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Structure of shells in complex networks

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Structure of shells in complex networks
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  Structure of shells in complex networks Jia Shao, 1 Sergey V. Buldyrev, 2,1 Lidia A. Braunstein, 3,1 Shlomo Havlin, 4 and H. Eugene Stanley 1 1  Department of Physics and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA 2  Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA 3  Departamento de Física, Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata–CONICET, Funes 3350, 7600 Mar del Plata, Argentina 4  Department of Physics and Minerva Center, Bar-Ilan University, 52900 Ramat-Gan, Israel  Received 11 March 2009; published 9 September 2009  We define shell    in a network as the set of nodes at distance    with respect to a given node and define  r   as the fraction of nodes outside shell   . In a transport process, information or disease usually diffuses from arandom node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the studyof the transport property of networks. We study the statistical properties of the shells of a randomly chosennode. For a randomly connected network with given degree distribution, we derive analytically the degreedistribution and average degree of the nodes residing outside shell    as a function of   r   . Further, we find that r    follows an iterative functional form  r   =    r   −1  , where     is expressed in terms of the generating function of the srcinal degree distribution of the network. Our results can explain the power-law distribution of thenumber of nodes  B   found in shells with    larger than the network diameter  d  , which is the average distancebetween all pairs of nodes. For real-world networks the theoretical prediction of   r    deviates from the empirical r   . We introduce a network correlation function  c  r    r   /    r   −1   to characterize the correlations in thenetwork, where  r    is the empirical value and     r   −1   is the theoretical prediction.  c  r    =1 indicates perfectagreement between empirical results and theory. We apply  c  r     to several model and real-world networks. Wefind that the networks fall into two distinct classes:   i   a class of   poorly connected   networks with  c  r     1,where a larger   smaller   fraction of nodes resides outside   inside   distance    from a given node than inrandomly connected networks with the same degree distributions. Examples include the Watts-Strogatz modeland networks characterizing human collaborations such as citation networks and the actor collaboration net-work;   ii   a class of   well-connected   networks with  c  r     1. Examples include the Barabási-Albert model andthe autonomous system Internet network.DOI: 10.1103/PhysRevE.80.036105 PACS number  s  : 89.75.Hc, 87.23.Ge, 64.60.aq I. INTRODUCTION AND RECENT WORK Many complex systems can be described by networks inwhich the nodes are the elements of the system and the linkscharacterize the interactions between the elements. One of the most common ways to characterize a network is to de-termine its degree distribution. A classical example of a net-work is the Erd ő s-Rényi   ER   1,2   model, in which the linksare randomly assigned to randomly selected pairs of nodes.The degree distribution of the ER model is characterized bya Poisson distribution P  k    =  k   k  k  ! e −  k   ,   1  where   k    is the average degree of the network. Anothersimple model is a random regular   RR   graph in which eachnode has exactly   k   =    links, thus  P  k   =    k  −    . TheWatts-Strogatz model   WS   3   is also well studied, where arandom fraction     of links from a regular lattice with   k   =    are rewired and connect any pair of nodes. Changing    from 0 to 1, the WS network interpolates between a regularlattice and an ER graph. In the last decade, it has been real-ized that many social, computer, and biological networks canbe approximated by scale-free   SF   models with a broad de-gree distribution characterized by a power law P  k     k  −  ,   2  with a lower and upper cutoff,  k  min  and  k  max   4–9  . A para-digmatic model that explains the abundance of SF networksin nature is the preferential attachment model of Barabásiand Albert   BA   4  .The degree distribution is not sufficient to characterize thetopology of a network. Given a degree distribution, a net-work can have very different properties such as clusteringand degree-degree correlation. For example, the network of movie actors   4   in which two actors are linked if they playin the same movie, although characterized by a power-lawdegree distribution, has higher clustering coefficient com-pared to the SF network generated by the Molloy-Reed al-gorithm   10   with the same degree distribution.Besides the degree distribution and clustering coefficient,a network is also characterized by the average distance be-tween all pairs of nodes, which we refer to as the network diameter  d    11  . The diameter  d   depends sensitively on thenetwork topology. Random networks with a given degreedistribution can be “small worlds”   2  , d     ln  N  ,   3  or “ultrasmall worlds”   8  , d     ln  ln  N   .   4  Another important characteristic of a network is the struc-ture of its shells, where shell    is defined as the set of nodesthat are at distance    from a randomly chosen root node   12  .The shell structure of a network is important for understand-ing the transport properties of the network such as the epi-demic spread   13,14  , information diffusion and synchroni- PHYSICAL REVIEW E  80 , 036105   2009  1539-3755/2009/80  3   /036105  13   ©2009 The American Physical Society036105-1  zation processes   15,16  , where the information or virusspread from a randomly chosen root and reach nodes shellafter shell. The structure of the shells is related to both thedegree distribution and the network diameter. The shellstructure of SF networks has been recently studied in Ref.  12  , which have introduced a new term “network tomogra-phy” referring to various properties of shells such as thenumber of nodes and open links in shell   , the degree distri-bution, and the average degree of the nodes in the exterior of shell   .Many real and model networks have fractal propertieswhile others do not   17  . Recently, Ref.   18   reported apower-law distribution of number of nodes  B   in shell   d   from a randomly chosen root. They found that a largeclass of models and real networks, although not fractals onall scales, exhibits fractal properties in boundary shells with   d  . Here we will develop a theory to explain these find-ings. II. GOALS OF THIS WORK In this paper, we extend the study of network tomographydescribing the shell structure in a randomly connected net-work with an arbitrary degree distribution using generatingfunctions. Following Ref.   12  , we denote the fraction of nodes at distance larger than    as r      1 −1  N    m =0   B m   5  and the nodes at distances larger than    as the exterior  E    of shell   . Similarly, we define the “ r   exterior,”  E  r  , as the  rN  nodes with the largest distances from a given root node. Tothis end, we list all the nodes in ascending order of theirdistances from the root node. In this list, the nodes with thesame distance are positioned at random. The last  rN   nodes inthis list which have the largest distance to the root are called  E  r  . Notice that  E  r  =  E    if   r  = r   . Introducing  r   as a continuousvariable is a different step compared to Ref.   12  , whichallows us to apply the apparatus of generating functions tostudy network tomography.The behavior of   B   for    d   can be approximated by abranching process   19,20  . In shells with    d  , the network will show different topological characteristics compared toshells with    d  . This is due to the high probability to findhigh degree nodes   “hubs”   in shells with    d  , so there is adepletion of high degree nodes in the degree distribution in  E    with    d  . Indeed, the average degree of the nodes inshells with    d   is greater than the average degree in theshells with    d    12,18  .Here, we develop a theory to explain the behavior of thedegree distribution  P r   k    in  E  r   and the behavior of the aver-age degree   k   r    as a function of   r   in a randomly connectednetwork with a given degree distribution. Further, we deriveanalytically  r    as a function of   r   −1 ,  r   =    r   −1  , where     canbe expressed in terms of generating functions   20   of thedegree distribution of the network. Using these derived ana-lytical expressions, we explain the power-law distribution P   B    B  −2 for    d   found in   18  . Further, based on ourapproach, we introduce the network correlation function c  r    = r   /    r   −1   to characterize the correlations in the net-work. We apply this measure to several model and real-worldnetworks. We find that the networks fall into two distinctclasses:   i   a class of   poorly connected   networks with  c  r     1, where the network is less compact than its randomlyconnected counterpart with the same degree distribution;   ii  a class of   well-connected   networks with  c  r     1, in whichcase the network is more compact than its randomly con-nected counterpart.In a network with  c  r     1, more nodes reside in the ex-terior of shell    than in its randomly connected counterpart.Hence, the fraction of nodes residing inside and on shell   ,1− r   , is smaller than that of its randomly connected coun-terpart. Thus, the network is less compact than a randomlyconnected network with the same degree distribution, and wecall it poorly connected. The poorly connected networkshave high redundancy of their connections and high cluster-ing than their randomly connected counterpart. The well-connected networks are on the opposite side.In this paper we study RR, ER, SF, WS, and BA models,as well as several real networks including the actor collabo-ration network    Actor   4  , high energy physics citations net-work    HEP   21  , the Supreme Court citation network    SCC  22  , and autonomous system   AS   Internet network   DIMES   23  . As we will show below, WS, Actor, HEP, andSCC belong to the class of poorly connected networks  c  r     1  , while the BA model and DIMES network belongto the class of well-connected networks   c  r     1  .The paper is organized as follows. In Sec. III, we derive analytically the degree distribution and average degree of nodes in  E  r   and test our theory on ER and SF networks. InSec. IV, we derive analytically a deterministic iterative func- tional form for  r   . In Sec. V, we apply our theory to explain the power-law distribution of number of nodes in shells. InSec. VI, we introduce the network correlation function and apply it to different networks. Finally, we present a summaryin Sec. VII. III. DEGREE DISTRIBUTION OF NODES IN THE  r EXTERIOR  E  r A. Generating function for  P (  k ) In this section, we define the generating functions for thedegree distribution which will be used extensively in ourderivations. The generating function of a given degree distri-bution  P  k    is defined as   19,20,24,25  G 0   x      k  =0  P  k    x  k  .   6  It follows from Eq.   6   that the average degree of the net-work    k   = G 0   1  . Following a randomly chosen link, theprobability of reaching a node with  k   outgoing links   thedegree of the node is  k  +1   is P ˜   k    =   k   + 1  P  k   + 1  /  k  =0   k   + 1  P  k   + 1  .   7  Notice that SHAO  et al.  PHYSICAL REVIEW E  80 , 036105   2009  036105-2  the growth process, as more and more nodes are connected tothe aggregate, the degree distribution of the remaining nodeschanges. In this section, we will present and solve the differ-ential equations describing these changes.Let  A r   k    be the number of nodes with degree  k   in the  r  exterior  E  r   at time  t  . The probability to have a node withdegree  k   in  E  r   is given by   27  P r   k    =  A r   k   rN  .   11  When we connect an open link from the aggregate to a freenode   case   i  ,  A r   k    changes as  A r  −1 /  N   k    =  A r   k    − P r   k   k   k   r    ,   12  where   k   r   =  P r   k   k   is the average degree of nodes in  E  r  .In the limit of   N  →  , Eq.   12   can be presented in terms of the derivative of   A r   k    with respect to  r  , dA r   k   dr    N    A r   k    −  A r  −1 /  N   k    =  N P r   k   k   k   r    .   13  Differentiating Eq.   11   with respect to  r   and using Eq.   13  ,we obtain−  r dP r   k   dr  =  P r   k    − kP r   k   k   r    ,   14  which is rigorous for  N  →  .In order to solve Eq.   14  , we make the substitution  f     G 0−1  r   .   15  We find by direct differentiation that P  f   k    =  P 1  k    f  k  G 0   f   ,   16  and  k    f    =  fG 0    f   G 0   f    17  is the solution satisfying Eq.   14  . Notice that  P 1  k   P  k   .Equations   16   and   17   are, respectively, the degree dis-tribution and the average degree in  E  r   as functions of   f  . Oncewe know the explicit functional form for  G 0   x   , we can in-vert  G 0   x    to find  f  = G 0−1  r    and find analytically both  P r   k   and   k   r   : P r   k    =  P  k   G 0−1  r   k  r  ,   18  k   r    = G 0−1  r   G 0  „ G 0−1  r   … r  .   19  In a network with minimum degree  k  min  2, we find by Tay-lor expansion that  k   r    =  k  min + P  k  min + 1  P  k  min  1+    r    +  O  r  2    ,   20  where     1 / k  min .For ER networks, using Eqs.   10   and   17  , we find  k   r    = ln  r   +   k   ,   21  where  r    r   1. The value of   r    is presented in Eq.   33  .Note that  r    0 for ER networks. Equation   16   can be re-written as P r   k    =  P  k   ln  r  /  k    + 1  k  r  =  e −  k   r    k   r   k  k  !,   22  which implies that the degree distribution in the distantnodes remains a Poisson distribution but with a smaller av-erage degree   k   r   .Next, we test our theory numerically for ER networkswith  N  =10 6 nodes and different values of    k   . To obtain P r   k   , we start from a randomly chosen root node and findthe nodes in  E  r   and their degree distribution  P r   k   . This pro-cess is repeated many times for different roots and differentnetwork realizations. The results are shown in Fig. 2  a  . Thesymbols are the simulation results of the degree distributionin  E  r   for  r  =1, 0.5, and 0.05. The analytical results   full lines  are computed using Eq.   22  . As can be seen, the theory 0 5 10 15 20 25 30  k 00.10.2       P     r       (      k      )  r=1 r=0.5 r=0.05 ER <k>=6  (a) ER 10 0 10 1 10 2 10 3  k 10 -6 10 -4 10 -2 10 0       P     r       (      )  r=1 r=0.5 r=0.1 (b) SF λ=3.5 FIG. 2.   Color online   Comparison between the simulation re-sult and the theoretical prediction for the degree distribution,  P r   k   ,in  E  r  .   a   ER network with  N  =10 6 ,   k   =6 and  r  =1, 0.5, and 0.05.The simulation results   symbols   agree very well with the theoreti-cal predictions   lines   of Eq.   22  .   b   SF network with   =3.5, k  min =2 and  N  =10 6 ,  P r   k    with  r  =1, 0.5, and 0.1. The simulationresults shown by symbols fit well with the theoretical predictions of Eq.   16  . For a SF network, we compute Eq.   16   numerically usingthe  P  k    obtained from the generated network.SHAO  et al.  PHYSICAL REVIEW E  80 , 036105   2009  036105-4  agrees very well with the simulation results for both  r  =0.5and 0.05. We compared our theory with the simulations alsofor other values of   r   and   k    and the agreement is also excel-lent.For SF networks,  G 0   x    and  G 1   x    cannot be expressed aselementary functions   19  . But for a given  P  k   , they can bewritten as power series of   x   and one can compute the expres-sions in Eqs.   16   and   17   numerically. In order to reducethe systematic errors caused by estimating  P  k   , we write G 0   x    and  G 1   x    based on the  P  k    obtained from the simu-lation results instead of using its theoretical form.We built SF networks using the Molloy-Reed algorithm  10  . In Fig. 2  b  , the symbols represent the simulation re-sults for  P r   k    obtained for  E  r   of SF network with  =3.5 and r  =1, 0.5, and 0.1. The lines are the numerical results calcu-lated from Eq.   16  . Good agreement between the simulationresults and the theoretical predictions can be seen in Fig.2  b  . Other values of   r   and  have also been tested with goodagreement.In Fig. 3  a  , we show the average degree   k   r    in  E  r   as afunction of   r   for ER networks with different values of    k   .Lines representing Eq.   21   agree very well with the numeri-cal results   symbols   even for very small  r  . We note that Fig.3  a   shows different value of lower limit cutoff   r    for  r  ,when   k   r    is very small. As mentioned before,  r    is thefraction of nodes which are not connected to the aggregate atthe end of the process. In the next section, we will present anequation for  r   .In Fig. 3  b  , we present the numerical results of Eq.   17  for SF networks with different values of    . For a given  E  r  ,  k   r    is computed from the simulated network and the re-sults are averaged over many realizations. Good agreementbetween the theory   lines   and the simulation results   sym-bols   can be seen. IV. ITERATIVE FUNCTIONAL FORM OF  r  , THEFRACTION OF NODES OUTSIDE SHELL   In this section, we will derive a recursive relation between r    of two successive shells of a randomly connected network,which is the main result of this paper. Let  L  t    be the numberof open links belonging to the aggregate at step  t   and   t    L  t   /  N  . The number of open links belonging to shell   of the aggregate is defined as  L   t    and    t    L   t   /  N  .Afterwe finish building shell    and just before we start to buildshell   +1, all the open links in the aggregate belong to nodesin shell   , so at  t  = t   , we have     t   =   t    28  . In the pro-cess of building shell   +1,    t    decreases to 0.Next we show that both   t    and    t    can be expressedas functions of   r  . In analogy with Eq.   9  , we define thebranching factor of nodes in the  r   exterior  E  r   as k  ˜   r    =  k  2  r    −   k   r   k   r    =  k  =0  k  2 P r   k   k   r    − 1.   23  Using Eqs.   17   and   23  ,  k  ˜   r    can be rewritten as a functionof   f   as k  ˜    f    =  fG 0    f   G 0    f    .   24  Appendix A shows that    r    and     r    obey differentialequations d    r   dr  = −  k  ˜   r    + 1 +2   r   r   k   r   ,   25  d     r   dr  = 1 +   r   r   k   r    +    r   r   k   r   .   26  Equations   25   and   26   govern the growth of the aggre-gate. To solve them, we make the same substitution  f  = G 0−1  r    Eq.   15   as before. The general form of the solu-tion for Eq.   25   is    f    = −  G 0    f    f   +  C  1  f  2 ,   27  where  C  1  is a constant. At time  t  =0,  r  =  f  =1, and    1  =0.With this initial condition, we obtain  C  1 = G 0   1  =  k   . UsingEq.   27  , the general solution of Eq.   26   is     f    =  G 0   1   f  2 +  C  2  f  ,   28  where  C  2  is a constant. When  r  = r   , the building of shell    isfinished. At that time, all the open links of the aggregate 10 -5 10 -4 10 -3 10 -2 10 -1 10 0  r 0123456789      <       k      (    r      )     > <k>=4<k>=6<k>=8<k>=10(a) ER 0 0.2 0.4 0.6 0.8 1  r 23456      <       k      (    r      )     > λ=2.5λ=3.0λ=3.5λ=4.0 (b) SF FIG. 3.   Color online   Comparison between the simulation re-sult and the theoretical prediction for average degree   k   r    of thenodes in  E  r  .   a   Four ER networks with different values of    k    and  b   four SF networks with  k  min =2 and different values of    . Thesymbols represent the simulation results for ER and SF networks of size  N  =10 6 . The lines in   a   represent Eq.   21  . The lines in   b   arethe numerical results of Eq.   17   using the degree distribution ob-tained from the networks.STRUCTURE OF SHELLS IN COMPLEX NETWORKS PHYSICAL REVIEW E  80 , 036105   2009  036105-5
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