Nicolas J Gotelli
email: ngotelli@uvm.edu

Werner Ulrich
email: ulrichw@umk.pl

Franck Jabot
email: franck.jabot@irstea.fr

Competitive interactions change the pattern of species co-occurrences under neutral dispersal

Keywords: Ecological species-sites matrix, competition, C-score, nestedness, beta diversity, ecological drift, meta-community

Competitive interactions change the pattern of species co-occurrences under neutral dispersal

Introduction

Ecologists have devoted much effort to understanding the role of competitive interactions in shaping ecological communities (reviewed in Weiher and Keddy 1999[START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF][START_REF] Chave | Comparing classical community models: theoretical consequences for patterns of diversity[END_REF][START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF]. Within the framework of competitive exclusion [START_REF] Gause | The struggle for existence[END_REF], a simple dominance hierarchy of competitive strengths (species A > species B > species C…) should eventually lead to a monoculture of the competitively superior species. But this is rarely seen in nature [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF]. Instead, most communities are characterized by a small number of common species (which may be competitively dominant), and a large number of rare species (which may be competitively inferior).

How are inferior competitors able to persist in a community? Proposed mechanisms include niche segregation [START_REF] Chesson | Mechanisms of maintenance of species diversity[END_REF], environmental heterogeneity [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: a synthesis[END_REF], abiotic stress [START_REF] Bowker | Competition increases with abiotic stress and regulates the diversity of biological soil crusts[END_REF], disturbance [START_REF] Watt | Pattern and process in the plant community[END_REF][START_REF] Grime | Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory[END_REF], and limited dispersal [START_REF] Hurtt | The consequences of recruitment limitation: reconciling chance, history and competitive differences between plants[END_REF][START_REF] Kerr | Local dispersal promotes biodiversity in a real-life game of rock-paperscissors[END_REF][START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF]. These mechanisms may also explain the observed high diversity and co-existence of ecologically similar species [START_REF] Fox | The intermediate disturbance hypothesis should be abandoned[END_REF]. In this respect, [START_REF] Grime | Competitive exclusion in herbaceous vegetation[END_REF] highlighted the importance of context-dependent competitive strength, in which the ordering of species in a competitive hierarchy changes in different environments (Chamberlain et al. 2014, Gioria and[START_REF] Gioria | Resource competition in plant invasions: emerging patterns and research needs[END_REF].

If competitive strength is context-dependent, species richness and abundance should differentially co-vary with environmental factors that most limit reproduction, leading to segregated occurrences of competing species along the environmental gradient. [START_REF] Diamond | Assembly of species communities[END_REF] used examples of perfectly segregated species pairs ("checkerboard pairs") as evidence for competitive exclusion (Diamond 1975, p 387), although he did not explicitly invoke context-dependent competitive interactions. Subsequent null model analyses of species cooccurrence have frequently detected individual species pairs and assemblages in which there is less co-occurrence than expected by chance [START_REF] Gotelli | Species co-occurrence: a meta-analysis of J.M. Diamond's assembly rules model[END_REF][START_REF] Ulrich | Null model analysis of species associations using abundance data[END_REF], 2013). Following [START_REF] Diamond | Assembly of species communities[END_REF] approach, many authors have inferred past or present competitive exclusion from spatially segregated co-occurrence patterns (e.g. [START_REF] Pitta | Significant pairwise co-occurrence patterns are not the rule in the majority of biotic communities[END_REF][START_REF] Kennedy | Missing checkerboards? An absence of competitive signal in Alnusassociated ectomycorrhizal fungal communities[END_REF], but see [START_REF] Connor | The checkered history of checkerboard distributions[END_REF]. Comparing communities at different times, [START_REF] Zaplata | Increasing structure and species-driven phases in plant communities during early succession[END_REF] and [START_REF] Ulrich | Species interactions and random dispersal rather than habitat filtering drive community assembly during early plant succession[END_REF] found that local plant assemblages became increasingly spatially segregated during early succession, and that these changes were associated with spatial variability in soil attributes.

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100.

The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

Although empirical and statistical support for segregated and aggregated species pairs is widespread [START_REF] Lyons | Holocene shifts in the assembly of terrestrial plant and animal communities implicate increasing human impacts[END_REF], these community patterns are hard to generate from theoretical models. Only a few studies have so far evaluated whether and to what degree competition alone (without additional habitat effects like filtering) is able to produce a spatially segregated pattern of species occurrences (e.g. [START_REF] Wootton | Causes of species diversity differences: a comparative analysis of Markov models[END_REF]. With appropriate parameter settings, simple two-species [START_REF] Levin | Dispersion and population interactions[END_REF]) and three-species (Caraco and Whitham 1984) interaction models can predict aggregated or segregated occurrences. However, patterns of species aggregation and segregation are more difficult to generate for models of diffuse competition in multi-species assemblages [START_REF] Hastings | Can competition be detected using species co-occurrence data?[END_REF].

A number of theoretical (Allesina andLevine 2011, Ulrich et al. 2014) and empirical [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF][START_REF] Ulrich | Species interactions and random dispersal rather than habitat filtering drive community assembly during early plant succession[END_REF]) studies suggest that competitive intransitivity is an important mechanism that allows species to coexist within a single community in spite of strong competitive interactions. Intransitive competitive networks [START_REF] Gilpin | Limit cycles in competition communities[END_REF]) are formed by loops in the hierarchy of competitive strength. For example, in the he rock-scissors-paper game, the competitive hierarchy species A> species B> species C> species A forms a loop that can theoretically promote coexistence [START_REF] Huisman | Towards a solution of the plankton paradox: the importance of physiology and life history[END_REF][START_REF] Kerr | Local dispersal promotes biodiversity in a real-life game of rock-paperscissors[END_REF][START_REF] Laird | Competitive intransitivity promotes species co-existence[END_REF][START_REF] Ulrich | A consumer's guide to nestedness analysis[END_REF]. [START_REF] Ulrich | Matrix models for quantifying competitive intransitivity from species abundance data[END_REF] demonstrated that transitive and intransitive competitive hierarchies in ecological communities can be unequivocally translated into a stable state distribution of abundances by means of a Markov chain model [START_REF] Horn | Markovian properties of forest succession[END_REF]. This model predicts constant abundance distributions within a homogeneous environment and no spatial segregation of species occurrences among sites [START_REF] Ulrich | Species interactions and random dispersal rather than habitat filtering drive community assembly during early plant succession[END_REF]. However, if competitive hierarchies differ among sites because of environmental conditions, species abundance distributions and co-occurrence patterns change. In this scenario, species segregation among sites is solely linked to environmental heterogeneity and not caused by the underlying competitive hierarchy [START_REF] Ulrich | Species interactions and random dispersal rather than habitat filtering drive community assembly during early plant succession[END_REF].

Models of context-dependent competition among sites and intransitive competitive

hierarchies within sites make different assumptions about equilibrium conditions. Context dependency explicitly includes environmental spatial and temporal variability [START_REF] Chamberlain | How context dependent are species interactions?[END_REF]) and thus applies to both equilibrium and non-equilibrium conditions. It does not make precise predictions about changes in species abundances and dominance orders in space.

In contrast, models of intransitivity are most relevant to equilibrium conditions and have so far been applied only to closed assemblages in which species compete locally and are not affected by migration (e.g., [START_REF] Allesina | A competitive network theory of species diversity[END_REF]. But some local communities are organized as an open metacommunity, which is defined by [START_REF] Gilpin | Metapopulation dynamics: empirical and theoretical investigations[END_REF] and [START_REF] Leibold | The metacommunity concept: a framework for multi-scale community ecology[END_REF] as a set of interacting local communities that are linked by the dispersal of multiple, potentially interacting species. For open metacommunities, it is unclear whether models of intransitivity will predict constancy in the richness (and abundances) of local assemblages. For example, [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF] reported that local dryland and grassland plant communities often contained intransitive loops, but that these competitive hierarchies explained little of the spatial variation in species richness. Interestingly, the frequency of intransitivity decreased with increasing habitat heterogeneity [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF] suggesting that species richness might be controlled by both intransitive networks and context-dependent competition.

In this study, we ask how competitive transitivity and intransitivity translate into dominance orders in a meta-community that incorporates dispersal and non-equilibrium dynamics at local scales. The neutral model framework [START_REF] Hubbell | The unified neutral theory of biogeography and biodiversity[END_REF] allows us to generate predictions of local abundance and species composition from a set of first principles (birth/death processes, dispersal, and speciation). These predictions can then be compared to patterns in real assemblages [START_REF] Gotelli | Null model analysis of species co-occurrence patterns[END_REF]McGill 2006, Rosindell et al. 2012).

In their original formulation, neutral models were based on assumptions of random dispersal and the ecological equivalence of species [START_REF] Hubbell | The unified neutral theory of biogeography and biodiversity[END_REF][START_REF] Chave | Neutral theory and community ecology[END_REF] (2) Does competition change the spatial or temporal variability in local richness?

(3) Does competitive intransitivity alone lead to species segregation across sites within a meta-community?

(4) Do diffuse and direct competitive interactions predict different patterns of species coexistence?

Materials and methods

The dispersal-limited competition model

The present study is based on a simulation platform for neutral modelling that was previously used to show that appropriately parameterized neutral models are able to generate segregated spatial distributions of species co-occurrence [START_REF] Ulrich | Species co-occurrences and neutral models: reassessing J. M. Diamond's assembly rules[END_REF]) and that ground beetle meta-community structures are indistinguishable from neutral predictions [START_REF] Ulrich | Are ground beetles neutral?[END_REF]. In our original simulations, we use a square grid of 100 patches initially populated randomly by a total of 10,000 individuals belonging to 30 species. This placement procedure leads to a grid of patches with different maximum numbers of individuals per patch (carrying capacities) and a Poisson distribution of species richness. The grid of occupied patches represents the metacommunity, whereas each occupied patch represents a local community. In the following, we will interchangeably use the terms grid/metacommunity and patch/local community. The subsequent dynamics in each patch follows a zero-sum rule [START_REF] Hubbell | The unified neutral theory of biogeography and biodiversity[END_REF], meaning that each local birth, death, immigration, or emigration (all probabilities set to 0.01) is immediately counterbalanced by a corresponding death, birth, emigration, or immigration. Any grid-wide species extinction is counterbalanced by a single point mutation speciation in a randomly selected patch. In contrast to [START_REF] Hubbell | The unified neutral theory of biogeography and biodiversity[END_REF] original formulation, this point mutation speciation ensures that the total number of species within the meta-community remains constant.

In this study, we added two features to this neutral model. First, we followed Jabot and Chave ( 2011) and introduced death rates that are species-specific and incorporate interspecific density-dependence. The local death probability π i of an individual of species i in a community of j species is given by:

= ∑ (1)
where d i is the density of species i and δ is the density-dependence parameter. For δ = 0, death probabilities are proportional to the observed abundance distribution and thus equal for all individuals. For δ < 0, the model penalizes abundant species (diffuse A) by higher local

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

death rates, and if δ > 0, the model penalizes rare species (diffuse R). For δ = 1 species mortality rates are identical irrespective of abundance. This modification incorporates effects of diffuse (indirect) competition from the entire assemblage.

Second, we incorporated the effects of direct pairwise competition on birth rates using the Markov chain approach of [START_REF] Ulrich | Matrix models for quantifying competitive intransitivity from species abundance data[END_REF]. These authors showed that any j x j matrix C of pairwise species interaction effects can be translated into a unique column stochastic transition matrix P (cf. Fig. 1). The inner product PA 0 =A 1 provides the vector of expected (2) An intransitive competitive hierarchy generates a non-absorbing ergodic transition matrix and therefore predicts coexistence of species (Fig. 1). Intransitivity might either increase or decrease the equilibrium species richness compared to the predictions of the simple neutral model.

(3) Diffuse competition that penalizes death rates of less abundant species has the same effect as strong competitive hierarchy in favouring abundant species and accentuating dominance orders. Therefore it should increase the tendency of the model to generate a monoculture of the strongest competitor.

Using stochastic simulations, we ask whether these predictions still hold in a spatially explicit model of an open metacommunity that incorporates dispersal.

Simulation protocol

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

To assess the influence of interspecific competition on otherwise neutral communities, we created a factorial design of five binary model parameters. We crossed two levels of dispersal limitation (unconstrained = all cells are equally likely to be the target of dispersal, limited = only the adjacent cells (the Moore Neighbourhood) can reached in a single dispersal step), with two levels of migration rates (low, high), with two levels of diffuse competition penalizing either rare ( δ = 0.5) or abundant (δ = -0.5) species, and two levels of competitive interactions (intransitive, transitive; cf. electronic supplement S1A and S1B, respectively).

Together with the four neutral scenarios of δ = 0 and P = I (I being the identity matrix), we considered a total of 2 4 + 4 = 20 parameter combinations. Because variability in the model output within parameter sets was low, we replicated each parameter combination only 10 times.

Transitive competitive interactions led to monocultures of the best competitors, although the time to complete competitive exclusion was very long for some parameter combinations (> 90,000 time steps). To ensure that the slowest-running model (diffuse competitive interactions and high dispersal rates; Table 1) reached equilibrium, we ran all models for 92,000 time steps, which incorporated 1,010,000 birth/death, immigration/emigration, speciation/extinction events, that is approximately 100 cycles of complete turnover in species composition. Equilibrium conditions were defined by a change of < 1 species per single time step of the moving average of species richness in the grid. We further traced the decrease in average species richness among sites (S M ) from the initial 27 to 30 species per site with the slope z of the semi-logarithmic regression model = -( ) where t denotes the time step of the model and S 0 the initial species richness. The semi-logarithmic model provided the best linear fit to the decay of species richness through time.

Analysing community structure and co-occurrences

For each grid, we quantified the degree of species segregation (negative species associations) with the common C-score of species co-occurrences (a normalised count of the number of pairwise mutual exclusions among sites; Stone and Roberts 1990). Species spatial aggregation was quantified by the clumping score, which is a normalised count of the number of pairwise co-occurrences among sites [START_REF] Ulrich | Pattern detection in null model analysis[END_REF]. Nestedness measures the ordered loss of species along a focal environmental or ecological gradient [START_REF] Patterson | Nested subsets and the structure of insular mammalian faunas and archipelagos[END_REF]Atmar 1986, Ulrich et al. 2009) and is therefore distinct (although not mutually exclusive) from species turnover [START_REF] Ulrich | Pattern detection in null model analysis[END_REF]. We quantified the degree of nestedness using the standard NODF (nestedness from overlap and decreasing fill) metric, which is a Metrics of species co-occurrences depend on matrix row (species) and column (sites) totals and cannot be compared directly. Therefore, we used a null model approach and compared observed scores with those obtained from 200 matrices randomly resampled by two different null algorithms. First, we resampled species incidences where placement probabilities were uniform for all grid cells (the equiprobable null model algorithm). In the second null model, placement probabilities were proportional to observed marginal occurrence totals (the proportional -proportional null model, Ulrich and Gotelli 2012). We did not use the popular fixed-fixed algorithm [START_REF] Gotelli | Null model analysis of species co-occurrence patterns[END_REF] because it preserves the marginal totals of the matrix, which would lead to low variation in the NODF and lack of variation in the beta metrics.

Neutral models of limited dispersal (e.g. [START_REF] Babak | A neutral model of edge effects[END_REF] and biogeographic models of the mid-domain effect [START_REF] Colwell | The mid-domain effect: geometric constraints on the geography of species richness[END_REF] predict that random processes can lead to a reduction of species richness near the boundaries of spatial domains. To estimate the size of this effect, we calculated the difference ∆S in richness between the 12 cells at each of the four grid corners and the 12 cells in the centre of the grid. Increases in species richness towards the centre of the spatial grid will yield a negative ∆S. To assess the spatial variability in species richness, we used [START_REF] Lloyd | Mean Crowding[END_REF] variance -mean ratio = -+ 1, with µ and σ 2 being the mean species richness and its variance, respectively. I = 1 is the expected value in the case SES values of simulated assemblages were moderately correlated with species richness of the meta-community for both the proportional null model (all r 2 > 0.47, P < 0.001) and the equiprobable null model (all r 2 > 0.20, P < 0.001). NES values performed better except for the clumping score -equiprobable null model (r 2 = 0.67, P < 0.001) and the C-scoreproportional null model (r 2 = 0.47, P < 0.001) combinations.

In both cases, the low equilibrium average richness of one to two species per cell generated by the transitive and diffuse R competition models were responsible for these correlations. Therefore, we used linear models and covariance analysis with NES as the dependent variable and average species richness and squared average richness per site as covariates to assess the effect of competition and dispersal on patterns of species cooccurrences. We note that the standardized effect sizes (proportional null model) of the Cscore and β sim were strongly positively correlated (r = 0.86), whereas the standardized effect sizes of the C-score and the clumping score were strongly negatively correlated (r = -0.76).

The standardized effect size of NODF was negatively correlated with the C-score (r -0.53), β sim (r =-0.64), and the clumping score (r = -0.30). The complete raw data used in the present study are contained in the electronic supplement S1B.

Results

In the pure neutral model, limited dispersal and low migration probability reduced the time to species equilibrium (Tab. 1, Fig. 2a,b). The logarithmic decay model explained on average 85% of the variability in richness and the model fit was independent of dispersal strength (Fig. 2b). Competitive effects reduced the fit of the exponential decay model (Fig. 2b) but edge effects were of minor importance (Fig. 2c) and did not significantly change between model settings (Tab. 1). Dispersal limitation significantly decreased average species richness in the pure neutral communities (Fig. 3a, Tab. 1). Spatial patterns of richness within the grid (Fig. 3b) matched a Poisson distribution.

Irrespective of the degree of dispersal limitation, transitive pairwise competition severely decreased average species richness per patch in comparison to the neutral expectation (Fig. 3a). The strongest competitor, as defined by the transition elements of the competition matrix P, often excluded all other species, resulting in a monoculture. Slopes of the species loss function were comparably steep (Fig. 2b), and edge effects of minor importance (Fig. 2c).

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

In contrast, intransitive competitive interactions significantly increased species richness in comparison to the neutral expectation (Fig. 3a) and decreased the species richness decay slopes (Fig. 2a). Consequently, a generalized linear model (Tab. 2) identified the type of competition as being the most important driver of species richness. Transitive competition decreased the variability in species richness among grids, leading to a segregated pattern of richness (Fig. 3b). Neither transitive nor intransitive competition altered edge effects on species richness (Fig. 2c).

Pure neutral communities were more spatially aggregated, with lower spatial turnover in species richness than predicted by the equiprobable null model (Fig. 4a, b, c, electronic supplement Fig. S2Aa), but less spatially aggregated with more species turnover than predicted by the proportional null model (Fig. 5a, b, c, Fig. S2Ba). Neutral community dynamics did not generate significant patterns of nestedness (Figs. 4d, 5d, Fig. S2Ad,S2Bd).

Pairwise (Tabs. 2, 3) and diffuse (Tab. 2) competition significantly altered species cooccurrences compared to the neutral expectation. The C-score and clumping indices, but not β sim , were most sensitive to competition. Dispersal and competition explained between 5% and 75% of variance in co-occurrences depending on the two types of null expectation (Tab.

2). Standardized effect sizes of the co-occurrence metrics (Figs. S2A, B) were highly significant for the majority of competition-dispersal combinations with respect to the equiprobable null model, while only 33 of the 2160 comparisons with the proportional null model were significant at the 5% error level (1.5%).

Type of competition and dispersal limitation interacted and caused specific patterns of co-occurrences (Tab. 3,Figs. 4,5). High dispersal caused intransitive competitive communities to be significantly (P < 0.001) more segregated (C-score) than neutral ones when compared to an equiprobable null model and less segregated when compared to the proportional null model expectation (Tab. 3,Figs. 4,5). At low dispersal both null models detected trends towards aggregation in intransitive communities (Tab. 3). Irrespective of the null model transitive communities tended to have an aggregated and/or nested structure compared to their pure neutral counterparts (Tab. 3,Figs. 4,5). This is in line with a higher species turnover among sites (β sim ) at high dispersal rates (Tab. 3, Fig. 5) compared to the neutral expectation (proportional null model).

Diffuse competition penalizing rare species (diffuse R) had on average similar effects on the spatial distribution of species than transitive competition (Figs. 4. 5), while diffuse A communities equalled intransitive ones qualitatively in behaviour.

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

Discussion

In the tradition of the competitive exclusion principle [START_REF] Gause | The struggle for existence[END_REF], a large number of coexisting species is often attributed to weak competitive interactions [START_REF] Gilpin | Limit cycles in competition communities[END_REF][START_REF] Wootton | Causes of species diversity differences: a comparative analysis of Markov models[END_REF][START_REF] Liao | Species interactions determine the spatial mortality patterns emerging in plant communities after extreme events[END_REF]. However, recent theoretical models (Huisman et al. 2001, Rojas-Echenique andAllesina 2011), pointed to the possibility that intransitive competitive hierarchies [START_REF] Vandermeer | Intransitive loops in ecosystem models: From stable foci to heteroclinic cycles[END_REF][START_REF] Hillerislambers | Rethinking community assembly through the lens of coexistence theory[END_REF]) might be an important mechanism allowing for species co-existence, although there is little empirical evidence so far [START_REF] Reichenbach | Mobility promotes and jeopardizes biodiversity in rock-paperscissors games[END_REF][START_REF] Kraft | Plant functional traits and the multidimensional nature of species co-occurrence[END_REF][START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF]. Our simulation corroborates these predictions. Even a weak degree of intransitive competition significantly increased average local richness above the pure neutral expectation (Tab. 2, Fig. 3). Therefore we argue that competitive intransitivity might be a neglected factor that increases local richness [START_REF] Kraft | Plant functional traits and the multidimensional nature of species co-occurrence[END_REF][START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF]. Further, our results suggest that increased species richness can occur purely from intransitive competition, and does not require environmental variability and associated differential habitat filter processes [START_REF] Keddy | Assembly and response rules: two goals for predictive community ecology[END_REF].

Neutral community dynamics did not cause a modular pattern of species occurrence with regions of higher and lower richness (Fig. 3). We speculated (question 2) that competitive interactions in combination with low dispersal might cause a respective spatial patterning leading to a richness landscape within a homogeneous environment. This was not the case (Fig. 3). Neutral as well as communities governed by intransitive and diffuse A competitive hierarchies (Fig. 3b) retained a Poisson random distribution of species richness among the grid cells. In contrast, transitive and diffuse R competition had a significant tendency of equalizing richness among cells within the overall species poor landscape (Fig. 3b). We note that this results might stem, at least partly, from a statistical bias due to the low number of species per cell. We also note that at equilibrium, spatial and temporal variability in richness are equivalent. Thus our results do also show that competition does not lead to increased temporal fluctuations of species richness in single grid cells.

Since the seminal work of [START_REF] Diamond | Assembly of species communities[END_REF], replicated patterns of negative species association (segregation) are often seen as evidence for interspecific competition (reviewed in Götzenberger et al. 2012) although many other reasons for species segregation are known [START_REF] Blois | A framework for evaluating the influence of climate, dispersal limitation, and biotic interactions using fossil pollen associations across the late Quaternary[END_REF]. Starting with [START_REF] Grime | Competitive exclusion in herbaceous vegetation[END_REF], several authors (reviewed in [START_REF] Chamberlain | How context dependent are species interactions?[END_REF][START_REF] Gioria | Resource competition in plant invasions: emerging patterns and research needs[END_REF][START_REF] Gioria | Resource competition in plant invasions: emerging patterns and research needs[END_REF] argued that context-dependent competitive strength in heterogeneous environments might be the major driver of species segregation among habitats, whereas trait differentiation and small-scale environmental variability allows for local coexistence (e.g. [START_REF] Adler | Trait-based test of coexistence mechanisms[END_REF].

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

However, many studies of community assembly rules do not refer to these mechanistic models of species interactions and often treat species spatial segregation as sufficient evidence for competition (e.g. [START_REF] Price | Small-scale grassland assembly patterns differ above and below the soil surface[END_REF]. Here, we focused on the question of whether competitive effects alone permit species coexistence and generate non-random species associations. A related question is whether habitat heterogeneity and habitat filtering [START_REF] Zobel | Competitive strength matrices used in the present study, B: Complete raw data. Electronic supplement S2: Standardized effect sizes (SES) of the co-occurrences metrics[END_REF]) are primarily responsible for non-random species associations.

We found strong evidence that competitive effects alone are indeed able to influence the geometry of species occurrences (Tab. 3, Figs. 4, 5, S2A, B). However, there was no simple rule or consistent directional change caused by species interactions. Instead, the spatial pattern depended on both the type of species interaction and the level of dispersal. Specifically, intransitive competition increased species turnover and decreased clumping when compared to the predictions of a neutral model with unlimited dispersal (Tab. 3). However, the opposite pattern -decreased species turnover and increased clumping -emerged when compared to the predictions of a neutral model with limited dispersal (Tab. 3). It seems that dispersal limitation and intransitivity are both able to generate small scale clusters of communities with distinct species composition. That means from co-occurrence analysis alone we cannot draw simple (simplified) conclusions about the effect of competitive interactions on the patterns of species co-occurrences (cf. [START_REF] Kraft | Plant functional traits and the multidimensional nature of species co-occurrence[END_REF] for a similar conclusion). [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF] recently reported a pattern of nested community structure associated with intransitive competitive hierarchies in dryland plant assemblages. However, it remained unclear whether this effect was due to environmental heterogeneity or due to the internal dynamics of species interactions. Our results partly corroborate [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF]: transitive competition increased the degree of meta-community nestedness relative to a neutral assemblage (Tab. 3,Figs. 4,5). This trend is not biased by low species richness because the NES transformation effectively removed the richness effect on the NODF metric.

The influence of intransitive competition on the degree of nestedness is less clear. Although we observed the trend [START_REF] Soliveres | Intransitive competition is widespread in plant communities and maintains species richness[END_REF] reported, statistical corroboration was weak (Tab. 3). Nevertheless, our results suggest that negative species interactions alone might suffice for a trend towards nested community structure.

Magnitude and direction of effect sizes in null model analyses depend on the choice of the algorithm and therefore on the underlying assumption about the constraints applied to randomization [START_REF] Ulrich | A null model algorithm for presence -absence matrices based on proportional resampling[END_REF]. Different null model approaches frequently lead to contradictory effect sizes making the interpretation of pattern challenging. In the present case the effect sizes of the C-score of the two null models were partly contradictory (Figs. 4,5),

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100. The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392 yielding a pattern of aggregation when compared to the equiprobable expectation, but a pattern of segregation when compared to the proportional null model.

In the present case, we used the liberal equiprobable null model, the more conservative (proportional) null model, and the pure neutral model [START_REF] Rosindell | The case for ecological neutral theory[END_REF]) as standards for comparison. Specifically, we compared patterns generated by three neutral mode ls with and without competition. The three different null models served to control for differences in species abundances, species richness, and matrix fill. Only this combination allowed for an unequivocal interpretation of the results.

Recently, [START_REF] Rosindell | The case for ecological neutral theory[END_REF] argued that neutral models provide process based adequate standards for ecological patterns. However, the major drawback of neutral modelling is the sensitivity to parameter settings, and the fact that those parameters cannot be estimated in a non-circular way from the presence-absence matrix. The simplest neutral models are based on at least four free parameters: species pool size, dispersal limitation, birth rate, and speciation rate. Additionally, total spatial extent and associated edge effects might influence the outcome [START_REF] Babak | A neutral model of edge effects[END_REF]. Thus we argue that process-based neutral models may be just as sensitive to model structure as traditional null model analysis. Whether empirical data are compared to a neutral model or a null model, a thorough sensitivity analysis with different model variation may be necessary to fully understand the results.

One useful distinction that emerged here is the idea that the null model reveals nonrandom patterns above and beyond those generated by matrix constraints such as row and column totals, matrix size, or matrix fill. To tease apart mechanisms of habitat filtering, species interactions, or dispersal limitation requires additional data and additional tests on the spatial pattern of occupied and unoccupied sites, and on the habitat structure of those sites [START_REF] Blois | A framework for evaluating the influence of climate, dispersal limitation, and biotic interactions using fossil pollen associations across the late Quaternary[END_REF]. In contrast, the neutral model formulation explicitly models random dispersal and a lack of species interactions, though often with a zero-sum constraint imposed.

If the parameters for such a model can be estimated independently of the co-occurrence data, the neutral model can also be used to generate an expectation for comparison with real data.

Alternatively, the neutral model predictions can themselves be compared to the predictions of a null model (Ulrich 2004, Gotelli and[START_REF] Gotelli | Null versus neutral models: what's the difference?[END_REF]. Neither approach by itself is complete, but the combination of null and neutral modelling may be the best way forward.

Our work has influence on the interpretation of observed patterns of species cooccurrences in field studies. Many authors (e.g. [START_REF] Gotelli | Species co-occurrence: a meta-analysis of J.M. Diamond's assembly rules model[END_REF], Götzenberger et al. 2012[START_REF] Connor | The checkered history of checkerboard distributions[END_REF] associations (aggregation) have usually been interpreted in terms of habitat filtering and facilitation (e.g. Götzenberger et al. 2012[START_REF] Vaz | Plant species segregation in dune ecosystems emphasises competition and species sorting over facilitation[END_REF]. Our results identified clear tradeoffs between the type of competitive interactions and the degree of dispersal: competitive interactions can generate species aggregation, and segregation may stem from dispersal limitation alone. Possibly, some reported effects of competition on the geometry of species occurrences might require reassessment. Consequently, future empirical work on the spatial structure of meta-communities needs to include independent information on dispersal ability and resource utilization, as well as information on habitat heterogeneity, for a proper interpretation of co-occurrence patterns. Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100.

The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

species abundances A 1

 1 after one time step, given initial abundances A 0 . Within a neutral model framework, birth probabilities are proportional to current abundances. vector Q, which (after normalization) contains the local birth probabilities of an individual in the community. In this way, our model incorporates effects of direct (pairwise) competition on birth rates and effects of indirect (diffuse) competition on death rates. In the absence of dispersal this model of competition yields three qualitative predictions for isolated local communities:(1) A fully transitive competitive hierarchy modulates abundances in favour of the stronger competitors by increasing their fecundity while leaving death probabilities unchanged (Fig1). Because the respective transition matrix (P) generated from the matrix of competitive strength (C) describes an absorbing state (Fig.1)[START_REF] Ulrich | Matrix models for quantifying competitive intransitivity from species abundance data[END_REF]), this Markov model predicts that the final result of transitive competition is a monoculture of the strongest competitor, independent of the model settings.

  of a Poisson random distribution, I < 1 indicates equitability in richness across the grid, and I > 1 indicates clumping. For comparison among model settings, we used the normalised effect sizes (NES = (observed -expected scores) / expected scores) and standardized effects sizes [SES = (observed -expected scores)/standard deviation of expectation]. Under the assumption of a normal distribution of errors, |SES score | > 1.96 indicates approximate statistical significance at

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Fully transitive (a) and intransitive (b) competitive strength matrices (as defined by Laird and Schamp 2006 and Ulrich et al. 2014) of 12 five species can be unequivocally transformed into respective column stochastic transition matrices (entries denote probability levels of 13 transition) by the algorithm derived in Ulrich et al. (2014). The latter provide estimates of temporal changes in abundances (A0, A1, …) and 14 equilibrium abundances (the dominant eigenvector EV1 of the transition matrix). 1s in the competitive strength matrix indicate competitive 15 superiority. For example, in panel (b) species a (in rows) is superior to all species except species e. Matrix multiplication of the associated 16 transition matrix with the abundance vectors now returns species abundances in the next generations. The dominant eigenvector of this matrix 17 predicts species abundances at equilibrium. In this model of an isolated local community, intransitivity predicts increased equilibrium species 18 richness. 19 20

  Author-produced version of the article published inOikos, 2017, 126 (1), 91-100.The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392 normalized count of the degree of species overlap among the sequence of plots ordered according to decreasing species richness[START_REF] Almeida-Neto | A consistent metric for nestedness analysis in ecological systems: reconciling concept and quantification[END_REF]. NODF ranges from zero (perfect species turnover) to 1 (perfect nestedness). Following the method of Baselga (2010),we assessed the degree of spatial species turnover among cells by the additive partitioning of

	the Sørensen metric β sor (a metric of dissimilarity in community composition) into a
	component representing the difference in species richness among sites (β nest ) and a component
	representing the spatial turnover of species (β sim ). Below we focus on this turnover component
	because it represents the compositional variation of communities after controlling for
	differences in richness.

  Author-produced version of the article published inOikos, 2017, 126 (1), 91-100.The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392 p < 0.05 (two-tailed test). These measures of effect size allowed for comparisons among different model results, but they did not completely remove the influence of species richness:

  have interpreted non-random segregation as evidence for competitive interactions (but seeUlrich and Gotelli 2010, Blois et al. 2014), whereas positiveAuthor-produced version of the article published inOikos, 2017, 126 (1), 91-100.The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

Table 1 .

 1 In the pure neutral model migration probability and the degree of dispersal limitation influenced final mean species richness, the spatial variability in richness, the edge effect of the lattice, and the slope of the logarithmic decrease model. Given are partial η 2 values of a general mixture linear model. Significant parametric p(F) < 0.001 in bold.

			Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100.
				The original publication is available at https://onlinelibrary.wiley.com/
							doi:10.1111/oik.03392
	Factor	Degrees of freedom	Mean species richness	Lloyd	Edge	Slope
	Migration probability	1	0.61	0.39	0.00	0.89
	Dispersal limitation	2	0.31	0.42	0.01	0.88
	Migration×Dispersal	2	0.46	0.30	0.03	0.87
	Mean species richness	1	-	0.34	0.01	0.81
	Error	53
	r	2 (whole model)		0.74	0.51	0.08	0.93

Table 2 .

 2 Main effects general linear modelling of normalized effect sizes of C-score, clumping β sim , and NODF (partial η 2 scores) identified 1 particularly pairwise competitive interactions to influence average species richness, spatial variability in richness(Lloyd index), and patterns of 2

									Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100.
										The original publication is available at https://onlinelibrary.wiley.com/
													doi:10.1111/oik.03392
	3	co-occurrence. Significant parametric p(F) < 0.001 in bold.
	4
				Degrees	Mean
		Factor	of	species	Lloyd	Equiprobable null model			Proportional null model
				freedom	richness
							C-score	Clumping	β sim	NODF	C-score	Clumping	β sim	NODF
		Migration probability	1	0.07	0.04	0.06	0.02	0.01	0.01	0.03	<0.01	<0.01 <0.01
		Dispersal limitation	2	0.03	0.02	0.04	0.01	<0.01 <0.01	0.02	0.01	0.01	<0.01
		Diffuse competition	2	0.46	0.20	0.10	0.01	<0.01	0.05	0.01	0.02	<0.01	0.01
		Pairwise competition	2	0.84	0.01	0.42	0.06	0.01	0.02	0.03	0.08	<0.01	0.04
		Mean species richness	1	-	0.54	0.05	0.11	0.01	0.30	0.24	0.04	0.04	0.09
		Squared mean species richness	1	-	0.49	<0.01	0.08	0.01	0.25	0.24	0.05	0.03	0.13
		Error	530
		r	2 (whole model)		0.86	0.86	0.75	0.61	0.06	0.65	0.66	0.24	0.23	0.23
	5

Table 3 .

 3 Tukey post hoc significances p(F) for pairwise competitive model comparisons (unequal slope covariance analysis with average final 6 species richness and squared richness as covariates) for models without diffuse competition (δ = 0) with high migration probability and unlimited 7 dispersal (HU) and low migration probability and limited dispersal (LL). Normalized effects sizes (NES) entered the models as dependent 8 variable. First NES < second NES in white (black letters), first NES > second NES in grey with white letters.

	9
	10
	Comparison	Equiprobable null model			Proportional null model
		C-score	Clumping	β sim	NODF C-score Clumping	β sim	NODF
						HH
	Neutral -Transitive	0.02	0.001	<0.001 <0.001 <0.001	0.87	<0.001 0.005
	Neutral -Intransitive	<0.001	0.84	<0.001	0.11	<0.001	<0.001	0.21	0.002
	Transitive -Intransitive	<0.001	<0.001	<0.001 <0.001 <0.001	<0.001	<0.001	0.94
						LL
	Neutral -Transitive	<0.001	<0.001	0.007 <0.001 <0.001	<0.001	0.11	<0.001
	Neutral -Intransitive	<0.001	0.18	0.95	0.82	0.01	<0.001	0.80	0.07
	Transitive -Intransitive	<0.001	<0.001	0.004 <0.001 <0.001	0.28	0.04	0.02

Author-produced version of the article published in Oikos, 2017, 126 (1), 91-100.The original publication is available at https://onlinelibrary.wiley.com/ doi:10.1111/oik.03392

Acknowledgements

W.U. was supported by a grant from the Polish National Science Centre (2014/13/B/NZ8/04681).


P Jehel

A look into uncertainty in structural seismic performance arising from additional Rayleigh damping in inelastic time history analysis

Estimating the vulnerability of a structure subjected to a certain hazard, that is the probability that it would not meet a given performance objective for a given hazardous event, is a key step in an overall seismic risk management process. Structural performance indicators, coupled to hazard analysis, directly provide decision-makers with readable information: e.g. the risk for a structure to collapse within a certain number of years. This work focuses on the vulnerability assessment of reinforced concrete moment-resisting frame structures subjected to seismic hazard, based on inelastic time history structural analyses. Structural analyses entail numerous uncertainties that can compromise the reliability of its outcomes and, in turn, the assessment of the structural performance. The aim of this paper is to provide insight into the uncertainty in the structural seismic performance analysis that arises from the addition of viscous damping in the inelastic time history analyses and to compare it to the variability that comes from the ground motion. Latin hypercube samples of random variables are generated and the uncertainty is propagated to the outcomes of interest using Monte Carlo simulations.

INTRODUCTION

Background

In seismic performance-based design, engineering demand parameters (EDP) are computed for a set of ground motions and of structural models sometimes too, and then expressed in terms of performance criteria which can be easily interpreted by decisionmakers or other stakeholders. The concept of structural fragility, viz. the probability that a performance criterium will not be met for a given loading, is also useful for structural performance assessment [START_REF] Tantala | Earthquake loss estimation for the New York City Metropolitan Region[END_REF]. The reliability of a performance analysis depends on the uncertainty in the EDPs, and thus on the uncertainty in the variables involved in the structural analyses. [START_REF] Hwang | Generation of hazard-consistent fragility curves[END_REF] build seismic fragility curves of buildings accounting for several sources of uncertainty in the earthquake-site-structure system, among which the Rayleigh viscous damping ratio. The relative impact of each uncertainty source on the fragility curves is not discussed. Yet, estimating the sensitivity of the EDPs to the uncertain variables is particularly useful to identify which of them should be focused on to effectively reduce the overall uncertainty in the performance assessment.

Therefore, more recent studies have been dedicated to the analysis of the sensitivity of the EDPs to several uncertain variables pertaining to both the ground motion and the structural model. [START_REF] Porter | Sensitivity of building loss estimates to major uncertain variables[END_REF] present a simple methodology to evaluate the relative contribution of each variable to the overall performance uncertainty based on assembly-based vulnerability studies. What the authors call "pure" -also referred to as "intrinsic" -damping, that is damping not generated by hysteretic energy dissipation, is one of the uncertain variables. According to experimental data corrected to eliminate the effects of hysteretic damping in the measures, a coefficient of variation (COV) of the damping ratio is expected to be around 0.3 to 0.4. Then in the structural model, Rayleigh damping is used with uncertain critical damping ratio described by a Gaussian distribution centered at 5% and with a COV of 0.4 (there is no mention of whether the damping matrix is built from the initial or the tangent stiffness). It is finally concluded, for the high-rise nonductile reinforced concrete momentresisting frame building considered, that the most important contributions to the overall uncertainty in the model performance outputs are the fragility of the building components and the ground motion characteristics, whereas the contribution of viscous damping is shown to be significant too.

There are other studies of the sensitivity of performance outputs to structural and ground motion parameters: with focus on local output such as curvature [START_REF] Lee | Seismic demand sensitivity of reinforced concrete shear-wall building using FOSM method[END_REF] or on collapse assess-ment [START_REF] Liel | Incorporating modeling uncertainties in the assessment of seismic collapse risk of buildings[END_REF], in the context of incremental dynamic analysis [START_REF] Vamvatsikos | Incremental dynamic analysis for estimating seismic performance sensitivity and uncertainty[END_REF], where different methods for propagating the uncertainties are investigated (FOSM, Monte Carlo simulation), and where damping is sometimes considered as a source of uncertainty. Those papers contains other useful references which cannot all be cited here.

Objectives

This paper specifically focuses on Rayleigh damping because it is very commonly used in earthquake engineering. It is often expressed as a damping ratio ξ associated to two modal frequencies at certain times in the structural history. A first objective of this paper is to assess how the uncertainty on ξ propagates to the outcomes of interest of inelastic time history analysis (ITHA). In particular, we aim at assessing the sensitivity of the model performance outputs to ξ, and at quantifying the contribution of an uncertain ξ to the overall dispersion in the outputs. The other source of uncertainty considered is the ground motion, with record-to-record and peak acceleration variability.

When performing ITHA, Rayleigh damping not being added to solely convey intrinsic structural properties ("pure" damping) yields another objective for this paper. As stated in FEMA P692 (2009, §6.4.4), "additional viscous damping may be used to simulate the portion of energy dissipation arising from both structural and nonstructural components that is not otherwise incorporated in the model". For instance, in [START_REF] Zona | Nonlinear seismic response analysis of steel-concrete composite frames[END_REF], Rayleigh damping is added in the nonlinear seismic analysis of steel-concrete composite frames to account for energy dissipation due to friction between steel beams and concrete slabs, which indeed is not explicitly introduced in the inelastic structural model due to a lack of quantitative information about this mechanism. Another objective is thus to investigate the sensitivity to the variable ξ of the energy absorbed by either the structural model or the additional Rayleigh damping model.

Outline

In the next section, we clarify the dual very nature of the structural and additional damping models and recall some basics of the state of the practice for using Rayleigh damping in ITHA. Section 3 is devoted to the description of the random variables we introduce to account for uncertainty in both the seismic loading and the Rayleigh damping model. Then, in a fourth section, the deterministic test structure used in the estimation of the effects of the uncertain variables in the ITHA outputs is presented. This is a reinforced concrete moment-resisting frame that was tested on a shaking table. Its numerical inelastic structural model is briefly presented and a deterministic analysis is run to assess the capability of this model to retrieve displacement-based performance as well as energy quantities observed during the experimental test. Then, Latin hypercube samples of the random variables are generated and Monte Carlo simulations are performed to propagate the uncertainty. Results of various uncertainty and sensitivity analyses are shown and discussed in section 5. The main results of this work are summarized in the concluding section.

RAYLEIGH DAMPING IN INELASTIC TIME HISTORY ANALYSIS

Rational definition for damping

For earthquake engineering applications, a general form of the models used to predict a set of outputs b from both a set of inputs a and a set of parameters θ can be given as (see figure 1):

b = M(a, θ) = K(a K , θ K ) + D(a D , θ D ) (1)
where K represents the structural model and D is an additional damping model commonly used in practice to convey, at the structural level, the effects of all the energy dissipative mechanisms not otherwise explicitly accounted for in K. The discretized structural model K takes the form:

(K) f kin ( dn ) + f int (d n ) = f sei (ü sei n ) + f sta
where f kin , f int , f sei and f sta are the inertia, internal, seismic, and static forces vectors. d is the discrete displacements vector.

Adding damping to K defines the model M as:

(M) f kin n + f int n + f dam n (?) = f sei n + f sta (2)
where f dam is the additional damping force vector.

In most of the practical applications, the damping is hysteretic and/or viscous, thus ? = d n or ? = ḋn , but it could take any other form, as long as equation ( 2) can be effectively resolved. Among the models K and D, only the structural model K can genuinely be qualified as "physical". Indeed, it is developed in consistence with the theory of continuum mechanics. However, there is no such a well-established theory which damping model D could be developed in. Nevertheless, equation (2) defines a relation of duality between models K and D which serves as the baseline for the following definition of a physical damping model: a model that completes the structural model for a reliable prediction of the model M outcomes of interest.

Rayleigh damping model

One now has to choose a damping model that would satisfy the rational definition above. In most of the applications in earthquake engineering, Rayleigh damping is chosen, that is, in the most general form:

f dam = C(t) ḋ(t) with C(t) = α(t)M + β(t)K(t) (3)
where M and K are the mass and stiffness matrices and (α, β) are computed according to both two modal frequencies at certain times in the structural history and a targeted damping ratio ξ. Whether the initial or the tangent stiffness matrix is used and whether α and β are taken as time-dependent or not has to be decided regarding the consequences those assumptions imply (Charney 2008, among others).

Some other practical aspects of Rayleigh damping: (i) Adding a small amount of damping, say ξ < 0.5%, is often necessary to guarantee the good convergence of the overall resolution algorithm; (ii) Usual practice for reinforced concrete low-rise frames consists in targeting a damping ratio of about 5% for the dominant structural modes. This is larger than the 0.5% to 1.5% of "pure" damping classically observed for low-amplitude oscillations, which implies that using Rayleigh damping for representing mechanisms "not otherwise incorporated in the model" or poorly described by the structural model is common practice in ITHA.

RANDOM VARIABLES

Ground motion dataset

To account for record-to-record variability of the seismic loading, we select a dataset in the PEER ground motion database with its Web application (PGMD 2011), so that the selected ground motions are consistent with i) the likely earthquake scenarios and ii) the soil properties in the region where the design ground motion was recorded, viz. the Cascadia subduction zone, and iii) exhibit good match with the target design spectrum in the most important period range.

The Cascadia subduction zone is an active seismic area that has been observed for several decades [START_REF] Silva | Assessing earthquake hazards and reducing risk in the Pacific Northwest[END_REF], Wiest et al. 2007, among others) and it has been reported that three types of earthquakes are produced in this zone: (i) Shallow crustal earthquakes due to surface faults in the American continental plate with magnitude M w larger than 7.0 and hypocenter depth less than 30 km;

(ii) Thrust interplate or interface earthquakes generated by differential motion in the interface between the two plates. Such earthquakes happen offshore with surface hypocenter, generally with depth less than 30 km. In the Cascadia subduction zone, a large event of this kind is likely to be produced with magnitude M w = 8.3 ± 0.5; (iii) Intraplate or intraslab earthquakes occur deep within the Cascadia subduction zone (depth > 40 km). For depth larger than 100 km, no seismic activity has been observed.

As far as the soil properties in this region are concerned, the geologic profile of the station where the design ground motion was recorded is detailed in table 1 [START_REF] Silva | Assessing earthquake hazards and reducing risk in the Pacific Northwest[END_REF]. The PEER database is currently limited to recorded time series from shallow crustal earthquakes. The initial acceptance criteria used for the search of ground motions are as follows: moment magnitude M w ∈ [7.0, 9.0], Joyner-Boore distance R JB ∈ [0, 150] km, closest distance to rupture plane R rup ∈ [0, 150] km, and average shear wave velocity in the top 30 meters of the site v s30 ∈ [0, 200] m/s. With these criteria, approximately 20 earthquakes were identified. To increase the number of different recoding stations, we proceeded to a second search with the range of allowed moment magnitudes extended to M w ∈ [6.0, 9.0] and finally selected 20 ground motions.

Equal probability is attributed to every ground motion in the dataset.

Peak acceleration of the ground motions

Displacement-based EDPs such as the interstory drift generally strongly depends on the horizontal peak acceleration of the ground motion. Besides, probabilistic seismic hazard often is expressed in term of PGA or spectral acceleration in a given geographical region. In other words: uncertainty in PGA is both an accessible data and expected to cause uncertainty in the EDPs. We therefore add PGA as another source of uncertainty in our analysis.

Every ground motion is first normalized to PGA = 1 g and then scaled to a value between 0.1 and 0.4. The scale factor is random and computed as:

λ a = 0.1 + 0.05(N a -1) with N a ∼ dU(1, 7) (4)
where dU(n), n ∈ N, is the discrete uniform probability density function.

If our objective were to perform a seismic risk analysis and not only a structural performance study, seismic hazard curves should be used here instead of a uniformly distributed scale factor. For the sake of illustration, we mention that according to the 1985 seismic zoning map of the National Building Code of Canada, ground motions with horizontal PGA between 0.16 g and 0.23 g is likely to be observed with a probability of exceedance of 10% in 50 years in the Canadian Cascadia subduction zone.

Target damping ratio ξ

The last random variable considered is the target damping ratio ξ characterizing the amount of Rayleigh damping in the simulation (sub-model D). This variable can both have an influence on the performance EDPs (outcomes of model M) and on the behavior of the interrelated structural sub-model K.

The two damping ratios needed to build the Rayleigh damping matrix are taken as equal in the range [1%, 10%] without assuming any knowledge on the actual value:

ξ[%] = 1 + 9 50 (N ξ -1) with N ξ ∼ dU(1, 50) (5)
Rayleigh damping can be introduced either at the level of the whole structure or at the level of the structural elements. The second option should be retained because there is no reason to assign the same damping ratio to structural elements which only have few in common. In other words, the hypothesis of perfect spatial correlation of the damping ratios does not hold. Indeed, as already mentioned, we do not limit our analysis to the sub-model D solely representing "pure" damping but also as an ad hoc source of energy dissipation in addition to the energy dissipation coming from the nonlinear mechanisms in the sub-model K. If only "pure" damping were at stake, this would not be worth considering this correlation except for composite structures composed by elements made of different materials. The analysis outcomes can be affected by this obvious correlation between damping ratios pertaining to different structural elements. As previous evidence of the potential effects of space correlations in ITHA performance outputs, one can cite the work of [START_REF] Stefanou | Nonlinear dynamic analysis of frames with stochastic non-gaussian material properties[END_REF].

DETERMINISTIC NUMERICAL INELASTIC SIMULATION

Description of the structure

The test structure considered to estimate the effects of the selected random variables on the performance outputs of ITHA is depicted in figure 2. It was designed at a reduced scale of 1/2 according to the provisions of the National Building Code of Canada and of the Canadian concrete standard. The structure was assumed to have a nominal ductility, that is a design base shear computed with a reduction factor R = 2. Four concrete blocks were used to simulate concentrated gravity loads in every beam span; they were designed so that their center of gravity coincide with the center of gravity of the supporting beam. These latter additional masses induced service cracks. The total weight of the frame was 95 kN, its fundamental period T 1 = 0.36 s was measured preliminary to the seismic excitation. Mode 1 is preponderant in the sense that it accounts for 91% of the total mass. The detailed presentation of the design assumptions and parameters can be found in [START_REF] Filiatrault | Seismic performance of ductile and nominally ductile reinforced concrete moment resisting frames. I. Experimental study[END_REF]. The N04W component of the accelerogram recorded in Olympia, Washington on April 13, 1949 was selected for the test program and scaled to a peak ground acceleration PGA = 0.21 g. The feedback record of the acceleration measured on the shaking table during the test is shown in figure 3 along with the corresponding elastic response spectrum with 5% viscous damping ratio.

Structural model K

Fiber beam/column elements implemented in the framework of a displacement-based finite element method with Euler-Bernoulli kinematics are used to model the response of the structural elements of the frame. The finite element mesh of the RC frame is shown in figure 4. The structure is assumed fixed at its base. Rigid end zones are added to model the beamto-column connections and rebar slip in surrounding concrete is not represented. These later hypotheses are questionable because the beam-to-column connections exhibited inelastic behavior during the test.

The uniaxial constitutive laws used to model steel and both confined and unconfined concrete fibers are also shown in figure 4. The inelastic model used to represent the salient features of the materials cyclic behavior has been developed in the framework of thermodynamics with internal variables and is presented in [START_REF] Jehel | Towards robust viscoelastic-plastic-damage material model with different hardenings / softenings capable of representing salient phenomena in seismic loading applications[END_REF]. The set of parameters has been identified to fit experimental monotonic curves. From the computational point of view, because hardening and softening laws are modeled as linear, there is no need for local iteration when the internal variables -the memory of the materials -are updated, except for transitions between hardening and softening regimes, which makes the resolution procedure robust and efficient. The loading time history consists in two phases: i) static dead load is first applied step by step and then kept constant; ii) the seismic loading is applied. A first validation check of the inelastic structural model is carried out by simulating a free vibration test. Before dead load is applied, the elastic fundamental period of the structure is computed as T ela 1 = 0.28 s; then, due to the inelastic behavior of the structure, the elongated period is evaluated as T ini 1 = 0.36 s when dead load is completely applied. Both T ela 1 and T ini 1 coincide with the experimental values.

Rayleigh damping model D

For all the ITHA that will be run in the following, the Rayleigh damping model is build with tangent stiffness matrix and designed according to the procedure presented in [START_REF] Jehel | Initial vs. tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses[END_REF]) so that the effective modal damping ratios remains as close as possible to a targeted damping ratio ξ while structure deteriorates.

Energy quantities

The following energy quantities will be considered:

Hysteretic energy: E H = T 0 Ω σ : ǫ dΩ dt; Damping energy: E D = T 0 f dam • u dt; Total energy: E tot = E H + E D .

Numerical vs. experimental results

For this modeling of the experimental test, a targeted damping ratio ξ = 2% is used. For this simulation as well as for all the simulations in the next section, Newmark β-method with γ = 0.5 and β = 0.25 with a time step of 5.10 -3 s is employed. Computed displacement-based performance outcomes and energy quantities are summarized and confronted to experimental data in table 2 and figure 5. num.

E H + E D exp. E H E D

UNCERTAINTY AND SENSITIVITY ANALYSES

The structural model will remain unchanged all along the following uncertainty and sensitivity analyses and, consequently, all the probabilistic quantities hereafter discussed have to be understood as conditional to the model K.

Uncertainty analysis

Two Latin hypercube samples S 1 and S 2 of size N = 1000 of the random variables are generated from the probability spaces P 1 and P 2 [START_REF] Mckay | A comparison of three methods for selecting values on input variables in the analysis of output from a computer code[END_REF]).

The random parameter for ground motion recordto-record variability is denoted as GM#. The events in P 1 , resp. in P 2 , are the triplets (GM#, PGA, ξ), resp. the (N e +2)-tuple (GM#, PGA, [ ξe , e = 1, .., N e ])

where N e is the number of elements in the finite element mesh of the modeled structure. The random variables are considered as independent, although attenuation laws could for instance provide indication about how the distance to rupture plane -the GM#, in turn -is correlated to PGA. Also, P 1 describes perfectly spatially correlated damping ratios, whereas P 2 accounts for spatially independent damping ratios.

Uncertainty is propagated to the model outputs with Monte Carlo simulations. The uncertainty analysis of the maximum interstory drift ratio (MISDR) is shown in figures 6 for sample S 1 and in figures 7 for sample S 2 (figures on next page). The distributions are slightly different and, in the following, only the results obtained with sample S 1 will be discussed.

MISDR sensitivity

From figure 8 (figure on last page), the following trends can be observed: i) The absolute variability increases with PGA; ii) The variability in MISDR due to the Rayleigh damping ratio is generally much less significant than the variability due to PGA, except for some records (e.g. GM# = 8 or GM# = 17).

Figures 9, 10 and 11 also provide insight into the sensitivity of MISDR to the three random variables considered in this study. The clear linear correlation between MISDR and P GA, which is usually assessed in the preliminary study to structural vulnerability analysis, is also observed here in figure 9. Indeed, the mean of MISDR linearly increases with PGA, and the relatively small COV(MISDR|PGA) -compared to the overall COV observed in figure 6 -provides good confidence in the computed means. COV(MISDR|PGA) being approximately half COV(MISDR) also shows that PGA is an important contributor to the overall uncertainty in MISDR. Figures 10 and11 show that GM# and ξ less importantly contribute to the uncertainty in MISDR. Same conclusions are drawn in the studies of [START_REF] Porter | Sensitivity of building loss estimates to major uncertain variables[END_REF] and [START_REF] Lee | Seismic demand sensitivity of reinforced concrete shear-wall building using FOSM method[END_REF].

Focussing now on figure 11, one may jump to the following intuitive conclusion when looking at the mean E(MISDR| ξ): the larger the Rayleigh damping ratio, the smaller the MISDR. Although actually a general trend, this is however to be taken with caution for two reasons: i) the number of data computed for a given ξ is around 20 only, and ii) the large dispersion of the data (COV(MISDR| ξ) is large) entails small confidence in the mean.

Energy sensitivity

The observation of energy quantities is used to provide insight into the sensitivity of the interrelation between models K and D to the random variables. The mean values E(E tot | ξ) in figure 12 shows that the total amount of energy absorbed by the model M is weakly correlated to the Rayleigh damping ratio (E(E tot ) = 2408 N.m for sample S 1 and to 2364 N.m for S 2 ). This again has to be taken with caution because the means are computed with only 20 values that, besides, exhibit large scatter (COV(E tot | ξ) is to be compared to the overall COV(E tot ) which is equal to 0.98 for sample S 1 and to 1.01 for S 2 , which also shows that the main contribution to the overall variability of E tot comes from the ground motion). Nevertheless, the small influence of ξ on E tot is in accordance with the observations of [START_REF] Léger | Seismic-Energy Dissipation in MDOF Structures[END_REF]. With much less scatter (approximately 5% < COV(DER| ξ) < 25%), figure 13 shows that the share of the total energy E tot absorbed by model D tends to increase with ξ.

CONCLUSIONS

In this paper, the impact of random variables on the performance outcomes of seismic inelastic time history analyses of a reinforced concrete momentresisting frame is investigated. The random variables considered describe uncertainty in the seismic ground tio can be a very important contributor to the variability in MISDR; iii) the correlation between the total amount of energy absorbed by the model and the Rayleigh damping ratio tends to be weak; iv) the share of the total energy absorbed by the damping model tends to increase with the Rayleigh damping ratio.

Figure 1 :

 1 Figure 1: System for earthquake engineering applications. The model M allows for predicting a set of outcomes b from a set of inputs a and of parameters θ. M results from the interaction between structural and damping sub-models K and D.

Figure 2 :

 2 Figure 2: RC frame structure tested on the shaking table at École Polytechnique in Montreal. Dimensions are in [mm].

Figure 3 :

 3 Figure 3: Acceleration time history recorded on the shaking table during the test and corresponding elastic response spectrum with a critical viscous damping ratio of 5%.

Figure 4 :

 4 Figure 4: Finite element mesh (dimensions in [mm]) and material constitutive laws for the inelastic structural modeling. [bottom left] Confined and unconfined concrete behavior laws. [bottom right] Steel constitutive law.

Table 2 :

 2 Experimental and numerical results. M RD is the maximum roof displacement, M ISDR the maximum inter-story drift ratio, and E tot the total energy absorbed by the system. MRD (mm) MISDR (%) E tot (N.m

Figure 5 :

 5 Figure 5: Comparison of experimental and numerical results. E H is the energy absorbed by the sub-model K, E D is the energy absorbed by the additional Rayleigh damping sub-model D designed with targeted damping ratio is ξ = 2%.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: M ISDR w.r.t. the 7 PGA levels considered in the analysis. Scatter is due to the variability in GM and ξ.

Figure 6 :Figure 7 :

 67 Figure 6: Distribution of the maximum interstory drift, as well as its mean and COV convergence assessment for sample S 1 .

Figure 12 :

 12 Figure 12: Total energy absorbed by the model M throughout ITHA w.r.t. increasing Rayleigh damping ratio. Scatter is due to the variability in GM and PGA.

Figure 8 :Figure 13 :

 813 Figure8: M ISDR w.r.t. the 140 ground motions considered in the analysis. The figure can be read as follows: GM# is indicated on the x-axis; between two successive GM#, a right shift indicates a PGA increase of 0.05 g; then, the scatter following any vertical line is due to the sole variability in ξ.

Table 1 :

 1 Geologic profile for the Olympic Highway Test Lab strong-motion recording site in Washington. v s is the shear wave velocity and ρ the density.

	Depth Geology	Description	v s	ρ
	(m)			(m/s) (kg/m 3 )
	0-3	Fill	Loose sand.	165	1500
			Medium dense
	3-12 Deposits	fine to	220	1500
			medium sand.
			Interbedded
			very stiff to hard
	12-20 Deposits	sandy silt and	270	1500
			very dense silty fine
			to medium sand.
	20-41 Deposits	(Like just above.)	330	1500

ACKNOWLEDGMENTS

This research is supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme (proposal No. 275928). I thank Prof. Deodatis for valuable discussions during the preparation of this work.


Joyee Sourya

Abdessamad De

Imine

On Consent in Online Social Networks: Privacy Impacts and Research Directions

Keywords: Online Social Networks (OSN), privacy, consent, GDPR, privacy risk

The EU General Data Protection Regulation (GDPR) recognizes data subject's consent as a legitimate ground of data processing. At present, consent mechanisms in OSNs are either non-existent or not GDPR compliant. While the absence of consent means a lack of control of the OSN user (data subject) on his personal data, non-compliant consent mechanisms can give them a false sense of control, encouraging them to reveal more personal data than they would have otherwise. GDPR compliance is thus the only way to obtain meanigful consents, thereby protecting user privacy. In this paper, we discuss the characteristics of valid consent as per the GDPR, analyze the present status of consent in OSNs and propose some research directions to arrive at GDPR compliant consent models acceptable to users and OSN providers (data controller). We observe that evaluating privacy risks of consents to data processing activities can be an effective way to help users in their decision to give or refuse consents and hence is an important research direction.

Le consentement dans les réseaux sociaux : Impacts sur la vie privée et axes de recherche

Résumé : En vigueur récemment dans l'Union Européenne, la Réglementation Générale de la Protection de Données (RGPD) stipule que le consentement d'un utilisateur est un préalable à tout traitement de données le concernant. Actuellement, les mécanismes de consentement dans les Réseaux Sociaux (RS) sont soit inexistants, soit non conformes à la RGPD. L'absence du consentement signifie clairement la perte de contrôle de l'utilisateur sur ses données personnelles. Quand le consentement est non conforme, il donne un faux-semblant de contrôle et encourage l'utilisateur à divulguer plus d'informations sensibles que nécessaire. La conformité à la RGPD est donc le seul moyen de manipuler des consentements clairs et contribuant à la protection de données personnelles. Dans ce rapport, nous discutons des caractéristiques du consentement valide selon la RGPD, nous analysons la conformité des consentements existants dans les RS et nous proposons des axes de recherche pour concevoir des modèles de consentement conformes et acceptables par les usagers et les fournisseurs des RS. Pour assister l'utilisateur à prendre les « bons » consentements aux différents traitements de données dans les RS, l'évaluation des risques sur la vie privée est un important axe de recherche.

Mots-clés : Réseaux sociaux en ligne, vie privée, consentement, RGPD, Risques sur la vie privée

Introduction

The EU General Data Protection Regulation (GDPR) [4], which has come into force across Europe from 25th May 2018, recognizes consent of data subject as a legitimate ground of data processing. The central aim of promoting the notion of consent is to provide data subjects control over their personal data. Today, users reveal a wide variety of personal data in Online Social Networks (OSNs), not only to the OSN provider (the data controller) but also to third party applications and other users. OSNs involve a wide variety of data processing activities, such as face recognition and friend suggestion. At present, consent mechanisms for these processing activities are either nonexistent or not GDPR compliant. This situation, in the absence of other legitimate reasons for data processing, is undesirable. The absence of consent means a lack of choice and consequently, a lack of control of the data subject on how his data is processed. Non-compliant consent mechanisms can give users a false sense of control, encouraging them to reveal more personal data than they would have otherwise [START_REF] Solove | Privacy self-management and the consent dilemma[END_REF]. Data controllers, third parties and/or governments may misuse the huge data repository accumulated by OSNs for surveillance, profiling leading various forms of discrimination and other privacy harms. Other malicious entities can misuse the data for identity theft, stalking, shaming, defamation and bullying, to say the least. On top of this, non-compliant data controllers are subject to hefty fines by the GDPR and may suffer loss of reputation among an increasingly privacy-conscious population when privacy harms come to light.

Quitting OSNs to protect one's privacy is not an effective measure. OSN providers can still possess the data of past members and collect information about non-users. Moreover, users derive various social benefits from OSNs, such as establishing new friendships and reviving and strengthening existing ones [START_REF] De | To Reveal or Not to Reveal -Balancing User-Centric Social Benefit and Privacy in Online Social Networks[END_REF]. Therefore, the right approach to ensure user privacy is for data controllers to be compliant to privacy regulations and compliance of consent mechanisms constitute a major step in this direction. Achieving consent compliance is not an easy task in OSNs. In OSNs, often, user privacy depends both on the data subject's consent and other users' actions. Users face many cognitive and structural challenges in their decision to give or refuse consent [START_REF] Solove | Privacy self-management and the consent dilemma[END_REF]. Moreover, a compliant consent mechanism must have certain characteristics and to be practical, must be acceptable to both users and OSN providers. These issues open up new research directions to be explored. Privacy risk evaluation of consents for data processing activities can be an effective way to help users to decide whether to give or refuse consents and is an important research direction. The GDPR encourages informing users about the privacy risks of personal data processing.

In this paper, we first discuss what constitutes a "valid consent" according to the GDPR in Section 2 and then, in Section 3, we analyze the current state of consent in OSNs, with a focus on examples from Facebook, the leading OSN provider. Finally, in Section 4, we propose some research directions to arrive at GDPR compliant consent models acceptable to both OSN providers and users.

In the context of OSNs, users are the "data subjects", the OSN service provider is the "data controller " and third party application providers are the "third parties" as defined in the GDPR. We also use the terms "personal data" and "data processing" in the sense of the GDPR.

Valid Consent as in the GDPR

A valid consent, as in Article 4 of the GDPR, must be 1) freely given, 2) specific, 3) informed and 4) obtained by a clear affirmative action of the data subject. Below, we briefly describe these characteristics.

Freely given. The WP29 guideline on consent [START_REF]Article 29 Data Protection Working Party[END_REF] describes how to assess whether a consent is indeed freely given using the following criteria: 1) the relationship between the data controller and the user, 2) the conditionality and 3) the granularity of the consent and 4) if the withdrawal of consent is detrimental for the user. It is unlikely that consent is freely given if there is a power imbalance between the data controller, such as an employer, and the data subject who may fear significant negative consequences if he does not give consent. If the performance of a contract is conditional on the consent to data processing not necessary for the execution of the contract, then consent is not freely given. If a data controller seeks the consent for several purposes bundled together, then it lacks granularity as the data subject does not have the freedom to give or deny consent for each purpose. Refusal or withdrawal of consent must not be detrimental to the user. If a user gives consent to Facebook's facial recognition feature meant for tag suggestions, detection of fake accounts etc., it is not "freely given" as the data subject has to accept all purposes even if he finds only one of them acceptable. Specific. When several data processings have the same purpose, consent may be sought for all of them together. However, if a data processing has multiple purposes, then consent must be sought for each of them. Specificity of consent promotes transparency as the data subject knows about each purpose of data processing, increases his control over these purposes and safeguards against function creep. Facebook's facial recognition feature does not allow users to give "specific" consent as the provider does not ask for consent for each purpose. Informed. To really enable data subjects to understand what they are consenting to and to exercise their rights, such as the right to withdraw consent, the data controller must provide, in plain and clear language, a minimal set of information including its own identity, the purposes of processing and the data that are to be collected and used. Informed consent thus promotes transparency. Clear affirmative action. Data subjects must give consent in an active motion or declaration. Thus, opt-out and pre-ticked opt-in boxes in consent forms are invalid under GDPR. For example, Twitter requires users to uncheck pre-ticked boxes to opt-out of targetted advertising, making the resulting consent invalid. Silence, inactivity or simply proceeding with a service without any action are not consents.

Consent in Online Social Networks: Present Status

In OSNs, personal data is transferred from the data subject to the data controller (OSN provider), or third party application providers or other data subjects. These data transfers can be summarized into: 1) data subject-data controller interaction, 2) data subject-third party interaction and 3) data subject-data subject interaction. The last type of interaction is unique to OSNs. Several data processing activities can be related to each interaction and, according to the GDPR, all such activities require user consent to be legitimate. At present, such consent mechanisms in OSNs are either non-existent or if they exist, are mostly not GDPR compliant. In the following sub-sections, for each interaction, we discuss the current state of consent in OSNs. Facebook is a leading OSN provider, which has been questioned over the years by regulators and privacy advocates about its privacy practices [START_REF]Mark Zuckerberg Testimony: Senators Question Facebook's Commitment to Privacy[END_REF]. It is yet to be seen how Facebook manages to comply with the GDPR. With this in mind, we focus mainly on Facebook for examples of consent mechanisms.

Data subject-data controller interaction. Targetted advertising, based on personal data available from user profiles and activities or from external sources, is the primary source of revenue for OSNs today. These advertising platforms can be exploited to cause privacy problems, such as inferring users' full phone numbers just from the knowledge of their e-mail addresses [START_REF] Venkatadri | Privacy Risks with Facebook's PII-based Targeting: Auditing a Data Broker's Advertising Interface[END_REF].

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In response to the GDPR, Facebook has recently started seeking consent on whether to include data from its partners such as other websites to show advertisements, but it gives no options to users to say no to targetted advertisements. In contrast, Snapchat and Twitter have enabled their users to opt-out of targetted advertising [START_REF] Sweeney | GDPR and the major social networks: what you need to know[END_REF]. However, opt-out and pre-ticked opt-in boxes in consent forms, under the GDPR, do not constitute valid consent.

Facebook's face recognition feature can be used to scan the profile picture and other photos and videos of a user to compute a template which can then be matched with other photos and videos in Facebook. It serves a few purposes: 1) suggest other users to tag the data subject in photos they have uploaded, 2) notify the data subject about photos and videos they appear in but have not been tagged, 3) detect fake profiles that use the data subject's photos and 4) identify people in photos they are not tagged in for people who use a screen reader. The current consent mechanism allows the user to give consent for all purposes or none at all, thus lacking in specificity and granularity. Users may be interested to enable the facial recognition feature for detecting fake profiles but not for photo tagging. So, refusal of consent leads to losing out on otherwise useful service(s) and hence is detrimental to the user.

Data subject-third party interaction. When using third party applications on the OSN platform, users often have little comprehension of how much data they are sharing, with whom and that they are also responsible for sharing their friends' information [START_REF] Hull | Contextual Gaps: Privacy Issues on Facebook[END_REF]. Recently, an app called "thisisyourdigitallife" was used to gather the personal data of its users and their friends in the disguise of a paid psychological test on Facebook. This data was then shared with Cambridge Analytica which may have used it to influence choices in elections. Such data sharing, without valid user consent, may lead to many harms like loss of jobs and insurance and suppression of free speech [START_REF] Fruchter | Facebook/Cambridge Analytica: Privacy lessons and a way forward[END_REF]. To use third party gaming applications on Facebook, users must agree to disclose personal data such as their public profile, name, e-mail address, date of birth, friend list etc., by clicking on a "Play Now" button. Although Facebook provides an explanation below this button that clicking it implies that the user agrees to disclose a list of personal data, the positioning of this explanation and the bundling of the actions of playing the game and giving consent may be misleading for data subjects. It is questionable whether clicking on the "Play Now" button is indeed a "clear, affirmative action" of the data subject. Moreover, satisfactory information about the purposes of data processing is not always available to the users. Therefore, the consent given in this case appears to be neither fully "informed" nor "specific".

Data subject-data subject interaction. In Facebook, even if a user does not consent to sharing some personal data, the actions of other users can reveal this data. Wall posts or comments by one user may contain personal data of others. A friend may make public a user's wall posts originally meant only for friends. Another relevant scenario is where user A wants to upload a photo including his friend user B on Facebook, but user B does not. It is also possible to infer personal data, not disclosed by the data subject, from that revealed by his network, i.e., his friends, friends-of-friends and groups [START_REF] Zheleva | To Join or Not to Join: The Illusion of Privacy in Social Networks with Mixed Public and Private User Profiles[END_REF]. Facebook privacy settings, which is a consent mechanism meant to enable users to control the personal data they share with their network, cannot deal with these scenarios.

'People You May Know" is an important Facebook feature that opens up the scope of data subject-data subject interaction by allowing one user to easily access the personal data of another. It suggests a user A's profile to another user B based on their mutual friends, common groups, networks (such as school, university, work) and uploaded contacts. In its current form, Facebook does not seek consent from A before suggesting him as a potential friend to B, increasing the accessibility of A's profile to B without A's consent. Like NewsFeed, this feature does not make more information publicly available, but makes it easier for a potential misuser to get to this information [START_REF] Hull | Contextual Gaps: Privacy Issues on Facebook[END_REF].

Research Directions Towards Consent Compliance

While the lack of a consent mechanism deprives users of control over their personal data, a non-compliant consent mechanism can give users a false sense of control encouraging them to reveal more data than they would have otherwise. So, the research community must focus on converging towards GDPR compliant consent models that are acceptable to both users and service providers. To this end, we propose some research directions in this section. These research efforts would be multi-disciplinary in nature, involving the contributions of computer scientists, legal experts and psychologists.

Privacy Risk Evaluation for "Well-informed" Consent. A consent mechanism presents users with a set of choices. To help them make the right decision, the data controller should provide information about the data processing activity, its purpose and the personal data involved. However, there is a trade-off between in-depth, meaningful information and short, simple information that can be easily read and understood by an ordinary data subject. Users often do not read long privacy notices, yet they need deeper understanding and background to make informed choices. Even if they read, they lack the expertise to understand the consequences of their consents and are often ready to give up on privacy for small, immediate benefits [START_REF] Solove | Privacy self-management and the consent dilemma[END_REF]. In the Cambridge Analytica scandal, people gave up their own personal data and that of their friends in exchange for small monetary benefits. These data ended up being used to influence election results, a long-term harm for the society.

One way to address these problems in OSNs could be to design a tool that can assess on behalf of the user both long-term and immediate privacy risks of each consent. Usability surveys can help to construct the right interface for communicating these privacy risks to users. This risk information will be short, simple but concrete enough for the user to get a view of the consequences of their consent and help them arrive at a decision. In other words, it greatly enhances the "informed" characteristic of a valid consent. The GDPR, in its Recital 39, promotes this idea of making users aware of risks of personal data processing. Already, privacy risk evaluation has been utilized for other services to help users make meaningful choices of privacy settings [START_REF] De | Privacy risk analysis to enable informed privacy settings[END_REF].

Inter-provider and Intra-provider Risk Evaluation. Several pieces of data, aggregated over time, can lead to various privacy harms [START_REF] Solove | Privacy self-management and the consent dilemma[END_REF]. An average user may engage with several OSNs (not to mention other services) each of which usually involves several data processing activities.

To truly avoid all privacy harms, the decision to consent for a data processing activity must depend on the consents already given to other data processing activities. Thus, researchers should focus on designing intra-provider and eventually inter-provider privacy risk evaluation mechanisms that take into account personal data revealed for all data processing activities for a given OSN and those for all OSNs that a data subject uses, respectively.

Balancing Privacy Risks and Social Benefits. Users have to deal with several cognitive and structural hurdles to arrive at meaningful decisions regarding consents to data processing activities [START_REF] Solove | Privacy self-management and the consent dilemma[END_REF]. Apart from privacy risks, they must also consider various social benefits, such as building new friendships and reviving and maintaining existing ones [START_REF] De | To Reveal or Not to Reveal -Balancing User-Centric Social Benefit and Privacy in Online Social Networks[END_REF], for which they participate in OSNs. Automating the decision-making process of balancing privacy risks and social benefits for all data processing activities can take the burden off users. A similar approach was adopted recently to help users manage the privacy settings of OSN attributes [START_REF] De | To Reveal or Not to Reveal -Balancing User-Centric Social Benefit and Privacy in Online Social Networks[END_REF].

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Collaborative Consent and Risk-based Friend Selection. While the OSN provider is obligated to obtain valid consent to process data in the absence of another legitimate basis, other users are under no legal obligations to obtain the data subject's consent before sharing his personal data. Friends of a user may publicly share data that were meant to be seen only by friends or post photos of the user without the latter's consent. Potential misusers may also infer personal data of the user from the data revealed by his friends about themselves.

The design of collaborative consent mechanisms could be a fruitful research direction in this context. Using such mechanisms, a user and his friends can together decide which personal data they share and to what extent (i.e., keep it private, or share with friends, friends-of-friends or strangers) so that it is difficult to infer with high confidence any personal data that has been kept private. These mechanisms could also be used to resolve data disclosure scenarios where the user sharing some data is not the data subject (for example, a photo of A shared by B) or is only one of the data subjects (for example, a photo of A and B shared by B) [START_REF] Squicciarini | CoPE: Enabling collaborative privacy management in online social networks[END_REF]. Another solution approach could be to design mechanisms that enable users to choose friends based on the privacy risks they pose, both in terms of unintended disclosure and the inference of personal data from those revealed by the friends about themselves.

Conclusion

At present, consent mechanisms in OSN are either absent or are not compliant to the newly enforced GDPR. Such a scenario poses severe privacy problems for data subjects as the latter have no true control on their personal data. Consent compliance is an important step towards protecting user privacy. In this paper, we discussed the characteristics of valid consent according to the GDPR, analyzed the present status of consent in OSNs and proposed some research directions to arrive at GDPR compliant consent models that are acceptable to OSN users and OSN providers.

RR n°9197

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