Competition among networks highlights the unexpected power of the weak

Transcription

1 SUPPLEMENTARY INFORMATION Cometition among networks highlights the unexected ower of the weak J. Iranzo, J. M. Buldú and J. Aguirre Sulementary Figures Eigenvector centrality max Galesburg min Quincy Peoria Sulementary Figure 1. Collaboration network among hysicians in Peoria, Galesburg and Quincy (Illinois, US). Network arameters of each city are summarized in Sulementary Table 2. In the figure, the size of the nodes is roortional to the their number of connections, while colours reflect the eigenvector centrality of each node inside its city. Note that centralities are calculated before introducing connections between cities.

2 a!" $" -,.)/0"+1'!" #" ()*+#,' %"!"#$%&'!" b! $ c! $ d! $ "# "# "# Sulementary Figure 2. Cometition for centrality among rofessional networks. (a) Collaboration networks of hysicians working in Peoria (A), Galesburg (B) and Quincy (C) (Illinois, US), from [10]. Creating connections between hysicians leads to a network-of-networks T, whose centrality is distributed among the cities. In (b), (c) and (d) we show the centrality retained at each network deending on different connection strategies. The radius of each circle is roortional to the centrality accumulated by each network. Network strengths are λ A = 6.16, λ B = 5.78 and λ C = In (b), when one connection between hysicians is allowed (l = 1), the unique Nash equilibrium leads to a centrality distribution of C A = 0.78, C B = 0.14 and C C = When networks are allowed to create two connections (l = 2), two solutions coexist: networks B and C obtain weak outcomes (c) when connecting to A (X : C A = 0.65, C B = 0.21 and C C = 0.14), but their best strategy is to create the links between them, thus, forcing network A to join them (X 0: C A = 0.33, C B = 0.39 and C C = 0.28). In summary, the final result of the contest is largely deendent on the reached solution.

3 b STRONGEST a c WEAKEST Sulementary Figure 3. Cometition for centrality in scientific collaboration networks. (a) Giant comonent of the collaboration network of researchers of the University of Oxford who investigate on Ebola. The width of the links is roortional to their weights, which account for the strength of the collaboration between two authors based on the aearance as co-authors in the same aer (as exlained in the text). Node size is roortional to the eigenvector centrality. Before connecting to other centres, Oxford needs to evaluate its strength comared to the other co-authorshi networks, in order to choose the otimum strategy. (b) Oxford connects with other two centres whose collaboration networks are weaker, being X the most convenient Nash equilibrium. (c) Now, Oxford connects with centres with stronger collaboration networks and X 0 turns out to be its most convenient Nash equilibrium.

4 Airline A (layer A)? Airline B (layer B)?? Airline C (layer C) Sulementary Figure 4. Cometition for centrality between commercial airlines. In this illustrative examle, each network i.e. layer corresonds to a different commercial airline. The nodes of the networks reresent airorts, which are linked together if a flight (with a certain airline) between two airorts exists. Colours red, green and blue indicate those airorts where a given airline has flights, while uncoloured nodes are the airorts not reached by the airlines. Two layers are connected when two airlines decide to make an agreement and connect flights at a given airort. In this case, each layer of the multi-layer network corresonds to a subnetwork that is cometing for acquiring the highest ossible centrality. Note that the adequate election of a artner airline can lead to a higher increase of the centrality of a given comany (or a air of them) in the global air transortation network.

5 5 CC PP X!! a B CC" CC" C" A b C B PP" PP" A c d B X 0!! CC" C B CC" A A C PP" B PP" A C(X ) equilibrium C(X 0 ) equilibrium Strategy: CC PP CC PP Network A Network B Network C Sulementary Figure 5. Differences in the distribution of centrality for central-central (CC) and eriheral-eriheral (PP) connection strategies. The circles reresent the loan networks of three indeendent Indian villages (see Fig. 1 of the main manuscrit for details). Creating connections between villages would lead to a network-of-networks T, whose centrality is distributed among the villages. Each icture shows the centrality retained at each village (network) for (a) solution X = {A(0), B A, C A} and CC connection, (b) solution X and PP connection, (c) solution X 0 = {A B, B C} and CC connection, and (d) solution X 0 and PP connection. The radius of each circle is roortional to the centrality accumulated by each network, and circles of radius close to zero are lotted in dashed line (see the table for the actual values). Networks strengths are λ A = 4.27, λ B = 4.05 and λ C = 3.38.

6 Sulementary Figure 6. Difference between the total strength of solutions X 0 and X, λ 1 = λ 1 (X 0 ) λ 1 (X ), as a function of the strength of the strong network in the cometition for centrality among 3 networks. Data oints (error bars) corresond to averages (standard deviations) over all realizations whose λ A,1 lies in the corresonding X-axis interval. Note that the strength of solution X 0, the cooerative equilibrium, is always larger than the strength of solution X, the equilibrium based on weak networks connecting to the strong one. The realizations are develoed with the networks generated with the Barabási-Albert model [27] used in Fig. 2 of the main text: each cometitor makes use of as much as l = 1 link to connect with the rest of the networks, for each choice of B and C the system is solved for 20 series of A, and results are an average of more than 500 sets of A, B, and C. The X-axis has been rescaled to allow for comarisons among different realizations. 6

7 Sulementary Figure 7. Cometition for centrality among 3 networks as a function of the strength of the strong network, when the centrality is defined as C i = λ T,1C i. Each cometitor makes use of as much as l = 1 connector link. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C (see cation of Fig. 2 of main manuscrit for more details). (a) Number of coexisting Nash equilibria er realization. The radius of each circle is roortional to the fraction of realizations. (b) Relative occurrence of different configurations in the set of solutions, averaged over all realizations: (i) Equilibrium X 0 = {A B, B C} (yellow), (ii) equilibrium X = {A(0), B A, C A} (blue) and (iii) other equilibria (grey). (c) Centrality of networks A ( ), B ( ) and C ( ) for a articular choice of B and C (λ B = 5.25 and λ C = 5.2). The results are shown for solutions X 0 and X (colour code as in (b)). (d) Relative centrality variability C among different Nash equilibria. Data oints (error bars) corresond to averages (standard deviations) over all realizations whose λ A lies in the corresonding X-axis interval. Network symbols as in (c). 7

8 Sulementary Figure 8. Cometition for betweenness centrality among 3 networks as a function of the strength of the strong network. Each cometitor makes use of as much as l = 1 link to connect with the rest of the networks. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C (see cation of Fig. 2 of main manuscrit for more details). The centrality of each network is measured as its betweenness centrality. (a) Number of coexisting Nash equilibria er realization. The radius of each circle is roortional to the fraction of realizations. (b) Relative occurrence of different configurations in the set of solutions, averaged over all realizations: (i) Equilibrium X 1 = {A B, B C, C A} (dark orange), (ii) equilibrium X 2 = {A C, C B, B A} (ale orange), (iii) X = {A(0), B A, C A} (blue) and (iv) other equilibria (grey). (c) Centrality of networks A ( ), B ( ) and C ( ) for a articular choice of B and C (λ B,1 = 5.25 and λ C,1 = 5.2). The results are shown for solutions X 0 (blue) and X 1 + X 2 (orange). (d) Relative centrality variability C among different Nash equilibria. Data oints (error bars) corresond to averages (standard deviations) over all realizations whose λ A,1 lies in the corresonding X-axis interval. Network symbols as in (c). It is clear that the benefit for migrating from one solution to another is negligible. 8

9 9 Sulementary Figure 9. Cometition for centrality among 3 Erdo s-re nyi networks as a function of the strength of the strong network. Each cometitor makes use of as much as l = 1 link to connect with the rest of the networks. We modify the size and/or mean degree of network A in order to increase its strength from λa,1 = λb,1 to λa,1 λb C,1. The X-axis has been rescaled to allow for comarisons among different realizations. (a) Number of coexisting Nash equilibria er realization. The radius of each circle is roortional to the fraction of realizations. (b) Relative occurrence of different configurations in the set of solutions, averaged over all realizations: (i) Equilibrium X0 = {A B, B C} (yellow), (ii) equilibrium X = {A(0), B A, C A} (blue) and (iii) other equilibria (grey). (c) Centrality of networks A ( ), B ( ) and C (4) for a articular choice of B and C (λb,1 = 5.25 and λc,1 = 5.2). The results are shown for solutions X0 and X (colour code as in (b)). (d) Relative centrality variability C among different Nash equilibria. Data oints (error bars) corresond to averages (standard deviations) over all realizations whose λa,1 lies in the corresonding X-axis interval. Network symbols as in (c).

10 Sulementary Figure 10. Cometition for centrality among 3 networks as a function of the strength of the strong network, when both ure and mixed strategies are allowed. In this examle l = 1, which means that each network can connect through weighted connector links to all its cometitors as far as the addition of such weights is equal to or less than 1. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C (see cation of Fig. 2 of main manuscrit for more details). (a) Number of coexisting Nash equilibria er realization. The radius of each circle is roortional to the fraction of realizations (no cases with more than four coexisting solutions were found). (b) Relative occurrence of different configurations in the set of solutions, averaged over all realizations: (i) Pure equilibrium X 0 = {A B, B C} (yellow), (ii) ure equilibrium X = {A(0), B A, C A} (blue), (iii) other ure equilibria (dark grey), and (iv) mixed equilibria (ale grey). Note that (a) and (b) corresond to Figs. 2a and 2b, but where mixed equilibria are allowed. (c) Mixed-vs-ure relative centrality, C. Here, C = 1 and C = 0 corresond to the centralities of a network under its most and least desirable ure equilibria, resectively. The lot shows the mean values of C for the strong network ( ), the coalition of the small networks ( ), and the 5/95 ercentiles (error bars). 10

11 11 Sulementary Figure 11. Solution rofiles for a cometition in which each network can create u to two connector links, as a function of the strength of the strong network. Each cometitor makes use of as much as l = 2 links to connect with the rest of the networks. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C. Bars indicate the relative occurrence of different configurations in the set of solutions, averaged over all realizations, as a function of the strength of A. Solutions are classified in three grous (yellow: equilibrium X 0 = {A B, B C}, blue: equilibrium X = {A(0), B A, C A}, grey: other equilibria). Sulementary Figure 12. Comarison between the solution rofiles for a cometition among 3 networks relative to the cases where each network has different maximum number of connector links and different weights in the connector links. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C. Bars indicate the relative occurrence of different configurations in the set of solutions, averaged over all realizations, as a function of the strength of A. Two different cases are studied: (a) Different maximum number of connector links: l A = l C = 2 and l B = 3, with w A = w B = w C = 1. (b) Different weights in the connector links: w A = w C = 2 and w B = 3, with l A = l B = l C = 1. Solutions are classified in three grous (yellow: equilibrium X 0 = {A B, B C}, blue: equilibrium X = {A(0), B A, C A}, grey: other equilibria). Note that both lots are identical.

12 Sulementary Figure 13. Solution rofiles for a cometition among 3 networks where one network increases the weight of their connector links, as a function of the strength of the strong network. Each cometitor makes use of as much as l = 1 link to connect with the rest of the networks. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C. Bars indicate the relative occurrence of different configurations in the set of solutions, averaged over all realizations, as a function of the strength of A. (a) w A = w B = w C = 1 (equivalent to Fig. 2 in the main text). Increase in w A : (b) w A = 1.2, w B = w C = 1 and (c) w A = 1.5, w B = w C = 1. Increase in w B : (d) w A = w C = 1, w B = 1.2 and (e) w A = w C = 1, w B = 1.5. Increase in w C : (f) w A = w B = 1, w C = 1.2 and (g) w A = w B = 1, w C = 1.5. Solutions are classified in three grous (yellow: equilibrium X 0 = {A B, B C}, blue: equilibrium X = {A(0), B A, C A}, grey: other equilibria). 12

13 13 Sulementary Figure 14. Solution rofiles for a cometition among 3 networks A, B, and C of size 10 nodes as a function of the strength of the strong network. Each cometitor makes use of as much as l = 1 link to connect with the rest of the networks. Results are an average of 100 sets of A, B, and C. Bars indicate the relative occurrence of different configurations in the set of solutions, averaged over all realizations, as a function of the strength of A. Bars indicate the relative occurrence of different configurations in the set of solutions as a function of the strength of A. Solutions are classified in three grous (yellow: equilibrium X 0 = {A B, B C}, blue: equilibrium X = {A(0), B A, C A}, grey: other equilibria). Sulementary Figure 15. Solution rofiles for a cometition among 3 networks A, B, and C of size 1 node as a function of the strength of the strong network. The strength of each 1 node-network λ i,1 is equal to the weight of the auto-loo associated with each node. Bars indicate the relative occurrence of different configurations in the set of solutions as a function of the strength of A. Solutions are classified in three grous (yellow: equilibrium X 0 = {A B, B C}, blue: equilibrium X = {A(0), B A, C A}, grey: other equilibria).

14 14 Sulementary Figure 16. Solution rofiles for a cometition among 4 networks A, B, C and D as a function of the strength of the strong network. λa,1 > λb,1 > λc,1 > λd,1. Each cometitor makes use of as much as l = 1 link to connect with the rest of the networks. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C. Results are shown for two ossible sizes of the weakest network D: much smaller than C (λd,1 = 0.6λC,1, uer anel) and similar to C (λd,1 = 0.9λC,1, lower anel). Bars indicate the relative occurrence of different configurations in the set of solutions according to the following classification: Yellow: equilibrium X000 = {A B, B C, D B}; dark blue: 00 = {A(0), B A, C A, D A}; light blue: equilibrium equilibrium X cannot and X,P X,P = {A D, B A, C A}; grey: other equilibria. Note that equilibria X aear together in the same realization and the latter can be considered a erturbation of the former. Sulementary Figure 17. Cometition for centrality among 3 scale-free networks as a function of the strength of the strong network when the ayoff is that of the central nodes. Each cometitor makes use of l = 1 connector link. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results are an average of more than 500 sets of A, B, and C (see cation of Fig. 2 of main manuscrit for more details). Relative occurrence of different configurations in the set of solutions, averaged over all realizations, when the ayoff is the connector nodes centrality divided by the addition of the centralities of the rest of nodes: (i) 0 Equilibrium X0 = {A B, B C} (yellow), and (ii) equilibrium X = {A C, B A, C A} (light blue).

15 Sulementary Figure 18. Cooeration and control of the game by weak networks (ower of the weak) when the ayoff is a combination of the collective centrality and the individual centrality of connector nodes. Each cometitor makes use of l = 1 connector link. For each choice of weak networks B and C, the system is solved for 20 series of the strong network A, and results reresent fractions over more than 100 sets of A, B, and C scale-free (Barabási-Albert) networks [27] (black), such that λ B C < λ A < λ B C (see cation of Fig. 2 of main manuscrit for more details). The Y-axis reresents the fraction of realizations in which (i) weak networks share the same referred equilibrium, (ii) such equilibrium involves cooeration between weak networks, and (iii) weak networks can imose their referred equilibrium on the strong network (migrations towards cooeration are ossible). α = 0 is the situation in which the ayoff is that of the connector node individually, and α = 1 is the case in which the ayoff is that of the whole network. 15

16 16 Sulementary Tables Network N L λ 1 k C d k max max{u 1 } Village A Village B Village C Sulementary Table 1. Summary of the main arameters of the loan networks of villages A, B and C (in [1], villages 55, 15 and 14, resectively). Secifically, the arameters shown in the table are: size of the networks N, number of connections within each village L, largest eigenvalue of the connectivity matrix λ 1, average degree (i.e., number of connections) of the network k, clustering coefficient C, average shortest ath of the network d, highest degree k max and eigenvector centrality of the most central node max{u 1 }. Network/City N L λ 1 k C d hub k max max{u 1 } Peoria Galesburg Quincy Sulementary Table 2. Summary of the main arameters of the collaboration networks of Peoria, Galesburg and Quincy [10]. Secifically, the arameters shown in the table are: size of the networks N, number of connections within each city L, highest eigenvalue of the connectivity matrix λ 1, average degree (i.e., number of connections) of the network k, clustering coefficient C, average shortest ath of the network d, node with the highest degree (hub), highest degree k max and eigenvector centrality of the most central node max{u 1 }.

17 17 (A) Oxford co-authorshi network being the strongest Institution N L λ 1 C(X ) C(X 0 ) Oxford Network B Network C (B) Oxford co-authorshi network being the weakest Institution N L λ 1 C(X ) C(X 0 ) Network A Network B Oxford Sulementary Table 3. Summary of the main arameters of the Oxford co-authorshi network before and after connecting to other two research centres. The cometing networks have been generated with the model described in the text in order to fulfil that: (A) Oxford is the strongest network and (B) Oxford is the weakest network. C(X ) accounts for the Nash equilibrium where the two weak networks attach to the strongest one, while C(X 0 ) is the Nash equilibrium where the two weaker network are linked together, forcing the strong one to connect to them. The arameters shown in the table are: size of the networks N, number of connections within each research centre L, highest eigenvalue of the connectivity matrix λ 1. The last two columns account for the total centrality accumulated for each network in the C(X ) and C(X 0 ) equilibria.

18 18 Sulementary Note 1. Nash equilibria in network-of-networks: Alications to real cases In this Sulementary Note we overview a series of cases in which the methodology introduced in the main text could be alied. Imortantly, all these examles, where networks with a certain identity interact creating a network-of-networks, must accomlish two fundamental requirements: Every node should belong to a single network or community, that is, networks must not overla. All the nodes of a network must behave as a whole cometing entity. In ractice, this means that the strategies followed by each network during cometition ursue the imrovement of the whole network imortance, not that of certain individuals. This requirement can be fulfilled either by (i) all members of the community (i.e., nodes of the network) sharing a common objective, or (ii) by the existence of a single agent controlling the strategies of the network. Once these two constraints are satisfied, the methodology roosed here can be alied, regardless of the underlying system. Thus, a diversity of social, technological and biological networks can be insected under the roosed framework. Certainly, social networks with some degree of hierarchy are the most suitable candidates, since their modular structure combined with the existence of agents that romote coordinated actions (local authorities, funding agencies, rofessional associations) make these two constraints easily fulfilled. As we will see, loan networks, scientific collaboration networks or hysician networks are good examles. In addition, two ossible alications to technological and biological systems will be discussed. Social networks Loan networks in Indian villages Loan networks are a good examle of how our methodology could imrove our understanding of the dynamics of interaction among networks. In the year 2006, a series of surveys were carried out at 75 villages in rural southern Karnataka, a state in the south of India [1]. The objective of these surveys was to understand the social interactions among inhabitants within each village, with the aim of carrying out a microfinance rogram consisting of granting small credits to develo the economy of the villages. Besides their economical interest, the surveys reorted very relevant information about the social behaviour of the inhabitants of the villages, how they interacted with each other and which the main actors of such relations were. In our study, we construct loan networks making use of the answers to the questions who would you lend money to? and from whom would you borrow money? of such surveys. As in [1], when the answer to both questions links two inhabitants of the village, we create a connection between them, leading to a binary (0/1) connectivity matrix G. In turn, we consider the matrix to be symmetric, since when a loan is granted, money moves in both directions of the link as sooner or later it will have to be aid back. From the whole dataset of 75 villages, we selected three of them (from now on, villages A, B and C) and used them as an examle to analyse how the connections between villages would affect the resulting loan network-of-networks. Details about the size, i.e. number of eole (nodes) and interactions between them (links) can be found in Sulementary Table 1. Villages were named according to the value of the largest eigenvalue of their adjacency matrix, such that λ 1,A > λ 1,B > λ 1,C. As the strength of each network (village) is given by its largest eigenvalue, network A is the strongest and network C the weakest cometitor. Before investigating the consequences of connecting networks, we identified which are the most relevant nodes inside each village by means of the eigenvector centrality [2] (see Fig. 1a of the main manuscrit).

19 19 Local loan networks constructed as exlained above are based on real confidence among inhabitants of a community, and for this reason rovide much information about some critical financial roerties associated to such community [3, 4]. In articular, they cast light on the financial resilience of a region, that is, its caacity to overcome financial difficulties and systemic risks [5 7]. In our examle, imagine the usual case of several close villages that base their economy on different roducts (e.g., one village grows mainly tobacco while the neighbours raise sugar cane, the two leading cros in Karnataka see [8] for a detailed exlanation on the agricultural techniques and challenges of such region ). If a lague or disease got into the cros and caused a lot of damage, the most affected village would benefit critically from the connections to other villages that would allow them to obtain (i) an alternative to that cro to ensure the feeding of its oulation and animals, and (ii) a way to obtain an effective financial backing to overcome the crisis. On the other hand, the rest of villages would rofit from finding a new client for their roducts and from offering their financial assistance. While this is just a simle examle, it clarifies that increasing the loan network of a village or region by romoting connections to other regions is doubtlessly an attractive strategy to successfully face unexected climatic, natural or financial challenges. How would the local authorities romote such connections to other villages? This fact will critically deend on the cultural, economical and social roerties of the regions to connect. In develoed regions useful ways are imroving transortation and communications, or in case the nodes reresented comanies, by regional Chambers of Commerce romoting the connections between different businesses, through bilateral investment lans, etc. However, in our Indian case, the authorities would surely focus on romoting the social contact between villages as loan networks are based on ersonal confidence. To achieve this objective, in this context the local administrations frequently organize inter-regional social activities, such as feasts or festivals, that romote friendshi and marriages between eole of different villages. In articular, the connection between two esecially wealthy families the central nodes in our model, would be very favourable for the economy of both villages. Secifically, we can investigate in our articular examle which village would accumulate more centrality/imortance in the loan network, thus having a comarative advantage when comared to their interacting artners. As exlained in the main text (see Fig. 1 and related text), once the three villages have been allowed to lay the game of connecting to other villages, the centrality accumulated by each network strongly deends on the new connections. This way, villages B and C can overcome the imortance of village A by creating connections between them. What is more, village A should not remain disconnected from B and C, since in the long run, it would lose centrality comared to the case of linking to the cluster formed by the two weakest networks. Furthermore, any of the weaker villages can lead the strategies in the connection rocess, snatching the strong village from the control of the whole network-of-networks. Finally, note that similar qualitative results would have been obtained with many other combinations of the 75 villages.

20 20 Professional collaboration networks: The case of Illinois hysicians A similar examle can be found in the context of rofessional collaboration networks. In such networks, nodes are individuals working in a secific field and the links of the network account for collaborations among them. The study of collaborations among American hysicians in the middle fifties carried out by Coleman et al. [9,10] is a seminal examle of the analysis of this kind of networks. In that work, authors were concerned about the sreading of innovation along social networks. Between November 1953 and February 1955 a series of surveys were carried out in different cities of Illinois, with the aim of obtaining how the rofessional interactions between hysicians led to the adotion of a new treatment. We focused on the art of the Coleman s datasets that evaluated the collaboration network among hysicians. Secifically, we analysed the answers to the question who are the three or four hysicians with whom you most often find yourself discussing cases or theray in the course of an ordinary week last week for instance? [9]. The rocedure was simle: each answer creates a bidirectional link between two hysicians. This way, we obtained the collaboration networks of 3 different cities: Peoria, Galesburg and Quincy (see Sulementary Fig. 1). In all cases, cities were disconnected from each other, i.e., none of the medical doctors had weekly connections with counterarts in any of the other two cities. As in the examle of the Indian villages, the imortance of the individuals belonging to each hysician network was quantified by means of the eigenvector centrality: Sulementary Fig. 1 shows a qualitative descrition of such collaboration networks, with the node size roortional to the eigenvector centrality of the nodes. The main arameters of the collaboration network of each city are summarized in Sulementary Table 2. At this oint we can address the consequences of connecting two cities. Let us suose that a city through its local Medical Council, for examle would like to romote the collaboration of their hysicians with hysicians of other cities (by means of a funding rogram, for examle), with the aim of increasing their own knowledge and leadershi in a certain field. What connections should the city romote? Note that if each network reresented a hosital instead of a city, this model would simulate quite recisely the interconnection between hositals through fellowshi rograms, a very established ractice in some (mostly Euroean) countries. Many hositals tyically finance the temoral stay of their medical doctors in other hositals, with the aim of acquiring exerience in a new medical or surgical technique. Once the stay is finished usually lasts between one and two years, the visitors go back to their original deartment and transmit the new technique to their colleagues, imroving the know-how of the whole grou and the caacity of the hosital of attracting atients. In this way, if two hysicians from different cities started to work together, the two collaboration networks would connect forming a network-of-networks and the imortance of each city (i.e. network) could be measured by the imortance accumulated by its hysicians. When establishing these connections, could the cities comete for imortance, thus romoting secific collaborations to increase their outcome? Which city would benefit the most from the knowledge exchanged by the connecting hysicians? As exlained in the main text, the key metric to understand how the centrality is redistributed after networks creating new connections is the largest eigenvalue of the connectivity matrix λ 1. In turn, it allows to rank the strength of the cities leading to λ P eoria > λ Galesburg > λ Quincy. Therefore, Peoria would be network A (the strongest), Galesburg would be network B and Quincy would be network C (the weakest). Sulementary Figures 2a-d show the cometition for centrality between these three cities. Allowing a single connector link (l = 1) between them, only one Nash equilibrium is obtained {A(0), B A, C A}, and the strongest network drastically outerforms its cometitors (Sulementary Fig. 2b), i.e. Peoria (A) would strongly benefit from the interaction with hysicians from any other city. However, if two interactions between hysicians are allowed (l = 2), two coexisting solutions aear (see Sulementary Note 5 for more details on cometitions where more than one connector link is acceted). Now, connecting to the strong city (solution X = {A(0), B A, C A}, Sulementary Fig. 2c) leads to much worse

21 21 results for B and C than connecting with each other (X 0 = {A B, B C}, Sulementary Fig. 2d). In the latter case, the strong city must, in turn, connect to B to retain at least a small art of the centrality of the whole system, thus deending comletely on the strategy adoted by the weak cities. And remind that any weak network can control the migration towards more beneficial equilibria (from X to solution X 0), while the strong network lacks the caacity to do the oosite. Scientific collaboration networks: Investigating Ebola Collaboration networks can also be studied in the framework of scientific co-authorshi [11, 12]. In this case, the nodes of the network stand for researchers, who are linked when aearing together as co-authors of the same scientific ublication [12]. Once again, for determining the imortance of scientists from the co-authorshi network we can use the eigenvector centrality, since it has become a traditional metric for quantifying the imortance of a erson inside a social network [13, 14]. Nevertheless, as exlained above, we need to identify (or define) clusters or communities of researchers that, somehow, work with a common goal or, at least, belong to a community of some sort. For examle, we can think of a research institute interested in imroving its imortance and its scientific roduction. Reasonably, the committee of the institute could ursue this goal by romoting the career of its scientists and establishing a collaboration visiting rogram. This way, encouraging rofessional connections to other centres in a way that the increase of centrality/imortance of the research institute as a whole is larger, would be an interesting strategy for the institute s decision-makers. Under this framework, research centres reresented by the networks of their scientists internal collaborations would romote collaborations with other centres to generate a network-of-networks. As an illustrative examle, we selected the co-authorshi network of those scientists who investigate Ebola at the University of Oxford one of the world leading centres in the subject. Secifically, we retrieved all ublished aers that aear at the Web of Science [15] containing the word Ebola in either the title or the abstract. Next, we selected only the connections between scientists belonging to the University of Oxford who have remained active during the last five years (eriod ). If authors i and j articiated in a aer of n authors, the weight of their connection would be w ij = 1/(n 1), since the time devoted to collaborate is finite [11]. Additionally, the more aers authors i and j wrote together, the more weight was added to their link (following the same exression). In this way, we obtained the Oxford co-authorshi network, which has a giant comonent of N = 34 scientists (see Sulementary Fig. 3 for details). Imortantly, desite the links of this network are weighted, the methodology defined in the main text also alies. Once a network of this tye is constructed we can address how to evaluate the efficiency of tentative Collaboration Programs organized by the University of Oxford. Suose that a certain amount of grants is given by the institution to romote collaborations with other research centres (e.g., covering travel exenses). When creating new links to other institutions, how would the University maximize its centrality comared to the others? Should the centre favour connections to weaker institutions or to stronger ones? To investigate these issues, an analysis of the distribution of centrality and the corresonding Nash equilibria would lead to helful information for the Exchange Program Coordinators of the University. Sulementary Figure 3 shows the Ebola co-authorshi network of the University of Oxford, where node sizes and link widths are roortional to the eigenvector centralities and weights, resectively. Let us test now, by means of a simle model, how the creation of links with other institutions would modify the centrality of the Oxford co-authorshi network. Note that, for simlicity, the link roerties are based only on the number of ublished aers, but a much finer network could be built taking into account the imact factor of aers, their number of citations, etc. This would lead to an increase in the alicability and recision of the results. To obtain the most roductive connection strategies, we constructed a series of artificial co-authorshi networks of different sizes. All of them were created with a model that generates aers with a random distribution of authors and links them with a weight inversely roortional to the number of authors (as

22 22 exlained above). In this way, we obtained networks of different size N, number of links L and strength λ 1, which have structures similar to those of real co-authorshi networks. Note that if we want to use networks associated to real research centres, these would be obtained following the stes exlained above to create the network associated to the University of Oxford. Imortantly, when evaluating the consequences of connecting to other centres, the first issue is to determine whether the Oxford network is strong or weak in the grou of otential cometitors. Suose that Oxford has the oortunity to collaborate with 2 other institutions whose co-authorshi networks are weaker (i.e., λ Oxford > λ B > λ C ). Our methodology identifies which are the ossible Nash equilibria of this situation and allows to quantify the gain of the University of Oxford in the collaboration network. Similarly, the same study can be carried out by considering the University of Oxford as the weakest network (i.e., λ A > λ B > λ Oxford ), allowing to assess how to connect to other weak networks to overcome a strong one. Sulementary Table 3 summarizes the outcome, for the existing Nash equilibria, of the Oxford coauthorshi network cometing against two other test networks under two alternative scenarios (i.e., being the strongest and the weakest). As we can see, for the articular networks shown in Sulementary Table 3, the most roductive strategy is to let weaker networks connect to Oxford and reach the X solution. However, according to the results resented in this manuscrit, any of the weak networks could be able to ush the whole system towards X 0 solution, and the centrality of the University of Oxford would decrease from to In that sense, the Centre Committee should decide whether they refer this situation or the connection to stronger networks (see (B) in Sulementary Table 3), where it would obtain a slightly lower centrality (0.224) but could benefit from other otential advantages of being in connection to strong research grous. Note that this is just a toy model and real cases should be evaluated taking into account more recise co-authorshi networks, as well as other socio-economical asects. Nevertheless, we believe that this methodology could be an additional tool to hel a research centre making a correct decision. Finally, let us remark that similar studies could guide national olicy makers when romoting collaborations between scientists of different countries, where, in this case, each cometing network accounts for collaborations among researchers of a articular country and the creation of links is suorted by grants romoting collaboration among countries (e.g. the bilateral rograms for imroving scientific collaboration among members of the Euroean Union). Technological networks Multi-layer air transortation networks Although social networks are ossibly the ones that can benefit the most from the methodology roosed in this aer, alications to technological networks can also be of interest. For examle, air transortation networks rely on the coordination of different flying comanies, which have the ossibility of creating connections among them in order to attract a higher number of tentative customers. Recently, air transortation networks have been interreted as multi-layer networks, where layers account for different commercial airlines, showing that the rojection into a multi-layer structure describes more recisely the toological roerties of air transortation networks [16]. From this oint of view, we can create a network-of-networks where airlines are subnetworks interacting through an exchange of assengers at certain airorts, being the latter the fundamental nodes of the air transortation network and a flight between two airorts giving rise to a link between two nodes. As in [16], this oint of view leads to a multi-layer network, where each layer accounts for the connections inside a given airline. Next, we can create the inter-layer connections when two comanies decide to sign an agreement, exchanging assengers at certain airorts (see Sulementary Fig. 4 for a qualitative descrition). Now the roblem is how an airline could choose the most adequate airline to connect to, taking into account that other comanies are cometitors that also try to maximize their own benefit. Using the eigenvector centrality as a

23 23 measure of imortance for the airorts [17], our methodology could reveal the most adequate artners to deal with. Imortantly, the rules and equilibria redicted by the methodology roosed in this aer fully aly to this articular case and to multi-layer networks in general. Biological networks Connecting fragmented habitats The alication of the roosed methodology to biology is more limited than in the case of social or technological networks, since most biological networks, such as rotein interaction or gene regulatory networks, lack the ability to choose among different strategies of connection. However, here we resent one secial case to motivate the reader to search for other alications in this field. We focus on the roblem of fragmented habitats, in which certain secies live in close but sarsely connected locations (also known as atches). The connectivity of a landscae is a critical factor for the ersistence of flora and fauna, the maintenance of high levels of genetic variability, and the adatability of secies to environmental changes [18]. Within this framework, network theory is often alied to evaluate the degree of fragmentation of a habitat and to design efficient conservation strategies [19]. In a network reresentation of a landscae the nodes corresond to atches of conserved habitat where a secies can survive, and the links are routes (e.g. ecological corridors, wildlife crossings... ) through which individuals can diserse from one atch to another. Entities engaged in secies conservation and landscae management can mitigate the negative effects of habitat fragmentation by creating new connections among atches. However, it is not unusual that the distribution range of a secies sans over lands administered by different entities, which calls for coordinated actions for the conservation of the secies. An examle of this situation could be the intensive work develoed during the last decades to reintroduce and rotect the brown bear (Ursus arctos) in the Pyrenees, a challenge addressed in coordination among the governments of France, Sain and Andorra. Other examle would be the acquisition of forest lands in endangered areas of the African equatorial forest or the Amazonia by different non-rofit organizations with the aim of fighting deforestation. Because the creation and maintenance of ecological corridors requires a significant investment, how should areas managed by different countries/organizations be connected, if each articiant asires to maximize the outcome (e.g. biodiversity) in the area it manages? This could be a good examle to use the alternative ayoff exlained at the beginning of Sulementary Note 5, where cometitors try to maximize the eigenvector centrality multilied by the maximum eigenvalue. In this case, the eigenvector would be roortional to the oulation of the secies and its associated eigenvalue would be a roxy for the growth rate of the secies, a quantity that in this case should also be considered. As we show in that Sulementary Note, the henomenology is the same as in the general case studied in the manuscrit, and therefore deending on the size and ecological otential of each ortion of the habitat (i.e. network) the stable ways to connect them would be: (i) the connection of weak networks with the strong one (X ), or (ii) the collaboration between the weak (X 0 ). Note that, as mentioned in the manuscrit and shown at the end of Sulementary Note 3, the final global eigenvalue of solution X 0 is always larger than that of X, which means that the final growth rate of the endangered secies after the connection would be otimized if the different networks decided to collaborate.

24 24 Sulementary Note 2. Basic asects of the cometition among networks for eigenvector centrality The election of eigenvector centrality as the metric to be otimized The quantification of the imortance of a node within a network is an issue with a long tradition, and the eigenvector centrality is the most extended metric due to its ractical alications. To name a few, it is behind the Google Pagerank algorithm, which is an indicator of the imortance of a given webage and, in turn, it is used to elaborate the lists of webages dislayed by Google when doing a certain search [20]. The idea of using the eigenvector centrality to evaluate the scientific imact of scientists has also been roosed [21] and the eigenfactor roject [22] has recently adated the eigenvector centrality to be an effective measure for quantifying Journal imortance. Other illustrative (real) alications of the eigenvector centrality are the detection of imortant regions in the brain [23] or the, more classical, evaluation of imortance of an individual within a social grou [24]. In addition, the eigenvector centrality is also related with a diversity of dynamical rocesses such as disease or rumour sreading (see [2] for an overview). Relevant definitions and analytical solution of the two-network roblem The cometition for eigenvector centrality between two networks A and B that are connected through a small number of connector links was established and analysed in [25]. We review here the main conclusions. Networks A and B have N A and N B nodes resectively. We connect them through L connector links to create a new network T of N T = N A + N B nodes. For simlicity, let us suose that A and B are weighted but undirected networks. Let us call λ A,i, and λ B,i the i eigenvalues of the adjacency matrices G A and G B resectively. Let us suose λ A,1 > λ B,1. We call u A,i to the N A eigenvectors of network A, and u B,i to the N B eigenvectors of network B. The eigenvector u T,1 determines the outcome of the cometition between networks A and B. Centrality A (C A ) and centrality B (C B ) are the fractions of the total centrality that remain in the nodes of network A and B after the connection, and are obtained as NA i=1 C A = ( u T,1) i NT i=1 ( u, (1) T,1) i C B = 1 C A. (2) This way, the goal of the cometition between networks is to increase C as much as ossible. The first eigenvalue and eigenvector of the total network T formed by the networks A and B connected, exressed as quantities that are only deendent on the isolated networks A and B, can be aroximated to first order by u T,1 λ T,1 = u A,1 + ɛ N B u A,1 P u B,k k=1 λ A,1 λ B,k u B,k + o(ɛ 2 ), (3) = λ A,1 + ɛ 2 N B ( u A,1 P u B,k ) 2 k=1 λ A,1 λ B,k + o(ɛ 3 ). (4) Note that P ij = P ji = 1 for ij {cl} and P ij = 0 elsewhere, being {cl} i=1,...,l the set of airs lm corresonding to the connector links that attach the connector nodes l of network A with nodes m of network B to form T. Furthermore, the terms k = 1 of both summations are the most relevant ones because λ A,1 λ B,k > λ A,1 λ B,1 for k > 1. Furthermore, u A,1 P u B,1 = cl ( u A,1) i ( u B,1 ) j, that is, the sum of the roducts of the eigenvector centralities of all connector nodes measured when the networks are disconnected.