A STOCHASTIC MODEL FOR THE SPREADING OF AN IDEA IN A HUMAN COMMUNITY

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Vol. 17 (2012) 83–93

World Scientific Publishing Companyc DOI:10.1142/S2010194512007970

A STOCHASTIC MODEL FOR THE SPREADING OF AN IDEA IN A HUMAN COMMUNITY

FREDERIK W. WIEGEL

Institute of Theoretical Physics, University of Amsterdam Science Park 904, 1090 GL Amsterdam, Netherlands

We call an idea “alive” in a human community of N individuals, if at least one of them accepts it. Such an individual will create “copies” of this idea; any such free-floating copy has a probability to be accepted by any other individual. In this way the idea can spread through the community. The opposite processes can also occur: somebody can drop a previously held notion; any free-floating copy of an idea can be annihilated (newspapers get thrown away, for example). We present a simplified stochastic model for these processes. The various transition probabilities combine into a single, dimensionless constant. A generating function technique is used to formulate the dynamics of the model in terms of a partial differential equation, which is first order in the time variable and second order in the auxiliary variable. Using this equation we calculate exactly the average life-time of an idea (the time between the moment the idea is “born” in a single individual, and the moment the idea goes extinct because it is dropped by the last person who still accepted it). This average life-time is a function of the dimensionless constant and N. It has a quite different form for the three basic cases in which the product of the dimensionless constant and N-1 is larger than, equal to, or smaller than unity. In the last section various extensions and relevant questions are discussed in a qualitative way.

1. Examples

In order to introduce the topic of this paper we give some examples of an “idea”

spreading through a “human community”.

(a) A Boarding School. The number of students is of order 10 to 1000; so a typical value of N would be 10². An idea spreading through this community could be a speciﬁc topic of gossip or a rumour. So one can divide the N individuals at any time into two groups: a group of n individuals who are aware of this rumour, and a group of (N− n) individuals who are not aware of it.

A second important distinction is the one that discriminates between students who know about this gossip, and what we shall call “free-ﬂoating copies” of the rumour. In the present case one could think of somebody writing an article in the school’s magazine in which the rumour is mentioned. If this magazine is then printed in 200 copies one has 200 “free-ﬂoating” copies of the idea. There is a probability that any of the (N− n) unaware students picks up the magazine, reads the relevant article and accepts it too. A loose way to phrase this is to say that an

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idea can occur in two states: as a “localized idea” bound to the mind of a speciﬁc student, or as a “free-ﬂoating” idea that is carried by the spoken word, the printed word, or any of the media.

(b) A Large Corporation. The number of employees might be in the range 10³ to 10⁵. Most of them will be active in the day-to-day running of the company.

Only a relatively small number of them will participate in the generation of new ideas that might help the company to adapt to the changes in society that come with time. Call this number N . I am not aware of any systematic way to estimate N . A very simple organizational argument suggests the proportionality

N ∼

(number of employees). (1.1)

This would give values of N in the range 30 to 400; roughly the same as for the boarding school. This example shows that one is especially interested in the spreading of relatively new ideas throughout the corporation.

(c) The Army of a Sovereign Nation. The number of personnel might be of order 10⁴to 10⁶. As an army is even more hierarchical than a commercial corpora- tion one would expect that the number N in this case would be smaller than the estimate (1.1); perhaps N is in the range 50 to 500.

It is of interest to note the diﬀerence between a regular army and a terrorist organization. The latter (probably) consists of small groups of individuals, so N could then be as small as 5 or 10 for a single group.

(d) A City. The number of inhabitants is in the range 10³to 10⁷. The “creative output” of cities has recently been studied by Bettencourt, Lobo, Helbing, K¨uhnert and West [1]. They found a remarkable pattern in which the various forms of creative output followed scaling laws with very similar exponents. We cannot explain these scaling exponents, but at least the model which will be analysed in the following pages gives a rough estimate for the number of ideas that are alive in a city, as a function of the size of the city. As modern civilization is the product of the city, this last example is the most important one to try to model in a quantitative way.

2. The Model

Follow one speciﬁc idea, which is spreading stochastically through a community of N individuals. At time t let n denote the number of individuals that are aware of the idea; hence (N− n) is the number of individuals that are unaware of it.

The model to be analysed here assumes that any individual that is aware of the idea will generate α free-ﬂoating copies of the idea in the environment that supports the community. Hence the total number of free-ﬂoating copies equals αn.

(In a more detailed model one could describe the population of these copies in a statistical way: they are created stochastically by the aware individuals, and they Int. J. Mod. Phys. Conf. Ser. 2012.17:83-93. Downloaded from www.worldscientific.com by 37.44.196.120 on 06/10/16. For personal use only.

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can be destroyed stochastically by various processes in the environment. We shall not pursue these details here.)

Let p(n, t) denote the probability that n individuals in the community will be aware of this idea, at time t. This is also the probability to find αn free-floating copies of the idea, or to find (N−n) unaware individuals, at time t. The probability p(n, t) can change, during the time interval (t, t + dt), as a result of four transitions.

• An individual who was aware of the idea at time t, has dropped it. This happens with probability

n p(n) k₃dt (2.1)

where we suppress t in the notation as it is the same everywhere.

• At time t there where (n + 1) aware individuals and one of them dropped the idea during time dt. This has a probability

(n + 1) p(n + 1) k₃dt. (2.2)

• One of the (N −n) unaware individuals has encountered one of the αn free- ﬂoating copies, and accepted it. Mass-action tells you that this happens with a probability

αn(N − n) p(n) k₀dt. (2.3)

• At time t there where (n−1) aware individuals, hence α(n−1) free-ﬂoating copies, and (N− n + 1) unaware individuals. The probability that an en- counter between them generates one more aware individual equals

α(n− 1)(N − n + 1) p(n − 1)k0dt. (2.4) Combination of these four contributions, with the proper (+) sign for gain and (-) sign for loss, gives the equation

∂

∂t p(n) = +k₀α (n− 1) (N − n + 1) p(n − 1)

−k₀α n (N− n) p(n) +k₃ (n + 1) p (n + 1)

−k₃ n p (n). (2.5)

This is the basic equation for the dynamics of the spreading in the present model.

Of course you always have

p(n, t) = 0 f or n < 0 or n > N. (2.6) Using the last equation one easily veriﬁes the relation

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∂

∂t

N n=0

p(n, t) = 0 , (2.7)

which shows that the probability distribution is always properly normalized if it starts that way at time zero.

3. The Generating Function

The probabilities p(n, t) can in principle be solved from (2.5), using the initial condition

p(n, 0) = 1 f or n = n₀

= 0 otherwise (3.1)

which expresses the fact that at time zero one has n₀ aware individuals (hence, by assumption, αn₀ free-ﬂoating copies of the idea). We are especially interested in n₀= 1 and in the average life-time of this idea.

This calculation is simpler if one uses the generating function

G(z, t)≡

∞ n=0

p(n, t) zⁿ. (3.2)

It is easy to verify that G has the following properties:

∞ 0

n p(n) zⁿ = z ∂G

∂z , (3.3a)

∞ 0

(n + 1) p (n + 1) zⁿ= ∂G

∂z , (3.3b)

∞ 0

n²p (n) zⁿ =

z ∂

∂z

z ∂

∂z

G , (3.3c)

∞ 0

(N− n) n p (n) zⁿ = N z∂G

∂z −

z ∂

∂z

z ∂

∂z

G. (3.3d)

It is also straightforward to show that, if

∞ 0

a (n) p (n) zⁿ= A (z) (3.4a)

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then

∞ 0

a (n− 1) p (n − 1) zⁿ= zA (z) . (3.4b)

Multiply (2.5) with zⁿand sum n from 0 to∞. With (3.2 − 4) one ﬁnds the following equation for the time-development of the generating function:

∂G

∂t = k₃(1− z)∂G

∂z + k₀αN (z− 1) z∂G

∂z +k₀α (1− z)

z ∂

∂z

z ∂

∂z

G. (3.5)

Note that, if G (1, 0) = 1 then G (1, t) = 1 for all later times t, which guarantees the normalization of the probabilities. We shall be interested in the behavior of G (z, t) in the interval 0 z 1. Actually, because N is ﬁnite, eq.(3.2) shows that G (z, t) is a polynomial of order N , and hence “well-behaved” mathematically speaking.

A first application of (3.5) is the analysis of the t→ ∞ equilibrium distribution Gêq(z). As ∂Gêq/∂t = 0 one can set the left-hand side equal to zero. It is straight- forward to verify that the only solution which is finite for all z, and which obeys the boundary condition Gêq(1) = 1, is

G^eq(z) = 1. (3.6)

A glance at the deﬁnition (3.2) shows that this implies:

t→∞lim p (n, t) = 1 f or n = 0

= 0 f or n > 0. (3.7)

In other words: after a suﬃciently long time any idea will be extinct (as long as the total number N of individuals is ﬁnite).

In order to calculate time-dependent properties of the model one writes (3.5) in a dimensionless form by introducing the new variables

τ = k₃t, (3.8)

a dimensionless time, and

γ = αk₀

k₃ , (3.9)

a dimensionless parameter. The function

H (z, τ ) = 1− G (z, τ) , (3.10)

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which measures how far G (z, t) is located below G^eq(z), is then the solution of the equation

∂H

∂τ ={1 − γ(N − 1)z} (1 − z)∂H

∂z + +γz²(1− z)∂²H

∂z², (3.11)

with initial condition

H (z, 0) = 1− zⁿ⁰

= 1− z if n₀= 1. (3.12)

Note that H (1, τ ) = 0 for all τ , and lim

τ→∞H (z, τ ) = 0 for all z.

4. The Average Life-Time of an Idea

The deﬁnition (3.2) shows that G (0, t) equals the probability that the idea is extinct at time t. The deﬁnition (3.10) shows, therefore, that H (0, τ ) equals the probability that the idea is still alive (i.e. not yet extinct) at scaled time τ . Hence, the probability that the idea went extinct during the scaled time interval (τ, τ + dτ ) equals

H (0, τ )− H (0, τ + dτ) = −dτ ∂

∂τH (0, τ ) .

This is also the probability that the idea had a scaled life-time in the interval (τ, τ + dτ ). The average scaled life-time thus equals

−

∞

0

τ∂H (0, τ )

∂τ dτ =

∞

0

H (0, τ ) dτ,

provided H (0, τ ) goes to zero faster than ¹_τ for τ → ∞.

In order to calculate this quantity one considers the function

W (z)≡

∞

0

H (z, τ ) dτ. (4.1)

The average life-time then equals k⁻¹₃ W (0). The function W (z) is the solution of an ordinary second order diﬀerential equation which follows from (3.11) by integration over τ from τ = 0 to τ =∞. The left-hand side gives

∞

0

∂H

∂τ dτ = H (z,∞) − H (z, 0) . Int. J. Mod. Phys. Conf. Ser. 2012.17:83-93. Downloaded from www.worldscientific.com by 37.44.196.120 on 06/10/16. For personal use only.

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But H (z,∞) = 0, and H (z, 0) = 1 − z for the case of interest (3.12). In this way one ﬁnds the equation

− 1 =

1− z

z₀

dW

dz + γz²d²W

dz² (4.2)

where we wrote

1

z₀ = γ (N− 1) (4.3)

for simplicity of notation. The equation has to be solved with the boundary condi- tions

W (1) = 0; W (0) =−1. (4.4)

The second boundary condition follows automatically from (4.2). Equation (4.2) implies a ﬁrst order equation for the derivative

F (z)≡dW

dz , (4.5)

namely,

− 1 =

1− z

z₀

F + γz²dF

dz, (4.6)

which must be solved subject to the boundary condition

F (0) =−1. (4.7)

Again, this boundary condition follows automatically from (4.6), provided F (z) is ﬁnite for all z.

It is straightforward to verify that the solution of (4.6) is given by the integral

F (z) =

−1 γ

z^N−1exp

1 γz

^z

0

y^−N−1exp

−1 γy

dy. (4.8)

This is the case because the right-hand side is a solution of the inhomogeneous equation. In principle one can add any solution of the homogenous equation; however it is easy to verify that any non-zero solution of the homogeneous equation diverges in the limit z↓ 0, and should therefore be rejected.

In order to proceed with the calculation of the average life-time, k₃⁻¹W (0), one integrates (4.5) from z = 0 to z = 1 :

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W (1)− W (0) =

1

0

F (z) dz. (4.9)

As W (1) = 0 you ﬁnd from the last two equations:

W (0) = 1 γ

1

0

dz z^N−1exp

1 γz

×

z

0

y^−N−1exp

−1 γy

dy. (4.10)

Using eq. 2.231.2 of ref.[2] the y-integral is found explicitly:

z

0

y^−N−1exp

−1 γy

dy = γ exp

−1 γz

1 z

_N−1

+

N−1

=1

(N− 1) (N − 2) · · · (N − ) γ z^N−1−

.

(4.11) Substitution into (4.8), and taking the limit z↓ 0, gives F (0) = −1, which shows that (4.8) obeys the boundary condition (4.7). Substitution into (4.10) gives, after a trivial integration:

W (0) = 1 +

N−1

=1

(N− 1) (N − 2) · · · (N − ) γ

( + 1). (4.12) As the average life-time of an idea equals k⁻¹₃ W (0) this formula is the basic result of the present model. In the next section we shall analyse it for the three parameter regimes γ (N− 1) < 1, γ (N − 1) = 1, γ (N − 1) > 1. We shall always be interested in the regime where

0 < γ  1 and 1 N < ∞. (4.13) The artiﬁcial limiting case

γ→ 0, N → ∞, γN = fixed (4.14)

is only of mathematical interest (in this case the model can be solved exactly, but it has lost most of its relevance to understanding what happens in actual human communities).

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5. Basic Cases

5.1. The case γ(N − 1) < 1

In this case the terms in (4.12) are decreasing with increasing . An upper bound to the ﬁnite sum would be

W (0) 1 +^N−1

=1

{(N − 1) γ}

 + 1 (5.1)

which is smaller than the inﬁnite series, so:

W (0) 1 +

∞ 1

{(N − 1) γ}

 + 1 . (5.2)

The series is easy to sum because 1 + 1

2ξ +1 3ξ²+1

4ξ³+· · · = −1

ξln (1− ξ) (5.3)

for|ξ| < 1. In this way one ﬁnds the upper bound

W (0) 1

(N− 1) γln

1

1− (N − 1) γ

. (5.4)

One concludes that in the case γ (N− 1) < 1 the average life time is given by a ﬁnite expression; when γ (N− 1) ↑ 1 it becomes large, but it grows slower than ln

1

1−γ(N−1)

.

5.2. The case γ (N − 1) = 1.

In this case the small- terms in the series in (4.12) can be approximated by ( + 1)⁻¹. This leads to the upper bound estimate

W (0) 1 +^N−1

=1

( + 1)⁻¹. (5.5)

For (N− 1) 1 the right-hand side approaches the value ln N + 0.5772 . . . , where the constant is sometimes called for Euler, sometimes for Mascheroni.

5.3. The case γ (N − 1) > 1.

In this case the terms in the series in (4.12) will ﬁrst increase, then decrease with increasing . Denoting the -th term by

w≡ γ ( + 1)

(N− 1)!

(N− 1 − )! (5.6)

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the maximum term obtains for = ₀which is the solution of w₀

w

0−1

= (N− 0) γ ₀

(₀+ 1) = 1. (5.7)

This gives

₀= N−1

γ, (5.8)

which is 1 because of (4.13); this is the reason why the factor ₀/ (₀+ 1) in (5.7) could be replaced by unity.

For the magnitude of the maximum term combination of (5.6) and the last equation gives

w₀ =

γ(^N−^γ¹) N−_γ¹+ 1

(N− 1)!

1 γ − 1

!

. (5.9)

This is very large as compared to unity (unless γ (N− 1) is very near to unity) so the bounds

1 + w₀< W (0) < 1 + (N− 1) w0 (5.10) show that for some applications W (0) could be replaced by the maximum term.

6. Discussion

In this section we bring together various comments and interpretations, each of which could serve as the starting point for future research.

(a) First of all it is important to realise that the model analysed in the preceding pages is very similar to a model for the infection of the cells of the human immune system by the HIV virus [3]. The notion that (some) ideas can be viewed as diseases that infect large parts of humanity will not surprise anybody who has watched humanity for some length of time.

(b) The analysis in this paper concerns the spreading of a single idea that is characterized by the parameters γ and k₃. We found for the average life-time of this idea the expression W (0) /k₃ where W (0) is given by (4.12). If the community is in a state of equilibrium one can extend the model by assuming that each member of the community will create a new idea with a probability β per unit of time. If, for simplicity, one assumes that all these various new ideas are characterized by the same values of γ and k₃, then the average number of ideas that are still alive at any time equals:

number of “living” ideas = βNW (γ, N) /k3 , (6.1) Int. J. Mod. Phys. Conf. Ser. 2012.17:83-93. Downloaded from www.worldscientific.com by 37.44.196.120 on 06/10/16. For personal use only.

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where W (γ, N ) is just another notation for W (0) which makes its dependence on γ and N more obvious. A more sophisticated variant of the model would assume that each member of the community will create a new idea of “type” j with a probability β_j per unit of time. All new ideas of type j are characterized by the same values γ_j and (k₃) j. The average number of new ideas that are still alive at any time, now equals:

number of “living” ideas = N

j

β_jW (γ_j, N ) / (k₃)_j . (6.2)

Note that the expressions (6.1,2) give the number of ideas that are alive in, for example, a city with N inhabitants; they do not tell you anything about the number of inhabitants that are actually aware of any of these ideas.

(c) Questions about the number of individuals that are aware of a certain new idea could be answered only by fully solving the dynamical equation (2.5) or its equivalent (3.5). As an individual’s awareness of one new idea is independent of his/her awareness of another new idea, any member of the community could be

“host” to 0,1,2,... diﬀerent new ideas. It is only when an individual is aware of two (or more) new ideas simultaneously that they can jointly give birth to another - perhaps more intricate - new idea. In this way the aware individuals in a community can function like reaction vessels in which the chemistry of ideas proceeds to transmute simple ideas into more complicated ones.

Acknowledgment

I am indebted to Dr. Alan S. Perelson for our continuing collaboration, and to the Santa Fe Institute for its hospitality.

References

1. L. M. A. Bettencourt, J. Lobo, D. Helbing, Ch. K¨uhnert and G. B. West, Proc.

Nat. Acad. Sci. USA 104, 7301 (2007).

2. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Aca- demic Press, 1980).

3. A. S. Perelson and F.W. Wiegel, to be published.