A message-passing approach for threshold models of behavior in networks
AA message-passing approach for threshold models of behavior in networks
Munik Shrestha
1, 2 and Cristopher Moore Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (Dated: October 30, 2018)We study a simple model of how social behaviors, like trends and opinions, propagate in net-works where individuals adopt the trend when they are informed by threshold T neighbors who areadopters. Using a dynamic message-passing algorithm, we develop a tractable and computationallyefficient method that provides complete time evolution of each individual’s probability of adoptingthe trend or of the frequency of adopters and non-adopters in any arbitrary networks. We validatethe method by comparing it with Monte Carlo based agent simulation in real and synthetic net-works and provide an exact analytic scheme for large random networks, where simulation resultsmatch well. Our approach is general enough to incorporate non-Markovian processes and to includeheterogeneous thresholds and thus can be applied to explore rich sets of complex heterogeneousagent-based models. I. INTRODUCTION
Mathematical modeling of epidemics has attracted theinterest of researchers from diverse academic disciplines[1–17]. Epidemics range from outbreaks of infectious dis-ease to the contagion of social behaviors such as trends,memes, fads, political opinions, rumors, innovations, fi-nancial decisions, and so on. In an early study, sociologistMark Granovetter [18, 19] proposed a threshold model,where individuals adopt a behavior when they are in-formed by at least T of their neighbors.We consider a stochastic model similar to Granovet-ter’s with a trend propagating on a network. At eachtime, an individual has integer valued awareness of atrend ranging from 0 to T . Each time an individual isinformed by one of its neighbors, this awareness is incre-mented until it reaches the threshold T . At that point,that individual adopts the trend, and starts informingits neighbors about it. We will assume that the networktopology is fixed, but our model of information flow (or“contagion”) is probabilistic. Each adopter informs eachof its neighbors at a rate r ( τ ), where τ is the time elapsedsince it became an adopter. Since r ( τ ) may depend on τ , the resulting dynamics can be non-Markovian.Given an initial condition, where some individuals havealready become adopters, or have done so with someprobability, our goal in this paper is to calculate the prob-ability that any given individual i is an adopter (or notan adopter) as a function of time. We can do this by firstcalculating the probability P ia ( t ) that i has awareness a at time t . The probability that i is an adopter is then P iT .Calculating the time evolution of the probability P ia ( t )is non-trivial as a result of intrinsic nonlinearities in thedynamics. The heterogeneous network interactions be-tween individuals make it even harder. One simple way toestimate these probabilities is to put on a computational-frequentist hat, simulate the model many times indepen-dently by a Monte Carlo agent-based method, and mea-sure in what fraction of these runs each vertex becomes an adopter. Doing this is computationally costly, how-ever, as we are required to perform many independentruns of the simulationWe thus consider the dynamic message passing algo-rithm (DMP), where we evolve the probabilities P ia ( t )directly according to certain update equations. Com-pared to a Monte Carlo simulation that requires manyindependent runs, we only need to run the DMP algo-rithm once. In the special case where T = 1, DMP wasrecently formulated by Karrer and Newman [20] to an-alytically study non-Markovian dynamics of the Suscep-tible, Infected, Recovered (SIR) epidemic model of thenetworks. In an analogy with the SIR model, we some-times refer to a vertex as susceptible if it is not yet anadopter, infected if it is an adopter, and recovered if itis an adopter but the rate r ( τ ) at which it informs itsneighbors has dropped to zero.The underlying idea of dynamic message passing issimilar to belief propagation [21, 22], where we use thenetwork structure to update posterior probabilities of thevertices’ states. However, unlike belief propagation wherewe update posterior distributions according to Bayes’rule, the causal structure of information flow is captureddirectly by the time iteration of DMP. As in belief propa-gation, the DMP algorithm assumes that the neighbors ofeach vertex are conditionally independent of each other.As a result, like belief propagation, DMP is exact on treesand approximate on networks with loops, where the con-ditional independence assumption cannot capture higherorder correlations.However, as we will see, DMP gives good approxi-mations to the probabilities even on real networks withmany loops. We will show this by implementing it in areal social network, specifically Zachary’s karate club net-work [23]. Although the Zachary’s club network containsmany loops, the probabilities computed by DMP com-pare well with those from the Monte Carlo simulation.We present this in Section III.In the limit of large random networks in the Erd˝os-R´enyi model, or networks with a given degree distribu-tion, DMP is asymptotically exact because these net- a r X i v : . [ phy s i c s . s o c - ph ] D ec works are locally treelike. In Section IV, we use DMPto obtain the exact results for such random networks inthe thermodynamic limit. A. Related Work
There are many related studies that consider whatfraction of vertices eventually become adopters if eachneighbor informs them with probability p . The set ofeventual adopters are the ones who have at least T neigh-bors who are also adopters. This is reminiscent of themodel commonly studied in statistical physics as k -core(or bootstrap) percolation. The k -core is the maximalinduced subgraph in the network, such that each vertexhas at least k other neighbors in the subgraph.By deleting each edge with probability 1 − p indepen-dently, we can ask whether the resulting diluted networkin the thermodynamic limit contains an extensive k -corein the ensemble of similarly prepared networks. Inter-estingly for k ≥
3, the emergence of a k -core in randomnetworks is a first-order (discontinuous) phase transitionin the sense that when it first appears it covers a finitefraction of the network [24]. An early work on k -core per-colation was on the Bethe lattice in the context of mag-netic systems [25]. Recently, it has been used in studiesof the Ising model and nucleation [26, 27], analysis of zerotemperature jamming transitions [28], and in a bootstrappercolation model in square lattices and random graphs[13, 29–32]. II. MESSAGE PASSING APPROACH
We now formulate the dynamic message passing(DMP) technique for the threshold model described inSection I. We define the message U i ← j ( t ) as the proba-bility that vertex j has not informed i about the trendby time t . If we have U i ← j ( t ) for all neighboring pairs i , j , we will be able to calculate the marginal probability P ia ( t ) that i has awareness a at time t , i.e. that it hasbeen informed by a of its neighbors. We focus on initialconditions where each vertex is either an adopter or hasawareness zero. So given that i is not an initial adopter, P ia ( t ) = (cid:88) Θ ⊆ ∂i | Θ | = a (cid:89) j ∈ Θ (1 − U i ← j ( t )) (cid:89) j ∈ ∂i \ Θ U i ← j ( t ) . (1)Here, ∂i is the set of i ’s neighbors, and Θ ranges overall subsets of ∂i of size a . Note the conditional inde-pendence assumption in Equation (1), where we assumethat the events that j has informed (or not informed) i are independent. Given that i is not an initial adopter,the probability P iS ( t ) that the vertex i is susceptible attime t , i.e. its awareness is less than T at time t , is then P iS ( t ) = T − (cid:88) a P ia ( t ) . (2) Equivalently, P iS ( t ) = (cid:88) Θ ⊆ ∂i | Θ |
The message passing formulation in Section II is exactonly on trees, since we assumed that the probabilities P ia ( t ) are independent. However, typical networks con-tain many loops. Thus, the independence assumption ofthe message passing approach is an approximation in realnetworks. Our goal in this section is to see how accurateDMP is in real networks by comparing it with MonteCarlo simulations of the actual stochastic process.To compare the results between DMP and Monte Carlosimulations, we show the infection probability of each in-dividual calculated through both methods in a scatterplot. In Fig. 1, we compare the eventual infection (adop-tion) probability of each individual in Zachary’s karateclub network. Each point in the scatter plot refers tothe eventual infection probability of an individual in theclub. If the DMP were exact, all points in the figurewould lie exactly on the dotted diagonal line.Here, each individual’s threshold T is set to 2. Fourvertices labeled { , , , } in Fig. 1 (left) are the ini-tially infected individuals. We assume f ( τ ) = βe − ( γ + β ) τ with a transmission rate β = 0 . γ = 0 .
3. We simulate the actual stochastic process us-ing a continuous-time Monte Carlo method algorithm.Events are maintained in a priority queue using a heapdata structure to sort the events in the model: specifi-cally, sort the edges ( i, j ) according to the time at which j will inform i . The probabilities are then averaged over10 independent runs.In Fig. 2, using the same parameters and initial con-ditions as Fig. 1, we compare the infection probabilityof each individual at a particular finite time t = 2. Wechose this time because this is when the average numberof infected individuals is at its maximum.In Fig. 3, we again use the same parameters as Fig. 1,but with different initial conditions. Each individual isinitially infected with probability 0 .
2. There are now twosources of randomness in the model: the dynamics and
FIG. 1. Comparison (right) with a scatter plot of individuals eventual infection probability in the Zachary club (left), wherethreshold T = 2. Horizontal axis is the eventual infection probability calculated by the DMP, whereas vertical axis is the resultfrom the Monte Carlo simulation. Each point refers to the eventual infection probability of one of the individuals in the club.Here, four initially infected individuals are { , , , } . Simulation is averaged over 10 runs. Transmission rate β = 0.6, andrecovery rate γ = 0.3. Vertices on the left are colored according to their eventual infection probability from the DMP.FIG. 2. Same parameters and initial conditions as Fig. 1, except that we are comparing the infection probability at time t = 2. the set of initial adopters. This again forces us to domany independent runs of the Monte Carlo simulationto estimate the infection probabilities. By setting P iS (0)= 0.8 in Equation (3) however, we can calculate the in-fection probability with the same computational cost asbefore where the initial infectors were fixed. Accordinglyin Fig. 4, we show the density plot of the probability thateach individual (horizontal axis) is eventually infected,when each of them is initially infected with increasingprobability (vertical axis).Checking the scatter plot of the results computed fromDMP and Monte Carlo simulation in Figures 1 - 3, wefirst see that the results computed from DMP do not match perfectly with those from the simulation. Aspointed out in [20], where T = 1 the probability esti-mated by DMP is always an upper bound on the trueprobability, since the events that two or more neighborsbecome infected are positively correlated.However, for T > { } in Fig. 1, { , , , } in Fig. 2,and { , , } in Fig. 3.To see why this happens, suppose i has two neighbors, FIG. 3. Same as Fig. 1, where we compare individuals probability of eventually getting infected. Here the initial condition issuch that each is infected with probability 0.2.FIG. 4. We show the eventual infection probability of each individual (horizontal axis) in the Zachary karate club network atincreasing uniform probability (vertical axis) of getting infected initially. Here, threshold T = 2, transmission rate β = 0 .
6, andrecovery rate γ = 0 .
3. On the left is the result calculated through the DMP. Whereas, on the right, we show the result fromthe Monte Carlo simulations, where the probabilities are averaged over 10 runs for each initial infection probability. j and k . Let P [ i ] denote the probability that i becomesinfected, and let P [ j ] and P [ k ] denote the probabilitiesthat j and k inform i respectively. If T = 1, then P [ i ] = P [ j ∨ k ] = P [ j ] + P [ k ] − P [ j ∧ k ] . Let’s assume that DMP computes the right marginals,so that P DMP [ j ] = P [ j ] and P DMP [ k ] = P [ k ]. However,DMP ignores correlations, and assumes that these eventsare independent. Thus P DMP [ i ] = P [ j ] + P [ k ] − P [ j ] P [ k ] . However, j and k are positively correlated if they have acommon neighbor that may have infected them both, orif they are neighbors of each other. That is, P [ j ∧ k ] > P [ j ] P [ k ] . Then P [ i ] < P DMP [ i ], and DMP overestimates P [ i ]. Onthe other hand, if T = 2, then P [ i ] = P [ j ∧ k ] > P [ j ] P [ k ] = P DMP [ i ] , and DMP underestimates P [ i ].Similarly, suppose i has three neighbors, j , k , and (cid:96) .Again taking T = 2, we have P [ i ] = P [ j ∧ k ] + P [ j ∧ (cid:96) ] + P [ k ∧ (cid:96) ] − P [ j ∧ k ∧ (cid:96) ] , whereas, DMP gives P DMP [ i ] = P [ j ] P [ k ]+ P [ j ] P [ (cid:96) ]+ P [ k ] P [ (cid:96) ] − P [ j ] P [ k ] P [ (cid:96) ] . In this case, DMP can either underestimate or overesti-mate P [ i ], depending on the strength of the correlationsbetween its neighbors. For example, if (cid:96) is independentof j and k , then P [ i ] = P [ j ∧ k ] + P [ j ] P [ (cid:96) ] + P [ k ] P [ (cid:96) ] − P [ j ∧ k ] P [ (cid:96) ]= P [ j ∧ k ](1 − P [ (cid:96) ]) + ( P [ j ] + P [ k ]) P [ (cid:96) ] . If j and k are positively correlated so that P [ j ∧ k ] >P [ j ] P [ k ], then DMP underestimates P [ i ] if P [ (cid:96) ] < / P [ (cid:96) ] > / IV. EXACT SOLUTION IN NETWORKS WITHARBITRARY DEGREE DISTRIBUTIONS
In this section, we consider the message passing ap-proach in the ensemble of random networks in the ther-modynamic limit. Our goal is to show that DMP canbe applied to large random networks just as well as to aparticular finite network.In random networks, we are interested in the expectedbehavior of the dynamics rather than the dynamics ina single realization of the network. So, instead of com-puting messages for individual vertices, we assume thatthese messages are drawn from some probability distri-bution, and update this distribution based on their aver-age behavior. We can then compute the distribution ofmarginals as well.We consider random networks with a given degree dis-tribution, specifically an ensemble of networks called the configuration model [33]. Each of n vertices is first as-signed an integer degree from a specified degree distri-bution, say p k . We think of a vertex with degree k ashaving k “spokes” or half-edges coming out of it. Wethen choose a uniformly random matching of these 2 m spokes with each other, where m is the number of edgesin the network. The key fact is then that, in the ther-modynamic limit, i.e. n → ∞ , following an edge fromany given vertex connects with a vertex of degree k withprobability proportional to kp k . Strictly speaking, thismodel generates random multigraphs. But, the averagesize of such graphs is a constant as n → ∞ , as a re-sult of which the density of self-loops and multiple edgesvanishes when n is large.Now, consider the message U i ← j ( t ) from Equation (9).Recall that this is the probability that j has not informed i by time t . In the configuration model however, differentindividuals j are connected to i in different realizations ofthe network. But, edges are now statistically identical inthe sense that each edge identically connects to a vertexbased on its degree. So, we consider a single averagemessage U ( t ).This average message U ( t ) then has the following inter-pretation. It is the average probability that by followinga random edge, the neighbor we reach has not informedthe vertex we came from by time t . This in turn will tellus the probability P a ( t ) that a randomly chosen vertexhas awareness a at time t . However, this probability de-pends on the degree of the vertex: specifically, if it hasdegree k , then P a ( k, t ) = P S (0) (cid:18) ka (cid:19) U ( t ) k − a (1 − U ( t )) a . (11)Averaging over p k , we get P a ( t ) = P S (0) ∞ (cid:88) k p k (cid:18) ka (cid:19) U ( t ) k − a (1 − U ( t )) a . (12)It is useful to write this in terms of the generating func- tion G ( x ) of the degree distribution and its derivatives: G ( x ) = (cid:88) k p k x k , (13) G ( a ) ( x ) = d a G ( x ) dx a . (14)Then P a ( t ) can be written as P a ( t ) = P S (0) (1 − U ( t )) a a ! G ( a ) ( U ( t )) . (15)Thus the probability P S ( t ) that a randomly chosen vertexis susceptible at time t is P S ( t ) = T − (cid:88) a =0 P a ( t ) . (16)Equivalently, P S ( t ) = P S (0) T − (cid:88) a =0 (1 − U ( t )) a a ! G ( a ) ( U ( t )) . (17)So, we see that given U ( t ), computing P a ( t ) and P S ( t ) inthe configuration model reduces to knowing G ( a ) to someorder.To capture the information flow that U ( t ) represents inthe configuration model, we define the cavity probability Q ( t ) by simplifying Equation (4). This is the probabilitythat a randomly chosen edge leads to a vertex that has not been infected by time t , if the vertex we came fromis assumed to be absent from the network. Equivalently, Q ( t ) is the probability that if we follow a random edgefrom a vertex i , the vertex j it leads to has been informedby at most T − i . Thisprobability also depends on j ’s degree. Namely, if it hasdegree k + 1, then Q ( k, t ) = T − (cid:88) a =0 (cid:18) ka (cid:19) U ( t ) k − a (1 − U ( t )) a , (18)where k is the number of neighbors that j has other than i . As discussed above, a random edge leads to a ver-tex with degree k with probability proportional to kp k .Therefore, the probability that j has k neighbors otherthan i is q k = ( k + 1) p k +1 (cid:80) k kp k = ( k + 1) p k +1 G (1) (1) . (19)Averaging Q ( k, t ) over q k , we obtain Q ( t ) = (cid:88) k q k T − (cid:88) a =0 (cid:18) ka (cid:19) U ( t ) k − a (1 − U ( t )) a . (20)Similar to Equation (17), we can write Q ( t ) in terms ofthe generating function as Q ( t ) = 1 G (1) (1) T − (cid:88) a =0 (1 − U ( t )) a a ! G ( a +1) ( U ( t )) . (21)We now calculate U ( t ) by simplifying (i.e. averaging)Equation (9) for the configuration model. But, note theright-hand side of (9) consists of products of U ( t ), andthe average of products is not always the product of av-erages. In the limit n → ∞ however, the network islocally treelike in the sense that the typical size of theshortest loops diverges as O (log n ). As a result, U ( t ) isasymptotically independent, and the average of productsis equal to the product of averages. So, the self-consistentrelation for U ( t ) becomes U ( t ) = 1 − (cid:90) t dτ f ( τ ) + P S (0) (cid:90) t dt (cid:48) f ( t − t (cid:48) ) Q ( t (cid:48) ) . (22)To numerically integrate this equation in time, we differ-entiate it with respect to t , dU ( t ) dt = − f ( t ) + P S (0) f (0) Q ( t )+ P S (0) (cid:90) t dt (cid:48) Q ( t (cid:48) ) df ( t − t (cid:48) ) dt . (23)It is also possible to get this from Equation (8). We canfurther simplify this to an ordinary differential equationin some cases. For example, if f ( τ ) = βe − ( β + γ ) τ , we canwrite it as dU ( t ) dt = − βU ( t ) + γ (1 − U ( t )) + βP S (0) Q ( t ) . (24)So, given the initial conditions U (0) = 1 , P S (0), and G ( a ) ( x ), we can calculate P S ( t ) using Equation (17).Similarly, the fraction of infected and recovered verticesat time t can be calculated. Note that, in general, we canlet f ( τ ) depend on the degree of the vertex by followinga degree dependent transmission method formulated byNewman [3]. Similarly, we can allow for the case wherethe probability P T (0) = 1 − P S (0) of getting initiallyinfected depends on the degree of the vertex.In Fig. 5 (left), we show the time evolution of the frac-tion of susceptible (blue), infected (red), and recovered(green) vertices in the configuration model, where the de-grees are drawn from the Poisson distribution with mean c , or equivalently the Erd˝os-R´enyi graphs G ( n, p = c/n ).For Poisson distribution, G ( a ) ( x ) are given by c a e − c (1 − x ) .We take c = 9, T = 3, f ( τ ) = βe − ( β + γ ) τ , where β = 0 . γ = 0 .
2, and the initial fraction of adopters/infectedsis P T (0) = 0 . vertices averaged over100 runs. Similarly, Fig. 5 (right) gives the fraction P a ( t )of vertices with awareness a , where the continuous linesare obtained by using Equation (15).In Fig. 6, we show the fraction P T ( t ) of adopters as afunction of time for the same parameter values as Fig. 5,except where T is 1 (green square), 2 (blue circle), 3(magneta triangle), and 4 (black diamond). Root Mean Square deviations in the simulation are provided whenthey are larger than the the markers.Using the same framework, we can calculate theasymptotic probability u = U ( ∞ ) that the infection hasnot been transmitted along a random edge. This in turnwill tell us the asymptotic probability that a randomlychosen vertex ever becomes infected.We can think of the long time behavior as k -core perco-lation. Either the edge is closed in the sense that its otherendpoint fails to inform the vertex we came from, whichhappens with the probability 1 − p = 1 − (cid:82) ∞ f ( τ ) dτ . Inthis case, it does not matter if the neighbor gets infectedby its other neighbors, since it fails to inform the vertexwe came from. Or, it can be the case that the edge isopen (with probability p ), but the vertex we reach is itselfnot infected eventually by its other neighbors. This hap-pens when the neighbor we reach by randomly followingthe edge is informed by at most T − u : u = 1 − p + pP S (0) ∞ (cid:88) k q k T − (cid:88) a =0 (cid:18) ka (cid:19) u k − a (1 − u ) a = 1 − p + pP S (0) G (1) (1) T − (cid:88) a =0 (1 − u ) a a ! G ( a +1) ( u ) . (25)Note that we could have written this equally by takingthe limit t → ∞ in Equation (22). Similarly, the prob-ability P S that a randomly chosen vertex never gets in-fected, i.e. the fraction of susceptible vertices is P S = P S (0) T − (cid:88) a =0 (1 − u ) a a ! G ( a ) ( u ) . (26)For Erd˝os-R´enyi networks G ( n, p = c/n ), or equivalentlythe Poisson distribution with average degree c , we havethe following self-consistent relation for u : u = 1 − p + pP S (0) e − c (1 − u ) T − (cid:88) a =0 c a (1 − u ) a a ! . (27)We can also obtain this expression by following [32]. Sim-ilarly, P S in Erd˝os-R´enyi networks is P S = P S (0) e − c (1 − u ) T − (cid:88) a =0 c a (1 − u ) a a ! . (28)Equations (25) and (26) have a nice interpretation interms of well-studied problems in random graphs, includ-ing percolation and the emergence of the k -core. We saythat Equation (25) is the generating function in P S (0) ofthe size of the connected component of susceptible ver-tices by following a random edge in the long time limit.Similarly, Equation (26) is the generating function of thesize of the connected susceptible component of a ran-domly chosen vertex. FIG. 5. On the left is the dynamics in the Erd˝os-R´enyi graphs G ( n, p = c/n ) where individuals have threshold T = 3, averagedegree c = 9, initial fraction of adopters/infecteds P T (0) = 0.1. The fractions of infected, recovered and susceptible verticesare red, green, blue respectively. Continuous lines are analytic results calculated using our DMP approach, by numericallyintegrating Equation (24), whereas dots are from the Monte Carlo based simulations with 10 vertices averaged over 100 runs.Transmission rate β = 0.8, and recovery rate γ = 0.2. On the right is the time evolution of P a ( t ), where continuous lines arecalculated using Equation (15). Root Mean Square deviations in the simulation are provided when they are larger than themarkers.FIG. 6. Same parameters and initial conditions as Fig. 5,except we are computing the fraction P T ( t ) of adopters, i.e.either infected or recovered vertices, as a function of timewhen the threshold T is 1 (green square), 2 (blue circle), 3(magneta triangle), and 4 (black diamond). V. CONCLUDING REMARKS ANDGENERALIZATIONS
In this paper, we have considered the dynamicmessage-passing (DMP) technique to study a simplethreshold model of behavior in networks. In doing so,we are able capture how each individual’s probability ofbecoming an adopter evolves in time in an arbitrary net-work with far less computational cost than Monte Carlosimulations. Although DMP is exact only on trees, we observe that it compares well with simulations even in areal social network where there are many loops. Interest-ingly, unlike in the SIR model, or equivalently the case T = 1, there are cases where DMP can either underesti-mate or overestimate the probability of infection.In addition, we have used the DMP equations to giveanalytical results in the thermodynamic limit of largerandom networks. We have provided an exact analyticresult for calculating the time dependence of the proba-bilities, thereby learning something about the dynamicsof bootstrap percolation.The message-passing dynamics we have consideredhere can be generalized in many ways, including lettingthe transmission probability and the threshold vary arbi-trarily across edges and vertices. Because the transmis-sion rate r ( τ ) may depend on the elapsed time τ since anindividual became an adopter, our study can be imple-mented in networks where some non-Markovian assump-tions are warranted, as we pointed out in Section II.We can include so-called “rumor spreading” modelswhere, rather than setting r ( τ ) = 0 until an individual’sawareness reaches a threshold as we have done here, anindividual starts telling its neighbors about the rumoreven if it has only heard about it once. Such models wererecently applied to the diffusion of microfinance [17]. Wecan also let the rate at which an individual receives newinformation depend on its own awareness. An interestingcase is to consider a unimodal function.We can also consider a model where j can transmitrepeatedly to i , raising i ’s awareness each time. We sim-ply replace each directed edge ( j, i ) with T multi-edges.So, each message U i ← j ( t ) would now be mapped to T identical copies of itself. The update equations and ex-pressions are the same as above, but now we sum overall these multi-edges accordingly.In Section IV, we focused on random networks in theconfiguration model. However, the DMP equations canbe easily generalized to many other families of randomgraphs, including interdependent networks [16], scale-freenetworks [34], small-world networks [6, 7], and bipartitenetworks [15] to name a few. In some cases this is amatter of plugging in a different degree distribution, andallowing for a finite number of types of vertices. How-ever, for preferential attachment networks the topologyis correlated with the vertices’ ages, so we would have tolet the messages U ( t ) depend on the age of the verticessending them.We can also extend this study to a network that hascommunity structures such as the stochastic block model.We can then study how trends move through commu-nities, and how the distribution of initial adopters (for instance, whether they are concentrated in one commu-nity, or are spread across many communities) affects theeventual fraction of the network that adopts the trend.Community structures can be driven by socio-economic,ethnic, religious and linguistic separations. So, it wouldbe useful to gain some perspective on how the structuresof communities contribute to the norms and social pref-erences that prevail in real populations, and in turn howdifferences in these norms drive the division of social net-works into communities. VI. ACKNOWLEDGMENTS
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