3.1.1 Definitions and notation

First, we must define the individuals and their corresponding skill sets. Let $S$ be a set of all individuals $i$ , where $i=1,2,\ldots ,N$ . Here, $N$ represents the total number of individuals within the data set and we define $M$ as the total number of individuals on each team. Let $C$ represent a matrix of corresponding skills, where $c_{i}$ is the competency vector of individual $i$ . Each individual also has a corresponding trait vector $T_{i}$ to be utilized in the development of communication links between individuals. Each design team is represented as a network graph $G(V,E)$ , where $V$ represents the set of individuals and $E$ represents the set of edges between individuals for each design team.

Graph theory allows for the observation of pairwise connections between various components (Wilson Reference Wilson1996). In an unweighted graph, a connection between node $v_{i}$ and node $v_{j}$ is represented by $e_{ij}=1$ and an absence of a connection is given by $e_{ij}=0$ . Throughout this work, all graphs represented are unweighted. Weighted graphs apply a stronger or weaker connection between nodes for instances where the amount of information traveling among the edges depends on which nodes they pass between. However, since this work supports the development of newly formed design teams, previous communication history is disregarded. Throughout this work, nodes represent individual actors within the networks and the edges between them represent information flows, such as shared design variables or design components.

When an edge exists between two nodes, those nodes are considered to be adjacent. Each graph has an associated adjacency matrix represented by a binary $n\times n$ matrix $A$ (Borgatti & Everett Reference Borgatti and Everett2006). Within the adjacency matrix node, $v_{i}$ is adjacent to node $v_{j}$ if both $a_{ij}=1$ and $a_{ji}=1$ . For undirected graphs, the adjacency matrix is symmetric. Graphs can be either directed or undirected; however, for this work, undirected graphs will be utilized, as the information flowing between two members is assumed to be mutually available.

Non-adjacent nodes can pass information between one another by traversing through intermediary nodes. This information can follow a walk, a path or a trail. A walk from node $v_{i}$ to node $v_{j}$ follows a sequence of adjacent nodes, beginning with node $v_{i}$ and ending on node $v_{j}$ . A trail is similar to a walk; however, each edge can only be used once; a path is similar to a trail; however, each node and edge cannot be repeated (Borgatti & Everett Reference Borgatti and Everett2006). A geodesic represents a walk between two nodes in which the length of the walk is the shortest possible between these two nodes. Different modes of travel are beneficial for specific pieces of information and measures of centrality (Borgatti Reference Borgatti2005).

Within a crowd based design project, relations between members are very ill-defined as the crowd is theoretically composed of thousands of individuals who have most likely not worked together on previous design projects. Because of this, communication among members of the crowd must be simulated to develop network groupings. For this work, random intersection models are employed.

3.1.2 Random intersection model

A random intersection graph is utilized to develop modes of communication between members (Karonski, Scheinerman & Singer-Choen Reference Karonski, Scheinerman and Singer-Choen2013). As the members of an open-innovation project are unknown to one another at initial conception, this method of network generation is used to generate connections based on matching personality traits of individuals within the crowd. Random intersection graphs can provide purely random connections between members, as well as guided connections based on individual preferences or previously known characteristics (Deijfen & Kets Reference Deijfen and Kets2009). This concept is explored by developing communication flows between members based on three network development techniques.

First, a random network is developed, where each individual is randomly assigned a set of 12 possible traits. These traits are represented by binary values, with one indicating that an individual exhibits the specific trait and zero indicating that they do not possess that specific trait. Twelve traits are used as the network pool for this work consists of six unique disciplines, thus allowing for a mapping of two traits to each discipline. Additional traits allow for a greater decomposition of each individual personality, which is a topic of future work. Second, a guided network is developed, where there exists a higher probability for individuals of similar backgrounds to share similar traits. Finally, the third developed network consists of connections where individuals only communicate with others who have identical assigned skill vectors.

It is important to note that the random nature of the developed networks is a result of the trait assignment and not edge allocation. The edge placement between nodes is a result of the similarity between members, which, in the context of open design, supports the notion that two individuals with common interests are more likely to work with one another. In alternative random network generation methods, such as the Erdös–Rényi model, the random nature of the network comes directly from the assignment of edges (Erdös & Rényi Reference Erdös and Rényi1960).

All three models work on the principle of developing information flows between members with similar interests; however, the difference lies in the assignment of those interests. Once the interests have been individually assigned, we look for commonality of interests between members to determine whether a connection should be made between these two individuals based on a threshold of common interests. Threshold values represent how many traits individuals must share in order for a link to be drawn between them. The effects of different threshold values are studied further in Section 4.2. The specific degree of commonality is variable, allowing for multiple combinations of network connections. The explicit development of the three models is described below.

Full random model. The full random model assigns 12 discrete traits for each individual, with equal probability of that individual having a specific trait. This process ensures complete randomness, with respect to their design abilities, when considering which individuals are able to communicate with one another. During this process, all traits are treated equally, with no explicit regard to their corresponding meaning. The result of this model when considering five individuals with 12 traits leads to the creation of Table 1 which exhibits no obvious pattern between individuals and their corresponding traits.

From Table 1, it is observed that individuals 1 and 2 share three trait assignments. Assuming a threshold of three, a communication link would be generated in this case. Individuals 4 and 5 only share two common traits, indicating that with a threshold of three, a communication link would not be generated.

Probabilistically guided model. The probabilistically guided model allows for members of similar skill sets to have a higher probability of similar traits. For this, each trait must exhibit a correlation between the individual’s skill set and their specific traits. As this work focuses on six areas of background knowledge, further described in Section 4, traits will be mapped to the following: development, marketing, production, quality management, research and sales. While these areas are not all inclusive to a design process, they do provide a broad look at the functional components of a design team. Further decomposition of the functional processes of a team and trait mapping can be performed for networks of different skill compositions. The mapping of these traits is shown in Table 2.

Directed model. The directed model creates communication connections between members of identical disciplines. This model leads to the network with the greatest segregation as only members with identical background knowledge communicate with one another.

Following the completion of the trait matrix, individual communication links can be formed based on the similarity of traits between individuals. These links are shown through the development of the adjacency matrix.

3.1.3 Forming the adjacency matrix

Each trait matrix only gives insight into which traits each individual possesses. To develop connections within the network, individuals with similar interests and traits are connected with respect to an overall threshold of similar interests. As this threshold increases, fewer connections become present, as individuals must be extremely similar to one another in order to develop communications. With a lower threshold, networks begin to increase in density, as the required trait similarity between individuals looking to form a connection decreases. All three aforementioned models of trait distribution follow the same process of adjacency matrix formation shown below, with the only difference being that the directed model must have a threshold of two, as each level of background information is only mapped onto two traits.

The application of required trait similarity variances is used to observe the effects of homophily. The principle of homophily is based on the idea that individuals with similar characteristics are more likely to develop network connections (McPherson et al. Reference McPherson, Smith-Lovin and Cook2001). Individual characteristics such as age, religion, education or occupation have all been observed to impact the dynamics of networks. For example, the directed model exhibits high levels of occupational homophily, as individuals are connected solely on their product development abilities. As the number of required trait similarities increases, the homophily within the network also increases.

Upon creation of the adjacency matrix for each network formation mode, we can visually observe the formed team. The sample network, shown in Figure 1, is developed using the probabilistically guided network, in which individuals with similar competencies have a higher probability of sharing similar traits. We observe clusters of individuals indicated by their node color; however, unlike the directed network, we also observe cross-departmental network communications.

Figure 1. Sample network graph.

3.3.1 Notations and definitions

For this work, let $c_{i}$ represent the competency vector of each individual. The competency vector, as introduced in Takai (Reference Takai2010), presents the ability of each individual based on a variety of distinguishing factors. The impact of the competency vector is presented through the use of a value added metric, $V$ , where each individual, $i$ , contributes to the value of the design through $V_{(i)}$ . The total value added metric is represented as $V_{D,I,M,Q}$ , representing the specific KPI of added value ( $D=$ demand, $I=$ innovation, $M=$ manufacturing, $Q=$ quality). The specific design under review may have alternate or additional motivations, dependent upon the design or stakeholders involved, to determine the success of each design. However, the components of the design must be able to map to the abilities of the individuals within the crowd. Each component of the added value also has a corresponding weight, represented by $w_{D,I,M,Q}$ .

During the calculation of the value added, each individual has a base score which is denoted by $\unicode[STIX]{x1D6FC}$ . The base score of each individual represents their overall ability based on previous experience, while the competency vectors correspond to their specific domain knowledge. Various values for $\unicode[STIX]{x1D6FC}$ were tested, including 10, 50, 100 and 1000 and while the overall score scaled up proportionately, the observed trends remained identical. Since the overall score is used for comparative purposes only, the precise assignment of $\unicode[STIX]{x1D6FC}$ proved to be insignificant. Before beginning the development of the design scores, a few core assumptions must be introduced.

Assumptions. Given that the objective of this work is to study the relationship between design team formation and design success, some underlying assumptions must be understood. These assumptions are based on observations from the existing literature, but do not represent fully validated claims from an empirical perspective.

1. (i) This simulation framework considers design improvements in a specific discipline to occur with the greatest effect when two members of the same discipline collaborate. Connections between dissimilar skill sets also improve the design, but with less overall impact. When there is no connection between members, the collaborative impact on the design score is not affected (Haque, Pawar & Barson Reference Haque, Pawar and Barson2000; Feng et al. Reference Feng, Jiang, Fan and Fu2010).

2. (ii) Individuals with higher levels of experience will have a more profound impact on the overall design (McDaniel, Schmidt & Hunter Reference Mcdaniel, Schmidt and Hunter1988). Their experience is measured by their overall time spent at the company or working in a specific discipline. This consideration is represented through the use of each individual’s base score.

3. (iii) Additional individuals of the same discipline will follow a power law distribution of diminishing returns on design improvements. The order in which each individual of a specific discipline is added to the team determines the potential value that said individual can provide. Power law distributions have been shown to consistently prevail when observing social network structures where users are allowed to contribute openly (Barabási & Albert Reference Barabási and Albert1999; Barábsi Reference Barabási2003). The use of the power law distribution causes members who initially work on a design component to generate the greatest impact on that component when compared against subsequent members.

Each team member has pre-assigned abilities and expertise based on their current job title and department, as shown in Table 4. This work evaluates six distinct and broad scope abilities. However, further decomposition of these abilities can lead to a model with higher resolution. Because of the current limitation of precise ability mapping, the competency matrix for this work is of size ( $6\times 6$ ), representing six dimensions of skill decomposition.

Each column of this matrix shows the title of each member, while each row represents the corresponding ability. These columns form a competence vector for each member where each skill is assigned from 1 to 10. In this work, skills are assigned based on each individual’s working title, allowing for a slight overlap of abilities between disciplines. The development of a more complete skill decomposition is currently an active research topic (Kramer, Agogino & Roschuni Reference Kramer, Agogino and Roschuni2016); however, it is outside the scope of this work.

The creation of the competency vectors allows for the evaluation of the collaborative ability of each individual. This process accounts for the similarity of abilities to develop the overall collaborative effect of individuals with different backgrounds. The degree of collaboration is calculated based on an adaptation of Takai’s degree of complementarity (Takai Reference Takai2010), as follows:

The degree of collaboration, $l$ , is calculated from the degree of orthogonality, $d$ , between the competency vectors of individuals one, $c_{1}$ , and two, $c_{2}$ . While Takai’s original work looked at the degree to which each individual was able to complement the other individual’s skill set, this work takes $(1-d)$ to indicate the ability of two individuals to collaborate on a specific design component. These values are in the range $[0{-}1]$ . If two members have identical competency vectors, then they receive a degree of collaboration of $l=1$ , indicating that they have full ability to collaborate. If two members have zero common abilities, then they will have a degree of collaboration of $l=0$ , indicating that a collaboration effort on a specific component would not provide any additional design improvement. The final component, $e$ , indicates whether a connection exists between members. This component can take on a value of $e=1$ when a communication link exists and $e=0$ when a communication link does not exist.

In the context of a design project, when two connected individuals share project attributes, the level of commonality between each individual’s abilities is directly related to their capacity to support the work of one another. If two members have identical abilities, the information shared can be worked on collaboratively by both parties, while if two members share zero common abilities, the design component would be worked on independently by each member, corresponding to a degree of collaboration of $l=0$ .

3.3.2 Value added calculation

To calculate the overall value added, each individual is assigned a competency vector based on their working title. The direct mapping of individual background to design component is shown in Figure 2. As collaborations are formed through network connections, we begin to observe overlap in these mappings accounting for the collaborative abilities of individuals.

Figure 2. Direct mapping of ability to design component.

Each individual member receives a value added score based on how many members of that discipline exist within that design team, following a power law distribution, and their overall experience. The initial individual of a specific discipline receives the most impactful score, while each subsequent individual increases the score, with a decrease in impact, following the value added function

Here, $x$ is the number of the individual in their corresponding discipline relative to when they were added to the team and $y$ is the normalized days of employment. Days of employment are normalized between the most senior and the newest members in the pool of individuals to account for the experiential effects of individuals. The combination of value added across each individual and their collaborations is given as

The inner loop calculates the sum of all collaborations for one individual, $i$ , as they collaborate with the other individuals, represented by $j$ . The outer loop repeats this process for each individual value, $V_{(i)D,I,M,Q}$ , until all members of the team and all collaborations have been accounted for within team size $T$ . It is worth noting that the index of the inner loop begins at $j=i$ as the networks developed are undirected, causing reciprocal collaborations to be identical.

The overall design improves as a result of the product of their individual value added function and their degree of collaboration for each discipline. This simulation framework supports strong collaborations between similar disciplines while still accounting for added benefit from complementary skill sets. However, connections between individuals with zero commonality are not supported as their resulting contribution reduces to zero. The specific design components that increase in value are based on the title of the individual being analyzed and are proportional to their collaborative skill set.

These values are then combined in a weighted sum function to calculate the overall design score. For this work, all design attributes are weighted equally as the specific nature of the design and desired outcomes is not explicitly specified, as shown in Equation (6). For designs requiring higher aspects of innovation or quality, their weights, $w_{D,I,M,Q}$ , could be adjusted accordingly.

Following this calculation of each team’s design score, the team structure is then reviewed as a network graph and social network metrics are calculated. At this point, each team can be evaluated for its composition and projected design ability. Any conclusions are specific to the group of individuals being considered and the assumptions being made, and the authors acknowledge the uncertainty in mapping individual abilities to design outcomes. However, the intent of this work is to develop a fundamental understanding of team dynamics in mass collaboration product development to be leveraged in practical applications.

4.1.1 Design score

To demonstrate that the team generation captured a wide variety of team compositions, Figure 3 shows the distribution of the design scores.

Figure 3. Histogram of design scores.

The distribution of the simulated design scores follows a normal distribution with an average of 936.5 and a standard deviation of 159.6. The top score from this set of teams is 1415.5 and the worst performing team scores a 526.6. It is important to obtain a complete coverage of design team potentials as this allows for teams of widely varying abilities to be studied. While the sample of teams only represent a limited portion of the potential combinations, given the approximate normal distribution, the authors believe that the results found can be considered characteristic of the pool and model being utilized. In the following sections, overall trends in network composition are discussed.

4.1.2 Network characteristics

Closeness centrality, shown in Figure 4(a), represents the number of information channels necessary for one individual to reach another individual and not the physical closeness of each individual. High physical closeness in a distributed mass collaboration context is rarely possibly as members of the crowd are widely dispersed and they must have the ability to work with other individuals regardless of their location. Average closeness centrality follows a positive linear correlation with respect to design score.

Figure 4. Network graph centrality indicators.

This result yielded an $R$ -squared value of 62.3%, indicating a reasonably strong correlation. Another thing to note is that this result excludes teams with a closeness value of zero, indicating the presence of isolates or incomplete network graphs. From this result, we conclude that increased levels of closeness centrality, indicating a shorter geodesic path between members, generate design teams with higher design potential. This idea points to the development of new information channels for members who currently experience individual levels of low closeness centrality, following the notion that direct lines of communication between members help to improve the collaborative efforts of the design team.

It is also worth noting that closeness centrality assumes that the item being passed between the edges follows the shortest path between nodes. Because of this assumption, the correlations found for this metric are most applicable to items that are spread in series, such as CAD models or shared design variables, as opposed to concepts or ideas, which adhere to more of a parallel duplication process.

Betweenness centrality, shown in Figure 4(b), exhibits a negative linear correlation. The correlation between design score and average betweenness exhibits an $R$ -squared value of 40.7%; however, it still has a noticeable effect on design performance.

Higher levels of betweenness centrality are known to indicate critical nodes, as they fall within the information paths of multiple adjacent nodes (Borgatti & Everett Reference Borgatti and Everett2006). From these results, higher potential design scores have teams consisting of a lower number of critical nodes. This result is promising as teams with better design potential do not rely on any individual node to transfer large flows of information. The removal of any individual node would not have a significant impact on the entirety of the team.

As with closeness centrality, betweenness centrality assumes that information follows the shortest path between nodes. In the context of an open design initiative, this may not always hold true for concepts or ideas, indicating that nodes of high betweenness generally control the flow of specific items as opposed to ideas.

Eigenvector centrality, as shown in Figure 4(c), exhibits a positive correlation to design score with an R-squared value of 56.4%. Higher levels of eigenvector centrality lead to members of the design team having strong influences on other members of the design team. This consideration can also point to ‘group think’, reducing the variety of ideas and limiting innovation, as members of the design team can be persuaded to agree with influential members so as to follow the sentiment of the group.

Unlike closeness and betweenness, eigenvector centrality does not rest on the assumption of shortest path flows. Because of this, the results shown for eigenvector centrality apply to the transfer of ideas and concepts regarding designs. This implies that the observation of a positive correlation supports the diffusion of design concepts within the team.

Degree centrality, shown in Figure 4(d), shows a positive linear correlation between the average degree of the design team and the potential design score, with an $R$ -squared value of 68.3%, indicating a relatively strong relationship.

Higher degree for each individual member, however, increases the amount of information flow each individual must be responsible for. As the degree increases, the level of individual involvement increases as they now receive information from additional members. This metric must be carefully leveraged so as to increase the degree of members where it is most advantageous for the entire network.

Degree centrality is also an indication of the immediate influence of a given node. In the context of a design effort, immediate influence is shown through the direct sharing of design variables or CAD models between two collaborating individuals. Since this metric only considers nodes directly incident upon one another, increased levels of degree centrality lead to more connected designs.

Three discrete values for diameter are observed, as shown in Figure 5(a). Networks of lower diameter provide stronger design potential. It is noted that the top performing design team had the lowest potential diameter while the highest values of network diameter all contained design scores that fell below the average. While diameter does not appear to be a strong indicator of design score, it can be concluded that networks where information channels between members must include multiple intermediary nodes perform worse.

The final network characteristic reviewed is the density of each network. From Figure 5(b), another strong positive linear correlation is observed as teams with greater density tend to generate higher design score potentials.

Figure 5. Network graph characteristics.

The density of a network is in direct relation to the average degree centrality of that network. The overall impact of density on design scores is a result of the same network dynamics as when average degree centrality was considered. However, density is an overall network characteristic while degree is individualized.

A multiple linear regression was also performed with respect to the design score and the four centrality metrics. However, due to the high multicollinearity between predictors, measured by their variance inflation factors (Craney & Surles Reference Craney and Surles2002), the results were excluded from this work.

4.1.3 Top performing network

The top performing design team received a design score of 1415.45. This network was very well connected, with strong connections between members of similar disciplines allowing for increased collaboration efforts, as shown in Figure 6. This network had a density of 0.833, indicating that approximately 83% of the potential connections between nodes were utilized. It also had a diameter of two, allowing for a close flow of information between all members.

As shown in Figure 6, node size is proportional to its degree, as it was previously determined that degree held the highest statistical significance when considering levels of network centrality. The top performing team has multiple nodes of high individual degree, supporting the spread of information within the network.

The colors for each node represent the cluster in which they belong. This network developed two distinct clusters of individuals represented by the pink and green nodes using a community detection algorithm based on a heuristic optimization approach to find high-modularity partitions, outlined in Blondel et al. (Reference Blondel, Guillaume, Lambiotte and Lefebvre2008).

Figure 6. Top performing team network structure.

Table 5 highlights all network metrics attributed to the network graph of the top performing design team. Each of the network parameters outlined falls well above the average values determined from the entire simulation of the 1,000 teams.

Another consideration regarding the top performing team is related to the number of members from each discipline. This team is composed of a relatively even distribution of members, with the exception of only one marketing specialty and two research members, as shown in Table 6. The wide variety of disciplines allows for a well distributed design effort. The exact combination of individuals leading to the most successful designs would depend on the design task being considered.

4.1.4 Worst performing network

The worst performing design team received a design score of 526.64. From the network graph shown in Figure 7, it is observed that there were multiple members that had a degree of one or two, indicating that they were partially removed from the design effort. This led to poor information sharing from these members, thus decreasing their design score. This network also had a density of only 0.397, indicating that only approximately 40% of all possible connections were utilized.

When reviewing the network graph for the worst performing team, it is observed that there are fewer nodes of individually high degree, as the average degree for this team was significantly lower. It is also observed that there are four clusters that have formed, one of which is the isolated development engineer represented by the blue node. This lack of connectivity negatively impacts the team’s performance.

Figure 7. Worst performing team network structure.

Table 7 highlights all network metrics attributed to the network graph of the worst performing design team. Each of these network properties presented falls below the average values determined from the complete simulation of the 1,000 teams.

When reviewing the worst performing team for the distribution of individual members, it is observed that this team heavily consists of development engineers, with zero quality assurance engineers, as shown in Table 8. Because of this breakdown, the team does not sufficiently capture the entire design process, creating a poor overall design.

Comparison of the top performing team with the worst performing team furthers the idea that teams with greater connectivity, increased skill distribution and increased levels of information flow tend to create higher potential design scores. Another characteristic that is significantly different between the top and the worst teams is the experience level and variety of individuals on the team. The top performing team has 20 senior members, indicating greater ability in their respective disciplines, while the worst performing team only has 14 senior members.

4.1.5 Network generation comparisons

Next, we consider the impact of the communication link generation method, considering randomly formed networks, probabilistically guided networks and directed networks.

Figure 8. Network generation method statistics.

As shown in Figure 8(a), the network formed from random trait assignment consistently provided the most effective design team, with the probabilistically guided network receiving the second highest average marks and the directed network performing the worst of all three. These results, however, must be carefully interpreted as the average density, Figure 8(b), indicates that directed networks were also the least connected networks. It was previously identified that there exists a strong positive correlation between design score and network density, potentially leading to the variation in design scores observed.

The decreased overall design scores can be explained by the limited potential for collaboration efforts, expressed through a decreased network density, as the probability of communication links was decreased. The probabilistically guided and directed networks limit the overall amount of potential connections, as these are now dependent on the variety of members within the design team. The limited potential for communication links can also be quantified by the average degree of each network, as it was only 19.3 for the partially random network compared with 28.9 for the fully random network generation. The directed network is the most restrictive as it only allows for individuals with the same background knowledge to communicate. The decreased design score is primarily attributed to the decrease in the probability of collaboration.

Figure 9 illustrates a summary of the network metrics across the three types of network formations. Random trait assignment lead to teams with the highest ability and networks of the greatest density. Partially random networks have the highest levels of betweenness centrality and largest average diameter.

The directed network received average closeness and betweenness values of zero as there were isolated groups within the networks. These isolated groups meant that no member within the network could communicate with any other members.

4.2.1 Network characteristics

Closeness centrality, shown in Figure 11(a), is studied relative to varying team size at constant lines of threshold values. Threshold values of one allow for networks of much greater density as individuals require only one trait similarity before links are formed, while threshold values of six require individuals to share half of their traits before they collaborate. As the team size increases, the average closeness centrality for each group decreases for threshold values of one and two. When looking at higher threshold values, the closeness centrality remains constant at zero due to incomplete network graphs, with the minor exception of insignificant closeness levels for a threshold of three with team sizes of under 30 members.

Figure 9. Summary of network statistics.

Figure 10. Design scores with 95% confidence intervals.

Focusing on threshold levels of one and two, it can be concluded that network closeness decreases as team size increases. This is a result of increased team size leading to greater geodesic paths between individuals. Due to the addition of team members, the average distance between each node increases as communication between members now spans across a greater number of individuals. This result indicates that as teams are formed with significantly different individuals, their networks become less centralized.

When observing the betweenness centrality, shown in Figure 11(b), it is evident that as team size increases, the betweenness centrality of each network also increases. As additional members are introduced to the design team, connections between members have a greater chance of passing through other members, causing the average betweenness of the entire network to increase. Based on this, caution must be taken when developing networks of large team size as these contain additional critical nodes that control large information flows.

The impact of threshold level illustrates a curious result, as a threshold of three creates the most impactful change across varying team sizes, while a threshold of one has a less profound effect. Lower threshold levels lead to a greater overall probability of developing connections between individuals. Because of this increased probability, we observe that with a threshold of one, the betweenness does not increase as quickly as when looking at thresholds of two, three and four since the increase in connections also supports the development of direct lines of communication. When considering higher thresholds, the probability of an increased number of additional communication links decreases, also decreasing the number of direct paths between individuals, forcing information flows to pass through intermediary nodes.

We observe that at 25 individuals, the betweenness centrality of networks with a threshold of three begins to increase above that found for a threshold of two. This phenomenon can most likely be attributed to the decreasing probability of additional direct lines of communication in higher thresholds, causing networks with a threshold of three to increase betweenness at a greater rate as more individuals are added to the network. It is also observed that thresholds of five and six only exhibit a very minor impact as the probability of new communication links remains low, not significantly impacting each individual’s betweenness level.

Figure 11. Social network metrics with 95% confidence intervals.

An increase in the team size creates a decrease in the eigenvector centrality, as shown in Figure 11(c), for each design team following a decreasing power regression line. Because this correlation follows a power regression, changes to team size for smaller teams have a much greater impact on eigenvector centrality when compared with larger team sizes. When team sizes remain small, eigenvector centrality is high due to the limited pool of potential member connections. At smaller team sizes, the probability of influential members connecting other influential members is higher due to the decreased pool of potential member connections. As team sizes increase, this probability decreases, causing the decrease in eigenvector centrality to a point where adding additional team members creates a negligible effect.

The correlation between average network degree and team size is shown in Figure 11(d). Degree centrality increases linearly with respect to team size as the number of potential connections increases due to the increased probability of other team members sharing the required number of traits for a connection.

Comparison of team size with network diameter, as shown in Figure 12(a), reveals results that do not show much discernible pattern across both team size and communication thresholds. It is also observed that these values had much greater variability, notably limiting any statistically supportable conclusions. One observation that can be noted, although cautiously, is that the average values of diameter appear to increase asymptotically toward a constant. This constant also appears to increase with increasing threshold value. Thresholds of one and two reach their constant values, diameters of 2 and 3 respectively, immediately. A threshold of three requires a team size of approximately 25 before a constant diameter of 4 is reached and a threshold of four reaches a constant value of approximately 5 at a team size of 55. It also appears that the deviation in the results is a function of team size as a threshold of one has a near-zero level of deviation for a team size of 60 or above. Based on this trend, the authors believe that if additional larger teams were considered, all threshold values would begin to settle at a constant diameter with decreasing levels of variability.

Networks generated with thresholds of three and four both follow increasing patterns along a positive power regression. Networks of threshold level three smooth out at a network diameter of four and networks of threshold level four smooth out at a network diameter of five. The cause of the increasing diameter level for smaller team sizes for these two threshold values is due to incomplete network graphs for low team sizes. With a low number of team members and a higher threshold value, isolates and disconnected clusters form, causing network diameters of zero. When taking the average across these networks, the incomplete graphs begin to disappear, as networks become more connected, causing the average diameter to level out around the constant value for each threshold. For example, this phenomenon occurs at 20 individuals for a threshold of three.

For networks with thresholds of five and six, the impact of these incomplete graphs is much more prominent. These values appear to increase linearly with respect to team size; however, they are expected to smooth out, in a similar fashion to thresholds of three and four, as the number of incomplete graphs decrease.

Figure 12. Network graph characteristics with 95% confidence intervals.

When reviewing team size against network density, Figure 12(b), it is observed that there is no discernible effect of team size on network density. As the team size increases, the network density across constant threshold lines remains relatively constant. It is observed that network density decreases significantly with increased threshold level. This is due to the increased potential for communication links when the threshold value is low.

Based on these results, it is preferable to support each design team with additional lines of communication to allow for greater sharing of design activities, especially for teams of greater size. When considering crowdsourced design, this could come in the form of using members with similar traits and complementary abilities. With the traits of individuals being used to generate connections between them, crowdsourced networks would benefit from the combination of individuals who share common interests to support collaboration efforts. Thus, it is advantageous to develop additional modes of communication between members, potentially through increased content sharing or trait matching.

While this work allows for the initial analysis of simulated design teams, there exist further possibilities to extend this simulation framework to generate a more robust and adaptive model, allowing for the theoretical performance validation of the framework. Currently, individual abilities are restricted to broad estimations of their overall competencies. Further understanding of the specific attributes of each individual and how these map to design improvements is required before self-organizing mass collaboration efforts can expand.

Additionally, it is important to note that increased connections between individuals are not penalized, supporting the trend of increased communication between members. In practice, additional communication links can increase time of development and costs. This work reviews a design initiative under constant iteration, similar to open source projects, such that development time is assumed to not significantly impact the design score.