aa r X i v : . [ phy s i c s . s o c - ph ] F e b August 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi
Advances in Complex Systemsc (cid:13)
World Scientific Publishing Company
On the evolution of a social network
TIMOTEO CARLETTI
D´epartement de math´ematiqueFacult´es Universitaires Notre Dame de la Paix, rempart de la Vierge 8Namur, B5000, [email protected]
DUCCIO FANELLI
Dipartimento di Energetica and CSDCUniversit`a di Firenze, and INFNvia S. Marta, 3, 50139 Firenze, Italyduccio.fanelli@unifi.it
SIMONE RIGHI
D´epartement de math´ematiqueFacult´es Universitaires Notre Dame de la Paix, rempart de la Vierge 8Namur, B5000, [email protected]
Received (received date)Revised (revised date)In this paper we show that the small world and weak ties phenomena can spontaneouslyemerge in a social network of interacting agents. This dynamics is simulated in the frame-work of a simplified model of opinion diffusion in an evolving social network where agentsare made to interact, possibly update their beliefs and modify the social relationshipsaccording to the opinion exchange.
Keywords : Opinion dynamics; social network; small world; weak ties.
1. Introduction
Modeling social phenomena represents a major challenge that has in recent yearsattracted a growing interest. Insight into the problem can be gained by resorting,among others, to the so called
Agent Based Models , an approach that is well suitedto bridge the gap between hypotheses concerning the microscopic behavior of indi-vidual agents and the emergence of collective phenomena in a population composedof many interacting heterogeneous entities.Constructing sound models deputed to return a reasonable approximation ofthe scrutinized dynamics is a delicate operation, given the degree of arbitrariness inassigning the rules that govern mutual interactions. In the vast majority of cases,data are scarce and do not sufficiently constrain the model, hence the provided ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi answers can be questionable. Despite this intrinsic limitation, it is however impor-tant to inspect the emerging dynamical properties of abstract models, formulatedso to incorporate the main distinctive traits of a social interaction scheme. In thispaper we aim at discussing one of such models, by combining analytical and nu-merical techniques. In particular, we will focus on characterizing the evolution ofthe underlying social network in terms of dynamical indicators.It is nowadays well accepted that several social groups display two main features:the small world property [18] and the presence of weak ties [12]. The first propertyimplies that the network exhibits clear tendency to organize in large, densely con-nected, clusters. As an example, the probability that two friends of mine are also,and independently, friends to each other is large. Moreover, the shortest path be-tween two generic individuals is small as compared to the analogous distance com-puted for a random network made of the same number of individuals and inter-linksconnections. This observation signals the existence of short cuts in the social tissue.The second property is related to the cohesion of the group which is mediated bysmall groups of well tied elements, that are conversely weakly connected to othergroups. The skeleton of a social community is hence a hierarchy of subgroups.A natural question arise on the ubiquity of the aforementioned peculariar as-pects, distinctive traits of a real social networks: how can they eventually emerge,starting from an finite group of randomly connected actors? We here provide an an-swer to this question in the framework of a minimalistic opinion dynamics model,which exploit an underlying substrate where opinions can flow. More specifically,the network that defines the topological structure is imagined to evolve, coupledto the opinions and following a specific set of rules: once two agents reach a com-promise and share a common opinion, they also increase their mutual degree ofacquaintance, so strengthing the reciprocal link. In this respect, the model thatwe are shortly going to introduce hypothesize a co-evolution of opinions and socialstructure, in the spirit of a genuine adaptive network [13, 19].Working within this framework, we will show that an initially generated randomgroup, with respect to both opinion and social ties, can evolve towards a final statewhere small worlds and weak ties effects are indeed present. The results of thispaper constitute the natural follow up of a series of papers [3, 9, 8], where thetime evolution of the opinions and affinity, together with the fragmentation vs.polarization phenomena, have been discussed.Different continuous opinion dynamics models have been presented in literature,see for instance [10, 11], dealing with the general consensus problem. The aim is toshed light onto the assumptions that can eventually yield to fixation, a final mono-clustered configuration where all agents share the same belief, starting from aninitial condition where the inspected population is instead fragmented into severalgroups. In doing so, and in most cases, a fixed network of interactions is a prioriimposed [2], and the polarization dynamics studied under the constraint of theimposed topology. At variance, and as previously remarked, we will instead allowthe underlying network to dynamically adjust in time, so modifying its initiallyugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi
On the evolution of a social network imposed characteristics. Let us start by revisiting the main ingredients of the model.A more detailed account can be found in [3].Consider a closed group of N agents, each one possessing its own opinion ona given subject. We here represent the opinion of element i as a continuous realvariable O i ∈ [0 , N − α ij , whose entries are real numberdefined in the interval [0 , α ij , the more reliablethe relation of i with the end node j .Both opinion and affinity evolve in time because of binary encounters betweenagents. It is likely that more interactions can potentially occur among individualsthat are more affine, as defined by the preceding indicator, or that share a closeopinion on a debated subject. Mathematically, these requirements can be accom-modated for by favoring the encounters between agents that minimizes the socialmetric D tij = | ∆ O tij | (1 − α tij ) + N j (0 , σ ), where ∆ O tij = O ti − O tj is the opinions’difference of agents i and j at time t , and the last term is a stochastic contribution,normally distributed with zero mean and variance σ . For a more detailed analysison the interpretation of σ as a social temperature responsible of a increased mixingability of the population, we refer to [3, 9, 8].Once two agents are selected for interaction they possibly update their opinions(if they are affine enough) and/or change their affinities (if they have close enoughopinions), following: ( O t +1 i = O ti − ∆ O tij Γ (cid:0) α tij (cid:1) α t +1 ij = α tij + α tij (1 − α tij ) Γ (cid:0) ∆ O tij (cid:1) , (1)being:Γ ( x ) = tanh( β ( x − α c )) + 12 and Γ ( x ) = − tanh( β ( | x | − ∆ O c )) , (2)two activating functions which formally reduce to step functions for large enoughthe values of the parameters β and β , as it is the case in the numerical simulationsreported below.Despite its simplicity the model exhibits an highly non linear dependence on theinvolved parameters, α c , ∆ O c and σ , with a phase transition between a polarizedand fragmented dynamics [3].A typical run for N = 100 agents is reported in the main panel of Fig. 1, for achoice of the parameters which yields to a consensus state. The insets represent threesuccessive time snapshots of the underlying social network: The N nodes are theindividuals, while the links are assigned based on the associated values of the affinity.The figures respectively refer to a relatively early stage of the evolution t = 1000,to an intermediate time t = 5000 and to the convergence time T c = 10763. Timeis here calculated as the number of iterations (not normalized with respect to N ).The corresponding three networks can be characterized using standard topologicalindicators [1, 5] (see Table 1), e.g. the mean degree < k > , the network clusteringugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi coefficient C and the average shortest path < ℓ > . An explicit definition of thosequantities will be given below.In the forthcoming discussion we will focus on the evolution of the networktopology, limited to a choice of the parameters that yield to a final mono cluster. Table 1. Topological indicators of the so-cial networks presented in Fig. 1. Themean degree < k > , the network cluster-ing C and the average shortest path < ℓ > are reported for the three time configura-tions depicted in the figure. < k > ( t ) C ( t ) < ℓ > ( t ) t = 1000 0 .
073 0 .
120 3 . t = 5000 0 .
244 0 .
337 2 . t = T c .
772 0 .
594 1 . α c = 0 .
5, ∆ O c = 0 . σ = 0 .
01. Theunderlying network is displayed at different times, testifying on its natural tendency to evolvetowards a single cluster of affine individuals. Initial opinions are uniformly distributed in theinterval [0 , α ij are randomly assigned in [0 , /
2] with uniform distribution. ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi
On the evolution of a social network
2. The social network
The affinity matrix drives the interaction via the selection mechanism. It hence canbe interpreted as the adjacency matrix of the underlying social network , i.e. thenetwork of social ties that influences the exchange of opinions between acquain-tances, as mediated by the encounters. Because the affinity is a dynamical variableof the model, we are actually focusing on an adaptive social network [13, 19] : Thenetwork topology influences in turn the dynamics of opinions, this latter providinga feedback on the network itself and so modifying its topology. In other words,the evolution of the topology is inherent to the dynamics of the model because ofthe proposed self-consistent formulation and not imposed a priori as an additional,external ingredient, (as e.g. rewire and/or add/remove links according to a givenprobability [14, 15] once the state variables have been updated).
Remark 1. (Weighted network)
Let us observe that the affinity assumes pos-itive real values, hence we can consider a weighted social networks, where agentsweigh the relationships. Alternatively, one can introduce a cut-off parameter, α f :agents i and j are socially linked if and only if the recorded relative affinity is largeenough, meaning α ij > α f . Roughly, the agent chooses its closest friends among allhis neighbors.The first approach avoids the introduction of non–smooth functions and it issuitable to carry on the analytical calculations. The latter results more straightfor-ward for numerical oriented applications.As anticipated, we are thus interested in analyzing the model, for a specificchoice of the parameters, α c , σ and ∆ O c , yielding to consensus, and studying theevolution of the network topology, here analyzed via standard network indicators:the average value of weighted degree , the cluster coefficient and the averaged short-est path . These quantities will be quantified for (i) a fixed population, monitoringtheir time dependence; (ii) as a function of the population size, photographing thedynamics at convergence, namely when consensus has been reached. Time evolution of weighted degree
The simplest and the most intensively studied one–vertex (i.e. local) characteristicis the node degree a : the total number of its connections or its nearest neighbors.Because we are dealing with a weighted network we can also introduce the weightednode degree , also called node strength [4], namely s i ( t ) = P j α tij / ( N − a Let us observe that the affinity may not be symmetric and thus the inspected social network willbe directed. One has thus to distinguish between
In–degree , k in , being the number of incomingedges of a vertex and Out–degree , k out , being the number of its outgoing edges . In the followingwe will be interested only in the outgoing degree, from here on simply referred to as to degree. ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi value averaged over the whole network reads: < s > ( t ) = 1 N N X i =1 s i ( t ) . (3)Let us observe that the normalization factor N − N agents, self-interaction being disregarded, < s > belongs hence to the interval [0 , ddt < s > = 1 N ( N − N X i,j =1 ddt α tij . (4)Let us observe that the evolution of affinity and opinion can be decoupled when∆ O c = 1. For ∆ O c <
1, this is not formally true. However on can argue for anapproximated strategy [9], by replacing the step function Γ by its time averagecounterpart γ , where the dependence in ∆ O tij has been silenced. In this way, weobtain form (4) ddt < s > = γ N ( N − N X i,j =1 α tij (1 − α tij ) = γ ( < s > − < s > ) , (5)where < s > = P α ij / ( N ( N − γ is of the order of 1 /N times, a factor taking care of the asynchronous dynamics [9].In [6] authors proved that (5) can be analytically solved once we provide theinitial distribution of node strengths (see Appendix A for a short discussion of theinvolved methods). Assuming s i (0) to be uniformly distributed in [0 , / < s > ( t ) = e γ t e γ t − − e γ t ( e γ t − log (cid:18) e γ t + 12 (cid:19) , (6)Using similar ideas we can prove [6] that the variance σ s ( t ) = < s > − < s > is analytically given by σ ( t ) = 2 e γ t ( e γ t − ( e γ t + 1) − e γ t ( e γ t − (cid:20) log (cid:18) e γ t + 12 (cid:19)(cid:21) . (7)The comparison between analytical and numerical profiles is enclosed in Fig. 2,where the evolution of < s > ( t ) is traced. Let us observe that here γ will serveas a fitting parameter, when testing the adequacy of the proposed analytical curvesversus direct simulations, instead of using its computed numerical value [9]. Thequalitative correspondence is rather satisfying, so confirming the correctness of theanalytical results reported above.ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi On the evolution of a social network time < s > ( t ) Fig. 2. Evolution of < s > ( t ). Dashed line (blue on-line) refers to numerical simulations withparameters α c = 0 .
5, ∆ O c = 0 . σ = 0 .
3. The full line (black on-line) is the analyticalsolution (6) with a best fitted parameter γ = 1 . − . The dot denotes the convergence time inthe opinion space to the consensus state, for the used parameters affinities did not yet converge.Let us observe in fact that affinities and opinions do converge on different time scale [9]. Assume T c to label the time needed for the consensus to be reached. Clearly, T c depends on the size of the simulated system b . From the above relation (6), theaverage node strength at convergence as an implicit function of the population size N reads: < s > ( T c ( N )) = e γ ( N ) T c ( N ) e γ ( N ) T c ( N ) − − e γ ( N ) T c ( N ) ( e γ ( N ) T c ( N ) − log (cid:18) e γ ( N ) T c ( N ) + 12 (cid:19) , (8)where we emphasized the dependence of γ and T c on N . However, as alreadyobserved γ ( N ) = O (cid:0) N − (cid:1) and T c ( N ) = O ( N a ), with a ∈ (1 , γ ( N ) T c ( N ) → N → ∞ and thus < s > ( T c ( N )) is predicted to be adecreasing function of the population size N , which converges to the asymptoticvalue 1 /
4, a value identical to the initial average node strength (see Fig. 3), giventhe selected initial condition. In sociological terms this means that even when con-sensus is achieved the larger the group the smaller, on average, the number of local b In [3, 7] it was shown that T c scales faster than linearly but slower than quadratically with respectto the population size N . ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi acquaintances. This is a second conclusion that one can reach on the basis of theabove analytical developments. N < s > ( T c ) Fig. 3. Average node strength at convergence as a function of the population size. Parametersare ∆ O c = 0 . σ = 0 . α c have been used : ( ♦ ) α c = 0, ( △ ) α c = 0 .
25, ( (cid:3) ) α c = 0 . (cid:13) ) α c = 0 .
75. Vertical bars are standard deviations computed over 10 replicas ofthe numerical simulation using the same initial conditions.
Small world
Several social networks exhibit the remarkable property that one can reach an arbi-trary far member of the community, via a relatively small number of intermediateacquaintances. This holds true irrespectively of the size of the underlying network.Experiments [16] have been devised to quantify the “degree of separation”in realsystem, and such phenomenon is nowadays termed the “small world”effect, alsoreferred to as the “six degree of separation”.On the other hand several, models have been proposed [18, 17] to constructcomplex networks with the small world property. Mathematically, one requires thatthe average shortest path grows at most logarithmic with respect to the network size,while the network still displays a large clustering coefficient. Namely, the networkhas an average shortest path comparable to that of a random network, with the samenumber of nodes and links, while the clustering coefficient is instead significantlylarger.ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi
On the evolution of a social network In this section we present numerical results aimed at describing the time evo-lution of both the average shortest path and the clustering coefficient of the socialnetwork emerging from the model. As before, the parameters are set so to inducethe convergence to a consensus state in the opinion space.We will be particularly interested in their asymptotic solutions, terming theassociated values respectively < ℓ > ( T c ) and C ( T c ) once the consensus state hasbeen achieved.In Fig. 4 we report these quantities (normalized to the homologous values esti-mated for a random network with identical number of nodes and links) versus thesystem size. The (normalized) clustering coefficient is sensibly larger than one, thiseffect being more pronounced the smaller the value of α c . On the other hand the(normalized) average shortest path is always very close to 1.Based on the above we are hence brought to conclude that the social networkemerging from the opinion exchanges, has the small world property. This is a re-markable feature because the social network evolves guided by the opinions, as itdoes in reality, and not result from an artificially imposed recipe. N C ( T c ) / C r nd N < ℓ > ( T c ) / < ℓ > r n d Fig. 4. Normalized clustering coefficient (left panel) and normalized average mean path (rightpanel) as functions of the network size at the convergence time. Parameters are ∆ O c = 0 . σ = 0 . α c have been used : ( ♦ ) α c = 0, ( △ ) α c = 0 .
25, ( (cid:3) ) α c = 0 . (cid:13) ) α c = 0 .
75. Vertical bars are standard deviations computed over 10 repetitions. ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi
Weak ties
Social networks are characterized by the presence of hierarchies of well tied smallgroups of acquaintances, that are possibly linked to other such groups via “weakties”. According to Granovetter [12] these weak links are fundamental for the cohe-sion of the society, being at the basis of the social tissue, so motivating the statement“the strength of weak ties”.The smallest group in a social network is composed by three individuals sharinghigh mutual affinities, in term of network theory they form a clique [1], i.e. a maximalcomplete graph composed by three nodes. This can of course be generalized to largermaximal complete graphs, defining thus m -cliques.The degree of cliqueness of a social network is hence a measure of its cohe-sion/fragmentation: the presence of a large number of m -cliques together with veryfew, m ′ -cliques, for m ′ > m , means that the population is actually fragmented intosmall pieces, of size m not strongly interacting each other.We are interested in studying such phenomenon within the social network emerg-ing from the opinion dynamics model here considered, still operating in consensusregime. To this end we proceed as follows. We introduce a cut–off parameter α f used to binarize the affinity matrix, which hence transforms into a an adjacencymatrix a . More precisely, agents i and j will be connected, i.e. a ij = 1, if and onlyif α ij ≥ α f . Once the adjacency matrix is being constructed, we compute the num-ber of m –cliques in the network. Let us observe that this last step is highly timeconsuming, being the clique problem NP-complete. We thus restrict our analysis tothe cases m ∈ { , , } .For small values of α f the network is almost complete, while for large ones it canin principle fragment into a vast number of finite small groups of agents. As reportedin the inset of right panel of Fig. 5, for α f ∼ α f is lowered. On the other hand for α f ∼ .
98 few 4–cliques emerge while 5–cliques appear around α f ∼ .
73. This means that the socialnetworks is mainly composed by 3–cliques, i.e. agents sharing high mutual affinities,that are connected together to form larger cliques, for instance 4 and 5–cliques byweaker links, i.e. whose mutual affinities are lower than the above ones.Results reported in left panel of Fig. 5 show that for specific parameter values,still falling into the class deputed to the consensus dynamics, the model does notpresent the weak ties phenomenon: 3, 4 and 5-cliques are all present at the sametime for large values of α f . This is an important point that will deserve futureinvestigations. Let us observe here that the observed differences stem from thesocial temperature.
3. Conclusion
Social system and opinion dynamics models are intensively investigated within sim-plified mathematical schemes. One of such model is here revisited and analyzed.The evolution of the underlying network of connections, here emblematized by theugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi
On the evolution of a social network α f C li qu e s cliques 3cliques 4cliques 5 0.7 0.75 0.8 0.85 0.9 0.95 10123456x 10 α f C li qu e s cliques 3cliques 4cliques 5 Fig. 5. Number of 3, 4 and 5–cliques in the social network once consensus has been achieved.Parameters are N = 100, ∆ O c = 0 . α c = 0 .
5. Right panel, σ = 0 .
5, the network exhibits theweak ties property. Left panel, σ = 0 .
1, the network does not display the weak ties phenomenon. mutual affinity score, is in particular studied. This is a dynamical quantity whichadjusts all along the system evolution, as follows a complex coupling with the opin-ion variables. In other words, the embedding social structure is adaptively createdand not a priori assigned, as it is customarily done. Starting from this setting, themodel is solved analytically, under specific approximations. The functional depen-dence on time of the networks mean characteristics are consequently elucidated. Theobtained solutions correlate with direct simulations, returning a satisfying agree-ment. Moreover, the structure of the social network is numerically monitored, viaa set of classical indicators. Small world effects, as well weak ties connections, arefound as an emerging property of the model. It is remarkable that such proper-ties, ubiquitous in nature, are spontaneously generated within a simple scenariowhich accounts for a minimal number of ingredients, in the context of a genuineself-consistent formulation.
Appendix A. On the momenta evolution
The aim of this section if to present and sketch the proof of the result used to studythe evolution of the momenta of the affinity distribution. We refer the interestedreader to [6] where a more detailed analysis is presented in a general setting.For the sake of simplicity, let us label the N ( N −
1) affinities values α ij by x l , upon assigning a specific re-ordering of the entries. Hence ~x is a vector with M = N ( N −
1) elements. As previously recalled (5), we assume each x l to obeya first order differential equation of the logistic type, once time has been rescaled,namely: dx l dt = x l (1 − x l ) . (A.1)ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi
The initial conditions will be denoted as x l .Let us observe that each component x l evolves independently from the other.We can hence imagine to deal with M replicas of a process ruled by by (A.1) whoseinitial conditions are distributed according to some given function. We are interestedin computing the momenta of the x variable as functions of the initial distribution.The m -th momentum is given by: < x m > ( t ) = ( x ( t )) m + · · · + ( x M ( t )) m M , (A.2)and its time evolution is straightforwardly obtained deriving (A.2) and making useof Eq. (A.1): ddt < x m > ( t ) = 1 M M X i =1 dx ml dt = mM N X l =1 x m − l dx l dt = mM N X l =1 x m − l x l (1 − x l ) = m (cid:0) < x m > − < x m +1 > (cid:1) . (A.3)To solve this equation we introduce the time dependent moment generating func-tion , G ( ξ, t ), G ( ξ, t ) := ∞ X m =1 ξ m < x m > ( t ) . (A.4)This is a formal power series whose Taylor coefficients are the momenta of thedistribution that we are willing to reconstruct, task that can be accomplished usingthe following relation: < x m > ( t ) := 1 m ! ∂ m G∂ξ m (cid:12)(cid:12)(cid:12) ξ =0 . (A.5)By exploiting the evolution’s law for each x l , we shall here obtain a partialdifferential equation governing the behavior of G . Knowing G will eventually enablesus to calculate any sought momentum via multiple differentiation with respect to ξ as stated in (A.5).On the other hand, by differentiating (A.4) with respect to time, one obtains : ∂G∂t = X m ≥ ξ m d < x m >dt = X m ≥ mξ m (cid:0) < x m > − < x m +1 > (cid:1) , (A.6)where used has been made of Eq. (A.3). We can now re-order the terms so toexpress the right hand side as a function of G c and finally obtain the followingnon–homogeneous linear partial differential equation: ∂ t G − ( ξ − ∂ ξ G = Gξ . (A.7) c Here the following algebraic relations are being used: ξ∂ ξ G ( ξ, t ) = ξ X m ≥ mξ m − < x m > = X m ≥ mξ m < x m > , ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi On the evolution of a social network Such an equation can be solved for ξ close to zero (as in the end of the procedurewe shall be interested in evaluating the derivatives at ξ = 0, see Eq. (A.5) ) and forall positive t . To this end we shall specify the initial datum: G ( ξ,
0) = X m ≥ ξ m < x m > (0) = Φ( ξ ) , (A.8)i.e. the initial momenta or their distribution.Before turning to solve (A.7), we first simplify it by introducing G = e g namely g = log G , (A.9)then for any derivative we have ∂ ∗ G = G∂ ∗ g , where ∗ = ξ or ∗ = t , thus (A.7) isequivalent to ∂ t g − ( ξ − ∂ ξ g = 1 ξ , (A.10)with the initial datum g ( ξ,
0) = φ ( ξ ) ≡ log Φ( ξ ) . (A.11)This latter equation can be solved using the method of the characteristics , hererepresented by: dξdt = − ( ξ − , (A.12)which are explicitly integrated to give: ξ ( t ) = 1 + ( ξ (0) − e − t , (A.13)where ξ (0) denotes ξ ( t ) at t = 0. Then the function u ( ξ ( t ) , t ) defined by: u ( ξ ( t ) , t ) := φ ( ξ (0)) + Z t
11 + ( ξ (0) − e − s ds , (A.14)is the solution of (A.10), restricted to the characteristics. Observe that u ( ξ (0) ,
0) = φ ( ξ (0)), so (A.14) solves also the initial value problem.Finally the solution of (A.11) is obtained from u by reversing the relation be-tween ξ ( t ) and ξ (0), i.e. ξ (0) = ( ξ ( t ) − e t + 1: g ( ξ, t ) = φ (cid:0) ( ξ − e t + 1 (cid:1) + λ ( ξ, t ) , (A.15) and ξ∂ ξ G ( ξ, t ) ξ = ξ∂ ξ X m ≥ ξ m − < x m > = ξ X m ≥ ( m − ξ m − < x m > = X m ≥ ( m − ξ m − < x m > Renaming the summation index, m − → m , one finally gets (note the sum still begins with m = 1): ξ∂ ξ G ( ξ, t ) ξ = X m ≥ mξ m < x m +1 > . ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi where λ ( ξ, t ) is the value of the integral in the right hand side of (A.14).This integral can be straightforwardly computed as follows (use the change ofvariable z = e − s ): λ = Z t
11 + ( ξ (0) − e − s ds = Z e − t − dzz
11 + ( ξ (0) − z , (A.16)which implies λ = − Z e − t dz (cid:18) z − ξ (0) −
11 + ( ξ (0) − z (cid:19) = − log z + log(1 + ( ξ (0) − z ) (cid:12)(cid:12)(cid:12) e − t = t + log(1 + ( ξ (0) − e − t ) − log ξ (0) . (A.17)According to (A.15) the solution g is then g ( ξ, t ) = φ (cid:0) ( ξ − e t + 1 (cid:1) + t + log ξ − log(( ξ − e t + 1) , (A.18)from which G straightforwardly follows: G ( ξ, t ) = Φ (cid:0) ( ξ − e t + 1 (cid:1) ξe t ( ξ − e t + 1 . (A.19)As anticipated, the function G makes it possible to estimate any momentum(A.5). As an example, the mean value correspond to setting m = 1, reads: < x > ( t ) = ∂ ξ G (cid:12)(cid:12)(cid:12) ξ =0 = h Φ ′ (cid:0) ξ − e t (cid:1) e t ξe t ( ξ − e t + 1+ Φ (cid:0) ξ − e t (cid:1) e t ( ξ − e t + 1 − ξe t (1 + ( ξ − e t ) i(cid:12)(cid:12)(cid:12) ξ =0 = e t − e t Φ(1 − e t ) . (A.20)In the following section we shall turn to considering a specific application in thecase of uniformly distributed values of affinities. A.1.
Uniform distributed initial conditions
The initial data x l are assumed to span uniformly the bound interval [0 , . ψ ( x ) clearly reads d : ψ ( x ) = (cid:26) x ∈ [0 , / , (A.21)and consequently the initial momenta are: < x m > (0) = Z ξ m ψ ( ξ ) dξ = Z / ξ m dξ = 1 m + 1 12 m . (A.22) d We hereby assume to sample over a large collection of independent replica of the system underscrutiny ( M is large). Under this hypothesis one can safely adopt a continuous approximation forthe distribution of allowed initial data. Conversely, if the number of realizations is small, finitesize corrections need to be included [6]. ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi On the evolution of a social network Hence the function Φ as defined in (A.8) takes the form:Φ( ξ ) = X m ≥ m + 1 ξ m m . (A.23)A straightforward algebraic manipulation allows us to re-write (A.23) as follows: X m ≥ y m m + 1 = 1 y Z y X m ≥ z m dz = 1 y Z y z − z dz = − − y log(1 − y ) , (A.24)thus Φ( ξ ) = − − ξ log (cid:18) − ξ (cid:19) . (A.25)We can now compute the time dependent moment generating function, G ( ξ, t ),given by (A.19) as: G ( ξ, t ) = ξe t ( ξ − e t + 1 (cid:20) − − ξ − e t + 1 log (cid:18) − ( ξ − e t + 12 (cid:19)(cid:21) , (A.26)and thus recalling (A.5) we get < x > ( t ) = e t e t − − e t ( e t − log (cid:18) e t + 12 (cid:19) (A.27) < x > ( t ) = e t ( e t − + 4 e t ( e t − log (cid:18) e t + 12 (cid:19) + 2 e t ( e t − ( e t + 1) . Let us observe that < x > ( t ) deviates from the logistic growth to which all thesingle variable x i ( t ) does obey.For large enough times, the distribution of the variable outputs is in fact con-centrated around the asymptotic value 1 with an associated variance (calculatedfrom the above momenta) which decreases monotonously with time.Let us observe that a naive approach would suggest interpolating the averagednumerical profile with a solution of the logistic model whose initial datum ˆ x actsas a free parameter to be adjusted to its best fitted value: as it is proven in [6] thisprocedure yields a significant discrepancy, which could be possibly misinterpretedas a failure of the underlying logistic evolution law. For this reason, and to avoiddrawing erroneous conclusions when ensemble averages are computed, attention hasto be payed on the role of initial conditions. References [1] Albert, R. and Barab´asi, A.-L., Statistical mechanics of complex networks,
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PRE (2007)066105.[4] Barrat, A. e. a., The architecture of complex weighted networks, PNAS (2004)3747. ugust 14, 2018 0:45 WSPC/INSTRUCTION FILE CarlettiFanelliRighi T. Carletti, D. Fanelli and S. Righi [5] Boccaletti, S. e. a., Complex networks: Structure and dynamics,
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PRE (2004) 065102. .7 0.75 0.8 0.85 0.9 0.95 10100200300400500600700800900 α f c li qu e s cliques 3cliques 4cliques 5 0.8 101020304050cliques 3cliques 4cliques 5 0.8 101020304050