Neutral Theory for competing attention in social networks
Carlos A. Plata, Emanuele Pigani, Sandro Azaele, Violeta Calleja-Solanas, María J. Palazzi, Albert Solé-Ribalta, Sandro Meloni, Javier Borge-Holthoefer, Samir Suweis
NNeutral Theory for competing attention in social networks
Carlos A. Plata , Emanuele Pigani , Sandro Azaele ,Violeta Callejas , Mar´ıa J. Palazzi ,Albert Sol´e-Ribalta , Sandro Meloni , Javier Borge-Holthoefer , Samir Suweis Dipartimento di Fisica “G. Galilei”,Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy IFISC, Institute for Cross-Disciplinary Physics and ComplexSystems (CSIC-UIB), 07122, Palma de Mallorca, Spain Internet Interdisciplinary Institute (IN3),Universitat Oberta de Catalunya, Barcelona, Catalonia, Spain (Dated: June 17, 2020) a r X i v : . [ phy s i c s . s o c - ph ] J un bstract We used an ecological approach based on a neutral model to study the competition for attentionin an online social network. This novel approach allow us to analyze some ecological patterns thathas also an insightful meaning in the context of information ecosystem. Specifically, we focus onthe study of patterns related with the persistence of a meme within the network and the capacityof the system to sustain coexisting memes. Not only are we able of doing such analysis in anapproximated continuum limit, but also we get exact results of the finite-size discrete system.
I. INTRODUCTION
An online social network (OSN) is a virtual social structure made of individuals using theInternet as a communication medium for interacting, sharing contents and opinions. OSNsallow hundreds of millions of Internet users worldwide to produce and consume contents,providing access to a very vast source of information on an unprecedented scale.Nowadays, OSNs therefore constitute mainstream communication channels to interact,exchange opinions, and reach consensus. In recent years, it has increasingly become evidentthat competition significantly shapes the topology and the dynamics on these information-driven platforms [1–3]: users thrive for visibility, while memes resemble can be thought asentities that compete for users’ attention.Nevertheless, it is hard to disentangle the effects of limited attention from many concur-rent factors [4], such as the structure of the underlying social network [5, 6], the activity ofusers [7], the different degrees of influence of information spreaders [8], the intrinsic qualityof the information they spread [9], and the persistence of topics [10].Through the analysis of the OSNs big data, emergent properties, such as the emergenceof consensus [1], viral spreading [11], power-law distribution of memes popularity [12] andecho-chambers effects [13], have been observed even among different OSNs platforms. Nev-ertheless, availability of massive online social data streams does not give per se theoreticalinsights to understand these complex inter-plays among structure of OSNs, memes popular-ity and users attention dynamics.The aforementioned ubiquitous properties are signatures of the emergent simplicity char-acterizing complex systems. Therefore, it naturally calls for a statistical mechanics approach,i.e., the attempt to understand regularities at large scale as collective effects of the dynamics2t the individual/meme scale. The bridge between these scales would also allow to betterunderstand how these macroscopic (system-wide) patterns are affected by the microscopicdynamics of the interacting elements (users and memes) forming the OSN.One of the first attempts to unveil the emergence of fat-tailed power-law distribution ofmemes popularity starting from an interacting particle model is the work of Gleeson andcollaborators [14]. Therein, they studied a stochastic model with simple microscopic rulesto describe the evolution, that is, the spreading of memes competing for the limited resourceof users attention, on a Twitter-like OSN. In real life, each user in the OSN pays attentionto a finite number of memes constrained by the finite capacity of users. In the model,this picture is simplified assuming that each user can just pay attention to a single meme.Further generalizations can be considered [12].Different memes can be thought as different species. In addition to the spreading dynam-ics, new species are introduced in the system by innovations events (speciation processes).The type of stochastic processes introduced in [14] has been used for decades in populationgenetics [15] and in ecology [16]. In particular, these models, based on simple stochasticrules, which neglect species (genes) fitness, represent “null” neutral models in which nointrinsic advantage is ascribed to a particular type of species (genes). Therefore, the evolu-tion of the system through network depends only on the network structure and experiencesrandom demographic effects.Remarkably, the concept of neutrality and the so-called neutral models have attracted alot of interest in the communities of statistical physicists [17–19]. In fact, neutral theory hasbeen proven to be very successful when describing universal emergent patterns of ecosystems[20–23].The analogy between memes competing for attention in an OSN and species competingfor resources in an ecosystem suggests that the approaches introduced in the frameworkof ecological neutral theories can be exploited to extract novel relevant information aboutthe dynamics of users attention. In fact, such qualitative resemblance between natural and“information” ecosystems has already been explored towards more quantitative accounts.In this direction, Borge-Holthoefer et al. [24] have found evidence of nested structuralsignatures –a landmark feature in natural mutualistic systems [25–28] when analyzing time-resolved online communication discussions after external information “shocks”, e.g. breakingnews. Similarly, Lorenz and collaborators[29], using a mathematical model based on Lotka-3olterra equations, have been able to explain some empirical data patterns in differentOSNs. Moreover, they have suggested that faster exhaustion of limited attention is drivenby increasing production and consumption of the contents in the OSN, leading to higherturnover rates and shorter collective attention for individual topics.Here we propose an analytically tractable neutral theory for competing attention in OSN.In particular, we show that starting from the interacting particle model for social interactionoriginally introduced by Gleeson et al. [14], we can derive the equations for the evolution ofthe users attention to the different memes. The exact approach involves a Master Equationdescribing probabilistically the evolution of the users attention to a meme. In the limit of bigenough networks, the equation reshapes into a Fokker-Planck structure which resembles thedynamics of species abundance in ecological neutral theories [30]. This parallelism allow usto analytically compute several new quantities of interest such as the number of coexistingmemes, the distribution of user attention to these memes, and the average persistence timefor attention on a meme. We compare our theoretical predictions to numerical simulations ofthe model, and investigate how the emergent properties depend on the underlying networkof users interaction.The work is organized as follows. In the next section, we introduce the interacting particlemodel describing memes propagation. Then, we derive evolution equation for competingattention in social networks in section III. Therein, we present both (exact) discrete and(approximated) continuous description of the system, computing several relevant patternsof the OSN related to attention. As expected, the evolution of users attention depends onthe OSN structure. Finally, in section IV, we summarize and discuss the achievements ofthe proposed neutral model to the study of competing attentions in social networks.
II. A NEUTRAL MODEL FOR USER ATTENTION
Let us consider a directed OSN of N nodes where each of them represent a user. Thenetwork is solely characterized by its out-degree distribution p k , that is, a random user have k followers with probability p k . Each node can be thought as the screen of the individualdisplaying the meme of current interest for that user. For the sake of simplicity, we assumethat each screen has capacity for only one meme although some generalizations to thisrespect are possible [12, 14]. Therefore, the state of the system at time t is given by the list4 IG. 1. Sketch of the model. Each node represents a user in the network with the color denoting thedifferent memes on its screen. The directed edges stand for the relations of following. Specifically,our convention is that the edge starts in the node which is followed and ends in the follower. Thenode chosen to spread the meme is highlighted with a dotted circle. A time step is representedwith its two possible outcomes, either an innovation event (with probability ˜ µ ) or a spreading event(with probability 1 − ˜ µ ). of memes appearing in all the nodes.The dynamics is introduced in discrete time steps representing the subsequent times wherean action is carried out by any of the users. During each time step one node is picked atrandom and it either (i) (re)tweets the meme currently on its screen to all its followers withprobability 1 − ˜ µ while its own screen remains unchanged or (ii) innovates with probability ˜ µ generating a brand-new meme that appears on its screen and is tweeted to all the followers.An illustrative visualization of the dynamics in a one time step is shown in Fig. 1.The configurational state of the system in a given time is described by the correspondencelist of memes the users are paying attention to. We define the set of attention variables x m as the fraction of nodes in the system which is paying attention to meme m . Consequently,the normalization constraint (cid:80) m x m = 1 holds for all time.5nother relevant variable in the simulated OSN, as originally introduced by Gleeson andcollaborators [14] is the memes popularity . The popularity n m of meme m is defined asthe number of times that the meme m has been broadcast. Because of this definition, n m is a non-decreasing function of time, which increases by one every time m is (re)tweeted.On the contrary, the evolution of attention x m can either increase or decrease until thememe disappears from the system. Note that, in our model, once a meme disappears fromthe OSN, there is no mechanism to re-introduce it in the system (innovation brings alwaysbrand-new memes). Therefore, once a meme goes extinct, its popularity remains constantfor the remaining simulation time.In the aforementioned work by Gleeson et al. [14], they semi-analytically find, and com-pare to numeric simulations, asymptotic power-laws behaviour for the popularity distribu-tion. In particular they have investigated the cases where p k = δ k,k ∗ (constant number offollowers for all users), and p k ∝ k − γ , i.e. a power-law distribution for the number of follow-ers. In both cases, they have found that memes popularity has a fat tail distribution, butwith different exponents [14]. Popularity can be measured from real Twitter data streams,as the number of (re)tweets since a given hashtag appears in the OSN. Although a systematicanalysis of memes popularity distributions is still lacking, there are some results suggestingthat indeed they display asymptotic power-law behaviour [12], pointing out thus that theproposed model is a good candidate to elucidate at least some systemic patterns observedin empirical OSNs.The proposed neutral framework complement the above results by studying in detail aset of novel emergent properties of users competing attention to the memes in the OSN. III. NEUTRAL EMERGENT PATTERNS IN USERS ATTENTION.
We now present a set of novel emergent patterns in OSNs related to users attentionto memes, which can be analytically studied from the above presented interacting particlemodel.As previously explained, since the model is neutral, there is no meme fitter than other,and we can thus think our system as an effective two-memes system. We call meme A to thememe we are focusing on, whereas meme B is simply defined as non- A . For the variablesdescribing users attention, we introduce the notation x ≡ x A and hence x B = 1 − x . It6omes in handy to define the total number of users paying attention to meme A as ν , i.e., ν ≡ xN .The dynamics, as described in the previous section, is defined in discrete time steps andobeys the Markovian property, i.e. the transition probability to the next state depends onlyon the current state. Consequently, we can properly formulate the time evolution of theprobability P (cid:0) ν, ˜ t (cid:1) of having ν users paying attention to meme A at the time step ˜ t bymeans of a discrete Master Equation, which reads P (cid:0) ν, ˜ t + 1 (cid:1) − P (cid:0) ν, ˜ t (cid:1) = N − (cid:88) k =0 k (cid:88) j =0 (cid:110) W A,sk,j ( ν − j ) P (cid:0) ν − j, ˜ t (cid:1) + W A,ik,j ( ν + j + 1) P (cid:0) ν + j + 1 , ˜ t (cid:1) ++ W Bk,j ( ν + j ) P (cid:0) ν + j, ˜ t (cid:1) − (cid:104) W A,sk,j ( ν ) + W A,ik,j ( ν ) + W Bk,j ( ν ) (cid:105) P (cid:0) ν, ˜ t (cid:1)(cid:111) . (1)In the Master Equation (1), we have taken into account that there are three main differentclasses of transformations with different transition probabilities W . Namely, the selectednode to act in a particular time step may be either A or B , this is the the first superscript inthe transition probabilities W . The second superscript { s, i } denotes the type of event, i.e. ifthe selected node spreads the current meme with probability 1 − ˜ µ or first innovates withprobability ˜ µ instead. Note that, since B is defined as non- A , innovation of B generates alsoa B meme and thus there is no need to differentiate between spread or innovation when theselected node contains a B meme. The subscripts give information of the follower nodes ofthe selected node. We define k ∈ { , . . . , N − } as the number of the followers of the selectednode whereas j ∈ { , . . . , k } is the number of followers with a current meme different tothe final meme of the spreading node. The detailed form of the transition rates W and thephysical meaning of the Master Equation (1) are provided in Appendix A.Assuming the innovation rate scales as ˜ µ = µ/N and defining the continuous time variable t = ˜ t/N , it is possible to perform diffusive approximation of the Master Equation (1). Thesystem size (or van Kampen’s) expansion [23, 31] yields the Fokker-Planck equation ∂ t P ( x, t ) = − ∂ x [ A ( x ) P ( x, t )] + 12 ∂ x [ B ( x ) P ( x, t )] , (2)with respectively drift and diffusion coefficients A ( x ) = − ax, a = µ (1 + (cid:104) k (cid:105) ) , (3a) B ( x ) = bx (1 − x ) , b = (1 + (cid:104) k (cid:105) ) (cid:104) k (cid:105) + σ k . (3b)7he drift term is negative and drives the memes to extinction with a velocity that increaseswith the innovation rate and the mean number of followers in the network (cid:104) k (cid:105) = (cid:80) k kp k .The dependence of the diffusion term with x (1 − x ) is usually found in population dynamicswith demographic stochasticity with a carrying capacity [32]. The constant b increases thedemographic fluctuations with (cid:104) k (cid:105) and the variability of the network through the variance σ k = (cid:80) k ( k − (cid:104) k (cid:105) ) p k . Detailed derivation of the Fokker-Planck equation (2) associated tothe Master Equation (1) is provided in Appendix A.Thanks to the master equation (1) and the Fokker-Planck equation (2) we can now studythe emergent patterns in users’ attention in the proposed OSN neutral model. A. Mean persistence time for attention
The evolution’s dynamics for the users attention has an absorbing boundary in x = 0,i.e. the attention to a given meme will eventually go to zero. The time that a meme persistsin the OSN receiving attention is an insightful quantity to study, as it gives insights on thevirality of that meme, i.e., the longer it remains “active” the more probable it may go viral.In the ecological context, species persistence times [22] or lifetimes [33] have been thoroughlystudied in different ecological neutral models and proved to be able to describe and thuselucidate the power-law shape (with exponential cut-off) of species persistence patternsobserved in real ecosystems [34]. Due to the stochastic nature of the neutral dynamicsof the memes these persistence times are random variables identically distributed amongmemes, herein we focus on the computation of the mean persistence time and its variance.Above, we have derived the forward evolution equations that study the attention prob-ability density function P ( x, t ). The probability density function Q ( x , t ) of reaching theabsorbing boundary at zero at time t , departing from initial attention x , obeys the backwardFokker-Planck equation [35] ∂ t Q ( x , t ) = A ( x ) ∂ x Q ( x , t ) + 12 B ( x ) ∂ x Q ( x , t ) , (4)with the same A and B defined in (3).Multiplying (4) by t and integrating over all times, one obtains the differential equationthat govern the mean persistence time τ ( x ) to reach zero departing from x . That equationcan be analytically solved with the proper boundary conditions (see Appendix B for details)8iving the result τ ( x ) = − (1 − x ) F (cid:0) , ab ; 1 + ab ; 1 − x (cid:1) + ab (cid:2) γ + ψ (cid:0) ab (cid:1) + ln x (cid:3) a (cid:0) − ab (cid:1) , (5)where F stands for the ordinary hypergeometric function, γ is the Euler–Mascheroni con-stant, ψ denotes the digamma function, and the couple { a, b } is defined in (3). Studying thedependence on the parameters of the model one can check that τ is a monotonic decreasingfunction of µ , (cid:104) k (cid:105) and σ k , taking the rest of parameters fixed. This is not a surprisingbehavior. On the one hand, higher innovation rate favors the demise of living memes. Onthe other hand, higher connectivity of the OSN, or equivalently higher circulation of memesin the systems, also drives, on average , to a faster decay of attention on existing memes.Note that (5) is an approximated result since it comes out from the diffusive continuumlimit. To go beyond this approximation, it is possible to use the backward formalism towrite the backward Master Equation for the discrete system, describing the evolution of theprobability Q ( ν , ˜ t ) for a meme to reach zero users attention at time ˜ t , if at the initial timethe meme has ν users following it. This equation reads Q (cid:0) ν , ˜ t + 1 (cid:1) − Q (cid:0) ν , ˜ t (cid:1) = N − (cid:88) k =0 k (cid:88) j =0 (cid:110) W A,sk,j ( ν ) Q (cid:0) ν + j, ˜ t (cid:1) + W A,ik,j ( ν ) Q (cid:0) ν − j − , ˜ t (cid:1) + W Bk,j ( ν ) Q (cid:0) ν − j, ˜ t (cid:1)(cid:9) − Q (cid:0) ν , ˜ t (cid:1) . (6)Multiplying by ˜ t the above backward Master Equation (6) and summing for all times,leads to the system of linear equations whose solution is the vector of mean persistence time˜ τ , whose components are ˜ τ ( ν ) with ν ∈ { , . . . , N } . This equation written in matrix formyields ˜ τ = M ˜ τ + , (7)where the components of the matrix M are proper combinations of the transition probabili-ties W (see Appendix C for the details) and is a N -dimensional vector of ones. Therefore,the exact result for the mean persistence times is given by˜ τ = ( I − M ) − , (8)with I being the identity matrix in dimension N . Furthermore, one can obtain higher ordermoments using (6), multiplying it for powers of ˜ t and summing over all times. Specifically, we9ave done so in order to obtain the meme attention variance σ τ = (cid:104) ˜ t (cid:105) − ˜ τ . The calculationof the mean squared attention persistence time results (cid:104) ˜ t (cid:105) = ( I − M ) − (2 ˜ τ − ) . (9)Detailed calculations are provided in Appendix C.In Fig. 2 we have compared the results obtained from numerical simulation of the modelregarding persistence time of memes with our theoretical predictions. We have done simu-lations for a system with N = 10 users. The OSN structure chosen here is homogeneous,i.e., p k = δ k,k ∗ with constant k ∗ = 10, and thus all users has the same number of followers.For further details on the simulation algorithm, see Appendix F.The value for k ∗ has been chosen so to avoid falling in a regime very far from the diffusiveapproximation. As seen in Fig. 2, the exact results perfectly agrees with the simulationsfor both mean value τ and standard deviation σ τ . Remarkably, in spite of the diffusiveapproximation performed, the prediction given by (5) still gives a very good estimation ofthe persistence time of the memes. This relative success of the diffusive approximation isreasonable in a system where there is not big chances of having burst events. Nevertheless,we expect the diffusive theory to fail when the distribution allows such a burst events. Infact, if k ∗ is very large, then in a single step it is possible that most of the users in thenetwork change their attention, and thus the dynamics regime for the attention is burstyrather than diffusive. This is the case for example in scale free OSNs, where p k presents longtails, i.e. few nodes have very large degree centrality. The role of power-law distributed p k and related effect on the attention dynamics is explored in Appendix E. B. Bio diversity patterns in information ecosystems.
In information ecosystem, as the one analyzed in this work, diversity of the memes playsan analogous role to biodiversity in natural ecosystems. Specifically, we focus on two differentaspects of meme diversity: the number of different memes coexisting in the system and howuser attention is distributed among the memes, i.e. how memes are distributed on usersscreens.The capacity of attention of users in an OSN is finite. Therefore, a given network cannotsustain an arbitrarily large number of memes. Interestingly, due to both neutrality and10
IG. 2. Mean persistence time for meme attention. The simulations (symbols) agrees perfectlywith the exact solution (blue line) given by (8). Moreover, the diffusive approximation in (4) (redline) gives also a quite good estimation. We have considered a system with N = 10 , an out-degreedistribution p k = δ k, , µ = 15. In the inset, the variance of the persistence time shows a perfectagreement with the exact solution (9). constant innovation rate ( µ ) assumption, it is possible to exactly calculate the probability P S of having S different memes coexisting in the OSN [36]. In fact, the stationary solutionfor P S reached in the large time limit is a Poisson distribution P S = e −(cid:104) S (cid:105) (cid:104) S (cid:105) S S ! (10)with an average number of memes given by (cid:104) S (cid:105) = ˜ µ N (cid:88) ν =1 p ν − ˜ τ ( ν ) . (11)This result was originally derived in the ecological context for a coarse-grain birth-deathmodel for species diversity [36]. In our model, the only difference is that memes may appear11ith any ν initial number of users paying attention to it, with a probability which isgiven by the out-degree distribution p ν − , whereas in the original model all species enteredthe system with only one individual. If we consider the particular case of a delta peakedout-degree distribution p k = δ k,k ∗ , then the average number of memes simply reduces to (cid:104) S (cid:105) = ˜ µ ˜ τ ( k ∗ + 1)Note that, equivalently, we can formulate equation (11) using the framework of con-tinuous time. It simply reads (cid:104) S (cid:105) = N µ (cid:82) dx τ ( x ) p ( x ), where we have used that, in thecontinuous description, innovation is a Poisson process with rate N µ and p ( x ) is the contin-uous counterpart of the out-degree distribution, giving the initial fraction of users attentionwhen the meme appears in the OSN. Consistently, the fraction (cid:104) S (cid:105) /N depends only on theparameters of the continuous model.In Fig. 3 we have compared the theoretically predicted Poisson distribution with thehistogram obtained in numerical simulations for long time in the system studied above,that is, we consider µ = 15, p k = δ k, and N = 10 . Moreover, we have considered inthe inset different values for the innovation rate µ to study the dependence of the averagenumber of memes. On the one hand, we obtain a perfect agreement when we use the exactcalculation for ˜ τ as given by (8). On the other hand, although there is not a quantitativeagreement, the prediction provided by the diffusive approximation (5) remarkably shares thesame qualitative behavior. As expected, both theories and simulations converge to (cid:104) S (cid:105) = 1for the limit case µ → P RMA that a meme is receiving anattention from a fraction x of the users in the OSN. We name this pattern as relative memesattention (RMA), which is completely analogous to the relative species abundance (RSA)pattern in ecology [17, 20, 23].Let us define φ ( x ) as the density function of the average number of memes receiving anattention x . Note that the only difference between φ ( x ) and P RMA ( x ) is the normalizationconstant, being the latter normalized to unity, while the former to the total number ofdifferent memes S . In the stationary state, the memes that contribute to the RMA arethose which have been generated by innovation a time t ago and they are still present in12 IG. 3. Histogram of number of different memes present in the whole system. Numerical simu-lations perfectly agrees with the theoretically predicted Poisson distribution. Note that there isno fitting parameter in the distribution since the mean is given by (11). We have considered thesame model parameters to those chosen in Fig. 2. Inset: Mean number of memes (species) (cid:104) S (cid:105) fordifferent values of the innovation rate µ . Apart from µ , the rest of parameters are equal to thosein the main figure. the OSN. They thus contribute to φ ( x ) with an amount (cid:82) dx p ( x ) P ( x, t | x ), which isintegrated for all possible initial configurations x weighted by the out-degree distribution.We have explicitly included as argument the initial condition in the solution of the forwardFokker-Planck equation (2). Since the number of species generated in a small time interval dt is simply µN dt , the RMA finally reads φ ( x ) = µN (cid:90) ∞ dt (cid:90) dx p ( x ) P ( x, t | x ) . (12)The result above is the generalization of the expression derived in the context of a neutralecological model [17] to the case in which the new species may start with a different number13f individuals.The application of Eq. (12) is not straightforward since we need the full time solution for P ( x, t | x ). Unfortunately, we have not been able to exactly solve the Fokker-Planck equation(2). Nevertheless, we have the solution of a very similar equation [30], that in a given limitis equivalent to P ( x, t | x ). Specifically, the two equations are the same if we neglect thequadratic term in x in the diffusion coefficient (3b). This approximation is particularlygood if one can assume that, during the time evolution, the invasion of most of the usersscreens by one single meme is unlikely.Plugging the full transient solution of P ( x, t | x ) [30] into (12) is neither immediate.An explicit analytical integration is not possible. However, if x is small enough, a Taylorexpansion of P ( x, t | x ) in x is suitable and makes the integration possible. Once the integralis carried out, we just need to take into account the normalization of P RMA to finally obtain P RMA ( x ) = exp (cid:0) − ab x (cid:1) xψ (cid:0) aNb (cid:1) , (13)where we recall that ψ denotes the digamma function. The normalization has been imposedfrom 1 /N that is the minimum non-vanishing value for attention fraction up to infinity.Note, that our assumptions legitimate the change of one by infinity in the upper bound. Fora detailed derivation of (13), see Appendix D.Remarkably, the outcome of our prediction for the relative meme attention is a log-series.This rekindles the classical result for RSA in ecology given by Fisher, who obtained a log-series RSA by assuming both a negative binomial distribution for the abundances and aninfinite number of species [37].We compare the log-series with the relative meme attention obtained from numericalsimulations of the interacting particle model in Fig. 4. As before, the result is shown for adelta like out-degree distribution peaked around k ∗ = 10. Surprisingly, in spite of the strongapproximation we have carried out, the log-series successfully explains the simulated pattern.We identify the fact of having a homogeneous degree distribution as a crucial property forthe fulfillment of our assumptions, and therefore the success of our prediction. In fact, whencomparing the log-series with the RMA for an OSN with power-law degree distribution, theapproximation fails as expected (see Appendix E for details).14 IG. 4. Relative memes attention. The probability P RMA ( x ) of finding a meme which receivea fraction x of attention is plotted. Theoretical prediction (blue line) given by (13) reproducesremarkably well the tail of the pattern obtained by numerical simulations (symbols). We haveconsidered the same model parameters to those chosen in Fig. 2. IV. DISCUSSION AND CONCLUSIONS
Having high diversity in information ecosystem is not less important than having a richbiodiversity in natural ecosystems. In fact, diversity and heterogeneity of ideas in OSNs isa crucial aspect for the quality of the deliberative process [38]. Online spaces dominated byone or few visions, i.e., with very low biodiversity, represent diseased information ecosystemswhere phenomena such as fake news, filter bubbles, and echo chambers crystallize beliefs andannihilate diverse opinions [39]. Therefore, being able to characterize diversity patterns ininformation ecosystems is not only a theoretically intriguing problem, but also an importantaspect to measure their “health”.In this work, we have proposed an analytically tractable neutral theory to describe thedynamics of user attention to competing memes in OSNs. In particular, we have shown that15e are able to compute several new quantities of interest such as the number of coexistingmemes, the distribution of user attention to these memes, and the average persistence timefor attention on a meme. All these emergent properties have an ecological analogy in naturalsystems, suggesting that an ecological approach to study information ecosystems can opennovel paths and understanding of the dynamics of memes in OSN.By comparing our theoretical predictions to numerical simulations of the model, we haveshown that the continuous approximation provides guaranteed results when the dynamics ofthe user attention x is diffusive rather than bursty. This, in turn, depends on the underlyingarchitecture of the user-user network. In fact, if the network is characterized by a scalefree degree distribution, then the diffusive approximation fails to quantitatively reproducethe biodiversity patterns, although the qualitative behavior is still well described. Usinga backward semi-analytical Master Equation approach allows to overcome this limitationand to correctly predict the mean number of different coexisting memes and their meanpersistence attention time. We note that having an analytical theory to calculate suchquantitative is of paramount importance, as it allows to easily understand the effect ofthe system parameters (such us the network connectivity or the innovation rate) to thepersistence of active memes in the OSN and their diversity. Moreover, we note that, becauseof very strong finite size effects and fluctuations in the dynamics, simulations may lead towrong conclusions especially when considering, as it typically is, large and heterogeneousOSN.Finally we note that, differently from popularity measures, user attentions to competingmemes is not a feature that can be directly measured from data but can only be evaluatedthrough some proxy. To this regard, a future perspective of this work is to connect meanpersistence attention time and RMA to actual measurable proxies. Doing so, we could alsounderstand if –and when– the neutrality of the dynamics is broken. For instance, strongexternal events, such as breaking news, may have an impact to the attention dynamics thatcannot be described in term of demographic stochasticity, but calls for incorporating in thisframework also environmental noise [40] and non-neutral effects [41].16 CKNOWLEDGMENTS
This project has been funded by Cariparo Foundation Visiting program 2019. SS andCP also acknowledge UNIPD Stats grant BioReact 2018.
Appendix A: Transition probabilities and diffusive approximation
We put forward explicitly, in this appendix, the functional form of the transition proba-bilities assumed in our model of OSN. As explained in the main text, there are three differentclasses of transformations that correspond to: • Spread of a node that holds the meme A and has k followers, j of the which hold thememe B . The probability of such a transition in a system with ν nodes carrying meme A is W A,sk,j ( ν ) = νN (1 − ˜ µ ) p k (cid:18) ν − k − j (cid:19)(cid:18) N − νj (cid:19)(cid:18) N − k (cid:19) . (A1) • Innovation of a node that holds the meme A and has k followers, j of the which holdthe meme A . The probability of such a transition in a system with ν nodes carryingmeme A is W A,ik,j ( ν ) = νN ˜ µp k (cid:18) ν − j (cid:19)(cid:18) N − νk − j (cid:19)(cid:18) N − k (cid:19) . (A2) • Action of a node that holds the meme B and has k followers, j of the which hold thememe A . The probability of such a transition in a system with ν nodes carrying meme A is W Bk,j ( ν ) = N − νN p k (cid:18) νj (cid:19)(cid:18) N − ν − k − j (cid:19)(cid:18) N − k (cid:19) . (A3)All expressions are obtained as the product of four probability factors. The first one is theprobability of choosing the node that acts. The second one is the probability of the typeof event either spread or innovation. The third one is the probability of having k followers.The last one is the hypergeometric probability coming from the distribution of the possible17tates of the k followers. Note that the “second” factor in (A3) is a factor 1 that is notexplicitly written.The Master Equation (1) is simply a balance of gained and lost probability due to tran-sitions between states. Note that ν − ∆ ν appears as the argument of P with ∆ ν being thechange in the number of users paying attention to node A .In order to perform the diffusive approximation, we consider the limit N → ∞ , with x = ν/N . In such a limit, we can approximate the hypergeometric distribution with abinomial one, which yields the continuous description of the transition rates W A,sk,j ( x ) = x (1 − ˜ µ ) p k x j (1 − x ) k − j , (A4a) W A,ik,j ( x ) = x ˜ µp k (1 − x ) j x k − j , (A4b) W Bk,j ( x ) = (1 − x ) p k x j (1 − x ) k − j . (A4c)Carrying out an expansion of the Master Equation (1) in powers of N − , introducing tran-sition probabilities as in (A4) and assuming scaling of time t = ˜ t/N and innovation rate µ = N ˜ µ as introduced in the main text, we achieve the diffusive approximation given by theFokker-Planck equation (2) with the coefficient reported in (3). Appendix B: Backward Fokker-Planck equation and lifetime distribution
We have obtained the forward Fokker-Planck equation ∂ t P ( x, t ) = ∂ x [ axP ( x, t )] + 12 ∂ x [ bx (1 − x ) P ( x, t )] . (B1)The backward version of this equation, ∂ t Q ( x , t ) = − ax ∂ x Q ( x , t ) + bx (1 − x )2 ∂ x Q ( x , t ) , (B2)governs the probability Q ( x , t ) of reaching the absorbing boundary placed at x = 0 depart-ing from x after an evolution of time t . Consequently, the differential equation that holdsfor the probability of reaching the absorbing point at any time Π( x ) = (cid:82) ∞ dt Q ( x , t ) is0 = − ax ∂ x Π( x ) + bx (1 − x )2 ∂ x Π( x ) (B3)where we have used that Q ( x ,
0) = lim t →∞ Q ( x , t ) = 0. Since, the only absorbing boundaryis that at x = 0, the meaningful solution for the previous equation is Π( x ) = 1. In otherwords, all the memes eventually will reach the extinction.18o study the mean lifetime τ ( x ) of the memes we need to multiply the backward equation(B2) by t and then integrate for all time. After assuming regular behavior for the boundaryterms and taking into account that Π( x ) = 1, we get1 = − ax ∂ x τ ( x ) + bx (1 − x )2 ∂ x τ ( x ) . (B4)Imposing the boundary conditions τ (0) = 0 and lim x → τ ( x ) < ∞ we obtain solution (5)presented in the main text. Appendix C: Backward Master Equation and lifetime distribution
The backward Master Equation that rules the dynamics of the probability Q ( ν , ˜ t ) ofreaching extinction departing from a initial attention of ν after ˜ t time steps is written in(6). The probability in the next time step is a linear combination of the probability in thecurrent one. Therefore, we can write a vector equation of dimension N + 1 using matrixnotation (cid:98) Q (˜ t + 1) = (cid:98) M (cid:98) Q (˜ t ) (C1)with (cid:98) Q (˜ t ) being the vector with components Q ( ν , ˜ t ), for ν = { , . . . , N } , and (cid:98) M the( N + 1) × ( N + 1) matrix with elements M ν ,ν (cid:48) = N − (cid:88) k =0 (cid:104) W A,sk,ν (cid:48) − ν ( ν ) + W A,ik,ν − ν (cid:48) − ( ν ) + W Bk,ν − ν (cid:48) ( ν ) (cid:105) . (C2)In the previous equation, to prevent from clutter our formulae, we assume that transitionprobabilities are equal to zero when the indexes involves a transformation with no physicalmeaning, e.g. second subindex does not belong to the interval [0 , k ]. Probability conservationis reflected in the property N (cid:88) ν (cid:48) =0 M ν ,ν (cid:48) = 1 . (C3)The absorbing boundary condition at ν = 0 makes that Q (0 , ˜ t ) = δ ˜ t, . Hence, in dimen-sion N , we have that Q (˜ t + 1) = M Q (˜ t ) + M δ ˜ t, (C4)where Q (˜ t ) is the vector (cid:98) Q (˜ t ) after removing the first component corresponding to ν = 0, M is the submatrix of (cid:98) M obtained after removing the rows and columns corresponding to value19, and M is a vector column that contains the transition probabilities to the extinctionstate.We define the total probability of arriving 0 departing from n as the time sum Π = ∞ (cid:88) ˜ t =0 q (˜ t ) , (C5)where the components of Π are Π( ν ). Summing the backward Master Equation (C4) forall time, and taking into account that Q (0) = lim ˜ t →∞ Q (˜ t ) = 0, we get Π = M Π + M . (C6)Since property (C3) holds, we have that the solution for the previous equation is simply Π = , where we have used the same notation for the vector of dimension N full of onesthat in the main text. Consistently with our results in the continuous description and withour physical understanding, we get, in this exact framework, that all the memes will reachextinction eventually.The mean first passage time ˜ τ ( ν ) to reach the absorbing point starting from the state ν is given by ˜ τ = ∞ (cid:88) ˜ t =0 ˜ t Q (˜ t ) . (C7)Multiplying (C4) by ˜ t and summing for all time, we obtain the equation for the meanpersistence time (7) presented in the main text with its corresponding solution (8).It is also possible to compute the average of the square passage time, which is useful forthe variance. We define (cid:10) ˜ t (cid:11) = ∞ (cid:88) ˜ t =0 ˜ t Q (˜ t ) . (C8)Multiplying (C4) by ˜ t and summing for all time, we obtain (cid:10) ˜ t (cid:11) − τ + = M (cid:10) ˜ t (cid:11) , (C9)which has the solution (9) presented in the main text. Appendix D: Derivation of the relative meme attention
Our starting point here is the approximated Fokker-Planck equation after neglecting thesquared term in the diffusion coefficient, that is, ∂ t P ( x, t | x ) = ∂ x [ axP ( x, t | x )] + 12 ∂ x [ bxP ( x, t | x )] . (D1)20rom now on, we consider that the equation is defined in the region x >
0. We understandthe neglect of the squared term as a rescaling of the attention that moves the boundary from x = 1 to infinity. The solution of (D1) submitted to an absorbing boundary at x = 0 andthe initial condition P ( x, t | x ) = δ ( x − x ) is [30] P ( x, t | x ) = 2 ab − e − at exp (cid:18) − ab x + x e − at − e − at (cid:19) (cid:16) x x e − at (cid:17) I (cid:18) ab √ x xe − at − e − at (cid:19) . (D2)A Taylor expansion of this solution up to linear order yields P ( x, t | x ) (cid:39) a exp (cid:8)(cid:2) − (cid:0) at + ab x − e − at (cid:1)(cid:3)(cid:9) b (1 − e − at ) x . (D3)With this approximation, valid for small x , we can carry out exactly the integration for alltime that results (cid:90) ∞ dtP ( x, t | x ) ∝ exp (cid:0) − ab x (cid:1) x , (D4)where we have explicitly written the dependence with x . Since the relative meme attention P RMA is proportional to the integral in (D4), we just need to impose the normalizationcondition. Taking into account that the minimum non-vanishing value for the attention is1 /N , and assuming, as said above, that the upper bound is at infinity, we end up with the P RMA reported in equation (13) of the main text.
Appendix E: Role of the network structure
In the main text, we have tested our prediction in a system with a fixed number offollowers. Obviously, this is not the general case. To investigate the role of the networkstructure, we consider in this appendix a power-law out-degree distribution which is usuallyassumed and has been also empirically found [12] for some OSN. More accurately, we assumethat p k is zero for k < k min and p k ∝ k − α for k ≥ k min similarly to the power-law case studiedin the seminal work by Gleeson [14]. Namely, we consider k min = 4 and α = 2 .
5. On the onehand, this choice for the parameters produces an average number of follower close to ourprevious k ∗ = 10, (cid:104) k (cid:105) ≈
10, as in the case analyzed in the main text. On the other hand,the variability and thus the variance, become quite big in strong contrast with the previouscase.In figure 5, we put forward the comparison for the prediction regarding the persistencetime for the memes attention. Clearly, the result of the exact approach using the backward21
IG. 5. Mean persistence time for meme attention for power-law out degree distribution. Thesimulations (symbols) agrees perfectly with the exact solution (blue line) given by (8). Converselyto what it is observed with a fixed number followers, the diffusive approximation in (4) (red line)ceases to be such a good estimation. We have considered a system with N = 10 , an out-degreedistribution p k ∝ k − . for k ≥ µ = 100. In the inset, the variance of the persistence timeshows a perfect agreement with its the exact solution (9). Master Equation matches accurately the values found in the numerical simulations. Thediffusive approximation instead, fails to quantitatively describe the observed pattern. Notethat, with a power-law distribution, bursting events with big changes in the configurationof the system are possible. It is reasonable that a diffusive theory fails to capture such adynamics.The results for the number of memes coexisting are presented in Fig. 6. Our prediction,the Poisson distribution, explains quite successfully the pattern observed in the numericalsimulations. The quality of the agreement is slightly lower than that observed in the maintext. Herein, we see an underestimation of the tails of the theoretical prediction, especiallybelow the average value. A proper sampling of power-law is always delicate. Therefore,22
IG. 6. Histogram of number of different memes present in the whole system for power-lawout degree distribution. Numerical simulations agrees quite well with the theoretically predictedPoisson distribution. Note that there is no fitting parameter in the distribution since the mean isgiven by (11). We have considered the same model parameters to those chosen in Fig. 5. it is not surprising that statistics obtained from scale-free distributions are not so clean ingeneral.Finally, we show the comparison between the numerical simulation of the relative memeattention and our theoretical prediction in Fig. 7. Unfortunately, our approximated diffusivetheory does not catch the nature of the pattern found by numerical simulation. This factevinces again how diffusive approximation cannot reproduce a pattern which is the result ofa dynamics partially burst. Interestingly, the observed pattern is compatible with a power-law P RMA with exponent 1.5 in a broad range of attention fraction. This hints the powerof a scale-free out degree distribution to generate scale-free distributed patterns within thesystem. 23
IG. 7. Relative memes attention for power-law out degree distribution. The probability P RMA ( x )of finding a meme which receive a fraction x of attention is plotted. Theoretical prediction (red line)given by (13) fails to catch the behavior of the obtained in the numerical simulations (symbols).The observed patter is quite compatible with a power-law with exponent 1.5 (gray dashed line).We have considered the same model parameters to those chosen in Fig. 5. Appendix F: Simulation details
In this appendix, we cover in detail the algorithm used to carry out the numerical sim-ulations presented in this work. We have explicitly simulated the microscopic dynamicsdescribed in the main text. Specifically, we consider a system made by N nodes. Each ofthem carries just one meme. The dynamics is generated by repetition of the next recipe foreach time step: • We randomly choose a node with homogeneous probability 1 /N . • With probability ˜ µ , the node innovates the meme.24 The degree of the node k is chosen from the out degree distribution p k . • From the available N − k followers. • The meme of the firstly chosen node spreads to all the followers.Note that the connections in the network are not fixed, we work with a random networkwhich is dynamic. This random dynamic feature guarantees that the approach given bythe Master Equation used in the main text is the correct mathematical description of oursystem.For each set of parameters, we have run N S = 100 simulations of a total duration ˜ t f =5 · time steps. The measurements that correspond to the persistence time comes from theaverage of all registered times for those memes which reach extinction inside the simulationwindow. Recording the history of all memes, it is easy to recover the times to extinctionstarting from all possible initial attentions. We just need to subtract the time in whichthe meme has that initial attention to the extinction time of the corresponding meme. Wehave increased the statistics of the measurements of biodiversity, which are performed in thestationary state, taking different times of observation. We have taken not only ˜ t f , but also˜ t f − n ∆˜ t with n = 1 , . . . ,
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