document: today , with a vast amount of publications being produced in every discipline of scientific research , it can be rather overwhelming to select a good quality work ; that is enriched with original ideas and relevant to scientific community . 
   more often this type of publications are discovered through the citation mechanism . 
   it is believed that an estimate measure for scientific credibility of a paper is the number of citations that it receives , though this should not be taken too literally since some publications may have gone unnoticed or have been forgotten about over time . 
   knowledge of how many times their publications are cited can be seen as good feedback for the authors , which brings about an unspoken demand for the statistical analysis of citation data . 
   one of the impressive empirical studies on citation distribution of scientific publications @xcite showed that the distribution is a power - law form with exponent @xmath0 . 
   the power - law behaviour in this complex system is a consequence of highly cited papers being more likely to acquire further citations . 
   this was identified as a _ preferential attachment _ process in @xcite .    the citation distribution of scientific publications is well studied and 
   there exist a number of network models @xcite to mimic its complex structure and empirical results @xcite to confirm predictions . 
   however , they seem to concentrate on the total number of citations without giving information about the issuing publications . 
   the scientific publications belonging to a particular research area do not restrict their references to that discipline only , they form bridges by comparing or confirming findings in other research fields . 
   for instance most _ small world network models _ 
   @xcite presented in statistical mechanics , reference a sociometry article @xcite which presents the studies of milgram on the small world problem . 
   this is the type of process which we will investigate with a simple model that only considers two research areas and referencing within and across each other . 
   the consideration of cross linking also makes the model applicable to _ the web of human sexual contacts _ 
   @xcite , where the interactions between males and females can be thought of as two coupled growing networks . 
   this paper is organized as follows : in the proceeding section the model is defined and analyzed with a rate equation approach @xcite . in the final section discussions and comparisons of findings with the existing data are presented . 
   one can visualize the proposed model with the aid of fig . 
   ( [ coupled ] ) that attempts to illustrate the growth mechanism . 
   we build the model by the following considerations .    initially , both networks @xmath1 and @xmath2 contains @xmath3 nodes with no cross - links between the nodes in the networks . 
   at each time step two new nodes with no incoming links , one belonging to network @xmath1 and the other to @xmath2 , are introduced simultaneously . 
   the new node joining to @xmath1 with @xmath4 outgoing links , attaches @xmath5 fraction of its links to pre - existing nodes in @xmath1 and @xmath6 fraction of them to pre - existing nodes in @xmath2 . 
   the similar process takes place when a new node joins to @xmath2 , where the new node has @xmath7 outgoing links from which @xmath8 of them goes to nodes in @xmath2 and the complementary @xmath9 goes to @xmath1 . 
   the attachments to nodes in either networks are preferential and the rate of acquiring a link depends on the number of connections and the initial attractiveness of the pre - existing nodes . 
   we define @xmath10 as the average number of nodes with total @xmath11 number of connections that includes the incoming intra - links @xmath12 and the incoming cross - links @xmath13 in network @xmath1 at time @xmath14 . 
   similarly , @xmath15 is the average number of nodes with @xmath16 connections at time @xmath14 in network @xmath2 . 
   notice that the indices are discriminative and the order in which they are used is important , as they indicate the direction that the links are made . 
   further more we also define @xmath17 and @xmath18 the average number of nodes with @xmath12 and @xmath19 incoming intra - links to @xmath1 and @xmath2 respectively . 
   finally , we also have @xmath20 and @xmath21 to denote the average number of nodes in @xmath1 and @xmath2 with @xmath13 and @xmath22 incoming cross - links . 
   to keep this paper less cumbersome we will only analyse the time evolution of network @xmath1 and apply our results to network @xmath2 . 
   in addition to this , we only need to give the time evolution of @xmath23 , defined as the joint distribution of intra - links and cross - links . using this distribution 
   we can find all other distributions that are mentioned earlier . 
   the time evolution of @xmath23 can be described by a rate equation @xmath24\nonumber\\ & & + p_{ba}m_{b}[(k_{aa}+k_{ba}-1+a)n_{a}(k_{aa},k_{ba}-1,t)\nonumber\\ & & -(k_{aa}+k_{ba}+a)n_{a}(k_{aa},k_{ba},t)]\}+ \delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ] the form of the eq . 
   ( [ na ] ) seems very similar to the one used in @xcite . in that model 
   the rate of creating links depends on the out - degree of the issuing nodes and the in - degree of the target nodes . 
   here we are concerned with two different types of in - degrees namely intra- and cross - links of the nodes .    on the right hand side of eq . 
   ( [ na ] ) the terms in first square brackets represent the increase in the number of nodes with @xmath11 links when a node with @xmath25 intra - links acquires a new intra - link and if the node already has @xmath11 links this leads to reduction in the number . 
   similarly , for the second square brackets where the number of nodes with @xmath11 links changes due to the incoming cross - links . 
   the final term accounts for the continuous addition of new nodes with no incoming links , each new node could be thought of as the new publication in a particular research discipline . 
   the normalization factor @xmath26 sum of all degrees is defined as    @xmath27    we limit ourself to the case of preferential linear attachment rate@xcite @xmath28 shifted by @xmath29 , the initial attractiveness @xcite of nodes in @xmath1 , which ensures that there is a nonzero probability of any node acquiring a link . 
   the nature of @xmath30 lets one to obtain , as @xmath31 @xmath32 where @xmath33 is the average total in - degree in network @xmath1 . 
   ( [ mat ] ) implying that @xmath26 is linear in time . 
   similarly , it is easy to show that @xmath34 is also linear function of time . 
   we use these relations in eq . 
   ( [ na ] ) to obtain the time independent recurrence relation    @xmath35n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ]    the expression in eq . 
   ( [ arec ] ) does not simplify however , it lets us to obtain the total in - degree distribution @xmath36 writing @xmath37 and since @xmath38 then @xmath39 satisfies @xmath40n_{a}(k_{a})=<m_{a}>(k_{a}+a-1)n_{a}(k_{a}-1 ) \nonumber\\ + ( a+<m_{a}>)\delta_{k_{a}0}.\end{aligned}\ ] ]    solving eq . 
   ( [ narec ] ) for @xmath41 yields , @xmath42 with @xmath43    as @xmath44 eq . 
   ( [ nagamma ] ) gives the asymptotic behaviour of the total in - degree distribution in @xmath1 @xmath45 which is a power - law form with an exponent @xmath46 that only depends on the average total in - degree and the initial attractiveness of the nodes . 
   similarly , we can write the total in - degree distribution in network @xmath2 for the asymptotic limit of @xmath47 as @xmath48 again , the exponent depends upon the initial attractiveness @xmath49 of nodes and the average total incoming links @xmath50 . 
   we now move on to analyse @xmath17 , the distribution of the average number of nodes with @xmath12 intra - links in network @xmath1 . in citation network 
   one can think of these links being issued from the same subject class as the receiving nodes and in the case of human sexual contact network , they represent the homosexual interactions . since @xmath51 which can also be written as @xmath52 , a linear function of time . then summing eq . 
   ( [ arec ] ) over all possible values of @xmath13 @xmath53 we get @xmath54n_{aa}(k_{aa})+<m_{a } > \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+p_{aa}m_{a } \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+a)n_{aa}(k_{aa})+p_{ba}m_{b } \sum_{k_{ba}=0}^{\infty}(k_{ba}-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] for large @xmath12 eq . 
   ( [ aarec ] ) reduces to @xmath55n_{aa}(k_{aa})=\nonumber\\ p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+(a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] iterating former relation for @xmath56 yields    @xmath57    where @xmath58 in the asymptotic limit as @xmath59 eq . 
   ( [ aagamma ] ) has a power - law form @xmath60 that depends upon both @xmath5 and the coupling parameter @xmath61 . 
   similarly , the time independent recurrence relation for @xmath18 has the same form as eq . 
   ( [ aarec ] ) with the only difference being the parameters . 
   therefore we will simply give the power - law distribution @xmath62 where the other coupling parameter @xmath63 is revealed in the exponent . 
   finally , the distribution of average number of nodes with incoming cross - links @xmath20 in @xmath1 can be found by summing over @xmath23 for all its intra - links @xmath64 as before @xmath65 is also linear in time . when the cross links @xmath13 are large enough , then from eq . 
   ( [ arec ] ) we obtain    @xmath66    where @xmath67 in the asymptotic limit as @xmath68 the distribution @xmath69 has a power - law form and similarly for the network @xmath2 as @xmath68 @xmath70 unlike the case in intra - links , here the exponents are inversely proportional to the coupled parameters @xmath61 and @xmath63 respectively . 
   for the sake of simplicity , we set the number of outgoing links of the new nodes in either networks to be the same , i.e. @xmath71 . furthermore taking the rate of cross linking to be @xmath72 and the rate of intra linking @xmath73 , consequently we have @xmath74 , and @xmath75 as the coupling parameter .    in the weak coupling case , 
   the cross linking is negligibly small i.e. @xmath76 then the power - law exponent of the intra - link distribution is @xmath77 equal to total link distribution @xmath78 . 
   this gives a solution obtained in@xcite and when @xmath79 we recover the exponent @xmath80 , the empirical findings in @xcite . 
   thus , varying @xmath75 in @xmath83 yields any values of @xmath84 between @xmath85 and @xmath86 . on the contrary , 
   the exponent of cross - link distribution @xmath87 decreases from @xmath86 to @xmath88 , as @xmath75 increases from @xmath89 to @xmath90 . 
   taking @xmath79 gives @xmath91 supposing @xmath92 , which seems reasonable for consideration of citation networks , we find that @xmath93 and @xmath94 . 
   the former result coincides with the distribution of connectivities for the electric power grid of southern california @xcite . where the system is small and the local interactions is of importance hence there seems to be some analogy to the intra - linking process . 
   for the latter , as far as we are aware there is none empirical studies present in the published literature .    now , consider the web of human sexual contacts@xcite . 
   if we let @xmath1 to represent males and @xmath2 females that is @xmath95 and @xmath96 then @xmath97 are the power - law exponents of the degree distributions of the sexes . 
   where @xmath98 and @xmath49 denote the male and female attractiveness respectively and usually @xmath99 is considered @xcite . by setting @xmath100 , @xmath101 and @xmath81 that is , cross links are predominant then as in @xcite we obtain @xmath102 for males and @xmath103 for females . 
   the exponents @xmath104 and @xmath105 have been observed for the cumulative distributions in empirical study @xcite . 
   the model we studied here seems to have the flexibility to represent variety of complex systems . 
   we would like to thank the china scholarship council , epsrc for their financial support and geoff rodgers for useful discussions .
summary: we introduce and solve a model which considers two coupled networks growing simultaneously . 
   the dynamics of the networks is governed by the new arrival of network elements ( nodes ) making preferential attachments to pre - existing nodes in both networks . 
   the model segregates the links in the networks as intra - links , cross - links and mix - links . 
   the corresponding degree distributions of these links are found to be power - laws with exponents having coupled parameters for intra- and cross - links . in the weak coupling case the model reduces to a simple citation network . as for the strong coupling , it mimics the mechanism of _ the web of human sexual contacts_.