You will be presented with the full text of a science research paper. Please write an abstract for this paper. Your response should include the abstract and no additional text.

Paper text:
recently there has been much interest in random growing networks review , review1,review2 , both from the point of view of theoretical modelling , as well as the empirical study of real networks .
there is considerable evidence that while traditional erds - renyi random graphs have a poisson degree distribution @xcite , real graphs and random growing graphs have a power - law degree distribution .    of particular interest
are directed networks , which can be used to model systems in which directed flow takes place .
such networks include the world wide web @xcite , the phone - call graph @xcite and the citation graph 3 .    as a complement to computer simulations and exact solutions of simplified systems ,
thermodynamic formulations have been used to study a number of complex co - operative systems including granular media , econophysics , breaking phenomena and many others . in a recent paper @xcite , bianconi and
barabsi ( bb ) mapped the different behaviour of random growing networks with fitness onto the thermodynamically distinct phases of a free bose gas .
the fitness model predicts that , in the large network limit , the fittest node will have the most links .
this is called the _ fit - get - rich _ ( fgr ) phase . unlike the scale - free model ( sfm ) , in which the degree distribution of the network is power - law , the fgr behaviour ( phase in the thermodynamic language ) has nodes with a very large degree which dominate the network .
another phase revealed in @xcite is the bose - einstein condensation ( bec ) , when the fittest node acquires a finite fraction of the total degree .
in contrast to bec , in the fgr phase the richest node is not an absolute winner , since its share of the links ( _ i.e. _ the ratio of its degree and the total degree of the network ) decays to zero for large system sizes , whereas in bec the winner maintains its share irrespective of system size .    in this case , the fact that this ratio is constant corresponds to the extensivity property of a bose gas , when the gas keeps a finite fraction of its particles in the ground state . in this paper
we examine the phenomenon of bose - einstein condensation , which was considered for an undirected graph in @xcite on the random directed growing network introduced in @xcite .    to do this we work with the model introduced in @xcite , introducing a dependence between the vertex fitness and an energy .
as each vertex has two fitnesses ( for in - degree and out - degree ) then it is necessary to assign two different energy levels to represent each vertex .
this is detailed in the first sub - section , together with the description of edges in terms of particles assigned to two energy levels .
the necessary coexistence of two sub - systems forming the network and the formulation of an equilibrium condition in a formal way is discussed in detail in sections 3 and 4 .
the occurrence of bose - einstein condensation , and its interpretation in a physical framework makes it possible to describe the directed network in terms of canonical concepts of statistical physics , such as a phase diagram and first and second order phase transitions .
these are introduced in the following sections .    in the conclusions we emphasize the usefulness of this formulation in providing a clear and concise interpretation of different phenomena in networks .
so , the possibility of coexistence of phases in which the in - degree distribution exhibits a clear winner whereas the out - degree shows scale free behaviour , or the simultaneous existence of two different winners ( bipolarity ) and the conditions for its existence are predicted in thermodynamic language .
in bb @xcite , a correspondence between fitness @xmath8 and energy @xmath9 given by @xmath10was introduced . in our model
the vertices have , in general , different fitnesses @xmath11 and @xmath12 for the addition of an in- or out - degree . as in er , here the model
consists of the addition of bare vertices ( i.e. without edges , but with fitness @xmath11 for in - degree and @xmath12 for out - degree ) to the network with probability @xmath0 , and the creation of directed edges between vertices with probability @xmath13 .    a kinetic equation describing the process of directed networks must include the kinetics of both in - degree and out - degree simultaneously since they coexist and influence each other .
then , the kinetic equation for the mean number of vertices with in - degree @xmath4 and out - degree @xmath5 is , as in er ,    @xmath14+\frac{qb } { m_{2}}[jn_{i , j-1}-(j+1)n_{ij}]+p\delta _ { i0}\delta _ { j0}f(a , b ) .\ ] ]    where the fitnesses @xmath2 are chosen from the fitness distribution @xmath15 .
the normalisation factors @xmath16 and @xmath17 are given by @xmath18and @xmath19the first term in the first square brackets of eq.(2 ) represents the increase in @xmath20 when vertices with in - degree @xmath21 and out - degree @xmath5 gain an incoming edge and the second term represents the corresponding loss .
the second square brackets contain the analogous terms for outgoing edges and the last term ensures the continuous addition of new vertices with fitnesses @xmath22 .
the translation of this problem into an energetic formulation is straightforward .
we associate the addition of a vertex with fitnesses @xmath23 to the network with the creation of two energy levels ; one representing the fitness of the incoming edge and the other for the outgoing edge .
this means that the edge can be mapped into two separate isolated sub - systems .
then the creation of a directed edge corresponds to the creation of two particles , one in each sub - system , simultaneously .
the particles are created in the energy levels corresponding to incoming fitness of the receiving vertex and the outgoing fitness of the emitter vertex , as fig .
1 illustrates .
then , using the energy variables , rather that the fitness , we have @xmath24@xmath25and @xmath26+\frac{% qe^{-\beta _ { 2}\varepsilon _ { 2}}}{m_{2}}[jn_{i , j-1}-(j+1)n_{ij}]+p\delta _ { i0}\delta _ { j0}f(\varepsilon _ { 1},\varepsilon _ { 2}).\]]and the normalisations
are re - written as @xmath27and @xmath28    as we distinguish between incoming and outgoing edges , and their respective fitnesses , when going to the energy representation the edges and their fitnesses must also be distinguished as belonging to different isolated sub - systems of the same system .
let us denote the sub - systems as 1 and 2  so , the creation of a vertex in the network implies the simultaneous creation of an energy level in each sub - system , i.e. @xmath29and @xmath30 .
the creation of an edge joining two vertices implies the simultaneous creation of one particle in each sub - system in the energy levels corresponding to the fitnesses of the vertices gaining an in- and out - degree , as fig.1 illustrates .    it should be noted that we are doing a simultaneous description of two isolated sub - systems which compose the whole system ( the network ) .
these sub - systems do not exchange energy or particles , but are correlated in the sense that the creation of an energy level ( a particle ) in one of the sub - systems implies the simultaneous phenomenon in the other .    by definition of the model ,
the total number of particles increases linearly with time , so @xmath31    let us define the reduced moments @xmath32 and @xmath33 by @xmath34@xmath35and introduce @xmath36 as @xmath37    as shall be seen these magnitudes will be useful to calculate the characteristics of the network .
then , following the procedure in @xcite , we can show that @xmath38and @xmath39 which determines @xmath40 and @xmath41 as functions of @xmath0 , @xmath42 and @xmath43 .      in this representation ,
as we already pointed out , the addition of a vertex means the creation of two energy levels , each corresponding to its fitness . when an edge joining two vertices is created , we add one particle in each corresponding level .
the distribution of in- and out - degree then corresponds to the distribution of particles in systems 1 and 2    @xmath44    is the number of vertices with in - degree @xmath4 , and    @xmath45    is the number of vertices with out - degree @xmath5 . for large degree
their asymptotic expressions are @xmath46and @xmath47where we have introduced @xmath48for @xmath49 and @xmath50 as _ chemical potentials_. unlike bb , the chemical potential is introduced here as an exact expression , not as an asymptotic one .      the equations for @xmath32 and @xmath33 can be transformed by introducing the generating function @xmath51where @xmath52with @xmath53}\]]and @xmath54}.\]]then the equations for @xmath55 @xmath56 become @xmath57this means that @xmath58unlike bb , this is a new condition generated by the model .
let us denote it as a _ generalized equilibrium condition_. a _ strong equilibrium _
condition implies that @xmath59and @xmath60this condition will be discussed later . in this formalism
the occupation probability is , then , @xmath61for @xmath62 , representing the probability for a given edge to belong to a given vertex ( probability of a particle to belong to an energy level ) .
if , now , the fitness distribution is separable then @xmath63and @xmath64separately for each system @xmath62 . in each case , @xmath65 is the solution of this equation , which depends on the density of states @xmath66 .      to illustrate ,
let us start with a particular case , when @xmath67 with @xmath68 .
the normalisation @xmath69 as in bb .    then the integral in eq.(25 ) , @xmath70    but this integral reaches its maximum value when @xmath71
. then if for a given value of @xmath72 we have @xmath73 then we have a bose - einstein condensation ( bec ) .
if we calculate the temperature for bec in this particular case , we obtain @xmath74    unlike bb , the factor @xmath75 appears here . if @xmath76 we recover bb result .
even in this simple example , @xmath77 may be different for the sub - systems 1 and 2 , if the value of @xmath78 is different . in a more general form , we can state the condition for bose einstein condensation . then , as @xmath79for @xmath62 . , if @xmath80 then bec occurs . the conditions for bec depend on @xmath81      below @xmath77 there is bec .
above it the phase is fit - get - richer as in bb1 , but here both phases must be considered for both systems .
then , a phase diagram can be drawn with the temperatures @xmath82 = @xmath83 and @xmath84 = @xmath85 characterizing each sub - system ( fig .
2 ) .    imposing a given dependence of fitness with energy so that @xmath86means that the equilibrium is broken , ( i.e. in a computer simulation we would impose temperatures that do not satisfy eq . [ 26 ]  ) . then there can be a winner in one of the phases .
one of the vertices accumulates , say , most of the in - degrees , coming from the other vertices . if both phases are in bec there is a vertex with most of the incoming and another with most of the outgoing edges . then both vertices will be highly correlated .
though a more complicated situations are possible , let us illustrate the case @xmath87 .
the point a represents that both sub - systems are in fgr phase .
if , for example , we make the simulations with the parameters given by a1 then will be an fgr and a bec  phase coexisting , but not _ in equilibrium _ since one of them satisfies eq .
[ 35 ] but not the other .
there is a coexistence of phases out of equilibrium
. we might call this an _ out of equilibrium first order phase transition_. the motion along aa1a2 ... leads to the region in which bec occurs for both phases . here is hard to speak about equilibrium unless a belongs to the line @xmath88 .
all we know in bec region is that none of the sub - systems satisfies eq .
35    the concept of equilibrium in this model is purely formal since the sub - systems are not in contact , but it is a comfortable tool to make a map of directed networks into bose - einstein statistics .      if one of the systems is such that @xmath89 , then eq . (
[ 31 ] ) gives @xmath90then @xmath91with a corresponding expression for @xmath92 .
if @xmath76 the scale free model is recovered with a degree distribution @xmath93 . thus the model with directed edges gives the possibility of a scale free phase for both in the in - degree and the out - degree .      if in fig .
3 we move in such a way as to cross into the bec region through the point b then bec occurs simultaneously for both phases . no coexistence of phases will occur , the system goes from fgr to bec or vice - versa instantly .
this is a _
second order phase transition _ since the system as a whole changes its phase .
then the point b is the _
critical point _ for the transition .
bec is itself a second order phase transition since it effects the system as a whole , so that when sub - system 1 or 2 experiences bec , the sub - system itself suffers a second order phase transition , but our system contains both sub - systems 1 and 2 so that when they are in different phases we consider our whole system as composed of two phases .
we have studied bose - einstein condensation in a random growing network model . in this model , with a formulation considering incoming and outgoing edges , a formal analogy can be adopted to describe the change in the behaviour of the directed edges in terms of bose statistics .    in this formulation , as in that of bb , the temperature plays the role of a dummy variable .
real systems are characterised by the functional form of the distribution function of fitness and there is no @xmath94 to consider .
it only emerges when we translate from the language of fitness to that of energies .
but the formulation of networks in terms of quantum statistics and the introduction of temperature is an elegant and simple description of the behaviour of networks under different conditions . in this sense ,
the temperature plays its role in the simulations of the networks , determining the strength of the dependence between the fitness and its associated energy , and this dependence is reflected in the behaviour of the network by determining its position in the phase diagram .
though phase diagrams and phase transitions are not exclusive to thermodynamic formulations ( see , e.g. dorogovtsev and mendes @xcite ) , such a formulation is a very powerful tool to describe different phases of random directed networks , where the presence of directed edges make the description in terms of fitness difficult .
considering this network as a thermodynamic system , makes it simple to understand , provided we appropriately interpret the thermodynamic parameters .
we would like to thank the royal society , london and the london mathematical society for their financial support .
one of us ( o. s ) is greatly indebted to the staff of the department of mathematical sciences at brunel university for their warm hospitality .
\1 . correspondence between fitness and energy for directed networks .
we consider the phenomenon of bose - einstein condensation in a random growing directed network . 
 the network grows by the addition of vertices and edges . at each time 
 step the network gains a vertex with probabilty @xmath0 and an edge with probability @xmath1 . 
 the new vertex has a fitness @xmath2 with probability @xmath3 . 
 a vertex with fitness @xmath2 , in - degree @xmath4 and out - degree @xmath5 gains a new incoming edge with rate @xmath6 and an outgoing edge with rate @xmath7 . 
 the bose - einstein condensation occurs as a function of fitness distribution @xmath3 .