aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Solvable Metric Growing Networks
M O Hase and J F F Mendes
Departamento de F´ısica da Universidade de Aveiro, 3810-193 Aveiro, PortugalE-mail: [email protected]
Abstract.
Structure and dynamics of complex networks usually deal with degreedistributions, clustering, shortest path lengths and other graph properties. Althoughthese concepts have been analysed for graphs on abstract spaces, many networkshappen to be embedded in a metric arrangement, where the geographic distancebetween vertices plays a crucial role. The present work proposes a model forgrowing network that takes into account the geographic distance between vertices: theprobability that they are connected is higher if they are located nearer than farther.In this framework, the mean degree of vertices, degree distribution and shortest pathlength between two randomly chosen vertices are analysed.PACS numbers: 89.75.-k, 89.75.Fb, 89.75.Hc olvable Metric Growing Networks
1. Introduction
For some decades the paradigm of network was represented by the random graphs[1],which modelled graphs with a huge number of vertices and edges as a stochastic process.In the well - known Erd¨os - R´enyi model with N vertices, the occurrence of an edge, fromthe (cid:0) N (cid:1) possibilities, is determined independently with some fixed probability. Despitethe fact that the introduction of probability concepts in graph theory representeda progress in many areas – even besides mathematics –, it failed to describe manyproperties of “real networks” like Internet, air traffic system, social links between people,and so on. However, the 1990s witnessed many important progress in attempt toexamine these “real networks”. To cite some of them, the small - world network ofWatts and Strogatz[2] (see also [3]) is a model that can interpolate, by tuning a singleparameter, a random graph and a regular lattice. This model succeeded in characterizean interval for this parameter where the small - world property is displayed. Barab´asiand Albert, on the other hand, have proposed the growing network with preferentiallinking[4], which succeeded in display the scale - free properties, present in a myriadvariety of “real networks”. These models have been intensively analysed, with manyprogress.Nevertheless, the networks studied by the above authors lied on an abstract space,where the geometric structure on which the vertices are placed is not important. Thispicture revealed to be favourable for networks that can be analysed, for instance, bytools from spin glass theory or spin systems with finite connectivity that can be studiedwith similar techniques[5, 6, 7, 8]. Many “real networks”, however, have an additionalfeature: they are embedded in some geographic structure, which not only allows oneto distinguish “near” and “far” vertices from some point, but they may determine thedynamics of the evolution of networks. For instance, many cities grow starting from a“initial” region and expand away from this center gradually; as another example, it iscommon to see people making friendships with those that live nearer than farther. Thistheme – the influence of the geography on networks – has attracted attention, and thereare interesting researches being done in the context of computer science[9, 10, 11, 12],with recent progress also in the physical literature[13, 14].In the literature, networks on Euclidean structures have usually other propertiesattached together[15, 16, 17]. For instance, in [16], a random growing network wasconsidered on a two-dimensional Euclidean structure together with preferential linking;the authors showed, by numerical methods, that the strength of the interaction (thatdepends on the Euclidean distance between them) between vertices can change theprofile of the degree distribution from a power-law to a stretched exponential (similarresults were found by [17]), and the small-world property is preserved for any strengthof this interaction[17]. A related problem was studied in [18], where the crossoverbetween scale-free and spatial networks was considered. The authors of [19] havesucceeded in constructing a non-growing geographical small-world network with scale-free distribution. There are also works that investigated weight properties (differences olvable Metric Growing Networks R . To stress the metric structure of the model,a power - law form, with exponent α , was chosen to describe the probability that twovertices, separated by a metric distance r , are connected (the probability is proportionalto r − α ). The motivation to choose this particular form is twofold: firstly, the power -law form can tune, with a single paramenter ( α ), the strength of interaction betweenvertices (low values of α makes the network close to a random graph, while large valuesof α makes the network close to a regular lattice structure). In the literature, theexponential distribution has already been tested by Waxman[21] (moreover, the randomnetwork topology generator of Waxman yields a graph that is quite different from the oneproposed here, where the distance between vertices are “carefully” chosen). Moreover,power - law form is interesting since it yielded many non - trivial results to problemsthat are similar to ones that are discussed in this work[9, 10]. The model proposed hereis manageable to analytic treatments in some interesting cases, and an analysis of themean degree is carried out, as well as other graph properties.The next two sections present the model and fix notations. The mean degree of thenetwork is analysed in sections IV and V. The aim to discuss this quantity is to showthat the model proposed in this work exhibits a metric structure that shares the samenotion of distance present in models embedded in a Euclidean space like, say, R . Forsake of completeness and to stress the fact that the model does show metric structure,the sections VI and VII discuss, respectively, the degree distribution and an estimationof the shortest path length between two randomly chosen vertices in the graph. Somefinal comments are made in the section VIII.
2. Notations
Let x be a continuous variable that tends to infinity or some limiting value and let g bea positive function of x and consider f as any other function (of x ). Throughout thiswork, some symbols[22] will be extensively used, which are:I) O : If f = O ( g ), then | f | < Ag , where A does not depend on x , for all values of x in olvable Metric Growing Networks o : If f = o ( g ), then f /g → ≪ (or ≫ ): y ≪ z will always be understood in the sense that y/z = o (1) if nospecial conditions are specified.IV) O ( x, y ): Indicates max {O ( x ) , O ( y ) } ( x and y are assumed to be non - negative).In this work, a network will be defined on a metric structure, and the distinctionbetween “(metric) distance” and “path length” should be clarified. The “(metric)distance between two vertices, x and x , is the separation between them on the metricstructure, and no reference is made to the links between vertices. On the other hand,the “path length” between x and x is the minimum number of links that joins them.In the example given below (Figure 1), the (metric) distance between the vertices x a and x b is 5, but the path length connecting them is 2. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) x a x b Figure 1.
Example: the metric distance between each adjacent intersection of dashedlines is taken to be 1. The bold curves are links between vertices.
3. Model
Let 0 be an initial vertex (the “central node”) from which the network grows toward m branches. At each time step, taken as unity, m new vertices are added to the network,one for each branch. Each one of the new vertex (there are m of them: one for eachbranch) is added at distance 1 from the previous vertex in the same branch. Moreover,the distance between two vertices located at adjacent branches will be taken as being 1if they were born at the same time (this situation is valid for all m , except when m = 1).Choose now one branch and denote it by 1; the branches 2, 3, · · · , m are counted incounter clockwise orientation, as shown in Figure 2. olvable Metric Growing Networks (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) mm−112tt−1 s p(k,s,t) p (k,s,t) Figure 2.
Growing network with m branches at time t . The probability that thevertex born at time s (located at branch 1) has k links is p ( k, s, t ). Note that in thisexample, t = 3 and s = t − p ( k, s, t ) that a vertex located at the branch 2 (this vertex is indicated by an arrowin the picture) has k links should be the same: p ( k, s, t ) = p ( k, s, t ). Following thesame argument, one has p ( k, s, t ) = p ( k, s, t ) = · · · = p m ( k, s, t ). In this work, the probability p i ( k, s, t ) that a vertex, which was born at time s ( ≤ t ) at branch i , has k links at time t plays the major role. However, the symmetryof the model shows that the m functions p ( k, s, t ), p ( k, s, t ), · · · p m ( k, s, t ) are equal.Thus, for simplicity of notation, this work will deal with just one of them, say p ( k, s, t ).Now denote p ( k, s, t ) by p ( k, s, t ). The function p ( k, s, t ) corresponds to a particularchoice of one vertex (born at time s ) among m without losing generality. This selectedvertex will be considered to be located, by notation, at the “branch 1”. This choicedoes not affect the analysis, since the only important quantity involving branches aretheir differences.After t ( >
1) time steps, a vertex born at time s , with 0 < s < t , will have fournearest neighbors which are at distance 1 (except when m = 1 or m = 2): the vertices s − s + 1 (both of them belonging to the same branch), and the vertices thatwere born at the same time at the (two) adjacent branches. This sets the positions thatvertices will occupy in the network, but it does not say how they link to each other,which will be described below.Summarizing the model presented above, one has the following notations:a) p ( k, s, t ) denotes the probability that a vertex born at time s has k links at time t . This fixes a specific vertex (denoted as s ) among m vertices that have also born atthe same time.b) This fixed vertex s will be considered to belong at branch 1, from which thebranches are counted up to m in counter clockwise orientation.c) By “vertex s ” one should understand the vertex which was born at time s or,equivalently, a vertex which is at distance s from the central node. olvable Metric Growing Networks t refers to the “time” of the network growing process and alsoindicates the distance of the “border” of the network to the central node.Now consider a growing network with m branches and fix a vertex s (at branch 1,by convention). At time t ( > s ), each one of the m new vertices that joins the networkis made to connect to one of the m ( t −
1) + 1 old vertices. The probability that a newvertex (born at time t ), located at branch b , links to the selected vertex s (at branch 1)is denoted by w ( m ) b ( s, t ). This probability is made to decay algebrically with the distancebetween vertices, and its explicit form is shown during the next sections. The distance ismeasured according to the previously defined network structure: a vertex s (0 < s < t )is at distance 1 from its nearest neighbors along its branch and also at distance 1 withits nearest neighbors, born at the same time s , located at adjacent branches; see Figure2. These latter two neighbors are absent when m = 1 and m = 2 (when there are onlytwo vetices that join the network at each time). The power - law form of w ( m ) b ( s, t )enables one to observe the effects of the range of the interaction between vertices, ascan be seen below.In this work, rewiring and decimation of edges are not allowed.
4. One branch
Consider a network which grows along one direction only (as a “queue”), which is thecase m = 1. Each new vertex, t + 1 (henceforth, by convenience, new vertices willborn at time t + 1 instead of t ), is linked to just one old vertex, say s , with probability w (1)1 ( s, t + 1), which depends on the distance (along the line the vertices are placed)between them. For simplicity, w (1)1 ( s, t + 1) will be denoted as w ( s, t + 1) in this onebranch case; the notation w ( m ) b will be useful later, in the m branches case. w(0,3) w(1,3) w(2,3) Figure 3.
Rule for the link of the ( t + 1) born vertex (here, t + 1 = 3). The vertex 3links vertex 0, 1 and 2 with probability w (0 , w (1 ,
3) and w (2 , Denoting by p ( k, s, t ) the probability that a vertex s has k links at time t , themaster equation of the growing process of this network is casted as olvable Metric Growing Networks p ( k, s, t + 1) = w ( s, t + 1) p ( k − , s, t ) + w ( s, t + 1) p ( k, s, t ) , (1)where w ( s, t ) := 1 − w ( s, t ), and is subjected to the condition p ( k, s = t, t >
0) = δ k, ,which means that a new vertex has, initially, one (“own”) link.At time t + 1, the probability w ( s, t + 1) of a vertex s be connected by the newincoming vertex t + 1 is w ( s, t + 1) = ( t + 1 − s ) − α P tr =1 r − α , s < t + 10 , s ≥ t + 1 . (2)In particular, w ( s = t, t ) = 0 excludes the formation of “tadpoles” (a vertex that linksitself). In the one branch case, both the s vertex and the new born t + 1 vertex belongalways to the same branch. This may not be the case when more branches are present.Defining h k i ( s, t ) := ∞ X k =1 kp ( k, s, t ) (3)as the mean degree of the vertex s at time t , its equation of moviment can be casted as h k i ( s, t + 1) − h k i ( s, t ) = w ( s, t + 1) (4)by using the master equation (1).Taking into account the condition h k i ( s = t, t ) = 1 (which is derived from p ( k, s = t, t >
0) = δ k, ), the solution of this equation can be determined as h k i ( s, t ) = t X r = s +1 w ( s, r ) , s < t , s = t . (5)The behaviour of h k i ( s, t ) depends on the order of magnitude of both s and t . Theanalysis will focus on the regime 1 ≪ s , which may, at first, seem to ignore the region ofthe network “close” to the central node; however, for sufficiently large t (in other words,for sufficiently large network), the relative position s , initially chosen to be farther fromthe point 0, is translated to a region near to the central node. Therefore, for the purposesof this work, selecting s to be much larger than 1 is not a serious restriction. As a remark,the regime 1 ≪ s ≪ t enables one to cast the continuous version of the equation (4),although this procedure of transforming a difference equation to a differential equationis not necessary, since the solution (5) could be obtained.Before presenting the results, equipping this m = 1 model with preferential linkingterm (which makes highly connected vertices more likely to receive more links), leadsto an already studied work[23]. olvable Metric Growing Networks ≪ s ≪ t , which leads the relative position s on the network closerto the central node for sufficiently large t , one has h k i ( s, t ) = (1 − α ) ln (cid:18) ts (cid:19) + O (1) , ≤ α < (cid:18) ln t ln s (cid:19) [1 + o (1)] , α = 12 + O ( s − α , τ − α ) , α > . (6)The mean degree in a more general regime, 1 ≪ s < t , is presented in the AppendixA.1. For α ≤
1, the mean degree is always increasing with the size of the network.Nevertheless, when α becomes larger than 1, the contribution to the degree of a vertexcomes mainly due to the next vertex only. The calculations have assumed that s ≪ t inthe sense that ln( t/s ) ≫ t/ ln s ) ≫ ≤ α < α = 1, respectively(theses points are detailed in the Appendix A.1). m branches The general case, where the network grows toward m ( >
1) directions, shows that thenumber of branches plays a major role in the behaviour of the mean degree of a vertex.The probability that a new born vertex, at branch b (and time t + 1), links to anold one, born at time s (and located, by convention, at branch 1), is given by w ( m ) b ( s, t + 1) = d ( m ) ( s, t + 1 , b ) − α N ( t + 1 , m, α ) , (7)where N ( t, m, α ) := t − α + t − X u =1 m X h =1 d ( m ) ( u, t, h ) − α (8)is the normalization (see Appendix A.2) and d ( m ) ( s, t, b ) := t − s + min { s, min { b − , m − b + 1 }} (9)is the distance between these two vertices. This is a natural generalization of the previousprobability (2). Here, the distance should take into account the fact that the minimumpath (over the metric structure) between a new born vertex and some old vertex maycross the central node (this possibility is not seem for m ≤ { s, min { b − , m − b + 1 }} , where 2 s stands for a path (over the metric structure)that passes along the central node, and min { b − , m − b + 1 } is appliable when theshortest “metric” path realizes contourning the central node. For the model with m branches, the master equation is written as p ( k, s, t + 1) = p ( k − m, s, t ) m Y j =1 w ( m ) j ( s, t + 1) + p ( k, s, t ) m Y j =1 w ( m ) j ( s, t + 1) + olvable Metric Growing Networks m − X b =1 p ( k − b, s, t ) X q < ··· . (13) olvable Metric Growing Networks ≪ s ≪ t ≪ m (in the sense also that ln ( t/s ) ≫ h k i ( s, t ) = ln (cid:18) ts (cid:19) + O ( s − ) , α = 01 − α − α − (cid:18) ts (cid:19) + O (1) , < α < (cid:18) ts (cid:19) + O (1) , α = 1 α − − − α ln (cid:18) ts (cid:19) + O (1) , < α ≤ α − − − α ln (cid:18) ts (cid:19) + O (1) , α > m ≫ t α − O ( ms − α ) , α > m ≪ s α − (14)The results for the mean degree in other regimes are presented in the AppendixA.4 for completeness. Note that the asymptotic behaviour of the mean degree forlarge number of branches is distinct from the one branch (or relatively small number ofbranches – in the sense that m ≪ s ) case, showing that m does influence on the graphproperty of the network. Basically, one has h k i ( s, t ) ∼ − α − α − ln (cid:0) ts (cid:1) for any α for large m , except when the number of branches is large but not larger than s α − (see the α > m being large, it belongs tothe case where the interaction parameter α is strong enough to let each vertex to havetwo links only: one from its “own” and the other from the vertex born immediatelyafter it in the same branch; in other words, the interaction α overcomes the size of thenetwork and each vertex can see its neighborhood only (therefore, when α is sufficientlylarge, the network effectively behaves as a small m case).As a last remark, the number of branches became important for non-zero α only,which leads the model to “notice” the metric structure of the space. In the particularcase where s ≪ t , in the sense that ln( t/s ) ≫
1, one has h k i ( s, t ) = ln( t/s ) + O (1).These mean degrees recover results from recursive random graphs, where the metricstructure that the system lies on can be ignored – this is cleary seen from the fact that w ( m ) b ( s, t + 1) = ( mt + 1) − for α = 0.
6. Degree distribution
Consider first the case m = 1. From the master equation (1), it is possible to show that(see Appendix A.5 for technical details) p ( k, s, t ) = 1( k − " t X r = s +1 w (1)1 ( s, r ) k − exp " − t X u = s +1 w (1)1 ( s, u ) [1 + o (1)] olvable Metric Growing Networks (cid:12)(cid:12)(cid:12) w (1)1 ( s, u ) (cid:12)(cid:12)(cid:12) ≪ , for u ∈ ( s, t ] ⊂ Z + . (15)This condition can be satisfied for 0 ≤ α ≤ s ≫
1, but it does not hold for largervalues of α , except if α ∼ + . The form of the equation for p ( k, s, t ) given above showsa Poisson distribution with mean P tr = s +1 w (1)1 ( s, r ) = h k i ( t ) − s < t (see (5)). Thismeans that the probability of a vertex s , at time t , having k links is concentrated onthe mean h k i ( t ) −
1, and the structure of the network is homogeneous.Using the asymptotic form (15), one can show that the degree distribution P ( k ) :=lim t →∞ P ( k, t ), with P ( k, t ) := P ts =1 p ( k, s, t ) /t , behaves as P ( k ) = − α (cid:18) − α − α (cid:19) k − , ≤ α < e − ( k − , α ∼ + . (16)For α = 1, it is possible to evaluate the degree distribution at time t as P ( k, t ) = 2 − k ln tt k F k (2 , · · · ,
2; 3 , · · · ,
3; ln t )= ln tt ∞ X n =0 n + 2) k (ln t ) n n ! , α = 1 , (17)where k F k ( ., . ; . ) is the hypergeometric function.A simple calculation shows that for s ≫
1, one has ∂p/∂α < k . The degree distribution decays more rapidly as α increases. For instance, in the case0 ≤ α <
1, the decayment is exponential, and for α ∼ + the degree distribution decaysmore rapidly, as an inverse of a factorial. Moreover, in the limit of α → ∞ , P ( k ) tendsto δ k, .For the other case, when the number of branches is large in the sense that m ≫ t ,a heuristic approach will be adopted to estimate the form of the degree distribution. Ifone assumes that the probability that a vertex s has exactly h k i ( s, t ) links at time t – the“ δ -Ansatz” –, it is possible to show (using results for the mean degree h k i ( s, t )[24, 25])that P ( k ) has an exponential distribution P ( k ) ∼ K ( α ) e − k/K ( α ) , (18)whenever h k i ( s, t ) ∼ K ( α ) ln( t/s ), where K ( α ) is a factor that depends on α . Assumingthe “ δ -Ansatz”, which turns to be reasonable in the regime 1 ≪ s ≪ t , the exponentialform of the degree distribution suits the cases where m ≫ t ≫ s ≫ α ), except when α is sufficiently large (in the sense that ms − α ≪ P ( k ) tends to δ k, .When each vertex has just two links on average due to the high value of α , themodel indicates that each vertex has, apart from its “own link”, another that comesfrom its successor in the same branch. Thus, the network achieves a regular structure, olvable Metric Growing Networks m ≫ t . Despite the fact that the probability that decaysalgebrically with distance can tune the network between a random graph type one (lowvalues of α , typically 0 ≤ α <
1) and a regular lattice type (large values of α , when h k i ( s, t ) ∼ m and w ( m ) b ( s, t ) becomes “nearly” uniform. This is the origin of the observedrandomness in the network connections, which reflects in the exponential form of thedegree distribution (18).
7. Shortest path length
This section will provide an estimation for the mean shortest path length ℓ ( t ) betweentwo randomly chosen vertices, say, x a and x b . Since they are randomly chosen, theirdistance from the central node will be taken as O ( t ).In this work, the order of magnitude of the shortest path length between x a and x b will be considered to be the graph distance (path length) between x a and the centralnode 0 plus the graph distance between x b and 0. Denoting the (shortest) path betweentwo vertices, x and x by x ↔ x , the estimation of the shortest path between x a and x b proposed in this work stems in replacing x a ↔ x b by x a ↔ ↔ x b . This estimation isexpected to be reasonable at least in the case where the number of branches is sufficientlylarge, because then the probability of the paths x a ↔ x b ↔ x a ↔ x b includes,necesarilly, the vertex 0). It will be shown later that if m is sufficiently larger than t , theshortest path length is estimated as O (ln t ), which resembles the result from randomgraphs. Note that by construction, there are no two different shortest paths connectingtwo points in the model presented in this work, since a new born vertex links to onlyone old vertex.Next, define the function∆ ( m ) s ( r ) := r − X s =1 m X b =1 ( r − s ) w ( m ) b ( s, r ) = r − r − X s =1 m X b =1 sw ( m ) b ( s, r ) , (19)which is the measure of the mean (metric) distance covered by a link (starting from apoint at distance r from the origin) toward the central node. In other words, this is themetric length of the projection of a link on some fixed branch (see Figure 4).The evaluation of ∆ ( m ) s ( r ) will focus on two distinct limit situations for the numberof branches, and it will play a major role in the estimation of the shortest path length.As mentioned earlier, one should estimate the (order of the) number of paths thata randomly chosen vertex x a takes to achieve the central node (and the same for x b );furthermore, as stated in the beginning of this section, it will be assumed that x a = O ( t )(the distance between x a and 0) and by the same argument, one has also x b = O ( t ). olvable Metric Growing Networks (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) ∆ (x ) ∆ (m)s (x ) x x x Figure 4.
Let the path x ↔ x ↔ x ↔ · · · be the shortest path between x andthe central node. The projection of x and x (and also x ) onto the branch that joins x and the central node are represented by a filled ball in the figure. The quantities∆ ( m ) s ( x ) and ∆ ( m ) s ( x ) are also indicated. For sake of clearness, most of the brancheswere not drawn. This means that the path length x a ↔ ↔ x b (which is an estimation of the shortestpath length x a ↔ x b ) is of the order of the path length x a ↔ x a = x ↔ x ↔ · · · ↔ x n = 0 be the shortest path between x a = x and thecentral node 0 = x n . Using (19), one has n − X i =1 ∆ ( m ) s ( x i ) = distance between x a and 0, which is O ( t ) . (20)Remembering that n is of the order of the shortest path length ℓ ( t ) between x a and x b at time t , the relation (20) can be used to estimate ℓ ( t ).The details of the estimation is presented in Appendix A.6. For m ≪ s , when thenumber of branches is not large, one has α ℓ ( t )0 ≤ α ≤ O (ln t )1 < α < O ( t α − ) α = 2 O ( t/ ln t ) α > O ( t )On the other hand, if m ≫ t , one has α m , t ℓ ( t )0 ≤ α ≤ O (ln t ) α > mt − α ≫ O (ln t )2 < α < mt − α ≪ O ( t α − /m ) α = 3, mt − α ≪ O ( t / ( m ln t )) α > mt − α ≪ ≪ mt − α O ( t α − /m ) α > mt − α ≪ O ( t ) olvable Metric Growing Networks mt − α ≫ t .
8. Conclusions
The present work has introduced a network growing toward m branches embedded on ametric structure and analysed some graph properties. The mean degree of a fixed vertexdecreases as α increases, indicating, as expected, that the strength of the “interaction”confines the vertices to link to other vertices that are located nearer than farther. Toensure the fact that the number m of branches does influence on the graph properties likemean degree, two opposite conditions were then considered. For small m , the resultswere similar to the one branch case, while for large m , the mean degree was larger.In any case, for α sufficiently large, one has h k i ( s, t ) →
2, no matter the number ofbranches. The degree distribution was evaluated for 0 ≤ α < α ∼ + , and itshows, as expected, that they decay as rapidly as the magnitude of α increases. Finally,the shortest path length between two randomly chosen vertices was estimated as afunction of t . The metric structure of the model is better displayed if the number ofbranches are not overwhelmingy large; otherwise, the model behaves as a random graph,and this fact is supported by the results obtained for the degree distribution, shortestpath length and mean degree; this last one grows as ln( t/s ), when s ≪ t , for any α forsufficiently large m – see (14) –, as it is expected[25]. The model presented in this workwas successful in displaying properties that are common to networks that are embeddedin a Euclidean space (like R ) and is manageable to analytic treatment.
9. Acknowledgements
The authors are thankful to S. N. Dorogovtsev for comments and MOH is supported bythe project DYSONET.
A. Appendix
This appendix is devoted to present some technical observations and list results omittedin the main text for clearness.The present work has used extensively the fact that t − X r =1 r − α ∼ t − α − α + ζ ( α ) + O ( t − α ) , ≤ α < t + γ + O ( t − ) , α = 1 ζ ( α ) − t − α α − O ( t − α ) , α > , (21) olvable Metric Growing Networks ζ is the zeta function of Riemann and γ (= 0 . ... ) is the Euler - Mascheroniconstant. As a remark, ζ ( α ) > α > ζ ( α ) < α < A.1. One branch case
For 1 ≪ s < t , one has h k i ( s, t ) = − α ) Z s/t drr (1 − r ) α + O ( s α − ) , ≤ α <
11 + t Z s +1 dr ( r − s ) ln r + O (ln − s ) ,α = 11 + 1 ζ ( α ) τ X u =1 u − α + O ( s − α ) , α > , (22)where τ := t − s .The case α = 1 may deserve a short comment. Consider the case α = 1, m = 1and 1 ≪ s ≪ t such that t − s ≫ s δ for some positive δ = O (1). Using (21), the meandegree (5) is h k i ( s, t ) = 1 + t X r = s +1 ( r − s ) − ln r + O (1)= t − s Z dyy ln ( y + s ) + O , t − s Z dyy ln ( y + s ) . (23)However, since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t − s Z dyy ln ( y + s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e Z dzz ln s + t − s Z e dzz ln z = O (ln − s ) + ln( t − s ) Z duu = O (1) , (24)one is left with h k i ( s, t ) = t − s Z dyy ln ( y + s ) + O (1) olvable Metric Growing Networks s δ +1 Z dyy ln ( y + s ) + t − s Z s δ +1 dyy ln ( y + s ) + O (1) , (25)where the first term (last line) on the right hand is bounded ( δ = O (1)): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s δ +1 Z dyy ln ( y + s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s s δ +1 Z dyy = O (1) . (26)Thus, h k i ( s, t ) = t − s Z s δ +1 dyy ln ( y + s ) + O (1)= t/s Z s δ dzz ln( sz ) (cid:18) z − (cid:19) + O (1)= t/s Z s δ dzz ln( sz ) h O ( s − δ ) i + O (1) , (27)and the desired result follows.The estimation for α = 1 assumed that τ ≫ s δ for some positive δ = O (1). A.2. On the normalization (8)
The (asymptotic form of) normalization factor is evaluated by considering the cases m odd and m even separately. It is shown that both have the same general formula.For m odd, the normalization is N odd ( r, m, α ) = r − α + r − X s =1 ( r − s ) − α + 2 r − X s =1 m − X j =1 h r − s + min { s, j } i − α , (28)where the first term is the distance between a r born vertex to the central node; thesecond term counts the distance between this new vertex to the vertices located in thesame branch, and the last term the remaining ones.If m is even, one has N even ( r, m, α ) = r − α + r − X s =1 ( r − s ) − α + r − X s =1 h r − s + min { s, m/ } i − α ++2 r − X s =1 m − X j =1 h r − s + min { s, j } i − α , (29) olvable Metric Growing Networks m is odd.For 1 < m ≪ r , one has N { oddeven } ( r, m, α ) = mr − α − α + C ( m, α ) + O ( r − α ) , ≤ α < m ln r + C ( m,
1) + O ( r − ) , α = 1 C ( m, α ) − mr − α α − O ( r − α ) , α > , (30)where C ( m, α ) is defined in (37).For m ≫ r , one has N { oddeven } ( r, m, α ) == (cid:18) − α − − α (cid:19) mr − α − (cid:18) − α (cid:19) mr − α + 8 [(1 + α ) 2 − α − − α ) (2 − α ) r − α ++ O ( mr − α − , r − α ) , ≤ α < m ln 2 − mr − + 4 (2 ln 2 − r + O (ln r, mr − ) , α = 1 (cid:18) − − α α − (cid:19) mr − α − (cid:18) − α (cid:19) mr − α + 8 [1 − (1 + α ) 2 − α ]( α −
1) (2 − α ) r − α ++ O ( mr − α − , , < α < mr − − mr − + 2 ln r + 4 + 2 γ − ζ (2) − O ( mr − , r − ) , α = 2 (cid:18) − − α α − (cid:19) mr − α − (cid:18) − α (cid:19) mr − α + 2 ζ ( α − − ζ ( α )++ 8 [(1 + α ) 2 − α − α −
1) ( α − r − α + O ( mr − α − , r − α ) , α > . (31) A.3. On the m branch master equation After multiplying both sides of the master equation (10) by k and summing them, onegets h k i ( s, t + 1) = A h k i ( s, t ) + B ( s, t + 1) . (32) olvable Metric Growing Networks A := m Y j =1 w ( m ) j ( s, t + 1) + m Y j =1 w ( m ) j ( s, t + 1) ++ m − X b =1 X q < ··· Before analysing the mean degree for many cases not presented in the main text, it isconvenient to define C ( m, α ) := mζ ( α ) − m − X b =1 b X u =1 u − α , m odd mζ ( α ) − m/ X u =1 u − α − m − X b =1 b X u =1 u − α , m even α = 1 mγ − m − X b =1 b X u =1 u − , m odd mγ − m/ X u =1 u − − m − X b =1 b X u =1 u − , m even α = 1 . (37)This work has concentrated in two limiting regimes to stress the role played by m .The results for “small” number of branches (1 < m ≪ s < t ) was compared to the otherlimit case, where m is “large” (1 ≪ s < t ≪ m ). In both cases, s is chosen to be muchlarger than 1 and in this appendix, t ( > s ) is taken free, although simple analytic resultsare provided if s ≪ t , as shown in the section 5.For 1 < m ≪ s < t , one has h k i ( s, t ) = olvable Metric Growing Networks (cid:18) ts (cid:19) + O ( s − ) , α = 01 + (1 − α ) Z s/t dyy (1 − y ) α + O ( s α − ) , < α < 11 + t Z s +1 dy ( y − s ) ln y + O (ln − s ) , α = 11 + mζ ( α ) C ( m, α ) − C ( m, α ) (cid:20) m − X b =1 b X u =1 u − α + ∞ X u = τ +1 u − α + 2 m − X b =1 ∞ X u = τ + b +1 u − α (cid:21) ++ O ( s − α ) , m odd , α > 11 + mζ ( α ) C ( m, α ) − C ( m, α ) (cid:20) m − X b =1 b X u =1 u − α + m/ X u =1 u − α + 2 m − X b =1 ∞ X u = τ + b +1 u − α ++ ∞ X u = τ +1 u − α + ∞ X u = τ + m/ u − α (cid:21) + O ( s − α ) , m even , α > , (38)where τ := t − s . The calculations can not be always computed directly in the α > < m ≪ s < t is similar to the one branch case.On the other hand, for 1 ≪ s < t ≪ m , one has h k i ( s, t ) = olvable Metric Growing Networks (cid:18) ts (cid:19) + O ( s − ) , α = 01 + 1 − α − α − Z s/t dyy (1 + y ) α + O ( s − , t/m ) , < α < 11 + 1ln 2 ln (cid:18) t + s s (cid:19) + O (cid:16) s − , t/m, ( s/m ) ln t, ( τ /m ) ln s (cid:17) , α = 11 + α − − − α Z s/t dyy (1 + y ) α + O (cid:18) tm , s (cid:19) , < α < O (cid:18) t ln tm , s (cid:19) , α = 2 O (cid:18) t α − m , s (cid:19) , α > , (39)where it was assumed that s ln t ≪ m and τ ln t ≪ m for α = 1; t ln t ≪ m for α = 2;and t α − ≪ m for α > ≪ s < t ≪ m , the interplay between the free parameters should be examined,specially the competition between the strength of “interaction” α and the numberof branches m . The present work has considered the limiting cases t α − ≪ m and s α − ≫ m to show their differences. Since m ≫ t , note that the former condition(“large m ” – in contrast to the later, s α − ≫ m , the “large α ”) is always satisfied for0 ≤ α ≤ 2. Finally, the parameters mt − α and ms α − dictates the behaviour of themean degree in the case α > 2. If t α − ≪ m , one has the situation shown in (39), wherefor sufficiently large m (such that m ≫ t α − ), the mean degree grows with the size ofnetwork. On the other hand, if s α − ≫ m , or, equivalently, the α is sufficiently strong,one has (38), where the mean degree behaves as in the case m = 1. A.5. On the degree distribution ( m = 1 ) Performing the zeta transform˜ p ( K, s, t ) := ∞ X k =1 K k p ( k, s, t ) (40)on the master equation (1), one can show that˜ p ( K, s, t + 1) = ˜ p ( K, s, s ) t Y u = s ˜ p ( K, s, u + 1)˜ p ( K, s, u )= K t Y u = s [1 + ( K − w ( s, u + 1)] , (41) olvable Metric Growing Networks p ( K, s, u ) = 0 for any u ≥ s . From the condition p ( k, s, s ) = δ k, (or ˜ p ( K, s, s ) = K ), one has p ( k, s, t + 1) = 1( k − k − X q =0 (cid:18) k − q (cid:19) ×× ( d k − − q dK k − − q exp " ( K − t X u = s w ( s, u + 1) K =0 ) (cid:26) T q ( α, K ) (cid:12)(cid:12)(cid:12)(cid:12) K =0 (cid:27) , (42)where T q ( α, K ) (cid:12)(cid:12)(cid:12)(cid:12) K =0 = δ q, + O (cid:0) q ! s α − (cid:1) , ≤ α < O (cid:18) q !ln s (cid:19) , α = 1 α → + (43)for sufficielntly large s . Therefore, for large s (say, s ≫ e √ k ! ), one has p ( k, s, t ) = e − [ h k i ( s,t ) − ( k − h k i ( s, t ) − k − [1 + o (1)] , (44)where the main contribution comes from the q = 0 term (in the sum (42)) and h k i ( s, t )is given by (5). A.6. Shortest path length Firstly, the asymptotic form of ∆ ( m ) s ( r ), which is closely related to the estimation of theshortest path length, will be listed. The definition (37) will be used below.For 1 < m ≪ r , one has∆ ( m ) s ( r ) = (cid:18) − α − α (cid:19) r + O ( r α ) , ≤ α < r ln r + O (cid:18) r ln r (cid:19) , α = 1 mr − α C ( m, α ) (2 − α ) + O ( r − α , , < α < mC ( m, 2) ln r + O (1) , α = 2 (45)and∆ ( m ) s ( r ) = C ( m, α − C ( m, α ) + olvable Metric Growing Networks C ( m, α ) m − X u =1 (cid:18) m − 14 + u − u (cid:19) u − α −− ζ ( α ) C ( m,α ) (cid:16) m − (cid:17) , m odd1 C ( m, α ) m − X u =1 (cid:18) m − m u − u (cid:19) u − α ++ m m/ X u =1 u − α − ζ ( α ) − ζ ( α ) C ( m, α ) m (cid:16) m − (cid:17) , m even + O ( r − α ) (46)for α > 2. Note that ∆ ( m ) s ( r ) becomes smaller as it approaches the central node.On the other hand, for 1 ≪ r ≪ m , the parameter mr − α and mr − α play a majorrule to the asymptotic behaviour of ∆ ( m ) s ( r ). Firstly, one can show that∆ ( m ) s ( r ) = ( α + 2 − α − r (2 − α ) (2 − α − 1) + O (cid:18) , r m (cid:19) , ≤ α < (cid:18) − (cid:19) r + O (cid:18) , r m (cid:19) , α = 1( α + 2 − α − r (2 − α ) (2 − α − 1) + O (cid:18) , r m (cid:19) , < α < − r + O (cid:18) , r ln rm (cid:19) , α = 2 , (47)(48)where a stronger condition, m ≫ r ln r , was assumed for α = 2. For α > mr − α ≫ 1, one has∆ ( m ) s ( r ) = ( α + 2 − α − r (2 − α ) (2 − α − 1) + O (cid:18) , r α m (cid:19) , ≤ α < r + O (cid:18) , r m (cid:19) , α = 3( α + 2 − α − r (2 − α ) (2 − α − 1) + O (cid:18) , r α m (cid:19) , α > . (49)(50)In this regime, one sees that the leading term of ∆ ( m ) s ( r ) (for large r ) is proportional to r . Note that the condition mr − α ≫ ≤ α ≤ olvable Metric Growing Networks m ≫ r ). A different asymptotic behaviour is seen for mr − α ≪ 1; if 2 < α ≤ 3, one gets∆ ( m ) s ( r ) = ( α + 2 − α − − α ) (1 − α ) mr − α [2 ζ ( α − − ζ ( α )] ++ O ( m r − α , mr − α , r − α ) , < α < mr − [2 ζ (2) − ζ (3)] + O (cid:0) m r − , mr − , ln r (cid:1) , α = 3 , (51)and for α > 3, one has∆ ( m ) s ( r ) = ( α + 2 − α − − α ) (1 − α ) mr − α [2 ζ ( α − − ζ ( α )] ++ O ( m r − α , , mr − α ≪ ≪ mr − α ζ ( α − ζ ( α − − ζ ( α ) + O ( mr − α ) , mr − α ≪ mr − α ≪ . (52)From the results, (45) to (52) above, one sees that the asymptotic behaviour of∆ ( m ) s ( r ) is one of the following form:∆ ( m ) s ( r ) ∼ c rr ln rc r ω c ln rc , (53)where c (0 < c < c , c , c and ω do not depend on r . For instance, in the regime1 < m ≪ r and for 1 < α < 2, one has c = m/ [ C ( m, α )(2 − α )] and ω = 2 − α .Each asymptotic form for ∆ ( m ) s ( r ) leads to a different shortest path length ℓ .The shortest path length between two randomly chosen vertices, x a and x b , will beestimated to be of the order of the path length between x a and 0, as stated in section7. Symbolically, this implies ℓ X i =1 ∆ ( m ) s ( x i ) ∼ distance from x a = x to the central node, which is O ( t ) , where x is closer to the central node than x = x a by a distance ∆ ( m ) s ( x ) on average, x is closer to the origin than x by a distance ∆ ( m ) s ( x ) on average, and so on.Since x is at distance O ( t ) from the origin on average, it means that if ∆ ( m ) s ( r ) = O (1) (or ∆ ( m ) s ( r ) ∼ c ), then ℓ ( t ) = O ( t ) , when ∆ ( m ) s ( r ) ∼ c . (54) olvable Metric Growing Networks ( m ) s ( r ) ∼ c r , where c ∈ (0 , 1) is a constant. In thiscase, one has the following sequence: x = x − ∆ ( m ) s ( x ) = x − c x = (1 − c ) x x = x − c x = (1 − c ) x = (1 − c ) x x = x − c x = (1 − c ) x · · · x n = (1 − c ) n − x (55)From (55), one has n = 1 + ln x − ln x n ln (cid:16) − c (cid:17) , c ∈ (0 , . (56)Remembering that x ∼ O ( t ) and if n is of the order of ℓ , then one has ℓ ( t ) ∼ ln t .However, this kind of argument is suitable for n not so small (since the estimation of∆ ( m ) s ( r ) is made for large values of r ). Then, to estimate the shortest path length when∆ ( m ) s ( r ) ∼ c r ( c ∈ (0 , n will be taken as being of order of ℓ , which means that x n is close to the central node but not the central node itself; in other words, x n ≈ O (1)and n ≈ ℓ . For large values of t , one has ℓ ( t ) ∼ ln t , when ∆ ( m ) s ( r ) ∼ c r for c ∈ (0 , . (57)For technical reason, the shortest path length for the other three cases will beestimated by using h ∆ ( m ) s ( r ) i := P u ∆ ( m ) s ( u ) /t ≈ R r ∆ ( m ) s ( u ) du/t instead of ∆ ( m ) s ( r ).From this approximation, one has ℓ ( t ) ∼ ln t , when ∆ ( m ) s ( r ) ∼ r ln r , (58) ℓ ( t ) ∼ t − ω , when ∆ ( m ) s ( r ) ∼ c r ω , (59)and ℓ ( t ) ∼ t ln t , when ∆ ( m ) s ( r ) ∼ ln r . (60) References [1] Bollob´as B, Random Graphs (Academic Press, London, 1985)[2] Watts D J, Strogatz S H, 1998 Nature Phys. Rev. E Science J. Phys. C J. Stat. Mech. P09004[7] Zdeborov´a L, M´ezard M, 2006 J. Stat. Mech. P05003[8] Hase M O, Mendes J F F, 2008 J. Phys. A Nature olvable Metric Growing Networks [11] Berger N, Borgs C, Chayes J T, D’Souza R M, Kleinberg R D, Lecture Notes in Computer Science , 208 (2004)[12] Fabrikant A, Koutsoupias E, Papadimitriou C H, Lecture Notes in Computer Science , 781(2002)[13] Sen P, Manna P, 2003 Phys. Rev. E Phys. Rev. E Proc. Natl. Acad. 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