BBass-SIR model for diffusion of new products
Gadi Fibich ∗ Department of Applied Mathematics, Tel Aviv University, Tel Aviv 69978, Israel (Dated: September 22, 2018)We consider the diffusion of new products in social networks, where consumers who adopt theproduct can later “recover” and stop influencing others to adopt the product. We show that thediffusion is not described by the SIR model, but rather by a novel model, the Bass-SIR model, whichcombines the Bass model for diffusion of new products with the SIR model for epidemics. The phasetransition of consumers from non-adopters to adopters is described by a non-standard Kolmogorov-Johnson-Mehl-Avrami model, in which clusters growth is limited by adopters’ recovery. Therefore,diffusion in the Bass-SIR model only depends on the local structure of the social network, but noton the average distance between consumers. Consequently, unlike the SIR model, a small-worldsstructure has a negligible effect on the diffusion. Surprisingly, diffusion on scale-free networks isnearly identical to that on Cartesian ones.
PACS numbers: 89.65.Gh, 87.23.Ge, 02.50.Ey, 89.75.Hc, 64.60.Q-
Diffusion through social networks concerns the spread-ing of “items” ranging from diseases and computerviruses to rumors, information, opinions, technologiesand innovations, and has attracted the attention of re-searchers in physics, mathematics, biology, computer sci-ence, social sciences, economics, and management sci-ence [1–6]. In marketing, diffusion of new products is afundamental problem [7]. Ideally, firms would like to beable to predict future sales of a new product, its marketpotential, and the impact of various promotional strate-gies, based on sales data from the first few months.The first mathematical model of diffusion of new prod-ucts was proposed in 1969 by Bass [8]. The Bass modelinspired a huge body of theoretical and empirical research(in 2004 it was named one of the 10 most-cited papersin the 50-year history of Management Science [9]), indiverse areas such as retail service, industrial technol-ogy, agriculture, and in educational, pharmaceutical, andconsumer-durables markets [7]. In all these studies, how-ever, it was assumed that once a consumer adopts theproduct, he influences other non-adopters to adopt (ordisadopt) the product at all later times. More often thannot, however, adopters “ recover ” from influencing otherpeople after some time. For example, it was recently ob-served that new installations of solar photovoltaic (PV)systems are strongly influenced by the presence of nearbypreviously installed systems, but the effect of nearby PVsystems decays after several months [10].In this Letter, we study the diffusion of new productswhen adopters are allowed to recover. This problem can-not be analyzed using the SIR model [11], since in thismodel all the external adopters , i.e., those who were notinfluenced by previous adopters (“patients zero”), ex-ist at t = 0, which is not the case for new products.Therefore, we introduce a novel model, the Bass-SIRmodel, which allows for an on-going creation of external ∗ Electronic address: fi[email protected] adopters. We show that this difference in the generationof external adopters is not a technical issue, as it leadsto a completely different diffusion dynamics.To understand the effect of the network structurein the Bass-SIR model, we introduce a nonstandardKolmogorov-Johnson-Mehl-Avrami (KJMA) model forphase transitions, in which clusters growth is limited byadopters’ recovery. The KJMA model with recovery mayalso be relevant to other problems in physics, such as al-gorithmic self-assembly of DNA tiles [12], where “recov-ery” corresponds to an assembly error.
Discrete Bass-SIR model.—
Consider a new productwhich is introduced at time t = 0 to a market with M po-tential consumers. The consumers belong to a social net-work which is represented by an undirected graph, suchthat if consumers i and j are connected, they can influ-ence each other to adopt the product. As in the Bassmodel [8], if consumer j did not adopt the product bytime t , his probability to adopt (and thus become a con-tagious adopter) is [13]Prob (cid:18) j adopts in( t, t + ∆ t ) (cid:19) = (cid:18) p + q i j ( t ) k j (cid:19) ∆ t + o (∆ t ) (1a)as ∆ t →
0, where i j ( t ) is the number of contagiousadopters connected to j at time t , and k j is the numberof social connections (“degree”) of j . The parameters p and q describe the likelihood of a consumer to adopt theproduct due to external influences by mass media or com-mercials, and due to internal influences by contagiousadopters to which he is connected (“word of mouth”),respectively. In physical contexts, such influences cor-respond to an external source term and a drift term,respectively. In epidemics, such influences correspondto animal to human and human to human infections, re-spectively. The magnitude of internal influences increaseslinearly with the number i j of contagious adopters con-nected to j , and is normalized by k j so that regardlessof the network structure, the maximal internal influencethat j can experience (when all his social connections arecontagious adopters) is q . a r X i v : . [ phy s i c s . s o c - ph ] M a y Unlike previous Bass models, we do not assume thatadopters remain contagious forever. Rather, as in theSIR model [11], we assume that the probability of anadopter who was contagious at time t to become non-contagious (“recover”) in ( t, t + ∆ t ) isProb (cid:18) j recovers in( t, t + ∆ t ) (cid:19) = r ∆ t + o (∆ t ) (1b)as ∆ t →
0, where r is the recovery parameter. Since (1a)and (1b) come from the discrete Bass and SIR models, re-spectively, we refer to (1) as the discrete Bass-SIR model .We denote by S ( t ), I ( t ), and R ( t ) the fraction ofnon-adopters (“ susceptible ”), contagious adopters (“ in-fected ”), and non-contagious adopters (“ recovered ”) attime t , respectively. The fraction of adopters (contagiousand recovered) is denoted by f = I + R = 1 − S . Since theproduct is new, initially all consumers are non-adopters,and so S (0) = 1 and f (0) = I (0) = R (0) = 0. Non-spatial (complete) networks.—
When all M con-sumers are connected to each other, then i j ( t ) = M · I ( t )is the number of contagious adopters in the market and k j = M −
1. As M → ∞ , the aggregate (macroscopic)diffusion dynamics is governed by [21] S (cid:48) ( t ) = − S ( p + qI ) , I (cid:48) ( t ) = S ( p + qI ) − rI, R (cid:48) ( t ) = rI, (2a) S (0) = 1 , I (0) = 0 , R (0) = 0 , (2b)where (cid:48) = ddt . In the absence of recoveries ( r = 0), R = 0and f = I = 1 − S , and so eqs. (2) reduce to the originalBass model [8] f (cid:48) ( t ) = (1 − f )( p + qf ) , f (0) = 0 . Solving this equation yields the well-known Bass formula f Bass ( t ) = 1 − e − ( p + q ) t q/p ) e − ( p + q ) t . (3)Similarly, when p = 0, eqs. (2a) reduce to the SIRmodel [11] S (cid:48) ( t ) = − qSI, I (cid:48) ( t ) = qSI − rI, R (cid:48) ( t ) = rI, which is typically supplemented by the initial conditions S (0) = 1 − I , I (0) = I > , R (0) = 0 , where I is the fraction of contagious adopters at t = 0.Therefore, we refer to (2) as the continuous Bass-SIRmodel .Fig. 1A demonstrates the agreement between the dis-crete and continuous Bass-SIR models on a nonspatialnetwork. Fig. 1B shows the dependence of f ( t ), the frac-tion of adopters, on r . When r (cid:28) q , adopters have suf-ficient time to influence their social contacts before theybecome non-contagious. Hence, the effect of recovery issmall, and diffusion is only slightly slower than in the absence of recovery, i.e., f ( t ; p, q, r ) ≈ f ( t ; p, q, r = 0) = f Bass , see (3). As r increases, internal influences per-sist for shorter times, hence diffusion becomes slower.Therefore, f is monotonically decreasing in r . In par-ticular, when r (cid:29) q , adopters have little time to in-fluence their social contacts before they become non-contagious. Therefore, internal influences effectively dis-appear, and diffusion is driven by purely-external adop-tions, i.e., f ( t ; p, q, r ) ≈ f ( t ; p, q = 0) = 1 − e − pt . Inparticular, unless r (cid:28) q , neglecting recovery (i.e., usingthe Bass model and not the Bass-SIR model) leads toinaccurate results.. t * =qtf A continuous Bass−SIR discrete Bass−SIR 0 20 4000.51 t * =qtf B r=0r=0.1qr=qr=7qq=0 FIG. 1: Fraction of adopters in the Bass-SIR model on a non-spatial network, as a function of t ∗ = qt . Here p = 0 .
01 and q = 0 .
1. A) Agreement between the continuous model (2)[solid] and a single simulation of the discrete model (1)with M = 10 ,
000 [dashes]. Here r = 0 .
1. B) The contin-uous model with r = 0 , . q, q , and 7 q . Here r = 0 is f Bass ,see (3), and q = 0 is f = 1 − e − pt . Cartesian networks.—
To analyze the effect of anetwork with a spatial structure on the diffusion, wefirst consider periodic D -dimensional Cartesian networks,where each node (consumer) is connected to its 2 D near-est neighbors. In that case, relation (1a) readsProb (cid:18) j adopts in( t, t + ∆ t ) (cid:19) = (cid:18) p + q i j ( t )2 D (cid:19) ∆ t + o (∆ t ) . (4)Thus, when D = 1 the network is a circle and each con-sumer can be influenced by his left and right neighbors,when D = 2 the network is a torus and each consumercan be influenced by his up, down, left, and right neigh-bors, etc.Our simulations reveal that for given values of p , q ,and r , diffusion in a 2D network is faster than in a 1D net-work but slower than in a 3D network, which, in turn,is slower than in a nonspatial network (Fig. 2). Notethat this result is not obvious, since as a network getsmore connected, the effect of each connection decreases,so that the maximal internal influence remains q , see (1a).The differences among the four networks decrease with r .This is because a larger r means shorter internal effects,hence a weaker dependence on the network structure.A priori, it may seem that diffusion becomes fasterwith D , because for a Cartesian network with M con- f A nonspatial3D2D1D 0 4 800.51 r = 0.01 B f t * = qtC t * = qtD FIG. 2: Fraction of adopters in the Bass-SIR model on 1D(dots), 2D (dash-dot), 3D (solid), and nonspatial (dashes)networks. Here p = 0 . q = 0 .
1, and M = 10 , r = 0.B) r = 0 . q . C) r = q . D) r = 7 q . sumers, the average distance between consumers de-creases as M /D . If diffusion depends on the average dis-tance between consumers, however, then increasing thepopulation size should slow down the fractional adoption,so that lim M →∞ f ( t ) = 0. This, however, is not the case,since lim M →∞ f ≥ − e − pt . Kolmogorov-Johnson-Mehl-Avrami (KJMA) model.—
To understand the effect of the network structure, it isuseful to visualize the diffusion process as an on-goingrandom creation of external adopters (“seeds”). Oncecreated, each seed expands through internal adoptionsinto a cluster of adopters, and expanding clusters canmerge into larger clusters. This is nothing but the KJMAmodel for phase transitions [14–16] from non-adopters toadopters. Unlike its standard applications in physics (butas in algorithmic self-assembly of DNA tiles [22]), in theBass-SIR model the evolution of clusters can be morecomplex, because of the recoveries. To see that, in Fig. 3we simulate the evolution of single cluster in a 2D net-work, by placing a single contagious adopter at t = 0,and setting p = 0 in (1a) so that all subsequent adop-tions are purely-internal. When r is sufficiently small,clusters expand as squares/circles, whose radius increaseswith time (top row). As r increases, the cluster expandsmore slowly, the fraction of recovered adopters (out of alladopters) increases, and contagious adopters are mostlyconcentrated near the cluster surface (second raw). As r further increases (third row), some adopters on the clus-ter boundary recover before they lead to new adoptions.As a result, clusters evolve into irregular shapes. As r further increases (fourth row), the cluster ceases to ex-pand after some time, once all of its adopters becamenon-contagious.Since external adoptions are independent of the net-work structure, the KJMA model implies that the net-work structure affects the diffusion by affecting the av-erage rate at which clusters expand . We can use thisinsight to explain the results of Fig. 2, as follows. Theaverage radius ρ of clusters of size N scales as N D , hencetheir average surface area (i.e., the number of adopters onthe cluster surface which are in direct contact with non- FIG. 3: Typical evolution of a single cluster on a 2D network.Here p = 0, q = 0 .
1, and there is a single contagious adopterat t = 0. Each row corresponds to a different value of r .Contagious and recovered adopters are marked by orange andblack pixels, respectively. adopters) scales as ρ D − ∼ N D − D . Therefore, the clusterexpansion rate scales as qN D − D /D , see (4). Hence, thehigher D is, the faster the cluster expansion is [13]. Small-worlds network.—
The structure of real-life so-cial networks is different from that of Cartesian networks.Watts and Strogatz [17] suggested that social networkshave a small-worlds structure, whereby most connectionsare local, but there are also some random long-rangeconnections. They showed that even a small fractionof long-range connections can lead to a dramatic reduc-tion in the average distance between nodes. As a result,epidemics spread much faster on networks with a small-worlds structure.The acceleration of diffusion by a small-worlds struc-ture should be maximal in the 1D model, because fora given M , the average distance is maximal in the1D model. To induce a “
5% small-worlds structure ” inthe 1D model, we add a link between any two nodes withprobability 0 . /M , so that the average graph degree in-creases from 2 to 2.05. If, as a result, j is connectedto k j > j from q/ q/k j , inaccordance with (1a).Our simulations reveal that the addition of a small-worlds structure has a negligible effect on diffusion ofnew products (Fig. 4A–C). This is because a small-worldsstructure reduces the average distance between agents.This global network property has a negligible effect, how-ever, on diffusion of new products, which depends on thegrowth rate of a cluster, which, in turn, depends on localproperties of the network (such as the grid dimension D ).Indeed, roughly speaking, if in the absence of a small-worlds structure a certain cluster reaches at time t a sizeof N ( t ) = 20, then adding a “5% small-worlds structure”would increase N ( t ) at most to 21. f Bass−SIR A small worldsdeterministic 0 10 20 30 40 5005001000 n SIRD f B n E f t * = qtC n t * = qtF FIG. 4: Diffusion on a 1D network with (solid) and without(dashes) a small-worlds structure. Here q = 0 . M =10 , p = 0 .
01, and there are no adopters at t = 0. A) r = 0. B) r = 0 . q . C) r = q . (D)–(F) Number ofadopters in the discrete SIR model. Here p = 0, and diffusionstarts from 5 randomly-chosen contagious adopters at t = 0.D) r = 0. E) r = 0 . q . F) r = q . To show that our results are not inconsistent with [17],in Fig. 4D–F we repeat these simulations for the SIRmodel on the same 1D network with 5% small-worldsstructure, and with the same values of q and r . Thus, theonly differences from Fig. 4A–C is that we now set p = 0in (1a), and we let 5 randomly-chosen agents be conta-gious adopters at t = 0. In this case the small-worldsstructure has a major effect on diffusion, in agreementwith [17]. Interestingly, while in the absence of recov-ery a small-worlds structure always accelerates diffusion,in the presence of recoveries it may also slow it down(Fig. 4E). Scale-free networks.—
Another popular model for so-cial networks is that of a scale-free network. We con-structed scale-free networks using the Barab´asi-Albert(BA) preferential-attachment algorithm [18], in whicheach new node makes m new links with the existing net-work nodes, such that the probability of a new node toconnect to node i is k i / (cid:80) i k i , where k i is the degreeof node i . In the resulting scale-free network, if node j isconnected to k j nodes, the effect of each of these nodeson j is q/k j , see (1a).Our simulations of the discrete Bass-SIR model (1) onscale-free networks show that, as expected, the larger m is, the faster the diffusion (Fig. 5A). Surprisingly, the dif- fusion on a scale-free network with parameter m is nearlyidentical to that on a Cartesian network with D = m (Fig. 5B–E). This numerical observation is very surpris-ing, since these networks are different from each otherin almost any aspect. Yet, for some reason, the aver-age growth rate of clusters is nearly identical in thesenetworks. t * = qtf AAAA m=4m=3m=2m=10 5 1000.51 t * = qtf E scale−free (m=4)Cartesian (D=4)0 5 1000.51 t * = qtf D scale−free (m=3)Cartesian (D=3) 0 5 1000.51 t * = qtf C scale−free (m=2)Cartesian (D=2)0 5 1000.51 t * = qtf B scale−free (m=1)Cartesian (D=1) FIG. 5: Fraction of adopters in the Bass-SIR model (1) onnetworks with p = 0 . q = 0 . r = 0 . q , and M = 50 , m = 1 , ,
3, and 4. (B)–(E) Scale-free (solid) and Cartesian (dashes) networks. B) m = D = 1.C) m = D = 2. D) m = D = 3. E) m = D = 4. Summary.—
Two fundamental models of diffusion insocial networks are the Bass model for new products,and the SIR model for epidemics [2]. To the best of ourknowledge, these models have not been combined into asingle model until now.The Bass-SIR model is fundamentally different from ei-ther of these models. Indeed, since the Bass model doesnot allow for recovery, it cannot be used for products forwhich recovery affects the diffusion (e.g., solar PV sys-tems). The SIR model does allow for recovery. However,in the SIR model all external adopters exist at t = 0,whereas in the Bass-SIR model there is an on-going gen-eration of new external adopters. This difference in thegeneration of external adopters is not a technical issue,as it leads to a completely different diffusion dynamics.The key difference between these models is that inthe SIR model there is a threshold value of r , abovewhich the epidemics will peter out. In contrast, inthe Bass-SIR model everyone eventually adopts, since f ( t ; p, q, r ) ≥ f ( t ; p, , r ) = 1 − e − pt [23].The effect of the social network structure in these twomodels is also very different. Thus, in the SIR model,diffusion occurs through the expansion of a single clusterof internal adopters around “ patient zero ”. Therefore,the key determinant of diffusion speed is the average dis-tance from patient zero (or more generally, the averagedistance between individuals), which is a global propertyof the network. In contrast, diffusion in the Bass-SIRmodel occurs through the expansion of numerous clus-ters. Therefore, the diffusion speed is determined by thegrowth rate of clusters, which depends on local proper-ties of the network. It is because of these differencesthat, e.g., (i) A small-worlds structure has a large effecton diffusion in the SIR model, but a negligible one inthe Bass-SIR model. (ii) Doubling the population sizeroughly doubles the time T / for half of the populationto adopt in the SIR model, but has a negligible effecton T / in the Bass-SIR model. The choice between the Bass-SIR model and the SIRmodel depends on the initiation of the diffusion process.Diseases and rumors that start from a “patent zero” callfor the SIR model. External adoptions of new productsand external infections from mosquito bites are on-goingprocesses, and thus call for the Bass-SIR model.Some remaining open questions concern the effect ofthe network structure.For example, is it true that as M → ∞ , the diffusion on a scale-free network becomesidentical to that on a Cartesian network with D = m ?Can we derive macroscopic (averaged) equations for dif-fusion in Cartesian and scale-free networks? How doesrecovery affect the phase transition kinetics in the KJMAmodel? Acknowledgments.—
We thank G. Ariel and O. Razfor useful discussions. This research was conducted whilethe author was visiting the Center for Scientific Compu-tation and Mathematical Modeling (CSCAMM) at theUniversity of Maryland, and was partially supported bythe Kinetic Research Network (KI-Net) under NSF GrantNo. RNMS [1] E. Rogers,
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