Heterogeneity Improves Speed and Accuracy in Social Networks

How does temporally structured private and social information shape collective decisions? To address this question we consider a network of rational agents who independently accumulate private evidence that triggers a decision upon reaching a threshold. When seen by the whole network, the first agent’s choice initiates a wave of new decisions; later decisions have less impact. In heterogeneous networks, first decisions are made quickly by impulsive individuals who need little evidence to make a choice but, even when wrong, can reveal the correct options to nearly everyone else. We conclude that groups comprised of diverse individuals can make more efficient decisions than homogenous ones.

Figure 1

Waves of collective decisions. (a) The first in a clique of identical agents gathers sufficient private evidence but decides incorrectly (red). (b) The first decision convinces a few agents to agree. Since this wave is small, it reveals to undecided (blue) agents that the first decision was likely wrong. (c) The difference between decided and undecided agents leads the remaining agents to choose correctly (green). (d) The first wave increases with $N$ (red) but comprises a smaller fraction of the population [blue; Eq. (5)]. Here, the first decision is correct. (e) The time to the first decision decreases with network size (red), allowing each agent less time to accumulate private information [Eq. (4)]. Information provided by an individual first wave decision also decreases [$R_{+}$, blue; Eq. (6)]. $\theta =0.7$ in (d) and (e). Here, and below, solid and dashed lines represent simulations and theory, respectively, and shaded regions capture one standard deviation around the mean.

Figure 2

Decision statistics for homogeneous and dichotomous cliques. (a) Belief increment ${\widehat{c}}_1^{\pm }$. (b) Probability the full clique decides after the second wave. Chebyshev’s inequality provides an upper bound on $N$ by which the probability is reached ([24], Sec. XI). Inset: threshold $\theta$ at which ${\widehat{c}}_1^{\pm }=2\theta$ as $N$ varies. (c) First decision time for dichotomous threshold cliques for various $\gamma$ [Eq. (S10)]. (d) Fraction of accurate deciders in dichotomous threshold cliques under consensus bias [Eqs. (S11) and (S12)]. (e) Belief increment of agents in the second wave in dichotomous threshold cliques under consensus bias [Eq. (S12)]. Clique size $N=15000$ in panels (c)–(e).

Figure 3

Balancing hasty and deliberate decisions in dichotomous cliques. (a) With few low-threshold agents, the remaining agents receive insufficient information to decide after the first wave. (b) With many low-threshold agents, a wrong first decision sways much of the network. (c) With the right number of low-threshold agents, a few hasty agents follow an incorrect decision, but the difference between agreeing and disagreeing low-threshold agents drives the rest to choose correctly. (d) Fraction of the clique choosing accurately for a dichotomous threshold clique. White line represents Eq. (8). (e) Fraction of the clique deciding by the end of the second wave. Isoclines indicate time to first decision. $N=15000$ in (b) and (c).

Figure 4

Improved accuracy for heterogeneous cliques. (a) Mean fraction of clique choosing accurately after two waves for different threshold distributions in omniscient populations. For the dichotomous case, $\gamma$ is chosen to maximize accuracy for each ${\theta }_{\min }$ value. (b) Over a range of possible first decision times, heterogeneous cliques give better accuracy than homogeneous ones with omniscient social updating ([24], Sec. XXII for simulation details). Also shown: accuracy of a single agent with access to private information of all agents. $N=15000$ and ${\theta }_{\max }=1$.