasing @xmath5 . this decay follows a power - law as well , with an exponent of about @xmath104 . this decay is closer to the results found in @xcite for a subset of the 1998 internet data and the fitness model developed therein ( which decays with @xmath105 ) . however , the decay is observed in our results for much higher degree @xmath5 than in @xcite , which has data limited to @xmath106 . we speculate that this difference is due to the size of the network studied , although it might be possible that the evolution of the www in the last 10 years is responsible for the structural change .    , where @xmath107 is the alexa index and @xmath5 the corresponding in - degree of the domain . the density is given in log scale , as well as both axes . the solid line shows the average @xmath108 , as a function of @xmath107 . the probability is normalized , such that @xmath109.,scaledwidth=45.0% ]      we have analyzed the correlations of the alexa rank @xcite with respect to the in - degree @xmath5 . the alexa rank is arguably one of the most popular measures of the traffic received by an internet site , and so , its relevance . the ranking is proprietary , so the general public does not have access to the specifics of its calculation , although according to the official information , it is derived from the traffic observed , with data partly retrieved from users who installed the alexa add - on to their web browser @xcite . in this ranking , the site with the most traffic has rank @xmath110 , the following largest rank @xmath111 and so on . the rank does not provide any information about the precise amount of traffic , such that a larger @xmath107 index does not give any indication of how much less traffic does that site receives , but rather only that it receives less traffic than the sites with smaller @xmath107 .        in fig . [ fig : alexavsk ] , we present the density of domains as a function of its in - degree @xmath5 and alexa rank @xmath107 . we only analyze the alexa rank for sites having an in - degree @xmath112 , with a few exceptions , due to constraints in retrieving the alexa rank data . we observe a distribution limited from above and below by two power laws with exponents @xmath113 and @xmath114 . the lower limit is however less pronounced , due to the lack of samples .    the solid line in fig . [ fig : alexavsk ] shows the average in - degree @xmath108 , for sites having and alexa rank @xmath107 . we observe a very marked power law decay @xmath115 the exponents are not the inverse of each other , since @xmath116 and @xmath117 are distinct averages . there is a saturation at @xmath118 for the scaling regime , presumable due to our constraint @xmath112 for the alexa index . we find this scaling particularly interesting , since the alexa rank is not derived directly from the topology of the network but rather from the traffic generated by users . for site administrators the relatively weak decay ( [ eq : scaling_k - a ] ) implies that the traffic generated by in - coming hyperlinks can be a relevant contribution to the overall traffic volume . we have performed a visualization study of the data discussed hitherto by considering relative joint distribution functions , which are obtained by dividing a given joint probability distribution by the product of the respective marginal distributions , @xmath119 in the absence of correlations , viz when @xmath120 , the respective density plots would be homogeneous and flat .    in fig . [ fig : correl]a , we show the relative density @xmath121 for pairs of in - degree @xmath5 and local clustering coefficient @xmath75 . the distribution is quite homogeneous and the upper and lower limits of the distribution are exalted in comparison with the plot of the bar joint distribution presented in fig . [ fig : kvsc ] .    in fig . [ fig : correl]b , we present the relative joint density for the correlation between in- and out - degree . the distribution is considerably more homogeneous than the respective bare probability density shown in fig . [ fig : kvskout ] . however , a substantial enhancement remains for small in- and out- degrees .    from the shape of the joint nearest - neighbor degree distribution presented in in fig . [ fig : kvsknn ] , it would be tempting to think that its shape is mostly determined by the marginal distributions , i.e.  that @xmath122 , and that therefore @xmath5 and @xmath96 would be essentially decorrelated . this is , however , not the case , as we can see in fig . [ fig : correl]c . in this plot the correlations are seen more clearly in terms of the relative joint distribution we can observe that the resulting relative distribution still shows a stronger correlation when both in - degree and the in - degree of the neighbor are small .    in fig . [ fig : correl]d , we present the relative density distribution of for sites having an alexa rank @xmath107 and an in - degree @xmath5 . as for the case of the clustering coefficient @xmath75 , the distribution maintains the very marked upper and lower limits , being otherwise essentially only slightly more uniform than the orginal data shown in fig . [ fig : alexavsk ] . the present study shows that many properties of the www are characterized by non - trivial correlations . we observe that the joint probability distributions , for several of the properties tested , follow power - law scaling for the respective averages . additionally , the distributions have , in many instances , density distributions which are limited by power laws . the power law limiting functions do not share exponents neither with that of the marginal distributions , nor with the respective average value of the property studied . interestingly one also observes power - law scaling for a seemingly unrelated quantity , the alexa traffic rank , which decays as a function of in - degree and also the average in - degree decays moderately weakly as a function of the alexa rank . we found that the statistical properties of the world wide web differ remarkably for domains receiving more / fewer than about @xmath1 hyperlinks from different domains . the change in behavior is observed for the correlations between in- and out - degree , between in - degree and local clustering coefficient and between in - degree and the in - degree of neighbors . this observation points towards an underlying hierarchical structure of the www , with the `` elite '' of the internet domains , receiving links from more than one thousand different domains , being made - up by about @xmath123 sites . we would like to acknowledge the support of the german science foundation ( dfg ) .
we performed a large - scale crawl of the world wide web , covering @xmath0 million domains , including all high - traffic sites of the internet . 
 we present a study of the correlations found between quantities measuring the structural relevance of each node in the network ( the in- and out - degree , the local clustering coefficient , the first - neighbor in - degree and the alexa rank ) . 
 we find that some of these properties show strong correlation effects and that the dependencies occurring out of these correlations follow power laws not only for the averages , but also for the boundaries of the respective density distributions . 
 in addition , these scale - free limits do not follow the same exponents as the corresponding averages . in our study 
 we retain the directionality of the hyperlinks and develop a statistical estimate for the clustering coefficient of directed graphs . 
 we include in our study the correlations between the in - degree and the alexa traffic rank , a popular index for the traffic volume , finding non - trivial power - law correlations . 
 we find that sites with more / less than about @xmath1 links from different domains have remarkably different statistical properties , for all correlation functions studied , indicating towards an underlying hierarchical structure of the world wide web .