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since the pioneering papers by watts and strogatz on small - world networks @xcite and barabsi and albert on scale - free networks @xcite , complex networks , which describe many systems in nature and society , have become an area of tremendous recent interest @xcite . in the last few years , modeling real - life systems has attracted an exceptional amount of attention within the physics community . while a lot of models have been proposed , most of them are stochastic @xcite . however , because of their advantages , deterministic networks have also received much attention  @xcite . first , the method of generating deterministic networks makes it easier to gain a visual understanding of how networks are shaped , and how do different nodes relate to each other  @xcite ; moreover , deterministic networks allow to compute analytically their properties : degree distribution , clustering coefficient , average path length , diameter , betweenness , modularity and adjacency matrix whose eigenvalue spectrum characterizes the topology  @xcite . the first model for deterministic scale - free networks was proposed by barabsi _ _ in ref . @xcite and was intensively studied in ref . @xcite . another elegant model , called pseudofractal scale - free web ( psw )  @xcite , was introduced by dorogovtsev , goltsev , and mendes , and was extended by comellas _ .  @xcite . based on a similar idea of psw , jung _ et al . _ presented a class of recursive trees  @xcite . additionally , in order to discuss modularity , ravasz _ et al . _ proposed a hierarchical network model  @xcite , the exact scaling properties and extensive study of which were reported in refs . @xcite and  @xcite , respectively . recently , in relation to the problem of apollonian space - filing packing , andrade _ et al . _ introduced apollonian networks  @xcite which were also proposed by doye and massen in ref . @xcite and have been intensively investigated @xcite . in addition to the above models , deterministic networks can be created by various techniques : modification of some regular graphs  @xcite , addition and product of graphs  @xcite , edge iterations  @xcite and other mathematical methods as in refs . @xcite .    as mentioned by barabsi _ _ , it would be of major theoretical interest to construct deterministic models that lead to scale - free networks  @xcite . here we do an extensive study on pseudofractal scale - free web  @xcite . the psw can be considered as a process of edge multiplication . in fact , a clique ( edge is a special case of it ) can also reproduce new cliques and the number of the new reproduction may be different at a time . motivated by this , in a simple recursive way we propose a general model for psw by including two parameters , with psw as a particular case of the present model . the deterministic construction of our model enables one to obtain the analytic solutions for its structure properties . by adjusting the parameters , we can obtain a variety of scale - free networks . before introducing our model we give the following definitions on a graph ( network ) . the term _ size _ refers to the number of edges in a graph . the number of nodes in a graph is called its _ order_. when two nodes of a graph are connected by an edge , these nodes are said to be _ adjacent _ , and the edge is said to join them . complete graph _ is a graph in which all nodes are adjacent to one another . thus , in a complete graph , every possible edge is present . the complete graph with @xmath1 nodes is denoted as @xmath2 ( also referred in the literature as @xmath1-_clique _ ) . two graphs are _ isomorphic _ when the nodes of one can be relabeled to match the nodes of the other in a way that preserves adjacency . so all @xmath1-cliques are isomorphic to one another .     and @xmath3 . only the first three steps are shown.,width=491 ]    the network is constructed in a recursive way . we denote the network after @xmath4 steps by @xmath5 , @xmath6 ( see fig . [ recursive ] ) . then the network at step @xmath4 is constructed as follows : for @xmath7 , @xmath8 is a complete graph @xmath9 ( or @xmath10-clique ) consist of @xmath11 @xmath1-cliques ) , and @xmath8 has @xmath11 nodes and @xmath12 edges . for @xmath13 , @xmath5 is obtained from @xmath14 by adding @xmath15 new nodes for each of its existing subgraphs isomorphic to a @xmath1-clique , and each new node is connected to all the nodes of this subgraph . in the special case @xmath16 and @xmath3 , it is reduced to the pseudofractal scale - free web described in ref . @xcite . in the limiting case of @xmath16 , we obtain the same networks as in ref . @xcite . however , our family is richer as @xmath15 can take any natural value . there is an interpretation called ` aggregation ' @xcite for our model . as an example , here we only explain them for the case of @xmath16 and @xmath3 . figure [ pseudofractal ] illustrates the growing process for this particular case , which may be accounted for as an ` aggregation ' process described in detail as follows . first , three of the initial triangle ( @xmath7 ) are assembled to form a new unit ( @xmath17 ) . then we assemble three of these units at the hubs ( the nodes with highest degree ) in precise analogy with the step leading from @xmath7 to @xmath17 to form a new cell ( @xmath18 ) ( see fig . [ aggregation ] ) . this process can be iterated an arbitrary number of times . moreover , an alternative explanation of our model which is often useful is that of ` miniaturization ' ( see ref . @xcite ) .     and @xmath3 ) , exhibiting the first three steps.,width=453 ]     to @xmath19 , which is obtained by adjoining of three copies of @xmath20 at the hubs.,width=453 ] below we will find that the tunable parameters @xmath15 and @xmath1 control some relevant characteristics of the network @xmath5 . because @xmath3 is a particular case , for conveniences , we treat @xmath3 and @xmath21 separately . _ order and size . _ in the case of @xmath3 , we denote @xmath5 by @xmath22 . let us consider the total number of nodes @xmath23 and total number of edges @xmath24 in @xmath22 . denote @xmath25 as the number of nodes created at step @xmath4 . note that the addition of each new node leads to two new edges . by construction , for @xmath26 , we have @xmath27 and @xmath28 considering the initial condition @xmath29 and @xmath30 , it follows that @xmath31 then the number of nodes increases with time exponentially and the total number of nodes present at step @xmath4 is @xmath32 thus for large @xmath4 , the average degree @xmath33 is approximately @xmath34 . _ degree distribution . _ let @xmath35 be the degree of node @xmath36 at step @xmath4 . then by construction , it is not difficult to find following relation : @xmath37 which expresses a preference attachment @xcite . if node @xmath36 is added to the network at step @xmath38 , @xmath39 and hence @xmath40 therefore , the degree spectrum of the network is discrete . it follows that the degree distribution is given by @xmath41 and that the cumulative degree distribution @xcite is @xmath42 substituting for @xmath38 in this expression using @xmath43 gives @xmath44 so the degree distribution follows the power law with the exponent @xmath45 . for the particular case of @xmath16 , eq . ( [ gamma1 ] ) recovers the result previously obtained in ref . @xcite . _ second moment of degree distribution . _ let us calculate the second moment of degree distribution @xmath46 . it is defined by @xmath47^{2},\end{aligned}\ ] ] where @xmath48 is the degree of a node at step @xmath4 , which was generated at step @xmath38 . this quality expresses the average of degree square over all nodes in the network . it has large impact on the dynamics of spreading @xcite and the onset of percolation transitions @xcite taking place in networks . when @xmath46 is diverging , the networks allow the onset of large epidemics whatever the spreading rate of the infection @xcite , at the same time the networks are extremely robust to random damages , in other words , the percolation transition is absent @xcite .    substituting eqs . ( [ nv1 ] ) , ( [ nt1 ] ) and ( [ ki1 ] ) into eq . ( [ ki21 ] ) , we derive @xmath49\nonumber\\ & \approx&\frac{8\,(m+1)^{2t+1}}{m(2m+1)}\rightarrow\infty \qquad\hbox{for large $ t$.}\end{aligned}\ ] ] in this way , second moment of degree distribution @xmath50 has been calculated explicitly , and result shows that it diverges as an exponential law . so the networks are resilient to random damage and are simultaneously sensitive to the spread of infections . _ degree correlations . _ as the field has progressed , degree correlation @xcite has been the subject of particular interest , because it can give rise to some interesting network structure effects . an interesting quantity related to degree correlations is the average degree of the nearest neighbors for nodes with degree @xmath51 , denoted as @xmath52 @xcite . when @xmath52 increases with @xmath51 , it means that nodes have a tendency to connect to nodes with a similar or larger degree . in this case the network is defined as assortative @xcite . in contrast , if @xmath53 is decreasing with @xmath51 , which implies that nodes of large degree are likely to have near neighbors with small degree , then the network is said to be disassortative . if correlations are absent , @xmath54 . we can exactly calculate @xmath55 for the networks using eq . ( [ ki1 ] ) to work out how many links are made at a particular step to nodes with a particular degree . except for three initial nodes generated at step 0 , no nodes born in the same step , which have the same degree , will be linked to each other . all links to nodes with larger degree are made at the creation step , and then links to nodes with smaller degree are made at each subsequent steps . this results in the expression @xmath56 for @xmath57 . here the first sum on the right - hand side accounts for the links made to nodes with larger degree ( i.e.@xmath58 ) when the node was generated at @xmath38 . the second sum describes the links made to the current smallest degree nodes at each step @xmath59 .    substituting eqs . ( [ nv1 ] ) and ( [ ki1 ] ) into eq . ( [ knn1 ] ) , after some algebraic manipulations , eq . ( [ knn1 ] ) is simplified to @xmath60^{t_i}-\frac{2(m+1)}{m}+\frac{2m}{m+1}\,(t - t_i).\end{aligned}\ ] ] thus after the initial step @xmath55 grows linearly with time .    writing eq . ( [ knn2 ] ) in terms of @xmath51 , it is straightforward to obtain @xmath61^{t}\,\left ( \frac{k}{2}\right)^{-\frac{\ln\left [ \frac{(m+1)^{2}}{2m+1}\right ] } { \ln(m+1)}}\nonumber\\ \qquad\qquad\qquad\qquad-\frac{2(m+1)}{m}+\frac{2m}{m+1}\,\frac{\ln(\frac{k}{2})}{\ln(m+1)}.\end{aligned}\ ] ] therefore , @xmath52 is approximately a power law function of @xmath51 with negative exponent , which shows that the networks are disassortative . note that @xmath52 of the internet exhibit a similar power - law dependence on the degree @xmath62 , with @xmath63 @xcite . _ clustering coefficient . _ the clustering coefficient defines a measure of the level of cohesiveness around any given node . by definition , the clustering coefficient  @xcite @xmath64 of node @xmath36 is the ratio between the number of edges @xmath65 that actually exist among the @xmath66 neighbors of node @xmath36 and its maximum possible value , @xmath67 , i.e. , @xmath68 . the clustering coefficient of the whole network is the average of all individual @xmath69 . next we will compute the clustering coefficient of every node and their average value . obviously , when a new node @xmath36 joins the network , its degree @xmath70 and @xmath71 is @xmath72 and @xmath73 , respectively . each subsequent addition of a link to that node increases both @xmath70 and @xmath71 by one . thus , @xmath71 equals to @xmath74 for all nodes at all steps . so one can see that , there is a one - to - one correspondence between the degree of a node and its clustering . for a node with degree @xmath51 , the exact expression for its clustering coefficient is @xmath75 . therefore , the clustering coefficient spectrum of nodes is discrete . using this discreteness , it is convenient to work with the cumulative distribution of clustering coefficient  @xcite as @xmath76 it is worth noting that for the special case of @xmath16 , this result has been obtained previously  @xcite . the clustering coefficient of the whole network at arbitrary step @xmath4 can be easily computed , @xmath77}.\end{aligned}\ ] ] in the infinite network size limit ( @xmath78 ) , @xmath79 . thus the clustering @xmath80 is high and increases with @xmath15 . moreover , similarly to the degree exponent @xmath81 , @xmath80 is tunable by choosing the right value of parameter @xmath15 : in particular , @xmath80 ranges from @xmath82 ( in the special case of @xmath16  @xcite ) to limit of 1 when @xmath15 becomes very large . _ diameter . _ the diameter of a network is defined as the maximum of the shortest distances between all pairs of nodes , which characterizes the longest communication delay in the network . small diameter is consistent with the concept of small - world and it is easy to compute for our networks . below we give the precise analytical computation of diameter of @xmath22 denoted by @xmath83 . it is easy to see that at step @xmath84 ( resp . @xmath85 ) , the diameter is equal to 1 ( resp . 2 ) . at each step @xmath86 , one can easily see that the diameter always lies between a pair of nodes that have just been created at this step . in order to simplify the analysis , we first note that it is unnecessary to look at all the nodes in the networks in order to find the diameter . in other words , some nodes added at a given step can be ignored , because they do not increase the diameter from the previous step . these nodes are those that connect to edges that already existed before step @xmath87 . indeed , for these nodes we know that a similar construction has been done in previous steps , so we can ignore them for the computation of the diameter . let us call `` outer '' nodes the nodes which are connected to a edge that did not exist at previous steps . clearly , at each step , the diameter depends on the distances between outer nodes .    at any step @xmath86 , we note that an outer node can not be connected with two or more nodes that were created during the same step @xmath88 . indeed , we know that from step @xmath72 , no outer node is connected to two nodes of the initial triangle @xmath89 . thus , for any step @xmath86 , any outer node is connected with nodes that appeared at pairwise different steps . now consider two outer nodes created at step @xmath86 , say @xmath90 and @xmath91 . then @xmath90 is connected to two nodes , and one of them must have been created before or during step @xmath92 . we repeat this argument , and we end up with two cases : ( 1 ) @xmath93 is even . then , if we make @xmath15  jumps " , from @xmath90 we reach the initial triangle @xmath89 , in which we can reach any @xmath91 by using an edge of @xmath89 and making @xmath15 jumps to @xmath91 in a similar way . thus @xmath94 . ( 2 ) @xmath95 is odd . in this case we can stop after @xmath15 jumps at @xmath96 , for which we know that the diameter is 2 , and make @xmath15 jumps in a similar way to reach @xmath91 . thus @xmath97 . it is easily seen that the bound can be reached by pairs of outer nodes created at step @xmath4 . more precisely , those two nodes @xmath90 and @xmath91 share the property that they are connected to two nodes that appeared respectively at steps @xmath87 , @xmath92 .    hence , formally , @xmath98 for any @xmath99 . note that @xmath100 , thus the diameter is small and scales logarithmically with the number of network nodes .      in these cases , the analysis is a little difficult than those of the last subsection . an alternative approach has to be adopted , although it may also holds true for the first case in some situations . the method of the last subsection is relatively easy to generalize to these cases , and below we will address it , focusing on order , size , degree distribution , clustering coefficient and diameter . _ order and size . _ let @xmath25 , @xmath101 be the number of nodes and edges created at step @xmath4 , respectively . denote @xmath102 as the total number of @xmath1-cliques in the whole network at step @xmath4 . note that the addition of each new node leads to @xmath1 new @xmath1-cliques and @xmath1 new edges . by construction , we have @xmath103 , @xmath104 and @xmath105 . thus one can easily obtain @xmath106 ( @xmath99 ) , @xmath107 ( @xmath108 ) and @xmath109 ( @xmath108 ) . from above results , we can easily compute the order and size of the networks . the total number of nodes @xmath23 and edges @xmath24 present at step @xmath4 is @xmath110}{q}\end{aligned}\ ] ] and @xmath111 respectively . for infinite @xmath4 , the average degree @xmath33 is approximately @xmath112 . _ degree distribution . _ when a new node @xmath36 is added to the graph at step @xmath38 , it has degree @xmath1 and forms @xmath1 new @xmath1-cliques . let @xmath113 be the total number of @xmath1-cliques at step @xmath4 that will created new nodes connected to the node @xmath36 at step @xmath114 . so at step @xmath38 , @xmath115 . by construction , we can see that in the subsequent steps each new neighbor of @xmath36 generates @xmath116 new @xmath1-cliques with @xmath36 as one node of them . let @xmath35 be the degree of @xmath36 at step @xmath4 . it is not difficult to find following relations for @xmath117 : @xmath118 and @xmath119 from the above two equations , we can derive @xmath120n_q(i , t-1)$ ] . considering @xmath115 , we obtain @xmath121^{t - t_i}$ ] and @xmath122^{t - t_i-1}$ ] . then the degree @xmath123 of node @xmath36 at time @xmath4 is @xmath124^{t - t_i}+q^{2}-2q}{q-1}.\end{aligned}\ ] ] since the degree of each node has been obtained explicitly as in eq . ( [ ki ] ) , we can get the degree distribution via its cumulative distribution @xcite , i.e. @xmath125 , where @xmath126 denotes the number of nodes with degree @xmath127 . the analytic computation details are given as follows . for a degree @xmath51 @xmath128^{t - j}+q^{2}-2q}{q-1},\ ] ] there are @xmath129 nodes with this exact degree , all of which were born at step @xmath130 . all nodes with birth time at @xmath130 or earlier have this and a higher degree . so we have @xmath131}{q}. \nonumber\\\ ] ] as the total number of nodes at step @xmath4 is given in eq . ( [ nt ] ) , we have @xmath132^{t - j}+q^{2}-2q}{q-1}\right]^{1-\gamma}\nonumber\\ = \frac{\frac{(q+1)[(mq+1)^{j}+q-1]}{q}}{\frac{(q+1)[(mq+1)^{t}+q-1]}{q}}.\nonumber\\\ ] ] therefore , for large @xmath4 we obtain @xmath133^{t - j}\right ] ^{1-\gamma}=(mq+1)^{j - t}\ ] ] and @xmath134}.\ ] ] for the special case @xmath16 , eq . ( [ gamma ] ) recovers the results previously reported in ref . @xcite . _ clustering coefficient_. the analytical expression for clustering coefficient @xmath135 of the individual node with degree @xmath51 can be derived exactly . when a node is created it is connected to all the nodes of a @xmath1-clique whose nodes are completely interconnected . its degree and clustering coefficient are @xmath1 and 1 , respectively . in the following steps , if its degree increases one by a newly created node connecting to it , then there must be @xmath116 existing neighbors of it attaching to the new node at the same time . thus for a node of degree @xmath51 , we have @xmath136 which depends on degree @xmath51 and @xmath1 . for @xmath137 , the @xmath135 is inversely proportional to node degree . the scaling @xmath138 has been found for some network models  @xcite , and has also observed in several real - life networks  @xcite .    using eq . ( [ ck ] ) , we can obtain the clustering @xmath139 of the networks at step @xmath4 : @xmath140 where the sum is the total of clustering coefficient for all nodes and @xmath141^{t - r}+q^{2}-2q}{q-1}$ ] shown by eq . ( [ ki ] ) is the degree of the nodes created at step @xmath142 .     on @xmath1 and @xmath15.,width=340 ] it can be easily proved that for arbitrary fixed @xmath15 , @xmath139 increases with @xmath1 , and that for arbitrary fixed @xmath1 , @xmath139 increases with @xmath15 . in the infinite network order limit ( @xmath143 ) , eq . ( [ ac ] ) converges to a nonzero value @xmath144 . when @xmath3 , for @xmath16 , 2 , 3 and 4 , @xmath144 equal to 0.8000 , 0.8571 0.8889 and 0.9091 , respectively . when @xmath145 , for @xmath3 , 3 , 4 and 5 , @xmath144 are 0.8571 , 0.9100 , 0.9348 and 0.9490 , respectively . therefore , the clustering coefficient of our networks is very high . moreover , similarly to the degree exponent @xmath81 , clustering coefficient @xmath144 is determined by @xmath1 and @xmath15 . figure [ cc ] shows the dependence of @xmath144 on @xmath1 and @xmath15 . _ diameter . _ in what follows , the notations @xmath146 and @xmath147 express the integers obtained by rounding @xmath148 to the nearest integers towards infinity and minus infinity , respectively . now we compute the diameter of @xmath5 , denoted @xmath149 for @xmath150 ( @xmath3 is a particular case that is treated separately in the last subsection ) :    _ step 0_. the diameter is @xmath73 . _ steps 1 to @xmath151_. in this case , the diameter is 2 , since any new node is by construction connected to a @xmath1-clique forming a @xmath10-clique , and since any @xmath10-clique during those steps contains at least @xmath152 ( @xmath1 even ) or @xmath152 + 1 ( @xmath1 odd ) nodes from the initial @xmath10-clique @xmath8 obtained after step 0 . hence , any two newly added nodes @xmath153 and @xmath154 will be connected respectively to sets @xmath155 and @xmath156 , with @xmath157 and @xmath158 , where @xmath159 is the node set of @xmath8 ; however , since @xmath160 ( @xmath1 even ) and @xmath161 + 1 ( @xmath1 odd ) , where @xmath162 denotes the number of elements in set @xmath163 , we conclude that @xmath164 , and thus the diameter is 2 . _ steps @xmath152 to @xmath1_. in any of these steps , some newly added nodes might not share a neighbor in the original @xmath10-clique @xmath8 ; however , any newly added node is connected to at least one node of the initial @xmath10-clique @xmath8 . thus , the diameter is equal to 3 .    _ further steps_. similar to the case of @xmath3 , we call `` outer '' nodes the nodes which are connected to a @xmath1-clique that did not exist at previous steps . clearly , at each step , the diameter depends on the distances between outer nodes . now , at any step @xmath165 , an outer node can not be connected with two or more nodes that were created during the same step @xmath166 . moreover , by construction no two nodes that were created during a given step are neighbors , thus they can not be part of the same @xmath1-clique . therefore , for any step @xmath165 , some outer nodes are connected with nodes that appeared at pairwise different steps . thus , if @xmath167 denotes an outer node that was created at step @xmath4 , then @xmath167 is connected to nodes @xmath168s , @xmath169 , where all the @xmath36s are pairwise distinct . we conclude that @xmath167 is necessarily connected to a node that was created at a step @xmath170 . if we repeat this argument , then we obtain an upper bound on the distance from @xmath167 to the initial @xmath10-clique @xmath8 . let @xmath171 , where @xmath172 . then , we see that @xmath167 is at distance at most @xmath173 from a node in @xmath5 . hence any two nodes @xmath167 and @xmath174 in @xmath8 lie at distance at most @xmath175 ; however , depending on @xmath176 , this distance can be reduced by 1 , since when @xmath177 , we know that two nodes created at step @xmath176 share at least a neighbor in @xmath8 . thus , when @xmath178 , @xmath179 , while when @xmath180 , @xmath181 . one can see that these bounds can be reached by pairs of outer nodes created at step @xmath4 . more precisely , those two nodes @xmath167 and @xmath174 share the property that they are connected to @xmath1 nodes that appeared respectively at steps @xmath182 .    based on the above arguments , one can easily see that for @xmath183 , the diameter increases by 2 every @xmath1 steps . more precisely , we have the following result , for any @xmath150 and @xmath13 ( when @xmath7 , the diameter is clearly equal to 1 ) : @xmath184 where @xmath185 if @xmath186 , and 1 otherwise . when @xmath4 gets large , @xmath187 , while @xmath188 , thus the diameter grows logarithmically with the number of nodes . it is easy to see that these cases of @xmath21 have very similar topological properties to the case @xmath3 . additionally , for the cases of @xmath21 , the networks will again be disassortative with respect to degree because of the lack of links between nodes with the same degree ; the second moment of degree distribution @xmath50 will also diverge , which is due to the fat tail of the degree distribution . to sum up , we have proposed and investigated a deterministic network model , which is constructed in a recursive fashion . our model is actually a tunable generalization of the growing deterministic scale - free networks introduced in ref . @xcite . aside from their deterministic structures , the statistical properties of the resulting networks are equivalent with the random models that are commonly used to generate scale - free networks @xcite . we have obtained the exact results for degree distribution and clustering coefficient , as well as the diameter , which agree well with large amount of real observations @xcite . the degree exponent can be adjusted , the clustering coefficient is very large , and the diameter is small . therefore , out model may perform well in mimicking a variety of scale - free networks in real - life world . moreover , our networks consist of cliques , which has been observed in variety of the real - world networks , such as movie actor collaboration networks , scientific collaboration networks and networks of company directors @xcite . this research was supported in part by the national natural science foundation of china ( nnsfc ) under grant nos . 60373019 , 60573183 , and 90612007 . lili rong gratefully acknowledges partial support from nnsfc under grant nos . 70431001 and 70571011 . the authors thank the anonymous referees for their valuable comments and suggestions .
we propose a general geometric growth model for pseudofractal scale - free web , which is controlled by two tunable parameters . 
 we derive exactly the main characteristics of the networks : degree distribution , second moment of degree distribution , degree correlations , distribution of clustering coefficient , as well as the diameter , which are partially determined by the parameters . 
 analytical results show that the resulting networks are disassortative and follow power - law degree distributions , with a more general degree exponent tuned from 2 to @xmath0 ; the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order ; the diameter grows logarithmically with the number of network nodes . 
 all these reveal that the networks described by our model have small - world effect and scale - free topology . 
 complex networks , scale - free networks , disordered systems , networks