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during the last years , many systems have been analyzed from the perspective of graph theory @xcite .
it turns out that seemingly diverse systems such as the internet , the world wide web ( www ) , metabolic and protein interaction networks and food webs , to mention a few examples , share many topological properties @xcite . among these properties ,
the fact that one can go from one node ( or element ) of the network to another node passing by just a few others is perhaps the most popular property , known as `` six degrees of separation '' or small - world ( sw ) property @xcite .
the sw feature has been shown to improve the performance of many dynamical processes as compared to regular lattices ; a direct consequence of the existence of key shortcuts that speed up the communication between otherwise distant nodes and of the shorter path length among any two nodes on the net @xcite .
however , it has also been recognized that there are at least two types of networks fulfilling the sw property but radically different as soon as dynamical processes are ran on top of them .
the first type can be called `` exponential networks '' since the probability of finding a node with connectivity ( or degree ) @xmath0 different from the average connectivity @xmath1 decays exponentially fast for large @xmath0 @xcite .
the second kind of networks comprises those referred to as `` scale - free '' ( sf ) networks @xcite .
for these networks , the probability that a given node is connected to @xmath0 other nodes follows a power - law of the form @xmath2 , with the remarkable feature that @xmath3 for most real - world networks @xcite .
the heterogeneity of the connectivity distribution in scale - free networks greatly impacts the dynamics of processes that they support .
one of the most remarkable examples is that an epidemic disease will pervade in an infinite - size sf network regardless of its spreading rate @xcite .
the change in the behavior of the processes is so radical in this case that it has been claimed that the standard epidemiological framework should be carefully revisited .
this might be bad news for epidemiologists , and those fighting natural and computer viruses .
on the other hand , in a number of important technological and commercial applications , it is desirable to spread the `` epidemic '' as fast and as efficient as possible , not to prevent it from spreading .
important examples of such applications are epidemic ( or rumor - based ) protocols for data dissemination and resource discovery on the internet @xcite , and marketing campaigns using rumor - like strategies ( viral marketing ) .
the above applications , and their dynamics , have passed almost unnoticed @xcite to the physics community working on complex networks despite the fact that they have been extensively studied by computer scientists and sociologists @xcite .
the problem here consists of designing an epidemic ( or rumor - mongering ) algorithm in such a way that the dissemination of data or information from any node of a network reaches the largest possible number of remaining nodes .
note that in this case , in contrast to epidemic modeling , one is free to design the rules of epidemic infection in order to reach the desired result , instead of having to model an existing process .
furthermore , in a number of applications , such as peer - to - peer file sharing systems @xcite built on top of the internet and grid computing @xcite , the connectivity distribution of the nodes can also be changed in order to maximize the performance of such protocols .    in this paper
we study in detail the dynamics of a generic rumor model @xcite on complex scale - free topologies through analytic and numerical studies , and investigate the impact of the interaction rules on the efficiency and reliability of the rumor process .
we first solve the model analytically for the case of exponential networks in the infinite time limit and then introduce a stochastic approach to deal with the numerical solution of the mean - field rate equations characterizing the system s dynamics .
the method @xcite is used to obtain accurate results for several quantities when the topology of random sf networks is taken into account , without using large and expensive monte carlo ( mc ) simulations .
the rest of the paper is organized as follows .
section ii is devoted to introducing the rumor model and to derive the mean - field rate equations used throughout the paper . in section
iii we deal with the stochastic approach , and compare its performance with analytical and mc calculations in homogeneous systems . we extend the method to the case of power - law distributed networks and present the results obtained for this kind of networks in section iv and v. finally , the paper is rounded off in the last section , where conclusions are given .
the rumor model is defined as follows .
each of the @xmath4 elements of the network can be in three different states .
following the original terminology and the epidemiological literature @xcite , these three classes correspond to ignorant , spreader and stifler nodes .
ignorants are those individuals who have not heard the rumor and hence they are susceptible to be informed .
the second class comprises active individuals that are spreading the rumor .
finally , stiflers are those who know the rumor but that are no longer spreading it .
the spreading process evolves by directed contacts of the spreaders with others in the population .
when a spreader meets an ignorant the last one turns into a new spreader with probability @xmath5 .
the decay of the spreading process may be due to a mechanism of `` forgetting '' or because spreaders learn that the rumor has lost its `` news value '' .
we assume this latter hypothesis as the most plausible so that the contacting spreaders become stiflers with probability @xmath6 if they encounter another spreader or a stifler .
note that as we are designing our rumor strategy in such a way that the fraction of the population which ultimately learns the rumor be the maximum possible , we have assumed that contacts of the type spreader - spreader are directed , that is , only the contacting individual loses the interest in propagating the rumor further .
therefore , there is no double transition to the stifler class .    in a homogeneous system ,
the original rumor model due to daley and kendall @xcite can be described in terms of the densities of ignorants , spreaders , and stiflers , @xmath7 , @xmath8 , and @xmath9 , respectively , as a function of time . besides , we have the normalization condition , @xmath10 in order to obtain an analytical insight and a way to later test our numerical approach , we first study the rumor model on top of exponentially distributed networks .
these include models of random graphs as well as the watts and strogatz ( ws ) small - world model @xcite .
this model produces a network made up of @xmath4 nodes with at least @xmath11 links to other nodes .
the resulting connectivity distribution in the random graph limit of the model @xcite takes the form @xmath12 which gives an average connectivity @xmath13 .
hence , the probability that a node has a degree @xmath14 decays exponentially fast and the network can be regarded as homogeneous .
the mean - field rate equations for the evolution of the three densities satisfy the following set of coupled differential equations : @xmath15 , \label{eq3}\\   \frac{d r(t)}{d t } & = & \alpha { \langle k \rangle}s(t ) [ s(t)+r(t ) ] , \label{eq4}\end{aligned}\ ] ] with the initial conditions @xmath16 , @xmath17 and @xmath18 . the above equations state that the density of spreaders increases at a rate proportional to the spreading rate @xmath5 , the average number of contacts of each individual @xmath1 and to the densities of ignorant and spreader individuals , @xmath7 and @xmath8 , respectively
. on the other hand , the annihilation mechanism considers that spreaders decay into the stifler class at a rate @xmath19 times the density of spreaders and of non - ignorant individuals @xmath20 .
the system of differential equations  ( [ eq2]-[eq4 ] ) can be analytically solved in the infinite time limit when @xmath21 . using equation  ( [ eq1 ] ) , we have that @xmath22 . introducing the new variable @xmath23 we obtain the transcendental equation
, @xmath24    equation  ( [ eq5 ] ) always admits the trivial solution @xmath25 , but at the same time it also has another physically relevant solution _ for all _ values of the parameters @xmath5 and @xmath6 .
this can be easily appreciated since the condition , @xmath26 reduces to @xmath27 .
that is , there is no `` rumor threshold '' contrary to the case of epidemic spreading @xcite .
this strikingly different behavior does not come from any difference in the growth mechanism of @xmath8 @xmath28the two are actually the same@xmath28 , but from the disparate rules for the decay of the spreading process .
on the other hand , this result also points out that a mathematical model for the spreading of rumors can be constructed in many different ways .
the results of this paper , however , indicate that the presence of spreader annihilation terms due to spreader - spreader and spreader - stifler interactions is very relevant for practical implementations @xcite .
we shall come back to this point later on .
recently @xcite , we have introduced a numerical technique @xcite to deal with the mean - field rate equations appearing in epidemic - like models .
it solves the differential equations by calculating the passage probabilities for the different transitions .
the main advantage of this method , as compared to mc simulations , is its modest memory and cpu time requirements for large system sizes . besides
, we do not have to generate any network .
instead , we produce a sequence of integers distributed according to the desired connectivity distribution @xmath29 .
the numerical procedure here proceeds as follows . at each time step until the end of the rumor spreading process , the following steps are performed :    .density of stiflers at the end of the rumor spreading process .
results are shown for 5 different values of @xmath6 for each method considered .
monte carlo ( mc ) simulations were performed in a ws network with @xmath30 and @xmath31 nodes .
the same system size was used in the stochastic numerical approach ( sna ) . [ cols="^,^,^,^",options="header " , ]     in figs .
[ figure4 ] and  [ figure5 ] we have represented the time ( in units of @xmath32 ) profiles of @xmath33 and @xmath34 for each decay term and several values of @xmath6 . from the figures ,
it is clear that while the final density of stiflers increases when modifying the original decay rules , the time needed to reach such high levels of reliability also increases .
this is due to the fact that the tails of the densities of spreaders decay more slowly than before . in particular , it is noticeable that when only spreader - spreader interactions are taken into account in the decay mechanism , the lifetime of the propagation process is more than two times longer than for the other two settings .
this means that this implementation is not very suitable for practical applications as the costs associated to the process rise as well . on the other hand
, the performance of the spreader - stifler setting seems to depend on the value of @xmath6 in such a way that it is more efficient at both the reliability level and time consumption for a large @xmath6 , but not in the middle region of the parameter space . in summary
, the present results support that the original model works quite well under any condition , while other variants can be considered depending on the value of @xmath6 used and the type of applications they are designed for .
in this paper , we have analyzed the spreading dynamics of rumor models in complex heterogeneous networks .
we have first introduced a useful stochastic method that allows us to obtain meaningful time profiles for the quantities characterizing the propagation process .
the method is based on the numerical solution of the mean - field rate equations describing the model , and contrary to monte carlo simulations , there is no need of generating explicitly the network .
this allows to save memory and a fast exploration of the whole evolution diagram of the process .
the kind of processes studied here are of great practical importance since epidemic data dissemination might become the standard practice in multiple technological applications .
the results show that there is a fragile balance between different levels of reliability and the costs ( in terms of time ) associated to them . in this sense ,
our study may open new paths in the use of rumor - mongering process for replicated database maintenance , reliable group communication and peer to peer networks @xcite .
besides , as shown here , the behavior and features of the different algorithms one may implement are not trivial and depend on the type of mechanisms used for both the creation and the annihilation terms .
it is worth noting here that we have studied the simplest possible set of rumor algorithms , but other ingredients such as memory must be incorporated in more elaborated models @xcite .    of further interest
would be a more careful exploration of the possibility of using dynamical shortcuts for a more efficient spreading of the updates .
our results suggest that it would be more economic to start from hubs and then kill off the updating process when a given level of reliability is reached than starting at random and letting the process dies out by itself .
preliminary studies of more elaborated models aimed at implementing a practical protocol confirm our results @xcite .
this feature is specially relevant for the understanding and modeling of social phenomena such as the spreading of new ideas or the design of efficient marketing campaigns .
we would like to thank a.  vespignani for many useful discussions and comments .
y.  m.  thanks m.  vzquez - prada for helpful comments and the hospitality of bt exact , u.k . where parts of this work were carried out .
y.  m.  acknowledges financial support from the secretara de estado de educacin y universidades ( spain , sb2000 - 0357 ) . this work has been partially supported by the spanish dgicyt project bfm2002 - 01798 .
a. j. demers , d. h. greene , c. hauser , w. irish , and j. larson , _ epidemic algorithms for replicated database maintenance_. in proc . of the sixth annual acm symposium on principles of distributed computing , vancouver , canada , 1987
.                            note that for the @xmath35 variant , the initial conditions should include the existence of at least one stifler .
this has been taken into account by randomly selecting a node among all the remaining ignorants .
hence the initial conditions are now @xmath36 , @xmath37 , and @xmath38 .
we derive the mean - field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks . 
 these equations are solved numerically by means of a stochastic approach . 
 first , we present analytical and monte carlo calculations for homogeneous networks and compare the results with those obtained by the numerical method . 
 then , we study the spreading process in detail for random scale - free networks . 
 the time profiles for several quantities are numerically computed , which allow us to distinguish among different variants of rumor spreading algorithms . 
 our conclusions are directed to possible applications in replicated database maintenance , peer to peer communication networks and social spreading phenomena .