1 Introduction
Collaboration can be widely observed in different social and economic systems, where agents strive to reach a common goal. Scientists collaborate to write joint publications (Katz and Martin, 1997), firms collaborate to file joint patents (Hoang and Rothaermel, 2005; Kim and Song, 2007), and software developers collaborate to create joint software products (Bitzer and Geishecker, 2010; Lakhani and Wolf, 2003). To explain collaboration, economic research has traditionally focused on different aspects of labor division (Durkheim, 2014) and productivity of teams (Scholtes et al., 2016). However, in the wake of technologydriven economic growth, the question how to boost collaboration to foster knowledge transfer and innovation has become more important (Frenz and IettoGillies, 2009).
The current research about the dynamics of R&D networks can be seen as a major contribution to better understand how firms collaborate in patenting activities. In this network representation, nodes depict the economic agents, i.e. the firms, and links between nodes their collaboration. Specifically, firms formally declare this collaboration in publicly announced alliances, which can involve more than two partners. So, it makes sense to ask how the size of alliances, i.e. the number of partners involved, can be explained by means of an agent based model, which is the aim of the current paper.
To address this questions, we can build on a number of empirical studies about R&D alliances. It was shown that, because firms are involved in different alliances at the same time, their collaboration results in a large network component, in which even firms not directly collaborating are still connected through other firms (See Figure 1). At the same time, a large number of small firm alliances exist that are not connected to the rest of the network. These coexisting subnetworks are called components in the following.
The formation of a strongly connected component can be seen as an emergent property of the economic network because it is not planned top down, but emerges during the process of alliance formation, if (some) firms become engaged in more than one alliance. Once such a strongly connected component exists, it greatly enhances the transfer of knowledge and the diffusion of innovations even between distant firms, so it is beneficial from a policy perspective.
(Tomasello et al., 2014, 2017) have already proposed an agentbased model that is able to reproduce most of the properties of the observed R&D network. These properties include (i) the distribution of component sizes, i.e. the number of components of a given size plus the size of the largest connected component, (ii) the distribution of local clustering coefficients, i.e. the fraction of firms in a component that form triads (closed triangles) in their collaboration, (iii) the distribution of the lengths of shortest paths that connect any two firms in the network, and (iv) the distribution of degrees, i.e. the number of partners of a firm.
This agentbased model, while successfully reproducing network features along different dimensions, takes two empirical distributions as an input: (a) the distribution of agent’s activities, i.e. their propensity to engage into a collaboration, and (b) the distribution of alliance sizes, i.e. the number of partners involved in an alliance. The latter has been investigated empirically (see Hagedoorn, 2002; Tomasello et al., 2016)
. Remarkably, one finds a broad and rightskewed distribution of alliance sizes (see Figure
2). The same distribution was found even for different industrial sectors (including manufacturing, research, financial and service sectors), that exhibit substantial differences otherwise.Previous modeling attempts in this field have, to the best of our knowledge, limited themselves only to general features of the R&D network, such as the degree distribution or small world properties. With the current paper, we want to move the agentbased modeling one important step forward, by explaining the distribution of alliance sizes as an emergent feature of an underlying agentbased model instead of taking it from observations. This requires us to explicitly model how agents form alliances, which implies to consider why agents form alliances. But given the empirical work on alliance sizes, we have some ground truth to later judge the performance of our agentbased model in reproducing the distribution of alliance sizes.
2 Empirical findings
2.1 The network of R&D alliances
The dataset.
We build our empirical R&D network using the SDC Platinum database,^{1}^{1}1http://thomsonreuters.com/sdcplatinum/ that reports approximately 672,000 publicly announced alliances in all countries, from 1984 to 2009, with a granularity of 1 day, between several kinds of economic actors (including manufacturing firms, investors, banks and universities) for which we commonly use the term “firm” in the following. We then select all the alliances characterized by the “R&D” flag; after applying this filter, a total of 14,829 alliances, connecting 14,561 firms, are listed in the dataset. An R&D alliance is defined as an declared partnership between two or more firms. This can range from formal joint ventures to more informal research agreements, specifically aimed at research and development purposes. Note that we do not have any information about the firm that initiated the alliance, nor about the sequence in which firms joined an alliance.
The analysis of the data set, as well as all the network analyses and plots, are done by means of the R software for statistical computing.^{2}^{2}2http://www.rproject.org/
Reconstructing the collaboration network.
In the present study, we investigate the R&D network aggregated over all years and all industrial sectors, which has to be reconstructed from the data set. Firms are represented as nodes in the network and R&D alliances as undirected links between nodes. Isolate nodes, i.e. firms not taking part in any R&D partnership, are simply excluded from our network representation.
When an R&D alliance involves more than two firms, we assume that all the corresponding nodes are connected in pairs, forming a fully connected clique. A “standard” twopartner alliance is then a fully connected clique of size 2. The choice of the fully connected clique – rather than less interconnected network architectures – derives from the fact that alliances of more than two partners, although representing only a minority, require great coordination and resources. Therefore, they have to be associated with a higher number of links in the corresponding collaboration networks. By following this procedure, the 14,829 R&D alliances listed in the dataset result in a total of 21,572 links. The resulting network is shown in Figure 1.
Distribution of alliance sizes
A salient feature of the R&D alliances in the SDC dataset is the variable number of partners they involve. Most of the collaborations (93%) are stipulated between two partners, the remaining ones involve three or more partners. In the following, we denote by the size of the alliance, whereas indicates the number of firms and the number of alliances.
We report the empirical distribution of the alliance size, in the R&D network in Figure 2. As clearly visible, it spans one order of magnitude and is rightskewed. It should be noted that an identification of the functional form of the distribution (e.g., powerlaw, exponential, lognormal and so on) is outside of the scope of this study. Our aim instead is to develop an agentbased model to reproduce this distribution, as described below.
2.2 Defining a fitness measure for agents
Our agentbased model requires an attribute, called “fitness”, that is assigned to each agent. It describes how attractive an agent is for the other agents, to form an alliance. To keep the model as a general as possible, we decide to proxy fitness by a measure which is not system specific, such as the operational value of a firm. We choose the socalled agents’ activity (Perra et al., 2012), which has been already successfully used on various data sets, such as online microblogging, actor networks, R&D and coauthorship networks (Tomasello et al., 2014). The empirical activity of an agent at time , over a time window , is defined as the number of alliances that involve agent in the time window ending at time , divided by the total number of alliances involving any agent in the network during the same time period:
(1) 
It was found that activity distributions in most collaboration networks are right skewed and dispersed over several orders of magnitude, as in many other social and technological systems (Barabasi, 2005; PastorSatorras et al., 2001). This is confirmed also for the case of R&D networks, where the empirical activity values range from low 0.002 to the maximum value of . Applying this to the fitness of agents, this means that the agent with the highest fitness has a value of 23 orders of magnitude larger than the agents with the lowest fitness. Indeed, the vast majority of the agents has a fitness equal to the minimum value, which is also the median value, and the average fitness is only slightly higher than that. Only one agent has a fitness equal to 1 (the highest possible value).
Contrary to most network indicators that display strong variability and dependence on time, especially in R&D networks (see Tomasello et al., 2016)
, activity is a stable attribute that can be assigned to firms to effectively estimate their propensity to engage in new collaborations, as well as their attractiveness to potential new collaborators. Empirical activities are robust with respect to (a) the time
at which they are measured, (b) the length of the selected time window , (c) the sectoral classification of firms or authors, as shown by Tomasello et al. (2014). Such a stability makes activity a perfect empirical proxy for our fitness attribute.Given the robustness with respect to the time window, we decide to compute the fitness values using the longest possible window, i.e. the entire observation period, therefore . This considers the full information from the data set and results in activities that are always strictly greater than 0 because, by definition, all firms in our network must be involved in at least 1 alliance. In Figure 3 we report the empirical distribution of activity, i.e. of fitness, , for the analyzed R&D network. Further, in Figure 1, we have used the empirical fitness values of agents to scale their size in the collaboration network. Agents with higher fitness obviously form the core of the empirical R&D network.
3 The modeling approach
3.1 Agentbased model of alliance formation
In the following, we develop an agentbased model to reproduce the observed size distribution of consortia, shown in Figure 2. This distribution is the result of a dynamic process in which agents decide to initiate or to join an alliance, i.e. it can only be understood by modeling the growth of the collaboration network.
Fitness of agents and initiation of an alliance.
Our model is a considerable extension of the network fitness model first proposed by Bianconi and Barabási (2001). Each agent is assigned a fitness which is fixed and independent of time. The values for the fitness are obtained from the empirical distribution , shown in Figure 3.
In our model, all agents can become active with a uniform probability, which is chosen to be , independent of their fitness. The sampling occurs with replacement, i.e. agents can also be chosen more than once to become active. Activity means here that an agent initiates
a new alliance; hence we refer to her as the “initiator”. We do not have empirical information about the agent that initiated an alliance, hence the assumption of a uniform probability for the activation is reasonable.
For our simulations, we choose a discrete time which measures the time to form an alliance. I.e. each time a new initiator is selected we start with and the maximum time for alliance formation is denoted as . The newly created alliance can grow only if new collaborators join. This process is reflected in two steps: (i) the initiator invites new collaboration partners, one per time step, (ii) the invitees accept or reject to join the alliance.
Utility of consortia.
A number of agents form an alliance of size which can change over time as new agents join the alliance. There can be many consortia of different sizes coexisting over time. The utility function, , of the alliance combines the benefits, , and the costs, , of the collaboration of the agents, i.e. both and depend on the current alliance size, . Further, the benefits should be a monotonous function of the fitness values of the currently involved agents, i.e. , whereas the costs should reflect the coordination effort of the alliance and thus should be a monotonous function of the size of the alliance. For simplicity, we assume linear dependencies for the monotonous functions, i.e. the utility of an alliance is defined as
(2) 
where the parameter allows to scale costs against benefits. We note that the costs scale with the number of alliance partners rather than with the number of their possible connections, which would be quadratic, i.e. , if an alliance is seen as a fully connected clique. The latter would account for a superlinear increase in the coordination effort between partners which sets strong limitations to larger consortia. Here, instead we assume some sort of administration cost based on the number of parties.
Invitation of alliance partners.
As the alliance grows, its utility will change in a nonmonotonous manner. Precisely, according to Equation 2, the utility will grow only if the fitness of the new alliance member is larger than the scaling constant . To ensure that this condition is met, an initiator preferably invites agents with a high fitness. Precisely, similar to the fitness model of Bianconi and Barabási (2001), the initiator chooses potential alliance partners , one at a time, with a probability proportional to their fitness .
Different from the mentioned model, it is however left to the agents to decide whether they want to accept this invitation, i.e. to join the alliance. The initiator repeats the invitation procedure until a number of invited partners refuse the invitation to the alliance. I.e. is a parameter of our agentbased model. If the current number of rejections, , reaches , we assume that the alliance is fully formed and stops to grow in size. At the same time the initiator loses its “active” status. If the initiator receives rejections already from the first selected partners, then no alliance is formed.
Formation of collaboration links
The second step in the formation of the alliance is the decision of the invitee to accept, or to not accept, the invitation. An agent decides to join an alliance at time if the utility of the alliance, is larger than a certain threshold, , which is assumed to be heterogeneous across agents.
Specifically, we argue that the threshold of an agent to join an alliance increases
with her fitness. The rationale behind this is that agents with a high fitness are very attractive for initiators of consortia and thus receive invitations very often. On the other hand, because of scarce resources, agents cannot simply accept all invitations, they have to be selective. Therefore, the higher the own fitness and the attractiveness for consortia, the higher the threshold to accept an invitation. Conversely, agents with a low fitness are not invited very often for an alliance, therefore they will be more inclined to accept invitations, i.e. their threshold is lower because of the lower fitness. Hence, it is reasonable to argue that
, i.e. is simply proportional to the fitness, where is a parameter of the model, to be determined later.This results in the following condition for agent to join the alliance in the next time step
(3) 
Implications.
Our agentbased model builds on an interesting tension between the attractiveness and the willingness to become an alliance member, which is a novel point in the discussion of fitness models. Hence, in our model the link formation process does not follow a simple preferential attachment rule. First, because agents do not increase their individual fitness by accepting an invitation. Secondly, because agents become the more selective, the fitter they are.
It should be noted that, even though lowfit agents are more common in the network, highfit nodes agents a much higher chance to be selected as potential partners and to establish new consortia. It is also clear from the setup of the model that agents with a low fitness would not be able to establish a larger alliance. They can likely not attract agents with high fitness, nor can they overcome the costs inclined in the formation of an alliance. Hence, larger consortia depend on the initiation by agents with a high fitness and their ability to attract other agents with high fitness.
Eventually, our agentbased model combines probabilistic elements, such as the activation of agents and the invitation of partners, with deterministic elements, such as the decision of agents to join the alliance. This decision differs from a “best response” rule, because agents do not decide based on complete (or global) information about all existing consortia. Instead, they base their decision only on the (local) information of the current offer.
Once the formation of an alliance is finished, this alliance is added to the existing network as a clique, i.e. a fully connected cluster of size , which is in line with our procedure to reconstruct the network from empirical observations (see Section 2). This again differs from the mentioned fitness model in that the network grows with the sequential addition of cliques, and not of single links. The addition of a single edge linking two nodes, as argued in Section 2, can be thought of as the addition of a fully connected clique of size 2.
3.2 Analytic description of the alliance size distribution
We now proceed in two different directions. First, we run stochastic simulations by implementing the agentbased rules described above. Hence, at each time step we choose an agent to initiate an alliance. Dependent on her success or failure, we add new cliques to the collaboration network and continue with choosing a new agent in the next time step. This procedure is followed to obtain the results discussed in the subsequent sections.
However, we also formalize the model in a more analytic way to obtain an expression for the distribution of consortia sizes, . We start from the fitness distribution , which we take as given from data. In accordance with the empirical distribution , we consider as discrete because of the binning given by the observations. We assume that does not change during the formation of consortia. That means even if agents with a given have accepted an invitation and can thus not be invited again to join the same alliance, we assume that the distribution of the remaining agents is as before. This assumption is justified if it is unlikely that an alliance grows to a significant proportion of the whole system, as it is the case in our studies.
With this, we derive an analytic proxy for the consortia size distribution . It saves considerable computational effort and allows better insights into the model evolution. The distribution gives us the probability to find an alliance of final size . This formation happened during the time steps . At , an initiator is picked at random and thus has a fitness with probability . The alliance size at that time is . In each following time step, another agent, which has the fitness with probability , is invited to join the alliance. She either accepts so that the alliance size increases, . Or she rejects which leads to an increase in the number of rejections, . Hence, for the next time step in the alliance formation process always holds.
To evaluate the probability for both cases, we have to keep track of the alliance utility. For this, we introduce a timedependent distribution . It represents the joint probability that at time step an alliance has reached the size , while offers were rejected times.
is the benefit of the alliance, i.e. the sum of the fitness values of the alliance partners according to Equation (2). The initial conditions for the timedependent distribution are because the alliance consists only of the initiator, and otherwise.
The arguments of can only have the following values: the size ranges from with as the total number of agents, the number of rejections from where is the maximum number of rejections that are still tolerated, and for the time where is the maximum time in which the formation of a single alliance is possible. The benefit is bound to . Furthermore, only combinations satisfying the condition are reasonable, because, in each time step, either or are incremented. Otherwise, the growth of the alliance stops. Hence, can be set for unreasonable combinations of , , and .
Following these considerations, we start from the initial conditions given above and iteratively deduce from at the previous time step. For this, we have to take three different constellations into account: (1) the alliance stops growing at time because the maximum number of rejections was reached, (2) the alliance could potentially grow, however the invited agent rejects the offer, which leads to and , and (3) the alliancea actually grows because the invited agent accepts the offer, which leads to and .
For each of these three constellations we have to express the probability of its occurrence and the fact that the boundary conditions for the given arguments , , and are met. For the latter, we use the indicator function, which can be either zero or one. means that the value of is within the range of and and otherwise, whereas means that the value of is precisely the value of and otherwise. This is more convenient and more compact than if and else to express different cases. We need two indicator functions because we have constraints on two variables, and , which determine . With this, we can capture the three different constellations as follows:
(4) 
The product in the first line of Eq. (4) refers to the case that too many rejections have happened already and the alliance stopped growing. Hence, , and . With this, , but still within the maximum time allowed for alliance growth, . , on the other hand, gives the probability of such a constellation at time .
The second line of Eq. (4) counts all cases in which an invited agent rejects to join the alliance because its fitness is too high in comparison with the alliance, , see Eq. (3
). The probability that this happens is given by the complementary cumulative distribution function,
, with(5) 
Because the rejection happens at time , the current rejection value is , while the size and the alliance benefit stay constant. The indicator function ensures that is still in the possible range of 1 (if that was the first rejection) and . Otherwise, this would have been captured in constellation (1). The second indicator function just reflects the boundary condition . Because the agent has rejected the offer, we have .
The last line of Eq. (4) eventually considers all cases in which an agent accepts to join the alliance at time . In this case, , and . The probability of this occurrence has to be multiplied by the probability to find agents of fitness . The summation goes over all possible benefit values, , for which agents accept to join the alliance, which follow from Eq. (3). This condition defines the value introduced above. I.e., agents join the alliance if their fitness is between . We can express this condition with respect to instead of , i.e. .
The indicator function eventually makes sure that the acceptable number of rejections is not already exceeded, i.e. the agent can still join the alliance. Otherwise, it had been considered in constellation (1). We note that our approach also applies to more general forms of cost functions than just . Only the bound would need to be adjusted correspondingly.
The iterations of Equation (4) occur until is reached, which is the maximum time possible for forming an alliance. This ensures that all consortia formations are counted in. The final alliance size distribution is then simply the marginal distribution
(6) 
This distribution depends, for a given number of agents, on the three parameters , , and further on the empirical fitness distribution , which is given. Hence, , , have to be determined in the following.
We emphasize that with our analytical solution we follow a probabilistic approach. That means, we take all possible constellation into account. This is equivalent to running a large number of computer simulations and averaging the results, at the end.
4 Results
4.1 Calibration of the agentbased model
Our first task is to calibrate the agentbased model introduced above. We take as model input the fitness distribution, , which is proxied by the empirical distribution shown in Figure 3. It then remains to determine the set of parameters of the model, to scale the costs of the consortium, Equation (2), to scale the individual threshold for accepting invitations, Equation (3), and which is the maximum number of rejections an initiator receives to stop the formation of an alliance. In order to determine these parameter values, we use a standard maximum likelihood approach. This can be based either on computer simulations of the agentbased model or on the numerical solution of the analytic expressions given in Equation (4). Both lead to the same results for all analyzed parameter combinations. Thus, we report the results from the numerics which can be obtained much faster.
We are interested in the distribution of the alliance size, , given the set of parameters . In a first step, we have to renormalize this distribution to alliance sizes because we have no observations about . This renormalized distribution reads
(7) 
with . The likelihood of each parameter combination is then determined by the probability to observe our data which consists of alliances with sizes , given these parameters:
(8) 
For our simulations, as well as for the numerical solution of Equation (4), we will choose the set of parameters that maximizes this likelihood:
(9) 
The exponent avoids the comparison of too small values and guaranties thus numerical stability. The results are depicted in Figure 4. We note that there exists a region where all the points with high goodness score are concentrated and that there is a sharp transition between the red and the blue region. This region corresponds to definite values of and , but there is no definite value for . In fact the “optimal” region is a line, along which can vary between 1 and 20. Thus, in the following we choose the maximum value .
4.2 Validation of the agentbased model
To challenge the validity of our agentbased model with the calibrated parameters, we need to reject the Null hypothesis
that the parameters , are correct.The assumption of our model is that all alliance formations are independent. This implies that our data has been generated by a multinomial distribution , where the probabilities result from our agentbased model. We use this multinomial distribution to construct a region of probability coverage, which we estimate by sampling times from . In Figure 5 we represent this region as bands around the alliance size distribution obtained from the maximum likelihood approach. One band corresponds to the quantile, the second one to the quantile for the probabilities .
A comparison with the empirical distribution reveals the very good results. We see that in most of the empirical values are within the confidence region. The four data points outside of that region are at least very close to the lower band. They also represent alliance sizes of only very small probabilities, about , and thus cannot be considered as important deviations from the model. Hence, we conclude that our null hypothesis that the data are generated by the calibrated agentbased model cannot be rejected.
5 Conclusions
Our goal in this paper is to explain the empirically observed alliance size distribution, Figure 2, of R&D networks by means of an agentbased model that explicitly models the alliance formation process.
Our model builds on heterogeneous agents characterized by one individual parameter, their fitness . The distribution of fitness values is proxied by the empirical activity distribution, Figure 3, as the only model input. Activity describes how often an agent was engaged in an alliance during the observation period, which is 26 years in our case. This can be seen as an indication of the agent’s attractiveness for other agents to collaborate with, and fitness should be interpreted in the same manner. We have shown that the fitness distribution obtained this way is right skewed and very broad.
Further, our agentbased model uses three free parameters that need to be determined in comparison with empirical data. The calibration process is based on a maximum likelihood estimation that returns those parameter values that match best the target, which is the empirical distribution of alliance sizes.
It is interesting to note that only two of these parameters, the scaling factors for the cost of the consortium and for the individual threshold to accept an invitation obtain a stable value in the maximum likelihood estimation, whereas the third parameter , the number of rejections to stop forming an alliance does not reach a definite value. Instead, we observe that equally good likelihoods are obtained for a larger range of between 1 and 20. Hence, our model works without assuming a specific value of . In other words, can vary across time, industrial sectors or even alliances without questioning the validity of our model.
For our model validation we used a high number of rejections, . This is definitely realistic for a system such as the global, intersectoral R&D network that we analyze. Here, firms have to search for their partners among a huge number of potential candidates, making the establishment of an R&D alliance potentially costly and risky. We argue that this leads to a very long and cautious selection process, from the side of both the initiator and the invited firm. Therefore, firms have to be willing to accept a high number of rejections, if they want to gain access to external knowledge and eventually establish R&D collaborations with other firms.
Regarding the other two parameters, and , we note from Equation (5) that actually their ratio matters, as it determines the range of fitness values for which agents join an alliance. The definite value of should be interpreted as rather large. I.e. when multiplied with the size of the alliance, the cost in Equation (2) is rather high in comparison with the benefit of the alliance, which is the sum of the fitness values of the agents. This has two consequences. First, it restricts the maximum size of an alliance to values below 20. Second, it restricts the maximum number of alliances with sizes larger than 2, because most agents in the system have a rather low fitness and are thus not able to overcome the considerable cost of forming an alliance. To illustrate this, an alliance of two agents with median fitness values exhibits a benefit of 0.004 and a cost of 0.04; or an alliance of four agents with medianfitness agents exhibits a benefit of 0.008 and a cost of 0.12, i.e. almost an order of magnitude larger.
This reflects the intention of our agentbased model. Agents with high fitness (typically incumbent firms) are the ones that are most likely to receive an invitation. At the same time, they will most often refuse the invitation, if an alliance consists of only agents with medium or low fitness nodes (typically midsize firms or startups). This leads to the high value of rejections obtained from the maximum likelihood estimation. But if agents with a high fitness initiate an alliance, agents with medium or low fitness are likely to accept this invitation. On the other hand, agents with low fitness are not very selective to refuse any invitation because their threshold utility is rather low, also as a consequence of the small value of .
The good match of our agentbased model with the empirical observations allows us to draw some conclusions about the formation of real R&D alliances, for which no data is available. As we have seen, alliances are more likely initiated by an incumbent firm of high fitness which directs its interest toward a midsize company or a startup. At the same time, the “bottleneck” in establishing new alliances is probably on the initiator’s side, which has to take rejections and keep looking for new partners until it finds the right one.
Our agentbased model was developed to reproduce the empirical distribution of alliance sizes. One could be interested to know whether this model, using the parameter from the maximum likelihood estimation, is also able to reproduce other features of the observed topology of the R&D network. This is not the aim of the paper, but we can comment at least on the degree distribution which was analyzed already by Tomasello et al. (2014). Degree refers to the number of collaboration partners of an agent, not to the number of alliances the agent is involved. As such, degree is not independent of the size of an alliance, and indeed the empirical degree distribution was also shown to be right skewed and very broad.
However, we argue that the degree distribution cannot simply be obtained from our agentbased model because this does not take degreedegree correlations into account. Assortativity reflects the tendency of agents with high degree to form alliances with other agents with high degree, whereas dissortativity would indicate that agents with high degree have the tendency to form alliances with agents of low degree. Such degreedegree correlations have been detected by (Tomasello et al., 2016) both for sectoral R&D networks and for the aggregated R&D network used in this paper. They play a role in particular for agents with high degree. Therefore, we can assume that our agentbased model will be able to reproduce the right skewed and broad degree distribution, but becomes increasingly worse in the range of larger degrees.
To conclude, our agentbased model provides a considerable step forward in identifying the real mechanisms for alliance formation (Ahuja, 2000). In particular, with the distribution of alliance sizes we are able to reproduce a feature that has received some attention in the existing literature, but never a conclusive explanation. Our model can be used for stochastic agentbased simulations, it also provides an analytical solution that considerably reduces the computational effort. We emphasize that it is rather rare to obtain an analytic description of an agent based model. Our derivations also apply to cases with different cost functions and are thus quite general. This should inspire further agent based modelling approaches.
Our agentbased model is fully calibrated and validated against real data from the global interfirm R&D network. It shows the emergence of a broad, rightskewed distribution of alliance sizes, taking into account a heterogeneous fitness distribution of agents. On the methodological side, our study provides an approach to infer the correct parameter values for the agentbased model, to interpret them and check their consistency with reality. Like for any agentbased model approach, we cannot conclude that our model is the only one able to explain and reproduce the alliance size distribution. However, the very good match with reality is a clear sign of plausibility for the set of agent rules that we propose, thus providing us with new insights into the micro dynamics of alliance formation.
Acknowledgements
M.V.T. and F.S. acknowledge financial support from the Swiss National Science Foundation, through grant 100014_126865, “R&D Network Life Cycles”. M.V.T. and F.S. acknowledge financial support from the Seed Project SPRC 0115 “Performance and resilience of collaboration networks”, granted by the ETH Risk Center of ETH Zurich. The authors thank Ryan Murphy for comments on an early version of this paper.
References
 Ahuja (2000) Ahuja, G. (2000). The duality of collaboration: Inducements and opportunities in the formation of interfirm linkages. Strategic management journal 21(3), 317–343.
 Barabasi (2005) Barabasi, A.L. (2005). The origin of bursts and heavy tails in human dynamics. Nature 435(7039), 207–211.
 Bianconi and Barabási (2001) Bianconi, G.; Barabási, A. (2001). Competition and multiscaling in evolving networks. EPL (Europhysics Letters) 54, 436.
 Bitzer and Geishecker (2010) Bitzer, J.; Geishecker, I. (2010). Who contributes voluntarily to OSS? An investigation among German IT employees. Research Policy 39(1), 165–172.
 Csardi and Nepusz (2006) Csardi, G.; Nepusz, T. (2006). The igraph software package for complex network research. InterJournal, Complex Systems 1695(5).
 Durkheim (2014) Durkheim, E. (2014). The division of labor in society. Simon and Schuster.
 Frenz and IettoGillies (2009) Frenz, M.; IettoGillies, G. (2009). The impact on innovation performance of different sources of knowledge: Evidence from the UK Community Innovation Survey. Research Policy 38(7), 1125–1135.
 Fruchterman and Reingold (1991) Fruchterman, T.; Reingold, E. (1991). Graph Drawing by Forcedirected Placement. Software Practice and Experience 21(11), 1129–1164.
 Hagedoorn (2002) Hagedoorn, J. (2002). Interfirm R&D partnerships: an overview of major trends and patterns since 1960. Research policy 31(4), 477–492.
 Hoang and Rothaermel (2005) Hoang, H.; Rothaermel, F. T. (2005). The effect of general and partnerspecific alliance experience on joint R&D project performance. Academy of Management Journal 48(2), 332–345.
 Katz and Martin (1997) Katz, J.; Martin, B. R. (1997). What is research collaboration? Research Policy 26(1), 1–18.
 Kim and Song (2007) Kim, C.; Song, J. (2007). Creating new technology through alliances: An empirical investigation of joint patents. Technovation 27, 461–470.
 Lakhani and Wolf (2003) Lakhani, K.; Wolf, R. G. (2003). Why Hackers Do What They Do: Understanding Motivation and Effort in Free/Open Source Software Projects. SSRN Electronic Journal .
 PastorSatorras et al. (2001) PastorSatorras, R.; Vazquez, A.; Vespignani, A. (2001). Dynamical and Correlation Properties of the Internet. Physical Review Letters 87.
 Perra et al. (2012) Perra, N.; Goncalves, B.; PastorSatorras, R.; Vespignani, A. (2012). Activity driven modeling of time varying networks. Scientific Reports 2, 469.
 Scholtes et al. (2016) Scholtes, I.; Mavrodiev, P.; Schweitzer, F. (2016). From Aristotle to Ringelmann: A largescale analysis of team productivity and coordination in Open Source Software projects. Empirical Software Engineering 21(2), 642–683.
 Tomasello et al. (2016) Tomasello, M. V.; Napoletano, M.; Garas, A.; Schweitzer, F. (2016). The Rise and Fall of R&D Networks. ICC  Industrial and Corporate Change 26(4), 617–646.
 Tomasello et al. (2014) Tomasello, M. V.; Perra, N.; Tessone, C. J.; Karsai, M.; Schweitzer, F. (2014). The Role of Endogenous and Exogenous Mechanisms in the Formation of R&D Networks. Scientific Reports 4, 5679.

Tomasello et al. (2017)
Tomasello, M. V.; Vaccario, G.; Schweitzer, F. (2017).
Datadriven modeling of collaboration networks: A crossdomain
analysis.
EPJ Data Science
6, 22.
Comments
There are no comments yet.