Fragile, yet resilient: Adaptive decline in a collaboration network of firms

The dynamics of collaboration networks of firms follow a life-cycle of growth and decline. That does not imply they also become less resilient. Instead, declining collaboration networks may still have the ability to mitigate shocks from firms leaving, and to recover from these losses by adapting to new partners. To demonstrate this, we analyze 21.500 R\&D collaborations of 14.500 firms in six different industrial sectors over 25 years. We calculate time-dependent probabilities of firms leaving the network and simulate drop-out cascades, to determine the expected dynamics of decline. We then show that deviations from these expectations result from the adaptivity of the network, which mitigates the decline. These deviations can be used as a measure of network resilience.


Introduction
Resilience denotes the ability of a system to withstand shocks and to recover from them [7,20]. Hence, it combines two di erent dimensions: robustness against shocks and adaptivity to overcome states that result from a shock [8,27]. Interestingly, most research has only focused on the rst aspect, robustness. Much less attention is paid to the second one, which is more di cult to quantify and to forecast. Therefore, in this paper we aim at better understanding the adaptive capacity of systems.
One reason for the biased research interest comes from the fact that robustness is strongly related to concepts like stability which are easier to assess. However, if robustness or stability are used as synonyms for resilience [6], the temporal aspects of recovery are neglected [9,17]. To quote Abraham Lincoln: "It's not important how many times you fall, but how many times you get back up". If we want to improve the resilience of systems, the solution is not to simply avoid situations that may lead to a breakdown, by increasing the robustness of a system [1]. Very often such breakdowns cannot be avoived or even controlled [19,28]. The real problem is how to enable systems to cope with these situations and to recover from them [12,21].
This requires us to develop a systemic view that takes the eigendynamics of a system into account [4,14]. To address this, we need an appropriate system representation. The complex systems 2 The collaboration network of firms 2

.1 Data and networks
Firms with a focus on research and development (R&D) activities continuously establish new collaborations with other rms, to exchange knowledge and to leverage synergies. Because rms have to declare their R&D alliances, we have access to a large data set of more than 14.500 rms and 21.500 collaborations over a time interval of 25 years (1984-2009), covering six di erent industrial sections, e.g., Pharmaceuticals or Computer Hardware. The details of the data set are described in our di erent publications [22,23,25]. Figure 1 shows an example of such a collaboration network in two di erent years.
An empirical investigation of the evolution of these sectoral collaboration networks has revealed a rise and fall dynamics, also illustrated in Figure 1. Two periods in the evolution of these collaboration networks have been distinguished: From 1984 to 1995, we see a steady growth of the networks, both Fragile, yet resilient: Adaptive decline in a collaboration network of rms with respect to the number of rms and the number of links. From 1995 to 2009, on the other hand, we observe that these networks continuously shrink. This holds despite the mentioned fact that rms continue to establish new alliances. But theses activities do not break the declining trend.

Leaving probability
In the following, we seek to quantify the tendency of rms to leave. Our considerations start from the question why rms stay in a collaboration network. As other economic actors, rms try to maximize their utility, i.e., the di erence between bene ts and costs. Hence, rms stay as long as their bene ts exceed their costs. But even if rms leave, they can still return to the network later to start new R&D collaborations with the same or with other partners.
While this dynamics seems reasonable, we have to overcome the problem that there is only data available about the starting date when rms establish a new alliance, but no data about the ending date. Thus, we rst need to estimate the life time of an R&D alliance. This problem was solved in a subsequent study [24]. We have estimated that the mean life time of an R&D alliance is about 3 years. We build on this result here, assuming that the life time of an alliance is randomly drawn from a normal distribution with a mean of 3 years and a standard deviation of 1 year.
This life time estimation has enabled us to reconstruct the evolution of the collaboration network as detailed in [22,23]. We use the starting date of each alliance and the information about its collaboration partners. Then, we sample a life time of the alliance from the mentioned distribution to determine its 4/14 (Submitted for publication) ending date, at which we remove all collaboration links related to that alliance. The end of an alliance does not imply that rms leave the collaboration network. In the meantime they may have used their presence to establish new alliances with other rms. Only if rms have no active alliances in a given year, they will leave. This information is aggregated for each year .
Once we know which rms stayed and which rms left, we calculate the leaving probability as follows. For each rm, we use a time dependent state variable, ( ) = 1 if rm is present in year and ( ) = 0 if it is absent. The probability to leave is then de ned as: ( ) = [ ( + 1) = 0| ( ) = 1], i.e., it is the probability that a rm present in year is absent in the following year. The probability to stay is Our aim in this paper is to estimate how the leaving probability depends on the bene ts of a rm. We argue that, given the aim of the collaboration is knowledge exchange, the bene ts of rm crucially depend on its number of active partners in the collaboration network. We conjecture, the better the rm's embeddedness in the network, i.e., the more active partners, the less the probability to leave. To obtain a quantitative relation, we rst measure, for each year , the number of active partners ( ) of each rm present in the network. Then, we determine for the same year its leaving probability ( ) as introduced above and de ne the relation to the number of active partners as: As a reference, we rst want to estimate for the period of network growth ending in 1995. Therefore we aggregate all data for the period from 1984-1995, and then do a logistic regression on the log odds (or logit), ln [ /(1 − )] = + . The regression results are shown in Figure 2 together with the empirical leaving probabilities (yellow marks) for the six di erent industrial sectors.
We clearly see the monotonous decrease of the leaving probabilities with the number of active partners. The plotted 95% con dence intervals of the estimates indicate that the results are indeed reliable. At the same time, we also notice the di erences between industrial sectors, in particular regarding the number of active partners.

Time dependent leaving probability
Our task is now to model the cascades of rms leaving the collaboration network. We start from the network at its maximum size in 1995 and only consider rms that are present there in 1995, which we call the "class of 95" in the following. If the total number of rms in the network is tot ( ) and the number of rms remaining from the class of 95 is ( ), by construction, in our reference year 1995 = tot . Afterwards, ( ) will decrease because of rms leaving, and the question is how fast this decline happens. Because we lack information about leaving dates, we have to generate our empirical observations for the class of 95 from data about their new alliances and about the life time of their established alliances, as before. Firms that are no longer part of any active alliance in a given year are assumed to leave. This way, we obtain reference data about ( ) that are plotted in Figure 4 (yellow marks). Now we have to compare these data with our results from simulating the cascades. Instead of the information about active alliances, we now consider the probabilities of rms to leave the network. The estimated leaving probabilities for the di erent sectors shown in Figure 2 denote a lower bound because they were obtained considering the growth phase of the network. To simulate the decline of the collaboration network, we need to adjust them over time, i.e., ( ). For 1995, this probability is given in the plots shown in Figure 2. For each year after 1995, i.e., from 1996 to 2009, we then recompute the probability that a rm which is present in year − 1 leaves in the coming year . For this recalculation, we take the information about the network at time − 1 into account, in particular about the number of active partners. I.e., we recompute the plots shown in Figure 2 for every year : We note that with this incremental update we are far from just tting the leaving probabilities to a 6/14 (Submitted for publication) given year. Instead, we consider the history of the collaboration network, as well as the actual situation for rms regarding their active partners.
The regression results are shown in Figure 3 for all industrial sectors. For all years after 1995 we have plotted the di erences ( ) − , where refers to the values of 1995, as shown in Figure 2.
From the results, we notice that rms with a fewer active partners are more a ected by the timedependent leaving probabilities than rms with many partners. Speci cally, for rms with less than 5 active partners the constant leaving probability underestimates their chances to leave, i.e., in reality they have left more often. Only for two sectors, Computer Hardware and Computer Software, in few years rms with 5 or more active partners leave less often, i.e., the constant leaving probability overestimates their chances to leave. With these adjusted probabilities, we make a prediction about the network in year + 1. Precisely, we calculate the expected number ( ) of rms from the class of 95 that will leave. These rms are then removed together with their links and the collaboration network of the class of 95 is updated: ( + 1) = ( ) − ( ). This way we obtain each year a small cascade of rms leaving, which sum up to the considerable decline of the network. Our prediction for ( + 1) is plotted in Figure 4 as the blue  Figure 4 shows both the empirical and the simulated network sizes for the six di erent industrial sectors. We want to point out two observations. First, it is remarkable how well our simulations of the network decay match the empirical network sizes for 4 out of 6 industrial sectors. We note that this holds irrespective of the di erent industrial sectors and the di erent size of the networks. Arguably, Computer Software and Electronic Components refer to very di erent industries and to larger or smaller collaboration networks. So, the agreement found lends evidence to the conclusion that the cascade dynamics we assumed indeed captures an essential mechanism of the observed decline.
Second, it is as interesting to note the two cases where the simulated cascade dynamics does not match the empirical decline: in Pharmaceuticals and Medical Supplies we clearly distinguish two phases of the decline. In an earlier phase, from 1995 to 2000, our simulations still agree with the empirical sizes. But in the last phase, from 2000 to 2005, they signi cantly deviate from the real evolution. Our model would predict that the cascades are further ampli ed and even more rms from the class of 95 have left, whereas the empirical dynamics shows a remarkable stabilization. The trend towards decline is stopped, instead the network size of the class of 95 remains almost constant until the end of the observation period.
This second observation motivates the discussion in the subsequent sections. In a rst step, we want to analyze how to improve the estimates for the leaving probabilities, to better reproduce the observed network sizes for the two cases of Pharmaceuticals and Medical Supplies. In a second step, we discuss what determines these improved leaving probabilities.

Adaptive leaving probability
What additional information do we have available to further improve the estimates for the leaving probabilities? So far, we have used only information from rms of the class of 95. But the collaboration network changes not only because of the exit of the established rms from the class of 95, there is also the entry of new rms. Despite the overall "rise and fall" trend, where decline dominates after 1995, a considerable number of newcomers enter the existing networks each year. Figure 5 shows the total number of rms in the network, tot ( ), the number of rms remaining from the class of 95, ( ), and the number of new rms entering the network in each year after 1995, entr ( ). By construction, in our reference year 1995 = tot , entr = 0.   As we observe, in most sectors the total network size tot ( ) declines over time, even though we have the entry of new rms. So this does not break the overall trend. The exceptions are the two sectors Medical supplies and Pharmaceuticals, and in these we are mostly interested. The di erence results from two combined e ects, as Figure 5 indicates: (a) the very pronounced increase in the number of newcomers, and (b) the slowed down decrease in the number of rms from the class of 95. We see that in all sectors already from 1998 on the number of newcomers exceeds the number of those rms from the class of 95 that still remained in the network. But for the two sectors Medical supplies and Pharmaceuticals the year 1998 is the time when the number of rms from the class of 95 stops plunging and exhibits a slower decline instead. The most suitable interpretation for this observation is in fact that these newcomers basically prevent the established rms from leaving the network.
We will discuss this argument in more detail in Section 4. Before, we want to check whether information about the newcomers would allow us to improve the estimates about the leaving probabilities. We repeat the procedure to calculate ( ), Eqn. (2), but now we correct the values for ( ) to take the newcomers into account. Then we repeat the simulations of the cascades shown in Figure 4 with the adaptive leaving probabilities.
The results are shown in Figure 6. They demonstrate that with the adaptive leaving probabilities we can accurately model the network decline of the rms from the class of 95 for each year. This now holds for all industrial sectors, even for Pharmaceuticals and Medical Supplies. Thus, our cascade model with adaptive leaving probabilities that takes the impact of newcomers into account is able to reproduce the empirical network sizes.

Improved adaptivity
The very good agreement between the empirical and simulated network sizes shown in Figure 6 lends evidence to our methodology to estimate the leaving probabilities of rms. In particular, it supports our underlying assumption, Eqn. (2), that the number of active partners is a main constituent for their decision to stay or to leave the network. But we learned that we have to correct this number to accommodate for the entry of new rms, to obtain the good results.
This leaves us with the task to explain why in some cases the newcomers have such a remarkable in uence. As already mentioned, we argue that these newcomers prevent the established rms from leaving the network because they provide new opportunities to collaborate and often also bring innovative knowledge to the collaboration network (start-ups). Instead of relying on the collaboration with established rms, the rms from the class of 95 now adapt to the situation. They form new R&D alliances with the newcomers, increasing their bene ts and have no further reason to leave. Figure 7 shows two snapshots from the collaboration networks in Medical Supplies and Pharmaceuticals, to il- lustrate this interpretation. These insights can be turned into a novel argument about the resilience of networks. What we observe is the adaptive ability, or adaptivity, of established rms to cope with the new situation. Instead of following the trend to leave the network, they nd new ways of leveraging the situation. This dynamics is precisely what the term resilience shall describe: the capacity to withstand shocks generated by the leave of active partners, and the ability to recover from these shocks, by establishing relations to new partners. What sounds reasonable for a personal life (and has inspired early de nitions of resilience in a psychological context), can be observed also for rms, as our analysis reveals.

Fragile, yet resilient
It is the reason for the time-dependent change of the leaving probabilities, ( ), that the collaboration networks have adapted to the situation of continuous decline. We note that this decline has not completely stopped. Compared to the golden age of 1995, all networks have become much more fragile. Many rms have left, established collaborations ceased to exist. But some networks are still resilient in the sense described above. Those rms that managed to stay in the network after the "fall" trend took over, are indeed the seed for this resilience. They o er newcomers possibilities to integrate into the, this time much smaller, collaboration network and they "connect the dots", as the backbone of the network. Thus, the decline of the network has o ered the chance, more correctly it increased the pressure, for the network to adapt to a changing environment of R&D collaborations.
This leaves us with the question whether our ndings could simply be reduced to the fact that newcomers enter the network. This assumes that a high entry rate would be su cient to make a network resilient. We can refute this argument with reference to an earlier study about the "autopsy" of the social network Friendster [3]. This network collapsed despite a size of 113 million users. New users always entered the network until the very end. But it was shown that after the network has reached a size of 80 million users, the more than 30 million new users still entering became less integrated into the network. Hence, what matters is not the network growth, i.e., the rate at which new nodes enter the network. Whether or not the network becomes resilient depends on the intergration of these new nodes into the network. Friendster failed in this respect and collapsed despite a steady growth.
As the two snapshots of Figure 7 and the dynamics in Figure 4 show, the R&D collaboration networks for Medical Supplies and Pharmaceuticals were successful in integrating newcomers. That's why the established rms continued to stay. This does not mean that the network has to be as dense as for Pharmaceuticals. As we have already shown in Figure 2, each sector is characterized by a di erent cost-bene t relation which determines the conditions for rms to leave. In case of Medical Supplies, it is obviously su cient that established rms start collaborating with 1-2 newcomers, whereas for Pharmaceuticals the critical number of active partners has to be higher.
To conclude, after 1995 all collaboration networks have become fragile, indicated by the global decline trend. To some degree,they are yet resilient dependent on their ability to integrate newcomers. The phrase "fragile, yet resilient" makes reference to an early study about the robustness of infrastructure networks, such as the internet, which were dubbed as "robust, yet fragile" [2]. There the term "fragile" referred to the fact that networks with a very broad degree distribution are vulnerable against the removal of nodes with a high degree. Such nodes are rare, therefore a random removal of nodes would most likely hit one of the many nodes with a very low degree. But a targeted attack, if focused on the high-degree nodes, can easily destroy the network. This insight, however, refers to the expected properties of an ensemble of scale-free networks and cannot be applied to all individual realizations. The internet, in particular, has a low probability to occur at random. It is carefully designed for robustness and therefore much less fragile than random realizations. A similar discussion also applies here. On the one hand, we observe cascades of rms leaving the network because they have less active partners, which in turn increases the trend. This denotes the expected behavior of a network break-down. The double feedback that ampli es this cascade, namely that over time more and more rms have less and less active partners for collaborations, is also known from other cascade models, e.g., from the so-called ber bundle model [11,13]. On the other hand, because collaboration networks are adaptive, they have in principle the ability to deviate from this expected behavior. Even more, we can turn this deviation from the expected behavior into a measures of the adaptivity of the system and, because it prevents the breakdown, as a measure of resilience.
As our results illustrate, not all collaboration networks in the di erent industrial sectors show this adaptive behavior to the same degree. Hence, their decline has continued as expected. We can only speculate why the networks in Medical Supplies and Pharmaceuticals seem to be more adaptive and thus more resilient. Two arguments come into play. One refers to the large number of newcomers which o er ample new opportunities. Economically, this points to low barriers for rms to enter the market, but also to an increased dependency of the industry on external innovations. In Pharmaceuticals, for instance, start-up rms provide a large share of new technologies, substances, etc. The second argument, however, is as important, namely the ability of established rms to integrate these newcomers into their own R&D activities. This largely depends on legal constraints, such as compliance or protection of intellectual properties, but also on the economic pressure to exploit innovative knowledge earlier than the competitors. Indeed, empirical evidence [25] shows that in the sector Pharmaceuticals established rms have a higher probability to form alliances with newcomers (30% of newly formed alliances) than in all other studied sectors, which exhibited probabilities ranging from 10% (Communications Equipment) to 25% (Computer Software).
With this discussion we have provided an interface toward economics, in a truly interdisciplinary manner, which can be explored in the future. But the research presented in this paper also o ers a general insight for network science, where studies about declining networks are still rare. Obsessed with network growth and stability, one should try to avoid the premature focus on the general trend.
Decline is not a synonym for instability and a precursor of collapse. As often it is part of a life-cycle dynamics, where decline should be expected rather than feared. As we have demonstrated, rms, as individuals in a social setting, have the ability to cope with this trend, this way making the system more resilient than expected. Hence, quantitative measures for resilient networks cannot be simply taken from the evolution of the network size. It needs a deeper re ection about the problem of resilience in face of a life cycle, which we just started to provide here.