Daniel Scherrer
email: daniel.scherrer@unil.ch

Heidi K Mod
email: heidi.mod@unil.ch

Julien ; Pottier
email: julien.pottier@inra.fr

Anne Litsios-Dubuis

Loïc ; Pellissier
email: loic.pellissier@usys.ethz.ch

Pascal Vittoz
email: pascal.vittoz@unil.ch

Lars Gotzenberger
email: lars.goetzenberger@gmail.com

Martin ; Zobel
email: martin.zobel@ut.ee

Antoine ; Guisan
email: antoine.guisan@unil.ch

Org

Lars ; Götzenberger

Disentangling the processes driving plant assemblages in mountain grasslands across spatial scales and environmental gradients

Keywords: community assembly, environmental gradient, habitat filtering, limiting similarity, niche

published or not. The documents may come

Introduction

Community assembly is assumed to work as a hierarchy of constraints with potentially varying strengths at different scales [START_REF] Keddy | Assembly and response rules: Two goals for predictive community ecology[END_REF][START_REF] Zobel | The relative of species pools in determining plant species richness: An alternative explanation of species coexistence?[END_REF][START_REF] Belyea | Assembly rules within a contingent ecology[END_REF][START_REF] Lortie | Rethinking plant community theory[END_REF][START_REF] Mcgill | Matters of scale[END_REF]. Community dynamics and structure are therefore the outcome of a succession of filters that limit the set of potentially coexisting species from larger pools. This concept integrates many mechanisms of species coexistence [START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Barot | Mechanisms promoting plant coexistence: Can all the proposed processes be reconciled?[END_REF][START_REF] Wilson | The twelve theories of co-existence in plant communities: The doubtful, the important and the unexplored[END_REF].

However, empirical investigations addressing actual coexistence in nature have shown little consensus over the years [START_REF] Miles | Temporal and spatial patterns of vegetation dynamics[END_REF][START_REF] Diaz | Plant functional traits and environmental filters at a regional scale[END_REF][START_REF] Firth | Competition and habitat suitability: Small-scale segregation underpins large-scale coexistence of key species on temperate rocky shores[END_REF][START_REF] Mason | Niche overlap reveals the effects of competition, disturbance and contrasting assembly processes in experimental grassland communities[END_REF][START_REF] Götzenberger | Ecological assembly rules in plant communities -approaches, patterns and prospects[END_REF][START_REF] Concepción | Contrasting trait assembly patterns in plant and bird communities along environmental and human-induced land-use gradients[END_REF][START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF][START_REF] Price | Withincommunity environmental variability drives trait variability in species-rich grasslands[END_REF], despite the prime importance for realistic forecasts of current and future communities [START_REF] Guisan | Sesam -a new framework integrating macroecological and species distribution models for predicting spatio-temporal patterns of species assemblages[END_REF][START_REF] D'amen | Spatial predictions at the community level: From current approaches to future frameworks[END_REF]. More research is required to detect and disentangle the signatures of assembly processes based on assemblages observed at different scales.

When focusing on local plant communities at fine spatial scales (i.e., excluding the roles of phylogeographic and dispersal assembly filters), many studies provide evidence of habitat filtering, an assembly process that allows only the species exhibiting suitable convergent traits to coexist in particular environmental conditions [START_REF] Watkins | Local texture convergence: A new approach to seeking assembly rules[END_REF][START_REF] Firth | Competition and habitat suitability: Small-scale segregation underpins large-scale coexistence of key species on temperate rocky shores[END_REF][START_REF] Chalmandrier | Spatial scale and intraspecific trait variability mediate assembly rules in alpine grasslands[END_REF][START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF]. Although habitat filtering is often implicitly linked to the physical constraints of the environment, the habitat is also defined by its biotic components, including competitors. Therefore, the convergence of traits at the community level may also be driven by the exclusion of weaker competitors by the stronger ones [START_REF] Bengtsson | Competition and coexistence in plant communities[END_REF][START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Mayfield | Opposing effects of competitive exclusion on the phylogenetic structure of communities[END_REF][START_REF] Kunstler | Competitive interactions between forest trees are driven by species' trait hierarchy, not phylogenetic or functional similarity: Implications for forest community assembly[END_REF]. However, when plant competition is at work, the coexistence of species can be empowered by mechanisms that counteract competitive exclusion [START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Barot | Mechanisms promoting plant coexistence: Can all the proposed processes be reconciled?[END_REF][START_REF] Wilson | The twelve theories of co-existence in plant communities: The doubtful, the important and the unexplored[END_REF]: Opposite to the trait convergence caused by habitat filtering, trait divergence results from the fact that species must differentiate to compete for different resources (corresponding to different α-niche attributes; [START_REF] Stubbs | Evidence for limiting similarity in a sand dune community[END_REF][START_REF] Wilson | Evidence for assembly rules: Limiting similarity within a saltmarsh[END_REF], usually resulting in a range of distinct traits in the community [START_REF] Macarthur | The limiting similarity, convergence, and divergence of coexisting species[END_REF][START_REF] Johansson | Intensity and asymmetry of competition between plant pairs of different degrees of similarity: An experimental study on two guilds of wetland plants[END_REF][START_REF] Wilson | The twelve theories of co-existence in plant communities: The doubtful, the important and the unexplored[END_REF][START_REF] D'andrea | Challenges in linking trait patterns to niche differentiation[END_REF]. Such trait divergence is therefore expected to limit the trait similarity of coexisting species through αniche differentiation [START_REF] Wilson | The twelve theories of co-existence in plant communities: The doubtful, the important and the unexplored[END_REF][START_REF] Götzenberger | Ecological assembly rules in plant communities -approaches, patterns and prospects[END_REF]. Simultaneous functioning of these counteracting processes (i.e., habitat filtering and limiting similarity) is assumed to be enabled by hierarchical organisation [START_REF] Weiher | Assembly rules, null models, and trait dispersion: New questions from old patterns[END_REF][START_REF] Holdaway | Assembly rules operating along a primary riverbed-grassland successional sequence[END_REF][START_REF] Bernard-Verdier | Community assembly along a soil depth gradient: Contrasting patterns of plant trait convergence and divergence in a mediterranean rangeland[END_REF]: abiotic constraints are expected to be more effective at coarser scales than plant interactions, whereas resource competition should mostly work at finer "neighbourhood" scales [START_REF] Huston | Local processes and regional patterns: Appropriate scales for understanding variation in the diversity of plants and animals[END_REF].

In addition to habitat filtering and limiting similarity, both assuming that communities are spatially homogeneous, assemblages of plant species may be driven by within community spatial partitioning [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: A synthesis[END_REF]. Such processes account for the heterogeneous nature of communities and may be endogenous (e.g., generated by ecology of the plants themselves ;[START_REF] Tilman | Competition and biodiversity in spatially structured habitats[END_REF]Pacala & Levin, 1997) or exogenous (e.g., generated by environmental micro-heterogeneities or disturbances ;[START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF][START_REF] Price | Withincommunity environmental variability drives trait variability in species-rich grasslands[END_REF]. Spatial partitioning is at the core of several mechanisms promoting species coexistence [START_REF] Bolker | Combining endogenous and exogenous spatial variability in analytical population models[END_REF][START_REF] Barot | Mechanisms promoting plant coexistence: Can all the proposed processes be reconciled?[END_REF][START_REF] Adler | Trait-based tests of coexistence mechanisms[END_REF]. An example of coexistence mechanisms promoted by endogenous heterogeneity is the competition/colonisation trade-off [START_REF] Levins | Regional coexistence of species and competition between rare species[END_REF] in which better colonisers escape competitive exclusion because they can occupy patches that are not yet occupied by the better competitors (e.g., [START_REF] Wildová | Fine-scale dynamics of rhizomes in a grassland community[END_REF][START_REF] Moora | Understory plant diversity is related to higher variability of vegetative mobility of coexisting species[END_REF]. Additionally, environmental heterogeneities are well recognised to develop at a wide range of scales [START_REF] Levin | The problem of pattern and scale in ecology: The robert h. Macarthur award lecture[END_REF], including the very fine ones of the rhizosphere [START_REF] Raynaud | Soil characteristics play a key role in modeling nutrient competition in plant communities[END_REF][START_REF] Scherrer | Elevational species shifts in a warmer climate are overestimated when based on weather station data[END_REF]. Although they cannot be considered as a mechanism of coexistence sensu stricto [START_REF] Wilson | The twelve theories of co-existence in plant communities: The doubtful, the important and the unexplored[END_REF], such environmental heterogeneities still increase opportunities for species to co-occur.

Combined endogenous and exogenous fine-scale spatial heterogeneities are expected to segregate plant functional syndromes leading to fine-scale spatial niche partitioning [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: A synthesis[END_REF][START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF][START_REF] Price | Withincommunity environmental variability drives trait variability in species-rich grasslands[END_REF].

Whereas the roles of different assembly processes are expected to vary in relation to spatial scale, they may also vary along environmental gradients [START_REF] Mason | Niche overlap reveals the effects of competition, disturbance and contrasting assembly processes in experimental grassland communities[END_REF][START_REF] Carboni | Changes in trait divergence and convergence along a productivity gradient in wet meadows[END_REF][START_REF] Mudrák | Fine-scale coexistence patterns along a productivity gradient in wet meadows: Shifts from trait convergence to divergence[END_REF][START_REF] Chalmandrier | Spatial scale and intraspecific trait variability mediate assembly rules in alpine grasslands[END_REF][START_REF] Bowman | The roles of stochasticity and biotic interactions in the spatial patterning of plant species in alpine communities[END_REF]. Based on the hypotheses related to stress gradients, abiotic constraints would be expected to be decisive for cooccurrences under harsh environmental conditions, such as in cold and dry areas, whereas competition would define species assemblages in more productive habitats [START_REF] Louthan | Where and when do species interactions set range limits?[END_REF]. Conversely, the role of spatial partitioning should be emphasised in environments prone to patchiness in vegetation and spatial heterogeneity [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: A synthesis[END_REF][START_REF] Stark | Does environmental heterogeneity drive functional trait variation? A test in montane and alpine meadows[END_REF]. Variations in importance of different assembly processes along environmental gradients demonstrate that interpretations derived from one habitat cannot necessarily be generalised to communities elsewhere. Thus, quantifying the role of different assembly processes across different spatial scales and along environmental gradients is essential for better understanding community assembly [START_REF] Weiher | Advances, challenges and a developing synthesis of ecological community assembly theory[END_REF][START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF].

In this study, we aimed to disentangle the signatures of different processes defining local plant assemblies (summarised in Table 1) across spatial scales (plots of 2 m × 2 m and subplots of 20 cm × 20 cm) and environmental gradients. Our analyses were based on a nested sampling design to record the within-community heterogeneity of vegetation (Fig. 1), including information on plant functional traits combined with environmental data in abiotic habitats. We applied different null models for trait reshuffling to assess the importance of habitat filtering (interpreted from trait convergence) and limiting similarity (interpreted from trait divergence) at two different spatial scales (i.e., plot and subplot level) and spatial partitioning among subplots (i.e., interpreted from β-diversity of traits within each plot). We further assessed the variations in the investigated assembly processes along environmental gradients.

Materials and Methods

Data collection

The study area covers approximately 700 km 2 of a mountain region located in the western Swiss Alps (46°23' N, 7°05' E). This region is characterised by a large elevation gradient with strong variation in climatic, edaphic and land use factors. A set of 298 plots of 2 m × 2 m, each with five nested subplots of 20 cm × 20 cm, were distributed across non-forested areas within the study area from 820 to 3045 m a.s.l. according to a stratified-random sampling (using elevation, slope and aspect as strata) designed to evenly cover the range of grassland habitat conditions (Fig. 1). At the 2 m plot scale, the cover of different plant species was estimated as modified ordinal classes [START_REF] Braun-Blanquet | Pflanzensoziologie: Grundzüge der vegetationskunde[END_REF], which were then subsequently converted to percentage cover values: 0 = absent; r = 0.01 %; + = 0.2 %; 1 = 3 %; 2 = 15 %; 3 = 37.5 %; 4 = 62.5 %, 5 = 87.5 % [START_REF] Pottier | The accuracy of plant assemblage prediction from species distribution models varies along environmental gradients[END_REF]. At the 20 cm subplot level, the percentage cover of species was visually estimated in the field. All the field data were collected during the peak vegetation period between June and August 2009, starting from lower elevation sites and moving to higher elevation sites.

Plant trait information was collected from a previous study [START_REF] Dubuis | Predicting current and future spatial community patterns of plant functional traits[END_REF] for the 244 most frequent and locally abundant species (of 722 vascular plant species recorded in total). Four traits associated with the performance of plant species during the persistence phase of their life cycle [START_REF] Westoby | Plant ecological strategies: Some leading dimensions of variation between species[END_REF][START_REF] Wright | The worldwide leaf economics spectrum[END_REF] were measured (hereafter, growth traits): average vegetation height (VH in cm), which is associated with the ability of a plant to compete for light, and specific leaf area (SLA in mm 2 mg -1 ), leaf dry matter content (LDMC in mg mg -1 ) and leaf nitrogen content (LNC in % of dry leaf mass), which indicate the resource-use strategy of a species [START_REF] Wilson | Specific leaf area and leaf dry matter content as alternative predictors of plant strategies[END_REF][START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: A traits manifesto[END_REF]. These traits were measured for 4 to 20 individuals (generally 10) per species within the study area, evenly distributed within the bioclimatic ranges of the species as defined in [START_REF] Pottier | The accuracy of plant assemblage prediction from species distribution models varies along environmental gradients[END_REF]. Trait measurements followed the basic recommendations of [START_REF] Cornelissen | A handbook of protocols for standardised and easy measurement of plant functional traits worldwide[END_REF] and are detailed in [START_REF] Dubuis | Predicting current and future spatial community patterns of plant functional traits[END_REF]. For the details on trait data, see Supplementary material Appendix 1 Fig. A1.

For the trait-based detection of assembly processes, we discarded the plots in which trait-assigned species accounted for less than 80 % of the relative vegetation cover resulting in 269 plots for the final analyses. With the threshold of 80 %, most of the plots were retained for the analyses (e.g., a 95 % threshold would have retained only 147 plots).

The spatial variation of different assembly processes was examined against the environmental conditions of each plot. Based on preliminary analyses, elevation (m a.s.l.), growing degree-days (GDD; average of 1961-1990 with a 0 °C threshold), moisture index over the growing season (average values of 1961-1990 from June to August) and topographic position (positive values express convex topographies such as ridges, peaks, and exposed sites, whereas negative values indicate concave surfaces such as valley bottoms or lower ends of slopes) were chosen as the most influential predictors (see Supplementary material Appendix 1 Fig. A2). These variables were extracted from digital maps at a 25 m × 25 m resolution following [START_REF] Zimmermann | Final report of the mlp climate and biophysical mapping project[END_REF].

Detecting the signatures of assembly processes

The detection of assembly processes was based on a trait-based approach and the comparison of observed patterns against patterns derived from null models. All analyses were conducted in R 3.4.1 (R Core Team, 2017; see Appendix 4 for details of the packages and functions used).

Habitat filtering and limiting similarity were tracked at both plot and subplot levels to examine scale dependency of these processes. Habitat filtering was inferred in the case of convergence of growth traits in which observed within-plot (or within-subplot) functional diversity was lower than that expected under null assembly. Correspondingly, limiting similarity was inferred in the case of divergence of growth traits in which observed within-plot (or within-subplot) functional diversity was greater than that expected under null assembly. Fine-scale spatial partitioning was inferred from spatial segregation of growth traits among subplots within each plot in which the β-component of functional diversity was greater than that under a null assembly. We measured plot and subplot functional diversity using a modified version of the α-Rao quadratic entropy index following de Bello, Lavergne, Meynard, [START_REF] De Bello | The partitioning of diversity: Showing theseus a way out of the labyrinth[END_REF]. This metric is based on the functional dissimilarity among species weighted by their abundance. The β-component of functional diversity within each plot was measured with the β-Rao quadratic entropy index [START_REF] De Bello | The partitioning of diversity: Showing theseus a way out of the labyrinth[END_REF].

We considered five different representations of trait diversity: one multidimensional space considering the four growth traits simultaneously (multidimensional space combining all growth traits, CGT), and four one-dimensional spaces each consisting of one separate growth trait. All traits were log transformed and standardised (for details, see Supplementary material Appendix 1 Fig. A1).

A critical aspect of testing for assembly processes is the use of an appropriate null model that focuses only on the ecological mechanisms under study [START_REF] Harvey | Null models in ecology[END_REF][START_REF] Gotelli | Statistical challenges in null model analysis[END_REF][START_REF] Götzenberger | Which randomizations detect convergence and divergence in trait-based community assembly? A test of commonly used null models[END_REF]. Detecting habitat filtering or limiting similarity requires testing whether the trait values of species make a difference in terms of community structuring. In this respect, we built null models in which trait values of a species were reshuffled as a unit (i.e., all traits of one species, hereafter called trait unit) among species to prevent unrealistic/unobserved combinations of traits and to preserve fundamental trade-offs between traits [START_REF] Mouillot | Is the abundance of species determined by their functional traits? A new method with a test using plant communities[END_REF][START_REF] Hardy | Testing the spatial phylogenetic structure of local communities: Statistical performances of different null models and test statistics on a locally neutral community[END_REF]. With this approach, the species abundance patterns and species richness of plots and subplots were fixed, as were the species frequencies among the plots and subplots.

Therefore, this reshuffling procedure also fixed the frequency of the functional syndromes over the study area. The range of trait units considered for this reshuffling procedure differed when seeking the signature of habitat filtering or limiting similarity. Whereas the tests of habitat filtering considered the total range of trait units (i.e., based on all species recorded in the study area) at the plot level and the local range of trait units (i.e., based on all species recorded in a plot) at the subplot level, the tests of limiting similarity were based on a restricted range of trait unit values. For limiting similarity, only trait units falling into the observed range of trait values were considered, which allowed detection of limiting similarity independent of habitat filtering [START_REF] Bernard-Verdier | Community assembly along a soil depth gradient: Contrasting patterns of plant trait convergence and divergence in a mediterranean rangeland[END_REF].

For example, if the observed range of vegetation height in a plot or a subplot was 10-50 cm, only trait values of the total (for plot level analysis) or local plot range (for subplot analysis) between 10 and 50 cm were considered. Using different species pool sources for trait units, habitat filtering and limiting similarity could be detected for a given plot/subplot simultaneously.

An additional question is whether the reshuffling of trait units among species should be free or constrained allowing only the trait units of species with similar abundances (i.e., vegetation cover) to be reshuffled. To test whether reshuffling had any effect on our results, we ran three different nullmodels for habitat filtering, limiting similarity and fine-scale spatial partitioning: (1) model allowing trait units to be reshuffled freely independent of abundances; (2) model only allowing trait units to be reshuffled within predefined abundance classes (see Appendix 2 for more details; [START_REF] Hardy | Testing the spatial phylogenetic structure of local communities: Statistical performances of different null models and test statistics on a locally neutral community[END_REF][START_REF] Wilson | Evidence for assembly rules: Limiting similarity within a saltmarsh[END_REF]; and (3) model constraining the reshuffling by weights according to the differences in abundance between species (i.e., reshuffling between two species with similar abundances is much more likely than between species with highly different abundances; see Appendix 3 for more details). Because the results of these three null models were similar, we only report the results for the first one here. Detailed descriptions and all results of the other two null models can be found in Appendices 2 and 3.

The null models were run 10,000 times for each test. Based on our null models, we then calculated the standard effect size (SES) of α-Rao or β-Rao for each plot and subplot as

𝑆𝐸𝑆 𝑅𝑎𝑜 = (𝑅𝑎𝑜 𝑝𝑙𝑜𝑡 -𝜇 𝑅𝑎𝑜 𝑛𝑚 )/𝜎 𝑅𝑎𝑜 𝑛𝑚
where 𝑅𝑎𝑜 𝑝𝑙𝑜𝑡 is the observed α-/β-Rao and 𝜇 𝑅𝑎𝑜 𝑛𝑚 is the mean and 𝜎 𝑅𝑎𝑜 𝑛𝑚 is the standard deviation of α-/β-Rao of the null model simulations. With negative SES of α-Rao for the habitat filtering test, the functional diversity was lower than that expected under the null assembly (i.e., signal of habitat filtering). Conversely, positive SES of α-Rao for the limiting similarity test indicated that the functional diversity was higher than that expected under the null assembly (i.e., signal of limiting similarity).

Positive SES of β-Rao for the spatial partitioning test indicated that plant traits were spatially segregated among subplots within a plot.

To test whether an overall effect of habitat filtering or limiting similarity occurred at the plot and subplot levels, and whether the effect was significantly different between these spatial scales, we used a Wilcoxon test comparing α-Rao at the plot level and the mean α-Rao at the subplot level (i.e., the mean of the five subplots nested within a plot). To test for an overall effect of fine-scale partitioning, we used a Wilcoxon test of the β-Rao among subplots.

Analysing changes in assembly processes along environmental gradients

To explore potential changes in the strength of habitat filtering and limiting similarity (at the plot and subplot levels) and spatial partitioning processes (among subplots) along environmental gradients (elevation, GDD, moisture index, topographic position), generalised linear models (GLM) for the SES (plot/subplot α-Rao, subplot β-Rao) of growth traits were fitted. Because we were not interested in building a single meaningful and accurate statistical model per assembly process but in interpreting the variation along environmental gradients, we fitted one model for each assembly process and environmental factor. Environmental factors were provided to the models as second order polynomials, and the optimal model was selected based on stepwise AIC. Because all of the strong environmental gradients in the study area were highly correlated with elevation (Pearson correlations: GDD = -0.99, moisture index = 0.88 and topographic position = 0.59; see Supplementary material Appendix 1 Fig. A2 for details), only the results for elevation as the principal stress gradient are reported in the main manuscript.

Results

The relative importance of assembly processes

The mean SES of α-Rao with total trait range reshuffling was significantly lower than zero at the plot level (p < 0.01, Wilcoxon test) for all growth traits except LDMC, but not at the subplot level (Fig. 2).

This result indicated the predominance of habitat filtering at the coarse spatial scale of 4 m². The mean SES of α-Rao with restricted trait range reshuffling was significantly greater than zero for all growth traits (except VH at the plot level) at both spatial scales (p < 0.01, Wilcoxon test) but was significantly higher at the subplot level (p < 0.01, Wilcoxon test), indicating the predominance of limiting similarity at the fine spatial scale of 0.04 m² (Fig. 2). Trait convergence from the total range of trait values was strongest for VH and LNC, whereas trait divergence with trait reshuffling restricted to the observed range of the local plot/subplot was strongest for SLA and LDMC. Based on comparison of SES of α-Rao values with total and restricted trait range reshuffling, habitat filtering and limiting similarity at the plot level were negatively linked, indicating that these two processes acted as opposing forces (i.e., plots with strong habitat filtering showed weak limiting similarity, and plots with weak habitat filtering showed strong limiting similarity; R 2 ≈ 0.8; Fig. 3), despite the independent null-tests. These functional patterns were not related to plot species richness (Fig. 3).

Overall, SES of β-Rao among subplots did not deviate from zero (Fig. 2). However, 3.3 to 10.3 % (depending on the studied trait) of the plots showed much greater β-Rao than that expected under null assembly, indicating that fine-scale spatial partitioning was a relevant process in our study area.

Most importantly, SES β-Rao among subplots was significantly positively correlated with plot level SES α-Rao with restricted trait range reshuffling (Fig. 4), indicating strong connection of limiting similarity at the plot level with within plot (i.e., among subplots) spatial partitioning. Neither process was linked to species richness of a plot (Fig. 4).

Assembly processes along environmental gradients

Of the four growth traits (i.e., community weighted mean of the plots), only vegetative height (R 2 = 0.6) and SLA (R 2 = 0.42) varied with elevation, whereas LNC and LDMC were largely independent of elevation (R 2 < 0.15; Fig. 5). Overall, the explanatory power of environmental gradients (i.e., elevation) for SES α-Rao with total trait range reshuffling (i.e., habitat filtering; R 2 = 0.05-0.26) and with restricted trait range reshuffling (i.e., limiting similarity; R 2 = 0.02-0.17) at the plot level was weak and negligible at the subplot level (R 2 <0.04; Fig. 5; for more detailed results on the other environmental gradients, see Supplementary material Appendix 1 Fig. A3). Our results for the CGT (combined growth traits) showed that habitat filtering was strongest in productive low elevation habitats; whereas limiting similarity was strongest in cold climates at high elevations. These signals were mostly driven by only two traits, with habitat filtering by SLA (R 2 = 0.26) and limiting similarity by VH (R 2 = 0.17). Nevertheless, habitat filtering for VH remained the strongest assembly process observed, although the importance was relatively constant with elevation (Fig. 5). Fine-scale spatial partitioning did not show any significant signal along environmental gradients (R 2 < 0.02; Fig. 5 and Supplementary material Appendix 1 Fig. A3).

Discussion

Variations of assembly processes with scale and traits

Our study revealed the signatures of several assembly processes across different spatial scales, with habitat filtering (interpreted from trait convergence) the dominant effect at the large scale (i.e., among plots) and limiting similarity (interpreted from trait divergence) the dominant effect at the fine scale (i.e., within a plot). Consistent with the literature [START_REF] Weiher | Assembly rules, null models, and trait dispersion: New questions from old patterns[END_REF][START_REF] Holdaway | Assembly rules operating along a primary riverbed-grassland successional sequence[END_REF][START_REF] Götzenberger | Ecological assembly rules in plant communities -approaches, patterns and prospects[END_REF], we found a strong overall signal of habitat filtering for three of four growth traits (not LDMC) at the plot level and no or only a weak signal at the subplot level. In contrast to habitat filtering, limiting similarity was detected at both spatial scales with stronger effects for all growth traits at the subplot level.

At the subplot scale (0.04 m 2 ), limiting similarity acting at the neighbourhood scale within herbaceous plant communities was the dominant assembly process indicating species competition at fine scales (van der [START_REF] Van Der Maarel | Small-scale plant species turnover in a limestone grassland: The carousel model and some comments on the niche concept[END_REF][START_REF] Purves | Fine-scale spatial structure in a grassland community: Quantifying the plant's-eye view[END_REF][START_REF] Reitalu | Plant species segregation on different spatial scales in semi-natural grasslands[END_REF]. However, we also detected trait divergence at the plot scale indicating that competition effects might act not only at centimetre but also at metre scales in grassland communities. Additionally, the plot level trait divergence could result from spatial partitioning processes. This interpretation was indeed supported by the positive correlation between plot level limiting similarity and spatial partitioning, which is a pattern that was also found in a recent study conducted by [START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF]. Consequently, similar to [START_REF] Mayfield | Opposing effects of competitive exclusion on the phylogenetic structure of communities[END_REF], who demonstrated that trait convergence could be due to different mechanisms, our results indicated that trait divergence could also be due to different mechanisms operating at different scales.

Despite the positive relationship between plot level trait divergence and SES of β-Rao, we detected fine-scale spatial partitioning of plant functional traits less frequently than habitat filtering and limiting similarity. Nevertheless, even if the size of our plots (4 m²) was sufficiently small to assume spatial homogeneity of the investigated environmental factors, we might still encounter either biologically generated micro-heterogeneities (e.g., very short-distance dispersal and vegetative reproduction) or fine-scale patchiness in other environmental factors (e.g., edaphic factors, spatially restricted disturbances). The observed spatial partitioning further suggests that the metacommunity concept should be accounted for at very fine scales in future studies (e.g., [START_REF] Kneitel | Trade-offs in community ecology: Linking spatial scales and species coexistence[END_REF][START_REF] Leibold | The metacommunity concept: A framework for multi-scale community ecology[END_REF][START_REF] Alexander | Metapopulations and metacommunities: Combining spatial and temporal perspectives in plant ecology[END_REF][START_REF] Laliberté | Contrasting effects of productivity and disturbance on plant functional diversity at local and metacommunity scales[END_REF]. This could be accomplished, for example, by combining our approach based on β-diversity patterns and fine-scale measurements of the physical environment [START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF][START_REF] Stark | Does environmental heterogeneity drive functional trait variation? A test in montane and alpine meadows[END_REF] with spatially explicit tests (Siefert 2012).

Additionally, our results revealed that assembly processes worked on different traits with habitat filtering mostly observed for VH and LNC and limiting similarity for SLA and LDMC. These results corroborated the observation of a strong decrease in VH with elevation: apart from the length of the growing season, fertilised lowland grasslands select tall species able to compete for light, whereas alpine/nival landscapes select strongly for small stature plants with thermic decoupling from atmospheric conditions [START_REF] Körner | Alpine plant life: Functional plant ecology of high mountain ecosystems[END_REF]. LNC varied across habitats depending on environmental conditions, with plants with high LNC selected for at high elevations in natural environments [START_REF] Han | Biogeography and variability of eleven mineral elements in plant leaves across gradients of climate, soil and plant functional type in china[END_REF]. We also found high LNC values at low elevations, presumably related to farming-based fertilisation. By contrast, SLA and LDMC are mostly associated with the ability to compete for nutrients [START_REF] Wilson | Specific leaf area and leaf dry matter content as alternative predictors of plant strategies[END_REF] and therefore are expected to show trait divergence (i.e., limiting similarity).

Here, we only used four growth traits, but other traits are also expected to show strong patterns of habitat filtering (e.g., leaf form or anatomy, flower pollination, diaspore morphology; [START_REF] Pellissier | Plant traits co-vary with altitude in grasslands and forests in the european alps[END_REF] or limiting similarity (e.g., root depth; [START_REF] Cornwell | Community assembly and shifts in plant trait distributions across an environmental gradient in coastal california[END_REF].

Indeed, recent studies concentrating on functional diversity reveal a tendency to combine traits into one functional space (e.g., in multi-traits; [START_REF] Cornwell | A trait-based test for habitat filtering: Convex hull volume[END_REF][START_REF] Laughlin | The intrinsic dimensionality of plant traits and its relevance to community assembly[END_REF][START_REF] Carmona | Traits without borders: Integrating functional diversity across scales[END_REF]. However, with this approach, or by concentrating only on one or few traits, important mechanisms depicting trait patterns to underlying assembly processes can be hidden [START_REF] Mason | Niche overlap reveals the effects of competition, disturbance and contrasting assembly processes in experimental grassland communities[END_REF][START_REF] Conti | Environmental gradients and microheterogeneity shape fine scale plant community assembly on coastal dunes[END_REF]. To discriminate between different sources of fine-scale spatial partitioning, one should consider more traits related to plant regenerative strategies and ability to pre-empt or explore space. In perennial grasslands in which regeneration is mostly vegetative, a signature of fine-scale spatial partitioning could therefore be detected by trait divergence of clonal traits/syndromes [START_REF] Moora | Understory plant diversity is related to higher variability of vegetative mobility of coexisting species[END_REF][START_REF] Pottier | Spatial pattern and process at the plant neighbourhood scale: Insights from communities dominated by the clonal grass elymus repens (l.) gould[END_REF][START_REF] Klimešová | Clonal and bud bank traits: Patterns across temperate plant communities[END_REF].

Although null model approaches are widely used and tested in studies of community assembly, their outcomes, similar to any other observational method, do not imply causation. The ecological interpretation that trait convergence is driven by habitat filtering and that divergence results from competition is certainly an over-simplification [START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF]. For example, competitive exclusion can also result in trait convergence, by clustering species of similar competitive ability (similar height, shade tolerance, or other competition related traits; [START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Mayfield | Opposing effects of competitive exclusion on the phylogenetic structure of communities[END_REF][START_REF] Herben | Community assembly by limiting similarity vs. Competitive hierarchies: Testing the consequences of dispersion of individual traits[END_REF]. Although we attributed some of the changes in functional diversity to such processes, overall, the exclusion of weaker competitors was unlikely: SES values for all traits were higher at the subplot level than those at the plot level. If competitor exclusion was effective at the subplot scale, the opposite should be observed.

For community assembly studies that compare observed trait diversity patterns with randomised ones, another concern is that simultaneously acting processes may lead to random patterns, which could mask the processes that create opposing patterns (e.g., trait convergence vs. divergence) [START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF][START_REF] Götzenberger | Which randomizations detect convergence and divergence in trait-based community assembly? A test of commonly used null models[END_REF]. Nonetheless, the deviation of SES value distributions from 0 to either positive or negative values can at least indicate the prevailing process acting for a given trait at a given scale. Moreover, in this study, we particularly addressed the simultaneously acting processes of habitat filtering and limiting similarity using different species pools over which trait data were randomised (Figures 3 and4, see also [START_REF] De Bello | Functional species pool framework to test for biotic effects on community assembly[END_REF].

Although such an approach still relies on the assumption that habitat filtering and limiting similarity can be interpreted from convergence and divergence, an indication of those sites in which simultaneous processes are potentially at play is nevertheless provided.

Despite the limitations of the null model and more generally of the empirical-approach, our study supports the conclusion of Münkemuller et al. ( 2012) that the inference of assembly processes from diversity patterns is more relevant when based on a set of criteria rather than on a unique one.

Variations of assembly processes along environmental gradients

Although we found a strong influence of elevation on the mean trait values themselves (i.e., VH and SLA), the strength of the processes creating different trait dispersion patterns only varied marginally with elevation. VH showed the strongest overall signal of habitat filtering but was relatively constant across elevation (R 2 = 0.07). This result indicated that a strong selection for VH (trait convergence) occurred at both high and low elevations, but for opposite sizes (i.e., small at high elevation, tall at low elevation). Dominance of the best competitors for light in productive habitats is often reported [START_REF] Perronne | How to design trait-based analyses of community assembly mechanisms: Insights and guidelines from a literature review[END_REF], and the advantage of short stature in the alpine/nival belt (high elevation) to allow thermic decoupling from atmospheric conditions is well documented [START_REF] Körner | Alpine plant life: Functional plant ecology of high mountain ecosystems[END_REF]. For SLA, the primary influence of elevation on habitat filtering was for selecting high SLA species towards lower elevations and low SLA species towards higher elevations.

At low elevation, the detected pattern was most likely not a direct effect of the physical environment but an indirect effect of favourable conditions in productive habitats favouring species with quick growth and high SLAs (e.g., Crepis biennis, Holcus lanatus; [START_REF] De Bello | Hierarchical effects of environmental filters on the functional structure of plant communities: A case study in the french alps[END_REF]. This trait convergence due to biotic interactions (i.e., competitive exclusion of weaker traits; [START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Mayfield | Opposing effects of competitive exclusion on the phylogenetic structure of communities[END_REF] might also explain why we observed less habitat filtering at higher than at lower elevations (see also [START_REF] Mudrák | Fine-scale coexistence patterns along a productivity gradient in wet meadows: Shifts from trait convergence to divergence[END_REF]. This finding further highlights the difficulty of null model approaches to distinguish trait convergence resulting from abiotic (i.e., physiological limits) and biotic (i.e., competitive exclusion) habitat filtering, because both processes might work simultaneously and on the same traits [START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF].

The signal of limiting similarity was only weakly linked to elevation with the strongest effect visible on SLA and VH for which limiting similarity increased with elevation. This result is apparently contradictory to the stress-gradient hypothesis, which predicts an increase in limiting similarity in productive lowlands in which plant competition is expected to be more intense [START_REF] Bertness | Positive interactions in communities[END_REF][START_REF] He | Global shifts towards positive species interactions with increasing environmental stress[END_REF][START_REF] Chamberlain | How context dependent are species interactions?[END_REF][START_REF] Chalmandrier | Spatial scale and intraspecific trait variability mediate assembly rules in alpine grasslands[END_REF]). An alternative interpretation, as presented above, is that the observed patterns of trait divergence at the plot level were linked to habitat heterogeneity within plots rather than the direct effects of competition-driven limiting similarity. With increasing elevation, the plots (4 m 2 ) were likely to become environmentally more heterogeneous (soil depth, percentage of rock cover) leading to the coexistence of species with varying SLA and VH, such as cushion plants, succulents (on rocks) and grasses. Nevertheless, spatial partitioning did not vary significantly with any of the environmental factors tested. This suggested that fine-scale partitioning might result from a variety of mechanisms (endogenous/exogenous heterogeneity) that did not necessarily respond similarly to environmental variations, or that the analyses and underlying data could not capture the true drivers of spatial partitioning within the communities.

Our findings support the general view that topo-climatic conditions act as strong drivers of community assemblages in a mountain range, such as the one studied here, with implications for spatial modelling of plant species assemblages [START_REF] Guisan | Sesam -a new framework integrating macroecological and species distribution models for predicting spatio-temporal patterns of species assemblages[END_REF][START_REF] D'amen | Using species richness and functional traits predictions to constrain assemblage predictions from stacked species distribution models[END_REF][START_REF] D'amen | Spatial predictions at the community level: From current approaches to future frameworks[END_REF][START_REF] D'amen | Improving spatial predictions of taxonomic, functional and phylogenetic diversity[END_REF]. However, mountain grasslands are often used as grazing grounds for livestock, with intensity varying in relation to elevation. In previous studies, added fertilisation and mowing has led to observed trait convergence and divergence depending on the attribute examined and the type/intensity of treatment [START_REF] Mason | Niche overlap reveals the effects of competition, disturbance and contrasting assembly processes in experimental grassland communities[END_REF][START_REF] Bloor | Grazing and spatial heterogeneity: Implications for grassland structure and function[END_REF][START_REF] Louault | Complex plant community responses to modifications of disturbance and nutrient availability in productive permanent grasslands[END_REF]. Therefore, further investigations of assembly processes in mountain grasslands should also integrate descriptors of land management.

Conclusions

The dominating assembly process at the coarse scale across elevation was habitat filtering, particularly for VH, selecting for small plants at high and tall plants at low elevation sites. Limiting similarity was detected at plot and subplot levels, particularly for resource-acquisition traits (SLA and LDMC), but was stronger and dominating at the subplot level. Plot level trait divergence occurred particularly at high elevations, presumably indicating spatial partitioning rather than limiting similarity as the underlying mechanism. Our findings suggest that approaches studying and modelling assembly processes, such as stacked species distribution models (S-SDMs; [START_REF] Dubuis | Predicting spatial patterns of plant species richness: A comparison of direct macroecological and species stacking modelling approaches[END_REF] or mechanistic models of community assembly [START_REF] Shipley | From plant traits to plant communities: A statistical mechanistic approach to biodiversity[END_REF][START_REF] Demalach | Size asymmetry of resource competition and the structure of plant communities[END_REF][START_REF] Lohier | Predicting stochastic community dynamics in grasslands under the assumption of competitive symmetry[END_REF], should consider the interaction of biotic and abiotic factors along environmental gradients, particularly when examining communities at fine spatial scales. Further, the detection of spatial partitioning in this study calls for high-resolution abiotic data to allow understanding and forecasting of community and biodiversity patterns at fine scales. Overall, our study highlights the importance of analysing the roles of several assembly processes simultaneously across different spatial scales and in combination with abiotic gradients to allow a deeper understanding of the complex interaction of abiotic and biotic drivers shaping natural grassland communities. Table 1. List of the plant assembly processes that are hypothesised to drive community patterns. In this study, the coarse scale level refers to plots of 4 m 2 and the fine scale level to "subplots" each of 0.04 m 2 nested within the 4 m 2 plots (see Fig. 1). VH = vegetation height, SLA =specific leaf area, LDMC = leaf dry matter content, LNC = leaf nitrogen content, CGT = multidimensional space combining all growth traits.

  Münkemuller et al. (2012) combined α-and β-indices for taxonomic, functional and phylogenetic diversity metrics. Here, we further highlighted the importance of combining several species pool definitions and tests of assembly rules at nested scales.

Figure 1 .

 1 Figure 1. The general framework of the study was based on a nested sampling design to test for the signatures of different assembly processes. Habitat filtering and limiting similarity were analysed at two scales (2×2 m plots and 20×20 cm subplots) using null model approaches to compare observed trait structure with null expectations. The among subplots spatial partitioning process was analysed in each plot using a null model designed to test for spatial segregation of plant functional traits or syndromes.

Figure 2 .

 2 Figure 2. Distribution of SES of α-Rao at the plot level and average (of the five subplots) α-Rao at the subplot level for (a) total trait range (indicating habitat filtering) and (b) restricted trait range (indicating limiting similarity). (c) Distribution of SES of β-Rao among subplots (indicating fine-scale spatial partitioning). Black asterisks above a boxplot mark significant (p < 0.01) differences from zero, and red asterisks mark significant differences (p < 0.01) between plot and subplot level α-Rao. The boxes span from the 25 th to the 75 th percentile, and the whiskers indicate ± 2 SD. CGT= multidimensional space combining all growth traits, VH = Vegetation height, SLA = Specific leaf area, LDMC = Leaf dry matter content, LNC = Leaf nitrogen content, ns = Non significant.

Figure 3 .

 3 Figure 3. Correlation of strength of limiting similarity and habitat filtering measured by SES of α-Rao at the plot level. Different coloured areas indicate the dominant assembly process, and the colour gradient (red to blue) indicates the species richness of the plots. Please note that the x-axis (habitat filtering) is inversed, i.e., increasing effect to the right. See Fig. 2 for abbreviations.

Figure 4 .

 4 Figure 4. Correlation of strength of limiting similarity and spatial partitioning measured by SES of α-Rao at the plot level and SES of β-Rao among subplots. Different coloured areas indicate the dominant assembly process, and the colour gradient (red to blue) indicates the species richness of the plots. See Fig. 2 for abbreviations.

Figure 5 .

 5 Figure 5. Generalised linear models fitted for different traits, habitat filtering, limiting similarity and spatial partitioning along the elevation gradient. Models were fitted as second-degree polynomials with stepwise AIC to select the best model. Please note that the y-axis for habitat filtering is inversed. See Fig. 2 for abbreviations.

differentiation, spatial partitioning, trait convergence, trait divergence

Acknowledgements

This study was initiated in the framework of the ECOCHANGE project funded by the Sixth European Framework Programme (Grant GOCE-CT-2007-036866) and of the BIOASSEMBLE project funded by the Swiss National Science Foundation (SNF Grant Nr 31003A-125145 to AG) and finalised during the SESAM'ALP project (SNF grant Nr. 31003A-1528661 to AG). IUT 20-28 and Globaalmuutuste Tippkeskus financed MZ. Computations were performed at the Vital-IT Centre for high-performance computing at the Swiss Institute of Bioinformatics (http://www.vital-it.ch).

AG and JP, with LG and MZ, conceived the initial idea. AG, ALD, LP, JP and PV led the collection of the data. JP and DS analysed the data. HM, DS and JP led the writing, with all authors contributing and helping in interpreting the results.

Data accessibility

All species, environmental and trait data are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.2dq836f [START_REF] Guisan | Data from: Disentangling the processes driving plant assemblages in mountain grasslands across spatial scales and environmental gradients[END_REF].


Christelle Braun
email: braun@lix.polytechnique.fr

Konstantinos Chatzikokolakis

Catuscia Palamidessi
email: catuscia@lix.polytechnique.fr

Quantitative notions of leakage for one-try attacks

Keywords: Information-hiding, hypothesis testing, probability of error, Rényi min-entropy

Recent research in quantitative theories for information-hiding topics, such as Anonymity and Secure Information Flow, tend to converge towards the idea of modeling the system as a noisy channel in the information-theoretic sense. The notion of information leakage, or vulnerability of the system, has been related in some approaches to the concept of mutual information of the channel. A recent work of Smith has shown, however, that if the attack consists in one single try, then the mutual information and other concepts based on Shannon entropy are not suitable, and he has proposed to use Rényi's min-entropy instead. In this paper, we consider and compare two different possibilities of defining the leakage, based on the Bayes risk, a concept related to Rényi min-entropy.

Introduction

Information-hiding refers to a large class of problems including Secure Information Flow and Anonymity. There has been a growing interest in developing quantitative theories for this class of problems, because it has been recognized that non quantitative (i.e. possibilistic) approaches are in general too coarse, in the sense that they tend to consider as equivalent systems that have very different degrees of protection.

Concepts from Information Theory have revealed quite convenient in this domain. In particular, the notion of noisy channel has been used to model protocols for information-hiding, and the flow of information in programs. The idea is that the input of the channel represents the information to be kept secret, and the output represents the observable. The noise of the channel is generated by the efforts of the protocol to hide the link between the secrets and the observable, often achieved by using randomized mechanisms.

Correspondingly, there have been various attempts to define the degree of leakage by using concepts based on Shannon entropy, notably the mutual information [START_REF] Zhu | Anonymity vs. information leakage in anonymity systems[END_REF][START_REF] Clark | Quantitative information flow, relations and polymorphic types[END_REF][START_REF] Malacaria | Assessing security threats of looping constructs[END_REF][START_REF] Malacaria | Lagrange multipliers and maximum information leakage in different observational models[END_REF] and the related notion of capacity [START_REF] Moskowitz | Quasi-anonymous channels[END_REF][START_REF] Moskowitz | Covert channels and anonymizing networks[END_REF][START_REF] Chatzikokolakis | Anonymity protocols as noisy channels[END_REF].

In a recent work, however, Smith has shown that the concept of mutual information is not very suitable for modeling the information leakage in the situation in which the adversary attempts to guess the value of the secret in one single try [START_REF] Smith | Adversaries and information leaks (tutorial)[END_REF]. He shows an example of two programs in which the mutual information is about the same, but the probability of making the right guess, after having observed the output, is much higher in one program than in the other. In a subsequent paper [START_REF] Smith | On the foundations of quantitative information flow[END_REF], Smith proposes to use a notion based on Rényi min-entropy.

We look at the problem from the point of view of the probability of error : the probability that an adversary makes the wrong guess. We propose to formalize the notion of leakage as the "difference" between the probability of error a priori (before observing the output) and a posteriori (using the output to infer the input via the so-called MAP rule). We argue that there are at least two natural ways of defining this difference: one, that we call multiplicative, corresponds to Smith's proposal. The other, which we call additive, is new. In both cases, we show that it is relatively easy to find the suprema, which is nice in that it allows us to consider the worst case of leakage. The worst case is also interesting because it abstracts from the input distribution, which is usually unknown, or (in the case of anonymity) may depend on the set of users.

Preliminaries

Noisy channels and Hypothesis Testing

In this section we briefly review some basic notions about noisy channels and hypothesis testing that will be used throughout the paper. We refer to [START_REF] Cover | Elements of Information Theory[END_REF] for more details.

A channel is a tuple X, Y, p(•|•) where X, Y are random variables representing, respectively, the input with possible values X = {x 1 , x 2 , . . . , x n } (the secrets or hypotheses) and the output with possible values Y = {y 1 , y 2 , . . . , y m } (the observables). The distribution on X, π = (π 1 , . . . , π n ) is called a priori input distribution. We will also use the notation p(x i ) and p(y j ) to indicate the probabilities of the input x i (i.e. p(x i ) = π i ) and the output y j , respectively. We will denote by p(y j |x i ) the conditional probability of observing the output y j when the input is x i . These conditional probabilities constitute what is called the channel matrix, where p(y j |x i ) is the element at the intersection of the i-th row and j-th column.

The a posteriori probability p(x i |y j ) is the probability that the input is x i , given that we observe the output y j . The a priori and the a posteriori probabilities of x i are related by Bayes theorem:

p(x i |y j ) = p(y j |x i ) p(x i ) p(y j )
In hypothesis testing we try to infer the true hypothesis (i.e. the value that was really given in input) from the observed output value. In general, it is not possible to determine the right hypothesis with certainty. We are interested, then, in minimizing the probability of error, i.e. the probability of making the wrong guess.

We assume that the process of guessing the hypothesis is represented by a decision function f : Y → X , i.e. the function which gives, for every output y j , the guessed input x i .

The (average) probability of error associated to f is given by the sum of the probabilities of guessing a wrong hypothesis for each given output, averaged by the probabilities of the outputs. Since the probability of making the wrong guess, when the output is y j , is given by 1 -p(f (y j )|y j ), the average probability of error is:

PE f = j p(y j )(1 -p(f (y j )|y j )) = 1 -j p(y j )p(f (y j )|y j )
It is easy to see that a decision function f B minimizes the probability of error if and only if it satisfies the MAP (Maximum A Posteriori probability) criterion, namely, for each output y j it chooses an input x i for which p(x i |y j ) is maximum. Formally:

f B (y j ) = x i ⇒ ∀k p(x i |y j ) ≥ p(x k |y j )
It is easy to see that the probability of error associated to f B is then given by

PE B = 1 -j p(y j ) max i p(x i |y j )
By using Bayes theorem, we can rewrite PE B as:

PE B = 1 -j max i (p(y j |x i ) π i )
PE B is also called the Bayes risk. Note that it is a function of π, so we will also write PE B ( π) when we need to emphasize its dependency on π.

Rényi entropies, Shannon entropy, and mutual information

Rényi entropies [START_REF] Rény | On measures of entropy and information[END_REF] are a family of functions representing the uncertainty associated to a random variable. The Rényi entropy of order α, with α ≥ 0 and α = 1, is defined as

H α (X) = 1 1 -α log i p(x i ) α
In the case of a uniform distribution all the Rényi entropies of are equal to log n. Otherwise the entropies are weakly decreasing as a function of α. The following are some particular cases:

α = 0 H 0 (X) = log |X| = log n Hartley entropy α → 1 H 1 (X) = -i p(x i ) log p(x i ) Shannon entropy α → ∞ H ∞ (X) = -log max i p(x i ) min-entropy
Shannon conditional entropy of X given Y represents the average residual entropy of X once the value of Y is known, and it is defined as

H 1 (X|Y ) = - y p(y)H 1 (X|Y = y) = ij p(x i , y j ) log p(x i |y j ) = H 1 (X, Y )-H 1 (Y )
where H 1 (X, Y ) represents the entropy of the conjunction of X and Y .

The mutual information of X and Y represents the correlation of information between X and Y . It is defined as

I(X; Y ) = H 1 (X) -H 1 (X|Y ) = H 1 (X) + H 1 (Y ) -H 1 (X, Y )
It is possible to show that I(X; Y ) ≥ 0, with I(X; Y ) = 0 iff X and Y are independent.

Convexity and corner points

We recall here some basic notions of convexity. Let R be the set of real numbers. The elements

λ 1 , λ 2 , . . . , λ k ∈ R constitute a set of convex coefficients if, for every i ∈ {1, . . . , k}, λ i ≥ 0 and k λ k = 1. Given a vector space V , a convex combination of v 1 , v 2 , . . . , v k ∈ V is any vector of the form i λ i v i where λ 1 , λ 2 , . . . , λ k ∈ R are a set of convex coefficients. Definition 2.1 (Convex set) A subset S of a vector space is convex if every convex combination of vectors in S is in S.
In the following we will denote by D (n) the domain of probability distributions of dimension n. It is easy to see that, for every n, D (n) is convex.

We give now the definition of convex function.

Definition 2.2 (Convex function)

Given a convex subset S of a vector space V , and a function f : S → R, we say that f is convex if for any v 1 , v 2 , . . . , v k ∈ S and any set of convex coefficients λ 1 , λ 2 , . . . , λ k ∈ R, we have f

( i λ i v i ) ≤ i λ i f ( v i ). A function f is concave if its opposite -f is convex.
We now introduce (with a slight abuse of terminology) the concept of convex base.

Given a subset S of V , the convex hull of S, which we will denote by ch(S), is the smallest convex set containing S. Since the intersection of convex sets is convex, it is clear that ch(S) always exists. Definition 2.3 Given two vector sets S and U , we say that U is a convex base for S if U ⊆ S and S ⊆ ch(U ).

In the following, for a given vector v = (v 1 , v 2 , . . . , v n ), we will use the notation ( v, f ( v)) to denote the vector (with one additional dimension

) (v 1 , v 2 , . . . , v n , f ( v)).
Similarly, given a vector set S in a n-dimensional space, we will use the notation (S, f (S)) to represent the set of vectors

{( v, f ( v)) | v ∈ S} in a (n + 1)-dimensional space. The notation f (S) represents the image of S under f , i.e. f (S) = {f ( v) | v ∈ S}.
Definition 2.4 Given a vector set S, a convex base U of S, and a function f : S → R, we say that U is a set of corner points of f if (U, f (U )) is a convex base for (S, f (S)). We also say that f is convexly generated by f (U ).

In other words, if U is a set of corner points of f , then for every v ∈ D (n) , there are elements u 1 , u 2 , . . . , u k in U and

λ 1 , λ 2 , . . . , λ k in R such that v = i λ i u i and f ( v) = i λ i f ( u i ).

Mutual Information and Capacity

In [START_REF] Smith | Adversaries and information leaks (tutorial)[END_REF] Smith proved that the notion of mutual information (based on Shannon entropy) is not suitable to express the information leakage when the adversary tries to guess the value of the input in one single try. In fact, he showed an example of systems with the same mutual information, the same a priori uncertainty, and for which the probabilities of guessing the right input a posteriori (i.e. after observing the output) are very different.

The systems in Smith's example are deterministic, i.e. have the property that the input determines univocally the output. For such systems, it turns out that the discrepancy observed for the mutual information does not arise in the case of the capacity. Surprisingly, indeed, Smith showed in [START_REF] Smith | On the foundations of quantitative information flow[END_REF] that, under a uniform input distribution, the capacity is equal to the logarithm of the ratio between the a posteriori and the a priori probability of making a right guess, which coincides with his proposal for the notion of leakage. We will come back on this point in the next section.

Unfortunately, this coincidence does not carry out to the more general case of probabilistic channels, and, worse yet, the notion of capacity suffers (in the general case) from the same problem as the mutual information. The following example illustrates the situation.

Example 3.1 Consider the following channels:

y 1 y 2 y 3
x 1 2/3 1/6 1/6

x 2 1/6 2/3 1/6

x 3 1/6 1/6 2/3 It is easy to see that the Shannon capacity of C is 1/3, while the one of D is 2/3. However, under the uniform input distribution, the ratio between the a posteriori and the a priori probability of making the right guess is the same (2). 4 Towards a more suitable notion of leakage

y 1 y 2 y 3 x 1 2/3 1/3 0 x 2 0 2/3 1/3 x 3 1/3 0 2/3
In the following, we are interested in quantifying the leakage of a security protocol, i.e. the amount of information about the input that an adversary can learn by running the protocol and observing the resulting output.

Probabilities of a right guess

Before running the protocol, the probability that a given input x i occurs depends only on the a priori distribution π, and a rational adversary will therefore assume that the most probable input, called the a priori probability of a right guess PR i ( π), will be the input having the maximum a priori probability, i.e.:

Definition 4.1 The a priori probability of a right guess is defined as

PR i ( π) = max i π i
After running the protocol and seeing the output, the adversary may revise his guess. An adversary applying the MAP rule, when observing output y j , will choose as most probable input x i the one for which the a posteriori probability p(x i |y j ) is the highest. The average of this value on all possible outputs gives the a posteriori probability of a right guess PR o ( π), which is the complement of the Bayes risk.

PR o ( π) = j max i (p(y j |x i )π i )
In the rest of this paper, we will consider only adversaries applying the MAP rule since this is the rule that gives the best result (from the point of view of the adversary).

Leakage and uncertainty

Intuitively, the leakage is the amount of information learnt by the adversary by observing the output of the protocol. Following [START_REF] Smith | On the foundations of quantitative information flow[END_REF], it seems natural to define it as the difference between the uncertainty about the input before observing the output, and the remaining uncertainty afterwards:

Information Leaked = Initial Uncertainty -Remaining Uncertainty (1)
Now, the question is how to measure information, and (correspondingly) what do we actually mean by uncertainty. We consider here two possibilities. The first leads to a multiplicative notion of leakage, and it follows the proposal of Smith [START_REF] Smith | On the foundations of quantitative information flow[END_REF]. The second leads to an additive notion, and it is new.

Multiplicative leakage

In relation to Equation (1), Smith [START_REF] Smith | On the foundations of quantitative information flow[END_REF] measures the information in bits, and proposes to define the initial uncertainty as the min-entropy of X, H ∞ (X), the instance of Rényi entropy [START_REF] Rény | On measures of entropy and information[END_REF] obtained for α = ∞. As for the remaining uncertainty, it would be natural to use the conditional min-entropy of X given Y . Unfortunately there is no agreement on what Rényi's generalization of Shannon's conditional entropy should be, even though there seem to be a consensus towards y p(y)H α (X|Y = y) [START_REF] Cachin | Entropy Measures and Unconditional Security in Cryptography[END_REF]. Smith however proposes to use the definition of H ∞ (X|Y ) equivalent to the one given in [START_REF] Dodis | Fuzzy extractors: How to generate strong keys from biometrics and other noisy data[END_REF], which is

H ∞ (X|Y ) = -log PR o ( π)
In this way, Smith obtains a definition of leakage similar to the definition of mutual information, except that Shannon entropy is replaced by H ∞ :

L(X; Y ) = H ∞ (X) -H ∞ (X|Y ) = log PR o ( π) PR i ( π)
We consider a similar definition for leakage, namely the ratio between PR o ( π) and PR i ( π), which coincides with Smith's notion apart from the absence of the logarithm. Furthermore, in general we want to abstract from the a priori distribution, and consider the worst case, hence we are particularly interested in the supremum of such ratio.

Definition 4.3 We define the multiplicative leakage as

L × ( π) = PR o ( π) PR i ( π)
We will also use the notation ML × to represent the supremum of this quantity:

ML × = max π (L × ( π))
Note that PR i ( π) > 0 for every π, hence L × ( π) is always defined.

Additive leakage

Another possible interpretation for Equation ( 1) is to consider the uncertainty as the probability of guessing the wrong input. The leakage then expresses how much the knowledge of the observable helps decreasing such probability. This leads to define the leakage as the difference between the probabilities of error before and after observing the output. As usual, we are particularly interested in the supremum of this difference.

Definition 4.4

We define the additive leakage as

L + ( π) = PR o ( π) -PR i ( π)
We will also use the notation ML + to represent the supremum of this quantity:

ML + = max π (L + ( π))
5 Properties of the mutiplicative leakage

In this section we consider the multiplicative leakage and we study its supremum. It turns out that the supremum is very easy to compute. In fact, it coincides with the value of the leakage in the point of uniform distribution, and it is equal to the sum of the maxima of the columns. This property was also discovered independently by Geoffrey Smith and Ziyuan Meng (personal communication).

Proposition 5.1

ML × = L × (π u ) = j max i p(y j |x i )
where π u is the uniform distribution.

Proof

L × (π 1 , . . . , π n ) = 1 max i π i j max i (p(y j |x i )π i ) ≤ 1 max i π i j (max i p(y j |x i ))(max i π i ) = j max i p(y j |x i ) = n j max i (p(y j |x i ) 1 n ) = L × ( 1 n , . . . , 1 n ) ✷

Properties of the additive leakage

We turn now our attention to the additive leakage. We will see that the supremum is not always in the point of uniform distribution. However, we prove that it is in one of the corner points of PR i . Since PR i has a finite set of corner points, and their form is known, also the additive leakage is relatively easy to compute. First we prove a general property concerning the relation between suprema, convexity, and corner points: Proposition 6.1 Consider two functions f, g : D (n) → R where f has a set of corner points U , and g is convex. Define h :

D (n) → R as h( v) = f ( v) + g( v).
Let w ∈ D (n) be a point in which h has a maximum, i.e. for every v ∈ D (n) , h( v) ≤ h( w). Then there exists u ∈ U such that h( w) = h( u). Namely, the maximum value of h is in a corner point of f . Proof By contradiction. Suppose that for every u ∈ U , h( u) < h( w). Since w ∈ D (n) , there are elements v 1 , v 2 , . . . , v k in U and c 1 , c 2 , . . . , c k in R such that w = i c i v i

( x) = i c i f ( v i ). Then h( w) = h( i c i v i ) = f ( i c i v i ) + g( i c i v i ) = i c i f ( v i ) + g( i c i v i ) ≤ i c i f ( v i ) + i c i g( v i ) since g is convex = i c i h( v i ) < i c i h( w) since v i ∈ U = h( w) since c i = 1

✷

An example of Proposition 6.1 is illustrated in Figure 3. We now show that -PR i and PR o satisfy the hypotheses of Proposition 6.1. The necessary property for -PR i comes from a result in [START_REF] Chatzikokolakis | On the bayes risk in information-hiding protocols[END_REF].

Proposition 6.2 ([3], Proposition 3.9) The function PR i on D (n) is convexly generated by (U, f (U )) with U = U 1 ∪ U 2 ∪ . . . ∪ U n
where, for each r, U r is the set of all vectors that have value 1/r in exactly r components, and 0 everywhere else. Remark 6.3 The function -PR i has the same corner points as PR i .

We now prove that PR o satisfy the necessary property. Proposition 6.4 PR o is convex.

Proof

Let z be the convex combination i λ i z i where the dimension of z, z 1 , . . . , z m corresponds to the number of input variables and z 1 , . . . , z m is a set of corner points.

The j th component z k j of any corner point z k corresponds to the input variable x k j chosen according to the MAP rule when the output variable y j is obtained.

PR o ( i λ i z i ) = j max k {p(y j |x k )( i λ i z i ) k } = j p(y j |x k j )( i λ i z i ) k j = j p(y j |x k j )( i λ i z i,k j ) = j i λ i p(y j |x k j )z i,k j = i λ i j p(y j |x k j )z i,k j ≤ i λ i j max k p(y j |x k )z i,k = i λ i PR o ( z i )
✷ Corollary 6.5 ML + is reached on one of the corner points of PR i .

Proof

Since L + = PR o -PR i and PR o is convex, Proposition 6.1 shows that if ML + exists, it must be reached on a corner point of -PR i , which correspond to the corner points of PR i . ✷ Remark 6.6 In general ML + is not reached on the point of uniform distribution.

Example 6.7 Consider the channel whose matrix is given in Figure 4. The calculation of L + on the distributions corresponding to the corner points gives:

y 1 y 2 y 3 x 1 1 0 0 x 2 0 1 -e e x 3 0 1 -2e 2e
Corner points PR o PR i L +
(1, 0, 0), (0, 1, 0), (0, 0, 1) Let π = (1/r, 1/r, . . . , 1/r, 0, . . . , 0) be the a priori distribution with an equal probability of 1/r for the r first inputs. This distribution is a corner point of the matrix, and since the matrix is symmetric, any other corner point corresponding to a distribution containing r non-null probabilities of 1/r will give the same results for PR i , PR o and L + .

1 1 0 (1/2,
PR i ( π) = 1/r PR o ( π) = 1,...,11 (1/r)(1/10) = 11/(10r) L + ( π) = PR o ( π) -PR i ( π) = 1/(10r)
Therefore ML + is reached when r has the smallest value, i.e. when r = 2. This corresponds to the distribution (1/2, 1/2, 0, . . . , 0) and gives ML + = 1/20, while L + (1/11, . . . , 1/11) = 1/110.

Comparison

In this section, we compare the two notions of leakage. We first compare them with respect to a specific distribution, and then we consider the comparison of their worst cases.

If we consider a specific distribution, it comes out that the two notions are equivalent, in the sense that a program is better with respect to the additive notion if and only if it is better with respect to the multiplicative notion. Proposition 7.1 Consider two programs P and P ′ , and let L + and L + ′ be the additive measures of leakage for P and P ′ , respectively. Analogously, let L × and L × ′ be the multiplicative measures of leakage for P and P ′ , respectively. We have that, for every π

L + ( π) ≤ L + ′ ( π) ⇔ L × ( π) ≤ L × ′ ( π)

Proof

Let PR o and PR o ′ be the a posteriori probability of a right guess for P and P ′ , respectively. Analogously, let PR i and PR i ′ be the a priori probability of a right guess for P and P ′ , respectively. Observe that PR

i ( π) = max i π i = PR i ′ ( π). Hence L + ( π) ≤ L + ′ ( π) ⇔ PR o ( π) -PR i ( π) ≤ PR o ′ ( π) -PR i ′ ( π) ⇔ PR o ( π) -PR i ( π) max i π i ≤ PR o ′ ( π) -PR i ′ ( π) max i π i ⇔ PR o ( π) -PR i ( π) PR i ( π) ≤ PR o ′ ( π) -PR i ′ ( π) PR i ′ ( π) ⇔ PR o ( π) PR i ( π) -1 ≤ PR o ′ ( π) PR i ′ ( π) -1 ⇔ L × ( π) ≤ L × ′ ( π)

✷

Another criterion of comparison is the worst case. We consider the two notions on some examples. Example 7.2 Let us consider a 2 k × 2 k channel with the 2 k first natural numbers as inputs and outputs, i.e. X = Y = {0, . . . , 2 k -1}. Consider a random input variable X with values ranging in {0, 2 k -1}.

Consider the following program:

Program P(X) 1 ✄ Input X 2 if X = 0 or X = 1 3 then Output X 4
else Output one of the values {2, . . . , 2 k -1} chosen randomly according to the uniform distribution This program corresponds to a channel whose matrix is given in Figure 6. Let us consider first ML + . Because of Corollary 6.5, we know that ML + is reached on a corner point, i.e. a distribution of the form (q 1 , . . . , q 2 k ) where each q i is either 0 or 1/r, and there are r elements with value 1/r in the distribution.

For every corner point π we have PR i ( π) = 1/r, thus maximizing L + for a given r is equivalent to maximizing PR o . From the channel matrix, one can see that the maximum value of PR o is reached on an input distribution where the two first elements are as high as possible.

Therefore, we can restrict our study to distributions of the form (1/r, . . . , 1/r, 0 . . . , 0), i.e. distributions where the elements with value 1/r are the r first elements.

For r = 1, we have: Since p 1 > p 2 , L × increases when r increases, and thus ML × = 3 is obtained for r = 2 k (on the uniform distribution, which confirms Proposition 5.1).

The programs P and P ′ have therefore the same worst-case multiplicative measures of leakage ML × = 3, but the worst case of the additive measure of leakage ML + is equal to 2/3 for P and equal to 1/2 k-1 for P ′ .

Conclusion

We have considered two notions of leakage related to the Bayes risk. One of them, which we call multiplicative, corresponds to the notion recently proposed by Smith based on Renyi min-entropy. The other, which we call additive, is new. We have shown that the two notions are equivalent in all distributions. If we consider the distributions that give the worst case for the leakage, however, then the two notions are different. In particular, the multiplicative one has the worst case always in correspondence of the uniform distribution, while this is not the case for the additive one. So we can consider the new notion as a criterion, in addition to the one of Smith, to help assessing the degree of protection offered by a protocol or a program.

Fig. 1 .

 1 Fig. 1. Channel matrix C

Fig. 2 .

 2 Fig. 2. Channel matrix D

Definition 4 . 2

 42 The a posteriori probability of a right guess is defined as

Fig. 3 .

 3 Fig. 3. An illustration of Proposition 6.1 and f( x) = i c i f ( v i ). Then

Fig. 4 .

 4 Fig. 4. Channel matrix (e ∈ [0, 1/2])

Fig. 5 .

 5 Fig. 5.

Fig. 6 .

 6 Fig. 6. Channel matrix (p = 1/(2 k -2) = 1/2 k-1 )

1 For r = 2 , 2 L 2 For r ≥ 3 , 2 = r(p 1 -p 2 ) + 2 k p 2

 122232122 PR i (1, 0, . . . , 0) = 1 PR o (1, 0, . . . , 0) = 1 Thus:L + (1, 0 . . . , 0) = 0 L × (1, 0, . . . , 0) = we have:PR i (1/2, 1/2, 0 . . . , 0) = 1/2 PR o (1/2, 1/2, 0, . . . , 0) = 1 Thus: L + (1/2, 1/2, 0 . . . , 0) = 1/× (1/2, 1/2, 0, . . . , 0) = we have: PR i (1/r, 1/r, 1/r, 0, . . . , 0) = 1/r PR o (1/r, 1/r, 1/r, 0, . . . , 0) = 1/r + 1/r + (2 k -2) * (1/r) * p = 3/rThus L + increases when r increases, and ML + = 1/2 k-1 is reached for r = 2 k (on the uniform distribution).L × ( π) = rp 1 + (2 k -r)p

  Even in case of symmetric matrices, in general ML + is not reached on the point of uniform distribution. Consider the channel whose matrix is given in Figure5.

	We have for every e ∈ [0, 1/2[,
	0 = L + (1, 0, 0) ≤ L + (0, 1/2, 1/2) < L + (1/3, 1/3, 1/3) < L + (1/2, 1/2, 0) = 1/2
	and L + (1/3, 1/3, 1/3) = L + (1/2, 1/2, 0) = 1/2 for e = 1/2. Therefore if e <
	1/2, ML + = 1/2, reached on distributions that are different from the uniform
	distribution (1/3, 1/3, 1/3).
	Moreover, this remark holds also for symmetric matrices:
	Remark 6.8 Example 6.9 y 1	y 2	y 3	. . .	y 10	y 11
	x 1	0	1/10 1/10 . . . 1/10 1/10
	x 2 1/10	0	1/10 . . . 1/10 1/10
	. . .	. . .	. . .	. . .	. . .	. . .	. . .
	x 10 1/10 1/10 1/10 . . .	0	1/10
	x 11 1/10 1/10 1/10 . . . 1/10	0
	1/2, 0), (1/2, 0, 1/2)	1		1/2	1/2
	(0, 1/2, 1/2)			(e + 1)/2 1/2	e/2
	(1/3, 1/3, 1/3)		(e + 2)/3 1/3 (e + 1)/3

Acknowledgement

We are grateful to Pasquale Malacaria and to Geoffrey Smith for the many fruitful discussions.

Thus:

L × (1/r, 1/r, 1/r, 0, . . . , 0) = 3

L + (1/r, 1/r, 1/r, 0, . . . , 0) = 2/r

We observe that for r ≥ 3, the value of L + decreases when r increases. Since L + (1/3, 1/3, 1/3, 0, . . . , 0) = 2/3 > L + (1/2, 1/2, 0, . . . , 0) = 1/2 > L + (1, 0, . . . , 0) = 0, we have ML + = 2/3 reached for r = 3.

In particular,

reached on any distribution (1/r, 1/r, 1/r, 0, . . . , 0) with r ≥ 3, and in particular on the uniform distribution, which confirms Proposition 5.1.

Example 7.3 Let us consider the following program: The symmetry of the matrix implies that we can restrict the study to the a priori distribution π = (1/r, 1/r, . . . , 1/r, 0, . . . , 0), where the r elements with value 1/r are the first elements in the distribution.

In this case, for r ≥ 1: Finally: