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Data collection: soil organic carbonThe process of data acquisition, selection and harmonisation is illustrated in the Supplementary Fig. S15 and in the Supplementary Tables S4–6. We conducted a systematic review for peer-reviewed journal articles, published before December 2018, from Web of Science, and Google Scholar with the search terms “(tree species OR forest) AND (soil organic carbon OR soil organic matter)”. We also used studies listed in two previously published meta-analyses16,17, or cited in already retained references (including references in English, French, Spanish, Portuguese, or Russian). For inclusion in the analysis we chose studies based on the following criteria: (1) the study reported soil organic carbon (SOC) or soil organic matter (SOM) concentrations or pools, at least in the topsoil layer and under at least two single-species forest stands; (2) the stands had to be older than 10 years81; (3) the stands had not experienced a major disturbance that differed between tree species, for at least 30 years (e.g., we rejected studies that compared natural forests with planted forests that were less than 30 years old); (4) The SOC concentration was More

Both in Polistes biglumis and P. gallicus in most of the inhabited nests all types of brood were present: eggs, larvae and pupae (Table S1), with the exception of one foundress nest of P. biglumis with only one egg. The size of thermographed nests was quite variable in both species, the number of cells ranging from 18 to 99 in P. biglumis (mean: 61.6 cells), and from 19 to 381 in P. gallicus (mean: 101.7 cells) (Table S1). The mean number of wasps on the thermographed nests was higher in P. gallicus (12.6 wasps) than in P. biglumis (7.1 wasps). All nests of Polistes biglumis we observed in this study were built on stone substrate or walls (Figs. 1c, 2a). Only recently we found one nest built on a pile of wood. The choice of the nest substrate was more diverse in P. gallicus (Figs. 1d, 2b). They chose stone, concrete, walls, window grilles, and metal of fences or doorframes.Figure 2Examples of nests and fieldwork set-up in Obergail (a) and Sesto Fiorentino (b). 1 = thermocouple wire; 2 = global radiation sensor, 3 = Peltier-element IR reference source.Full size imageDaily nest temperature coursePolistes biglumisFigure 3 shows a sequence of thermograms of a P. biglumis nest taken from dawn to dusk. Before sunrise the temperatures of the nest and of the wasps on it were quite low (mean ~ 15 °C) and uniform (~ 12 to 17.5 °C; Fig. 3a). The temperature of the stone substrate where the nest was built on was considerably higher (~ 20 °C). After sunrise (Fig. 3b,c) the nest temperature began to rise quickly. It only needed 13 min of sunshine (radiation) to heat the nest from ~ 17 to ~ 25 °C. Within one hour, temperature differences of almost 20 °C were measured within the nest. At 6:50, when the highest temperature on the nest was already at 36.2 °C, fast movements of the adults with inspections of the cells were observed (Fig. 3c). Soon afterwards the increasing temperature induced the wasps to start fanning (arrow in Fig. 3d). The wasps also began to gather water and spread it on and inside cells to cool the nest by evaporation (Fig. 3d,e). Towards late morning, some parts of the nest reached temperatures as high as 46 °C (Fig. 3e)! As soon as the nest was shaded by the substrate (~ 13:00) the nest temperature decreased according to the decrease in ambient temperature (Fig. 3f,g), reaching ~ 21 °C on average after dusk (Fig. 3h). At that time the substrate temperature (~ 25 °C) was still about 4 °C higher than the nest temperature.Figure 3Thermograms of a P. biglumis nest during a whole day (19.07.2017). (a) Before sunrise at 6:20; (b) during sunrise (06:33); (c) nest temperature increasing fast in sunshine; (d) with a fanner for convective nest cooling (arrow; see also Fig. S4); (e) with water drops for evaporative cooling when sunshine increased part of the nest to temperatures  > 45 °C; (f,g) after sunset (nest now in shade) in the afternoon; (h) at dusk with wasps sitting motionless on the nest. Time = CEST = UTC + 2 h.Full size imageThe nest and body temperatures of a complete 24 h cycle of a different nest are shown in Fig. 4a. At night the nest temperature and the wasps’ thorax temperature decreased slowly according to the decrease of the air temperature. The substrate temperature was always higher than the mean nest temperature, which surely helped to keep the nest temperature higher than the temperature of the surrounding air (Tanest). Variation of within-nest temperature (max–min) was low at night. As soon as solar radiation increased in early morning, the nest temperature and the body temperature of the wasps on it increased rapidly, and the variation of nest temperature (max–min) increased (see also Fig. 3b). Though the maximum nest temperature reached values as high as 46.9 °C, cooling measures of the wasps (fanning and spreading of water drops, see below) kept the mean nest temperature always below 38.5 °C. Cooling of the nest after sunset (at the nest) was much slower than the increase in the morning, following the decrease of ambient and substrate temperature (Fig. 4a,b).Figure 4Examples of daily temperature changes of nests and wasps of P. biglumis (a,b) and P. gallicus (c,d). Tthorax = mean thorax surface temperature of up to five adult individuals per time of measurement; gray ribbon: total range of nest temperatures (Tmax:Tmin) with mean; Tsubstrate = temperature beside the nest (see Fig. S1c,d); Tanest = ambient air temperature directly at the nest. Ta = ambient air temperature in shade 1–3 m away from nest; Radiation = global radiation hitting the nest; black bars = fanning events at the time of thermographic measurements: actually, many more fanning events were observed. (c) Fanning was never observed! See also Fig. S2 for another example of a P. gallicus nest in shade. Time = CEST = UTC + 2 h.Full size imagePolistes gallicusMost P. gallicus nests were built in locations with no or only little direct sunshine (Figs. 2b, 4c, Fig. S2). In their habitats temperatures in midsummer are often already quite high in the morning, and may increase to values higher than 40 °C during the day (Fig. 4d). Mean temperatures of the nest and of the imagines on it were usually higher than the air temperature close to the nest (Tanest). In most nests variation of within-nest temperature (max–min) remained small throughout the day. On hot days (Tanest  > 40 °C), however, maximum temperatures of empty cells in the nest margin sometimes reached values as high as 49.9 °C even in shade. Body temperature of the adults was mostly similar to the mean nest temperature (Fig. 4c, Fig. S2). At night, the nest temperature decreased according to the decrease of Tanest, similar to P. biglumis but at a higher level (Fig. 4d).The situation was different in one large nest which had been built in a location exposed to the morning sun (Figs. 4d, 5). On a hot day when Tanest increased to values higher than 42 °C, the body temperature of the adults increased to values up to 5 °C higher than the mean nest temperature. Nevertheless, though the combined effects of high air temperature and intense insolation increased part of the nest to a temperature of ~ 58 °C (Fig. 4d), mean nest temperature was kept below 41 °C. This was accomplished by cooling with many water droplets in the cells (dark spots in Fig. 5), and by the occurrence of fanning during the period when the sun was shining on the nest (Fig. 4d; see arrows in Fig. 5c). Fanning, however, was quite rare in all the other observed nests, even during the hottest time of the day! Water droplets were carried onto this nest until evening (Fig. 5h), as at that time the nest temperature was still at about 35–38 °C.Figure 5Thermograms of a large P. gallicus nest during a whole day (01.08.2017). Thermograms are rotated 90° clockwise (the upper part is on the right). (a) Before sunrise (6:36); (b) during sunrise (06:46) with the first water drops visible (dark spots); (c) with two fanners for convective nest cooling (arrows, see also Fig. 4d); (d) with more cooling drops; (e) after sunset at the nest site (nest now in shade); (f–h) after sunset in the afternoon and evening. Time = CEST = UTC + 2 h. For temperature evaluation see Fig. 4d.Full size imageBody and nest temperaturesFigure 6 shows a comparison of the dependence of body and nest temperatures on ambient air temperature and insolation between the two species. In the lower ranges of air temperature, usually at night, body temperature followed Tanest closely in both species. The exposition of the P. biglumis nests to the morning sun at ESE (Fig. 7) increased the wasp body temperature to values of often more than 15 °C higher than the surrounding air. However, body temperatures remained always below 40 °C (Fig. 6a). In P. gallicus, by contrast, the body temperature of the wasps increased considerably above 40 °C, to maximum values of about 46 °C, especially (but not exclusively) during intense insolation in the nest exposed to the morning sun (Fig. 6b).Figure 6Surface temperature of the thorax of adult wasps, of different stages of brood and of water drops of P. biglumis (left) and P. gallicus (right), in dependence on ambient air temperature close to the nest (Tanest) and global radiation (color scale). Egg f.n. = single egg on a foundress nest; diagonal lines = isolines. Regressions were calculated for shaded conditions (Radiation = 0–100 W/m2; black or gray solid lines) and sunshine (Radiation  > 100 W/m2; pink broken lines); P  More

Field collectionEndophytic fungi and insects were assessed from dormant twig samples from 155 tree species at 51 locations in 32 countries. Sampled tree species belonged to genera that are native to, and occur widely across, either the northern or southern hemisphere, since very few tree genera occur naturally in both hemispheres (e.g., in our study only Podocarpus appears in both hemispheres but has a limited distribution in the northern hemisphere). We sampled largely in botanical gardens and arboreta, which allowed us to sample native and non-native, congeneric and confamiliar, tree species at each location. At each location, one native and one to three non-native congeneric or confamiliar tree species were sampled.At each location, twenty 50-cm long asymptomatic twigs were collected from 1–5 individual trees per species, from different branches and different parts of the crown (Fig. 1). The number of individual trees per species depended on the number of trees available in the specific botanical garden or arboretum, which was often low (Table 1). All twigs per tree species and location were pooled and analysed as a single sample. On some occasions two samples of the same tree species at the same location are considered. Sampling was conducted in the month with the shortest day-length in the year (end of December 2017 in the Northern hemisphere, end of June 2018 in the Southern hemisphere). Samples originating from a tropical region (eleven samples from Tanzania) were collected in June 2018. Trees were sampled in winter to align with the timing of trade, i.e. most woody plants are traded in winter or early spring, as plants will be planted in the following spring, and to reduce the risk of introducing foliar pests in deciduous trees. Evergreen gymnosperm and angiosperm tree species, which were also considered, do not lose foliage during winter, and are thus sold with leaves/needles.Table 1 Site information for sampling locations included in this study.Full size tableFungal endophytesTo assess fungal communities, a total of 352 samples from 145 native and non-native tree species, belonging to nine families of angiosperms and gymnosperms, were collected. Sampling was done at 44 locations in 28 countries on five continents (Fig. 1, Table 1).From each twig in a sample, one bud, one needle/leaf and one 1 cm long twig segment were taken (Fig. 1). Needles from gymnosperms, and leaves from evergreen angiosperms were sampled to accurately assess the risk of trading these species. Twig segments were cut from the twig bases. The selected plant parts were surface sterilized by immersion in 75% ethanol for 1 min, 4% NaOCl for 5 min, and 75% ethanol for 30 s26. After air drying on a sterile bench, the following material from each of 20 twigs per sample was pooled: half of one bud, a 0.5 cm long piece of a needle (from gymnosperms) or a 0.25 cm2 leaf (for evergreen angiosperms) and a 0.5 cm long piece of twig.DNA extraction, PCR amplification and Illumina sequencingTotal genomic DNA was extracted from 50 mg of pooled, surface sterilized, and ground tissue (Fig. 1) using DNeasy PowerPlant Pro Kit (Qiagen, Hilden, Germany), following the manufacturer’s instructions. For a total of 31 out of 352 samples, DNA was extracted from different tissues separately, and DNA extracts were then pooled. DNA concentrations were quantified using the Qubit dsDNA BR Assay Kit (Thermo Fisher Scientific, Waltham, USA) on a Qubit 3.0 Fluorometer (Thermo Fisher Scientific) and DNA was diluted to 5 ng/μl. Samples that yielded less than 5 ng/μl were not diluted. The ITS2 region was amplified with the 5.8S-Fung and ITS4-Fung primers27. PCR amplifications were carried out in 20 μl reaction volumes containing 25 ng of DNA template, 1 mg/ml BSA, 1 mM of MgCl2, 0.4 μM of each primer, and 0.76 × JumpStart REDTaq ReadyMix Reaction Mix (Sigma-Aldrich, Steinheim, Germany). PCR was performed using Veriti 96-Well Thermal Cycler (Applied Biosystems, Foster City, CA, USA) as described in Franić et al. (2019). Each sample was amplified in triplicates and successful PCR amplification confirmed by visualization of the PCR products, before and after pooling the triplicates, on 1.5% (w/v) agarose gel with ethidium bromide staining. Pooled amplicons were sent to the Génome Québec Innovation Center at McGill University (Montréal, Quebec, Canada) for barcoding using Fluidigm Access Array technology (Fluidigm, South San Francisco, CA, USA) and paired-end sequencing on the Illumina MiSeq v3 platform (Illumina Inc., San Diego, CA, USA). Raw sequences obtained in this study are deposited at the NCBI Sequence Read Archive under BioProject accession number PRJNA70814822.Bioinformatics and taxonomical classification of ASVsQuality filtering and delineation into ASVs were done with a customized pipeline28 largely based on VSEARCH29, as described by Herzog et al.30. The output data available on Figshare show the abundances of fungal ASVs in the samples24. Taxonomic classification of ASVs was conducted using Sintax31 implemented in VSEARCH against the UNITE v.7.2 database32 with a bootstrap support of 80%. The data on the taxonomic classification of fungal ASVs is deposited in Figshare24.Quality filtering, delineation into ASVs, and taxonomical assignments were done on a larger data set (total of 474 samples), which increased the confidence in the selected centroid sequences. This data set consisted of (1) sequences obtained from 352 samples of pooled tree tissues that are presented here22, (2) sequences obtained from 33 samples of pooled tree tissues which were not included in this manuscript due to violation of the common protocol, (3) sequences from 21 contaminated samples (positive DNA extraction controls), including sequences from the two control samples (not presented here), and (4) sequences obtained from 66 samples of non-pooled tree tissues of Pinus sylvestris and Quercus robur that were collected from the subset of locations considered in this study, but for a different study, and are thus not presented here.Herbivorous insectsInsects were assessed from 227 samples of 109 tree species, collected at 31 locations and in 18 countries (Fig. 1, Table 1).The collected twigs (twenty 50 cm twigs per species per location) were brought to a laboratory close to each sampling location and inspected for the presence of insects that overwinter as adults. Twigs were kept at room temperature with the cut ends immersed in water to induce budding and to allow the development of insects that overwinter as larvae, pupae or eggs. Twigs from each sample were protected with gauze bags to prevent insects moving between samples (Fig. 1). Twigs were inspected for the presence of insects daily for 4 weeks and all collected insects were stored in 95% ethanol for further examination.Morphological and molecular identificationInsects were inspected using a stereo microscope and sorted to taxonomic orders and feeding guilds (i.e. herbivores, predators, parasitoids and other). The abundance of the different feeding guilds and taxonomic orders in the samples is presented in a file deposited on Figshare24. Herbivorous insects were further sorted into morphospecies and at least one specimen per morphospecies was stored at −20 °C for molecular analysis. The abundance of the different morphospecies in each sample is presented in a file deposited on Figshare24. Specimens for molecular analysis were photographed with a Leica DVM6 digital microscope and the Leica Application Suite X (LAS X). Depending on the size of the insects, the whole individual or parts (e.g. legs, head) were used for molecular analysis. Genomic DNA was extracted with a KingFisher (Thermo Fisher Scientific) extraction protocol suitable for insects (35 min incubation at RT, 30 min wash at RT with 3 different washing buffers, 13 min elution at 60 °C) in a 96-well plate. PCR for the COI was carried out in 25 µl reaction volume with 2 µl diluted DNA (1:10), 0.5 µM of each of the primers LCO1490 and HCO219833 and 1 x REDTaq ReadyMix Reaction Mix (Sigma-Aldrich) using a Veriti 96-Well Thermal Cycler (Applied Biosystems) with the following setting: 2 min at 94 °C, five cycles of 30 s at 94 °C, 40 s at 45 °C, and 1 min at 72 °C, 35 cycles of 30 s at 94 °C, 50 s at 51 °C, and 1 min at 72 °C, and a final extension step at 72 °C for 10 min. The success of amplification was verified by electrophoresis of the PCR products in 1.5% (w/v) agarose gel at 90 V for 30 min with ethidium bromide staining. A standard Sanger sequencing of the PCR products in both directions with the same primers was done at Macrogen Europe, Amsterdam, Netherlands. Sequences were assembled and edited with CLC Workbench (Version 7.6.2, Quiagen) and compared to reference sequences in BOLD34. If no conclusive results were found, sequences were compared to reference sequences in the National Centre for Biotechnology Information (NCBI) GenBank databases35. Specimens were assigned to species if the query sequence showed less than 1% divergence from the reference sequence. If two or more taxa matched within the same range, the assignment was ranked down to the next taxonomic level (i.e., genus). When no species match was obtained based on the above criteria, a genus was assigned with a divergence of less than 3%. For lower taxonomic groups the 100 nearest sequences were inspected on the Blast Tree (Fast Minimum Evolution Method) and the taxonomic relationship was evaluated based on that tree. If none of the approaches above revealed a conclusive taxonomic assignment, the morphological identification was taken as reference. The results of morphological and molecular identification of insect specimens are presented in a file deposited on Figshare24. Insect sequences are deposited in GenBank database under accession numbers MW441337-MW44176725.Sample metadataPairwise geographic distances (Euclidean distances) between sampling locations were calculated based on the geographic coordinates of the locations, with function “dist” in the R statistical programme36.Climate data, including mean annual temperature, mean annual precipitation, and temperature seasonality were obtained from the WorldClim database37, at a resolution of 2.5 min, and represent averages between 1970 and 2000.A host-tree phylogeny was constructed with the phylomatic function from the package brranching38 in R using the “zanne2014” reference tree39. One Eucalyptus sample collected in Argentina and two Eucalyptus samples collected in Tunisia were not identified to species. To place them in the phylogeny, we assigned them to different congeneric species that were not sampled in this study and that we considered as representative samples of phylogenetic diversity from across Eucalyptus genus (E. viminalis, E. robusta and E. radiata). Pairwise phylogenetic distances between study tree species were calculated using the “cophenetic” function in R36.The described sample metadata are available in a file on Figshare24. More

Pure coordination gameIn this section we study the Pure Coordination Game (PCG) (also known as doorway game, or driving game) in which (R=1), (S=0), (T=0), and (P=1), resulting in a symmetric payoff matrix with respect to the two strategies:$$begin{gathered} begin{array}{*{20}c} {} & {quad ; {text{A}}} &; {text{B}} \ end{array}hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & 0 \ 0 & 1 \ end{array} } right) hfill \ end{gathered}$$
(2)
There are two equivalent equilibria for both players coordinating at the strategy A or B (a third Nash equilibrium exists for players using a mix strategy of 50% A and 50% B). As the absolute values of the payoff matrix are irrelevant and the dynamics is defined by ratios between payoffs from different strategies, the payoff matrix (2) represents all games for which the relation (R=P >S=T) is fulfilled.In the PCG the dilemma of choosing between safety and benefit does not exist, because there is no distinction between risk-dominant and payoff-dominant equilibrium. Both strategies yield equal payoffs when players coordinate on them and both have the same punishment (no payoff) when players fail to coordinate. Therefore, the PCG is the simplest framework to test when coordination is possible and which factors influence it and how. It is in every player’s interest to use the same strategy as others. Two strategies, however, are present in the system at the beginning of the simulation in equal amounts. From the symmetry of the game we can expect no difference in frequency of each strategy being played, when averaged over many realisations. Still, the problem of when the system reaches full coordination in one of the strategies is not trivial. We address this question here.Figure 1Time evolution of the coordination rate (alpha) (in MC steps) in individual realisations for different values of the degree k in a random regular network of (N=1000) nodes, using (a) the replicator dynamics, (b) the best response, and (c) the unconditional imitation update rule.Full size imageFigure 2Coordination rate (alpha) and interface density (rho) vs degree k of a random regular network for (N=1000) using (a) the replicator dynamics, (b) the best response, and (c) the unconditional imitation update rule. Each green circle represents one of 500 realisations for each value of the degree k and the average value is plotted with a solid line, separately for (alpha >0.5) and (alpha le 0.5). Results are compared to the ER random network ((alpha _{ER})) with the same average degree.Full size imageFirst, we look at single trajectories as presented in Fig. 1. Some of them quickly reach (alpha =0) or 1, or stop in a frozen state without obtaining global coordination. Other trajectories take much longer and extend beyond the time scale showed in the figure. What we can already tell is that the process of reaching coordination is slower in the replicator dynamics where it usually takes more time than in the best response and unconditional imitation to reach a frozen configuration. For all update rules the qualitative effect of the connectivity is similar—for bigger degree it is more likely to obtain full coordination and it happens faster. For the UI, however, larger values of degree than for the RD and BR are required to observe coordination. For example, in the case of (k=10) or 20 the system stops in a frozen disorder when using UI, while for the RD and BR it quickly reaches a coordinated state of (alpha =0) or 1.To confirm the conclusions from observation of trajectories, we present the average outcome of the system’s evolution in the Fig. 2. The first thing to notice is that all plots are symmetrical with respect to the horizontal line of (alpha = 0.5). It indicates that the strategies are indeed equivalent as expected. In all cases there is a minimal connectivity required to obtain global coordination. For the RD and BR update rules this minimum value is (k=4), although in the case of BR the system fails to coordinate for small odd values of k due to regular character of the graph. This oscillating behaviour does not exist in Erdős–Rényi random networks. When nodes choose their strategies following the UI rule much larger values of k are required to obtain full coordination. Single realisations can result in (alpha = 0), or 1 already for (k=15). However, even for (k=60) there is still a possibility of reaching a frozen uncoordinated configuration.The important conclusion is that there is no coordination without a sufficient level of connectivity. In order to confirm that this is not a mere artefact of the random regular graphs we compare our results with those obtained for Erdős–Rényi (ER) random networks76,77 (black dashed line in Fig. 2). The level of coordination starts to increase earlier for the three update rules, but the general trend is the same. The only qualitative difference can be found in the BR. The oscillating level of coordination disappears and it doesn’t matter if the degree is odd or even. This shows that different behaviour for odd values of k is due to topological traps in random regular graphs78. Our results for the UI update rule are also consistent with previous work reporting coordination for a complete graph but failure of global coordination in sparse networks40.Figure 3Examples of frozen configuration reached under the UI update rule for small values of the average degree k in random regular networks (top row) and Erdős–Rényi networks (bottom row) with 150 nodes. Red colour indicates a player choosing the strategy A, blue colour the strategy B. Note the topological differences between random regular and ER networks when they are sparse. For (k=1) a random regular graph consists of pairs of connected nodes, while an ER network has some slightly larger components and many loose nodes. For (k=2) a random regular graph is a chain (sometimes 2–4 separate chains), while an ER network has one large component and many disconnected nodes. For (k=3) and (k=4) a random regular graph is always composed of one component, while an ER network has still a few disconnected nodes.Full size imageSince agents using the RD and BR update rule do not achieve coordination for small values of degree, one might suspect that the network is just not sufficiently connected for these values of the degree, i.e. there are separate components. This is only partially true. In Fig. 3, we can see the structures generated by random regular graph and by ER random graph algorithms. Indeed, for (k=1) and 2 the topology is trivial and a large (infinite for (k=1)) average path length23 can be the underlying feature stopping the system to reach coordination. For (k=3), however, the network is well connected with one giant component and the system still does not reach the global coordination when using RD or BR. For the UI update rule coordination arrives even for larger values of k. Looking at the strategies used by players in Fig. 3 we can see how frozen configuration without coordination can be achieved. There are various types of topological traps where nodes with different strategies are connected, but none of them is willing to change the strategy in the given update rule.We next consider the question of how the two strategies are distributed in the situations in which full coordination is not reached. Looking at the trajectories in Fig. 1 we can see that there are only few successful strategy updates in such scenario and the value of (alpha) remains close to 0.5 until arriving at a frozen state for (k=2) (also (k=7) for UI). This suggests that there is not enough time, in the sense of the number of updates, to cluster the different strategies in the network. Therefore, one might expect that they are well mixed as at the end of each simulation. However, an analysis of the density of active links in the final state of the dynamics, presented in Fig. 2, shows a slightly more complex behaviour. When the two strategies are randomly distributed (i.e. well mixed) in a network, the interface density takes the value (rho =0.5). When the two strategies are spatially clustered in the network there are only few links connecting them and therefore the interface density takes small values. Looking at the dependence of (rho) on k, we find that for the replicator dynamics the active link density starts at 0.5 for (k=1), then drops below 0.2 for (k=2) and 3 indicating good clustering between strategies, to fall to zero for (k=4) where full coordination is already obtained. When using the best response update rule the situation is quite different. For (k=1) there are no active links, (rho =0), and hardly any for (k=2). There is a slight increase of the active link density for (k=3), to drop to zero again for (k=4) due to full coordination. Because of the oscillatory level of coordination there are still active links for odd values of (kP) (otherwise we can rename the strategies and shuffle the columns and rows). What defines the outcome of a game are the greater than and smaller than relations among the payoffs. Therefore we can add/subtract any value from all payoffs, or multiply them by a factor grater than zero, without changing the game. Thus, the payoff matrix (1) can be rewritten as:$$begin{gathered} begin{array}{*{20}c} {} & {qquad {text{A}}} & {quad quad {text{B}}} \ end{array} ;; hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & {frac{{S – P}}{{R – P}}} \ {frac{{T – P}}{{R – P}}} & 0 \ end{array} } right) hfill \ end{gathered}$$
(3)
which, after substituting (S’=frac{S-P}{R-P}) and (T’=frac{T-P}{R-P}), is equivalent to the matrix: $$begin{gathered} begin{array}{*{20}c} {} &quad ;;{text{A}} &; {text{B}} \ end{array} ;quad quad quad quad quad quad begin{array}{*{20}c} {} & quad; {text{A}} & ;{text{B}} \ end{array} hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & {S^{prime}} \ {T^{prime}} & 0 \ end{array} } right)xrightarrow[{{text{apostrophes}}}]{{{text{skipping}}}}begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & S \ T & 0 \ end{array} } right) hfill \ end{gathered}$$
(4)
From now on we omit the apostrophes and simply refer to parameters S and T. This payoff matrix can represent many games, including e.g. the prisoner’s dilemma14,46 (for (T >1) and (S More

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