Emergence of opinion leaders in reference networks

Individuals often refer to opinions of others when they make decisions in the real world. Our question is how the people’s reference structure self-organizes when people try to provide correct answers by referring to more accurate agents. We constructed an adaptive network model, in which each node represents an agent and each directed link represents a reference. In every iteration round within our model, each agent makes a decision sequentially by following the majority of the reference partners’ opinions and rewires a reference link to a partner if the partner’s performance falls below a given threshold. The value of this threshold is common for all agents and represents the performance assessment severity of the population. We found that the reference network self-organizes into a heterogeneous one with a nearly exponential in-degree (the number of followers) distribution, where reference links concentrate around agents with high intrinsic ability. In this heterogeneous network, the decision-making accuracy of agents improved on average. However, the proportion of agents who provided correct answers showed strong temporal fluctuation compared to that observed in the case in which each agent refers to randomly selected agents. We also found a counterintuitive phenomenon in which reference links concentrate more around high-ability agents and the population became smarter on average when the rewiring threshold was set lower than when it was set higher.

agents in such class as shown below. Suppose that the agent is the l-th earliest to make the decision (l can be chosen from s + 1 to N with equal probability). Given that the agent is the l-th earliest, there are l − 1 agents who have already stated their answers. Thus, the number of stated referents of the l-th earliest agent follows the binomial distribution with parameters M and (l − 1)/N , since agents state their answers in a randomly determined order. This leads to the expression (S.1) for the mean number of agents who belong to class C s .
To derive an approximation formula for the mean performance, we assumed that all the agents who made their decisions the earliest to the c 0 -th earliest belong to class C 0 , i.e., they had no stated referent in making their decision; hence, they decided relying only on their own belief. Similarly, all the agents who made the (c s−1 + 1)-th earliest to the c s -th earliest are assumed to belong to class C s , (s = 1, 2, . . . , M ), i.e., put the answers of s referents together with their own choices in making their decisions. Here, we further assume that s referents of an agent of class C s are randomly chosen from the agents of class C 0 , C 1 , . . . , C s−1 in proportion to c 0 : c 1 : · · · : c s−1 .
Let π s (p) be the mean performance of an agent who has ability p and belongs to class C s .
Clearly, π 0 (p) = p. Let r * s be a random variable that represents the performance of a referent who is being referred by the agents in class C s . Let us consider an agent who is randomly sampled from the population, say agent i. We approximated the probability that a referent of agent i has ability p as g(p) ≡k(p)/(N M ) regardless of the number of followers of the agent, wherek(p) is the mean in-degree of the agent with ability p, as explained in Section 3.1 in the main text (Eq. (2)). Under this assumption, let p * be a random variable that represents a referent's ability. Thus, we assumed that the ability of a referent p * follows the distribution g. Using the above simplifying assumptions, r * 1 can be approximated as r * 1 ≈ π 0 (p * ) = p * since r * 1 is the performance of an agent in class C 0 .
Furthermore, we approximated the value of r * 1 = p * by its mean, The performance of an agent in class C 1 can be described as The first term on the right-hand side of (S.2) stands for the contribution to the mean performance when both the referent and the agent him/herself gave correct answers. The second term is the contribution when either the referent or the agent him/herself gave a correct answer (the two different opinions tie in this case, and the actual decision is made by tossing a coin; hence, the factor 1/2). Similarly, π 1 (p) can be approximated as follows: Now, we consider the mean performance of referents who are being referred by the agents in class C 2 , r 2 ≡ r * 2 , as follows. Under our assumption, an agent in class C 2 refers to an agent in class C 0 with a probability of c 0 /(c 0 + c 1 ) and refers to an agent in class C 1 with a probability of c 1 /(c 0 + c 1 ). Thus, Since π 0 (p) = p, and by equation (S.3), functions π 0 and π 1 can be assumed as linear functions-we can see that π 0 (p) and π 1 (p) also increase nearly linearly with ability p in our computer simulation ( Fig.S.1). Therefore, the mean performance of referents who are being referred by the agents in class C 2 , r 2 , can be approximated as where π 0 (p) = p and π 1 (p) = (p + p * )/2. Therefore, Similarly, π 2 (p), r 3 , π 3 (p), . . . , r M , and π M (p) can be derived sequentially as follows: Let Q(s, j) be the probability that j out of s stated referents of an agent in class C s give correct answers, as When the number of stated referents of an agent is even, there are s + 1 answers/choice including his/her own choice. The agent can give a correct answer by majority-rule when more than s/2 answers/choice are correct. Therefore, the mean performance of an agent with ability p in class C s , π s (p), can be described as when s is even. Similarly, when s is odd, The mean performance of referents who are being referred by the agents in class C s , r s , can be derived as can be inductively assumed to be a linear function of p for any s according to Equations (S.7)-(S.10), and from . Therefore, the mean performance of agents in class C s can be represented as where A g,s and B g,s do not depend on p. The mean performance depends linearly on the agent's ability.
Finally, the mean performance π(p) of an agent with ability p can be approximated as Since π s−1 (p) is a linear function of p for each s, π(p) is also a linear function of p and can be represented as The mean performance calculated by equation (S.13) agrees with the simulation results shown in The linearity of mean performance when self-loops and overlaps are allowed Here, we show that the mean performance can be described as a linear function of the ability even if there are self-loops and overlaps in reference links. First, we consider the case where there are a ii (≥ 1) self-loops of an agent i, where we regard that agent i's choice has a weight of (a ii + 1) to him/herself in i's majority-rule voting. We can calculate the mean performance of agent i as follows. Let s be a ii plus the number of the agent i's stated referents other than agent i. Let when s is even. Note that π s (p i ) shown above is a linear function of p i . Using the case that s is odd in Equation (S.9), we can again describe π s (p i ) as a linear function of p i when s is odd. Similarly, we can confirm that the mean performance of an agent can be described as a linear function of the ability even when there are overlaps in reference links.

B The relationship between the mean performance and the mean ability of referents
In Section A, we derived the approximation formula for the mean performance π(p) of an agent with ability p, which is expressed in terms of the mean ability of referents p * = N i=1 p ik (p i )/(N M ). As discussed in the main text, adaptive rewiring and a lower kick-off threshold lead to both a high mean ability of referents p * (FIG. 12) and a high mean performance of each agent (FIG. 9).
In this section, we show that the mean performance of each agent increases with the mean ability of referents according to the formula for the mean performance that we obtained in Section A. We can, therefore, say that adaptive rewiring and a lower kick-off increase the mean ability of referents, and this leads to a high mean performance of each agent.

C The formal derivation of the mean in-degree
Here, we assume that the number of agents N is infinitely large, and p is a continuous value. Let g(p) be the probability density function for the ability of an agent who is being referred to in the evolved network. In addition, we define f (Π) as the probability density function of the performance of the agent who is being referred to in the evolved network. ψ(p) and φ(Π) respectively denote the unconditional probability density functions of the ability and performance of the agents, who are either being referred to or not. Let T Π be the mean duration that the agent with performance Π is kept linked by a follower.
Let f t (Π) be the probability density function for the performance of the agent who is being referred to at iteration time t. The function f t (Π) satisfies the following equation by assuming that a link directing to an agent with performance Π is detached with a probability of 1/T Π in a unit time interval: The first term on the right-hand side of equation (S.20) corresponds to the probability that a reference link to an agent with performance Π remains without being rewired in a unit time interval.
The second term corresponds to the probability that a link is newly rewired to an agent with performance Π after it is discarded. Therefore, in the equilibrium state, the probability density function for the performance of the agent who is being referred to in the evolved network, f (Π), holds: Equation (S.21) can be calculated as follows.
and hence, Since Π(p) = A g p + B g from Equation (S.13) in Section A, f (Π) =g((Π − B g )/A g )/A g and φ(Π) = ψ((Π − B g )/A g )/A g are satisfied. Therefore, In the main text, we set ψ(p) as i.e., p follows the uniform distribution U(0.5, 0.75). For the finite number of agents N , we can approximate the probability g(p i ) that agent i with ability p i is being referred from an agent as p ig (p ′ )dp ′ . Thus the solid lines in FIG. 7 (b) are calculated as (S.25) D The numerical procedure to obtain the mean duration that an agent keeps a follower In this section, we explain how we solve the recurrence equations (5) and (6)  for the recurrence of T Π (y). Here T Π (y) represents the mean time until the evaluated performance Y t of a referent, whose actual performance is Π, hits the threshold θ first time in the stochastic where t is the time since it is linked by a follower.
Note that the evaluated performance Y t is always less than 1, because the right side of (1) in the main text represents the internally dividing point of I t , which is either 0 or 1, and the current value of Y t . Let b be (1 − α)θ, which is the infimum of the realization of Y t because the evaluated performance is updated to (1 − α)θ when a referent with its evaluated performance θ gives a wrong answer. We discretized the interval  The equation (S.32) can be solved as where E represents the identity matrix.
The I(y 0 )-th element in the vector x is the duration that an agent with the performance Π is kept linked from a follower, T , for the initial evaluated performance Y 0 = y 0 .

E Another distribution of ability
To check the validity of our way to derive g(p), we apply it to another probability density function (p.d.f) of agent's ability. The applied p.d.f. has a saw-toothed shape with the vertical tip at p = 0.5, declining linearly with p towards zero at p = 0.75: We calculated the p.d.f. of the ability of being referred agentsg(p) =ψ(p)T Ap+B / 1 0ψ (p ′ )T Ap ′ +B dp ′ as the same way for ψ(p). To compare with the calculation results, we conducted simulation under parameters N = 100, M = 5, y 0 = 0.625 and α = 0.1. We set the ability of agents as p ig (p ′ )dp ′ , see Fig. S.2. Note that we obtained the exponential tailed in-degree distribution again with this p.d.f.ψ (Fig. S.3).
As shown in Figure S.2, the mean in-degree also increases exponentially with the ability. The same as (a) except that the vertical axis is logarithmically scaled. We can see approximately exponential tails in the evolved networks. In both of these figures, the distribution of the abilities is given byψ(p).