Marc-Thorsten Hütt

Dieter Armbruster
email: dieter@asu.edu

Annick Lesne
email: annick.lesne@sorbonne-universite.fr

Predictable Topological Sensitivity of Turing Patterns on Graphs

Reaction-diffusion systems implemented as dynamical processes on networks have recently renewed the interest in their self-organized collective patterns known as Turing patterns. We investigate the influence of network topology on the emerging patterns and their diversity, defined as the variety of stationary states observed with random initial conditions and the same dynamics. We show that a seemingly minor change, the removal or rewiring of a single link, can prompt dramatic changes in pattern diversity. The determinants of such critical occurrences are explored through an extensive and systematic set of numerical experiments. We identify situations where the topological sensitivity of the attractor landscape can be predicted without a full simulation of the dynamical equations, from the spectrum of the graph Laplacian and the linearized dynamics. Unexpectedly, the main determinant appears to be the degeneracy of the eigenvalues or the growth rate and not the number of unstable modes.

INTRODUCTION

Turing's idea [START_REF] Turing | The chemical basis of morphogenesis[END_REF] of pattern formation through the interaction of chemical species via nonlinear reactiondiffusion equations had an enormous impact on the biology of morphogenesis [START_REF] Gierer | A theory of biological pattern formation[END_REF][START_REF] Green | Positional information and reaction-diffusion: two big ideas in developmental biology combine[END_REF]. It stood as an emblematic example of a dissipative structure in physics and chemistry [START_REF] Prigogine | Dissipative structures in chemical systems, in Fast reactions and primary processes in chemical kinetics[END_REF][START_REF] Walgraef | Spatio-temporal pattern formation: with examples from physics, chemistry, and materials science[END_REF][START_REF] Goldbeter | Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves[END_REF] and offered a mathematical laboratory for the investigation of spatially extended instabilities [START_REF] Satnoianu | Turing instabilities in general systems[END_REF]. It continues to serve as a prototypical example in studies of pattern formation on networks [START_REF] Nakao | Turing patterns in network-organized activator-inhibitor systems[END_REF].

Indeed [START_REF] Pastor-Satorras | Patterns of complexity[END_REF], in many interesting cases, diffusion acts along links in a network rather than in a continuous space. Mathematically, this implies swapping the Laplace operator for the graph Laplacian L in the diffusion terms of the evolution equations (see Methods). Defining vectors u = (u 1 ...u N ) and v = (v 1 ...v N ) for the activator and inhibitor species, the generic Turing reaction-diffusion model on a network of N nodes is thus written

du i dt = f (u i , v i ) + ε(Lu) i ( 1
)
dv i dt = g(u i , v i ) + σε(Lv) i i = 1..N ( 2
)
where ε is the diffusion coefficient of the activator and σε that of the inhibitor. We assume that the system has a uniform equilibrium solution (ū, v). A pattern will be defined as a non-uniform stationary state, reached after some transients.

Turing understood pattern formation as linear instability of the uniform equilibrium solution i.e. when a parameter crosses an instability threshold an eigenvector of the Laplacian operator exhibits exponential growth which eventually saturates due to nonlinear interactions. In [START_REF] Nakao | Turing patterns in network-organized activator-inhibitor systems[END_REF] the authors have shown that this mechanism still applies for reaction-diffusion dynamics on networks, where the eigenvectors are now those of the graph Laplacian L.

Since the beginning of the detailed study of complex networks within statistical physics in the late 1990s and early 2000s [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Barabási | Emergence of scaling in random networks[END_REF][START_REF] Albert | Statistical mechanics of complex networks[END_REF], a clear roadmap has emerged: from a topological characterization to the study of dynamical processes on network and, in this way, the search for universal relationships between network structure and network function. These investigations have revolutionized our understanding of complex systems [START_REF] Strogatz | Exploring complex networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF][START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF][START_REF] Rodrigues | The Kuramoto model in complex networks[END_REF] and have established themselves as a novel branch of interdisciplinary research.

Among the results in the statistical physics of complex networks the most striking ones are concerned with the emergence of self-organized, collective patterns in dynamics on graphs -with Turing patterns [START_REF] Nakao | Turing patterns in network-organized activator-inhibitor systems[END_REF] as a prominent example, but also synchronization patterns [START_REF] Arenas | Synchronization in complex networks[END_REF] (see also [START_REF] Zhang | Fluctuation-induced distributed resonances in oscillatory networks[END_REF] for a study of resonance patterns in oscillatory networks), growth patterns and clusters [START_REF] Dorogovtsev | Critical phenomena in complex networks[END_REF][START_REF] Nyczka | Inferring pattern generators on networks[END_REF], waves in networks [START_REF] Müller-Linow | Organization of excitable dynamics in hierarchical biological networks[END_REF][START_REF] Moretti | Link-usage asymmetry and collective patterns emerging from rich-club organization of complex networks[END_REF] and many more.

Theoretical studies reveal, how such patterns are enhanced by certain architectural features of the network [START_REF] Moretti | Link-usage asymmetry and collective patterns emerging from rich-club organization of complex networks[END_REF][START_REF] Lesne | Complex networks: from graph theory to biology[END_REF][START_REF] Hütt | Interplay between topology and dynamics in excitation patterns on hierarchical graphs[END_REF] and transdisciplinary investigations help understand the implications of such patterns for network-like infrastructures [START_REF] Armbruster | Autonomous control of production networks using a pheromone approach[END_REF][START_REF] Vespignani | The fragility of interdependency[END_REF][START_REF] Wang | Cascading failures and the emergence of cooperation in evolutionary-game based models of social and economical networks[END_REF][START_REF] Pastor-Satorras | Epidemic processes in complex networks[END_REF][START_REF] Blunck | The balance of autonomous and centralized control in scheduling problems[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Gallagher | A clarified typology of core-periphery structure in networks[END_REF].

Starting from [START_REF] Nakao | Turing patterns in network-organized activator-inhibitor systems[END_REF], several subsequent studies focused on the impact of network features on Turing pattern formation: In [START_REF] Muolo | Patterns of non-normality in networked systems[END_REF] the authors discussed how oriented networks explain subcritical patterns while [START_REF] Mimar | Turing patterns mediated by network topology in homogeneous active systems[END_REF] studied the impact of degree fluctuations in the network. Extensions to pattern formation for FitzHugh-Nagumo dynamics on networks [START_REF] Carletti | Turing patterns in a networkreduced FitzHugh-Nagumo model[END_REF], to predator-prey models [START_REF] Liu | Turing patterns in a predator-prey model on complex networks[END_REF] and to epidemic growth [START_REF] Putra | Turing patterns of non-linear SI model on random and real-structure networks with diarrhea data[END_REF] have been presented.

Our study is motivated by an observation for the Gierer-Meinhardt pattern formation model (see Methods) on the macaque cortical area network. In this network, nodes are anatomically defined cortical areas of the macaque brain and links represent experimentally verified connections among these areas (we only use connections reported as 'strong' in [START_REF] Markov | A weighted and directed interareal connectivity matrix for macaque cerebral cortex[END_REF]). Simulating Turing patterns on this network architecture we find that for a fixed set of dynamical parameters and a given graph, but random initial conditions, a large number of different patterns emerges. For other parameter values or different graphs, the dynamics converge to a single pattern. Moreover, one can change the pattern forming process from the former to the latter by small changes in the graph topology. Figure 1 summarizes this observation.

We study quantitatively how small changes in the graph topology can create such drastic changes in the attractor landscape. Numerical investigations were performed for a minimal network model and for random regular graphs. We identify basic spectral determinants of pattern diversity. Mainly, highly diverse pattern formation is observed in degenerate situations, created by almost symmetrical network structures or by nearly identical growth rates of neighboring eigenvectors of the graph Laplacian. We propose a general scheme for predicting the topological sensitivity of Turing pattern diversity, based on the movement on the dispersion curve of Laplacian eigenvalues under network perturbations.

METHODS

Network models

For the macaque cortical area network (Fig. 1) each node represents a cortical area built from the publicly available database Core-Nets (core-nets.org; see [START_REF] Morecraft | Cytoarchitecture and cortical connections of the anterior cingulate and adjacent somatomotor fields in the rhesus monkey[END_REF][START_REF] Morecraft | Cytoarchitecture and cortical connections of the anterior insula and adjacent frontal motor fields in the rhesus monkey[END_REF]) following a scheme described in [START_REF] Markov | A weighted and directed interareal connectivity matrix for macaque cerebral cortex[END_REF]. The Turing patterns obtained on this graph are compared to those obtained after removal of a link between the primary somatosensory cortex (area 2) and the ventral proisocortical area ProM (highlighted in red in Fig. 1A). Note that we use this network only as an illustrative example of the phenomenon of topological sensitivity of Turing patterns. A detailed analysis of pattern formation on this network architecture would require a more careful variation of kinetic parameters, as well as a stronger emphasis on the relevance of such results for neuroscience.

The ring graph has nearest neighbor connections and 111 nodes. We attach one end of a shortcut at node i 0 = 1 and the other end to nodes i s = 2...N successively. Random regular graphs with degree k = 4 are generated for 30 nodes in Fig. 4. Suppl. Fig. S6 shows a random (Erdős-Rényi network with 30 nodes and 100 links. The graph Laplacian is defined as L = A -D where A is the N × N adjacency matrix A. D is the diagonal matrix whose entries are the degree k i of node i. Since our networks are undirected, L is a symmetric matrix. L always has a zero eigenvalue and is negative-semidefinite (that is Λ i ≤ 0) [START_REF] Newman | Networks[END_REF]. Eigenvalues are labelled in decreasing order Λ N ≤ Λ 1 ≤ Λ 0 = 0. This definition is consistent with numerical discretization of the Laplace operator, describing diffusion in a continuum but not with standard graph theory where the opposite sign convention is generally adopted (L = D -A).

Pattern formation

The Gierer-Meinhardt model [START_REF] Gierer | A theory of biological pattern formation[END_REF] is a generic Turing model for an activator species u and a inhibitor species

v ∂u ∂t = a -bu + u 2 v + εLu ∂v ∂t = u 2 -v + σεLv. (3)
The reaction part of the dynamics involves two positive parameters a and b which determine the homogeneous steady state (ū, v) = ((a + 1)/b, (a + 1) 2 /b 2 . The parameter is the diffusion coefficient of the activator and σ is the diffusion coefficient of the inhibitor.

The dispersion relation λ(Λ) describes the functional dependence of the growth rate λ on the eigenvalues Λ of the graph Laplacian. We linearize eq. 3 around the homogeneous steady state and write the solution as a linear combination of the eigenvectors of the graph Laplacian. The growth rate for the Laplacian eigenvector V Λ associated with the eigenvalue Λ is [START_REF] Nakao | Turing patterns in network-organized activator-inhibitor systems[END_REF]

λ(Λ) = (1/2)[f u + g v + (1 + σ)εΛ + 4f v g u + (f u -g v + (1 -σ)εΛ) 2 (4)
where

f u = (∂f /∂u)(ū, v) is the partial derivative of f (u, v) at the equilibrium point (ū, v).
The expression for the continuum where the eigenmodes are sinusoids with wave-vector q is recovered by replacing Λ with -q 2 . We consider a steady state bifurcation and assume that λ(Λ) is real, and use σ as a bifurcation parameter. Thus a bifurcation occurs when the dispersion relation moves from negative to positive values, indicating a transition from damped modes to exponentially growing modes. For the Gierer-Meinhardt system eq. 3, this occurs for The dispersion curve is the plot of the growth rate λ as a function of ln(-Λ) (recall that Laplacian eigenvalues Λ are all negative), as displayed e.g. in Fig. 3. The structure of the network is reflected in the discrete sampling of the horizontal axis by the Laplacian eigenvalues.

σ c = a + 1 b(a -1) 2 (a + 3 + 2 √ 2 √ a + 1). (5)

Numerical simulations

For a given network and a given set of dynamical parameters ε, σ, a and b a numerical experiment entails 500 runs in Mathematica using randomly chosen initial conditions. Since transients appear to decay in less than 500 time steps, the duration of a run was typically 2,000 time steps.

The following parameter selection scheme, which we call growth-rate degeneracy (GRD), was used in the simulation: Parameters are chosen such that two succes-sive eigenvectors have approximately the same positive growth rate. This is achieved by selecting parameters in such a way that the maximum of the dispersion relation at σ c is exactly midway between the two corresponding eigenvalues of the graph Laplacian. To create a positive growth rate, σ is then increased slightly by an amount ∆σ = 0.02 beyond the instability threshold (see Fig. 4).

Pattern evaluation

The pattern wheel representation of multiple simulations of Turing patterns allows for a visual assessment of pattern diversity for the same graph and the same kinetic parameters. Each pattern, characterized through the asymptotic value of the dynamical variable u i for each node i is binarized using the average over all nodes as a threshold, retaining only the information whether the activator concentration is high or low. The pattern with the highest correlation to an unstable mode serves as the reference pattern. Each simulated pattern is multiplied by -1, if this increases the visual agreement to the reference pattern (accounting for the fact that a reflected pattern is still generated by the same eigenvector). These aligned and binarized patterns are plotted around a circular embedding of the graph under consideration. Averages over these binarized patterns for each node are then displayed as an outside ring (outer purple ring in the figures).

A pattern similarity index is developed to quantitatively assess the diversity of patterns: Let

u (k) = {u (k) i (t → ∞)} i=1,...,N
denote the pattern observed in the kth simulation run of a network of N nodes, with u (k) i (t → ∞) the asymptotic value of the dynamical variable at node i for run k. We then construct a pattern similarity graph, in which a node is a pattern u (k) and a link between two patterns is generated, if the similarity between these patterns exceeds a threshold θ, leading to the edge set E,

E = {(u (k) , u (l) ||corr(u (k) , u (l) )| > θ}, where corr(u (k) , u (l) ) is the Pearson correlation coefficient of two patterns u (k) and u (l) , corr(u (k) , u (l) ) = N i=1 (u (k) i -ū(k) )(u (l) i -ū(l) ) σ(u (k) )σ(u (l) )
.

Here ū(k) and σ(u (k) ) denote the average and the standard deviation of pattern u (k) over the components i, respectively. Specifically, an edge is drawn between two patterns if the absolute value of their Pearson correlation coefficient exceeds θ = 0.9. The pattern similarity index Π is then defined as the connectivity of this pattern similarity graph, Π = 2|E|/(N (N -1)). The correlation analysis in Figure 2C is based on the Pearson correlation coefficient C i between a pattern and the i-th eigenvector where the index i spans the set of unstable eigenvectors. The maximum correlation strength max i |C i | determines whether a pattern is weakly or strongly correlated with a single eigenvector. Histograms of the correlation strength over 500 patterns generated from random initial conditions reveal two classes: histograms with a marked peak at 1, or broad ones. We interpret a histogram with a broad shape as coming from a pattern that is generated by multiple modes whereas histograms with a peak near one represent patterns generated by a single eigenvector. Similarly, the maximum absolute value of the correlation coefficient between a pattern and a set of unstable modes characterizes how well a given pattern agrees with a single unstable mode, thus offering insight in the predictability of this pattern by eigenvectors. Across a whole set of patterns (same parameters, random initial conditions) the set of these maximal absolute values of correlation coefficients -the distribution of pattern predictabilities -is a suitable quantifier of pattern diversity.

The pattern diversity array shown in Fig. 4 allows to predict, for a given network, combinations of eigenvectors and links, where pattern diversity is particularly sensitive to the addition or removal of a link. The pattern diversity array -with links of the network enumerated on the horizontal axis and eigenvectors of the network (on which the parameter tuning scheme is centered) enumerated on the vertical axis -offers a color-coded representation of our prediction of changes in pattern diversity for each combination of an eigenvector and a link, for the GRD parameter selection scheme described above. Colors coding is summarized in Fig. 4.

A MINIMAL MODEL: THE RING GRAPH WITH ONE SHORTCUT

We first investigate the topological sensitivity of Turing pattern formation in a minimal network model, composed of a ring graph with one moveable shortcut.

Turing pattern diversity changes under small topological variations are shown in Figure 2 for a ring with a single shortcut. A similar model has been introduced in [START_REF] Hütt | Perspective: network-guided pattern formation of neural dynamics[END_REF] to study the influence of the topology of the underlying graph on pattern formation by a cellular automaton. The two pattern wheel representations, Figures 2A and2B, representing different shortcut positions, show very different pattern diversity. As this model is very simple, we can analyze it like a spatially extended system. Symmetric bifurcation theory [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF] states that eigenvalues of the ring graph Laplacian are degenerate with multiplicity 2 and the eigenvectors are sinusoids. Adding a single shortcut to the ring breaks the symmetry of the graph and thus the degeneracy of the eigenvalues. Supplementary Figure S1 provides an overview of spectra and eigenvectors for the ring graph and a ring graph with one shortcut.

As a consequence of the shortcut-induced symmetry breaking, one of the degenerate eigenvalues is independent of the shortcut leading to the same sinusoidal pattern as for the ring graph. The other eigenvalue changes.

Figure 2C shows that the other eigenvalue moves, returning periodically to the degenerate situation, as the shortcut is moved around the ring graph.

Since changing the graph does not change the dispersion relation λ(Λ) (i.e. the expression of the growth rate λ of an eigenmode as a function of the eigenvalue Λ, see Methods), the interval on the dispersion relation that leads to growing patterns is unaffected. Thus, we can set the dynamical parameters in such a way, that some eigenvectors of the Laplacian lead to growing pattern for all shortcut positions while other eigenvectors correspond to eigenvalues moving in and out of this interval, as in- dicated by blue (growing pattern) and red dots (stable homogeneous solution). Supplementary Figure S2 illustrates the movement of eigenvalues as a function of the shortcut position (Fig. 2C) via snapshots of the dispersion relation (see also Supplementary Movie 1). All of these findings can be analytically shown using an extension of the interlacing theorem [START_REF] Mohar | The Laplacian spectrum of graphs[END_REF][START_REF] Brouwer | Spectra of graphs[END_REF] and linear operator theory for spatially extended systems (see Appendix A).

The impact of the changes in the spectrum of the graph Laplacian on pattern diversity is measured in Fig. 2D. The top panel presents the similarity measure of the set of patterns observed for 500 random initial conditions as a function of the shortcut position. An al-ternative visualization is presented in the bottom panel, summarizing as a box-plot the correlation between the unstable eigenvectors and the patterns obtained by numerically solving the Turing dynamical system starting from randomly chosen initial conditions. Further analysis (see Appendix) shows that the shortcut position in Fig. 2(B) corresponds to almost exactly one wavelength of the pattern (within limits of the discretization due to the finite number of nodes) and hence does not break the rotational symmetry of the ring graph (more detail in Suppl. Figs. S3 andS4). As a result there is a free phase leading to a low pattern similarity and low pattern predictability. For shortcut position (A) the spectrum has a large gap between the central mode on the dispersion relation (the eigenvalue with the largest growth rate) and the neighboring unstable modes. In fact, although the top three eigenvalues are unstable and very close, since their growth rates are very weak, the central mode is very far from being degenerate and thus dominates the pattern formation (see also Suppl. Fig. S4, panels A and B). As a result, pattern similarity between patterns is high and the resulting pattern is almost deterministically predictable. Note that in this case four eigenvectors of the graph Laplacian are unstable whereas in shortcut position B only three are unstable suggesting that the driver of pattern diversity is not the number of unstable modes but the gap between the dominant mode and the neighboring modes. As a result, topological sensitivity of pattern diversity depends on the changes in the eigenvalue spectrum with eigenvalue degeneracy at maximal growth rate being most susceptible to topological changes. Suppl. Figs. S3 and S4 show additional quantitative details. Specifically, Suppl. Fig. S3 panel D shows high pattern diversity without an eigenvalue degeneracy. Instead the growth rate of two neighboring eigenvectors are very similar. This suggests that there may be a second mechanism involved in the generation and destruction of a high diversity of pattern which we call growth rate degeneracy.

PATTERN DIVERSITY ON RANDOM GRAPHS

Next, we explore the topological sensitivity of Turing pattern formation and the role of growth rate degeneracy (GRD) for random regular graphs. In random regular graphs, all nodes have the same degree but are otherwise randomly connected.

Emergence of a Turing pattern on a random regular graph follows the paradigm established for the minimal model: For a single unstable mode of a dispersion relation, starting from random initial conditions the asymptotic pattern agrees well with the unstable eigenvector.

No analytic results can be formulated for random regular graphs. However we developed predictive criteria for pattern diversity and for topological sensitivity based on the spectrum of the Laplacian. They have been validated in a large numerical experiment. Figure 3 shows the experimental setup: We adjust the dynamical parameters such that the growth rate for an eigenvector e k is the same as for the adjacent eigenvector e k+1 . Since the dispersion relation is locally quadratic, this is achieved by putting its maximum exactly in the middle between two neighboring eigenvalues of the Laplacian. We call this the GRDON parameter selection scheme (GRD in the original network). The GRDPN parameter selection scheme generates the reverse changes: we select dynamical parameters such that the maximum of the dispersion relation is between two eigenvalues of the perturbed network, i.e. after the deletion of a link in the original graph.

The GRDON parameter selection scheme is designed to search for cases, where link removal decreases pattern diversity (as we start from a situation -growth rate degeneracy -predicted to favor pattern diversity).

In order to systematically search for situations, where link removal can increase pattern diversity, we need to start from a non-degenerate setup and arrive at an approximate growth rate degeneracy. The purpose of the GRDPN scheme is to systematically search for such situations, as by construction upon link removal we arrive at an approximate growth rate degeneracy. Figure 3 shows the region of the dispersion relation near zero growth rate with all unstable modes and adjacent stable modes. The relevant measures for prediction of topological sensitivity are: (i) the growth rate of the unstable modes for the original network, λ u and after link deletion, λ * u and (ii) the weakest growth rate of the stable modes for the original network, λ s and after link deletion, λ * s . The relative size of these parameters and the number of unstable modes before (s) and after (s * ) dropping a link suggests to classify the pattern into six different types of pattern sensitivity, color coded in Fig. 4 with the formal criteria listed as Boolean expressions. There are three criteria for high sensitivity of pattern diversity to topological perturbations: (1) the perturbation should destroy or establish (approximate) growth rate degeneracy of the unstable modes; (2) no weakly stable modes close to the instability threshold and (3) the growth rates of the (approximately degenerate) unstable modes should not be too small. The topological perturbations producing the strongest signals thus trigger a transition from two unstable modes with degenerate growth rates to a single unstable mode, while avoiding all interfering factors (stable modes close to the instability threshold, too small growth rates of unstable modes). In Fig. 3 the applicable classification for this specific dispersion relation is shown, predicting slightly less pattern diversity (color coded as light blue). Note that our prediction scheme summarized in Fig. 3 and applied, e.g., in Fig. 4 is a heuristic derived from studying the influencing factors of pattern diversity for a large set of numerical examples. The factor 1/2 in the requirement of a small enough distance of λ * u to the maximum of the dispersion relation has been chosen arbitrarily and a broader study of influencing factors might lead to a refined heuristic.

Figure 4 illustrates the relevance of the classification scheme for a full analysis of Turing pattern diversity changes under link removal for a random regular graph of degree 4, with 28 nodes and 60 links. Panels (A) and (B) show the change in pattern diversity for a specific pattern characterized by an unstable eigenvector, upon removal of a single link in a regular random graph. They are similar to the ones we showed in Figures 1 and2 for the macaque cortical area network and for the ring graphs with one shortcut. While panels (A) and (B) show a specific case, panels (C) and (D) analyze the sensitivity of all possible patterns to the removal of any single link. Both, the GRDON and GRDPN parameter selection schemes are run for each eigenvector e k separately. In this way we explore systematically the whole variety of situations displaying growth rate degeneracy.

Summarizing Fig. 4 we note that for this choice of the dispersion relation shift (via the additive constant ∆σ to the diffusion constant ratio σ, see Methods; see Supplementary Table S1) eigenvectors with low index are not influenced by the link removal. Low eigenvector indices correspond by definition to eigenvalues of small absolute size and we observe sensitivity to link removal in this regime only for higher values of ∆σ. As discussed, strong changes in pattern diversity are triggered by a transition from two unstable modes with degenerate growth rates to a single unstable mode. We observe that for this parameter set and for random regular graphs, small eigenvalues of the graph Laplacian do not move much under link removal, making it unlikely that the associated patterns change stability. While we find this to be true for most of our parameter sets, we have not explored this in general for random regular graphs. However, note that this is also true (and intuitive) for the ring graphs: small eigenvalues correspond to long wavelength patterns. The distance between neighboring eigenvalues is much larger for long wavelengths than for short ones, making them more robust against perturbations.

We furthermore observe that the eigenvector determines whether the link removal leads to an increase, a decrease or no change in the pattern diversity. Specifically, eigenvectors that display growth rate degeneracy lead to changes of the pattern diversity.

DISCUSSION

Turing patterns in a continuum have been studied in great detail for many years, specifically their dependence on the parameters of the pattern-generating reaction diffusion system. Turing patterns on graphs offer a completely new way to influence the pattern formation by changing the underlying graph, e.g. adding or subtracting a small number of links. One important feature of Turing pattern formation on graphs evidenced in our study is the rarity of the emergence of a unique pattern. Even close to the instability threshold one often encounters a large number of simultaneously emerging stable patterns. They manifest themselves as a result of random sampling of initial conditions, a situation we call pattern diversity. Our study is a first step towards a full understanding of the mechanisms that generate such high diversity of patterns and its changes as the underlying graph undergoes small changes in its topology.

Our focus is on patterns determined by the unstable modes of the linear system, i.e. supercritical bifurcations. In particular, we do not consider the case of subcritical bifurcation where we are unable to relate pattern diversity to any topological features of the graph. In the supercritical case we can separate the influence of the reaction system from the influence of the underlying graph. The former is reflected in the dispersion relation determined by the parameters of the reaction system while the latter is a reflection of the eigenvalue spectrum of the graph Laplacian.

Our approach to disentangle the influence of the reaction parameters from the influence of the graph is based on the fact that the dispersion relation is locally quadratic (as in Fig. 3), and its width and location can only be changed by the reaction parameters. The part of the dispersion relation that lies above the zero growth rate determines the potential eigenvalues and associated eigenvectors that determine the emerging patterns. The number of eigenvalues and their spacing in this interval is a characteristic of the underlying graph. In that way the dispersion relation acts like a magnifying glass on the eigenvalue spectrum of the graph Laplacian.

Our analysis shows that the spacing of Laplacian eigenvalues and their movement on the dispersion curve under small topological changes of the graph translates into robustness or sensitivity of pattern diversity. In particu- lar, small gaps between the unstable (or nearly unstable) eigenvalues surrounded by larger gaps can lead to high pattern diversity, whereas sequences of large gaps tend to favor a unique pattern.

These results are independent of the specific choice of the dynamical model or its parameters, as they only rely on the placement and the corresponding discrete sampling of the dispersion relation by the eigenvalues of the graph Laplacian. This discrete sampling is the determi-nant of pattern diversity.

One extreme case is the case of multiplicity of eigenvalues, e.g. due to symmetries, like in the case of a ring graph. Here the gap between two neighboring eigenvalues is zero leading to a continuum of patterns. Reaction parameters have no influence on this feature, but small changes of the underlying graph generically create a non-zero gap between the formerly degenerate eigenvalues and subsequently reduce the diversity of the emerging patterns.

We extend the idea of small spectral gaps to the degeneracy of growth rates, i.e., situations where two adjacent eigenvalues of the graph Laplacian are not necessarily close in the spectrum but have the same growth rate. In order to systematically investigate these situations, we tune the reaction parameters such that the two adjacent eigenvalues are symmetrically distributed with respect to the maximum of the dispersion relation. A large number of numerical experiments for different types of graphs (small-world graphs [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], i.e., ring graphs with different numbers of shortcuts, ER graphs [START_REF] Erdős | On the evolution of random graphs[END_REF], random regular graphs and BA graphs [START_REF] Barabási | Emergence of scaling in random networks[END_REF]) show that pattern diversity changes significantly when small changes in the graph generate or destroy such approximate growth rate degeneracy.

CONCLUSION

Our study relates pattern forming capacities and spectral properties of networks. The variation in pattern diversity resulting from a change in network topology can be predicted by computing the change of the part of the eigenvalue spectrum of the graph Laplacian that is sampled by the unstable region of the dispersion curve. This link between the spectral properties of a graph and aspects pertaining to the robustness, predictability and controllability of self-organized patterns opens an avenue towards a better understanding of systemic vulnerabilities in network-like infrastructures relying on collective patterns.

Correspondence and requests for materials could be addressed to all authors.

Appendix A: Ring graph with shortcuts

We discuss two ways to generate topological sensitivity of pattern formation -one related to degenerate eigenvalues of the Laplacian created by symmetries, the other to degenerate growth rates, generated by two different but neighboring eigenvalues that are arranged in such a way that their growth rates are the same. To study the former, we consider a ring graph with a single shortcut as a minimal model. It offers the first step from spatially extended reaction-diffusion systems (here on a ring, as in the seminal work of Turing) to their counterpart on networks.

To simplify exposition we focus on a scalar reaction diffusion equation of the form

∂u ∂t = f (u) + Lu
where L is the Laplacian of the ring graph of N nodes and nearest-neighbor links given by

L i,j =    1 for j = i -1 1 for j = i + 1 -2 for j = i
where i and j are integers modulo N . The eigenvalues of L are Λ q = -2(1 -cos((k q )) where k q = 2πq/N for integer q. They are degenerate, each with multiplicity 2, and the associated eigenvectors are X q (i) = cos(k q i), i = 0...N -1 and (A1) Y q (i) = sin(k q i).

(A2)

Notice that any linear combination of these two eigenvectors (which is still an eigenvector with eigenvalue Λ q ) can be written in the amplitude-phase form ũk (i) = A cos((k q i) + φ), corresponding to the fact that any embedding of the graph on a circle is invariant under the symmetry group of the circle. Following equivariant bifurcation theory [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF] the generic steady state bifurcation can be described by a bifurcation on a 2-dimensional center manifold which leads to an invariant circle of fixed points. If the bifurcation is supercritical then the circle of fixed points is normally stable and has a zero eigenvalue in the tangential direction to the circle. Hence for uniformly randomly distributed initial conditions we expect uniform distributions of fixed points on the circle, representing patterns with a uniform distribution of phase shifts. We thus proved that the Gierer-Meinhardt model on the ring graph has a uniformly distributed ensemble of patterns around the circle. The average over patterns obtained with random initial conditions is therefore uniform, provided the number of patterns is large enough to avoid finite-sampling effects.

Laplacian eigenvalues for the ring graph with one shortcut

As discussed in the previous section, due to the symmetry of the ring graph, the eigenvalues Λ q of its Laplacian have multiplicity two. The interlacing theorem [START_REF] Mohar | The Laplacian spectrum of graphs[END_REF][START_REF] Brouwer | Spectra of graphs[END_REF] applied to the addition of edges of a graph can be formulated in the following way: Theorem: Let G be a graph with n vertices and let G * be obtained from G by adding an edge joining distinct vertices of G and let L(G * ), L(G) be the corresponding Laplacians. Then

Λ i-1 (L(G * ) ≤ Λ i (G) ≤ Λ i (G * ),
for all i = 1, ..., n. Thus, since the addition of an edge breaks the symmetry of the ring, the eigenvalues are nondegenerate and the eigenvalues of L(G) interlace with the eigenvalues of L(G * ). Since our variable on the horizontal axis for the dispersion curve is ln(-Λ), the eigenvalues of the ring graph with one shortcut move to the right on the dispersion curve, relative to the eigenvalues of the ring graph.

Suppl. Fig. S1 shows the spectrum and a typical eigenvector for the ring graph (panel A and C) and the ring graph with one shortcut (panel B and D), while Suppl. Fig. S2 and Suppl. Movie 1 show precisely how changing the shortcut position changes the number and eigenvalues of the unstable eigenvectors.

For the ring graph, the degenerate eigenvalues correspond to eigenvectors that are generated by even and odd sinusoidal functions with a free phase parameter. Thus for each shortcut, we can shift the phase parameter in such a way that one of the eigenvectors is even and the other is odd with respect to a reflection at the middle of the shortcut. Thus a theorem from graph theory called the edge principle [START_REF] Merris | Laplacian graph eigenvectors[END_REF][START_REF] Newman | The Laplacian spectrum of graphs[END_REF] can be used, which states that if X is eigenvector of the graph Laplacian with eigenvalue Λ satisfying X(i 0 ) = X(i s ), then the Laplacian of a graph with the shortcut (i 0 , i s ) has also X as an eigenvector with eigenvalue Λ. Thus for each pair of degenerate eigenvectors for the ring graph, the even eigenvector will be also an eigenvector with the same eigenvalue for the ring graph with a shortcut, i.e. a sinusoid with an appropriate phase shift as seen in Suppl. Fig. S1(D) (dashed line). As the eigenvector is the same, so is the eigenvalue. Hence, for each pair of degenerate eigenvalues of the Laplacian of the ring graph, one eigenvalue will be an eigenvalue for the ring graph with a shortcut and, following the interlacing theorem, the second one will move to the right but never cross the next eigenvalue of the ring graph.

Pattern diversity for the ring graph with one shortcut

We can recover these graph theoretic results and generate new insight by performing the analysis for eigenvec-tors of the Laplacian on the ring graph associated with the eigenvalue Λ q and the wavelength l = N/q and a shortcut from node i 0 to node i s . The physical intuition for the edge principle discussed in the previous section is the following: A ring graph has sinusoidal eigenvectors. Any link that connects two nodes exactly one wavelength apart will have no impact on the eigenvectors. This can easily be seen when we write out the the expression for the eigenvalue of the Laplacian for the node i 0 : j L 0j u j = u(i -1 ) -u(i 0 ) + u(i 1 ) -u(i 0 ) + u(i s ) -u(i 0 ). (A3) When u(i 0 ) = u(i s ) this expression becomes the same as in the ring graph, i.e. the shortcut has no influence. It is easy to see that the same is true for the time evolution of u(i s ). Hence, for any eigenvector with a wavelength that approximately fits into the spacing of the ring graph, there exist shortcuts that have no influence and therefore any linear combination of the two eigenvectors of the ring graph will still be invariant under any phase shift leading to a uniformly distributed ensemble of patterns around the circle and thus to maximal pattern diversity.

A generic shortcut cuts the ring into pieces which, as evidenced by the geometry has a reflection symmetry relative to a reflection on a line that is perpendicular to the shortcut through its midpoint. By representation theory of the Z(2) symmetry we will have an eigenfunction that is invariant under the reflection, i.e. even and another one that has a -1 representation, i.e. is an odd function. This is most easily illustrated for the eigenfunction with wavelength l = N , i.e. the cosine function cos(2πi/N ): If the shortcut connects node i to node N -i we have that cos 2π

N -i N = cos 2π(1 - 1 N = cos 2π i N ,
i.e. the value of the activator is the same at the endpoints of the shortcut and thus the eigenfunction is even. Thus for every shortcut, there is a reflection-symmetric pattern that is selected by the shortcut as the even eigenfunction from the continuum of eigenfunctions in the amplitudephase form of the ring graph. The associated eigenvalue is therefore the same as for the ring graph and, for N even (or N large), the amplitude of the node midway between the shortcut nodes is a maximum or minimum. The corresponding second eigenfunction of the ring graph will be anti-symmetric and thus will have that u(i 0 ) = -u(i s ). Inserting this into Eq. (A3) shows that this function is not an eigenfunction of the ring graph with a shortcut. Thus the eigenvalue for the antisymmetric eigenfunction of the ring graph with a shortcut will move and, for N even (or N large), the amplitude of the node midway between the shortcut nodes is zero (or approximately zero).

As we fix i 0 and move the endpoint i s of the shortcut from one node to the next, we therefore observe a splitting of the degenerate eigenvalues of the ring graph illustrated in Suppl. Fig. S1. Degenerate eigenvalues correspond to a continuum of patterns, while split eigenvalues show a small, finite number of patterns. This fully explains the diversity of the pattern formation in the ring graph with one shortcut. Suppl. Fig. S3 on the left panels shows the position of the eigenvalues on the dispersion relation as the shortcut moves and on the right panels shows the corresponding histograms of maximal correlations between the observed patterns in a set of 500 random initial conditions and the set of unstable eigenvectors. As the eigenvalues become degenerate (panel C and E), the growth rate for the associated eigenfunctions are very close. Hence, we expect that the final pattern is a linear combination of the two orthogonal eigenfunctions with about equal contributions. Measuring the correlation of that final pattern with one of the two eigenfunctions will lead to a projection of length 1/ √ 2, as one can see in panels D and F.

Numerical calculations of the anti-symmetric eigenvector shows, that the eigenvector corresponding to the moving eigenvalue evidently consists of two parts, each with the same wavelength but different amplitudes and phases. This suggests an Ansatz for an eigenfunction that is sinusoidal, with the same wavelength everywhere and where amplitude and phase of the two parts of the eigenfunctions have to be adjusted to satisfy continuity conditions and the eigenvalue is given by the relationship between the wavelength and the eigenvalue of the undisturbed ring graph Λ = Λ q = -2 (1 -cos(k q )) .

(A4)

Thus, the following calculation creates the eigenvalues as a function of the shortcuts like in Figure 2C. We assume without loss of generality that one end of the shortcut starts at node n = 1 and the shortcut ends at node y. The symmetry axis related to the Z(2) symmetry goes through the points i * = (1 + y) /2 and j * = (y + N + 1) /2 which are nodes of the ring, if i * and j * are integer and otherwise are just coordinates on the ring. Since we are looking for the odd eigenfunction, we have u(1) = -u(y). In general for finite number of oscillators our assumption implies that the eigenfunctions are of the form a sin(k(i -i * )) on i = 1...y and sin(k(i -j * )) on i = y...N . We have w.l.o.g. set the amplitude for the second part of the eigenfunction to one and leave the amplitude for the first part to be determined. The eigenvalue of the network Laplacian for the nodes that are involved with the shortcut at node i = 1 and i = y can be written as u(0) -u(1) + u(2) -u(1) + u(y) -u(1) = Λu(1), u(y -1) -u(y) + u(y + 1) -u(y) + u(1) -u(y) = Λu(y), which simplifies to u(0) -4u(1) + u(2) = Λu [START_REF] Turing | The chemical basis of morphogenesis[END_REF].

(A5)

Assuming sinusoidal eigenfunctions we can evaluate them at these nodes to get:

u(1) = a sin (k q (1 -i * ) = -a sin k q y -1 2 u(2) = a sin (k q (2 -i * ) = -a sin k q y -3 2 u(y) = a sin (k q (y -i * )) = a sin k q y -1 2 u(0) = u(N ) = sin (k(N -j * )) = sin k q N -y -1 2 .

Inserting these into Equation (A5) gives sin k q y + 1 -N 2 + a sin k q y -3 2 = (Λ + 4)a sin k q y -1 2 (A6)

for the three unknowns a, k, Λ. Finally there is a continuity condition, which is that at node i = 1 and at node i = y the two sinusoidal parts of the eigenvector have to have the same value. Hence u(y) = -a sin k q y -1 2

= sin k q N + 1 -y 2 . ( A7
)
We can thus solve Equations (A4, A6 and A7) for a, k q , Λ numerically. Note that these equations have N/2 solutions for k q , corresponding to the number of eigenvalues for the even eigenmodes of the ring graph. To find a particular one, we have to restrict the numerical search to the interval between two eigenvalues of the ring graph without a shortcut. Figure 5 shows the movement of an eigenvalue of the network Laplacian as a function of the shortcut position. We can also understand the primary bifurcations of this 2N -dimensional dynamical system based on the symmetries of the eigenfunctions: when we add a shortcut, we break the D n rotational symmetry of the ring graph. However for just one shortcut there is still a reflection symmetry at the midpoint of the shortcut. Thus we get an eigenfunction that is even with respect to that symmetry and one that is odd, corresponding to the two cases discussed above. The eigenfunction that is odd corresponds to a (-1)-representation of the Z(2) symmetry leading to pitchfork bifurcations [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF]. The Z(2) symmetry places no restrictions on the bifurcations for the even eigenfunction. However, since this eigenfunction inherits the sinusoidal structure of the ring graph, it generates a (-1)-representation under a translation along the ring graph by half a wavelength. Hence the primary bifurcation for both modes will be pitchfork bifurcations and thus no subcritical modes will appear in a ring graph with just one shortcut. This will change as the number of shortcuts increases [START_REF] Wolfrum | The Turing bifurcation in network systems: Collective patterns and single differentiated nodes[END_REF].

Pattern diversity analysis for the ring graph with two shortcuts

We observed the same set of phenomena when we add a second shortcut and move its end while the first shortcut is kept fixed. Supplementary Figure S5 displays the predictability scheme (analog of Fig. 4) for the ring graph with two shortcuts. Note that case A is again close to a degeneracy and therefore exhibits high pattern diversity.

FIG. 1 .

 1 FIG. 1. Topological sensitivity of Turing pattern diversity on the macaque cortical area network. (A) Representation of the macaque cortical area network (see Methods). (B) Example of a pattern arising for the Gierer-Meinhardt dynamical model from the sixth eigenvector of the Laplacian matrix of the graph, shown underneath. (C) We arrange the nodes in an arbitrary but fixed way on a circle. The links between the nodes are shown in blue in the center. Each concentric ring represents a binarized Turing pattern originating from a randomly chosen initial condition, in total 500 runs (see Methods). The surrounding purple ring shows their average. Panel (D) shows the same pattern wheel representation after removal of a single link (highlighted in red in (A) and (C) [upper left quadrant; see the dashed ellipse]).

FIG. 2 .

 2 FIG. 2. Topological sensitivity of Turing pattern diversity on the ring graph with one shortcut. (A) Pattern wheel representation for a ring graph of N = 111 nodes with a shortcut from node 1 to node 58. A global attraction to a single pattern independent of the initial condition is observed. (B) Same as (A), but for a shortcut from node 1 to node 68; now each observed pattern depends on the specific initial condition. (C) Evolution of eigenvalues for the ring graph with one shortcut as a function of the shortcut position (i.e., the endpoint of the shortcut, the starting point being kept fixed). Note that only eigenvalues which are unstable for some shortcut position are shown. (D) Top panel: Pattern similarity index (see Methods) as a function of the shortcut position. Bottom panel: Pattern predictability distributions (see Methods), depicting the correlation between patterns and unstable eigenvectors. The two shortcut positions for (A) and (B) are highlighted as dashed lines. The distribution of pattern predictability is shown as a sequence of box plots (white lines in the middle of the red bars indicating the median; size of the bar are the 25% and 75% quantiles; error bars ('whiskers') are the range of values covered with the exception of outliers, which are shown as additional light-red points).

FIG. 3 .

 3 FIG. 3. The dispersion curve for the parameter selection scheme GRDON. The top panel illustrates the quantities involved in the predictability criteria. The variable on the horizontal axis is the eigenvalue of the network Laplacian displayed in a logarithmic scale, ln(-Λ). Blue dots indicate the growth rates for the original network, red dots are the ones after link deletion. The dashed curve is the dispersion relation for ∆σ = 0. Positive values of ∆σ shift the dispersion relation upwards, yielding the dispersion relation used for evaluating pattern diversity (shown as a full line). Note that λ s and λ * s are meant as distances from zero and hence are positive.

FIG. 4 .

 4 FIG. 4. Predicting the impact of topological changes on pattern diversity. (A) Pattern wheel representation (see Methods) for 500 simulated patterns starting from random initial conditions under the GRD scheme of kinetic parameter selection (see Methods) for the original random regular graph and eigenvector 16. (B) Same as (A), but after removal of the link from node 4 to node 22. (C) Pattern diversity array (see Methods) depicting the expected change in Turing pattern diversity as a function of the deleted link index (horizontal axis) and the eigenvector label, eigenvectors being numbered at decreasing eigenvalues (vertical axis, recall that eigenvalues are negative). The parameter constellation investigated in (A) and (B) is highlighted in yellow. (D) Criteria defining the color code used in the pattern diversity array (C) and the corresponding prediction.

FIG. 5 .

 5 FIG. 5. Movement of the eigenvalue of the network Laplacian for a ring graph as a function of the endpoint of the shortcut. The starting point of the shortcut is always oscillator 1. The two eigenvalues at the top and bottom are the ones that do not depend on the shortcut.

ACKNOWLEDGMENTS

M. T. Hütt thanks LPTMC (Paris, Sorbonne Université and CNRS UMR 7600) for hospitality and the Physics Institute of CNRS (French National Center for Scientific Research) for funding his stays, during which part of this work has been performed. A. Lesne thanks Jacobs University, Bremen, for hospitality and CNRS for funding her stays.

AUTHOR CONTRIBUTIONS

All authors designed the study, carried out the analysis, and wrote the paper. D.A. performed the analytical computations presented in Appendix A. M.T.H performed numerical simulations.

COMPETING INTERESTS STATEMENT

The authors declare no competing financial interests. Supplementary information accompanies this paper.


II -Emotion at the heart of the tourist experience

Since the work of Pine and Gilmore (1999), the academic literature considers the tourism experience as a subject. It is not our intention to revisit this literature, which notably questions the design of experience [START_REF] Tussyadiah | Toward a Theoretical Foundation for Experience Design in Tourism[END_REF], the status of experience as a concept [START_REF] Vergopoulos | L'expérience touristique : une expérience des cadres de l'expérience touristique ? Via[END_REF], a process [START_REF] Aho | Towards a General Theory of Touristic Experiences: Modeling Experience Process in Tourism[END_REF], or as a fixing element of experiential marketing [START_REF] Knobloch | Experience, Emotion, and Eudaimonia: A Consideration of Tourist Experiences and Well-being[END_REF]. Experience is composite. [START_REF] Bastiaansen | Emotions as Core Building Blocks of an Experience[END_REF] identified various constituent elements: attention, involvement, engagement, immersion, cognitive absorption, and emotion. [START_REF] Skavronskaya | Cognitive psychology and tourism research: state of the art[END_REF] associate awareness and mindfulness, attention, memory, feelings, emotions, and a cognitive pattern with experience. We consider that emotion gives the experience its memorable character and, as such, is at the heart of the new tourism experience. After explaining the notion of emotion (1), we will see that for a tourist destination, being able to fix the emotion of visitors and to communicate around emotion can contribute to making it attractive (2) to the point of considering emotion as the new frontier of destination valorization (3).

What is emotion?

From an etymological perspective, emotion derives from the Latin motio 'to move' and 'to emote.' It has long been associated with a 'moral disorder,' making it an object of study in cognitive science and neuropsychology [START_REF] Moyle | The next frontier in tourism emotion research[END_REF]. In a figurative sense, emotion is felt individually or collectively to the point of generating a movement, a transport. An emotion is experienced, felt, generates sensations and feelings. Depending on the degree of sensitivity and receptivity of individuals and the circumstances, emotion can be strong, acute, intense, painful, happy, poignant, sentimental, etc. For a long time, repressed and judged sentimentalist, emotion is a total modification of the "the being-inthe-world" (l'être-dans-le-monde, Sartre, 1939: 66). Emotion can be of aesthetic, artistic, literary, musical, spiritual, or cultural origin. It is this informational and sentimental richness that interests tourism as well as its experiential content. For Gauducheau (2008: 390), "emotion is a complex reaction of the individual to a state of the environment or an event."

Emotion has been associated in the literature with various tourist practices such as festivals, shopping, leisure parks, heritage visits, adventure [START_REF] Hosany | Emotions in Tourism: Theoretical Designs, Measurements, Analytics, and Interpretations[END_REF]. Emotions permeate visitors' experiences and guide their responses to external stimuli, affecting their attitude, behavior, and memory. Emotions are "fleeting phenomena rather than stable attributes of an individual that come and go with the ebb and flow of daily life" (Kuppens, 2015: 297).

Emotions shape the tourism experiences (Kim & Fesenmeier, 2015;[START_REF] Volo | Emotions in tourism: from exploration to design[END_REF], and they contribute to creating a lasting memory, thus influencing future behavior and travel choices. Used by individuals to assess their interest in repeating specific experiences, emotions are also shared with others through storytelling, becoming powerful motivators of future behavior for self and others. Fleeting but powerful emotions are associated with individuals' past experiences. Thus, the direction and strength of emotional responses to stimuli can differ considerably from one tourist to another, resulting in various responses.

From a tourism perspective, emotion is often studied at the various stages of the decision cycle 'before -during -after' the trip [START_REF] Li | Current and potential methods for measuring emotion in tourism experiences: a review[END_REF][START_REF] Hosany | Emotions in Tourism: Theoretical Designs, Measurements, Analytics, and Interpretations[END_REF]). Before the trip, i.e., when choosing a destination, the image can generate a positive emotion, but this is passive and projected, often due to destination marketing. On the other hand, once there, the emotion becomes an "active" response to the visitor's experience. At this stage, emotion is plural. From the destination perspective, emotions generated during the trip, and fed after the trip, color the experience. They make it a memorable moment (or not) which will give rise to a revisit (or not) or a recommendation (or not) to friends and relationships. A visitor who associates positive emotions with a destination and displays satisfaction will have a positive experience. They are likely to be loyal to that destination.

Moreover, linking emotion and experience invites us not to reduce the experience to a product purchased from tourism providers and thus make the visitor more proactive [START_REF] Knobloch | Experience, Emotion, and Eudaimonia: A Consideration of Tourist Experiences and Well-being[END_REF]. It also allows the experience to be associated with a sentimental, affective dimension and meaning, sharing, and knowledge.

Capturing and communicating emotion

From a methodological point of view, how to capture and communicate the visitor's emotion? The measurement method is generally problematic because it borrows heavily from psychology [START_REF] Hosany | Emotions in Tourism: Theoretical Designs, Measurements, Analytics, and Interpretations[END_REF]. Indeed, many studies, particularly in psychology and neuromarketing, explore the neurotransmitters at the source of emotion via visual stimuli activated during tests, electroencephalograms, and questionnaires (Bastianen & al. 2016[START_REF] Moyle | The next frontier in tourism emotion research[END_REF][START_REF] Klonsky | The Multidimensional Emotion Questionnaire (MEQ): Rationale and Initial Psychometric Properties[END_REF]. Those analyses fall into two categories, those that seek to understand the relationship between stimulus and behavior (behaviorist approaches) and those that explore the mental process at work between stimulus and behavior (cognitive approaches). These approaches measure emotion and consider the link between emotion and destination based on images or rating scales, regardless of whether the individuals tested know the destination. The link between emotion and experience is independent of the on-site presence or the localized socio-cultural interactions. It depends on pre-or postvisit emotions, which poses a problem of memory distortion and emotion reconstruction in the latter case.

Moreover, this raises the question of the link between emotion and knowledge. It seems that emotion cannot be considered an independent variable with no link to the place visited. Capturing emotion is not simply a matter of measuring it on a scale of values.

For example, [START_REF] Hosany | Measuring Tourists' Emotional Experiences toward Hedonic Holiday Destinations[END_REF] developed the Destination Emotion Scale (DES) through two empirical studies. The DES measures the diversity and intensity of tourists' emotional experiences of destinations. The authors followed a rigorous process to develop the Destination Emotion Scale, following conventional guidelines. Data for the first study were collected face-to-face from 200 British people. Respondents must rate the intensity of their emotional experience of the destination they had visited (e.g., 'I felt a sense of pleasure'). An exploratory factor analysis with Varimax rotation initially identified a 3-dimensional measure of 23 items, and after establishing onedimensionality, the scale remains at 15 items. The three dimensions of the DES are: Joy (cheerfulness, pleasure, joy, enthusiasm, and delight); Love (tenderness, love, attention, affection, and warmth), and positive surprise (astonishment, surprise, fascination, inspiration, and surprise). The alpha coefficients and composite reliabilities exceed the recommended standards.

In the second study, [START_REF] Hosany | Measuring Tourists' Emotional Experiences toward Hedonic Holiday Destinations[END_REF] examined the dimensionality and validity of the DES. They were using the guidelines recommended by [START_REF] Dillman | Mail and Internet Surveys: The Tailored Design Method[END_REF] to collect the data via a postal survey in a town in the southeast of England (UK). The sample size for the second study was 520 respondents. Correspondence factor analysis (CFA) establishes the construct validity of the destination emotion scale. In addition, multiple regression analyses performed on the data with the emotion dimensions as independent variables, satisfaction, and intention to recommend as dependent variables confirm the scale's predictive validity. Although the CFA shows that the 15-item measure has solid psychometric properties, further evidence of the scale's validity is needed. Validity is the degree to which a construct achieves empirical and theoretical significance. The literature identifies six criteria for achieving construct validity: content validity, onedimensionality, reliability, convergent validity, discriminant validity, and nomological validity [START_REF] Garver | Logistics Research Methods: Employing structural Equation Modeling to Test for Construct Validity[END_REF][START_REF] O'leary-Kelly | The Empirical Assessment of Construct Validity[END_REF][START_REF] Ping | On Assuring Valid Measures for Theoretical Models using Survey Data[END_REF]).

Furthermore, in [START_REF] Hosany | Measuring Tourists' Emotional Experiences toward Hedonic Holiday Destinations[END_REF] study, the sample was limited to one culture (British nationals). To advance knowledge, [START_REF] Steenkamp | Optimum Stimulation Level and Exploratory Consumer Behavior in an Emerging Consumer Market[END_REF] emphasize the need for researchers to test new scales in different countries and environments, thereby establishing external validity. The scale's discriminant validity is assessed by using place attachment, a theoretically related but distinct construct [START_REF] Scannell | Defining Place Attachment: A Tripartite Organizing Framework[END_REF]. The "place attachment" conceptualization is the emotional connection between an individual and a particular spatial setting [START_REF] Williams | Beyond the Commodity Metaphor: Examining Emotional and Symbolic Attachment to Place[END_REF]). People's relationships with place can represent a range of positive emotions from love to contentment [START_REF] Manzo | For Better or Worse: Exploring Multiple Dimensions of Place Meaning[END_REF]).

If we abandon measurement for observation, the Internet and social networks seem to be vectors capable of fixing emotions and facilitating communication and, therefore, social interactions around emotions. Emotion manifests the mental state of the visitor, which, when expressed on social networks, calls for reactions in return and contributes to strengthening social links. One way of capturing this emotion is to search in the tone of posts, photos, and messages, the nature of the emotions of what is perceived and experienced by the visitor by analyzing verbal and non-verbal corpus.

Another way to do this is to focus on emoticons to express emotions, a state of mind, or a feeling in the digital age. An emoticon is a conventional representation of emotion through a typeface or image that substitutes for verbal expression. As Halté (2016: 442) points out, "The subjective elements that emoticons can indicate can be, for example, an emotion, an attitude, or even a practical instruction." Initially confined to ASCII 1 expressing smileys, they are now drawn and available in abundance and no longer necessarily concern facial expressions but also objects, monuments, animals, etc., to the point of being consigned to a universal library 2 . The primary emotions expressed are joy, humor, sarcasm, sadness, anger, laughter, amazement, sadness, etc. These emoticons can be positive, negative, relational. [START_REF] Hosany | Emotions in Tourism: Theoretical Designs, Measurements, Analytics, and Interpretations[END_REF], aware of the difficulty of measuring emotion and above all the need to adjust the measurement to the specificity of tourism, propose the "emotion apps model," which offers a reading of emotion combining, in a continuum, different criteria (joy, etc.), different configurations mixing positive and negative elements, consciousness and unconsciousness concerning the place visited, opening the way to a better consideration of emotion in research in tourism.

The communication of emotions is complex. Indeed, what distinction can we establish between emotion felt in situ and emotion expressed ex-post by the visitor? How can a destination understand why people feel emotions and tap into these emotions to enhance the territory?

Emotion as a destination enhancer: a new frontier

According to sociologists and psychologists, places are sources of identification and affiliation that give meaning and purpose to life [START_REF] Gustafson | Meanings of Place: Everyday Experience and Theoretical Conceptualizations[END_REF]Williams & Vaske 2003). Research in environmental psychology 3 establishes that people develop relationships with places [START_REF] Hidalgo | Place Attachment: Conceptual and Empirical Questions[END_REF]. Previous studies explore the nature and nuances of people's relationship with places, also known as place attachment [START_REF] Kaltenborn | Effects of Sense of Place on Responses to Environmental Impacts[END_REF]. Place attachment is the connection between individuals and places. Theorized as a multidimensional concept, place attachment "involves an interplay of affect and emotions, knowledge and beliefs, and behaviors and actions in reference to a place" (Altman & Low 1992: 5).

Two main conceptualizations of place attachment dominate the literature in environmental psychology [START_REF] Hidalgo | Place Attachment: Conceptual and Empirical Questions[END_REF][START_REF] Manzo | Relationships to Non-Residential Places: Towards a Reconceptualization of Attachment to Place[END_REF]Williams & Vaske, 2003) and tourism [START_REF] Gross | An Empirical Structure Model of Tourists and Places: Progressing Involvement and Place Attachment into Tourism[END_REF][START_REF] Lee | The Mediating Effect of Place Attachment on the Relationship between Festival Satisfaction and Loyalty to the Festival Hosting Destination[END_REF][START_REF] Yuksel | Destination Attachment: Effects on Customer Satisfaction and Cognitive, Affective and Conative Loyalty[END_REF]): place identity (emotional attachment) and place dependence (functional attachment). Place identity refers to the symbolic importance of a place as a repository of emotions that give meaning and purpose to life. Place dependence reflects functional or goaloriented connections with an environment. It indicates the extent to which physical 1 ASCII: American Standardized Code for Information Interchange. For exemple :) refers to the smiley 🙂. 2 This library is managed by a consortium of major Internet players (Amazon, Apple, Facebook, Google, Huawei, Microsoft, Netflix, Oracle, SAP, etc.) who decide after a long study process if the proposed emoticon can be included in the catalog (https://home.unicode.org) 3 According to Moser and Weiss (2003), environmental psychology studies the interrelations between the individual and his or her physical and social environment, from a spatio-temporal perspective.

environments provide conditions and characteristics that support a person's specific goals or desired activities [START_REF] Williams | Beyond the Commodity Metaphor: Examining Emotional and Symbolic Attachment to Place[END_REF]).

Beyond the iconic images used by territorial marketing to invite travel, how can a destination be enhanced through emotion? It comes down to asking how a destination can explore visitors' emotions during and after their journey and how to make their visits enriching experiences.

If emotion is a matter for visitors, we must not forget that this emotion must also concern the permanent residents of a territory, an emerging subject in tourism research. Indeed, residents also experience emotions concerning tourism development in their territory and its direct, indirect, and induced effects generated and sometimes suffered. The literature mainly explores reactions to the hosting of major sporting events [START_REF] Fabry | Pourquoi les villes ne veulent-elles plus accueillir les Jeux Olympiques ? Le cas des JO de 2022 et 2024[END_REF][START_REF] Ouyang | Role of Trust, Emotions and Event Attachment Residents' Attitudes toward Tourism[END_REF]), but more and more authors are interested in the stress of residents linked to tourism or the hosting of major sporting and cultural events [START_REF] Jordan | Tourism Impacts, Emotions and Stress[END_REF][START_REF] Zheng | The Role of Cognitive Appraisal, Emotion and Commitment in Affecting Resident Support toward Tourism Performing Arts Development[END_REF].

For a visitor to favorably associate emotions and experience, the place visited must provide most positive emotions. The visitor must also feel welcome and be able to interact with the permanent residents if they wish. It is, therefore, in a destination's interest to encourage these exchanges. In the digital age, a mobile application promoting the destination can also serve as an interface between the various population categories. Traditionally, applications promote access to information, digital products, and services. They can also support creative activities and local talent. Rarely do they allow emotions to be associated with places. Residents and visitors can share moments of life and values through geolocation. The originality lies in the fact that residents co-create and co-value the destination through a digital dialogue with visitors. By encouraging residents to share with visitors, tourism becomes more inclusive. We formulate the following proposals:

P1: Emotion is the basis of a renewed tourism valorization essentially carried by residents and visitors.

From the perspective of enhancing the value of a destination through digital technology, the destination can recover the application's data and try to explore and exploit it. This data would provide information not only on emotions but also on visitor journeys, visiting times, mobility, etc. This information would make it possible to improve and individualize the tourist offer. This data opens the way to an enhancement of the territory based on small data.

P2: The data generated by the application should, through its systematic analysis and exploitation, promote understanding visitors' needs and behavior to improve and individualize the tourism offer.

Emotions and the resulting data constitute an issue to explore in the management of tourist destinations and the ongoing work on the attractiveness of tourism on a more inclusive and even sustainable basis [START_REF] Barry | Attractivité territoriale, plus que du marketing touristique, Réseau de veille en tourisme[END_REF].

III. The use of an emotion-based application by a destination: some functional elements

Destination applications, initiated by the institutions in charge of tourism promotion, aim to make the territory in which visitors are temporarily staying intelligible [START_REF] Gavalas | Mobile recommender systems in tourism[END_REF]. At a minimum, they include characteristic tourist points, practical information (openings, prices, etc.), mobility (means of transport, timetables, prices, etc.), shops, restaurants and bars, places of entertainment, weather, the purchase of services, and any other element of interest to a person who does not intend to stay for the long term. In addition to these informative and transactional features, we propose to add an experiential dimension driven by emotion. The objective is not to measure emotion but to capture the sense of emotion expressed by visitors when they stay at the destination (1) to enable the destination to work on its attractiveness on a new basis (2).

Capturing the emotion of visitors via emoticons

As we saw in the first part, emotion is complex and composite. It reflects several categories of mental dispositions, including the six fundamental facial emotions identified by Eckmann. These are joy, anger, fear, sadness, surprise, and disgust. These emotions serve as the basis for the development of other derived emotions. They have a universal character (Eckmann 2002). For a territory that receives visitors from all over the world, this is an important characteristic. At the risk of oversimplification, the application can propose selecting emoticons deemed characteristic of a particular emotion that will be understood by all, regardless of the cultural base. By proposing to visitors to express their emotion(s) by selecting from a panel of emoticons, it then becomes possible to capture moments of satisfaction, contentment that is assimilated to a positive emotion, and moments of disappointment, sadness, frustration that refers to more negative emotions [START_REF] Derks | Emoticons and social interaction on the Internet: the importance of social context[END_REF]. The use of location data will make it possible to associate a place/point of interest with the emotions expressed.

Enhancing the value of the destination based on the emotions of its residents

A destination that attracts visitors is mainly concerned with enhancing its amenities and proposing places to visit and experience. Social networks and specifically dedicated platforms, for their part, allow individuals to communicate their experiences and rate certain services (Tripadvisor, Withlocals, etc.).

We can note that tourism involves a range of interactions between tourists and residents, which means that industrial development strategies need to consider both stakeholders.

In addition to tourists' perception, the local community's image of their place of residence and the destination is essential, as this audience acts passively and actively to shape the image of destinations in the minds of visitors. One of the biggest challenges for DMOs is to understand that they serve tourists and stakeholders directly around them and the local community.

Turning residents into tourism actors and even prescribers and promoters of the destination are growing (Mona 2021, Transat 2021 Chair). The idea we are proposing is to allow, through the destination application, the sharing of emotions between residents and visitors. It means allowing residents to attach fragments of their personal history to specific places in their city and communicate them to visitors. These stories could help to shape unusual tours or even new tourist services. These stories could help to shape unusual tours or even new tourist services. The addition of a dimension (text or photos) to the places visited helps arouse visitors' curiosity and attention off the beaten track.

Beyond the narrative dimension, this functionality of the application allows the destination not only to enhance its value by considering the emotions shared and crossed between residents and visitors but also to exploit small data (GPS, tracking via Wi-Fi terminals, connections to operators, RFID, etc.) and thus improve the image of the destination.

. Information for visitors produced by residents

Storytelling by residents about their attachment to the place in relation to their experiences, uses and practices.

Customization of tours and itineraries based on time at destination, budget, and selected individual stories.

Destination application

Traditional information functions

Small data Emotions Small data collection and exploitation: User generated content, device data, transaction data.

Improving the attractiveness of the destination through small data mining

Residents: Narratives related to places and their emotions

Enriching the visitor experience by sharing emotions with residents

Visitors: live like the locals and get off the beaten track

IV. Conclusion

Our proposal, helps to ensure that the destination and its permanent resident populations regain the ability to manage tourism rather than being impacted by it. This is what [START_REF] Haywood | A post COVID-19 future -tourism re-imagined and re-enabled[END_REF] proposes, when he suggests that residents make their destination 'special' and re-imagine future tourism on a smarter basis. As he points out, "Tourism becomes smarter by design when the revival and revitalization of all communities-asdestinations takes center stage and is taken seriously." (Haywood 2020, p.607).

The health context (Covid-19) also makes it possible to envisage the possibility of integrating other functionalities into the destination application, such as the number of visitors to a place. Visitors to a site could themselves indicate their feelings about the state of frequentation of a place. These are all means given to destinations to distinguish themselves and differentiate themselves.

Table 1 -

 1 Emoticon palette to capture visitors' emotions

	Emotions Emoticons	Natures	Manifestations	Expresses
	Joy	🙂	Satisfaction,	Happiness,	A positive emotion
			Success,	Good mood,
			Passion	Contentment,
				Interest
	Anger	😠	Frustration,	Closure,	A negative emotion
			Annoyance	Nervousness
	Fear	😱	Worry,	Apprehension,	A negative emotion
			Concern	Mistrust,
				Fear
	Sadness	🙁	Loss,	Unsatisfaction,	A negative emotion
			Disappointment	Affliction
	Surprise	😯	Unexpected,	Enthusiasm,	A positive emotion
			Revelation,	Wonder,
			Astonishment	Inspiration
	Disgust	🤮	Aversion,	Contempt,	A negative emotion
			Reject	Irritation
	Source : according to https://www.unicode.org/charts/PDF/U1F600.pdf

  Figure 1 -The place of emotions in the tourism experience managed by destinations

	Nature of the information	Category	Example of content (not exhaustive)
	Information for visitors	Points of interest	Museums,	archaeological	sites,
	produced	by	the		monuments, churches,
	territory's stakeholders		Parks and gardens
	and distributed by the	Ticketing and activities	Theaters,
	tourist offices				Exhibitions
					Guiding, visits, concierge service,
					Leisure
					Shopping, ....
				Local information	Local time
					Weather,
					Temperature
					Tourist office, ...
				Mobility to destination	Schedules and Lines
					Mobility solutions (walking, cycling, bus,
					streetcar, metro, etc.)
					Route calculation, ...
				Food and Lodging	Hotels
					Restaurants
					Bars
					Caterers, ..