Simon Redor

Maxim Avdeev

David Hrabovsky

Jean-Marie Tarascon

Gwenaëlle Rousse

Magnetic properties, magnetic structure, and possible magnetoelectric effect in orthorhombic

In Figures S1 to S8, the simulated patterns for the 8 Shubnikov space groups derived from the Pbcn nuclear space group are shown. For each magnetic space group, different peaks are observed depending on the number and the direction of the magnetic components taken into account in the calculations. For every simulated pattern M n was set to 1 µ B (n = x or y or z).

  Lattice parameters: a = 8.69523(8) Å, b = 5.06407(5) Å, c = 14.3451(18) Å, V = 631.66(12) Å 3 , Z = 4. Relaibility parameters: χ 2 = 2.65, R Bragg = 3.32 %. Phase composition: Pbcn Li 2 Ni 2 W 2 O 9 : 100.0(5) wt% / P2/c NiWO 4 : 0.02(1) wt%. Table S1 -Results of the Rietveld refinement for the room temperature XRD of the Li 2 Ni 2 W 2 O 9 sample used for the bulk magnetic measurements. Nuclear space group: Pbcn. Lattice parameters: a = 8.69663(7) Å, b = 5.06468(4) Å, c = 14.3514(11) Å, V = 632.117(8) Å 3 , Z = 4. Relaibility parameters: χ 2 = 2.11, R Bragg = 6.60 %. Phase composition: Pbcn Li 2 Ni 2 W 2 O 9 : 100.0 wt% / No other phase detected with XRD.

Figure S1 -

 S1 Figure S1 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pbcn Shubnikov space group (colored, bottom). Note that the (001) and (100) peaks appear if the magnetic moment results from two components along a and b.

Figure S2 -

 S2 Figure S2 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pb'c'n' Shubnikov space group (colored, bottom). Note that the (001) and (100) peaks appear if the magnetic moment results from two components along b and c.

Figure S3 -

 S3 Figure S3 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pb'cn Shubnikov space group (colored, bottom). Note that the (001) peak does not appear in any of the simulated patterns.

Figure S4 -

 S4 Figure S4 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pbc'n Shubnikov space group (colored, bottom). Note that neither the (001) peak, nor the (100) peak appear in any of the simulated patterns.

Figure S5 -

 S5 Figure S5 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pbcn' Shubnikov space group (colored, bottom). Note that the (100) peak does not appear in any of the simulated patterns.

Figure S6 -

 S6 Figure S6 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pbc'n' Shubnikov space group (colored, bottom). Note that the (001) peak does not appear in any of the simulated patterns.

Figure S7 -

 S7 Figure S7 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pb'cn' Shubnikov space group (colored, bottom). Note that neither the (001) peak, nor the (100) peak appear in any of the simulated patterns.

Figure S9 -

 S9 Figure S9 -Rietveld refinement of the neutron diffraction pattern of Li 2 Ni 2 W 2 O 9 powder at T = 3.5 K, using the Pbcn Shubnikov space groups for the calculations.

Figure S8 -

 S8 Figure S8 -Experimental neutron powder diffraction data at T = 3.5 K for Li 2 Ni 2 W 2 O 9 (black, top), and simulated patterns for k = (0, 0, 0) in the Pb'c'n Shubnikov space group (colored, bottom). Note that the (100) peak does not appear in any of the simulated patterns.

Figure S10 -

 S10 Figure S10 -Magnetic structure that would be displayed by Li 2 Ni 2 W 2 O 9 at T < T N if its Shubnikov space group was Pbcn. For the sake of clarity, only the nickel atoms are shown. The green arrows represent the magnetic moments.

Table S2 -

 S2 Results of the Rietveld refinement for the room temperature neutron powder diffraction of the Li 2 Ni 2 W 2 O 9 sample used for the magnetic structure study. Nuclear space group: Pbcn.

	Element	Wyckoff site	x/a	y/b	z/c	B iso (Å 2 )	Occupancy
	Li	8d	0.342	0.517	0.4939	1.80	1
	Ni	8d	0.6624(5)	0.497(4)	0.8153(18)	0.34	1
	W	8d	0.5237(12)	-0.004(12)	0.64356(5)	-0.01	1
	O1	4c	1/2	0.281	3/4	0.23	1
	O2	8d	0.5081	0.6793	0.9085	0.23	1
	O3	8d	0.6494	0.8519	0.7469	0.23	1
	O4	8d	0.8423	0.6635	0.8953	0.23	1
	O5	8d	0.3389	0.1542	0.9255	0.23	1
	Element	Wyckoff site	x/a	y/b	z/c	B iso (Å 2 )	Occupancy
	Li	8d	0.342(2)	0.517(4)	0.4939(7)	1.80(14)	1
	Ni	8d	0.6633(2)	0.5009(6)	0.81316(9)	0.34(17)	1
	W	8d	0.5240(3)	0.002(12)	0.6641(15)	-0.01(4)	1
	O1	4c	1/2	0.281(11)	3/4	0.23(14)	1
	O2	8d	0.5081(4)	0.6793(7)	0.9085(2)	0.23(14)	1
	O3	8d	0.6494(4)	0.8519(7)	0.7469(2)	0.23(14)	1
	O4	8d	0.8423(5)	0.6635(6)	0.8953(2)	0.23(14)	1
	O5	8d	0.3389(5)	0.1542(6)	0.9255(18)	0.23(14)	1


Claire Loiseau

Gabriele Sorci

Can microplastics facilitate the emergence of infectious diseases?

Keywords: antimicrobial resistance, biofilm, microbiota, plastic pollution, vectorial capacity

Plastic pollution is a major environmental problem. Small plastic particles (called microplastics) have been reported to have pernicious effects on human and wildlife health, by altering physiological functions (e.g., immunity, metabolism) and interfering with commensal microorganisms. However, in addition to these direct toxic effects, we suggest that microplastic pollution might also exert deleterious effects, modifying (i) the exposure to pathogens (e.g., multi-drug resistant bacteria) and (ii) the dynamics of vector-borne diseases. Therefore, we argue that microplastics should be considered as a ubiquitous environmental hazard, potentially promoting the (re)emergence of infectious diseases. The implementation of multi-and interdisciplinary research projects are crucial to properly evaluate if microplastic pollution should be added to the current list of global health threats.

INTRODUCTION

Since the middle of the 19th century, plastic materials have revolutionized our daily life, for better (e.g., in medical and digital sciences), and for worse (e.g., single-use plastic items), quickly becoming a dreadful visible pollutant [START_REF] Lavers | Exceptional and rapid accumulation of anthropogenic debris on one of the world's most remote and pristine islands[END_REF].

In addition to large waste, plastic pollution also involves small (<5 mm) particles (e.g., microfibers, microbeads and particles produced during the degradation of large plastic products), collectively called microplastics (Thompson et al., 2004). Microplastics are everywhere: in oceans and deep-sea [START_REF] Woodall | The deep sea is a major sink for microplastic debris[END_REF], in terrestrial ecosystems (Rilling and Lehmann, 2020), in the atmosphere [START_REF] Amato-Lourenço | An emerging class of air pollutants: Potential effects of microplastics to respiratory human health?[END_REF] and in organisms ranging from zooplankton [START_REF] Cole | The impact of polystyrene microplastics on feeding, function and fecundity in the marine copepod Calanus helgolandicus[END_REF] to vertebrates [START_REF] Carlin | Microplastic accumulation in the gastrointestinal tracts in birds of prey in central Florida, USA[END_REF][START_REF] Hou | A fish tale: a century of museum specimens reveal increasing microplastic concentrations in freshwater fish[END_REF], including humans [START_REF] Ragusa | Plasticenta: First evidence of microplastics in human placenta[END_REF][START_REF] Zhang | You are what you eat: microplastics in the feces of young men living in Beijing[END_REF]Fig. 1).

The effect of microplastic ingestion on organismal health has been evaluated and now there is evidence showing that microplastics can indeed exert deleterious effects on several physiological and life history traits in both aquatic and terrestrial species [START_REF] Blarer | Microplastics affect assimilation efficiency in the freshwater amphipod Gammarus fossarum[END_REF][START_REF] Cole | The impact of polystyrene microplastics on feeding, function and fecundity in the marine copepod Calanus helgolandicus[END_REF][START_REF] Lo | Negative effects of microplastic exposure on growth and development of Crepidula onyx[END_REF][START_REF] Salerno | Microplastics and the functional traits of fishes: a global meta-analysis[END_REF].

A detailed and exhaustive description of the underlying mechanisms is well beyond the scope of this discussion article; however, for instance, once ingested, microplastics can induce the mechanical disruption of the intestinal barrier integrity, causing (i) an upregulated expression of genes related to oxidative stress, (ii) inflammatory infiltration and production of pro-inflammatory cytokines such as TNF-, or (iii) the destruction of the structure of the intestinal mucosa (see [START_REF] Huang | Microplastic: a potential threat to human and animal health by interfering with the intestinal barrier function and changing the intestinal microenvironment[END_REF] for a recent review). Other direct toxic effects arise from the release of additives, plasticizers, trace elements and other chemicals that can provoke endocrine and inflammatory disorders once microplastics are ingested (see recent reviews: [START_REF] Bradney | Particulate plastics as a vector for toxic trace-elements uptake by aquatic and terrestrial organisms and human health risk[END_REF][START_REF] Callaghan | Plasticizers and cardiovascular health: role of adipose tissue dysfunction[END_REF][START_REF] Jiang | Health impacts of environmental contamination of micro-and nanoplastics: a review[END_REF][START_REF] Yilmaz | Endocrin disrupting chemicals: exposure, effects on human health, mechanism of action, models for testing and strategies for prevention[END_REF]. That said, a proper assessment of the potential deleterious effects of microplastic ingestion requires taking into account environmentally-relevant concentrations of plastic particles (i.e., those to which organisms are likely exposed in their natural environment during their lifespan).

Another important methodological issue when it comes to the assessment of the health consequences of microplastic pollution is related to the use of virgin microplastics versus aged particles, the reason being that microplastics released in the environment are readily colonized by a diversity of microorganisms. More specifically, several studies have reported that microplastics can carry different bacterial communities depending on their types [START_REF] Steinman | Persistent organic pollutants, metals, and the bacterial community composition associated with microplastics in Muskegon Lake (MI)[END_REF] and have also different composition and properties (antibiotic resistant bacteria and antibiotic resistance genes) compared to bacteria found in the surrounding environment [START_REF] Oberbeckmann | Environmental factors support the formation of specific bacterial assemblages on microplastics[END_REF][START_REF] Wang | Plastisphere enrich antibiotic resistance genes and potential pathogenic bacteria in sewage with pharmaceuticals[END_REF]Wu et al., 2019a;[START_REF] Zhang | Microplastics from mulching film is a distinct habitat for bacteria in farmland soil[END_REF]. Due to their persistence in the environment, microplastics might therefore play a role in spreading potentially harmful bacteria (both for humans and animals) across different environments (marine, freshwater, terrestrial). This would represent an additional, indirect, impact of microplastics on organismal health [START_REF] Bowley | Oceanic hitchhikers -Assessing Pathogen risks from marine microplastic[END_REF][START_REF] Prata | A One Health perspective of the impacts of microplastics on animal, human and environmental health[END_REF]. In addition, there are several reasons to think that microplastics might also alter the capacity of arthropod vectors to transmit a diversity of pathogens, through their effects on vector microbiota and immunity, or through the effects on interacting species (predators, competitors).

Despite the concern over the emergence and reemergence of infectious diseases at the local and global scales [START_REF] Morens | Emerging infectious diseases: Threats to human health and global stability[END_REF][START_REF] Plowright | Pathways to zoonotic spillover[END_REF], the possibility that microplastic pollution might facilitate the spread of pathogenic microorganisms and alter the dynamics of vector-borne diseases have been barely discussed and investigated. In this discussion article, we would like to highlight the link between two current global threats that are the increase in microplastic pollution and the acceleration of disease emergence. There are numerous unknown indirect effects of microplastics that need to be investigated to evaluate if and how microplastic pollution might promote the emergence of novel pathogens, potentially affecting human, livestock and wildlife health. Here, we indicate some research directions that might prove useful to assess the risk that microplastic pollution might exert as promoter of infectious disease (re)emergence. We focus on microplastics because of their ubiquitousness in the environment and their small size, which allows them to be ingested by a large diversity of aquatic and terrestrial organisms. We think they have the potential (i) to act as disease vehicles into organisms more easily than large plastic waste and (ii) to provoke physiological disorders into vectors of infectious diseases.

MICROPLASTICS CAN ACT AS VECTORS OF ANTIBIOTIC

RESISTANT BACTERIA

Increased exposure to antibiotic resistant bacteria

Virtually all human beings are today continuously exposed to plastic debris through ingestion and inhalation [START_REF] Zarus | A review of data for quantifying human exposures to micro and nanoplastics and potential health risks[END_REF]. A large amount of evidence has rapidly accumulated showing that we ingest microplastics on a daily basis: several food items and drinking water have been shown to be contaminated by microplastics [START_REF] Zhang | You are what you eat: microplastics in the feces of young men living in Beijing[END_REF]. Similarly, plastic debris with a size < 1m are suspended in the air and can be inhaled [START_REF] Amato-Lourenço | An emerging class of air pollutants: Potential effects of microplastics to respiratory human health?[END_REF][START_REF] Prata | Airborne microplastics: consequences to human health?[END_REF]. Estimates indicate that people living in highly urbanized areas inhale microplastic particles daily [START_REF] Wright | Atmospheric microplastic deposition in an urban environment and an evaluation of transport[END_REF] and this also applies to domestic indoor environments (Vianello et al., 2019).

The inhalation route has been corroborated by the finding of plastic particle contamination in human lung biopsies [START_REF] Amato-Lourenço | Presence of airborne microplastics in human lung tissue[END_REF].

Knowing all of this, an important parameter to take into consideration is that living beings get exposed through ingestion and inhalation not only to the microplastics but also to the microorganisms that live on them. The term "plastisphere" was coined by

Zettler and colleagues to describe the community of bacterial species that was found on marine plastic debris in the North Atlantic subtropical gyre [START_REF] Zettler | Life in the "plastisphere": microbial communities on plastic marine debris[END_REF]. The main result of their work, corroborated by several other studies, was that microorganisms living on microplastics were not a subsample of the surrounding environment, but rather showed a distinct diversity, composition and functional properties (Amaral- [START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF][START_REF] Mammo | Microplastics in the environment: Interactions with microbes and chemical contaminants[END_REF]Wang et al., 2020;Wu et al., 2019a;[START_REF] Xue | Increased inheritance of structure and function of bacterial communities and pathogens propagation in plastisphere along a river with increasing antibiotics pollution gradient[END_REF]. Microplastics have several features that promote bacterial colonization compared to the surrounding environment and natural substrates (Amaral- [START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF]Wu et al., 2019a). In particular, bacteria can readily form a biofilm on microplastics, which provides shelter and increases bacterial survival in the face of environmental stressors [START_REF] Laverty | Bacterial biofilms colonizing plastics in estuarine waters, with emphasis on Vibrio spp. and their antibacterial resistance[END_REF]Wang et al., 2020).

Antimicrobial resistance is one of the major health challenges that we will face in the upcoming years [START_REF] Cassini | Attributable deaths and disability-adjusted life-years caused by infections with antibiotic-resistant bacteria in the EU and the European Economic Area in 2015: a population-level modelling analysis[END_REF][START_REF] Collignon | Anthropological and socioeconomic factors contributing to global antimicrobial resistance: a univariate and multivariable analysis[END_REF]. Several studies revealed that microplastics selectively enrich antibiotic resistant bacteria and genes (ARBs and ARGs; [START_REF] Liu | Airborne microplastics from waste as a transmission vector for COVID-19[END_REF][START_REF] Sathicq | Are microplastic particles a hotspot for the spread and the peristance of antibiotic resistance in aquatic systems?[END_REF]Wang et al., 2020). Evidence shows that microplastics can harbor microbial communities carrying ARGs with abundances 100 to 5,000 times higher than planktonic bacteria [START_REF] Zhang | Potential risks of microplastics combined with superbugs: Enrichment of antibiotic resistant bacteria on the surface of microplastics in mariculture system[END_REF], although the process of ARGs enrichment depends on the features of the microplastics and their ageing (Su et al., 2021). Since biofilm formation favors horizontal gene transfer, microplastics can become hotspots for ARGs because resistance genes carried by mobile elements can spread more easily in the biofilm compared to planktonic bacterial communities [START_REF] Arias-Andres | Microplastic pollution increases gene exchange in aquatic ecosystems[END_REF]. Microplastics can also have affinity for antibiotics that are released in the environment [START_REF] Li | Adsorption of antibiotics on microplastics[END_REF] and for other chemical pollutants [START_REF] Scott | Perfluoroalkylated substances (PFAS) associated with microplastics in a lake environment[END_REF].

For example, antibiotics and trace metals can attach to microplastics and exert selection pressures on bacteria living on plastic debris, and may also have a synergistic effect on bacterial resistance due to co-selection [START_REF] Imran | Co-selection of multi-antibiotic resistance in bacterial pathogens in metal and microplastic contaminated environments: an emerging health threat[END_REF][START_REF] Yang | Plastics in the marine environment are reservoirs for antibiotic and metal resistance genes[END_REF]; Fig. 2).

Such synergistic effects can result in strengthened selection on resistant strains, and improve the resistome of microplastic-associated bacteria.

Although most studies have focused on microplastics in marine environments, the vast majority of plastic particles come from land-based human activities [START_REF] Xu | Are we underestimating the sources of microplastic pollution in terrestrial environment?[END_REF].

Therefore, the distribution of microplastics can be seen as a mosaic of pollution hotspots from which particles flow into other environments. Several hotspots of microplastic pollution have been identified such as urban wastewater treatment plants (WWTP) (Syranidou and Kalogerakis, 2022). These hotspots of microplastic pollution are however, also hotspots for antibiotics, trace elements, ARBs and ARGs [START_REF] Rizzo | Urban wastewater treatment plants as hotspots for antibiotic resistant bacteria and genes spread into the environment: A review[END_REF][START_REF] Zhang | Antibiotics and antibiotic resistance genes in landfills: A review[END_REF]. In these environments, microplastics might then serve as sinks for other pollutants and ARBs, and spread them in other environments via leachate, landfill leakage, effluents of WWTPs, sewage sludge [START_REF] Kelly | Wastewater treatment alters microbial colonization of microplastics[END_REF].

Current estimates indicate that 25% of microplastics found in marine environments come from WWTPs (Syranidou and Kalogerakis, 2022). However, although microplastics can successfully contribute to spread ARBs (homogenizing the risk of disease emergence), other environmental factors do contribute to the mosaic of disease emergence risk, including proximity to pollution hotspots and human population density. With this respect, urban WWTPs and landfills should be considered as particularly risky sites, especially when in proximity of megalopolis. Urban environments, highways and roads also produce large amounts of microplastics, such as tire particles that are subsequently found in the soil [START_REF] Sommer | Tire abrasion as a major source of microplastics in the environment[END_REF]. As other microplastic substrates, tire particles have been shown to harbor microbial communities that differ from the surrounding soil. Bacteria colonizing tire particles also have different sets of ARGs and, of more concern, the abundance of ARGs is enhanced when tire particles are exposed to soils containing other pollutants such as heavy metals and antibiotics [START_REF] Ding | Exposure to heavy metal and antibiotic enriches antibiotic resistant genes on the tire particles in soil[END_REF].

Risks associated with increased exposome and resistome

Microplastics have been reported to carry pathogens of aquatic organisms (fish, crustaceans, mollusks). Isolates of the fish pathogen Aeromonas salmonicida expressing virulence factors related to the secretion systems Type II and IV, and several toxins have been retrieved from plastic debris [START_REF] Radisic | Marine plastics from Norwegian West coast carry potentially virulent fish pathogens and opportunistic human pathogens harboring new variants of antibiotic resistance genes[END_REF]Viršek et al., 2017).

These isolates also carried antimicrobial resistance genes (lactamases and chloramphenicol acetyltransferase), which raise the question of the potential impact that microplastics might have as disease vectors in aquaculture [START_REF] Dong | Interactions of microplastics and antibiotic resistance genes and their effects on the aquaculture environments[END_REF]. In the same line, an incubation experiment showed that, compared to surrounding seawater, mariculture-derived microplastics harbor (i) distinct bacterial communities as the exposure time increases, (ii) a higher abundance of bacteria of the genera Vibrio, Pseudoalteromonas and Alteromonas (Sun et al., 2020). The idea that microplastics might be vehicles of pathogenic microorganisms in aquaculture systems is further supported by recent evidence showing that oysters (Oestra edulis) have a higher uptake rate of microplastics coated with bacteria compared to virgin particles. In addition, oysters exposed to bacteria-coated microplastics have increased oxygen consumption and respiration rate compared to oysters exposed to virgin microplastics [START_REF] Fabra | The plastic Trojan horse: Biofilm increase micoplastic uptake in marine filter feeders impacting microbial transfer and organism health[END_REF]. Therefore, benthic filter feeders are likely to be preferentially exposed to bacteria-carrying microplastics and as such transfer them over the trophic chain, up to humans.

Similarly, several human pathogens (including opportunistic ones) have been discovered on plastic debris in different geographical regions and environments (Table 1). For instance, enteropathogenic strains of Vibrio and Escherichia coli have been reported on microplastics sampled at the Guanabara Bay (Brazil) while water samples had low bacterial titers [START_REF] Silva | Dispersal of potentially pathogenic bacteria by plastic debris in Guanabara Bay, RJ, Brazil[END_REF]. Hence, microplastics can substantially contribute to spread such pathogens from areas with fecal bacteria contamination to preserved zones. If these preserved zones involve recreational environments, such as beaches and bathing waters, we might easily predict an enhanced human exposure to pathogenic organisms conveyed by microplastics [START_REF] Keswani | Microbial hitchhikers on marine plastic debris: Human exposure risks at bathing waters and beach environments[END_REF].

We suggest that the combined effect of microplastic-associated exposome and resistome might enhance the risk of emergence of novel bacterial pathogens. Although increased exposure and increased antimicrobial resistance alone represent risk factors for the emergence of bacterial diseases, microplastics might play the role of catalyzers favoring the exposure to resistant strains. In addition to this, microplastics carrying ARBs and ARGs might contribute to spread resistance genes over large distances, even in areas with no antibiotic pollution and to homogenize globally the risk of emergence of bacterial pathogens (Fig. 2). The example of polystyrene recovered in Antarctica, harboring microorganisms able to resist multiple drugs, including cephalosporin and quinolone [START_REF] Laganà | Do plastics serve as a possible vector for the spread of antibiotic resistance? First insights from bacteria associated to a polystyrene piece from King George Island (Antarctica)[END_REF] should act as a serious red flag.

Although most of the evidence supporting the role of microplastics as vectors of pathogens comes from aquatic environments, the possibility that particles suspended in the air might also serve as vectors for pathogens, including airborne viruses, cannot be discarded. For instance, it has been suggested that SARS-CoV-2 might have longer survival on air suspended plastic particles than in the aerosol, contributing to the transmission and the spread of the pathogen [START_REF] Liu | Airborne microplastics from waste as a transmission vector for COVID-19[END_REF]. ARGs have already been reported in airborne bacteria sampled in a WWTP in China, including the opportunistic human pathogens Comamonas and Moraxella [START_REF] Li | Bioaerosol emissions and detection of airborne antibiotic resistance genes from a wastewater treatment plant[END_REF]. If these airborne bacteria can use microplastics as aquatic microorganisms do, there might be a risk of environmental contamination well beyond the surroundings of WWTPs.

Summary

Here, we argued that microplastic pollution is likely to pose a threat to human and wildlife health, not only because of direct toxic effects due to the ingestion of plastic particles, but also due to the exposure to pathogenic microorganisms carrying antimicrobial resistance genes. Our argument is based on a series of well-established observations showing that microplastics harbor communities of microorganisms that 1) differ from those found on natural substrates or as planktonic forms, and 2) present a greater abundance of ARBs and ARGs. From these two observations, we consider that microplastics, by increasing the contact rate between hosts and pathogens, could facilitate the emergence of multi-drug resistant bacterial pathogens and pose a threat to humans, livestock and wildlife species. Actually, we might draw a parallel with the recent emergence of bacterial pathogens associated with the use of medical devices (such as prostheses and catheters). Bacteria can form biofilms on such devices [START_REF] Pelling | Bacterial biofilm formation on indwelling urethral catheters[END_REF][START_REF] Stickler | Bacterial biofilms in patients with indwelling urinary catheters[END_REF]Tande and Patel, 2014) as they do on microplastics and several new pathogens are now associated with the use of urinary catheters (Aerococcus spp., Actinobaculum schaalii, Alloscardovia omnicolens; Vouga and Greub, 2016). We suggest that the potential health and economic costs associated with the emergence of novel multi-drug resistant microorganisms is worth to stimulate more research in this area, as to better quantify the risk associated with microplastic pollution.

MICROPLASTICS CAN ALTER VECTOR-BORNE INFECTIOUS DISEASE TRANSMISSION

Emerging pathogens have been associated not only with antimicrobial resistance, but also with the spread of arthropod vectors (Kilpatrick and Randolph, 2012). In fact, in addition to parasites that are directly or environmentally transmitted, wildlife and humans are exposed to multiple vector-borne pathogens, many of them being considered as emerging or reemerging (e.g., Dengue, Zika, West Nile viruses; [START_REF] Jones | Global trends in emerging infectious diseases[END_REF][START_REF] Morens | The challenge of emerging and re-emerging infectious diseases[END_REF]. Historically, vector control has relied on synthetic insecticides and environmental management of breeding sites, but several novel vector control tools are under development, such as the gene-drive technology or bacterial infection of vectors (e.g., Wolbachia; [START_REF] Wilson | The importance of vector control for the control and elimination of vector-borne diseases[END_REF]. We suggest that microplastic pollution, as a ubiquitous environmental hazard, should be taken into account in vector control strategies since it could interfere either with life history traits of vectors and/or their microbiota.

Alteration of vector life history traits

Many medically important insect-vectors or intermediate hosts of infectious diseases (i.e., mosquitoes, black flies, snails) are freshwater organisms inhabiting lakes, rivers, swamps or coastal ponds that can be heavily polluted with microplastics [START_REF] Akindele | Freshwater insects of different feeding guilds ingest microplastics in two Gulf of Guinea tributaries in Nigeria[END_REF]Wagner and Lambert, 2018;[START_REF] Windsor | Microplastic ingestion by riverine macroinvertebrates[END_REF]). Microplastics contamination is not uniformly distributed around the globe and varies with freshwater ecosystems. Although lakes are generally more contaminated than rivers, and riverine and lacustrine sediments are more contaminated than the aquatic matrix [START_REF] Cera | Microplastics in freshwater: what is the news from the world?[END_REF], microplastics in lotic ecosystems are more studied than in lentic systems.

In addition, invertebrates are largely understudied compared to vertebrates (i.e., fish; [START_REF] Cera | Freshwater wild biota exposure to microplastics: A global perspective[END_REF]. Nonetheless, it has been shown that freshwater invertebrates have the capacity to ingest microplastics, the quantity of uptake depending on their feeding type and morphology [START_REF] Scherer | Feeding type and development drive the ingestion of microplastics by freshwater invertebrates[END_REF][START_REF] Setälä | Feeding type affects microplastic ingestion in a coastal invertebrate community[END_REF]. Interestingly, experimental work found that adult mosquitoes do not avoid water contaminated with microplastics when ovipositing, and that larvae do ingest microplastics that remain at the adult stage with, however, no impact on their survival rate [START_REF] Al-Jaibachi | Up and away: ontogenic transference as a pathway for aerial dispersal of microplastics[END_REF][START_REF] Al-Jaibachi | Examining effects of ontogenic microplastic transference on Culex mosquito mortality and adult weight[END_REF][START_REF] Cuthbert | The influence of microplastics on trophic interaction strengths and oviposition preferences of dipterans[END_REF].

Depending on the vector species and their ecological niches, microplastics might have variable influence on vector's life history traits and population dynamics. Even so, we can expect various outcomes of the microplastic ingestion by the vectors, either direct effects on development, growth or survival, or indirect effects on vectorial capacity (Fig. 3). Vectorial capacity (VC) can be defined as a measurement of the efficiency of vector-borne disease transmission (Garrett- [START_REF] Garrett-Jones | Prognosis for Interruption of Malaria Transmission through Assessment of the Mosquito's Vectorial Capacity[END_REF], corresponding to the basic reproductive rate of a vector-borne pathogen. Vectorial capacity can be assessed using the following equation:

𝑉𝐶 = 𝑚𝑎 2 𝑏𝑝 𝑁 -ln(𝑝)
where m refers to vector density, a to vector biting rate, b to vector competence (the susceptibility of the vector to become infected and transmit the pathogen), p to the probability of vector daily survival, and N to the pathogen extrinsic incubation period [START_REF] Cansado-Utrilla | The microbiome and mosquito vectorial capacity: rich potential for discovery and translation[END_REF]. Therefore, when vectors get easily infected, have high survival rates, high density, and bite at high rate, they effectively transmit the disease.

We propose that microplastics have the potential to affect all these parameters (e.g., impairing vector survival, or decreasing extrinsic incubation rate).

The effect of microplastic pollution on vector-borne diseases can be nonetheless particularly difficult to predict because of the complex ecological interactions that operate at different temporal and spatial scales. Microplastics might decrease parasite transmission if they have negative effects on vectors (reducing fecundity and/or survival). It has been well documented, for example, that decreasing the survival of adult Anopheles (the mosquitoes that transmit human malaria) reduced their vectorial capacity [START_REF] Brady | Vectorial capacity and vector control: reconsidering sensitivity to parameters for malaria elimination[END_REF]. Negative impacts of microplastics have also been reported on food intake, growth, reproduction or survival in freshwater invertebrates [START_REF] Foley | A meta-analysis of the effects of exposure to microplastics on fish and aquatic invertebrates[END_REF]. Therefore, snail-borne parasitic diseases, such as fascioliasis or schistosomiasis, could be less prevalent if microplastic pollution increases aquatic snail mortality and/or decreases cercariae release in water.

Modification of vector microbiota

Microplastics, and the associated 'plastisphere', might interact with the vector microbiota and immune system and consequently increase the vectorial capacity.

Recent work has focused on the effects of microplastics on the microbiota [START_REF] Fackelmann | Microplastics and the gut microbiome: How chronically exposed species may suffer from gut dysbiosis[END_REF]) and the immune system in model organisms -in mice [START_REF] Jin | Impacts of polystyrene microplastic on the gut barrier, microbiota and metabolism of mice[END_REF][START_REF] Lu | Polystyrene microplastics induce gut microbiota dysbiosis and hepatic lipid metabolism disorder in mice[END_REF] and zebra fish [START_REF] Jin | Polystyrene microplastics induce microbiota dysbiosis and inflammation in the gut of adult zebrafish[END_REF][START_REF] Limonta | Microplastics induce transcriptional changes, immune response and behavioral alterations in adult zebrafish[END_REF][START_REF] Qiao | Accumulation of different shapes of microplastics initiates intestinal injury and gut microbiota dysbiosis in the gut of zebrafish[END_REF] and invertebrates [START_REF] Ju | Effects of polyethylene microplastics on the gut microbial community, reproduction and avoidance behaviors of the soil springtail, Folsomia candida[END_REF][START_REF] Zhu | Exposure of soil collembolans to microplastics perturbs their gut microbiota and alters their isotopic composition[END_REF]. For example, it has been shown that ingestion of microplastics by dipteran larvae triggers the activation of phenoloxidase, one of the effectors of innate immunity (Silva et al., 2021a), but also increases oxidative damage (Silva et al., 2021b). Therefore, we might expect that insect vectors (such as black flies transmitting nematodes causing river blindness, or mosquitoes transmitting Plasmodium causing malaria) could have an altered vectorial capacity if the ingestion of microplastics affects their physiology (e.g., the immune response) and/or the microbiota. In fact, the microbiota has an important role in modulating the mosquito immune response and can alter susceptibility to infection through either immune priming or direct inhibition [START_REF] Caragata | Curious entanglements: interactions between mosquitoes, their microbiota, and arboviruses[END_REF]. It has been shown that the removal of the midgut bacteria through antibiotic treatment led to a greater susceptibility of Anopheles gambiae to Plasmodium falciparum and of Aedes aegypti to dengue virus infection [START_REF] Dong | Implication of the mosquito midgut microbiota in the defense against malaria parasites[END_REF][START_REF] Xi | The Aedes aegypti toll pathway controls dengue virus infection[END_REF]. Bacteria can also influence mosquito competence by impairing pathogen infection (Cirimotich et al., 2011a;Cirimotich et al., 2011b). On the contrary, the presence of members of the Enterobacteriaceae family in the mosquito midgut favors Plasmodium infection [START_REF] Boissière | Midgut microbiota of the malaria mosquito vector Anopheles gambiae and interactions with Plasmodium falciparum infection[END_REF], while another recent study reported that Aedes mosquitoes harboring a Serratia marcescens strain were more permissive to dengue virus infection than those free of S.

marcescens (Wu et al., 2019b). These few examples show the complexity of the interactions between pathogens and microbiota that could affect mosquito vectorial capacity. Considering that microplastics often lead to dysbiosis and alter immune responses, we suggest that mosquitoes, or other arthropod vectors, ingesting microplastics could see their susceptibility to pathogens decrease or increase, depending on the bacteria and contaminants that microplastics carry (Fig. 3). Finally, little is known regarding how the microbiome influences the extrinsic incubation period (EIP; [START_REF] Cansado-Utrilla | The microbiome and mosquito vectorial capacity: rich potential for discovery and translation[END_REF], but it has been shown that Wolbachia could increase the EIP of the dengue virus in Aedes aegypti [START_REF] Carrington | Field-and clinically-derived estimates of Wolbachia-mediated blocking of dengue virus transmission potential in Aedes aegypti mosquitoes[END_REF]. It cannot be excluded that on the opposite, the plastisphere could shorten the EIP and have a positive effect on pathogen transmission.

Changes in the trophic chain

We propose that microplastics might have even more subtle effects on the transmission of vector-borne diseases, by altering the population dynamics of interacting species (predator -prey, competing species). Even if microplastic pollution has no direct effect on vector species, it may have one on natural enemies that play an essential role to regulate the abundance of vectors, which is one of the factors, that influences the vectorial capacity. Pathogen transmission could drastically increase if natural enemies fail to regulate the vector proliferation (Fig. 3). For example, numerous aquatic insects or amphibians are regulating mosquito larvae density either by direct predation [START_REF] Quiroz-Martínez | Aquatic insects as predators of mosquito larvae[END_REF] or even by indirect stress caused by predators [START_REF] Bellamy | Mosquito responses to trait-and density-mediated interactions of predation[END_REF]. A study found that mosquito size, fecundity and longevity decreased with indirect effects of predation, which would result in a significant decrease of malaria transmission [START_REF] Roux | Evidence for carry-over effects of predator exposure on pathogen transmission potential[END_REF]. On the opposite, if predators are negatively affected by microplastic pollution, vectors would benefit from it and see their vectorial capacity enhanced. In agreement with this hypothesis, a recent meta-analysis showed that feeding activity of fish, such as the predatory performance or number of ingested prey, was consistently impaired in the presence of microplastic pollution [START_REF] Salerno | Microplastics and the functional traits of fishes: a global meta-analysis[END_REF]. Finally, human-driven alteration of top-down regulation of vector populations has also been reported. An emblematic example is the building of a dam on the Senegal River that was associated with an outbreak of schistosomiasis, due to the interference with the migration of prawns, one of the main predators of the snails that are the intermediate hosts of schistosome parasites. Reintroduction of prawns resulted in reduced abundance of infected snails and reduced prevalence of schistosomiasis [START_REF] Sokolow | Reduced transmission of human schistosomiasis after restoration of a native river prawn that preys on the snail intermediate host[END_REF].

Despite the manifold reasons to expect an effect of microplastic pollution on the dynamics of infectious diseases, studies on the relationship between the exposure of microplastics and risk of parasite infection in wildlife, livestock or humans are so to speak inexistent. Only two correlative studies found positive association between accumulation of microplastics and presence of parasites -in grey seals [START_REF] Hernandez-Milian | Microplastics in grey seal (Halichoerus grypus) intestines: are they associated with parasite aggregations?[END_REF] and fishes (Grazia [START_REF] Pennino | Ingestion of microplastics and occurrence of parasite association in Mediterranean anchovy and sardine[END_REF], and only one experimental study investigated the effects of microplastic ingestion in tadpoles and trematodes [START_REF] Buss | Effects of polyester microplastic fiber contamination on amphibian-trematode interactions[END_REF]. The latter study performed an experimental microplastic exposure during one day (i.e., 24h) and found that individuals that ingested microplastics did not show a significant increase in parasite infection risks. However, one can argue that the time of exposure might not reflect the likely experience of wild organisms, which are rather chronically exposed to this pollutant.

Summary

Here, we argued that microplastic pollution is likely to modify vector-borne disease dynamics via two pathways, either by altering component of the vectorial capacity or by modifying the interactions between species along the trophic chain.

First, our argument is based on studies showing that arthropods, either as larvae or adults, are capable of ingesting microplastics, which are known to interfere with the microbiota of organisms. Because infection dynamics of pathogens (viruses or parasites) are regulated by the microbiota, we hypothesize that microplastics could facilitate the emergence of vector-borne diseases by increasing the vectorial capacity.

Second, we believe that even if microplastics do not have a significant effect on the vector microbiota and immunity, they might have undesirable consequences on predators and/or competitors, which could fail to regulate vector populations.

To date, and to our knowledge, no study has investigated the impact of microplastics on vectorial capacity, which is surprising given the possible health and economic consequences. More research in this area should be performed to properly evaluate the emergence risk of vector-borne diseases associated with microplastic pollution.

RESEARCH DIRECTIONS

We argued here that microplastic pollution might promote infectious disease emergence through different pathways. Although there exists evidence supporting this view, we realize that the link between microplastics and the dynamics of infectious diseases (in particular vector-borne diseases) has been barely investigated. As such, we hope that this work will contribute to stimulate further investigation to better understand how microplastics may interfere with antimicrobial resistance and the (re)emergence of vector-borne diseases.

Whatever the aspect of microplastic contamination we want to address, it is worth reminding ourselves that whether in laboratory conditions or in the field, environmentally realistic concentration of microplastics should be used [START_REF] Lenz | Critical view on used microplastic concentrations[END_REF], while avoiding primary and pristine microplastics that are not representative of the contamination in situ. The age, shape, size and composition (additives and contaminants) of microplastics as well as their bacterial communities are parameters that should be better apprehended [START_REF] Waldman | Microplastic research should embrace the complexity of secondary particles[END_REF]. We realize however that identifying environmentally-relevant microplastic concentrations is already a challenging task because it does not only depend on the presence of microplastics in the environment but also on the actual exposure of organisms (spatially and temporally), which is linked to ecological and physiological traits of the studied species (e.g., feeding habits, lifespan). This suggests that environmentally-relevant concentrations might even differ between species living in the same areas.

Our knowledge on the many-fold impacts of microplastic pollution on virtually all levels of biological organizations (from gene expression to ecosystem functioning) is rapidly expanding. Research work has been built on a variety of species, environments, experimental contexts, and has provided indisputable evidence showing that microplastics do represent a major environmental concern. However, as we learn more, we think it is time to place greater emphasis on the possible role that microplastic pollution might also play as promoter of infectious diseases emergence.

Taking this into account, we would like to provide some suggestions for future research.

How does urbanization influence bacteria colonization on microplastics and infection risk?

We need to better understand how the process of microplastic colonization by microorganisms changes along the gradient of anthropogenic disturbance. Heavily polluted environments are hotspots for multi-drug resistant pathogens, however microplastics have the potential to spread them at very large geographical scales. Long term surveys of bacteria richness and diversity on microplastics should be performed in soil and aquatic environment along gradient from cities and intensive agricultural lands where the trace metals and antibiotics are most likely to occur, to wilderness areas that present low human population density. This might allow us to draw risk maps, providing a useful tool to guide decisions on where monitoring effort should be prioritized.

Can microplastics biofilm interfere and undermine mosquito control methods?

Future work should focus on the possible effects of microplastics on vector control strategies. Wolbachia-based strategy for controlling the transmission of dengue and other arboviral infections has been largely investigated and field releases have already occurred in different places [START_REF] Kamtchum-Tatuene | The potential role of Wolbachia in controlling the transmission of emerging human arboviral infections[END_REF]. However, microplastics might be able to modify bacterial communities and/or their interactions, and potentially wipe off the suppressive effects of Wolbachia on arboviroses. Biocontrol strategies should definitely take into consideration this increasing environmental pollution.

Does aquatic environment influence interaction between microplastics and population dynamics of vectors?

More effort should be put in the understanding of the environmental traits that might act in synergy with microplastic pollution. For instance, microplastics present in freshwater versus brackish water, or in small versus large body of water might affect differently the population dynamics of vectors. This is not a trivial point since many species of insect vectors or intermediate hosts have different ecological requirements and habitat preferences. We should investigate which ecological and biogeographical parameters are more likely to impact vector and intermediate host life history traits and vectorial capacity in the context of chronic exposure to microplastics.

Do microplastics change the population dynamics of vectors by reducing the abundance and/or behavior of natural enemies?

We advocate for a more holistic approach to the study of microplastics, that explores the effects at the ecosystem level. Up to now, most of the studies have focused on the direct effect of microplastic pollution on the species of interest. Nevertheless, the abundance of vectors and intermediate hosts of human infectious diseases can dramatically change if predators decline in number in polluted areas. We suggest that future work should implement a more integrative approach to assess the effect of microplastics on ecosystem functioning.

CONCLUSIONS

Microplastics are clearly not a classical pollutant, in the sense that they are ubiquitous and highly persistent, which make them today an accepted indicator of the anthropocene [START_REF] Zalasiewicz | The geological cycle of plastics and their use as a stratigraphic indicator of the Anthropocene[END_REF]. Evidence is rapidly accumulating showing that microplastics can have pernicious health effects on organisms that ingest/inhale them due to their direct toxicity. In this discussion piece, we draw the attention to an additional pernicious effect of microplastic pollution. We argue that microplastics can facilitate the emergence of infectious disease by increasing the exposure of bacterial pathogens to human, livestock and wildlife and by altering the dynamics of vector-borne infections.

We suggest that if microplastics do play a role promoting the (re)emergence of infectious diseases (both multi-drug resistant bacteria and vector-borne pathogens), they will pose a more serious threat in terms of public health than contaminants absorbed by microplastics (e.g., persistent organic pollutants; [START_REF] Amelia | Marine microplastics as vectors of major ocean pollutants and its hazards to the marine ecosystem and humans[END_REF]. Our argument is based on two lines of reasoning: first, once an infectious disease has emerged, the whole population of susceptible individuals is potentially at risk, irrespective of whether they are exposed to the source of the emergence or not. The COVID-19 pandemic perfectly illustrates this point. Although SARS-CoV-2 is supposed to have initially infected humans exposed to animals carrying the virus, at a very local scale, human-to-human transmission rapidly made the whole human population at risk (including those not exposed to SARS-CoV-2 reservoirs). Obviously, this argument does not apply to non-transmissible diseases induced by contaminants.

The second line of reasoning relies on the potential that microorganisms have to adapt to novel environmental conditions through evolutionary changes. These evolutionary changes might contribute to increase (or decrease) infectivity and virulence, depending on the selective forces operating on the pathogens and the epidemiological context.

Again this provides a very different scenario compared to inert toxic compounds that are not expected to become more toxic in response to natural selection.

We are convinced that interdisciplinary research projects are necessary to address all these questions and to propose global maps of emergence risks of infectious diseases associated with microplastic pollution, as it already exists with climate change. We are aware that such investigations are difficult to launch because it has to bring together experts from different biological fields, such as ecotoxicologists, microbiologists, ecologists, chemists, epidemiologists and modeling scientists. Because the framework that allow such holistic approach already exists into the 'One Health' concept [START_REF] Zinsstag | From "one medicine" to "one health" and systemic approaches to health and well-being[END_REF], researchers should be encouraged to develop studies both in controlled environments and in situ in order to anticipate future public health issues.

Table 1. A non-exhaustive list of bacterial species that can potentially cause human diseases (including opportunistic pathogens) found on microplastics. We report the location and the environmental type to illustrate the diversity of sites and environments where microplastics have the potential to carry pathogenic bacteria. However, the disruption of vector homeostasis due to plastic debris and/or the interference of the plastisphere with the vector's microbiota might also result in a better vectorial capacity, and increase the risk of disease transmission. Even in the absence of toxicity in vectors, microplastic pollution might affect their population dynamics through changes in the abundance of interacting species (i.e., predators and competitors), producing an increased risk of disease transmission.

Figure 3

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .Figure 3 .

 23 Figure 2.

ACKNOWLEDGEMENTS C.L. acknowledges support from the Portuguese National Funds through FCT (Foundation for Science and Technology) under the PTDC/BIA-EVL/29390/2017 Research Project. G.S. acknowledges support from the French Agence Nationale de la Recherche (project COINFECT).

AUTHOR CONTRIBUTIONS

C.L. and G.S. conceived and wrote the manuscript together. C.L. produced the paintings in the figures. Su, Y. et al., 2021. Microplastics act as vectors for antibiotic resistance genes in landfill leachate: The enhanced roles of the long-term aging process. Environ. Pollut. 270, 116278. Sun, X. et al., 2020. Impact of mariculture-derived microplastics on bacterial biofilm formation and their potential threat to mariculture: A case in situ study on the Sungo Bay, China. Environ. Pollut. 262, 114336. Sun, Y. et al., 2021. Selection of antibiotic resistance genes on biodegradable and nonbiodegradable microplastics. J. Hazard. Mat. 409, 124979.

Syranidou, E., Kalogerakis, N., 2022. Interactions of microplastics, antibiotics and antibiotic resistant genes within WWTPs. Sci. Total Environ. 804, 150141. Tande, A.J., Patel, R., 2014. Prosthetic joint infection. Clin. Microbiol. Rev. 27, 302-345. Thompson, R.C. et al., 2004. Lost at sea: where is all the plastic? Science 304, 838. Vianello, A. et al., 2019. Simulating human exposure to indoor airborne microplastics using a Breathing Thermal Manikin. Sci. Rep. 9, 8670. Viršek, M.K., Lovšin, M.N., Koren, S., Kržan, A., Peterlin, M., 2017. Microplastics as a vector for the transport of the bacterial fish pathogen species Aeromonas salmonicida. Mar. Pollut. Bull. 125, 301-309. Vouga, M., Greub, G., 2016 [START_REF] Gong | Microbial biofilm formation and community structure on low-density polyethylene microparticles in lake water microcosms[END_REF] * Environmental type refers to both in situ sampling and microplastic exposure to environmental samples under experimental 888 conditions. 889


David Knossow

Avinash Sharma

Diana Mateus

Radu Horaud

Inexact Matching of Large and Sparse Graphs Using Laplacian Eigenvectors

In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a low-dimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest non-null eigenvalues of the Laplacian matrices of two graphs, we project each graph onto its eigenenvectors. We estimate the histograms of these one-dimensional graph projections (eigenvector histograms) and we show that these histograms are well suited for selecting a subset of significant eigenvectors, for ordering them, for solving the sign-ambiguity of eigenvector computation, and for aligning two embeddings. This results in an inexact graph matching solution that can be improved using a rigid point registration algorithm. We apply the proposed methodology to match surfaces represented by meshes.

Introduction

Many problems in computer vision, shape recognition, document and text analysis can be formulated as a graph matching problem. The nodes of a graph correspond to local features or, more generally, to objects and the edges of the graph correspond to relationships between these objects. Solution to graph matching consists of finding an isomorphism (exact matching) or an optimal sub-graph isomorphism (inexact matching) between the two graphs. Spectral graph matching methods are attractive because they provide a framework that allows to embed graphs into isometric spaces and hence to replace the initial NP-hard isomorphism problem with a more tractable point registration problem.

An undirected weighted graph with N nodes can be represented by an N × N real symmetric matrix, or the adjacency matrix of the graph. Provided that this matrix has N distinct eigenvalues, the graph can be embedded in the orthonormal basis formed by the corresponding eigenvectors. Hence, an N -node graph becomes a set of N points in an N -dimensional isometric space. In [START_REF] Umeyama | An eigen decomposition approach to weighted graph matching problems[END_REF] it is proved that the eigendecomposition of the adjacency matrices provide an optimal solution for exact graph matching, i.e., matching graphs with the same number of nodes. The affinity matrix of a shape described by a set of points can be used as the adjacency matrix of a fully connected weighted graph [START_REF] Scott | An algorithm for associating the features of two images[END_REF][START_REF] Shapiro | Feature-based correspondence: an eigenvector approach[END_REF][START_REF] Carcassoni | Correspondence matching with modal clusters[END_REF][START_REF] Carcassoni | Spectral correspondence for point pattern matching[END_REF]. Although these methods can only match shapes with the same number of points, they introduce the heat kernel to describe the weights between points (nodes), which has a good theoretical justification [START_REF] Belkin | Laplacian eigenmaps for dimensionality reduction and data representation[END_REF].

Unfortunately, exact graph matching is not very practical, in particular when the two graphs have a different number of nodes, e.g., constructed from real data, such as visual sensors. Therefore one needs to combine spectral analysis with dimensionality reduction such that two graphs that are being matched are embedded in a common subeigenspace with lower dimension than to the original graphs. This immediately calls for methods that allow many-to-many point correspondences. A clever idea is to combine matching with agglomerative hierarchical clustering, as done in [START_REF] Caelli | An eigenspace projection clustering method for inexact graph matching[END_REF]. We also note that spectral matching has strong links with spectral clustering [START_REF] Ng | On spectral clustering: analysis and an algorithm[END_REF] which uses the Laplacian matrix of a graph [START_REF] Chung | Spectral Graph Theory[END_REF].

The analysis of the spectral methods cited above rely on the eigenvalues of the adjacency or Laplacian matrices. The strict ordering of the eigenvalues allows the alignment of the two eigenbases, while the existence of an eigengap allows the selection of a lowdimensional eigen space. In the case of inexact matching of large and sparse graphs, a number of issues remain open for the following reasons. The eigenvalues cannot be reliably ordered and one needs to use heuristics such as the ones proposed in [START_REF] Shapiro | Feature-based correspondence: an eigenvector approach[END_REF][START_REF] Zhang | Spectral methods for mesh processing and analysis[END_REF]. The graph matrices may have eigenvalues with geometric multiplicities and hence the corresponding eigenspaces are not uniquely defined. Dimensionality reduction relies on the existence of an eigengap. In the case of large graphs, e.g., ten thousands nodes, the eigengap analysis yields an eigen space whose dimension is not well suited for the task at hand. The sign ambiguity of eigenvectors is generally handled using simple heuristics [START_REF] Umeyama | An eigen decomposition approach to weighted graph matching problems[END_REF][START_REF] Caelli | An eigenspace projection clustering method for inexact graph matching[END_REF]. The link between spectral matching and spectral clustering has not yet been thoroughly investigated. Existing spectral matching algorithms put the eigenvectors on an equal footing; the particular role played in clustering by the the Fiedler vector [START_REF] Chung | Spectral Graph Theory[END_REF] has not been studied in the context of matching. We remark that the selection of strongly localized eigenvectors, which we define as the vector (eigenfunction) that spans over a small subset of the graph while being zero elsewhere, hence corresponding to subgraph clusters, has not been studied in depth. The only existing strategy for selecting such eigenvectors is based on eigenvalue ordering and the detection of an eigengap.

In this paper we propose an inexact spectral matching algorithm that embeds large graphs on an isometric space spanned by a subset of eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix. We claim that the tasks of (i) selecting a subset of eigenvectors, (ii) ordering them, (iii) finding a solution to the sign-ambiguity problem, as well as (iv) aligning two embeddings, can be carried out by computing and comparing the histograms of the projections of the graphs' nodes onto the eigenvectors. We postulate that the statistics of these histograms convey important geometric properties of the Laplacian eigenvectors [START_REF] Biyikoglu | Laplacian Eigenvectors of Graphs[END_REF]. In practice, we apply the proposed algorithm to match graphs corresponding to discrete surface representations of articulated shapes, i.e., mesh registration. Figure 1 shows a graph matching result obtained with our algorithm.

Laplacian embedding

We consider undirected weighted graphs. A graph G = {V, E} has a node set V = {V 1 , . . . , V N } and an edge set E = {E ij }. We use a Gaussian kernel to define the N × N weight and degree matrices:

W ij = exp(-d 2 ij )/σ 2 (1)
D ii = N j=1 W ij (2)
where d ij is the geodesic distance between two nodes and σ is a smoothing parameter.

In the case of meshes, a vertex is connected to its neighbors onto the surface. In practice we take the Euclidean distance between two connected vertices and D ii ≈ 6 which yields a very sparse graph. The Laplacian of a graph is defined as L = D -W. We consider the normalized graph Laplacian:

L = D -1/2 (D -W)D -1/2
. This is a semidefinite positive symmetric matrix with eigenvalues

0 = λ 0 ≤ λ 1 ≤ . . . ≤ λ N -1 .
The null space of this matrix is the constant eigenvector

U 0 = (1 . . . 1) ⊤ .
The eigenvectors of L form an orthonormal basis, U ⊤ 0 U i = 0, ∀i ∈ {1, . . . , N -1}. Therefore we obtain the following property: k U ik = 0. It is worthwhile to notice that L = I -W where W = D -1/2 WD -1/2 is the normalized adjacency matrix; matrices L and W share the same eigenvectors. Finally let L t = UΛU ⊤ be the truncated eigendecomposition where the null eigenvalue and constant eigenvector where omitted. Graph projections onto the eigenvectors corresponding to the smallest non-null eigenvalues of the Laplacian are displayed on the left side of figure 2. The right side of this figure shows the density of these projections being considered as continuous equivalent of histograms. These densities were estimated using the MCLUST software1 [START_REF] Fraley | MCLUST version 3 for R: Normal mixture modeling and modelbased clustering[END_REF].

Matching using Laplacian eigenvectors

We consider two graphs G x and G y , with respectively N and M nodes. Exact graph matching, i.e., the special case N = M , can be written as the problem of minimizing the Frobenius norm:

P * = arg min P W x -PW y P ⊤ 2 F (3)
over the set of N × N permutation matrices P. To solve this problem one can use the Laplacian embeddings previously introduced. Notice first that W x -

PW y P ⊤ = L x - PL y P ⊤ . Let L x = U x Λ x U ⊤
x and L y = U y Λ y U ⊤ y be the truncated eigendecompositions of the two Laplacian matrices. The columns of U = [U 1 , . . . , U N -1 ] correspond to the eigenvectors which form an orthonormal basis, and

Λ = Diag [λ 1 , . . . , λ N -1 ].
The spectral graph matching theorem [START_REF] Umeyama | An eigen decomposition approach to weighted graph matching problems[END_REF] states that if the eigenvalues of L x and L y are distinct and if they can be ordered, then the minimum of (3) is reached by:

Q * = U x SU ⊤ y , (4)
where S = Diag [s 1 , . . . , s N -1 ], s ∈ {+1; -1} accounts for the sign ambiguity in the eigendecomposition and where the domain of the objective function (3) has been extended to the group of orthogonal matrices. The entries of

Q * are Q * ij = x ⊤ i (s • y j ),
where a • b is the Hadamard product between two vectors. Since both U x and U y are orthonormal matrices, all entries Q * ij of Q * vary between -1 and 1. Therefore, Q * can be interpreted as a cosine node-similarity matrix. Several heuristics were proposed in the past to solve for the sign ambiguity and hence to recover node-to-node assignments uniquely. In [START_REF] Umeyama | An eigen decomposition approach to weighted graph matching problems[END_REF] the entries of U x and U y are replaced by their absolute values. The recovery of P * from Q * , i.e., exact matching, becomes a bipartite maximum weighted matching problem that is solved with the Hungarian algorithm.

In this paper we propose to perform the matching in a reduced space and let K < (N, M ) be the dimension of this space. We start with a solution provided by eigenvalue ordering followed by dimensionality reduction. This provides two sets of K ordered eigenvectors. However, ordering based on eigenvalues is not reliable simply because there may be geometric multiplicities that give rise, in practice, to numerical instabilities. To overcome this problem we seek a new eigenvector permutation which we denote by a K × K matrix P. Thus, equation ( 4) can be rewritten as:

Q = U x S P U ⊤ y , (5)
where U x and U y are (N -1) × K block-matrices. These matrices were obtained from U x and U y by retaining the first K columns and by re-normalizing each row such that The curves onto the right correspond to histograms of these graph projections. The first one of this vector, the Fiedler vector is supposed to split the graph along a cut but in this case such a cut is difficult to interpret. Other vectors, such as the second and fourth ones are very well localized which makes them good candidates for clustering and for matching. These histograms also reveal that not all these eigenvectors are well localized. This suggests that some of the eigenvectors shown here are not well suited for spectral clustering.

the N -1 row vectors lie on a hypersphere of dimension K, [START_REF] Ng | On spectral clustering: analysis and an algorithm[END_REF][START_REF] Caelli | An eigenspace projection clustering method for inexact graph matching[END_REF]. As above, S is a K × K sign-ambiguity diagonal form. Notice that, unlike Q * in (4), matrix Q is not an orthonormal matrix anymore (since U x and U y are no longer orthonormal) and it has rank K. Consequently, it can only define an inexact (many-to-many) matching.

Suppose now that matrices S and P are correctly computed. By extension of the spectral graph matching theorem mentioned above we obtain the following interpretation. The entries of Q in equation ( 5) can be written as Q ij = x ⊤ i y j , where x ⊤ i and y j are K-dimensional vectors corresponding respectively to the rows of U x and to the columns of of S P U ⊤ y . Because of the normalization performed onto the rows of U x and of U y , and because the two eigenvector bases are correctly aligned, the points x i and y j lie on the same hypersphere. This suggests that Q can be interpreted as a loose cosine node-similarity matrix, i.e., -1 ≤ Q ij ≤ 1, and that good matches x i ↔ y j may be chosen using a simple strategy such as Q ij > t > 0. As a consequence, manyto-many matches may be established by seeking for each point in one set its nearest neighbors in the other set, and vice-versa. This result is very important because it allows to bootstrap the matching of very large graphs through the alignment of two eigenbases of size K with K ≪ N . We return now to the more general case when the two graphs have different cardinalities. The best one can hope in this case is to find the largest isomorphic subgraphs of the two graphs. In terms of spectral embeddings, this means that one has to find the largest subsets of points of the two sets of K-dimensional points. This problem can be stated as the following minimization problem:

min R i,j x i -α ij R y j 2 (6)
This is an instance of the rigid point registration problem that can be solved by treating the assignments α ij as missing variables in the framework of the expectationmaximization algorithm. A detailed solution is described in [START_REF] Mateus | Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration[END_REF]. If matrices S and P are not correctly estimated, matrix R belongs to the orthogonal group, i.e., rotations and reflections. However, if S and P are correctly estimated, R belongs to the special orthogonal group, i.e., it is a rotation, which means that the two sets of points can be matched via an Euclidean transformation. The estimation of the latter is much more tractable than the more general case of orthogonal transformations.

Hence, the inexact graph matching problem at hand can be solved with the following three steps: (i) estimate matrices S and P using properties associated with the Laplacian eigenvectors, (ii) establish a match between the two sets of embedded nodes (K-D points) based on a nearest-neighbor strategy, and (iii) achieve point registration probabilistically by jointly estimating point-to-point assignments and a rotation matrix between the two sets of points.

Eigenvector alignment based on histograms

Each column U xi of U x (as well as U yj of U y ), 1 ≤ i, j ≤ K, corresponds to an eigenvector of dimension N -1 (and of dimension M -1). Hence, vector U xi (and U yj ) can be interpreted as function projecting the nodes of G x (and of G y ) onto the real line defined by U xi : R N -1 → R (and by U yj : R M -1 → R).

In the case of two isomorphic graphs, there should be a one-to-one match between the eigenfunctions of the first graph and the eigenfunctions of the second graph, provided that both S and P are known. Indeed, in the isomorphic case, the node-to-node assignment between the two graphs is constrained by [START_REF] Carcassoni | Correspondence matching with modal clusters[END_REF]. Unfortunately, as already explained in the previous section, the two eigenbases cannot be reliably selected such that they span the same Euclidean space. Alternatively, we consider the histograms of the eigenfuctions just defined, namely h([U xi ]) and h([U yj ]), where the notation h([U ]) corresponds to the histogram of the set of values returned by the eigenfunction U . The first observation is that these histograms are invariant to node permutation, i.e., invariant to the order in which the components of the eigenvectors are considered. Therefore, the histogram of an eigenfunction can be viewed as an invariant signature of an eigenvector. The second important observation is that h

([-U ]) = h(B -[U ]
), where B is the total number of bins used to build the histograms; Hence, the histograms can be used to detect sign flips. The third important observation is that the shape of the histogram is not too sensitive to the number of nodes in the graph and it is therefore possible to compare histograms arising form graphs with different cardinalities.

The problem of estimating matrices S and P can therefore be replaced by the problem of finding a set of assignments {U xi ⇔ ±U yj , 1 ≤ i, j ≤ K} based on the comparison of their histograms. This is an instance of the bipartite maximum matching algorithm already mentioned with complexity O(K 3 ). Since the eigenvectors are defined up to a sign (modeled by S), we must associate two histograms with each eigenfunction. Let C(h i , h j ) be a measure of similarity between two histograms. By computing the similarities between all pairs of histograms we can build a K × K matrix A whose entries are defined by:

Aij = min{C(h([Uxi]), h([Uyj])), C(h([Uxi]), h([-Uyj]))}
as well as another matrix whose entries contain the signs of U yj which are eventually retained. The Hungarian algorithm finds an optimal permutation matrix P as well as a sign matrix S.

Results

As a first example, consider a motion sequence of an articulated shape and their registration shown in figure 3. The articulated shape is described by a mesh/graph with 7,063 vertices and the degree of each vertex is approximately equal to six. The graphs were matched using the method described above, i.e., alignment of eigenvectors based on their histograms and naive point registration based on a nearest neighbor classifier.

On an average, the algorithm found 4,000 one-to-one matches and 3,000 many-to-many matches. Notice that in this case the two graphs are isomorphic.

Figure 4 shows two sets of eigenvector histograms (top and bottom) corresponding to the first pair of registered shapes of figure 3. The histograms shown on each row correspond to the five eigenvectors associated to the smallest non-null eigenvalues, shown in increasing order from left to right. There is a striking similarity between these histograms in spite of large discrepancies between the two shapes' poses. This clearly suggests that these histograms are good candidates for both exact and inexact graph matching. Figure 5 shows three more examples of inexact graph matching corresponding to different shape pairs: dog-horse, dancer-gorilla, and horse-seahorse. Each mesh in this figure is described by a sparse graph and there are notable differences in the number of nodes. For example, the horse graph has 3,400 nodes, the gorilla graph has 2,046 nodes, the dancer graph has 34,000 nodes, and the seahorse graph has 2,190 nodes. The top row of figure 5 shows the result of many-to-many inexact matching obtained with the method described in this paper. The bottom row shows the result of one-to-one rigid point registration obtained with a variant of the EM algorithm [START_REF] Mateus | Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration[END_REF] initialized from the matches shown on the top row.

Conclusion

In this paper, we proposed a framework for inexact matching of large and sparse graphs. The method is completely unsupervised, it does not need any prior set of matches between the two graphs. The main difficulty of the problem is twofold: (1) to extend the known spectral graph matching methods such that they can deal with graphs of very large size, i.e., of the order of 10,000 nodes and (2) to carry out the matching in a robust manner, i.e., in the presence of large discrepancies between the two graphs. We showed that it is possible to relax the graph isomorphism problem such that inexact graph matching can be carried out when the dimension of the embedding space is much smaller than the number of vertices in the two graphs. We also showed that the alignment of the eigenbases associated with the two embedded shapes can be robustly estimated using eigenvector's density instead of eigenvalue ordering. The method starts with an initial solution based on ordering the eigenvalues and then it finds the optimal subset of eigenvectors that are aligned based on comparing their density distribution. This selects both a one-to-one eigenvector alignment and the dimension of the embedding. We also pointed out localization as an important property of eigenvectors and presented initial results to support our observations.

Fig. 1 .

 1 Fig. 1. Two graphs (meshes) of a hip-hop dancer (top-left) with respectively 31,600 and 34,916 nodes (vertices). The matching (top-right), subsampled for visualization purposes, was obtained by computing the embeddings of the two graphs into a 5-dimensional space (bottom-left) and by registering the two embeddings (bottom-right).

Fig. 2 .

 2 Fig.2. A mesh with 7,063 vertices and with approximately six edges per vertex is shown here projected onto four Laplacian eigenvectors corresponding to the four smallest non-null eigenvalues. The curves onto the right correspond to histograms of these graph projections. The first one of this vector, the Fiedler vector is supposed to split the graph along a cut but in this case such a cut is difficult to interpret. Other vectors, such as the second and fourth ones are very well localized which makes them good candidates for clustering and for matching. These histograms also reveal that not all these eigenvectors are well localized. This suggests that some of the eigenvectors shown here are not well suited for spectral clustering.

Fig. 3 .

 3 Fig. 3. The result of applying the graph matching algorithm to a dance sequence.

Fig. 4 .

 4 Fig.4. Two sets of histograms associated with two sets of Laplacian eigenvectors. One may notice the striking similarity between these histograms that correspond to two isomorphic graphs.

Fig. 5 .

 5 Fig. 5. Examples of matching different shapes (dog-horse, dancer-gorilla, and horse-seahorse) corresponding to graphs of different size. The top row shows the results obtained with the algorithm described in this paper while the second row shows the results after performing point registration with a variant of the EM algorithm. While in the first two examples (dog-horse and dancer-gorilla) the graphs have the same global structure, the third example shows the result obtained with two very different graphs.

http://www.stat.washington.edu/mclust/

In future, we plan to investigate more thoroughly the link between graph matching and graph clustering. We believe localization property to be a promising direction to move forward.