Distinct types of eigenvector localization in networks
DDistinct types of eigenvector localization innetworks
Romualdo Pastor-Satorras and Claudio Castellano Departament de F´ısica, Universitat Polit `ecnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain Istituto dei Sistemi Complessi (ISC-CNR), Via dei Taurini 19, I-00185 Roma, Italy Dipartimento di Fisica, “Sapienza” Universit `a di Roma, P.le A. Moro 2, I-00185 Roma, Italy * [email protected] ABSTRACT
The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complexnetworks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding ofnodes centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principaleigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P ( q ) ∼ q − γ , localizationoccurs on the largest hub if γ > / ; for γ < / a new type of localization arises on a mesoscopic subgraph associated with theshell with the largest index in the K -core decomposition. Similar evidence for the existence of distinct localization modes isfound in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and ona recently proposed alternative measure of node centrality based on the non-backtracking matrix. Introduction
An issue of paramount significance regarding the analysis of networked systems is the identification of the most important(or central ) vertices. The centrality of a vertex may stem from the number of different vertices that can be reached from it,from the role it plays in the communication between different parts of the network, or from how closely knit its neighborhoodis. Following these approaches, different centrality measures have been defined and exploited, such as degree centrality,betweenness centrality, or the K -core index and associated K -core decomposition. Among those definitions, one of the mostrelevant is based on the intuitive notion that nodes are central when they are connected to other central nodes. This concept ismathematically encoded in the eigenvector centrality (EC) of node i , defined as the component f i of the principal eigenvector(PEV) f associated with the largest eigenvalue Λ of the adjacency matrix A i j . EC is the simplest of a family of centralitiesbased on the spectral properties of the adjacency matrix including, among others, Katz’s centrality and PageRank. Apart from providing relevant information about the network structure, the PEV and associated largest eigenvalue playa fundamental role in the theoretical understanding of the behavior of dynamical processes, such as synchronization andspreading, mediated by complex topologies. Considerable effort has thus been devoted in recent years to the study of the a r X i v : . [ phy s i c s . s o c - ph ] J a n pectral properties of heterogeneous networks. In this framework, Goltsev et al. (see also ) have considered the localization of the PEV, i.e., whether its normalization weight is concentrated on a small subset of nodes or not. More in detail,let us consider an ensemble of networks of size N , with a PEV f i normalized as a standard Euclidean vector, i.e. ∑ i f i = localized on a subset V of size N V if a finite fraction of the normalization weight is concentrated on V ( ∑ i ∈ V f i ∼ O ( ) ) despite the fact that V is not extensive, i.e., N V is not proportional to N . This includes the case of localizationon a finite set of nodes (i.e. N V independent of N , N V = N V ∼ N β with β <
1. Otherwise, the eigenvector is instead delocalized , and a finite fraction of the nodes N V ∼ N contribute to the normalization weight, implying that their componentsare f i ∼ N − / .In this context, Goltsev et al. study the localization in power-law distributed networks, with a degree distribution scalingas P ( q ) ∼ q − γ , for which the leading eigenvalue Λ is essentially given by the maximum between (cid:104) q (cid:105) / (cid:104) q (cid:105) and √ q max , where q max is the largest degree in the network. For γ > /
2, where Λ ∼ √ q max , Goltsev et al. find that the PEV becomeslocalized around the hub with degree q max . On the other hand, they argue that, for γ < /
2, when Λ ∼ (cid:104) q (cid:105) / (cid:104) q (cid:105) , the PEV isdelocalized. These observations are relevant in different contexts. Firstly, they point out a weakness of EC as a measure ofcentrality for heterogeneous power-law networks ( γ > / Onthe other hand, in the so-called quenched mean-field approach to epidemic spreading on networks, the density of infectedindividuals in the steady state can be related to the properties of the PEV. The localization occurring for large γ implies thatthe density of infected individuals in the steady state in those processes might not be an extensive quantity, casting doubts onthe validity of this theoretical approach and on the actual onset of the endemic infected state.Here we show that the localization properties of the adjacency matrix PEV for heterogeneous (power-law distributed)networks are described by a picture much more complex than previously believed. In fact, we provide strong numerical evidencethat the EC in heterogeneous networks never achieves full delocalization. In the case of uncorrelated synthetic networks witha power-law degree distribution, we obtain, by means of a finite-size scaling analysis, that for mild levels of heterogeneity(with γ > / γ < / K -core decomposition of the network. The paper of Goltsev et al. is perfectly correct for what concernsthe case γ > / γ < /
2. In order to overcome the localization effects intrinsic of the EC, a new centrality measure,based on the largest eigenvalue of the Hashimoto, or non-backtracking, matrix, has been recently proposed. We observe thatthis new centrality is not completely free from localization effects. Thus, while it almost coincides with the EC for γ < / γ > / EV is either localized on the hubs, or effectively localized on the maximum K -core of the network. Results
Eigenvector localization and the inverse participation ratio
A full characterization of an undirected network of size N is given by its adjacency matrix A , whose elements take the value A i j = i and j are connected by an edge, and value A i j = Λ i , and associated eigenvectors f ( Λ i ) , i = , . . . , N , defined by A f ( Λ ) = Λ f ( Λ ) . (1)Since the adjacency matrix is symmetric all its eigenvalues are real. The largest of those eigenvalues Λ , is associated with theprincipal eigenvector (PEV) which we denote simply by f .The concept of the localization of the PEV f translates in determining whether the value of its normalized components isevenly distributed among all nodes in the network, or either it attains a large value on some subset, and is much smaller inall the rest. While this concept is quite easy to grasp, assessing it in a single network instance is a delicate issue because anyquantitative definition involves some degree of arbitrariness. The task becomes however straightforward when ensembles ofnetworks of different size can be generated. In such a case, the localization of the eigenvector f associated with the eigenvalue Λ can be precisely assessed by computing the inverse participation ratio (IPR), defined as, Y Λ = ∑ i f i ( Λ ) . (2)In the absence of any knowledge about the localization support, it is possible to determine whether an eigenvector is localized(on some subset in the network) by studying its inverse participation ratio, as a function of the system size N and fitting itsbehavior to a power-law decay of the form Y Λ ( N ) ∼ N − α . (3)If the eigenvector is delocalized, i.e. for f i ∼ N − / , the exponent α is equal to 1. An exponent α < N V independent of the network size N , the corresponding components of the PEV are finite and this implies Y loc Λ ∼ O ( ) , i.e., α = N → ∞ . Finally, if localization takes place over a subextensive set of nodes of size N V ∼ N β , we expect Y Λ ( N ) ∼ N V ∼ N − β , (4)leading to a decay exponent α = β . Eigenvector localization in synthetic networks
We study the localization properties of the PEV computed for synthetic power-law distributed networks of growing size,generated using the uncorrelated configuration model (UCM), a modification of the standard configuration model igure 1. (a) Inverse participation ratio as a function of the network size for the adjacency matrix of synthetic networks withdifferent degree exponent γ . For large γ , the IPR tends to saturate to a constant value for sufficiently large value of N . For γ < /
2, on the other hand, the behavior of the IPR can be fitted to Y Λ ( N ) ∼ N − α , with α < /
2. The dashed line represents apower-law behavior ∼ N − , corresponding to a delocalized IPR. (b) Inverse participation ratio as a function of the degreeexponent γ for different network sizes N . The plot confirms the presence of transition in the behavior of the IPR, located in thevicinity of γ = / In order to explore the presence or absence of localization, we analyze the scaling of Y Λ ( N ) as function of N as discussed above. In Fig. 1(a) we apply this finite-size scaling analysis to synthetic networks withdifferent values of γ . In this and the following figures, statistical averages are performed over at least 100 different networksamples. Error bars are usually smaller than the symbol sizes. In the case of large γ we observe an IPR tending to a constant forlarge N , confirming the localization on the hubs predicted by. The situation is however surprisingly different for γ < / et al. , we should expect a delocalized PEV and an IPR decreasing as N − α with α =
1, weobserve instead power-law decays with N , with effective exponents α always smaller than 1 /
2. The change of behavior ofthe IPR can be further confirmed in Figure 1(b), where we plot the IPR as a function of the degree exponent γ , for differentvalues of N . While it is clear that for γ ≥ . γ for which the behavior changes. However, since thedependence of the largest eigenvalue on N changes for γ = / we expect the transition to take place exactly at γ = / γ < / Λ = (cid:104) q (cid:105) / (cid:104) q (cid:105) , coincides with the largest eigenvalue of the adjacency matrix in the annealed network approximation. Theannealed network approximation consists in replacing the actual, fixed, adjacency matrix by an average performed overdegree classes, taking the form¯ a i j = q (cid:48) j P ( q i | q (cid:48) j ) NP ( q i ) , (5)where P ( q | q (cid:48) ) is the conditional probability that a link from a node of degree q (cid:48) points to a node of degree q . For degree igure 2. (a) Rescaled scatter plot of f i [ N (cid:104) q (cid:105) ] / as a function of q i for a synthetic network with γ = . N = .Data fits the expectation for the PEV in the annealed network approximation, Eq. (7), with only small fluctuations. (b, main)Scatter plot of the squared PEV components as a function of the K -core index for the adjacency matrix of a power-lawsynthetic network with γ = . N = . (b, inset) Scatter plot of the squared PEV components as a function of thedegree q i in a synthetic network with γ = . N = .uncorrelated networks, with P ( q | q (cid:48) ) = qP ( q ) / (cid:104) q (cid:105) , we obtain an averaged adjacency matrix¯ a i j = q i q j N (cid:104) q (cid:105) . (6)The matrix ¯ a i j is semi-positive definite and therefore all its eigenvalues are non-negative. Then considering that Tr ( ¯a ) =[ Tr ( ¯a )] = ( (cid:104) q (cid:105) / (cid:104) q (cid:105) ) , where Tr ( · ) is the trace operator, we have that ¯ a i j has a unique non-zero eigenvalue Λ an = (cid:104) q (cid:105) / (cid:104) q (cid:105) ,with associated principal eigenvector f an i ∝ q i . Applying the normalization condition ∑ i f i =
1, we obtain the normalized form f an i = q i [ N (cid:104) q (cid:105) ] / . (7)Inserting the expression of f an i into Eq. (2) yields Y Λ ( N ) ∼ / N ( − γ ) / , (8)that is, a decay with an exponent smaller than 1 /
2, in agreement with the results in Fig. 1(b). Fig. 2(a) confirms that alsoquenched synthetic networks have PEV components proportional in average to the degree. Notice that Eq. (8) is approximatelytrue only in quenched networks for γ < /
2, since the condition leading to it, Eq. (7) fails at γ > /
2, see Fig. 2(b,inset).A more physical interpretation of the particular distribution of the PEV in power-law networks with γ < /
2, is that thePEV becomes effectively localized on the max(imum) K -core of the network, defined as the set of nodes with the largest coreindex K M in a K -core decomposition. The K -core decomposition is an iterative procedure to classify vertices of a networkin layers of increasing density of connections. Starting with the full graph, one removes the vertices with degree q =
1, i.e. withonly one connection. This procedure is repeated until only nodes with degree q ≥ K = K = q = K = K -core (of index K M ) is the set of vertices igure 3. (a) Total weight W K M of the PEV on the nodes of the max K -core in synthetic networks as a function of size N . (b)Inverse participation ratio as a function of the size of the max K -core N K M . The dashed line represents a power-law behavior ∼ N − K M . We can see the asymptotic behavior Y Λ ∼ N − K M , valid for large network sizes.such that one more iteration of the procedure removes all of them. The line of argument leading to this interpretation stemsfrom combining the results of Ref., in which it is proposed that, in epidemic spreading in complex networks, infectionactivity is localized on the PEV, with the observations in Ref., in which the maximum K -core is identified as a subset of nodessustaining epidemic activity for γ < / K -core in different ways. In the first place, in Fig. 2(b,main) we plotthe squared components f i of the PEV for all vertices against their corresponding K -core index. From this Figure we concludethat all nodes with the largest f i components belong to the max K -core. The size of this max K -core, N K M , grows sublinearly asa function of the network size as N K M ∼ N ( − γ ) / . However, despite this sublinear growth, a finite fraction of the total PEVweight is concentrated on this subset. We check this fact in Fig. 3(a): the total weight of the nodes in the max K -core, W K M = ∑ i | i ∈ K M f i , (9)tends to a constant in the limit of large network size, implying that more than half of the weight of the normalized PEVresides over the max K -core. Finally, the size dependence of the max K -core translates, from Eq. (4) in an IPR scaling as Y Λ ∼ N K M − ∼ / N ( − γ ) / , in agreement with the result obtained from the degree dependence of the PEV components, f i ∼ q i ,see Eq. (8). The relation between IPR and max K -core size is satisfactorily checked in Fig. 3(b), where we observe it to bevalid for large network sizes.For γ > /
2, instead, Figure 2(b,inset) confirms the localization of the PEV around the hub, displaying a disproportion-ately large component on the node with the largest degree. Notice that, irrespective of the value of γ , with high probability thehub belongs to the max K -core. What changes in the two cases is that for γ > / f i ∼ O ( ) ) while for γ < / K -core must beconsidered to have a finite weight W K M . The behavior for γ > / < γ < N . This effect is observed in Fig. 4, were we igure 4. Weight of the PEV as a function of the network size in power-law networks with degree exponent γ = .
8. Thedifferent functions correspond to: total weight of the nodes in max K -core, W K M ; total weight in the hub, W H ; total weight inthe max K -core, subtracting the hub, W K M − H .plot the total weight W K M of the nodes in the max K -core, Eq. (9), the total weight in the hub, W H , and the total weight in themax K -core, subtracting the hub, W K M − H . As we can observe from this Figure, the weight at the hub is small for network sizes N < , but it then starts to increase, to finally take over, for large network sizes N > . The non-backtracking centrality
The observations presented here, together with the arguments provided by Martin et al. , hint that the EC is problematic as auseful measure of centrality. For large values of γ , it is affected by an exceedingly strong localization on the hub, arising asa purely topological artifact: the hub is central because its neighbors are central, but those in turn are central only becauseof the hub. For small values of γ , on the other hand, the observed relation f i ∼ q i indicates that the eigenvector centralityprovides essentially the same information as the degree centrality. As an attempt to correct the flaws of the EC, Martinet al. propose a modified centrality measure, the non-backtracking centrality (NBTC), which is computed in terms ofthe non-backtracking matrix. The Hashimoto, or non-backtracking matrix (NBT), is defined as follows: an initiallyundirected network is converted into a directed one by transforming each undirected edge into a pair of directed edges, eachpointing in opposite directions. If the initial undirected network has E edges, the NBT matrix is a 2 E × E matrix with rowsand columns corresponding to directed edges i → j with value B i → j , l → m = δ i , m ( − δ j , l ) , δ i , j being the Kronecker symbol. Thecomponents of the principal eigenvector of the NBT matrix, f i → j measure the centrality of vertex i disregarding the contributionof vertex j . The NBT centrality of vertex j is given by the sum of these contributions for all neighbors of j : f NBTj = ∑ i A i j f i → j .The elements of the NBT matrix count the number of non-backtracking walks in a graph and hence remove self-feedback in thecalculation of node centrality, thus eliminating in principle the artificial topological enhancement of the hub’s centrality.As Figure 5 shows, however, the NBTC is not free from localization effects: For all values of γ the NBTC is not delocalized,i.e. Y Λ does not decrease as 1 / N when increasing N . This fact can be understood for γ < / K -core, which features many mutual interconnections: the centrality ofa node is only weakly affected by self-feedback, and removing the contribution of backtracking paths has therefore little igure 5. Inverse participation ratio as a function of the network size N for the NBTC for power-law synthetic networks withdifferent degree exponents γ . The dashed line has slope − f NBT i as a function of the corresponding components f i ofthe adjacency matrix PEV, computed for the same synthetic networks, Figure 6(a). For γ < / γ > / N more slowly than N − . This is indicative that also in this case a localization occurs on a mesoscopicsubset, whose size grows sublinearly. Figure 6(b) shows that this localization is not due to a strong correlation between theNBT centrality and the degree of nodes, contrary to what happens for the EC for γ < / Eigenvector localization in real networks
For real networks, which have fixed size and do not allow for a finite size scaling analysis, localization is necessarily a moreblurred concept. The value of Y Λ gauges how localized the PEV is, but it does not permit to unambiguously declare a networklocalized or not. However, also in this case it is possible to detect, as in synthetic networks, the existence of different localizationmodes. We consider here several real complex networks exhibiting large variations in size, heterogeneity and degree correlations(see Methods and Supplemental Material, SM, for details).The linear relation between f i and the degree q i is not fulfilled in real networks (see Supplementary Figure SF-1), probablydue to the presence of nontrivial degree correlations (see SM) which are absent in the synthetic networks. The effectivelocalization on the max- K core is however still present in some cases. In Fig. 7 we plot for these networks the squared PEVcomponent f i as a function of the K -core index. In some cases (HEP, Movies) all nodes in the max K -core have a comparableand large EC (as in synthetic networks for γ < / K -core. In other cases (Internet,Amazon) one or a few nodes have a disproportionately large value of f i , hinting at a localization around hubs, as in syntheticnetworks for large γ . igure 6. (a) Scatter plot of the NBTC centralities f NBT i as a function of the corresponding components of the PEV of theadjacency matrix f i , in synthetic uncorrelated networks with a power-law degree distribution. Network size N = . (b)Rescaled scatter plot of the NBTC centralities f NBT i [ N (cid:104) q (cid:105) ] / as a function of q i for a synthetic network with γ = . N = .To clarify the phenomenology we report in Table 1 for each of the real-world networks the values of the leading eigenvalue,and the factors (cid:104) q (cid:105) / (cid:104) q (cid:105) and √ q max . The analysis here is complicated by the presence of degree correlations (see SM), whichinvalidate the direct connection between Λ and the largest between √ q max and (cid:104) q (cid:105) / (cid:104) q (cid:105) . However, in some cases (Internet,Amazon) the leading eigenvalue is much closer to √ q max than to (cid:104) q (cid:105) / (cid:104) q (cid:105) : This suggests a localization around the hub andmatches well with Fig. 7. In others the opposite is true: Λ is very far from √ q max and relatively close to (cid:104) q (cid:105) / (cid:104) q (cid:105) , hintingat a localization on the max K -core, again in agreement with Fig. 7. In other cases (P2P, WWW), values are so close that noconclusion can be drawn.A further confirmation of this picture is provided by the analysis of the NBT centrality. When localization occurs on hubsone expects the elimination of backtracking paths to have a strong impact, as self-feedback effects are tamed. In this case weexpect the ratio between the IPR for the NBTC and the IPR for the adjacency matrix to be small. On the contrary, when thelocalization occurs on the max K -core, passing from the adjacency to the NBT matrix would not lead to a big change and weexpect the ratio to be close to 1. Table 1 confirms this expectation: the IPR ratio is small when the leading eigenvalue Λ isessentially given by √ q max (localization on hubs) while it is close to 1 when Λ is closer to the (cid:104) q (cid:105) / (cid:104) q (cid:105) factor (localization onthe max K -core). A visual representation of these results is provided in Figure 8, where we plot the IPR ratio as a function ofthe ratio between Λ and √ q max . As we can see, networks in which the PEV is localized in the max K -core are situated in theupper right corner of the panel, while the lower left corner shows the networks with localization occurring on the hubs. Discussion
The properties of the principal eigenvector (PEV), and associated largest eigenvalue, of the adjacency matrix defining a networkhave a notable relevance as characterizing several features of its structure and its effects on the behavior of dynamical processesrunning on top of it. Most important among these features is the role of the components of the PEV as a measure of a node’s igure 7.
Scatter plot of squared PEV components of the adjacency matrix of the real-world networks as a function of the K -core index.importance, the so-called eigenvector centrality. One of the properties of the PEV that has recently attracted the interest ofthe statistical physics community is its localization. In the case of networks with a power-law degree distribution P ( q ) ∼ q − γ ,initial research on this subject suggested that, for γ > /
2, the PEV is localized on the nodes with largest degree. On theother hand, for γ < /
2, the PEV should be delocalized.In this paper we have shown that eigenvector localization in heterogeneous networks is described by a more complexpicture. Thus, we present evidence that for all power-law distributed networks the PEV is always localized to some extent. Inthe case of synthetic power-law distributed networks, we observe that, while for mildly heterogeneous networks with γ > / γ < /
2, thePEV shows a peculiar form of localization, its components f i being proportional to the node’s degree, f i ∼ q i . This particularproportionality induces an effective localization on the maximum K -core of the network, defined as the core of maximum indexin a K -core decomposition. This max K -core concentrates a finite fraction of the normalized weight of the PEV, despite thefact that the size of the max K -core is sublinear with the network size. In the case of real world networks, the elucidation ofthe PEV localization is not so clearcut. We however provide evidence for an analogous scenario as that observed in syntheticnetworks, where the nature of the localization of the PEV is ruled by its associated largest eigenvalue Λ : When Λ is close tothe mean-field value (cid:104) q (cid:105) / (cid:104) q (cid:105) , localization on the max K -core is expected. On the other hand, when the largest eigenvalue isclose to √ q max , localization takes place on the hubs.The results presented here give a new perspective on complex topologies from several perspectives. Firstly, it is common igure 8. Ratio between the NBTC IPR and the IPR of the adjacency matrix as a function of the ratio between the largesteigenvalue and the square root of the maximum degree, for the real networks considered. The symbol codes are: square forlocalization on the max K -core; circle for localization on the hub; triangle up for networks in which √ q max is very close to (cid:104) q (cid:105) / (cid:104) q (cid:105) , so no conclusion can be drawn; triangle down for the rest of networks.knowledge that networks with γ > γ < < γ < points out that networkswith exponent γ < / γ > /
2. Secondly, our results point outthe weakness of eigenvector centrality as a measure of centrality for power-law networks. Indeed, for γ < /
2, eigenvectorcentrality does not provide more information than degree centrality, while for γ > / are also not free from localization effects. Finally, from a dynamical point of view, largest eigenvalues and theassociated eigenvectors are crucially related to the properties of processes on networks and their localization effectsshould be taken properly into account when developing theories relying on the structure of the adjacency matrix.The localization properties described here call for a revision of our present understanding of heterogeneous topologies.Other networks properties, such as degree correlations, clustering or the presence of a community structure, might play a rolein the localization of the PEV. The clarification of these effects, as well as the understanding of the nature of the mesoscopicsubgraph on which the NBTC is localized for γ > /
2, are still open questions, calling for further scientific effort.
Methods
Real networks analyzed
We consider in our analysis the following real networks datasets: • HEP : Collaboration network between authors of papers submitted to the High Energy Physics section of the onlinepreprint server arXiv. Each node is a scientist. Two scientists are connected by an edge if they have coauthored a reprint. • Slashdot : User network of the Slashdot technology news website. Nodes represent users, which can tag each other asfriends or foes. An edge represents the presence of a tagging between two users. • Amazon : Co-purchasing network from the online store Amazon. Nodes represent products, which are joined by edges ifthey are frequently purchased together. • Internet : Internet map at the Autonomous System level, collected at the Oregon route server. Vertices representautonomous systems (aggregations of Internet routers under the same administrative policy), while edges represent theexistence of border gateway protocol (BGP) peer connections between the corresponding autonomous systems. • Email : Enron email communication network. Nodes represent email addresses. An edge joins two addresses if they haveexchanged at least one email. • P2P : Gnutella peer-to-peer file sharing network. Nodes represent hosts in the Gnutella system. An edge stands for aconnection between two Gnutella hosts. • Movies : Network of movie actor collaborations obtained from the Internet Movie Database (IMDB). Each vertexrepresents an actor. Two actors are joined by an edge if they have co-starred at least one movie. • WWW : Notre Dame web graph. Nodes represent web pages from University of Notre Dame. Edges indicate the presenceof a hyperlink pointing from one page to another. • PGP : Social network defined by the users of the pretty-good-privacy (PGP) encryption algorithm for secure informationexchange. Vertices represent users of the PGP algorithm. An edge between two vertices indicates that each user hassigned the encryption key of the other. Some of this networks are actually directed. We have symmetrized them, rendering them undirected, to perform ouranalyses.
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Author contributions statement
R.P.-S. and C.C. designed the research. R.P.-S. performed the data analysis. R.P.-S. and C.C wrote the paper.
Additional information
Competing financial interests
The authors declare no competing financial interests. etwork N (cid:104) q (cid:105) (cid:104) q (cid:105) / (cid:104) q (cid:105) √ q max Λ Y ( Λ ) Y ( Λ NBT ) IPR RatioHEP 12006 19.74 129.94 22.16 244.93 0.003890 0.003887 0.9993Slashdot 82168 12.27 149.71 50.52 134.63 0.002174 0.002006 0.9228Amazon 403394 12.11 30.55 52.46 57.15 0.089122 0.005423 0.0608Internet 10790 4.16 259.46 48.34 59.58 0.066138 0.015783 0.2386Email 36692 10.02 140.08 37.19 118.42 0.003790 0.003446 0.9091P2P 62586 4.73 11.60 9.75 13.18 0.000921 0.000592 0.6429Movies 81860 89.53 594.92 61.55 817.36 0.000640 0.000638 0.9966WWW 325729 6.69 280.68 103.54 184.93 0.022726 0.008357 0.3677PGP 10680 4.55 18.88 14.32 42.44 0.016622 0.015989 0.9619
Table 1.
Relevant metrics for the various real-world networks with and the measured value of the IPR ratio between Y ( Λ NBT ) and Y ( Λ ) . Size and other information on the networks are provided in the Supplementary Information.. Size and other information on the networks are provided in the Supplementary Information.