Shared Culture Needs Large Social Networks

Abstract

We report on a model for the emergence of shared cultural traits in a large group of individuals that imitate each other and interact in random social networks. The average size of the social network is shown to be a crucial variable ruling the emergence of a prevalent cultural trait shared by a majority of individuals. We distinguish between the case of a social network that changes while individuals develop their culture through the influence of others and the case of social networks which change slowly so that individuals reach a stable trait value before their social network changes appreciably. In both cases, no shared cultural trait emerges unless social network size is larger than a critical value. We also discuss how the formation of a shared cultural trait depends on the strength of mutual imitation, random factors that affect individuals’ decisions, and external influences such as social institutions.

Appendix

1.1 Model Analysis

We are mainly interested in the expected value of the average trait defined in equation (5.1). This can be computed using the methods of statistical mechanics, a branch of theoretical physics (e.g., Thompson 1979). We summarize below a calculation given in full in De Sanctis and Guerra (2008). The statistical mechanical approach relies on calculating the so-called pressure of the model, from which group-level quantities can be easily computed. The average trait value m, for instance, can be computed as the derivative of the pressure with respect to the external influence h in equation (5.8).

The pressure is defined based on the function in equation (5.3). In the case of fixed social networks, the pressure is

$$A_N (\alpha,\,J,\,h) = \frac{1}{N} \mathbb{E}\,{\rm{ln}}\sum\limits_s {{\rm{exp}}(-H(s) - hNm(s))}$$

((5.10))

where $\mathbb {E}$ denotes expectation with respect to the random choice of individuals that interact and with respect to the values of the interactions; $\sum _s$ is the sum over all possible trait configurations. In the case of fast-changing social networks, the pressure is instead

$$\bar A_N (\alpha,\,J,\,h) = \frac{1}{N} \mathbb{E}_{\rm{v}} \,\ln \,\mathbb{E}_{\rm{I}} \mathop \sum \limits_s {\rm{exp}}(-H(s) - hNm(s))$$

((5.11))

where $\mathbb{E}_{\rm I}$ is the expectation with respect to the random choice of which individuals interact, and $\mathbb{E}_{\rm v}$ is the expectation with respect to the values of the interactions. The results discussed below for this model hold also if $\mathbb{E}_{\rm v}$ is moved inside the logarithm.

The case of fixed social networks is technically more challenging, and exact results are known only for sub-critical social network size or in the limit of large imitation strength. The model, however, is well understood by approximation techniques and numerical simulations. The case of fast-changing social networks is technically simpler, and we start from it to sketch the strategy for computing A and m. It is shown in De Sanctis and Guerra (2008) that equation (5.10) can be rewritten as

$$\bar A_N (\alpha ,\beta ) = \alpha \,{\rm{ln}}\,{\rm{cosh}}\,J + \frac{1}{N} \mathbb{E}_\nu \,{\rm{ln}}\mathop \sum \limits_s {\rm{exp}}(K\,{\rm{ln}}(1 + m^2 \,{\rm{tanh}}\,J)) + Nhm)$$

where K is the Poisson random variable of mean $\alpha N$. It is then convenient to define the function

$$f(m) = {\rm{ln}}(1 + m^2 \,{\rm{tanh}}\,\beta )$$

((5.12))

which is easily verified to be convex, that is

$$f(m) \ge f(M) + f'(M)(m - M)$$

for any given M. This allows to derive a lower bound for A by means of a trial function (easy to compute) depending on a trial value M, which is a fixed number and not a function of s. Notice also the following trivial identity:

$$\sum\limits_M {\delta _{mM} = 1}$$

which instead allows to derive an upper bound for A, thanks to the obvious $\delta _{mM} \le 1$, by means of the same trial function plus a correction which vanishes for increasingly large groups. Jointly, these bounds allow to compute A exactly for very large groups (formally, infinite). The result of this calculation is

$$\mathop {\lim }\limits_{N \to \infty } \bar A_N (\alpha,\,J,\,h) = \mathop {{\rm{sup}}}\limits_M \bigg\{ {\ln \,2 + \alpha \,{\rm{ln}}\,{\rm{cosh}}\,J + \alpha \,{\rm{ln}}(1 + M^2 \,{\rm{tanh}}\,J) - } $$

$$(2\alpha \,{\rm{tanh}}\,J)\frac{{M^2 }}{{1 + M^2 \,{\rm{tanh}}\,J}} + {\rm{ln}}\,{\rm{cosh}}\left[ {(2\alpha \,{\rm{tanh}}\,J\frac{M}{{1 + M^2 \,{\rm{tanh}}\,J}} + h} \right]\Big. {} \Bigg\}$$

for all values of α, J, h. The supremum condition implies

$$\begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}M}}\left[ {\alpha f(M) - \alpha f'(M)M + \ln \,{\rm{cosh}}\,\alpha f'(M) + h} \right] = \\ \alpha f''(M)[{\rm{tanh}}(\alpha f'(M) + h) - M] = 0 \\ \end{array}$$

and yields the mean trait value m = M, which has to satisfy

$$m = {\rm{tanh}}\left[ {(2\alpha \,{\rm{tanh}}\,J)\frac{m}{{1 + m^2 \,{\rm{tanh}}\,J}} + h} \right]$$

Let us consider for simplicity the case h = 0. It is easy to see that the only solution is m = 0 in the region

$$2\alpha \,{\rm{tanh}}\,J \le 1$$

while two opposite non-zero solutions exist in the complementary region (Fig. 5.2). The model is hence solved, in the sense that we have an expression for A and (at least implicitly) for the mean opinion. The case h > 0 is illustrated graphically in Fig. 5.2.

Regarding the case of fixed social networks, it can be proved that A and $\bar A$ coincide and are equal to ln 2 + ln cosh J in the region where 2α tanh $J \le 1$, where the average trait value is 0. This is interpreted noticing that for sufficiently low α, individuals interact too little to influence each other enough so as to develop a common opinion, irrespective of whether social networks can or cannot change. In particular, notice that this is the case if α < 1/2, for any strength of the mutual influence J. When 2α tanh J > 1, then the two expressions (5.10) and (5.11) differ from ln 2 + ln cosh J and from one another. In both cases, if m (the positive solution, for instance) is 0 in the interval [0, 1/2 tanh J], then it deviates from 0 and reaches a certain asymptotic value as $J \to \infty$, i.e., when only the imitative process counts and two connected individuals align their opinion with probability 1. Even in this case, however, not all individuals have the same trait value, i.e., m < 1 unless social networks are very large (formally $\alpha \to \infty$).

The expression for the average trait is different in the case of fixed social networks, but the behavior is qualitatively the same in both models. An implicit expression for the mean opinion in the rigid model is known in the limit $J \to \infty$, where m is such that

$$m(\alpha ) = 1 - {\rm{exp}}( - 2\alpha m(\alpha ))$$

which coherently exhibits a critical value a = 1/2, below which m is equal to 0, and above which it is different from 0.

Lastly, we note that if we let $\alpha \to \infty$, $J \to 0$ with $2\alpha \tanh J = J'$ kept constant, both models reduce to a very simple one, in which all individuals agree on the same trait value as $J' \to \infty$, and if $J' \le 1$, they do not agree at all (m = 0). This limiting model is in fact equivalent to the basic model of collective agreement mentioned in the main text [cf. Section 5.1 and text around equation (5.5)].