COMPARTMENTAL LIMIT OF DISCRETE BASS MODELS ON NETWORKS GADI FIBICH AMIT GOLAN AND STEVE SCHOCHET

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COMPARTMENTAL LIMIT OF DISCRETE BASS MODELS ON

NETWORKS

GADI FIBICH, AMIT GOLAN, AND STEVE SCHOCHET

DEPARTMENT OF APPLIED MATHEMATICS, TEL AVIV UNIVERSITY (FIBICH@TAU.AC.IL),

(AMITGOLAN33@GMAIL.COM), (SCHOCHET@TAUEX.TAU.AC.IL).

Abstract. We introduce a new method for proving the convergence and the

rate of convergence of discrete Bass models on various networks to their respec-

tive compartmental Bass models, as the population size Mbecomes inﬁnite.

In this method, the full set of master equations is reduced to a smaller system

of equations, which is closed and exact. The reduced ﬁnite system is embedded

into an inﬁnite system, and the convergence of that system to the inﬁnite limit

system is proved using standard ODE estimates. Finally, an ansatz provides

an exact closure of the inﬁnite limit system, which reduces that system to the

compartmental model.

Using this method, we show that when the network is complete and homo-

geneous, the discrete Bass model converges to the original 1969 compartmental

Bass model, at the rate of 1/M. When the network is circular, however, the

compartmental limit is diﬀerent, and the rate of convergence is exponential in

M. In the case of a heterogeneous network that consists of Khomogeneous

groups, the limit is given by a heterogeneous compartmental Bass model, and

the rate of convergence is 1/M . Using this compartmental model, we show

that when the heterogeneity in the external and internal inﬂuence parameters

among the Kgroups is positively monotonically related, heterogeneity slows

down the diﬀusion.

1. Introduction

Diﬀusion of innovations in networks has attracted the attention of researchers

in physics, mathematics, biology, computer science, social sciences, economics, and

management science, as it concerns the spreading of “items” ranging from diseases

and computer viruses to rumors, information, opinions, technologies, and innova-

tions [1,4,16,21,22,25]. In marketing, diﬀusion of new products plays a key

role, with applications in retail service, industrial technology, agriculture, and in

educational, pharmaceutical, and consumer-durables markets [19].

The ﬁrst quantitative model of the diﬀusion of new products was proposed in

1969 by Bass [5]. In this model, the rate of change of the number of individuals

who adopted the product by time tis

n0(t)=(M−n)p+q

Mn, n(0) = 0,(1)

where nis the number of adopters, Mis the population size, M−nare the remaining

potential adopters, pis the rate of external inﬂuences by mass media (TV, news-

papers,...) on any nonadopter to adopt, and q

Mis the rate of internal inﬂuences by

Keywords: Stochastic models, master equations, convergence, rate of convergence, compart-

mental models, diﬀusion in networks, ordinary diﬀerential equations, heterogeneity.

MSC[2020] Primary: 91D30, Secondary: 34E10, 60J80, 92D25.

arXiv:2210.04011v1 [math.CA] 8 Oct 2022

2 G. FIBICH, A. GOLAN, AND S. SCHOCHET

any adopter on any nonadopter to adopt (“word of mouth”, “peer eﬀect”). Internal

inﬂuences are additive, so that the overall rate of internal inﬂuences is proportional

to n.

The Bass model (1) is a compartmental model. Thus, the population is divided

into two compartments (groups), adopters and nonadopters, and individuals move

between these two compartments at the rate given by (1). The Bass model is one of

the most cited papers in Management Science [26]. Almost all its extensions have

also been compartmental models; given by a deterministic ODE or ODEs. The

main advantage of compartmental models is that they are easy to analyze. From

a modeling perspective, however, one should start from ﬁrst principles, and model

the adoption of each individual using a stochastic “particle model”. The macro-

scopic/aggregate dynamics should then be derived from this discrete Bass model,

rather than assumed phenomenologically, which is done in compartmental Bass

models. Moreover, the discrete Bass model allows us to relax the assumption that

all individuals are connected (i.e., that the social network is a “complete graph”)

and have any network structure. The discrete Bass model also enables us to re-

lax the assumption that individuals are homogeneous, and allows for heterogeneous

individuals, which is much more realistic.

At present, the only rigorous result on the relation between discrete and compart-

mental Bass models is by Niu [20], who derived the compartmental Bass model (1)

as the M→ ∞ limit of the discrete Bass model on a homogeneous complete net-

work. The approach in [20], however, does not extend to other types of networks,

nor does it provide the rate of convergence. Fibich and Gibori derived an explicit

expression for the macroscopic diﬀusion in the discrete Bass model on inﬁnite cir-

cles [13]. They did not prove rigorously, however, that this expression is the limit

of the discrete Bass model on a circle with Mnodes as M→ ∞, nor did they ﬁnd

the rate of convergence.

In this paper, we present a novel method for proving the convergence of dis-

crete Bass models.This method can be applied to various network types, and it also

provides the convergence rate. Since real networks are ﬁnite, the convergence rate

provides an estimate for the diﬀerence between a ﬁnite network and its inﬁnite-

population compartmental limit.

We ﬁrst use our method to provide an alternative proof to the convergence of

the discrete Bass model on a homogeneous complete network, and to show that the

rate of convergence is 1

M. We then use this method to prove the convergence of the

discrete Bass model on the inﬁnite circle, and to show that the rate of convergence

is exponential in M. Finally, we use this method to prove the convergence of a

discrete Bass model in a heterogeneous network. Speciﬁcally, we consider a het-

erogeneous population which consists of Kgroups, each of which is homogeneous.

We show that as M→ ∞, the fraction of adopters in the heterogeneous discrete

model approaches that of the compartmental model. Then, we analyze the qual-

itative eﬀect of heterogeneity in the heterogeneous compartmental Bass model. In

particular, we show that when the heterogeneity is just in {pj}, just in {qj}, or

when {pj}and {qj}are positively monotonically related, then heterogeneity slows

down the diﬀusion.

The main contributions of this paper are:

COMP. LIMIT OF DISCRETE BASS MODELS 3

(1) A new method for proving the convergence and the rate of convergence of

discrete Bass models as M→ ∞. This method is based on embedding a

system of ODEs with a varying number of equations in an inﬁnite system.

(2) A convergence proof for the discrete Bass model on the circle, and for a

heterogeneous network with Kgroups.

(3) Finding the rate of convergence of the discrete Bass model on a homoge-

neous complete network, a heterogeneous network with Kgroups, and a

homogeneous circle.

(4) An elementary proof that heterogeneity slows down the diﬀusion whenever

the heterogeneity is just in {pj}, just in {qj}, or when {pj}and {qj}are

positively monotonically related.

2. Discrete Bass model

We begin by introducing the discrete Bass model for the diﬀusion of new prod-

ucts. A new product is introduced at time t= 0 to a network with Mconsumers.

We denote by Xj(t) the state of consumer jat time t, so that

Xj(t) = (1,if jadopts the product by time t,

0,otherwise.

Since all consumers are initially nonadopters,

Xj(0) = 0, j = 1, . . . , M. (2)

Once a consumer adopts the product, she remains an adopter for all time. The

underlying social network is represented by a weighted directed graph, where the

weight of the edge from node ito node jis qi,j ≥0, and qi,j = 0 if there is no

edge from ito j. We scale the weights so that if ialready adopted the product

and qi,j >0, her rate of internal inﬂuence on consumer jto adopt is qi,j

dj(M), where

dj(M) is the number of edges leading to node j(the indegree of node j). This

scaling ensures that the maximal internal inﬂuence

qj:=

m=1

m6=j

qm,j

dj(M)(3)

on a nonadopter, which occurs when all her peers are adopters, will remain bounded

as Mtends to inﬁnity if the qm,j are bounded, and will equal their common value

when all the qm,j corresponding to edges leading to jare equal. In addition,

consumer jexperiences an external inﬂuence to adopt, at the rate of pj>0.

Hence, to ﬁrst order in ∆t,

Prob(Xj(t+ ∆t) = 1 |X(t)) = 









1,if Xj(t)=1,



pj+

m=1

m6=j

qm,j

dj(M)Xm(t)

∆t, if Xj(t)=0,

(4)

where X(t) := (X1(t), . . . , XM(t)) is the state of the network at time t. The

quantity of most interest is the expected fraction of adopters

fdiscrete(t;{pj},{qm,j }, M ) = 1

ME[N(t)] ,(5)

4 G. FIBICH, A. GOLAN, AND S. SCHOCHET

where N(t) := PM

j=1 Xj(t) is the number of adopters at time t.

Let [Sm1, . . . , Smn] := Prob(Xmi= 0, i = 1, . . . , n) denote the probability that

the nnodes {m1, . . . , mn}are nonadopters, where 1 ≤n≤M,mi∈ {1, . . . , M},

and mi6=mjif i6=j. These probabilities satisfy the master equations:

Lemma 1 ([14]).The master equations for the discrete Bass model (2,4)are

dt[Sm1, . . . , Smn](t) = −



i=1

pmi+

j=n+1

i=1

qlj,mi

dj(M)

[Sm1, . . . , Smn]

j=n+1 n

i=1

qlj,mi

dj(M)![Sm1, . . . , Smn, Slj],

(6a)

for {m1, . . . , mn}({1, . . . , M},where {ln+1, . . . , lM}={1, . . . , M}\{m1, . . . , mn},

and

dt[S1, . . . , SM](t) = − M

i=1

pi![S1, . . . , SM],(6b)

subject to the initial conditions

[Sm1, . . . , Smn](0) = 1,{m1, . . . , mn}⊆{1, . . . , M}.(6c)

In what follows, we will use equations (6) to analyze the limit of fdiscrete as

M→ ∞. Indeed, since E[Xj(t)] = 1 −[Sj](t), then

fdiscrete = 1 −1

j=1

[Sj].(7)

Therefore,

lim

M→∞ fdiscrete(·, M )=1−lim

M→∞

j=1

[Sj](·, M).

2.1. Relation between discrete and compartmental Bass models. From a

modeling perspective, the discrete Bass model is more fundamental than the com-

partmental model. The latter model, however, is much easier to analyze. Indeed,

the homogeneous compartmental Bass model (1) can be rewritten as

dtf(t) = (1 −f)(p+qf), f(0) = 0,(8)

where f:= n

Mis the fraction of adopters. This equation can be easily solved,

yielding the Bass formula [5]

fBass(t;p, q) = 1−e−(p+q)t

1 + q

pe−(p+q)t.(9)

The corresponding discrete network is complete and homogeneous, i.e.

pj≡p, qk,j ≡q, dj(M)≡M−1, k, j = 1, . . . , M, k 6=j. (10)

In that case, (4) reads

Prob(Xj(t+ ∆t) = 1 |X(t)) = (1,if Xj(t)=1,

p+q

M−1N(t)∆t, if Xj(t)=0.(11)

COMP. LIMIT OF DISCRETE BASS MODELS 5

The relation between the discrete Bass model on a homogeneous network and the

compartmental Bass model was established by Niu:

Theorem 1 ([20]).Let p, q > 0. Then the expected fraction of adopters in the

discrete Bass model (2,11)on a homogeneous complete network approaches that of

the homogeneous compartmental Bass model (8)as M→ ∞, i.e.,

lim

M→∞ fcomplete

discrete (t;p, q, M) = fBass(t;p, q).(12)

As far as we know, Theorem 1is the only previous rigorous proof of convergence

of any discrete Bass model as M→ ∞.

3. Homogeneous complete network

In this section we introduce a novel method for proving Theorem 1. This method

also provides the rate of convergence, and can be extended to other types of net-

works.

Theorem 2. Assume the conditions of Theorem 1. Then the limit (12)is uniform

in t. Moreover, the rate of convergence is 1

M, i.e.,

fcomplete

discrete (t;p, q, M)−fBass(t;p, q) = O1

M, M → ∞.(13)

Proof. Our starting point are the master equations (6). When the network is ho-

mogeneous and complete, see (10), then by symmetry, [Sm1, . . . , Smn]

= [Sk1, . . . , Skn] for any {m1, . . . , mn}and {k1, . . . , kn} ⊂ {1, . . . , M}. Hence, we

can denote by [Sn](t) the probability that any arbitrary subset of nnodes are non-

adopters at time t. Using this symmetry and (10), the master equations (6) reduce

dt[Sn](t;M) = −np+M−n

M−1q[Sn] + nM−n

M−1q[Sn+1], n = 1, . . . , M −1,

(14a)

dt[SM](t;M) = −M p[SM],(14b)

subject to the initial conditions

[Sn](0; M)=1, n = 1, . . . , M. (14c)

Moreover, by (7),

fcomplete

discrete = 1 −[S],[S] := [S1].(15)

If we formally ﬁx nand let M→ ∞ in (14), we get that

dt[Sn

∞](t) = −n(p+q)[Sn

∞] + nq[Sn+1

∞],[Sn

∞](0) = 1, n = 1,2,.... (16)

This does not immediately imply that limM→∞[Sn] = [Sn

∞], since the number of

ODEs in (14) increases with M, and becomes inﬁnite in the limit. In Lemma 2

below, however, we will prove that for any n≥1,

lim

M→∞[Sn](t;M) = [Sn

∞](t),uniformly in t. (17a)

Moreover,

[Sn](t;M)−[Sn

∞](t) = O1

M, M → ∞.(17b)

Therefore, we can proceed to solve the inﬁnite system (16).

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COMPARTMENTALLIMITOFDISCRETEBASSMODELSONNETWORKSGADIFIBICH,AMITGOLAN,ANDSTEVESCHOCHETDEPARTMENTOFAPPLIEDMATHEMATICS,TELAVIVUNIVERSITY(FIBICH@TAU.AC.IL),(AMITGOLAN33@GMAIL.COM),(SCHOCHET@TAUEX.TAU.AC.IL).Abstract.WeintroduceanewmethodforprovingtheconvergenceandtherateofconvergenceofdiscreteBassmodels...

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