An Agent-Based Model of Firm Size Distribution and Collaborative Innovation

ICT-basedCollaborative innovationhasa significant impacton theeconomyby facilitating technological convergence and promoting innovation in other industries. However, research on innovation suggests that polarization in firm size distribution, which has grown since the early 2000s, can interfere with collaborative innovation among firms. In this paper, I modelled firms’ decision-making processes that led to collaborative innovation as a spatial N-person iterated Prisoner’s dilemma (NIPD) game using collaborative innovation data from Korean ICT firms. Using an agent-based model, I experimented with the e ects of firm size heterogeneity on collaborative innovation. The simulation experiment results reveal that collaborative innovation in the industry increases as the size heterogeneity decreases. Findings suggest that policies promoting collaborative innovation should focus onmitigating structural inequalities in the industry.


Introduction
. Firms have actively participated in developing ICT-based convergent technologies to cope with the diversification of technology demands resulting from the rapid development of ICT technology since the early s (Nordmann ; Roco ). As a strategy for developing convergent technologies, the importance of collaborations among firms with di erent innovation capacities has increased (Cho et al. ; Nordmann ). Collaborative innovations enable firms to (a) respond to rapidly changing technology needs and (b) reduce innovation costs by implementing internal innovations (Williamson ). An ecosystem with collaborative innovation can improve e iciency associated with internal innovation by sharing innovation capabilities. Many developed countries have implemented policies to promote the participation of firms in collaborative innovation (Roco ).
. However, firms do not actively participate in cooperative innovation as much as the governments expect (Leiponen ). Some researchers argue that size asymmetry among partners can hinder the formation of strategic alliances (Kogut ). Other scholars have claimed that the larger the di erence in size between partners, the higher the administrative costs associated with collaboration (Park & Ungson ; Kelly et al. ). In an industry with a polarized firm size distribution, di erences in firm size can hinder collaborative innovation. .
The rapid growth of the ICT industry in the s caused polarization between large and small enterprises. In many countries, the firm size distribution (FSD) in the ICT industry is known to be highly skewed (Ceausu & Bourbonnais ; Henrekson & Johansson ; Johansson ; KISDI ). Gibrat ( ) argued that the FSD in a mature industry has a stable lognormal distribution. Industries in developed countries consist of a small number of large firms and a large number of SMEs and have an FSD with an asymmetrical structure skewed to the right (Axtell ). The claim that size heterogeneity appears within the industry has led many researchers to seek patterns of FSD distribution in an industry (Arrighetti et al. ; Babutsidze ; DâĂŹEste ; Na et al. ; Noda & Collis ).
. Firm size heterogeneity refers to size disparities in the FSD within an industry. Size heterogeneity depends on various economic and technological factors, but it can also a ect economic and technological change (Cantner & Hanusch ). Some researchers have suggested that heterogeneity in FSD may a ect the economic performance of the industry (Lee ; Malerba ). However, understanding the relationship between size heterogeneity and innovation pattern is primarily based on conceptual discussions (Cantner & Hanusch ; Malerba ). Therefore, direct evidence for the relationship between the two is lacking.
. This study explores how collaborative innovation patterns di er according to the size heterogeneity in FSD. If market-dominant large firms lead innovations, the innovation gap between large firms and SMEs increases (Na et al. ; Nelson & Winter ), the increase in the level of size heterogeneity can make the relationships between innovation-led firms and the rest of the industry more hierarchical, which can also negatively a ect the collaborative innovation between firms.

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However, several limitations restrict the empirical analysis. First, although firms regularly interact with one another in an industry, the mechanism of this interaction is hardly known (Yurtseven & Tandoğan ). Despite the di erences in the factors, mechanisms, and interactions that a ect innovation in each industry, there have been insu icient explorations of the industry-level variables that a ect innovation. Unlike firm-level research, industry-level research is challenging when attempting to control innovation mechanisms; it is also challenging to find comparable groups that are similar in all conditions and di er only in the FSD in reality. Second, since industry-level data is not standardized by country, it is also challenging to build datasets that contain a su icient number of industries to have statistical power. Therefore, there are restrictions on setting counterfactuals for econometric analysis.
. This study (a) produced counterfactuals in which all the conditions except the FSD are homogeneous, and (b) tried to establish theoretical propositions by comparing their collaborative innovation patterns through simulation experiments. To do so, this study focused on the Korean ICT industry, which is known to have a high level of heterogeneity in FSD. I used agent-based modelling (ABM), which is a proper methodology for modelling dynamic systems in which interactions among agents occur to analyse di icult real-world problems (Tesfatsion ). Innovation is a complex process, while ABM can model, explore and analyse the behaviour of heterogeneous agents with various mechanisms that are di icult to explain using mathematical models (Watts & Gilbert ). Thanks to these advantages, ABM has been used to analyse the formation of inter-firm networks (Özman ), inter-firm relationships (Squazzoni & Boero ), and the relationship between size heterogeneity and firm behaviour (Catullo ; Richiardi ). For instance, Cerulli ( ) explored the e ects of R&D funding policies using ABM based on game theory. Angelini et al. ( ) investigated the e ects of R&D subsidies and network topologies on innovation performance using network-based ABM. Here, I wanted to contribute to shedding light on the relationship between size heterogeneity and collaborative innovation between firms by illustrating the simple but essential patterns of complex interactions in the innovation ecosystem.

Modeling Collaborative Innovation in the Korean ICT Industry
Context: Korean ICT industry . This study focused on the Korean ICT industry. Korea has one of the most advanced ICT industries in the world, as well as one of the highest levels of ICT infrastructure. The R&D expenditure to GDP ratio is the highest among the OECD countries, indicating that the Korean ICT industry is highly innovation-intensive. Korea has reached the top position in the ICT Development Index (IDI) ranking, which indicates a high level of ICT development in a country (ITU ). Despite its short history of industrial development, the Korean ICT industry has innovative firms, such as Samsung and LG, which hold an essential position in the global market. .
The Korean ICT industry has its unique features represented by the term"Chaebol" (Steers et al. ). Chaebol refers to family-owned conglomerates consisting of large, individual firms. Korean firms have a hierarchical relationship focused on market-dominant firms (Jung & Hong ). Since some large firms have dominated innovation in the Korean ICT industry with high market concentrations, the gap between large firms and SMEs in innovation capacity is increasing in the Korean ICT industry.

Agents and landscape
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I built an agent-based model replicating the Korean ICT industry. To specify agents, I collected firm data from KISVALUE and the annual report published in the Financial Supervisory Service's (FSS) Digital Analysis, Retrieval, and Transfer System . A total of Korean firms were classified as ICT firms defined by the KIS-Industry Classification (KIS-IC). These firms operate in Korean provinces and cities. The firm data included annual sales and location information for Korean ICT firms. Sales distribution data skewed strongly to the right, suggesting a very high FSD (See Figure ). Location distribution was also concentrated in the metropolitan area, where % of the ICT firms were located in the Seoul metropolitan area, including Seoul and Gyeonggi Province. Inter-firm collaborative innovation data were collected from the patent data registered with the Korean Intellectual Property O ice published in the National Digital Science Library . The data included all joint patent data among Korean ICT firms from to and revealed that % of Korean ICT firms had registered patents, but only % had joint patents. .
Firms had three attributes, which included size, location, and strategy. First, the size distribution of the firms followed the sales distribution of . Second, I used GIS data to make a GIS-based virtual space of provinces and cities; the model was initialized with agents at their address location in South Korea (see Figure ). Third, firms had unique strategies, and they chose cooperation and defection according to their situation and strategy. I set firms that participated in at least one collaborative innovation during the past years, as of , firms that could potentially behave cooperatively. I assumed that firms were Pavlovian firms. firms that had not participated in collaborative innovation over the past years were set as "defective firms" that were not willing to cooperate. Also, firms that had not performed innovation activities in the last years among the defective firms were classified as non-innovative firms. Non-innovative firms did not engage in innovation activities and, therefore, did not participate in inter-firm interactions. This model included Pavlovian firms and defective firms and the defective firms included non-innovative firms. .
In the virtual space, firms interact with each other according to the spatial NIPD game and decide whether to perform collaborative innovation . Participation in collaborative innovation has been actively analysed using game theory since it has been perceived as a process by which firms choose the optimal strategy of collaboration or non-collaboration (Suzumura ). Since Axelrod ( ), researchers have modelled collaboration using Iterated Prisoner's Dilemma (IPD) games. Gulati et al. ( ) suggested that modelling assurance/coordination game with the Prisoner's dilemma is suitable because inter-firm collaboration, like a strategic alliance, is a social dilemma problem. Moreover, they claim that the Prisoner's dilemma game is the most popular model for inter-firm collaboration. Similarly, Arend & Seale ( ) argued that the formation of alliances among firms is essentially a prisoner's dilemma game. Note that many previous studies analysing strategic alliances and collaborative innovations were based on the Prisoner's dilemma game (Arend & Seale ; Axelrod ; McMillan et al. ; Yun et al. ).
. Not all collaborative innovation studies utilizes the same game model, with di erences according to the type of innovation in detail. Some studies have used non-cooperative games to analyse spillovers in R&D races (Martin ) or Nash bargaining games to analyse collaborative product development (Arsenyan et al. ). However, these studies either (a) assumed limited types of collaboration or (b) analysed interactions that were closer to competition rather than collaboration. Researchers have analysed various types of collaborative innovations such as knowledge sharing (Yang & Wu ), strategic alliances (Arend ; Arend & Seale ), open innovations (Yun et al. ), and collaborative R&D (Chiang ) using IPD or PD games. The model used in the present study was designed to observe changes in collaborative innovation patterns over time. An iterated Prisoner's dilemma game is the most suitable game model for this study because it focuses on observing the progress of the cooperation over time between innovators. An N-player game is also appropriate to implement interactions among multiple innovators. This study used a "spatial N-person iterated Prisoner's dilemma (NIPD) game." There are cases in which an NIPD game is used in strategic collaboration research as well as general cooperation ( Here, the Hu model has been used to determine the probability that two firms will establish a partnership. The equation for the Hu model was as follows: n: A set of competing firms; P ij : The probability that firm i will meet firm j; U ij : The utility of firm i for firm j; S j : The size of firm j; T ij : The distance between firm i and firm j; α, β: Parameters. . The utility of the firm is proportional to the firm size and inversely proportional to the distance between firms. Annual sales data from firms were used to measure each firm's size (Heshmati & Lenz-Cesar ). I also calculated P ij using the empirical annual sales data and the Euclidean distance between firms in the virtual model.

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In the Hu model, as the number of firms increased, the expected value of P ij decreased, because n j=1 = U ij increased under a condition in which U ij was constant. In other words, as the number of firms increased under the same conditions, the probability that firm i interacted with firm j decreased. This can cause an error that increases the probability of not interacting with the partner firm when implementing the Hu model to an agent-based model using a one-to-one matching algorithm. As the number of firms increases, the probability that firm i will not interact with anyone increases, which is unrealistic. I kept the maximum value of P iJ at at all times by assigning a weight to each tick to compensate for this discrepancy. Therefore, the probability distribution of this model satisfied the following condition: This correction can eliminate errors that occur when implementing the Hu model as an agent-based model.

Firm strategy and payo
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I set the Pavlov strategy as the default strategy for analysing the evolution of cooperation. The Pavlov strategy, also called the win-stay-lose-shi strategy, is a strategy in which an agent chooses one of the behaviours of collaboration or defection, and if the result is successful, it retains the behaviour; otherwise, it changes it to another behaviour. Pavlovian agents are the most realistic automata for analysing the evolution of cooperation (Szilagyi & Szilagyi ). The Pavlov strategy has been widely used in ABM as a tool for analysing the evolution of cooperation in rational choice (Power ). It is also known that the Pavlov strategy is e ective and superior to the TFT strategy when agents make mistakes (Nowak & Sigmund ).
. The Pavlov strategy assumes that an agent's behaviour depends on the payo gained from the action of his or her choice in the previous game. However, firms have all their past transaction records, and they predict the payo s from each action based on these records. Therefore, I assumed that Pavlovian firms calculated the expected value of payo s for each behaviour based on past transactions, and select a behaviour with high expectations for the payo at the current point in time. In addition to firms that used Pavlov strategies, some firms never considered cooperation with other firms because of the potential costs, such as transaction costs. Firms that followed this defective strategy will choose defective behaviour regardless of the expected value of the payo . There are also non-innovative firms that do not interact with other firms that are among those that use defective strategies. Therefore, only Pavlovian firms and defective firms, except non-innovative firms, are designed to interact. .
Based on the transaction records, each firm calculated the expected value by collecting the payo information from the past interaction. The expected value of cooperative behaviour at time t of firm i E(C) it and the expected value of defective behaviour E(D) it can be expressed by the following equation: γ k represented the discounting rate for monetary value in expected returns (Cerulli ). In this study, I set the discounting rate at .
, the annual average from to , based on the Consumer Price Index of Statistics Korea (see: http://kostat.go.kr). R was the payo obtained from mutual cooperation, and Rn was the number of mutual cooperation until t − 1. S was the payo obtained when the player cooperated and the opponent defected, and Sn was the number of S games played until time i − 1. T was the payo obtained when the player defected and the other cooperated, and T n was the number of T games played until time t−1. Finally, P was the payo obtained from mutual defection, and P n was the number of mutual defections until t − 1.
I also assumed that when E(C) it = E(D) it , the firm retained the behaviour selected in the immediately preceding tick based on path dependency. Next, firms that followed the defective strategy can be expressed by the following equation: In this model, only Pavlovian firms and defective firms except non-innovative firms were designed to interact. In a collaborative innovation game, the payo between firms follows a general payo of the Prisoner's dilemma game (Chiang ; Majeski ). Since the model of this study was based on the spatial NIPD game, it also followed the general payo of the Prisoner's dilemma game. T > R > P > S 2R > T + S .
The most common payo satisfying the condition of the PD game is T = , R = , P = , and S = . This study sets this payo as the default value for the simulation.  In this study, I validated the simulated output of the model by calibrating parameters using empirical data (Brenner ; Windrum et al. ). I performed validation through the optimization process, which minimizes the error between the empirical data and the agent-based model for the indicators that characterize the structure of the collaborative innovation network structure. Data for various network indicators in the Korean ICT industry, such as edges, linked firms, degree, betweenness centrality, closeness centrality, and clustering coe icient, were used to perform validation. The number of edges and the number of connected firms construct network density. The degree indicates the number of nodes directly connected to other nodes. Various centrality measurements are also important indicators (Butts et al. ). Betweenness centrality is measured as the degree to which one node is "between" the other points of the network (Freeman ). Nodes with high betweenness centrality are considered to play an important role in these interactions. Closeness centrality refers to the reciprocal of the sum of the minimum steps required from one node to another node (Freeman ). Nodes with high closeness centrality are considered to play an important role in spreading information between the nodes. Finally, the clustering coe icient is an indicator of the clustering tendency in the network. Nodes with a high clustering coe icient are more likely to belong to a cluster.

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The validation was performed by calibrating two parameters, α, and β, of the Hu model. It explored the most similar combination of empirical data among the combinations of α from . to and β from . to . The simulation was performed times each with the maximum tick = , according to the combinations of the coe icients in the initial setting without an edge, and then it was possible to confirm whether the network characteristic values were similar to those of the actual network. The range of each parameter was set from . to for α and from . to for β, since the edge was not generated su iciently at the maximum tick = , when the β value exceeded . The parameter settings for validation are shown in Table . Type Since the coe icients for validation were composed of α and β, the total number of possible combinations was . Each combination was simulated times until the average number of edges was (see Appendix A) . Therefore, the simulation was performed , times in total. The screenshot of the simulation run on Netlogo is shown in Figure   This study explored the combination of parameters with the lowest mean absolute percentage error (MAPE) between the simulation results and the observations for the above seven indicators. The equation for the MAPE is as follows:  This study adopted α = and β = . , which was the smallest combination of the MAPE ( . %). The validation results indicate that inter-firm collaborative innovation is associated more with firm size factors, such as sales, than by geographic distance. In other words, even if the distance between the firms is not large, it does not associate with collaborative innovation. However, if the firms' sales are high, the probability of collaborative innovation increases exponentially.

Experiments of FSD and Its E ect on Collaborative Innovation
. I experimented with the e ect of firm size distribution on collaborative innovation, using ( ) empirical distribution, ( ) exponential distribution, and ( ) normal distribution as independent variables. The mean values of these distributions were the same, but there was a di erence in the polarization level of the annual sales between firms.
. The empirical distribution was extremely polarized, which was similar to the exponential distribution but had a more skewed distribution. Most firms were located in the lowermost section of the tenth quartile, and the Gini coe icient was . , indicating the high level of size heterogeneity. The exponential distribution is a random exponential distribution with a mean equal to the empirical distribution but a smaller variance. The Gini coe icient was .
, and the level of size heterogeneity was medium. The normal distribution is the random normal distribution with the mean being equal to the empirical distribution and the standard deviation being % of the mean. The Gini coe icient was .
, and the level of size heterogeneity as low.  .
Collaborative innovation occurs when Pavlovian agents cooperate mutually. The simulations were performed up to ticks with an average edge count of in the empirical distribution, and iterations were performed for each distribution. In an undirected network, the network density (D) was calculated using nodes (N ) and the actual connections (A). Also, cooperative behaviour (CB) was calculated using the number of Pavlovian agents at time t(P A t ) and Pavlovian agents performing cooperative actions at time t(P OC t ). .
At the beginning of the simulation, the network was not formed but, as time passes, the edges were created due to cooperation between the firms. The red agents represent firms that have defected, and the blue agents represent firms that have cooperated -the size of the agent increases in proportion to its degree. At the endpoint of ticks, a small number of firms were found to be relatively larger than the other firms, indicating that the unequal distribution of degrees observed in the real world was successfully reproduced through this agent-based model.

Results
. In this experiment, I investigated changes in collaborative innovation according to changes in FSD. The simulation investigated whether there was a di erence between the firms' collaborative innovation by the level of size heterogeneity using the ANOVA for the sum of the frequencies of collaborative innovation performed by all firms until tick = .  Table : ANOVA results on collaborative innovation. Note: * * * p < 0.001.

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The results show that the e ect of firm size heterogeneity on collaborative innovation is statistically significant at the . % significance level. The e ects are presented in Figure . Figure : E ects of firm size heterogeneity on collaborative innovation.

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Collaborative innovation increased by about . times in the exponential distribution and . times more in the normal distribution than in the empirical distribution, indicating that it is sensitive to the level of firm size heterogeneity. Figure shows that as firm size heterogeneity decreases, the frequency of collaborative innovation increases. This finding suggests that collaborative innovation is performed more frequently when the FSD in the industry is relatively uniform than when the size heterogeneity is large. Therefore, as the firm size heterogeneity decreases, the tendency to engage in collaborative innovation among firms is relatively high. In conclusion, the results of this analysis suggest that, as the firm size heterogeneity decreases, the frequency of collaborative innovation in the industry becomes higher.

Sensitivity of behaviour
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I also performed a sensitivity analysis to check the robustness of the model. One of the most popular sensitivity analysis methodologies to assess robustness in ABM is the one-factor-at-a-time (OFAT) method (Ten Broeke et al. ). The OFAT method is a method of analysing the relationship between one independent variable and a dependent variable according to the change of the independent variable. Ten Broeke et al. ( ) called the OFAT method an important methodology for revealing the relationship between valid parameters and outputs. .
I performed OFAT sensitivity analysis for α and β, respectively. In all scenarios, iterations were performed.  .
The changes in collaborative innovation show a similar pattern until α decreases from to . When α is less than . , there is no significant observable change in collaborative innovation, according to FSD. This finding implies that the e ect of firm size distribution on collaborative innovation is not significant when the weight for firm size is or less. It is sensible that firm size distribution does not have a significant e ect on collaborative innovation if firms consider firms' size less critical. Conversely, a similar pattern is observed when weighing the firm size by more than , suggesting that this model is robust against α. Therefore, the sensitivity analysis results showed that the model is robust against α, as long as it weights the size when the firm seeks a collaboration partner. Next, Figure   . The changes in collaborative innovation show a similar pattern until β increases ten times from . to . . When β is or more, the pattern of change of collaborative innovation, according to FSD, reverses. When β is increased by about times to , collaborative innovation decreases with firm size heterogeneity. The results show that, in an industry in which geographical distance is critical for partner selection, the patterns of change of collaborative innovation according to firm size heterogeneity may di er. It is, therefore, not surprising that the e ect of firm size distribution decreases as companies weigh the distance between the firm size and the distance in selecting partners. Therefore, the sensitivity analysis results show that this model is robust against β as long as it does not weigh two times more geographical distance than the base model when selecting a collaboration partner. The result implies that the policy to reduce the constraint of geographical distance between firms, such as the R&D cluster, can be an e ective way to activate collaborative innovation, even in the case of firm size heterogeneity.

Sensitivity of strategy .
Fi een percent of innovative firms in the study have participated in collaborative innovation as Pavlovian firms and the rest as defective firms. This is a strong assumption that may a ect the results. Therefore, I performed the OFAT sensitivity analysis on the proportion of Pavlovian firms. In all scenarios, iterations were performed.  . Changes in collaborative innovation show almost the same pattern for all conditions, except for the " % scenario". No collaborative innovations are performed under the condition that the proportion of Pavlovian firms is %, and as the percentage of Pavlovian firms increased, the total frequency of collaboration innovations increased. This pattern indicates that the model works properly. Collaboration innovation increases as the heterogeneity in the FSD decreases under all conditions until the proportion of Pavlovian firms increases from % to %. The sensitivity analysis shows that the model is robust against firm strategy.

Discussion and Conclusions
. This study developed an agent-based model to analyse the e ect of firm size heterogeneity on collaborative innovation in the industry. The model was developed based on an NIPD game and was validated using collaborative innovation data of Korean ICT firms. The base model developed in this study successfully replicated the collaborative innovation patterns of Korean ICT firms. .
The simulation results indicate that a decrease in the firm size heterogeneity appears to have a positive e ect on collaborative innovation in the industry. This result implies that an increase in the firm size heterogeneity can reduce firms' collaborative innovation and further impede the open innovation ecosystem in the industry. Sensitivity analysis results showed that the model is robust as firms consider size as an important factor when selecting partners. .
This study contributes to the theoretical development of innovation research and confirms that firm size heterogeneity is an essential variable for collaborative innovation in industries. The FSD has previously been suggested by some researchers to be an important independent variable that a ects performance within an industry (Malerba ), but it has failed to advance further from the theoretical discussion due to practical limitations. This study attempts to mitigate the methodological constraints by using the agent-based model and report that firm size heterogeneity has significantly associated with collaborative innovation. Thus, it is necessary to shi the focus of the discussion beyond the current research trend, which focuses only on the firm, to the industry level. .
This study also contributes to the government policy to create an open innovation ecosystem in an industry. Collaborative innovation is not actively performed in an industry with size heterogeneity is attributable not only to the payo s (Chiang ; Majeski ) but also to size heterogeneity itself. The results show that collaborative innovation in the industry is sensitive to firm size heterogeneity. Therefore, governments need to mitigate the firm size heterogeneity to create an open innovation ecosystem in an industry. For example, governments can strengthen the protection of so ware intellectual property rights of SMEs to prevent unfair practices such as large firms' so ware imitation and technology the . Also, the government can expand SME-eligible industries and items in the field of ICT, such as so ware through pu blic procurement. In sum, it is possible to expand not only the regulation of large firms' unfair practices but also the improvement of the structure of large firms' labour force and the opportunities for SMEs entering the market through public procurement (Kim ).
. Finally, the model built in this study can be used for policy experiments. Chiang ( ) showed that it is possible to experiment with policy options that promote collaboration by adjusting T and R in PD games. Subsequent studies should explore the impact of policy alternatives using this model.

Model Limitations
. The model was built to explore the e ects of the skewness of the FSD on collaborative innovation; however, it does not consider the reverse causality. Firms can grow in the long run by participating in collaborative R&D projects to enhance their innovation capacity (Belderbos et al. ). However, the model assumes that the FSD is fixed over time. Since this model does not consider the reverse causality, the interpretation from the causal point of view is limited. Additionally, this model does not include firms' selection process driven by market mechanisms. The payo s by firms' behaviours are fixed, making changes in size distribution independent of market outcomes such as revenue. This assumption can hinder the model's expandability, and the size distribution can have a significant impact on endogenous innovation factors (Cabral & Mata ; Pagano & Schivardi ). Thus, it is reasonable to assume that changes in the size distribution a ect the payo s of the model, which may a ect firms' partner selection. Subsequent studies should aim to improve the model by replicating the success and failure process of firms with the changes in endogenous innovation factors caused by the changes in size distribution.

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Finally, this study replicated the collaborative innovation network through parameter calibration using the Hu model to simplify the model. However, preferential attachment rules based on various proximities can influence partner selection (Capone & Lazzeretti ). Subsequent studies should aim to develop an extended model that can analyse various collaborative characteristics, such as the persistence of collaborative innovation, by including these proximity factors.
value at iterations. The figures enable us to capture changes in network characteristics by each parameter. Changes in edges by α and β are shown in Figure   A significant change in the edge by α is not observed. However, when β is or higher, edges are generated at or less at the maximum tick, and, as β increases, the number of generated edges also decreases gradually. Changes in linked firms are presented in Figure   The number of linked firms tends to decrease slightly with an increasing α. The number of firms linked to β tends to decrease rapidly from and decreases to less than when β is . The number of edges and linked firms is more a ected by β than by α. Changes in maximum degree and mean degree are shown in Figure   The maximum degree tends to increase as α increases but decreases as β increases. The mean degree is more sensitive to β than it is to α and decreases rapidly when β is greater than . Changes in mean betweenness centrality and mean closeness centrality are shown in Figure   The mean betweenness centrality decreases as the value of α increases and decreases significantly when β is greater than . The mean closeness centrality does not change with α, but it increases when β is or higher. Changes in the mean clustering coe icient are shown in Figure .