In complex networks it is common to model a network or generate a surrogate
network based on the conservation of the network's degree distribution. We
provide an alternative network model based on the conservation of connection
density within a set of nodes. This density is measure by the rich-club
coefficient. We present a method to generate surrogates networks with a given
rich-club coefficient. We show that by choosing a suitable local linking term,
the generated random networks can reproduce the degree distribution and the
mixing pattern of real networks. The method is easy to implement and produces
good models of real networks.Comment: revised version, new figure

D.J. Watts

J.F. Nagle

M. Molloy

M.E.J. Newman

N. Masuda

R. Albert

R. J. Mondragón

R. Pastor-Satorras

R.J. Mondragón

S. Maslov

S. Zhou

V. Colizza

X.-K. Xu

English

arXiv

Crossref

The European Physical Journal B

Random networks with given rich-club coefficient

In complex networks it is common to model a network or generate a surrogate network based on the conservation of the number of connections of individual nodes. In this paper we analyse the ensemble of random networks that are defined by the conservation of the rich-club coefficient, which measures the density of connections among a group of nodes. We also present a method to generate such surrogate networks for a given network. We show that by choosing a suitable local linking term, the random networks not only preserve the rich-club coefficient but also closely approximate the degree distribution and the mixing pattern of real networks. Our work provides a different and complementary perspective to the network randomisation problem

Mondragon, RJ

Zhou, S

UCL Discovery

arXiv:1102.5407v2  [physics.soc-ph]  15 Aug 2011Random Networks with given rich–club coefficientRau´l J. Mondrago´n∗Queen Mary University of London, School of Electronic Engineeringand Computer Science, Mile End Road, London, E1 4NS, UKShi Zhou†University College London, Dept. of Computer Science, Gower Street, London WC1E 6BT, UK(Dated: December 9, 2013)In complex networks it is common to model a network or generate a surrogate network basedon the conservation of the network’s degree distribution. We provide an alternative network modelbased on the conservation of connection density within a set of nodes. This density is measure by therich–club coefficient. We present a method to generate surrogates networks with a given rich–clubcoefficient. We show that by choosing a suitable local linking term, the generated random networkscan reproduce the degree distribution and the mixing pattern of real networks. The method is easyto implement and produces good models of real networks.∗ r.j.mondragon@eecs.qmul.ac.uk† S.Zhou@cs.ucl.ac.uk2I. INTRODUCTIONMany complex networks are single networks in the sense that their structure is unique not one of several. Ingeneral, we do not have the equivalent of a physical law to verify if the statistical measures obtained from a singlenetwork are expected or exceptional. Instead, a common technique to analyze the properties of a single network is touse statistical randomization methods [1] to create a reference network which is used for comparison purposes. Theprocedure consists of using the observed network to generate an ensemble of networks via randomization, usually thereshuﬄing of the links. The reference network is generated from this ensemble and it is used to assess the significanceof a property of the network. To take into consideration the intrinsic structure of the network it is common to use arestricted randomization procedure. The restriction is to reshuﬄe the connections of the nodes without changing thedegree distribution of the nodes P (k) [2], where P (k) is the degree distribution, that is, the fraction of nodes in thenetwork with degree k.Many complex networks contain a small set of nodes that have large numbers of links, the so–called rich nodes.In some networks the rich nodes tend to be tightly interconnected between themselves, forming a rich–club [3]. Therich–club is an oligarchy in that it dominates the organization of the whole network. In scale–free networks [4]the connectivity of the rich–club plays an important role in the functionality of the network, for example in thetransmission of rumors in social networks [5] or the efficient delivery of information in the Internet [6]. The rich–clubcoefficient measures the density of connection among these group of nodes. This set of nodes can be defined by aranking scheme [3], by their degree [7] or by a network hierarchy [8]. If the nodes are ranked by non–increasing orderof their degree, i.e. the best-connected node is ranked as r = 1, the second best-connected node is r = 2 and so on,then the density of connections between the first r nodes is quantified by the rich–club coefficient [3]Φ(r) =2E(r)r(r − 1), (1)where E(r) is the number of links between the r nodes and r(r − 1)/2 is the maximum number of links that thesenodes can share. As a function of the nodes degree, the rich–club coefficient is [7]φ(k) =2EkNk(Nk − 1), (2)where Nk is the number of nodes with degree equal or higher than k and Ek is the number of links among these Nknodes. It is known that the rich–club coefficient φ(k) is a projection of the degree–degree correlation [7], is independentof its degree distribution [9] and it is non-trivially related to other properties like the clustering coefficient [10].It is possible to construct a surrogate network where the degree distribution P (k) is conserved [2, 11], the rich–clubcoefficient φ(k) is conserved or both of these properties are conserved [9]. Our aim here, is to build a network modeland surrogate random networks based on the rank rich–club coefficient Φ(r) and show that they are good models ofscale–free networks as they tend to reproduce key statistical properties.Before introducing the model, we would like to remark that the shape of the function Φ(r) depends on the way thatthe nodes are ranked. There is an ambiguity when labeling the nodes via a degree–dependent rank. For high degreenodes this is not a problem as the degree tend to be unique so the rank labels these nodes unambiguously. For lowerdegree nodes, there are many nodes with the same degree. In this case the labeling of the nodes is not unique. It ispossible to reduce greatly this redundancy of the labels by using a linear order relationship [12], where the degree ofthe second neighbors is used to disambiguate the labeling. This linear ordering has been used to visualize higher ordercorrelations in the network topology [13, 14]. We observed that using a degree–only ranking scheme or a linear orderranking has little effect on the statistical properties under consideration here. So in this communication we consideronly a degree–dependent ranking scheme.Figure 1 shows Φ(r) for the Scientists co–authorship (Scientists) [15], the protein interaction (Protein) [16], theAS–Internet (Internet) [17] and the power grid networks (Power) [18]. We selected these networks as they havebeen widely used when studying the topological properties of complex network as they have different topologicalcharacteristics (e.g. [19]). These networks have very different connectivities between the high degree nodes. TheInternet data has a fully connected core where the top seven nodes are fully connected (Φ(7) = 1) in contrast withthe power grid network where the top fourteen nodes do not share a link at all (Φ(14) = 0).3rScientist(a)10-1100100 101 102 10310-2Φ(r)rProtein(b)100 101 102 103 10410-210-310-1100Φ(r)rInternet(c)100 101 102 103 104 10510-210-310-110010-4Φ(r)rPower(d)100 101 102 103 10410-210-310-110010-4Φ(r)FIG. 1. The rich–club coefficient Φ(r) for the four networks under study.II. NETWORKS DEFINED BY THE RICH–CLUB COEFFICIENTFrom equation (1) we have that the number of links that node r shares with the r − 1 nodes of higher or equaldegree is∆E(r) = E(r) − E(r − 1) = Φ(r)r(r − 1)2− Φ(r − 1)(r − 1)(r − 2)2(3)where ∆E(1) = 0,∑Ni=1 ∆E(i) = L where L is the total number of links and N is the total number of nodes. The term∆E(1) is zero, because the node r = 1 is the top node. The number of links ∆E(r) is bounded by 0 ≤ ∆E(r) ≤ r− 1.If ∆E(r) is known then for r ≥ 2, Φ(r) is obtained from the recursive equationΦ(r) =2∆E(r) + (r − 1)(r − 2)Φ(r − 1)r(r − 1)(4)with Φ(1) = 0.To determine the network connectivity we need to distribute ∆E(r) links between node r and the r′ ∈ [1, r − 1]nodes, for all r. Let us assume that P (r′, r) is the probability that node r connects to node r′ < r, P (r, r) = 0 asself–loops are not allowed and two nodes can share only one link. Given ∆E(r) links for 1 ≤ r ≤ N , we constrain theconnectivity of a network by imposing the condition that the average number of links, ∆E(r), between node r andthe r − 1 nodes of lower rank is∆E(r) =r−1∑i=1P (i, r) = ∆E(r). (5)4This last equation defines an ensemble of networks where their average rich–club coefficient Φ¯(r) is defined by equa-tions (4) and (5). A network from this ensemble satisfies that the average degree k¯r of node r isk¯r =N∑r′=1P (r′, r) = ∆E(r) +N∑j=r+1P (r, j) (6)with standard deviation σ2kr =∑Nr′=1 P (r′, r)(1− P (r′, r)). The average number of links is L¯ =∑r∑r′ P (r′, r) withstandard deviation σ2L =∑r∑r′ P (r′, r)(1−P (r′, r)) and the average degree of the nearest–neighbors [20] of a nodewith degree k isknn(k) =kmax∑k′=1k′P (k′|k) ≈1NkN∑i=11kN∑j=1P (i, j)kj δki,k. (7)In the above equation, the Kronecker delta is introduce to consider only nodes with degree equal to k, Nk is thenumber of k–degree nodes and the term 1/k is a normalization factor.A. Local linking term T (r′, r)As an ansatz we assume that the probability that there is a link between node r and r′ can be factorized asP (r′, r) = T (r′, r)∆E(r), r′ < r, (8)where T (r′, r) is a local linking factor. From equation (5) we have that∑r−1r′=1 T (r′, r) = 1. In here the sample spaceis the set of all possible combination of links that the node has with the r− 1 nodes of lower rank, with the restrictionthat only one link is shared between two nodes. The set of events, used in the definition of the probability, is thedifferent combination that ∆E(r) links can be shared between node r and the r − 1 lower rank nodes. Notice that∆E(r) is bound by 0 ≤ ∆E(r) ≤ r − 1.1. Egalitarian linkingThe simplest case is when the ∆E(r) links that node r can share with the r− 1 nodes are evenly distributed, thenthe probability that node r connects to r′ isP (r′, r) =1r − 1∆E(r), r′ < r, (9)where T (r′, r) = 1/(r− 1). For example if node r has a link with all the r− 1 nodes of lower rank, i.e. ∆E(r) = r− 1then P (r′, r) = 1, r′ < r.2. Preferential linkingFor the case that node r prefers to connect to nodes with lower rank (i.e. higher degree), we propose the preferentiallinking term T (r′, r) = r′−α/S(r), where α > 0 is a constant and S(r) is a normalization factor. The probability thatthere is a link between r and r′ isP (r′, r) =r′−αS(r)∆E(r) =(r′−α∑r−1i=1 i−α)∆E(r), (10)where S(r) =∑r−1i=1 i−α to ensure that∑r−1i=1 P (i, r) = ∆E(r) (see equation 5). Note that it is possible that for somenetworks the probability function given by equation (10) may be larger than 1, in that case a more suitable T (r′, r)should be considered.5egalitarianr100100102104(a)k regalitarian100100 10310-4P(k)k(b)egalitarian100 103k(c)020k nn(k)FIG. 2. Topological properties of an egalitarian rich–club network constructed from the Scientists co–authorship network.B. Evaluation of the modelFigure 1(a) shows the rich–club coefficient Φ(r) as a function of node rank r for the Scientists network. From thedata we evaluated Φ(r) and ∆E(r). To build a model of the Scientists network we assume that the local linking termis egalitarian and evaluated P (r, r′), k¯r, and knn(k) using equations (9), (6) and (7). To obtain an integer value ofthe node’s degree we used the integer function |k¯r|. The degree distribution P (k) was obtained using |k¯r| for all r.Figures 2(a) to (c) show that our model resembles the real network, except that the top ranked nodes have similardegrees, which is not true in the real network (see r < 20 in Figure 2(a)). In fact these nodes have significantlydifferent degrees, which suggests that there is a preferential linking between the high degree nodes. To verify this laststatement we created a network model using the preferential linking defined in equation (10). The exponent α wasobtained by fitting the average degree of node k¯r using equation (6) against the degree kr from the original network,we did so by obtaining the value of α that minimizes the square of the error∑r(kr − k¯r)2 to a precision of 10−2 inα. Figures 3(a)–(c) show that a model based on the preferential linking is a good approximation of the Scientistsnetwork.We also created a network model based on preferential linking for the Protein interaction network [16], the AS–Internet [17] and the Power grid network [18]. Figures 3(d)–(l) demonstrate that our models closely resemble thedegree distribution and give a good approximation to the nearest-neighbors average degree for both assortative (co–authorship) and disassortative (Internet, Protein and Power grid) networks.III. GENERATING RANDOM NETWORKS WITH GIVEN RICH-CLUB COEFFICIENTFor a given network we can generate a surrogate random network which preserves the total number of nodes N ,links L, and the rich–club coefficient Φ(r) for all r. First the N nodes are ranked by their degree and the numberof links ∆E(r) is evaluated from Φ(r). To create a surrogate network, we create N stub nodes by assigning to eachnode r ∈ [2, N ], ∆E(r) links. From r = 2 to N , we connect the ∆E(r) links between the stub node r and thesmaller–ranked r′ < r nodes. The probability that node r connects to node r′ is defined by equation (8). We do notallow self–links or that two nodes share more than one link. Notice that∑Nr=1∆E(r) = L. All of these surrogatenetworks will have the same rich–club coefficient as the original network because all of them have the same ∆E(r)for all r.As an example, we illustrate the technique using the Scientists co–authorship collaboration network using prefer-ential linking. Figure 4(a)–(b) show the average node degree as a function of node rank, obtained from 40 surrogaterandom networks generated using the preferential linking with α = 0.28. The average degree of the random networksclosely (lines with error bars) approximates the degree of the original network (open circles). In the figure we alsoplotted the average degree obtained from equation (6) (filled diamonds). Notice that we expect a discrepancy betweenthe average degree evaluated from the random networks and the average degree obtained from equation (6). Thereason is that the restriction imposed by equation (5) means that the average number of links between node r and ther′ < r nodes is ∆E(r). However, the random surrogate networks are generated with the restriction that the number6102100100 104Scientistskrα=0.28 +/- 0.01r(a)P(k)10010-5100 103(b)kk nn(k)025100 103k(c)Proteinα=0.51 +/- 0.01103100k r100 104r(d)P(k)10010-5100 103(e)k 100 103k(f)040k nn(k)AS-Internetα=0.96 +/- 0.01104100k r100 104r(g)P(k)100100 103k10-5(h)0600k nn(k)100 104k(i)Powerα=0.25+/- 0.01102100k r100 104r(j)P(k)10010-4100 102(k)k010k nn(k)100 102k(l)FIG. 3. The node’s degrees kr, their degree distribution P (k) and average degree of the nearest–neighbors knn for the fournetworks under study. The properties of the original network are show in black and their models in gray. For these α values,using linear regression of (kr − k¯r) vs. r, the regression coefficient R of the networks is between 0.85 and 0.87.720601001400 20k rr(a)Scientistsα=0.28 05106000 6020k rr(b) (c)510152025300 40 80 120 160k nn(k)kFIG. 4. (a) Degree kr of the 20 smallest ranked nodes (i.e. 1 ≤ r ≤ 20), and (b) for 6000 ≤ r ≤ 6020. The open circle isthe original data, the filled diamonds is the average obtained from equation (6) and the line with the error bars obtained from40 realizations of the random networks. (c) The nearest–neighbors average degree knn(k) for the 40 random networks (dots)compared against the value of the original network (solid line).of links between node r and the r′ < r nodes is exactly ∆E(r).Figure 4(c) compares the nearest–neighbors average degree knn(k) of the original network (line) against the valueobtained by the 40 random networks (scatter dots). It is clear that the random networks resemble well the assortativeproperty of the real network.IV. CONCLUSIONWe investigated the properties of random networks defined by a rank–based rich-club coefficient. We show thatusing a preferential local linking term, we can approximate the degree distribution and the mixing pattern of a varietyof real networks. We also introduced a method to generate surrogate random networks conserving the network’srich–club coefficient. While existing surrogate network models have focused on connectivity of individual nodes, ourwork provides an alternative method based on the hierarchical structure of a network in terms of link density amonga group of nodes.ACKNOWLEDGMENTSWe thank the referees and Rui Carvalho for their useful comments and suggestions. SZ is supported by the RoyalAcademy of Engineering. Both authors thanks the UK Engineering and Physical Sciences Research Council (EPSRC)for support under grant number 10216/70 (SZ) and EP/H04812X (RJM).[1] B. Manly, Randomization, Bootstrap and Monte Carlo Methdos in Biology, 2nd ed. (Chapman & Hall, 1997).[2] M. Molloy and B. Reed, Comb. Probab. Comput. 7, 225 (1998).[3] S. Zhou and R. J. Mondrago´n, IEEE Communications Letters 8, 180 (2004).[4] R. Albert and A.-L. Baraba´si, Rev. Mod. Phys. 74, 49 (2002).[5] N. Masuda and N. Konno, Social Networks 28, 297 (2006).[6] S. Zhou and R. J. Mondrago´n, IEE Electronic Letters 2, 151 (2004).[7] V. Colizza, A. Flammini, M. A. Serrano, and A. Vespignani, Nature Physics 2, 110 (2006).[8] J. J. McAuley, T. S. Caetano, and L. da F. Costa, Applied Physics Letters 91 (2007).[9] S. Zhou and R. J. Mondrago´n, New Journal of Physics 9 (2007), doi:10.1088/1367-2630/9/6/E02.[10] X.-K. Xu, J. Zhang, and M. Small, Phys. Rev. E 82, 046117 (2010).[11] S. Maslov and K. Sneppen, Science 296, 910 (2002).[12] J. F. Nagle, Journal of Mathematical Physics 7, 1588 (1966).8[13] Y. Guo, C. Chen, and S. Zhou, Electronic Letters 43 (2007).[14] R.J.Mondrago´n, Philosophical Transactions A, Royal Society (2008), 10/1098/rsta.2008.0008.[15] M. E. J. Newman, Proceedings of the National Academy of Sciences of the United States of America 98, 404 (2001).[16] V. Colizza, A. Flammini, A. Maritan, and A. Vespignani, Physica A 352, 1 (2005).[17] P. Mahadevan, D. Krioukov, M. Fomenkov, B. Huffaker, X. Dimitropoulos, kc claffy, and A. Vahdat, Comput. Commun.Rev. 36 (2006).[18] D. J. Watts and S. H. Strogatz, Nature 440, 409 (1998).[19] A. Jamakovic, P. Mahadevan, A. Vahdat, M. Bogun˜a, and D. Krioukov, arXiv.0908.1143 (2009), arXiv:0908.1143.[20] R. Pastor-Satorras, A. Va´zquez, and A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001).

In complex networks it is common to model a network or generate a surrogate network based
          on the conservation of the number of connections of individual nodes. In this paper we
          analyse the ensemble of random networks that are defined by the conservation of the
          rich-club coefficient, which measures the density of connections among a group of nodes.
          We also present a method to generate such surrogate networks for a given network. We show
          that by choosing a suitable local linking term, the random networks not only preserve the
          rich-club coefficient but also closely approximate the degree distribution and the mixing
          pattern of real networks. Our work provides a different and complementary perspective to
          the network randomisation problem

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Random Networks with given Rich-club Coefficient

Random Networks with given Rich-club Coefficient

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