Universal dependence of distances on nodes degrees in complex networks

Universal dependence of distances on nodes degrees in complex networs Janusz A. Hołyst, Julian Sieniewicz, Agata Froncza, Piotr Froncza, Krzysztof Sucheci and Piotr Wójcici Faculty of Physics and Center of Excellence for Complex Systems Research, Warsaw University of Technology, Koszyowa 7, PL-00-66 Warsaw, Poland Abstract. We have studied dependence of distances on nodes degrees between vertices of Erdős- Rényi random graphs, scale-free Barabási-Albert models, science collaboration networs, biological networs, Internet Autonomous Systems and public transport networs. We have observed that the mean distance between two nodes of degrees and equals to l i j = A Blog( ). A simple heuristic theory for the appearance of this scaling is presented. Corrections due to the networ clustering coefficient and node degree-degree correlations are taen into account. Keywords: complex networs, distances, scaling, correlations PACS: 89.7.-, 0.0.-r, 0.0.+q The empirical analysis of many real complex networs (for a review see [1,,, ]) has revealed the presence of several universal scaling laws. The best nown scaling law appears for degree distributions P() γ [] and it is observed in a number of social, biological and technological systems. May other scaling laws in complex networs have been found, such as a dependence of clustering coefficient on node degree in hierarchical networs c() α [6], scale-free behavior of the connection weight [7, 8] and load [9] distributions, load dependence on degree [10] and others [11, 1, 1]. In [1] an analytical model for average path lengths in random uncorrelated networs was considered and it was shown that the shortest path length between nodes i and j possessing degrees and can be described as: l i j (, ) = ln + ln ( ) + lnn γ ln( / 1) + 1, (1) where γ = 0.77 is the Euler constant, whereas and correspond to the first and the second moments of node degree distribution P(). It follows that a mean distance between two nodes is lineary dependent on the logarithm of their degree product l i j = A Blog( ). () Below we show that the relation () can also be obtained from a simple model of branching trees exploring the space of a random networ [16, 17]. Let us notice that following a random direction of a randomly chosen edge one approaches node j with probability p j = /(E), where E = N is a double number of lins. It means that in average one needs M j = 1/p j = E/ of random trials to arrive at the node j. Now let us consider a branching process represented by the tree T i

x= l ij x= x=1 i j FIGURE 1. Tree formed by a random process, starting from the node i and approaching the node j. (Fig. 1) that starts at the node i where an average branching factor is κ (all loops are neglected). If a distance between the node i and the surface of the tree equals to x then in average there are N i = κ x 1 nodes at such a surface and there is the same number of lins ending at these nodes. It follows that in average the tree T i touches the node j when N i = M j i.e. when κ x 1 = N. () Since the mean distance from the node i to the node j is l i j = x thus we get the scaling relation () with A = 1 + log(n ) logκ and B = 1 logκ. () The result () is in agreement with the paper [18] where the concept of generating functions for random graphs has been used. One has to tae into account that in the above considerations we have assumed that there are no degree-degree correlations, we have neglected all loops and we have treated the branching level x as a continuum variable to fulfill the relation (). Assuming that the branching factor κ can be expressed as / 1 [], one can see that the differences between the results () and (1) are small, at least for the case when N and κ is finite. In general the mean branching factor κ is a mean value over all local branching factors and over all trees in the networ. In the first approximation for networs without degree-degree correlations [] it could be estimated as the mean arithmetic value of a nearest neighbor degree less one (incoming edge): κ = nn 1. Such a mean value is however not exact because local branching factors in every tree are multiplied one by another in (). The corrected mean value of κ should be taen as an arithmetic mean value over all geometric values from different trees, what is very difficult to perform numerically. To reduce this discrepancy we calculated arithmetic mean branching factor

TABLE 1. Basic properties of examined systems and comparison between experimental and theoretical data. Astro and Cond-mat are co-authorship networs, Silwood, Yeast and Ythan are biological networs. The number after the Internet Autonomous Systems means the year data were gathered, Gorzów Wlp., Łódź and Zielona Góra are public transport networs in corresponding Polish cities. N is the number of nodes, - mean degree value. A e and B e are mean experimental values (Fig. -) whereas A and B are given by (). networ N A e A B e B Erdős-Rényi random graph 1000 8.00..6 1.017 1.1 Erdős-Rényi random graph 10000 8.00 6.77 6.60 1.16 1.1 Barabási-Albert model 1000 8.00.. 0.81 0.80 Barabási-Albert model 10000 8.00.17.81 0.778 0.777 Astro 1986.6..0 0.707 0.9 Cond-mat 1701 9.6.90.09 0.908 0.786 Silwood 1.77..69 0.9 0.91 Yeast 186.9 7. 6.66 1.06 1. Ythan 1 8.8.9. 0.69 0.76 Internet Autonomous Systems 1997 01..99.9 0.6 0.96 Internet Autonomous Systems 1998 180.7.08.1 0. 0.7 Internet Autonomous Systems 1999 861.86.0. 0. 0.0 Internet Autonomous Systems 001 101.08.96. 0.71 0.81 Gorzów Wlp. 69.8.6 16.06 1.70. Łódź 10.8.01 11.67 8.61.08 Zielona Góra 1.98 10.0 8.96.908.68 over nearest neighborhood of every node m, i.e. κ (m) = (m) nn 1, and then averaged it geometrically over all nodes m, i.e. κ = κ (m) m. Figs. (-) present mean distances l i j between pairs of nodes i and j as a function of a product of their degrees in selected complex networs. We include data for Erdős-Rényi random graphs, Barabási-Albert evolving networs, biological networs [19, 0, 1] (Silwood, Ythan, Yeast), social networs [, ] (co-authorship groups Astro and Cond-mat), Internet Autonomous Systems [] and selected networs for public transport in Polish cities [, 6] (Gorzów Wlp., Łódź, Zielona Góra) (see Table 1. for characteristic parameters of these networs). One can observe, that the relation () is very well fulfilled over several decades for all our data. Let us stress that the networs mentioned above display a wide variety of basic characteristics. Among them there are both scale-free and single scale networs, with either negligible or very high clustering coefficient, assortative [7], disassortative or uncorrelated. The only apparent common feature of all above systems is the small-world effect. We have checed however that for the small-world Watts-Strogatz model [8], the scaling () is nearly absent and it is visible only for large rewiring probability, and only for highly connected nodes. Although the scaling () wors well for distances averaged over all pairs of nodes specified by a given product, there can be large differences if one changes while eeping constant. The Fig. presents the dependence of average path length l i j on, for a fixed product in the case of several networs from different classes. One can see that although the Astro networs assortative (see Table ) (short-range attraction), pairs of nodes with similar degrees are in average further away than different

6 (a) Erdos - Renyi (b) Barabasi - Albert <lij > 1 10 1 10 10 1 10 10 10 10 10 6 FIGURE. Mean distance l i j between pairs of nodes i and j as a function of a product of their degrees. (a) Erdős-Rényi random graphs: = 8 and N = 1000 (circles) N = 10000 (squares), (b)barabási- Albert networs: = 8 and N = 1000 (circles) N = 10000 (squares). Data are logarithmically binned with the power of. <lij > 8 7 6 1 10 0 10 1 10 10 10 (a) Biological 6 1 (b) Co-authorship 10 0 10 1 10 10 10 10 10 6 FIGURE. Mean distance l i j between pairs of nodes i and j as a function of a product of their degrees. (a) Biological networs: Silwood (triangles), Yeast (squares), Ythan (circles), (b) Co-authorship networs: Astro (circles), Cond-mat (squares). In (a) Data are logarithmically binned with the power of 1. and in (b) with the power of. degree pairs (long-range repulsion). For the disassortative networ AS [7] the behavior is opposite. For uncorrelated networs (Erdős-Rényi, Barabási-Albert), the average path length is constant if the product is fixed [16]. Table shows results for parameters A and B from real networs and from numerical simulations as compared expressions (). One can see that our approximate approach () fits very well to random Erdős Rényi graphs and BA models but the corresponding coefficients A and B for real networs are different from results of our simple theory. In fact, the relations () can be improved by taing into account effects of loops and

,0, (a) Autonomous systems 0 (b) Public transport,0 1 <lij >,,0 10 1, 1,0 10 0 10 1 10 10 10 10 10 6 10 7 0 10 0 10 1 10 FIGURE. Mean distance l i j between pairs of nodes i and j as a function of a product of their degrees. (a) Internet Autonomous Systems: Year 1997 (filled squares), Year 1998 (empty circles) Year 1999 (full circles), Year 001 (empty squares), (b) Public transport networs in Polish cities: Gorzów Wlp. (squares), Łódź (circles), Zielona Góra (triangles) In (a) data are logarithmically binned with the power of. In (b) data are not binned. TABLE. Comparison between experimental and theoretical data. ER stands for Erdős-Rényi random graph networ, BA for Barabási-Albert model and AS is an acronym for Internet Autonomous Systems, c is networ s clustering coefficient, r - assortativity coefficient and φ is the scaling exponent for degree correlations. A e and B e are mean experimental values (Fig. -) whereas A and B are given by (6), while A φ and B φ follow (9). networ c r φ A e A A φ B e B B φ ER N = 10 0.007 0 -..8-1.017 1.17 - ER N = 10 0.001 0-6.77 6.61-1.16 1.1 - BA N = 10 0.08 0 -..7-0.81 0.8 - BA N = 10 0.007 0 -.17.81-0.778 0.779 - Astro 0.609 0.0 1...98.1 0.707 0.786 0.7 Cond-mat 0.60 0.0 1.19.90 6.8.0 0.908 1.10 0.9 Silwood 0.1-0.16 0.71..78.19 0.9 1.00 0.668 Yeast 0.068-0.18 0.9 7. 6.87.71 1.06 1.69 0.916 Ythan 0.16-0. 0.61.9..81 0.69 0.8 0.66 AS 1997 0.18-0.9 0.6.99..8 0.6 0.69 0.7 AS 1998 0.0-0.00 0.8.08..6 0. 0.60 0.76 AS 1999 0.0-0.18 0.9.0.8. 0. 0.79 0.6 AS 001 0.89-0.18 0..96..0 0.71 0.18 0.17 Gorzów Wlp. 0.08 0.8 1..6 19.76 16.67 1.70 6.61 7.679 Łódź 0.06 0.070 1.19.01 1.70 11.89 8.61.89.670 Zielona Góra 0.067 0.8 1.1 10.0 9.6 9.6.908.917.781 degree-degree correlations. The influence of loops of the length three can be estimated as follows. Let us assume that in the branching process forming the tree T i two nodes from the nearest neighborhood of the node i are directly lined (the dashed line at Fig.1). In fact such a situation

lij 8 7 6 (a) Erdos - Renyi =1 =16 =0 =0 =60 =10,8,,0 (b) Barabasi - Albert =0 =0 =60 =10 lij 7, 7,0 6, 6,0,8,6 0 10 1 0 (c) Yeast, 0 6 8 10 1 (e) Autonomous systems = =6 =8 =10 =1 = =6 =8 =10 =1,6 0 10 1 0 0, (d) Astro =,0 i =6 =8 =1,8,6, 0 18 16 0 6 8 10 1 (f) Gorzów Wlp. = =6 =8 =10 =1 lij,, 0 6 8 10 1 1 1 1 10 0 6 8 FIGURE. Dependence of average path length on, for fixed product. The lines connecting the symbols are there for clarity. The bars show point weight, meaning relative numbers of pairs i j. The horizontal lines are weighted averages over, and are just average path lengths for given. Note: The very small shifts on axis between data for different are artificially introduced to mae the weight bars not overlap. can occur at any point of the branching tree T i. Since such lins are useless for further networ exploration by the tree T i thus an effective contribution from both connected nodes to the mean branching factor of the tree T i is decreased. Assuming that clustering coefficients of every node are the same, the corrected factor for the branching process equals to κ c = κ cκ where c is the networ clustering coefficient. This equation is not

< nn -1> 10 10 100 80 (a) Astro 80 70 60 0 0 (b) Cond-mat 60 0 0 10 0 10 1 10 10 (c) Gorzów Wlp. 0 10 0 10 1 10 (d) ód < nn -1> 1 1 10 1 1 10 FIGURE 6. Estimation φ value for assortative networs. (a) Astro: the slope corresponds to exponent φ 1 = 0., (b) Cond-mat: the slope corresponds to exponent φ 1 = 0.19, (c) Gorzów Wlp.: the slope corresponds to exponent φ 1 = 0.,(d) Łódź: the slope corresponds to exponent φ 1 = 0.19. valid for the branching process around the node i where κ i = κ c( 1). A similar situation arises around the node j. Replacing and with in κ i and κ j one gets [κ(1 c )] [κ(1 c)] x = N, () where c = c( 1)/κ. It follows that instead of () we have A = + log(n ) log[κ(1 c )], B 1 = log[κ(1 c)] log[κ(1 c)]. (6) The results (6) are presented in corresponding columns of the Table. One can observe a fairly good agreement between experimental data and (6) for co-authorship and biological networs as well as for the Internet Autonomous System and public transport networn Zielona Góra while for two other transport systems it leads to larger errors. Corrections due to clustering effects give a better fit for the coefficient A, while for some networs the coefficient B is closer to experimental value B e than B. Now, let us consider the presence of degree correlations. Such correlations mean that average degrees (nn) i of nodes in the neighborhood of a node i depend on the degree.

(a) Autonomous systems 1999 (b) Autonomous systems 001 10 10 < nn -1> 10 1 10 1 10 8 10 0 10 1 10 10 (c) Yeast 0 10 0 10 1 10 10 (d) Ythan 6 0 < nn -1> 0 10 10 0 10 1 10 0 10 1 10 FIGURE 7. Estimation φ value for disassortative networs. (a) Autonomous systems 1999: the slope corresponds to exponent φ 1 = 0.1, (b) Autonomous systems 001: the slope corresponds to exponent φ 1 = 0., (c) Yeast: the slope corresponds to exponent φ 1 = 0.1, (a) Ythan: the slope corresponds to exponent φ 1 = 0.9. Let us assume that this relation can be written as κ i (nn) i 1 = D φ 1 i (7) If φ is larger than one then the networs assortative, i.e. high degree nodes are mostly connected to other high degree nodes and similarly low degree nodes are connected to other low degree nodes. Such a situation occurs for example in networs describing scientific collaboration [7]. If φ is smaller than one, then the networs disassortative and high degree nodes are mostly connected with low degree nodes what is typical for the Internet Autonomous Systems [7]. If we neglect higher order correlations then Eq. should be replaced by κ i κ j κ x = N (8) Taing into account Eq. 7 we can replace parameters A and B given by the Eq. with A φ = A + BlogD and B φ = φb (9) The plots 6 and 7 show the degree correlations for several different networs and illustrate the estimation of φ coefficient. After obtaining the histogram of nn 1 for

1, 1, 1,0 0,8 0,6 0, -0, -0, 0,0 0, 0, r FIGURE 8. Dependency of φ on assortativity coefficient r. each node i and plotting the dependence of those values on the node degree we perform the linear approximation in the log-log scale. A slope calculated in this way corresponds to the exponent φ 1 (with accordance to the formula nn 1 φ 1 ). One can see, that the scaling (7) is not so obvious as for the relation (). We have compared the resulting φ coefficient with standard assortativity coefficient introduced in [7] (see fig.8). We need to point out, that positive values of r correspond to φ larger than one. The linear fit of all points at the diagram (r,φ) nearly crosses the point (r = 0,φ = 1), what means that our definition of φ parameter is correct. Results (9) are presented in corresponding columns of the Table. One can notice that the values of A φ and B φ are more accurate for the networs characterized by a φ coefficient above unity (assortative). In conclusions we have observed universal path length scaling for different classes of real networs and models. The mean distance between nodes of degrees and is a linear function of log( ). The scaling holds over many decades regardless of networ degree distributions, correlations and clustering. We would lie to stress, that the observed scaling holds over many decades even for strongly correlated networs with the correlation coefficients r > 0.. We expect that the observed scaling law is universal for many complex networs, with applicability reaching far beyond the quoted examples. A simple model of random tree exploring the networ explains such a scaling behavior and clustering effects have been taen into account to compare numerical data from model and data from real networs to theoretical predictions. We have also considered influence of first order degree-degree correlations on scaling parameters and we have found that inclusion of such correlations improves theoretical predictions for assortative networs, while it fails for disassortative ones. We suppose that to get better agreement between experimental data and theoretical results, higher order correlations should be taen into account.

ACKNOWLEDGMENTS We are thanful to S.N. Dorogovtsev and Z. Burda for useful comments. Authors acnowledge the State Committee for Scientific Research in Poland for financial support under grant No. 1P0B077. This wor has also been partially supported by European Commission Projects: CREEN FP6-00-NEST-Path-0186 and MMCOMNET FP6-00-NEST-Path-01999. JAH is thanful to the Complex Systems Networ of Excellence EXYSTENCE for the financial support that made possible his visit at MPI-PKS Dresden. A.F. acnowledges financial support from the Foundation for Polish Science (FNP 00). REFERENCES 1. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 7 7 (00).. S. Bornholdt and H.G. Schuster, Handboo of graphs and networs, Wiley-Vch (00).. S.N. Dorogovtsev and J.F.F. Mendes, Evolution of networs, Oxford Univ.Press (00).. R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet : A Statistical Physics Approach, Cambridge Univ. Press (00).. A.-L. Barabási and R. Albert, Science 86, 09 (1999). 6. E. Ravasz and A.-L. Barabási, Phys. Rev. E 67 0611 (00). 7. Ch. Li and G. Chen, e-print cond-mat/011 (00). 8. A. Barrat, M. Barthélemy, R. Pastor-Satorras and A. Vespignani, Proc. Natl. Acad. Sci. (USA) 101, 77 (00). 9. C.M. Ghim, E. Oh, K.-I. Goh, B. Khnag and D. Kim, Eur. Phys. J. B 8, 19 (00). 10. M. Barthélemy, Eur. Phys. J. B 8, 16 (00). 11. A. Arenas, L. Danon, A. Díaz-Guilera, P. M. Gleiser and R. Guimerà, Eur. Phys. J. B 8, 7 (00). 1. A. Rinaldo, I. Rodriguez-Itrube, R. Rigon, E. Ijjasz-Vasquez and R. L. Bras, Phys. Rev. Lett. 70, 8 (199). 1. J. Banavar, A. Maritan and A. Rinaldo, Nature 99, 10 (1999). 1. A. Froncza, P. Froncza and J.A. Hołyst, e-prints: cond-mat/0066 (00) and cond-mat/010 (00). 1. A. Froncza, P. Froncza and J.A. Hołyst, Phys. Rev. E 70, 06110 (00). 16. J.A. Hołyst, J. Sieniewicz, A. Froncza, P. Froncza and K. Sucheci, to be published. 17. J.A. Hołyst, J. Sieniewicz, A. Froncza, P. Froncza and K. Sucheci, e-print cond-mat/011160 (00). 18. A.E. Motter, T. Nishiawa and Y.C. Lai, Phys. Rev. E 66 0610(R) (00). 19. Data for Foodwebs (Silwood and Ythan) [0] have been taen from the website http://www.cosin.org /extra/data/foodwebs/ whereas data for protein interaction networ (Yeast) [1] has been taen from http://www.nd.edu/ networs/database/index.html. 0. D. Garlaschelli, G. Caldarelli and L. Pietronero, Nature, 16 (00). 1. H. Jeong, S. P. Mason, A.-L. Barabási and Z. N. Oltavi, Nature 11, 1 (001).. Scientific collaboration networ [] data have been collected by P. Wójcici from two publicly available databases of papers http://arxiv.org/archive/astro-ph (Astro) and arxiv.org/archive/cond-mat (Cond-mat) for the period 199-001.. M.E.J. Newman, Phys. Rev. E 6, 01611 (001).. Data for the Internet [] has been taen from www.cosin.org/extra/data/internet/.. In those networs vertices are bus- and tramstops while an edge exists if at least one public transport line crosses two stops. [6]. 6. J. Sieniewicz and J.A. Hołyst, unpublished. 7. M.E.J. Newman, Phys. Rev. Lett. 89, 08701 (00). 8. D.J.Watts and S.H.Strogatz, Nature 9 0 (1998).