To the best of our knowledge, the present study is the first to relate physiological, mechanical and specific performance parameters in basketball using the complex networks model. Thus, this study focused efforts on comparing, by means of centrality metrics (degree, betweenness, eigenvector, and pagerank), tests that can successfully establish power, velocity, and force results in a sport assessment typically related to the anaerobic system.

For each of these parameters, following <span style="color: #ff6600;"><a style="color: #ff6600;" href="https://www.nature.com/articles/s41598-020-65420-3">for running what Bar-Or11 validated for the cycle ergometer, it is possible to determine maximal, mean, minimum values and fatigue indexes13,14. The first (maximal results) is related to anaerobic alactic metabolism, the second (mean results) to anaerobic lactic metabolism and the fatigue index is closely linked to acidosis tolerance. </a></span>

Therefore, the all-out 30 s test is an anaerobic evaluation accomplished in the laboratory and especially robust in its accuracy. The RAST protocol, corresponding test to the all-out 30 s test in the field, consists of six maximal 35 m runs, separated by 10 s intervals.

For each of the sprints, it is possible to determine the values of the parameters of power, force and velocity, thus being able to establish the maximal, mean and minimum values and the respective fatigue indexes.
By submitting basketball athletes to both tests (AO30s and RAST), it was expected that the results of such mechanical parameters should be, at least, significantly correlated, since they are originally designed as corresponding assessments.

In Table 1, it was possible to verify that the dimension of the values of the parameters of power and force were greater in the AO30s (laboratory) than in the RAST (on-court). These results corroborate with previous studies, even for applications of the RAST in its original form, characterizing a protocol-dependency previously reported even when efforts involve accelerations and decelerations compared to continuous efforts15. Despite this difference in a dimensional way, our experimental design depended on the presence of significant correlations between the laboratory and on-court results, without which the proposal would strongly lose the possibility of comparing the scenarios. In this sense, and according to our hypothesis, all the results for power and force were highly correlated.

In the case of velocity, except for the maximal results, the values relating to the Vmean and Vmin obtained on-court were higher than the corresponding ones established in the laboratory (6% and 23%, respectively). Previous studies have shown lower values in the maximum and mean velocities in running performed in NMT when compared to the field, with variation around 20%, in efforts involving high-intensity sprints25,26,27 and endurance running28.

This response may be due to some factors such as the intrinsic resistance of the treadmill belt25,26,27, the shorter length and number of strides25, the friction between the track belt and the floor support structure during foot contact, as well as the inertia of the roller-belt system during strides and in the aerial phase of running26,28.

These factors involving inertial elements probably justify the greater variation in the minimum velocity (23%, laboratory vs. court) observed in the present study. The authors also report that athletes with greater body mass have advantages running on the NMT since the inertial elements cause, relatively (per kg), a greater effect on lighter athletes25,26,27,28.

This aspect may explain the prominent distribution of body mass in the four centrality metrics (Fig. 2) analysed in our study (in both scenarios), especially considering that body mass is one of the factors in determining force and power in the RAST.
Still regarding the velocity, our data also did not reveal significant correlations between the results in the AO30s and RAST protocols. Highton et al.27 investigated the 30 m performance times on the NMT and over ground, as well as splits of 10 m and 20 m.

Of the 12 velocity parameters obtained, the authors observed significant correlations between the laboratory and the field in only half, with the correlation coefficients considered moderate in five (r = 0.58–0.67) and high for one of the parameters (time of 30 m, r = 0.80). On the other hand, Stevens et al.28 reported strong correlations (r = 0.82) between times obtained in endurance running (5000 m) from the NMT and over ground athletics track.

In our case, the lack of significant correlations among velocities for the RAST and AO30s tests may be due to the development of the running velocity to be quite different between the tests. For the RAST, we used the adapted protocol for the distance of 35 m (i.e., 2 × 17.5 m), featuring six sprints (10 s rest among them), including phases of acceleration (concentric force), deceleration (eccentric force), 180° change of direction (isometric contraction) and new acceleration, characteristic of the shuttle run29.

Despite this, the RAST did not seem to generate a change in net energy cost when considering 17.5 m efforts, based on the report by Zamparo et al.5 involving young basketball athletes in shuttle runs with change of direction (180°). Considering that, in running on the NMT, there are no changes of direction, it is possible that this aspect, fundamentally methodological, is a strong candidate to explain the non-significant correlations for velocities between the AO30s and RAST protocols.

This point needs clarification and should be further investigated in future studies.
Regarding the results obtained from the centrality analysis in complex networks, the 25 specific and common scenario parameters (nodes), which resulted from Pearson correlations (with the criteria of being statistically significant) returned 104 edges for scenario 1 (AO30s - laboratory) and 74 edges for scenario 2 (RAST - on-court). It was possible to verify a huge similarity between the scenarios in the different centrality metrics. In this sense, BM and VJ-W were present as the most important nodes in all the metrics used (degree, betweenness, eigenvector and pagerank).

In a complex networks approach involving aerobic and anaerobic efforts in sprint athletes, Pereira et al.30 found a result of greater importance for BM in the measurement of betweenness centrality, and this parameter also ranked among the six most important in the other centrality metrics, leading the authors to highlight the relevance of anthropometric parameters in sports scenarios when analysed using this model. Delextrat and Cohen3 conducted a study involving basketball in an experimental design very similar to that of the present investigation, where they also related laboratory and field results, including vertical jump, the Wingate anaerobic test (on the cycle ergometer), 20 m sprint, agility test, suicide sprint, isokinetic (knee extensors) and one maximal repetition (1-RM, bench press).

These authors verified that, for the current rules of basketball, the VJ figured as one of the main performances for basketball, as well as others of anaerobic power, determined by different tests from the present work. Gallová et al.31 compared the performances of Slovak and Lithuanian adolescent basketball players using the CMV and 20 m sprints and assumed these as tests which relate well to the game demands. More consistently and corroborating this view, our results highlight these performances in a network analysis.
In the specific case of basketball, besides the values of BM and VJ-W, it was possible to find in both scenarios that the other vertical jump nodes (VJ-I and VJ-P) for the CMV test also presented high centralities for the metrics involved in the analysis.

Thus, all vertical jump nodes were among the main ones for degree and pagerank metrics in both scenarios, and for betweenness, the vertices VJ-W, VJ-P and VJ-I appear as main ones in the AO30s scenario, with corresponding VJ-W and VJ-I appearing in the RAST scenario. In the case of the eigenvector measure, VJ-W and VJ-P appear in the same prestigious positions in both scenarios. These results showed that the complex networks analysis was sensitive to these parameters and so important for basketball in the laboratory and field tests.

In a review study, Drinkwater et al.4 reported that, in a basketball match, an athlete performs around 50 jumps, and this skill should be understood as fundamental in match performance31,32, along with other high power activities, especially 10–20 m sprints4,33.
In this sense, our results add that the test models used revealed, besides the vertical jumps, the presence of the maximal and medium powers as parameters of high and intense connection, as well as of prestige of these parameters regarding the others analysed in the present study, as shown in both scenarios for the degree, eigenvector, and pagerank measures.

Since these vertices represent alactic and lactic anaerobic characteristics, respectively, the association with the specific profiles of the sport, played in 28 m courts, also reveals the extreme quality of the tests applied in both the laboratory and basketball court.

It has been discussed in the literature which are the best optimal sprint distances to test athletes of this sport, being found to be distances of 10 m to 27 m4.

Since it is suggested that acceleration, deceleration and change of direction are key components of the specific demands imposed on the basketball game34, power should be considered as a relevant parameter for the sport. Therefore, we can assume that both the all-out 30 s test and RAST adapted to the court dimensions generate extremely adequate results in terms of evaluation. The on-court test is especially interesting for this purpose for it includes, as already reported, sprints from repeated acceleration stimuli.
According to this definition, regardless of the distances suggested for the tests, the protocols from different scenarios were well associated regarding centrality metrics for these maximal and mean power parameters (except for betweenness), suggesting that the on-court test reveals results that, in an integrative way, are compatible to the technological tests performed in the laboratory.

This strongly indicates that the simpler on-court protocol gives, at least considering a network scenarios perspective, the same results as those obtained in the laboratory.

In terms of practical application, the field protocol is highly attractive, as it meets demands of interest that are widely usable and robust, such as a laboratory evaluation for these mechanical and metabolic profiles.
Figure 3 shows the results of the four metrics for the AT (aerobic capacity) and %SBPP (technical performance) nodes. This figure was especially presented to show that, although these parameters have no prominent representation in the analysis, the complex networks model was sensitive in showing differences between scenarios. The AO30s, as already reported, is a predominantly anaerobic exercise protocol, as is the execution of the maximal vertical jumps and basketball shots.

Not surprisingly, the participation of AT in this context was extremely small in the laboratory setting, but was much higher when participating in the basketball court network. In contrast, %SBPP had a higher representation in the laboratory scenario than in the on-court scenario.

Again, it is important to revisit the different protocols studied in order to discuss these data, which were provocatively included here. In the RAST, unlike the AO30s, athletes perform maximum 35 m (2 × 17.5 m in adapted form) back-and-forth runs with breaks of 10 s. Such intervals allow greater oxidative activity in this test as compared to the laboratory test. It is amazing to see that, in the complex networks model, this metabolic detail was sensitive enough in the four different centrality metrics. In contrast, since basketball shooting involves alactic anaerobic predominance, the %SBPP was much higher in the laboratory setting than in the on-court scenario for all centrality measures.

Viewed this way, our study also denotes the incredible sensitivity of the complex networks model in a task accomplishment analysis, which would hardly be possible in a conventional statistical analysis. Taken together, these results allow us to establish a broad perspective on the application of the metrics used in the sports scenario, in particular by showing that the possibilities of interpretation of the obtained values are extremely consistent with the physiological representations pertinent to the proposed evaluation protocols.
In summary, the results (maximal, mean, minimum values and fatigue index) for force, velocity and power obtained in the laboratory and on-court tests included other anthropometric and performance parameters, such as aerobic capacity, vertical jump, and specific technical (successful shot-to-basket), showed, in an incredible way, the correspondence of centrality metrics between the scenarios.

The parameters BM, VJ-W, VJ-I, Pmax and Pmean were the most prominent among the applied complex networks measurements (degree, betweenness, eigenvector and pagerank). Moreover, from these analyses, we can conclude that, at least in our experimental design, the RAST adapted for the basketball court is a protocol that fulfils its role very well, which for its simplicity on implementation should be attractively used by basketball teams.

Thus, in practical terms, basketball being a modality that involves high-intensity intermittent exercise in tasks that involve a high frequency of changes in direction, the RAST is confirmed as an important assessment tool for this sport, especially for young athletes who need muscle preparation capable of withstanding lower limb rotational demands, high anaerobic power and tolerance to acidosis.