Clustering knowledge and dispersing abilities enhances collective problem solving in a network

Diversity tends to generate more and better ideas in social settings, ranging in scale from small-deliberative groups to tech-clusters and cities. Implicit in this research is that there are knowledge-generating benefits from diversity that comes from mixing different individuals, ideas, and perspectives. Here, we utilize agent-based modeling to examine the emergent outcomes resulting from the manipulation of how diversity is distributed and how knowledge is generated within communicative social structures. In the context of problem solving, we focus on cognitive diversity and its two forms: ability and knowledge. For diversity of ability, we find that local diversity (intermixing of different agents) performs best at all time scales. However, for diversity of knowledge, we find that local homogeneity performs best in the long-run, because it maintains global diversity, and thus the knowledge-generating ability of the group, for a longer period.

An agent receives an itinerary as an array of n numbers representing each of the cities. The goal of the TSP is to minimize the distance of one complete round trip without returning to any city already visited on the trip. The round-trip distance is calculated as follows: Distance = ij + jk + kl + ... + xy + yz+ zi (1) If city i is the 1 st entry in the array, and the 2 nd entry is city j, then the distance traveled is contained in entry, ij. Next, if city j is the 2 nd entry in the array, and the 3 rd entry is city k, then the distance traveled is contained in entry, jk. From the 1st to the 2 nd city and from the 2 nd to the 3 rd , this process is repeated until the nth city is reached, where the final leg of the round-trip is computed as the distance from the nth city back to the 1 st city. Each of these individual legs is summed as the roundtrip distance for that particular itinerary.
In the TSP problem space, a given state of knowledge is represented as an ordered list of 12 cities in an itinerary, where every city appears once. The objective is to find the shortest roundtrip, visiting each city at least once. Here, the same setup holds, A and B agents are randomly assigned a non-overlapping search heuristic unique to each problem space, but held constant across all five intermixing distributions. The heuristics that agents use are similar to the setups used in the NK space, but with some slight alterations to accommodate the different structure between the problem spaces. As the TSP space is more limiting in terms of how states of knowledge can be manipulated (e.g., an agent cannot turn a city to a 1 or 0 like in the NK space), agents in the TSP space can engage in one of four behaviors: (1) swap two adjacent cities in the itinerary (e.g., city1 and city2 or city5 and city6, where city1 refers to the first city in the itinerary, city2 refers to the second, and so forth); (2) swap two cities that are not adjacent in the itinerary (e.g., city1 and city5 or city2 and city6); (3) randomly jumble some number (less than N-2) of adjacent cities (e.g., city3, city4, city5, city6, and city7); or (4) randomly jumble some number (less than N-2) of non-adjacent cities (e.g., city3, city6, city9, and city10). For the latter heuristics that manipulate more than two cities, one agent is randomly assigned a mask that includes anywhere between 2 and 12 cities in the itinerary (i.e., state of knowledge), while the other agent is randomly assigned a mask that includes between 2 and 12 cities NOT previously assigned to the first agent. In other words, if agent A's mask involved manipulating city2, city3, and city4, then agent B is assigned any of the remaining cities (e.g., city5, city6, city7, and city8; or city1, city6, city9, and city12; etc.). Agents iterate the mask in a stepwise fashion such that agents adopt the state of knowledge (out of the N=12 possible ones) that has the best solution. For instance, agent A would jumble city2, city3, and city4, record its score; then shift the mask over by one city (i.e., city3, city4, and city5), record its score; and continue in a similar fashion until the agent has stepped through all twelve potential itineraries, subsequently adopting the optimal itinerary. This setup offers a direct parallel to the NK setup, and allows agents to avoid getting stuck at local optima.
For our diversity of ability experiments, how we define ability is subject to a wide-array of approaches to operationalize how it becomes implemented in either the TSP or the NK spaces.
There are two ways that we address this. First, we test the opposite manifestation of mutually exclusive perspectives: namely, we run our experiments with agents whose perspectives are mutually nested within one another. Second, we tested a variety of mutually exclusive heuristics to ensure that no one heuristic was driving the results. In what follows, we outline both approaches in both spaces.
First, to test how opposite instantiation of mutually exclusive abilities, we define an A+ species agent that encompasses the search capability like those defined previously for the NK and TSP spaces. For instance, for the NK space search heuristics, agents are randomly assigned some mask and one of the four manipulation behaviors as previously outlined. Agents shift its mask through all N=20 possible solutions and selects the best state of knowledge out of the twenty. The A+ agents, by contrast, adopt the same mask as the other agent for that particular problem space, however it shifts its mask using all four manipulation behaviors and selects the best possible state of knowledge. In that way, A+ agents encompass the entire search capabilities of the other agent in that A+ agents can see everything the other agent sees, plus more. Similarly in the TSP space, A+ agents apply all four manipulation behaviors, move in a stepwise fashion across all N=12 possible solutions for each behavior, and adopts the best possible solution produced, allowing the A+ agents to see everything the other agent sees plus much more.
We extend our torus network model by rewiring ties to mimic a small world model 1 . Along with network structure, we also implement a complex contagion parameter in our model to test the robustness of preserving systems with intermixing inspired by Centola and Macy 2 . We use three small world settings based on the number of rewires: a torus network with no rewires; a torus network with 30 links randomly rewired; and a torus network with 100 links randomly rewired.
The networks are created before the simulations run and are held constant across each experiment.
We also built a complex contagion model (see Centola and Macy 2 ) where an agent copies a set of knowledge if the agent sees it adopted by at least two of its neighbors. However, we slightly deviate from the Centola and Macy 2 model because allowing some slight diffusion of information from single adopters, arguably, has greater verisimilitude with actual diffusion processes, as Centola's 5 follow up experiment demonstrated. In our updated model, an agent can see a set of knowledge with 100 percent probability if it is adopted by at least two of its neighbors. However, there is some ε probability (where ε << 100 percent) that an agent will see the single state of knowledge of a solitary neighbor. And the assumption of zero diffusion without multiple exposures typically results in no diffusion at all when the system starts with 100 different agents.
We implement three degrees of contagion: simple, moderately complex, and complex. A simple contagion is the absence of the aforementioned criteria, whereby an agent can see all of its neighbors' states of knowledge each round with 100 percent probability. A moderate contagion adopts our above contagion model, where an agent can see any individual neighbor's state of knowledge with a 50 percent probability each round; but can see two or more neighbors' states of knowledge if at least two of them have the same state of knowledge. A complex contagion operates in the same way but reduces to a 10 percent probability.
Our robustness checks are run for both the NK and the TSP problem spaces and for both diversity of ability and diversity of knowledge experiments. We find that for both types of diversity of ability (B and A+), across contagion and network structure, increasing intermixing is associated with improved performance and these results across time.
As the problem spaces use two different scales to measure performance (e.g., NK is on a zero to one scale, while TSP is the mean distance between cities on a set grid), we need to render network performance comparable. As such, the average network performance is measured for each of the five intermixing setups across problem spaces separately at each round of the simulation.
The network configuration (i.e., minimal, low, moderate, high, full intermixing, or random in the case of the diversity of knowledge simulations) with the best average result amongst its agents is counted as having the best solution for that particular problem space. In other words, this accounting for performance is setup as a winner-take-all or first-past-the-post approach: whichever intermixing setup found the best average solution in round t for a particular problem space, it "wins" this contest as having the best average solution among the five setups. As such, this setup also allows for two or more configurations to have settled on the same final average result, yielding "ties" among configurations having the best solution (and percentages that sum to mover 100%).
To that end, the y-axis captures the percent of these problem spaces that a particular network configuration found the best average result at equilibrium. While rewiring naturally removes the benefits of intermixing (i.e., everything is randomly connected to distant parts of the network), increased contagion seems to not have much of an effect on the relative performance between setups with higher intermixing and those that do not. High contagion networks with stronger "small-world" setups seem to negate the effects of the rewiring.
With more complex forms of contagion, or slower diffusion of knowledge, the system is driven more by exploration than exploitation. With more exploration, there are more opportunities to introduce more unique states of knowledge into the system. While we see that systems across the various distributions of diversity find optimal states of knowledge less frequently (i.e., fewer ties), the performance differential between the distributions of diversity across the three types of contagion slightly attenuate with more complex forms of contagion. The benefits of intermixing are not reaped and fewer unique states of knowledge are introduced into the system. Thus, the performance differential between different rates of intermixing diminishes with more rewiring. Higher degrees of small world-ness in the network undermine the buffer that non-intermixed states of knowledge afford, and the performance differential between intermixing systems attenuate. Just as with the diversity of ability experiments, more rewiring of the network prevents more unique states of knowledge to be introduced into the system.

Supplementary Figures
Supplementary Figure-1(a). The average NK score of networks across problem spaces, plotted by intermixing rates for the diversity of ability simulations with five diverse agents (NK Space). Error bars are the standard errors measured across NK spaces.

Intermixing Rate Average NK Score for Each Network Configuration
Supplementary Figure-1(b). The average NK score of networks across problem spaces, plotted over time for the diversity of ability simulations with five diverse agents by intermixing rates (NK Space). Error bars are the standard errors measured across NK spaces.

Intermixing Rate Percent of Problem Spaces Intermixing Network Found the Best Solution
Supplementary Figure-3. Performance over time of networks with various intermixing rates for the diversity of ability simulations.  Figure-4. Average number of unique states of knowledge by intermixing rate over time for the diversity of ability simulations. Error bars are the standard errors measured across NK and TSP spaces.   Figure-6(a). Performance of networks with various intermixing rates for the diversity of ability simulations (small world and contagion setups, NK space for A and A+).  Figure-6(b). Performance of networks with various intermixing rates for the diversity of ability simulations (small world and contagion setups, TSP space for A and A+).