Constructions and properties of a class of random scale-free networks
CConstructions and properties of a class ofrandom scale-free networks
Xiaomin Wang a,b , Fei Ma a, a. School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, Chinab. Key Laboratory of High-Confidence Software Technology, Peking University, Beijing 100871, China Abstract
Complex networks have abundant and extensive applications in real life. Re-cently, researchers have proposed a number of complex networks, in which some aredeterministic and others are random. Compared with deterministic networks, ran-dom network is not only interesting and typical but also practical to illustrate andstudy many real-world complex networks, especially for random scale-free networks.Here, we introduce three types of operations, i.e., type-A operation, type-B opera-tion and type-C operation, for generating random scale-free networks N ( p, q, r, t ).On the basis of our operations, we put forward the concrete process of producingnetworks, which constitute the network space N ( p, q, r, t ), and then discuss theirtopological properties. Firstly, we calculate the range of the average degree of eachmember in our network space and discover that each member is a sparse network.Secondly, we prove that each member in our space obeys the power-law distributionwith degree exponent γ = 1 + ln(4 − r )ln 2 , which implies that each member is scale-free.Next, we analyze the diameter, and find that the diameter may abruptly transformfrom small to large due to type-B operation. Afterwards, we study the clusteringcoefficient of network and discover that its value is only determined by type-C op-eration. Ultimately, we make an elaborate conclusion. Keywords:
Random network; degree distribution; diameter; clustering coefficient.
Over the course of the recent decades, complex network is an interdisciplinary study thathas attracted thousands of scholars from different fields, and is considered as a mathe-matical tool that connects the real world with theoretical research. Complex network isapplied across a multitude of disciplines ranging from natural and physical sciences tosocial sciences and humanities. In general, a network can be viewed as a graph consistingof vertices (or nodes) connected by edges (or links). An extensive example of complexnetwork contains the Internet, the World Wide Web (WWW), coauthorship network, ci-tation network, annotated network, musical solos network, protein network, informationnetwork, peer-to-peer network, manage network[1]-[11], and so forth. Corresponding author’s email: [email protected] a r X i v : . [ phy s i c s . s o c - ph ] J u l rd¨os and R`enyi, in 1960, defined a random network as n vertices connected by m edges, which are randomly chosen from the n ( n − / P ( k ) ∼ k − γ , < γ ≤ P ( k ) represents a probability of randomly selecting a vertex of degree k from theentire network[2]. On the basis of the degree distribution formula (1), Dorogovtsev et al.[21] gave the definition of cumulative degree distribution and the computational formulato explain topological structure of deterministic scale-free networks as follows: P cum ( k ) = (cid:88) k (cid:48) ≥ k N ( k (cid:48) , t ) n v ( t ) ∼ k − γ (2)where N ( k (cid:48) , t ) and n v ( t ) stand for the number of vertices with degree k (cid:48) and the networkorder N ( t ) at time step t , respectively. Currently, many empirical studies have shown thatscores of complex networks in society and nature have the scale-free property, namely, theyfollow either the degree distribution (1) or the cumulative degree distribution (2). In thispaper, we show that each member in our space obeys the power-law distribution withdegree exponent γ = 1 + ln(4 − r )ln 2 , by calculating the cumulative degree distribution.In other words, the degree distribution merely describe partial characteristics of net-work, it is not enough for ones who simply use the degree distribution to analyze complexnetworks. So, other statistical methods have been developed to depict the nature ofnetwork. Among of which, the prevalent statistical traits are diameter and clusteringcoefficient of network. If a network have smaller diameter and higher average clusteringcoefficient, it can be referred to as a small-world network which was discovered by Wattsand Strogatz in 1998[1]. The well-known six degrees of separation theory reveals thatthe diameter is short in comparison with its size. Actually, in our real-life networks, thediameter of some networks generally increase exponentially with network order[13], whileothers usually grow logarithmically[14, 15, 16]. Here, we discover the fact that the diam-eters of these members in our space that do not contain type-B operation grow linearlywith time t , otherwise, the diameters increase exponentially.The clustering coefficient of the network can be obtained by averaging the clusteringcoefficient of a vertex over all the vertices. The clustering coefficient of a vertex in networkrefers to the ratio of the number of actual existing edges between all vertices adjacent to it2nd the number of all possible edges between them. A mass of social and natural networkshave clustering characteristics, some networks have a low clustering coefficient[17], whilethe remaining have a high clustering coefficient[14, 15, 16]. With the aim of providinga deeper understanding of the underlying mechanism that leads to different clusteringcoefficient, we need to probe the clustering coefficient of network in more detail. Then,we show that the clustering coefficient of these members in our network space that donot contain type-C operation is always zero, otherwise the clustering coefficient is notequal to zero. Combined with the diameter and clustering coefficient, we reveal thatthese members of our network space not contain type-B operation but contain type-Coperation display better features. It may have smaller diameter and higher clusteringcoefficient, usually known as the small-world effect.The reminder of this paper is organized by the following several sections. In Section2, we introduce three types of operations in detail, namely, type-A operation, type-Boperation and type-C operation, to successfully construct a network space N ( p, q, r, t )where probability parameters p, q, r hold on p + q + r = 1 , ≤ p, q, r ≤
1. After that, inSection 3, we show the elaborate process of construction and discuss some fundamentaland typical topological structures of networks in our network space N ( p, q, r, t ). Aboveall, we discover that each member in the network space has sparsity due to averagedegree approaches to a constant. Secondly, we prove that each member of our space isscale-free because it follows a power-law distribution with γ = 1 + ln(4 − r )ln 2 . Hereafter,we analyze the diameter may abruptly transform from small to large due to the type-Boperation. Hence, we state that the diameters of these members in our space that donot contain type-B operation grow linearly with time t , otherwise, the diameters increaseexponentially. Afterwards, we study the clustering coefficient of the network and findtype-C operation has a decisive influence on the clustering coefficient. Finally, we makea detailed conclusion. In this section, we will construct a category of random networks with tuning parameterswhich are generated from three different typical growth operations and then applied tospan a network space, denoted as N ( p, q, r, t ). Here, the probability parameters p, q, r hold p + q + r = 1 with 0 ≤ p, q, r ≤
1, and t stands for time step. In other words, fortwo arbitrary members N ( p i , q i , r i , t ) and N ( p j , q j , r j , t ) randomly selected from networkspace N ( p, q, r, t ), they must have different “degree sequence”. To this end, we wantto introduce three classical operations, referred in this paper type-A operation, type-Boperation, type-C operation, which are explained in more detail, as follows Type-A operation
Add an edge xy on vertices x and y for a given edge uv with twovertices u and v , as well join vertex u with x and v with y producing two new edges,which generates a cycle C . Such a process is Type-A operation, which is also defined as3 ig.1. The diagram of type-A operation, type-B operation as well as type-C operation. a rectangle operation owing to each edge produces a rectangle [18], see Fig.1(a). Type-B operation
Create two vertices x and y for a given active edge uv with twovertices u and v , connect vertex x with two endpoints of edge uv by two new edges, joinvertex y with vertex pair u and v , and then remove the active edge uv , also obtain a cycle C . Such a process is Type-B operation, which is often defined as fractal operation becauseof each active edge applied fractal operation, or also known as a diamond operation dueto each active edge forms a diamond [18], see Fig.1(b). Type-C operation
Add a vertex w for a given edge uv with two vertices u and v ,and then connect vertex w with u and v , respectively. It produces a cycle C , also calleda triangle C . Such a process is Type-C operation, which is commonly defined as triangleoperation on account of each edge generates a triangle, see Fig.1(c).In fact, from a theoretical perspective, it is not necessary to distinguish two cycles C obtained from type-A operation and type-B operation described above. Obviously, C and C completely differ in clustering coefficient according to the former is always equalto 0 while the latter is 1. Nevertheless, in this paper, we prefer to call the first cycle C asa rectangle, the second cycle C as a diamond, and the third cycle C as a triangle. Thereason is that it is convenient for us to describe and organize the content of our article .Care is necessary in such a situation, for instance, the isomorphism between two cycles C does not guarantee the same topological structure, as we will describe the differencebetween two C cycles later. Remember this in mind and stay tuned on following.Generally, there exists two classical approaches to build the pre-existing network mod-els. One is to first construct networks using some rules, in addition to degree sequence,and then study their topological structures, including the degree sequence. Such as ER-model [12], WS-model [1], BA-model [2], NW-model [19]. The other is to generate anavailable network consistency with the degree sequence. From the perspective of theirappearance, the two processes mentioned above may be considered to be reversed. Thelatter is much more difficult theoretically than the former, it is likely for some networksto have identical degree sequence but to be in fact not isomorphic. In the previous work,some useful algorithms and methods have been provided to achieve this. According to4hree types of operations we defined, our goal is to construct growing random networks N ( p, q, r, t ), where p represents the probability of selecting rectangle operation, q standsfor the probability of choosing fractal operation, r denotes the probability of selectingtriangle operation. The degree of the vertex is the number of vertices of its neighbor set,represented by k . Let the | X | be the cardinality of set X .Taking useful advantage of three types of operations mentioned above, let us turn ourattention to constructing network space N ( p, q, r, t ). Above all, the initial network N (0) is a cycle C . Exploiting the three types of operationsaforementioned, one can easily get N ( p, q, r, t ) from N ( p, q, r, t −
1) for any time step t ≥ N ( p, q, r, t −
1) with probability p or employing type-B operation for each edge of network N ( p, q, r, t −
1) with probability q or applying type-C operation for each edge of network N ( p, q, r, t −
1) with probability r ,shown in Fig.2. As discussed in the previous literatures, after t time steps, there are manypossibilities for the member in our network space N ( p, q, r, t ). To make this paper muchself-contained, we simply find the upper and lower bounds of the topological structureof members in network space. In particular, our network space N ( p, q, r, t ) consists ofa subspace N ( p, q, , t ) at r = 0 or a single deterministic member at r = 1; at r = 0,namely p + q = 1 is always true, it becomes a subspace of N ( p, q, r, t ) whose certaintopological properties have been thoroughly investigated in [18]; at r = 1, that is to say, p + q = 0, it becomes a deterministic network whose several topological properties havebeen exhaustively studied in [20]. At 0 < r <
1, it contains numerous random networks,to which we will pay more attention.In order to better understand the properties of network N ( p, q, r, t ), we discuss certainrelated quantities that determine topological structures. On the basis of above descrip-tion, our goal is to research the topological structures of random networks with the tuningparameters, such as average degree, degree distribution, diameter, and clustering coeffi-cient. Firstly, we calculate two basic quantities such as the number of all vertices and edgesin N ( p, q, r, t ), named network order and size, respectively. According to three types ofoperations we defined, the order and size have the relationship in the below formula. | V ( p, q, r, t ) | = 2( p + q ) | E ( p, q, r, t − | + r | E ( p, q, r, t − | + | V ( p, q, r, t − || E ( p, q, r, t ) | = 4( p + q ) | E ( p, q, r, t − | + 3 r | E ( p, q, r, t − | ig.2. The diagram of operation implemented on an edge. In the process of constructing N ( p, q, r, t ) from N ( p, q, r, t −
1) ( t ≥ p represents the probability of applying type-Aoperation to each edge of N ( p, q, r, t − q represents the probability of applying type-Boperation to each edge of N ( p, q, r, t − r represents the probability of applying type-Coperation to each edge of N ( p, q, r, t − p + q + r = 1. It is not difficult to obtain the order and size of N ( p, q, r, t ) as follows | V ( p, q, r, t ) | = 4 × (2 p + 2 q + r ) (4 p + 4 q + 3 r ) t − p + 4 q + 3 r − × (2 − r ) (4 − r ) t − − r | E ( p, q, r, t ) | = 4 × (4 p + 4 q + 3 r ) t = 4 × (4 − r ) t The order and size of member N ( p, q, r, t ) in network space N ( p, q, r, t ) is a tuning valueby varying parameters t and r . Together with the definition of average degree, the averagedegree (cid:104) k (cid:105) of random network is the average value of all such vertex degree k i over theentire network, and is calculated as below (cid:104) k (cid:105) = 2 | E ( p, q, r, t ) || V ( p, q, r, t ) | = 2 × × (4 − r ) t × (2 − r ) (4 − r ) t − − r One can easily discover that (cid:104) k (cid:105) is also associated with t and r . In the limit of t , (cid:104) k (cid:105) = 2 × × (4 − r ) t × (2 − r ) (4 − r ) t − − r ≈ − r − r Obviously, it is not hard reveal that the (cid:104) k (cid:105) is independent of p, q , and completely deter-mined by r . 6 ig.3. The diagram of average degree with the probability parameter r , 0 ≤ r ≤
1, of networkspace N ( p, q, r, t ). It is clear for the eye that the curve illustrates the value of average degreechanging with probability parameter r . Because r has a value range from 0 to 1, put it another way, r ∈ [0 , r = 0,it means that N ( p, q, r, t ) only contains type-A operation ( p = 1) or merely containstype-B operation ( q = 1), or contains both of them ( p + q = 1 , < p, q < (cid:104) k (cid:105) of N ( p, q, r, t ), where (cid:104) k (cid:105) ≈ − r − r = − × − , is small andapproximately equal to 3. It should be mentioned that the network with same averagedegree have been discussed in[18]. We can note that when t is large enough, the resultingnetworks N ( p, q, r, t ) are sparse networks whose vertices have many fewer connectionsthan is possible.At r = 1, it indicates that N ( p, q, r, t ) contains neither type-A operation nor type-Boperation. In this particular case, the average degree (cid:104) k (cid:105) of N ( p, q, r, t ), where (cid:104) k (cid:105) ≈ − r − r = − × − , is close to 4. It is interesting to note that the identical average degree havebeen observed analytically in [13], Pseudofractal graphs [21], recursive graphs with q = 1[22], and Apollonian networks with d = 1 [23]. One can point out that when t tends toinfinite, the resulting network N ( p, q, r, t ) is a sparse network with fewer edges as possible.Based on our above analysis and Fig.3, we have the following proposition. Proposition 1
For any member N ( p, q, r, t ) of network space N ( p, q, r, t ), the averagedegree must hold the following inequality3 ≤ (cid:104) k (cid:105) = 6 − r − r ≤ . (3)From Eq.(3), it is worth noting that each r corresponds to a unique (cid:104) k (cid:105) . The Eq.(3),7lso, indicates that the value of average degree of the network with mix growth modeswill fall into a unique interval, where the lower and upper bounds of the interval are theminimum and maximum values of its constituent ingredients, respectively. Besides, theaverage degree of each member in network space is independent of the network size. Itimplies that the network order and network size both increases linearly with time t . Inother words, in the limits of t , each member of network space N ( p, q, r, t ) is a sparsenetwork independent of p, q , where r determines the sparsity. Degree distribution is one of most fundamental and important topological features of anetwork. It is a standard for judging whether the network is scale-free or not. Actually,our networks have scale-free property. The following analysis hold on this viewpoint.Now we analyze the degree distribution about random networks N ( p, q, r, t ), when anew vertex i is joined to the networks at a certain step t i ( t i ≥ i . It has beenfound that the vertex i has a degree of 2 no matter which operation is taken. We denoteby k i ( t ) the degree of vertex i at time t . From the construction process of network, thedegree k i ( t ) evolves with time as k i ( t ) = 2 k i ( t −
1) = 2 t +1 − t i , namely, the degree of vertex i is increased by a factor 2 at each time step.Considering the process of constructing network, we can see the degree spectrum ofnetwork in our space is a series of discrete real values. In order to relate the degreedistribution parameter of our network to the power-law exponent of a continuous degreedistribution for random scale-free networks, we take full advantage of the method proposedby Dorogovtsev[21], as follows P cum ( k ) = | V ( p, q, r, t i ) || V ( p, q, r, t ) | = 4 + 4 × (2 − r ) (4 − r ) ti − − r × (2 − r ) (4 − r ) t − − r where P cum ( k ) is the probability that the degree of a vertex in the network is greater than k i ( t ).So for large t , we omit the constant and get the formula as below. P cum ( k ) = 4 + 4 × (2 − r ) (4 − r ) ti − − r × (2 − r ) (4 − r ) t − − r ≈ (4 − r ) t i − t Due to k i ( t ) = 2 t − t i +1 , thus, substituting for t i using t i = t + 1 − ln k ln 2 , one can obtain P cum ( k ) ∼ k − γ where γ = 1 + ln(4 − r )ln 2 . 8 ig.4. The diagram of degree exponent with probability parameter, 0 ≤ r ≤
1, of networkspace N ( p, q, r, t ). Obviously, the curve indicates the degree exponent varies with theprobability parameter r . Therefore, the cumulative degree distribution P cum ( k ) of the random network followsa power-law form P cum ( k ) ∼ k − γ with the degree exponent γ = 1 + ln(4 − r )ln 2 . It is evidentthat γ is independent of p, q , as well associated with r .Thanks to r ∈ [0 , r = 0, it shows that N ( p, q, r, t ) not contain type-C operation.In this special circumstance, we can find that the network N ( p, q, r, t ) follows a scale-freerule P cum ( k ) ∼ k − γ with the degree index γ = 1 + ln(4 − r )ln 2 = 3. Note that the samedegree index has been obtained in the deterministic networks [13], classical BA networks[24] and typical random Sierpinksi networks [25]. However, our network is different fromBA networks and random Sierpinksi networks. It has been observed that our network isdeterministic while BA network and random Sierpinksi network are random.At r = 1, it displays that N ( p, q, r, t ) merely do type-C operation. In this particularcase, one can discover that the degree exponent γ of the network N ( p, q, r, t ) follows apower-law rule P cum ( k ) ∼ k − γ , where γ = 1 + ln(4 − = 1 + ln 3ln 2 . Apparently, 2 < γ =1+ ln 3ln 2 ≤
3, it indicates that the network N ( p, q, r, t ) is scale-free. It is notice that the samedegree exponent have been obtained in Pseudofractal graphs [21], recursive graphs with q = 2 [22], Apollonian networks with d = 2 [23], and deterministic scale-free networks[26],etc.Combining with previous analysis and Fig.4, it is easily to obtain the following propo-sition. Proposition 2
For any member N ( p, q, r, t ) of network space N ( p, q, r, t ), the power-9aw exponent of N ( p, q, r, t ) must satisfy the following inequality2 < ≤ γ = 1 + ln(4 − r )ln 2 ≤ . (4)From Eq.(4), we note that, γ = 1+ ln(4 − r )ln 2 , each r corresponds to a unique γ . Accordingto the expression of γ , the degree exponent is independent of p, q , and only affected by r , i.e., it illustrates that the value of γ varies with r . This result shows that the valueof power-law exponent of the network with mix growth patterns will fall into a restrictedscope, where the lower bound and the upper bound are the minimum and maximumvalues of its constituents, respectively. In addition, it is apparent that each member ofnetwork space N ( p, q, r, t ) is a scale-free network due to degree distribution is power-lawdistribution. It means that these members in network space most vertices have very lowdegrees and yet there are a few vertices having very high degrees. Consequently, thenumber of vertices that satisfy the condition k (cid:29) (cid:104) k (cid:105) is rare. This kind of complexnetworks is referred to as heterogeneous networks, where the high degree vertices are alsocalled hubs. It has been found that many real-world networks, including such typicalone as the Internet, WWW, metabolic networks, social networks, coauthorship networksetc., are power-law distribution with 2 < γ ≤
3. It is also worth remarking that thepresent framework can be included to networks with 2 < γ ≤
3, recovering qualitativelythe same results. In short, as t tends to infinite, all member of network space N ( p, q, r, t )are scale-free networks independent of p, q , where r determine the degree exponent. The small-world behavior describes the fact that there is a relatively short distance be-tween most pairs of vertices in most real-life networks. The distance between two verticesis the least number of edges to get from one vertex to other. The longest shortest path be-tween all pairs of vertices is called diameter, which is one of the major evaluation indexes.Diameter is itself a feature of graph structure and can be applied to characterize com-munication delay over a network. Typically, for coauthorship networks, diameter meansthat a social communication efficiency of author over the entire network. In general, thelarger diameter is, the lower communication efficiency is. Hence, computing the precisediameter can be done analytically and giving the solution process as below. The diameterof the network denoted by D ( p, q, r, t ), is defined to be the largest of all distances in thenetwork. Here, we will introduce the main idea of analysis.Before continuing, let us concentrate on three types of operations again, which help usto deduce iterative expression for the diameter of the network N ( p, q, r, t ).Clearly, according to the evolution of constructing the network, it is noticeable that thediameter D ( p, q, r, t ) of network N ( p, q, r, t ) must be based on the diameter D ( p, q, r, t − N ( p, q, r, t − D ( p, q, r, t − u u , ..., u D ( p,q,r,t − in network10 ( p, q, r, t − N ( p, q, r, t ), the key point is to chooseone of all operations. Now we discuss the calculation process of diameter in three cases.(a) If we choose the type-A operation, the diameter D ( p, q, r, t ) is obtained by D ( p, q, r, t − D ( p, q, r, t ) is equal to D ( p, q, r, t −
1) + 2, namely, it is availablefor all paths of various diameter in network N ( p, q, r, t − D ( p, q, r, t ) and D ( p, q, r, t − D ( p, q, r, t ) = D ( p, q, r, t −
1) + 2. Together with known term D ( p, q, r,
0) = 3, we shortly obtain aclosed-form formula of diameter D ( p, q, r, t ), i.e., D ( p, q, r, t ) = 2( t + 1) for all t ≥ | V ( p, q, r, t ) | ∼ ln 4 t +1 = ( t + 1) ln 4,we discover a connection between diameter D ( p, q, r, t ) and order | V ( p, q, r, t ) | under theapproximate relationship D ( p, q, r, t ) ∼ ln | V ( p, q, r, t ) | . Hence, more usually, as t tends toinfinite, the diameter D ( p, q, r, t ) has at most a logarithmic diameter with network order.(b) If we select the type-B operation, the diameter D ( p, q, r, t ) is derived from D ( p, q, r, t − D ( p, q, r, t ) and D ( p, q, r, t − D ( p, q, r, t ) = 2 D ( p, q, r, t − uv for a length 2 path which does contain the factor 2. Combining withthis initial condition D ( p, q, r,
0) = 3 and inferring a solution of diameter D ( p, q, r, t ), D ( p, q, r, t ) = 2 t +1 for all t ≥
2. It is unlikely that doing type-A operation on network N ( p, q, r, t − D ( p, q, r, t ) is not approximately equal to ln | V ( p, q, r, t ) | but fora squared value of order of network N ( p, q, r, t ), directly showing N ( p, q, r, t ) is large-scale.Obviously, when the order of network is large, the diameter is an exponential increasing.(c) If we take the type-C operation, one can deduce the diameter D ( p, q, r, t ) by D ( p, q, r, t − D ( p, q, r, t ) and D ( p, q, r, t − D ( p, q, r, t ) = D ( p, q, r, t −
1) + 2. Combined withinitial term D ( p, q, r,
0) = 3, we easily get a closed-form formula of diameter D ( p, q, r, t ),i.e., D ( p, q, r, t ) = 2( t + 1) for any t ≥
2. From another angle, there exists an resultthat ln | V ( p, q, r, t ) | ∼ ln 4 t +1 = ( t + 1) ln 4, we show an relationship between diameterand order, namely, D ( p, q, r, t ) ∼ ln | V ( p, q, r, t ) | . So, when t approaches to infinite, thediameter D ( p, q, r, t ) grows logarithmically with increasing network order.Thanks to the three cases of above analysis, we in theory compute the expected valueof D ( p, q, r, t ) of network N ( p, q, r, t ) on the basis of three cases. Because the probabilityof selecting each operation is p, q, r , respectively, it is easily to find that D ( p, q, r, t ) = p × ( D ( p, q, r, t −
1) + 2) + q × D ( p, q, r, t −
1) + r × ( D ( p, q, r, t −
1) + 2)= (1 + q ) × D ( p, q, r, t −
1) + 2 − q Together with initial condition, then, we have D ( p, q, r, t ) = (1 + q ) t × D ( p, q, r,
0) + (2 − q ) t Therefore, it is clear that D ( p, q, r, t ) is independent of p, r and relevant with q and t .11 ig.5. The diagram of diameter with constraints, 0 ≤ t ≤ ≤ q ≤
1, of network space N ( p, q, r, t ). It also demonstrates that the curve represents the value of diameter mainly relieson time t and probability parameter q . Because of q ∈ [0 , q = 0, it means that N ( p, q, r, t ) merely contains type-Aoperation, or just contains type-C operation, or contains both of them. In this case, wecan compute the diameter of networks N ( p, q, r, t ). D ( p, q, r, t ) = (1+0) t × D ( p, q, r, t ,then we get D ( p, q, r, t ) = 2( t + 1) which increase linearly with time for all t ≥
1. Notethat the same diameter has been obtained in the deterministic network[23]. It shouldbe mentioned that the diameters D ( p, q, r, t ) in our space that do not contain type-Boperation grow logarithmically with network order. This phenomena can be easily foundin a large quantity of networks, which indicates that network N ( p, q, r, t ) is like suchnetworks with small diameter.At q = 1, it implies that N ( p, q, r, t ) contain neither type-A operation nor type-Coperation. We obtain that D ( p, q, r, t ) = (1 + 1) t × D ( p, q, r, t − D ( p, q, r, t ) = 2 t +1 for all t ≥
2. It is noticeable that the diameter D ( p, q, r, t ) is increasingwith exponentially. It is different from case q = 0, diameter D ( p, q, r, t ) = 2 t +1 which is notlinear with network order but for a squared value of network order, directly indicating N ( p, q, r, t ) is large-scale. Apparently, when the network order is large, the diameterincrease exponentially.On account of aforementioned analysis and Fig.5, one can get the following proposition. Proposition 3
For any member N ( p, q, r, t ) of network space N ( p, q, r, t ), the diameterof N ( p, q, r, t ) must meet the following inequality2( t + 1) ≤ D ( p, q, r, t ) ≤ t +1 . (5)12rom Eq.(5), it is worth noticing that the diameter is mainly affected by q . This resultshows that the value of diameter of the network with mix growth modes will fall intoa restricted range, where the lower bound and the upper bound are the minimum andmaximum values of its constituent ingredients, respectively. For the member of networkspace, the diameter may abruptly transform a small value to a high value due to thetype-B operation. Together with our analysis, we discover that the diameters of thesemembers in our space that do not contain type-B operation grow linearly with time t , otherwise, the diameters increase exponentially. From another perspective, we canalso, say that the diameters of these members in our space that do not contain type-B operation grow logarithmically with network order, otherwise, the diameters increaseexponentially. It should be mentioned that diameter can be applied in many real-lifenetworks for characterizing the maximum communication delay over a network. Clustering is another vital property of a network, which provides measure of local structurewithin the network. The most immediate measure of clustering is the clustering coefficient C i for every vertex i . By definition, clustering coefficient of a vertex i is the ratio of thetotal number E i of edges that actually exist between all k i its nearest neighbors and thenumber of k i ( k i − / C i = 2 E i / [ k i ( k i − (cid:104) C (cid:105) of the entire network is average of all vertex C i . Now we willcompute the clustering coefficient of every vertex and their average value. As shown inprevious researches, most networks are highly transitive or clustered, i.e., a friend maybe two friends individually, who may then become acquainted with one another throughtheir common friend, and so end up friends themselves. “Transitivity” or “clustering” hasa different physical meanings in different networks, such as coauthorship networks andWWW. To better simulate the actual network, a subspace with non-clustering have beeninvestigated in our another paper[18]. Here, our goal is to research networks with tuningclustering coefficient by varying the value of r in the following.Due to type-A operation and type-B operation always forms a cycle C , while type-Coperation continuously generates a cycle C , it is evident that the clustering coefficient of C is always equal to 0 while the clustering coefficient of C is 1. It shows that merely type-C operation increase the clustering coefficient of the entire network. Put it another way, r has a decisive influence on the clustering coefficient of the whole network. Intentionally,there are two factors that have an impact on the average clustering coefficient of network N ( p, q, r, t ), the first is the value of r , and the second is that the total number of times ofperforming type-C operation at time t . Now we introduce the main ideas and the processin detail.Because of the three case of aforementioned description, we in theory compute theaverage degree of (cid:104) c (cid:105) of network N ( p, q, r, t ). We divide into 2 steps.Firstly, we find the clustering coefficient for each vertex in N ( p, q, r, t ), we can derive13 closed formula for the clustering coefficient C and list in the below tableTable-I the clustering coefficient of c i of vertices degree k i k i · · · t − t t +1 c i
13 314 · · · (2 t − − t − (2 t − −
1) 2 t − t − (2 t −
1) 2 t +1 − t (2 t +1 − Secondly, we calculate the proportion of the vertices with a clustering coefficient of c i innetwork N ( p, q, r, t ). In order to obtain the clustering coefficient (cid:104) c (cid:105) of the entire network N ( p, q, r, t ), it is necessary for us to give the degree distribution spectrum, namely, theprobability p i ( p i = n ki ( t ) | V ( p,q,r,t ) | ) of vertices degree k i is shown in the following tableTable-II the degree spectrum of degree k i k i · · · t − t t +1 p i − r − r − r (4 − r ) − r (4 − r ) · · · − r (4 − r ) t − − r (4 − r ) t − r (4 − r ) t On the basis of above detailed discussion, it is easily to obtain the clustering coefficient (cid:104) c (cid:105) of the entire network-model is (cid:104) c (cid:105) = r (cid:88) p i c i (6)Therefore, it seems that (cid:104) c (cid:105) is independent of p, q and have a relationship with r .Owing to r ∈ [0 , r = 0, this limiting case of network hasa self-similar structure that allows one to calculate the exact value C analytically. Sincethe network N ( p, q, r, t ) not contain type-C operation, there no exists triangle, and then N ( p, q, r, t ) have no clustering. So the clustering coefficient of every vertex in N ( p, q, r, t )is zero. In this situation, the average value of clustering coefficient in N ( p, q, r, t ) is alwaysequal to 0. The clustering coefficient of whole network is always zero.For second special case with r = 1, this limiting case of network N ( p, q, r, t ) only con-tains type-C operation. Substituting r into the Eq.(6), it is noticeable that the clusteringcoefficient is approximately to 0.7566. Apparently, when the parameters r is larger, theclustering coefficient is higher.As mentioned previously, it can be get the following proposition. Proposition 4
For any member N ( p, q, r, t ) of network space N ( p, q, r, t ), the cluster-ing coefficient must hold the following inequality0 ≤ (cid:104) c (cid:105) = r (cid:88) p i c i ≤ . ≤ r ≤
1, it is worthwhile to studythe processes taking place upon the network to discover the different effects on dynamicprocesses. As we all known, if we give a fixed value of r , there exists a unique cluster-ing coefficient (cid:104) c (cid:105) corresponding to r . According to our operations mentioned above, itshows that type-C operation always have an effect on clustering coefficient of network14 ( p, q, r, t ). This result shows that the value of clustering coefficient of the network withmix growth modes will fall within a restricted range, where the lower bound and theupper bound are the minimum and maximum values of its constituents, respectively. Formember of network space N ( p, q, r, t ), the clustering coefficient range 0 from 0.7566. We have introduced three types of operations to produce our network N ( p, q, r, t ). In termsof our operations, we have shown the detailed process of generating our network, whichconstitute the network space N ( p, q, r, t ), and then discussed their topological propertiessuch as, average degree, degree distribution, diameter, and clustering coefficient. Aboveall, we have computed the average degree of each member in our network space and dis-covered that all member are sparse network. Secondly, we have proven that each memberin our space follows the power-law distribution with degree exponent γ = 1 + ln(4 − r )ln 2 ,which means that all networks are always scale-free except that the degree exponent isdifferent.Hereafter, we have analyzed the diameters of several members in our network space,where the diameters of some members may suddenly transform from a small value toa large value. The reason is that the type-B operation is a fractal operation, if eachtype-B operation is performed, the value of the diameter increase exponentially. In thiscircumstances, the diameter is so large that the communication efficiency is low. Putit another way, by setting the parameters q = 0, one can make the diameter of certainnetworks in our space become relatively small.Eventually, we have studied the clustering coefficient in our network space, and dis-covered that the clustering coefficient of some member may abruptly leap to a high valuefrom 0, while others may gradually increase from 0. Because the type-C operation is atriangle operation, each time type-C operation is performed, the value of the clusteringcoefficient will increase, that is to say, by effectively varying the parameters r , one canmake the clustering coefficient of certain networks in our space become relatively higher.Therefore, small diameter network can be obtained with q = 0, while high clusteringcoefficient network can be obtained with larger r . Combined with small-world effect andour analysis, for the q = 0 and high r case, we have found that these network satisfyingthe above case have small diameter and high clustering coefficient, that is, it indicatesthat those networks in our network space are small-world networks.It is clear that the network in our network space serves as a better feature by chang-ing the probability parameters, for example, we can obtain a small-world network withsmall diameter and high clustering coefficient by setting q = 0 and higher r . Especially,the clustering coefficient may abruptly transform from zero to high value. Accomplish-ment notwithstanding, research on complex networks is far from enough and requireslong-term long-sustainable endeavor. New discoveries, developments, enhancements, and15mprovements are still needed. In the future, we will devote more efforts to investigatecomplex networks in order to better help people understand and apply it to explain somephenomena in real life. Acknowledgment
This research was supported by the National Natural Science Foundation of China undergrants No. 61163054, No. 61363060 and No. 61662066.