Matthieu Clement

Pierre Levasseur

Suneha Seetahul

Lucie Piaser

Does inequality have a silver lining? Municipal income inequality and obesity in Mexico

Keywords: Conceptualization, Investigation, Writing, Methodology, Data curation, Software, Does inequality have Mexico, income inequality, obesity, instrumental variables

Income inequality and obesity are both widespread socioeconomic issues, particularly salient in middle-income countries. This article seeks to detect the relationship between local income inequality and excess weight in Mexico, using robust municipal income inequality measures generated through small area estimation method and instrumental variable multilevel estimations. Our results emphasize a negative impact of municipal income inequality on individual bodyweight, especially for women. We also explore the potential channels through which income inequality may decrease bodyweight. Three-stage least squares estimations highlight that the social capital pathway, the public policy pathway and the psychological pathway help to explain the negative effect of inequality on excess weight.

Our results are fairly robust to alternative inequality measures and nutritional indicators.

Introduction

In the last decades, both within-country income inequality and obesity rates have alarmingly increased worldwide, especially in developing countries (NCD-Risk Factor Collaboration, 2016;[START_REF] Alvaredo | World Inequality Report 2018[END_REF]. Surprisingly, the potential link between these two issues remains poorly researched.

It is well-known that poverty has a harmful impact on lifestyles, nutrition and health [START_REF] Currie | Healthy, Wealthy, and Wise: Socioeconomic Status, Poor Health in Childhood, and Human Capital Development[END_REF]. However, because humans are social beings, absolute income and material deprivation are insufficient to predict health and nutritional outcomes [START_REF] Wagstaff | Income Inequality and Health: What Does the Literature Tell Us?[END_REF]. The latter may also be affected by the distribution of income within society.

Theoretically, income inequality creates a social environment that potentially affects health in several dimensions. The existing literature finds convincing damaging effects of income inequality on both mental and physical health, including psychological disorders, life expectancy reduction, and premature death [START_REF] Pickett | Income inequality and health: a causal review[END_REF].

The rare studies that analyze the role of income inequality on the alarming rise of worldwide obesity exhibit confusing results. Comparing 21 rich countries, [START_REF] Pickett | Wider income gaps, wider waistbands? An ecological study of obesity and income inequality[END_REF] report a positive association between income inequality and obesity. However, the inclusion of extreme cases such as the US and Japan shows a misleading positive link: while the US have the highest levels of inequality and obesity among rich countries, Japan has the lowest levels in both phenomena. Likewise, [START_REF] Su | Income inequality and obesity prevalence among OECD countries[END_REF] find that, in OECD countries, the positive association between income inequality and obesity rate disappears when extreme cases are excluded from the sample.

In addition to these cross-country studies, the literature focusing on the US context provides mixed evidence on the relationship between state/county income inequality and individual bodyweight outcomes. While some studies report a negative link [START_REF] Bjornstrom | An examination of the relationship between neighborhood income inequality, social resources, and obesity in Los Angeles county[END_REF][START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF] J o u r n a l P r e -p r o o f al., 2016), others conclude on a positive one [START_REF] Diez-Roux | A multilevel analysis of income inequality and cardiovascular disease risk factors[END_REF][START_REF] Robert | A multilevel analysis of race, community disadvantage, and body mass index among adults in the US[END_REF] or on the absence of a significant association [START_REF] Chang | Income inequality and weight status in US metropolitan areas[END_REF][START_REF] Mobley | Environment, obesity, and cardiovascular disease risk in low-income women[END_REF]. Furthermore, three limitations could bias the estimates from previous studies. First, most of existing studies are based on self-reported anthropometric data (namely height and weight) which is problematic since inequality-based psychosocial troubles potentially infer own body perception [START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF]. Indeed, individuals living in unequal societies might overstate their weight. Second, previous studies primarily use multilevel models that do not address potential endogeneity issues linked to reverse causality and omitted variables [START_REF] Pickett | Income inequality and health: a causal review[END_REF]. Third, although each study is highly informative on the relationship between inequality and obesity, potential pathways underlying this relationship have not been studied.

This study fills the literature gap by analyzing the nutritional impacts of municipal income inequality using objective anthropometric measurements and an endogeneity-correction methodology. Another input of the study is its original focus on a middle-income country where the link between local income inequality and obesity is unclear and potentially different from the US and other rich countries, due to specific institutional settings and sociocultural factors [START_REF] Karlsson | Income inequality and health: importance of a cross-country perspective[END_REF]. Nonetheless, country-specific studies focusing on developing countries demonstrate that provincial income inequality correlates with poorer self-rated health in Argentina [START_REF] Maio | Extending the income inequality hypothesis: Ecological results from the 2005 and 2009 Argentine National Risk Factor Surveys[END_REF], China [START_REF] Pei | Provincial income inequality and self-reported health status in China during 1991-7[END_REF] and India [START_REF] Rajan | Is wealthier always healthier in poor countries? The health implications of income, inequality, poverty, and literacy in India[END_REF], and with lower life expectancy in Brazil [START_REF] Rasella | Impact of income inequality on life expectancy in a highly unequal developing country: the case of Brazil[END_REF]. To our knowledge, only [START_REF] Larrea | Does economic inequality affect child malnutrition? The case of Ecuador[END_REF] focus on the nutritional consequences of inequality in the case of developing countries. They identify a positive link between provincial income inequality and childhood stunting in Ecuador, suggesting that income inequality threatens food security among vulnerable households. Finally, it should be noted that the literature examining the health consequences of inequality in developing countries J o u r n a l P r e -p r o o f only measures income inequality at a relatively aggregated scale (i.e. at the provincial level), thus ignoring the influence of more local social mechanisms.

In this study, we focus on the relationship between municipal income inequality and adult bodyweight in Mexico. Several reasons justify our choice. First, Mexico exhibits a very high level of income inequality with the fourth highest Gini index of 0.46 among OECD members in 2016 (OECD, 2019). Second, overweight and obesity have rapidly increased and are now predominant: one-third of Mexican adults are obese and two-thirds are overweight (NCD-Risk Factor Collaboration, 2016).

Our empirical investigation is based on the construction of an original database that merges municipality-level data from various sources and individual-level data from the 2016 Mexican Health and Nutrition Survey (Encuesta Nacional de Salud y Nutricion, ENSANUT).

To measure income inequality, we rely on small area estimation (SAE) and combine data from the 2015 inter-census survey (Encuesta Intercensal, EIC) and the 2016 National Survey of Household Income and Expenditure (Encuesta Nacional de Ingresos y Gastos de Hogares, ENIGH) to construct measures of income inequality that are representative at the municipallevel. Using this original dataset, we implement multilevel estimations to identify the link between municipal income inequality and individual nutritional outcomes. To limit potential reverse causality bias and the presence of unobserved heterogeneity, we also run a two-stage estimation model based on instrumental variables (IV), besides controlling for a comprehensive set of municipal and individual characteristics. The IV design uses meteorological data as instruments for municipal income inequality (i.e. yearly average amount of precipitation, temperature and altitude). We also investigate potential gender heterogeneity. Finally, we explore pathways through which municipal income inequality can affect individual nutritional outcomes, using three-stage least square estimations (3SLS).

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Surprisingly, our results show that municipal income inequality has a negative effect on adult bodyweight. This negative effect of inequality on individual weight is primarily driven by the results for women. Further investigations about potential pathways suggest that this counterintuitive effect particularly transits through the low levels of social capital and the deficit in public infrastructures associated with high municipal inequality. For instance, individuals living in unequal municipalities are less likely to have social and outdoor entertainments, including activities related to weight gain such as outdoor food and beverage intakes, and have lower access to public services such as piped water, increasing the risk of infectious diseases related to weight loss in addition to increase daily physical activity for fetching water. Moreover, we also detect a notable importance of the psychological pathway since income inequality tends to favor weight loss (or impede weight gain) through a poorer quality of sleep. These results point out the necessity for public policies aiming at reducing inequalities to be accompanied by nutritional policies and awareness campaigns.

The rest of the article structures as follows. In section 2, we illustrate the conceptual framework. In sections 3 and 4, we respectively describe the database and the empirical strategy. In section 5, we present the results. Finally, section 6 concludes and provides policy recommendations.

Literature review and conceptual framework

An extensive literature analyzes the impact of income inequality on health, considering both cross-country and country-specific studies [START_REF] Pickett | Income inequality and health: a causal review[END_REF]. Three pathways are consensually mentioned in order to explain how an unequal income distribution may influence health outcomes, especially mental and physical illness [START_REF] Kawachi | Income inequality and health: pathways and mechanisms[END_REF][START_REF] Subramanian | Income inequality and health: what have we learned so far?[END_REF]): (i) a psychological pathway (i.e. in unequal societies, the feeling of isolation and vulnerability increases and individuals are more often J o u r n a l P r e -p r o o f subject to mental illness such as chronic stress and depression); (ii) a social capital pathway (i.e. as unequal societies are prone to market-oriented policies and individualism, the structurally low levels of social cohesion, civic involvement, reciprocity and trust may affect health outcomes in several ways); (iii) a public policy pathway (i.e. public revenues from taxation being generally low in unequal areas, income inequality might interact with health through a reduction of local investments in human capital and public facilities).

Globally, empirical studies show that income inequality has negative impacts on health outcomes. In contrast, there is no consensus in the literature about its effects on nutritional outcomes. In fact, the nutritional impacts of income inequality might be different to those observed on other health outcomes insofar as the effects of each pathway are highly ambiguous.

First, individual psychological disorders induced by relative position in society might increase obesity risk. In Mexico, [START_REF] Esposito | The economic gradient of obesity in Mexico: Independent predictive roles of absolute and relative wealth by gender[END_REF] find that relative wealth deprivation predicts higher BMI and obesity risk. [START_REF] Pickett | Wider income gaps, wider waistbands? An ecological study of obesity and income inequality[END_REF] suggest that, by decreasing mental well-being, income inequality reduces the desire for physical exercise, outdoor social interactions and eating healthy. They also mention the potential physiological effects of chronic stress and depression such as snacking, overeating and risky non-food consumptions such as drinking alcohol and narcotic intake, which potentially lead to weight gain [START_REF] Brunner | Prospective effect of job strain on general and central obesity in the Whitehall II Study[END_REF]. However, the nutritional effects of stress and depression are ambiguous in the epidemiological literature since they can result in weight loss in some circumstances. In fact, food intake disorders associated with stress and depression depend on baseline bodyweight and eating status (dieter vs. non-dieter). [START_REF] Kivimäki | Work stress, weight gain and weight loss: evidence for bidirectional effects of job strain on body mass index in the Whitehall II study[END_REF][START_REF] De Wit | Depression and body mass index, a u-shaped association[END_REF] find that both stress and depression induce weight gain among already overweight subjects, but weight loss for initially thin individuals. Moreover, one's relative position in society may generate envy and social comparisons. As shown by [START_REF] Bjornstrom | An examination of the relationship between neighborhood income inequality, social resources, and obesity in Los Angeles county[END_REF] and [START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF] J o u r n a l P r e -p r o o f in the US context, the higher proportion of affluent and healthy residents in unequal localities, compared to localities where income is more homogenously distributed, can provide role models for mainstream social norms and health-related behaviors, therefore improving the nutritional outcomes of the poorest.

Second, the loss of social capital induced by income inequality is expected to have ambiguous effects on bodyweight. On the one hand, lower trust and cohesiveness in society may reduce the occurrence of outdoor entertainment and physical activity [START_REF] Broyles | Integrating Social Capital Into a Park-Use and Active-Living Framework[END_REF], which is a potential risk factor for weight gain. On the other hand, it might also induce a reduction of outings with friends and family, associated with junk food and beverage intakes, as well as outdoor non-food intakes associated with weight gain such as drinking alcohol [START_REF] Leyden | Social Capital and the Built Environment: The Importance of Walkable Neighborhoods[END_REF]. Furthermore, the lack of trust, reciprocity and transfers characterizing unequal societies can increase the risk of transitory food insecurity and infectious diseases when shocks occur and may lead to important individual weight loss [START_REF] Hadley | Seasonal food insecurity and perceived social support in rural Tanzania[END_REF].

Third, the nutritional impacts of the policy pathway are even more ambiguous. For instance, a lower availability of health services, social safety nets or sanitation-related infrastructures (e.g. drinkable water taps, drainage systems) intensifies household vulnerability to economic and demographic shocks, increasing the risks of nutritional deprivations and infectious diseases related to weight loss (e.g. diarrhea, cholera, intestinal worms) [START_REF] Gundry | A systematic review of the health outcomes related to household water quality in developing countries[END_REF].

Similarly, lower collective investments in public transportation may increase walking time to work and overall physical activity. In contrast, lower investments in urban amenities such as parks, street pathways and public lighting may decrease the occurrence of out-of-work outdoor physical activity, thus increasing body-mass [START_REF] Feng | The built environment and obesity: a systematic review of the epidemiologic evidence[END_REF]. However, [START_REF] Leyden | Social Capital and the Built Environment: The Importance of Walkable Neighborhoods[END_REF] finds a positive association between neighborhood walkability, social capital and outdoor social activities such as restaurant and pub attendance. This means that living in J o u r n a l P r e -p r o o f unequal areas, where public investments in pedestrian amenities are low, might protect residents against risky outdoor consumptions and related weight gain. 2019), we restrict our sample to adults aged from 18 to 65 (pregnant women excluded), accounting for more than 6,000 observations. As explained by [START_REF] Elia | Obesity in the Elderly[END_REF] Our main nutritional measure is the body mass index (BMI) defined as the weight (in kilograms) divided by the square of the height (in meters). The extreme values (below 15kg/m² and above 50kg/m²) have been excluded. It should be noted that in the ENSANUT J o u r n a l P r e -p r o o f surveys, weight and height are measured with an objective procedure (using stadiometers and weighting machines) by trained health professionals. Based on the BMI, we also identify clinical bodyweight categories using the WHO cut-offs that are officially approved in Mexico: underweight for BMI<18.5kg/m²; normal weight for BMI between 18.5 and 25kg/m², overweight for BMI between 25 and 30kg/m² and obesity for BMI>30kg/m². Lastly, a robustness test uses waist circumference (in centimeters) as an alternative nutritional measure.

Data

(Insert Figure 1 and Table 1)

Figure 1 presents the distribution of BMI for women and men respectively and Table 1 reports the proportion of population within each BMI clinical categories. As expected, there is a large diffusion of overweight and obesity within the Mexican society since approximately 70% of adults from our sample are overweight or obese. In contrast, underweight has become residual. In terms of gender, these results point out that there is a much higher prevalence of obesity among women (about 41% for women against 28.5% for men) while men are more concerned with overweight (41.5% for men against 37% for women).

Inequality variables

From a methodological perspective, analyzing the spatial distribution of intra-municipal inequality raises some important issues. Ideally, census data should be privileged to measure inequality at the municipal level, to the extent that doing so ensures representativeness at the municipal scale. However, censuses are not suited for the measurement of income inequality because of the absence of income data collection. Household surveys are better suited in this regard but fail to be representative at a disaggregated level, such as municipalities. This Despite many recent refinements in SAE methods, we adopt the standard approach developed by ELL because of its multiple applications in poverty and inequality analysis. This procedure relies on a welfare model estimated from the household survey to impute income to households in the census. The model includes income predictors that are common to both data sources and comparable across them. The methodology and its implementation are extensively described in the online Supplementary Material S1. From these SAE simulations, we generate our main measures of income inequality, calculated at the municipal level. We mainly use the Gini index but have also calculated the generalized entropy indices to test the robustness of our results.

We enrich the analysis using alternative measures describing the income distribution at the municipal scale. Interestingly, the National Council for the Evaluation of Social Development Policy (Consejo Nacional de Evaluación de la Política de Desarrollo Social, CONEVAL) has also proposed estimates of the municipal Gini index based on the 2015 EIC survey and on a different SAE methodology (i.e. the empirical best predictor methodology assuming heteroscedasticity) (CONEVAL, 2017). We use it as an alternative measure of municipal income inequality. Another measure calculated by CONEVAL for 2015 is also considered to account for the municipal-level income distribution: the extreme income poverty rate.

For these different inequality measures, Figure S2.3 reported in Supplementary Materials exhibits comparable distributions between the restricted sample of 214 municipalities and the J o u r n a l P r e -p r o o f whole set of municipalities, even though the restricted sample seems to be slightly more unequal when our own estimates of the Gini index are considered (average Gini respectively equal to 0.420 and 0.393).

Control variables

In the regression analysis, we consider control variables that are determinants of bodyweight in accordance with the empirical literature on the topic (e.g. [START_REF] Robert | A multilevel analysis of race, community disadvantage, and body mass index among adults in the US[END_REF][START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF]. First, we include individual covariates available in the ENSANUT-MC 2016 survey: age, squared age, gender, ethnicity (indigenous language spoken), working status (yes or no), education (tertiary education), and migration (at least one household member lives abroad). We also include the household's socioeconomic status. In the survey, the socioeconomic status is a multidimensional index constructed through principal components analysis and combining variables describing housing and living conditions: floor, wall and roof materials, number of bedrooms, water source, number of automobiles, number of domestic appliances, and number of electric appliances [START_REF] Morales-Ruán | Evolución de los J o u r n a l P r e -p r o o f programas de ayuda alimentaria en México a través de información de la Ensanut MC 2016[END_REF]. Quintiles of socioeconomic status are used as controls in the regression analysis.

Second, to account for the influence of the living environment on nutrition, we include municipal-level predictors collected from different sources. From INEGI datasets, we account for demography with municipal population density and control for the exposure to violence with the 2015 homicide rate per 100,000 inhabitants. From CONEVAL, we use a composite index of deprivation which measures the percentage of population who are deprived in at least three dimensions among six dimensions (educational lag, lack of access to health services, lack of access to social security, housing with inadequate quality or insufficient space, lack of basic housing services and lack of access to food) [START_REF] Coneval | Multidimensional poverty measurement in Mexico: An economic wellbeing and social rights approach[END_REF]. From the 2015 EIC survey, we calculate a proxy for migration prevalence defined as the proportion J o u r n a l P r e -p r o o f of household heads living in a different municipality in 2010. We control for municipal economic development by taking into account SAE estimates of the households' average annual income per capita and its square.

Variables accounting for potential pathways

As explained previously, three pathways may explain the bodyweight effect of income inequality: the social capital, psychological and policy pathways. To account for these transmission channels, we select three variables in the ENSANUT survey. For the social capital pathway, we consider adults' eating away from home practices. Given the strong diffusion of fast-food culture in Mexico, we suggest that accounting for fast-food attendance may be a relevant way to capture the importance of socialization behaviors [START_REF] Long | Food Culture in Mexico[END_REF]. The variable is defined as the number of times per week an adult consumes fast-food or snacks (at least 100 grams). For the psychological pathway, we consider sleep quality that is often used as a mediator of psychological stress. For instance, [START_REF] Hill | Neighborhood disorder, sleep quality, and psychological distress: Testing a model of structural amplification[END_REF] find that neighborhood disorder increases psychological distress through a poorer sleep quality. More precisely, our variable measures the bad quality of sleep on a 5-point Likert scale from "very good" to "very bad". For the policy pathway, although only limited information is available in the ENSANUT survey, we use a proxy for the quality of infrastructure based on water accessibility. This variable is defined as the number of days per week water is accessible for the adult's household. In case of limited accessibility, the quality of water provision can be viewed as of poor quality.

Descriptive statistics for the variables used in the study are reported in Table S3.1 in Supplementary Materials.

Methods

J o u r n a l P r e -p r o o f

Two of the main methodological challenges of this study are to control for both the multilevel structure of the data (individuals nested within municipalities) and the endogeneity of our variable of interest (potential reverse causality and unobserved heterogeneity). Addressing clustering in the analysis of hierarchical data is fundamental to ensure the validity of the results. If not, standard errors will be underestimated, leading to an overstatement of the statistical significance of coefficients, especially for higher-level variables. To take the hierarchical structure of our data into account, we use a multilevel modelling approach, which generates statistically efficient estimates of regression coefficients, and provides unbiased standard errors, confidence intervals and significance tests.

Dealing with endogeneity is another important issue. Indeed, we suspect that our different measures of inequality may be endogenous. The first reason is potential reverse causality. By reducing the chance of educational and professional success, obesity could contribute to lowering income and then to increasing income inequality at the municipal level. Although we control for many potential determinants of bodyweight, endogeneity may also arise from omitted variables plausibly correlated with income inequality and weight gain (e.g. local politico-economic changes such as austerity reforms or local natural disasters). To limit endogeneity issues, we adopt a multilevel model combined with an instrumental variables (IV) strategy based on a control function approach. In the first stage, we regress our endogenous variable on all exogenous variables defined at the municipal level and the selected instruments.

= + + + (1)
Equation ( 1) models inequality levels for each municipality j ( ). is a vector of municipality-level exogenous variables and is a vector of instruments. are municipal residuals.

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Then, we use a multilevel model to allow for clustering of adults' BMI by municipality (Equation 2). The BMI for individual i living in municipality j ( ) is regressed on the inequality index ( ) and the predicted residuals ( ̂ ) obtained from the previous stage. In this specification, we add control variables at the individual ( ) and municipal ( ) levels.

and refer to residuals fitted at the individual and municipal level, respectively. The error terms are assumed to be normally distributed. Identifying relevant instrumental variables is challenging as they have to satisfy two requirements: (i) being good predictors of the endogenous variable even after controlling for the exogenous regressors and (ii) having no direct effect on BMI other than through its influence on the endogenous variable. This challenge is even more important when focusing on a spatially disaggregated level such as the municipality for which little information is available. Moreover, as argued by Schneider et al. (2018: 501), "scholars have yet to identify a convincing instrument for income inequality, and it seems unwise to rule out any investigation (…) on those grounds".

= + + + + + + (
Following the pioneering work of [START_REF] Easterly | Inequality does cause underdevelopment: Insights from a new instrument[END_REF], and in particular its underlying intuition, we use meteorological and geographical data as instruments to tackle endogeneity. Sokoloff

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and [START_REF] Sokoloff | Institutions, Factor Endowments, and Paths of Development in the New World[END_REF] have demonstrated that factor endowments in Latin American colonies historically contributed to the emergence of wealth, human capital, and political power inequalities, which are still deeply rooted nowadays. Because these countries had soil and climate which were well suited for cash crops such as sugarcane, cocoa and coffee, settlers established large plantations relying on intensive slave labor. The resulting distribution of land, income and human capital was highly unequal. Even if Mexico was not historically known for high-scale sugarcane production relying on slavery, factor endowments played an important role in shaping inequality in the Mexican society [START_REF] Sokoloff | Institutions, Factor Endowments, and Paths of Development in the New World[END_REF].

At the time of colonization, Spanish authorities awarded property titles to the early settlers, allowing the implementation of large-scale agricultural exploitation and mines, concentrated in the hand of a local elite. This resulted in a highly unequal distribution of land and wealth.

After the independence, inequalities persisted as the elite maintained its dominant status and power.

Following this theory, [START_REF] Easterly | Inequality does cause underdevelopment: Insights from a new instrument[END_REF] uses measures of agricultural endowments to instrument inequality. In particular, he relies on geographical and meteorological data (such as soil, rainfall, temperature and altitude) to predict the percentage of agricultural land suitable for growing wheat versus sugarcane in a country. Furthermore, he argues that despite being less precise than real production data, relying on meteorological measures ensures the exogeneity of the instruments. As explained by Easterly (2007: 756) Finally, in order to explore the potential channels through which income inequality may affect bodyweight, we estimate a simultaneous equation model using three-stage least squares (3SLS). This model does not control for the hierarchical structure of the data but allows us to assess how the three potential pathways mediate the relationship between income inequality and nutrition. The use of 3SLS estimations to identify potential pathways is relatively common in the literature. Using a similar methodology as Seguino (2011) applied to our study context, the model structure can be expressed as follows, for a given pathway:

= + + + = + + + + ! = " + " + " + " + # (3)
The first equation is the instrumentation equation for the income inequality index, with vector being composed of the same instruments as in the multilevel modeling procedure. The second equation describes the effect of income inequality on the variable accounting for the considered pathway ( ). The third equation models the impact of the considered pathway on adults' BMI. Since we are interested in estimating the variation of BMI through variations in a given pathway imputable to municipal variations in structural income J o u r n a l P r e -p r o o f inequality, we voluntary exclude income inequality from the third equation, as it is commonly done in previous studies (e.g. [START_REF] Haveman | Market work, wages, and men's health[END_REF][START_REF] Seguino | Help or Hindrance? Religion's Impact on Gender Inequality in Attitudes and Outcomes[END_REF][START_REF] Passarelli | Evaluating the pathways from small-scale irrigation to dietary diversity: evidence from Ethiopia and Tanzania[END_REF].

Results

Effect of municipal income inequality on bodyweight (Insert Table 2)

To determine the impact of municipal-level inequality on bodyweight while controlling for the endogenous nature of this relationship, we instrument the municipal-level Gini index with municipal-level temperature, rainfall and altitude variables using a control function approach.

The first-stage regression of the IV approach (Table S3.2 in Supplementary Materials) shows positive and significant coefficients for these three instruments, suggesting that meteorological and altitude variations strongly affected farming specialties across Mexican municipalities in the past and then positively influenced local income inequality. Note that the R-squared and F-statistics on excluded instruments are both high, which confirms the strength and relevance of these instruments. Table 2 shows the results of our IV multilevel estimations on the continuous BMI variable and on binary weight thresholds (i.e.

underweight, overweight and obesity) for the whole sample and gender subsamples. Sargan-Hansen overidentification tests indicate that there is no correlation between the instruments and the error terms, suggesting that the instruments do not violate the exogeneity condition.

Table S3.3 in Supplementary Materials presents the complete results (including control variables) and also reports multilevel estimations without instrumentation. Broadly speaking, the control variables have the expected associations with bodyweight. Most of individuallevel control factors are significant and exhibit effects that are fully in line with the literature J o u r n a l P r e -p r o o f

analysing the determinants of obesity in Mexico (e.g. [START_REF] Levasseur | Causal effects of socioeconomic status on central adiposity risks: Evidence using panel data from urban Mexico[END_REF]. Moreover, although some municipal-level covariates are not significant, our results point out the crucial role of local socioeconomic factors in characterizing obesogenic environments in Mexico. We emphasize a U-inverted association between bodyweight and municipal average income indicating that adults living in municipalities with intermediate development levels are more at risk of excess weight. We also show that adult bodyweight is positively associated with municipal deprivation.

Concerning the influence of income inequality, the results reported in Table 2 show a negative effect of municipal-level Gini index on individual BMI (significant at the 1% level).

The disaggregation by gender reveals that this effect is driven by a strong negative effect among women (significant at the 1% level). Indeed, a one-point increase in the municipallevel Gini Index leads to a 0.52 kg/m² decrease in women's BMI. The alternative specifications, in which we detect the effect of inequality on BMI clinical categories, are in line with the results on the continuous BMI variable. A one-point increase in the Gini index decreases the probability for women to be overweight and obese by 1.8 and 3.2 percentage points, respectively. In contrast, the results showing the effect of the Gini index on the probability of female underweight are not significant. Likewise, income inequality has no significant effect on male nutritional outcomes. Tables S3.4 and S3.5 in Supplementary Materials respectively present IV binary logit regressions (marginal effects) and IV ordered logit regressions of BMI clinical classification (underweight, normal-weight, overweight and obesity) on income inequality. These results are fully consistent with those reported in Table 2.

Exploring potential pathways

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We explore three potential pathways through which the municipal-level Gini index is likely to have an impact on bodyweight: the social capital, public policy and psychological pathways. These pathways are respectively proxied by fast-food attendance, access to piped water, and bad sleep quality. To control for the endogeneity of the municipal-level Gini index, we implement a 3SLS model with the same instruments for the Gini index as in the previous step. The results are summarized in Table 3 (see Table S3.6 in Supplementary Materials for complete results).

(Insert Table 3)

Concerning the social capital pathway, the results for the whole sample show that the Gini index contributes to decreasing the number of visits to fast-foods but that, in turn, fast-food attendance is positively correlated to BMI, especially for women. For men, the results are non-significant. In other words, in more unequal municipalities, women tend to lower their attendance to fast-food institutions, thus lowering their bodyweight compared to women who live in less unequal areas.

Concerning the public policy pathway, the results show that the Gini index significantly and negatively affects access to piped water by approximately the same magnitude for the whole sample, men and women. This similarity is expected as the relationship between income inequality and access to piped water involves households and not specifically women or men.

However, and interestingly, access to piped water has a larger positive effect on women's bodyweight compared to men's bodyweight (0.80 vs. 0.48). These results suggest that inequality has a negative impact on access to water for all households but that this lower access particularly impairs female BMI.

Concerning the psychological pathway, the results show that inequality significantly lowers sleep quality, while sleep quality increases with adult BMI in the whole sample.

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Disaggregating the results on the lines of gender reveals specific patterns. Among women, the Gini index significantly increases reporting a bad reported sleep quality and BMI is negatively correlated to a bad quality of sleep. This pathway can explain why inequalities limit weight gain for women. For men, these results are less clear and significant: the Gini index does not affect reported sleep quality but BMI is positively correlated to sleep quality.

Finally, we also tested how each pathway affects the probability of being underweight, overweight and obese (Tables S3.7, S3.8 and S3.9 in Supplementary Materials). The results observed for overweight and obesity are fully in line with those for BMI. Indeed the negative influence of income inequality on the continuous BMI variable through the three pathways primarily reflects dynamics related to belonging to the overweight or obese categories.

Overall, the exploration of potential pathways yields interesting results. For women, the Gini index negatively affects fast-food attendance (social capital pathway), access to piped water (public policy pathway) and sleep quality (psychological pathway) which are all positively correlated to bodyweight. Through its effect on these pathways, income inequality prevents women's bodyweight from increasing. For men, the results are more ambiguous and do not allow to make clear conclusions.

Robustness checks

We propose to further explore the impact of inequality on bodyweight through several robustness checks. First, we estimate the effect of income inequality on BMI using alternative inequality measurements from the CONEVAL SAE estimations (Gini index and extreme income poverty rate). We also test the results using alternative inequality indices, namely the three entropy indices GE(0), GE(1) and GE(2), derived from our own SAE estimates. These estimations are respectively reported in Table S3.10 and S3.11 in Supplementary Materials and largely confirm our previous results. A notable exception concerns the CONEVAL Gini J o u r n a l P r e -p r o o f index that is never significant (even if the expected negative sign is confirmed). It should be noted that we tested the effect of the CONEVAL Gini index on waist circumference instead of BMI. These estimates (available upon request) exhibit a significant negative effect (at the 10%-level) in the female sample. Second, we also test the robustness of our results by replacing the BMI by waist circumference. The IV multilevel and 3SLS estimations are respectively presented in Tables S3.12 and S3.13 in Supplementary Materials and reveal consistent findings.

Discussion and conclusion

Using an original database combining different sources of information and an IV multilevel strategy, this article aimed to detect the effect of municipal income inequality on excess weight. Our results highlight a negative impact of municipal income inequality on individual BMI, waist circumference, overweight and obesity in Mexico, especially for women. This gendered result is consistent with previous empirical studies focusing on the US [START_REF] Bjornstrom | An examination of the relationship between neighborhood income inequality, social resources, and obesity in Los Angeles county[END_REF][START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF] and is robust to alternative inequality measures. Another contribution of this article relies on the identification of explanatory mechanisms behind our main findings. 3SLS estimations identify three potential pathways to explain why municipal income inequality leads to weight loss among Mexican women.

We find that income inequality contributes to reducing individual bodyweight by decreasing individual fast-food attendance (considered as a proxy of social capital). This result is consistent with the literature arguing that income inequality has destructive effects on social cohesion and related activities such as outings with friends [START_REF] Leyden | Social Capital and the Built Environment: The Importance of Walkable Neighborhoods[END_REF]. These social activities are associated with risky feeding behaviors such as fast-food, restaurant and bar attendance. Furthermore, women's sensitivity to the lack of social capital is not surprising considering that they are more dependent on mutual aid (a means of insurance against risks J o u r n a l P r e -p r o o f and shocks in developing countries) than men. In the absence of such informal mechanisms, women are more vulnerable to family and income shocks [START_REF] Perezneito | Gendered risks, poverty and vulnerability in Mexico[END_REF], hence more vulnerable to transitory food deprivations, infectious diseases and weight loss [START_REF] Hadley | Seasonal food insecurity and perceived social support in rural Tanzania[END_REF]. In a short digression, it is interesting to put our results into perspective with the worldwide lockdown that occurred in 2020 during the COVID-19 pandemic. This episode represents an unprecedented opportunity to study the nutritional impacts induced by a decrease of social capital and outside activities (initially related to weight-gain). Even if it is still too early to appreciate effects on overweight and obesity trends, several authors conclude on an improvement of diet quality, namely due to a reduction of outside food consumption.

For instance, [START_REF] Bracale | Changes in food choice following restrictive measures due to Covid-19[END_REF] Our estimates show that the lack of basic public services in municipalities such as daily water accessibility is a significant driver of the association between local income inequality and weight loss, at least for women. This result is consistent with the assumption that human capital-oriented investments are low in unequal municipalities because of poor taxation and redistributive policy [START_REF] Coburn | Income inequality, social cohesion and the health status of populations: the role of neo-liberalism[END_REF]. Two explanations may justify why the lack of public investments in water sanitation (and potentially health services) decreases female bodyweight: (i) it increases the prevalence of infectious diseases associated with weight loss such as diarrhea, cholera and tropical fevers [START_REF] Gundry | A systematic review of the health outcomes related to household water quality in developing countries[END_REF]; (ii) it increases the physical effort associated to fetching and transporting water daily (i.e. higher energy expenditures). This gender-specific result might be explained by different roles held by men J o u r n a l P r e -p r o o f and women regarding water-related tasks such as fetching water when water taps or wells are far from home [START_REF] Dewilde | An integrated method for evaluating community-based safe water programmes and an application in rural Mexico[END_REF], in addition to a higher vulnerability of women against infectious diseases related to poor sanitation systems [START_REF] Corburn | Slum Sanitation and the Social Determinants of Women's Health in Nairobi, Kenya [WWW Document[END_REF][START_REF] Perezneito | Gendered risks, poverty and vulnerability in Mexico[END_REF]. Indeed, women are potentially more dependent on sanitation and health services than men (e.g. pregnancy and delivery), and thus are more likely to get infectious diseases.

We also note some evidence of psychological disorders associated with income inequality.

Specifically, we find that women living in unequal municipalities report a lower quality of sleep compared to women from more equal municipalities, contributing to a significant weight loss. This result is consistent with the epidemiological literature that finds stronger damaging effects of income inequality on mental well-being among women [START_REF] Ribeiro | Income inequality and mental illness-related morbidity and resilience: a systematic review and meta-analysis[END_REF] and more specifically a significant association between psychosocial stress and impaired sleep (Âkerstedt, 2006). The higher psychosocial sensitivity of women regarding income inequality has two main explanations. Not only do women view wellbeing in a more relative than an absolute way [START_REF] Corazzini | Reign in hell or serve in heaven? A cross-country journey into the relative vs absolute perceptions of wellbeing[END_REF], but they are also more often affected by stress, depression and eating disorders than men [START_REF] Chen | Association between lifetime stress and obesity in Canadians[END_REF]. In contrast, [START_REF] Udo | Gender differences in the impact of stressful life events on changes in body mass index[END_REF] conclude on a positive link between stressful life events and female BMI.

However, there is no consensus in the literature about the nutritional consequences of stress.

Hence, more research should be conducted to understand why income inequality potentially creates a specific stressful atmosphere, which is associated with female sleep disorders and weight loss. In addition, the psychological pathway may also involve social comparisons and imitation behaviors in reaction to a high perceived level of inequality. In relatively unequal societies, there is an imitation of lifestyles from dominant social groups by the poorest, the latter considering the former as a model to follow [START_REF] Bjornstrom | An examination of the relationship between neighborhood income inequality, social resources, and obesity in Los Angeles county[END_REF][START_REF] Fan | Tract-and County-Level Income Inequality and Individual Risk of Obesity in the United States[END_REF]. In other words, people from underprivileged classes would aspire to resemble, in terms of J o u r n a l P r e -p r o o f physical appearance and taste, the dominant class (e.g. adopting thinness as a beauty standard). However, some evidence emphasizes that richer individuals are not so thin in Mexico, since their job-oriented and stressful lifestyles are associated with weight gain [START_REF] Levasseur | Causal effects of socioeconomic status on central adiposity risks: Evidence using panel data from urban Mexico[END_REF]. Hence, this mechanism is not likely to occur in Mexico. Nonetheless, as we only explored a limited number of pathways, more research would be needed to identify additional drivers.

To conclude, our results have important implications for public policy and nutrition in a country where obesity is a major public health concern. However, they should not in any case be considered as making a case for inequality. Although income inequality has a negative effect on female overweight and obesity, our pathway analysis shows that municipal inequality may have several damaging externalities on psychological health and social cohesion, and may decrease accessibility to basic public services such as health and water sanitation. Therefore, we encourage public policy to tackle inequalities by accompanying redistributive mechanisms with adequate nutrition-related components. For instance, the implementation of conditional cash transfers has already shown its beneficial impacts to fight simultaneously poverty and obesity, especially in Mexico [START_REF] Levasseur | Can social programs break the vicious cycle between poverty and obesity? Evidence from urban Mexico[END_REF]. Once again, targeting women as first beneficiary seems relevant to reduce the prevalence of obesity, since they are particularly vulnerable to local income inequality. Finally, our results call for an increased decentralization of policies fighting obesity. We suggest that the implementation of anti-obesity urbanization policies at the local level may help to reduce the spread of obesogenic built environments in Mexico.

Despite the new insights provided by this study, we argue that further research on the topic is required, especially to overcome some potential limitations. Future research applied to Mexico or other contexts should be based on larger samples of municipalities to improve the representativeness of the results. Likewise, despite constraints on data availability (especially J o u r n a l P r e -p r o o f in the context of developing countries), future research should also improve the specification of econometric models by not only controlling for omitted environmental determinants of obesity (e.g. occupational characteristics, time allocation), but also for genetic determinants.

Indeed, it is now widely acknowledged that BMI and obesity have a strong genetic basis [START_REF] Albuquerque | The contribution of genetics and environment to obesity[END_REF]. For this reason, the use of individual genetic information based on the collection and analysis of genetic samples is becoming more common in the social science literature addressing the determinants of obesity, at least in rich countries [START_REF] Willage | The effect of weight on mental health: New evidence using genetic IVs[END_REF]. Finally, relying on longitudinal datasets may also facilitate the identification of a causal relationship between inequality and bodyweight.

J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f 3 J o u r n a l P r e -p r o o f Municipal income inequality measures are generated through small area estimation.

Income inequality has a negative impact on bodyweight, especially for women.

Social capital, public policy and psychological pathways explain this effect.

J o u r n a l P r e -p r o o f

  Our data for the measurement of nutritional outcomes come from the Mexican Health and Nutrition Survey (ENSANUT), a nationally representative household survey coordinated by the National Public Health Institute (Instituto Nacional de Salud Pública, INPS). In this article, we use the 2016 mid-stage (medio camino) survey (ENSANUT-MC 2016). In line with the existing literature analyzing adult bodyweight indicators in Mexico (e.g. Levasseur,

  , older individuals are subject to several metabolic changes and are not physically comparable to working-age adults. Among the 2,457 Mexican municipalities, only 214 municipalities are covered by the ENSANUT-MC 2016 survey. However, the sampling procedure ensures the inclusion of municipalities with different sizes and socio-demographic characteristics in order to reflect the diversity of all Mexican municipalities and guarantee representativeness. Appendix S2 in Supplementary Materials presents descriptive evidence supporting the representativeness of the restricted sample. While Figure S2.1 suggests a quasi-normal distribution of observations across the 214 municipalities, Figure S2.2 shows that the 214 selected municipalities are representative of the distribution of municipality sizes in Mexico. Finally, for several variables of interest, Figure S2.3 displays similar distributions in both samples.

  is the reason why, in line with the pioneering work of Elbers et al. (ELL) (2003), we apply small area estimation (SAE) techniques. The main objective of SAE is to combine census and J o u r n a l P r e -p r o o f survey data in order to simulate representative inequality measures at a spatially disaggregated level. In this study, we provide our own SAE estimates based on the combination of the 2015 inter-census survey (EIC) and the 2016 National Survey of Household Income and Expenditure (ENIGH) implemented by the National Institute of Statistics and Geography (Instituto Nacional de Estadística y Geografía, INEGI).

  2)This multilevel model allows the intercept to vary randomly across municipalities. As a result, the residual variance is decomposed into a between-municipality component (variance of the municipal-level residuals) and a within municipality component (variance of the individual-level residuals). The standard errors of the second-stage estimates are adjusted via bootstrapping (500 replications) to account for the two-step estimation and obtain robust standard errors.

  find that households use homemade food as substitute for potentially more caloric snacks bought outside, at the beginning of the Italian lockdown. Likewise, comparing pre-COVID-19 intakes and lockdown-based intakes among adolescents in Italy, Spain, Chile, Colombia and Brazil, Ruiz-Roso et al. (2020) observe a significant increase in vegetable and fruit intakes against a significant fall in fast-food intakes.

Figure 1 :

 1 Figure 1: BMI distribution (Epanechnikov kernel).

  r n a l P r e -p r o o f Highlights We examine the effect of municipal income inequality on adult bodyweight in Mexico.

  . We were however able to collect weather data for 967 weather stations all over the territory. The data comes from the National Water Commission (Comisión Nacional del Agua, CONAGUA). For every station over the 1951-2010 period, it includes the yearly

	average amount of precipitation, temperature and altitude. Every municipality centroid is then
	matched with the nearest weather station based on latitude and longitude coordinates. These
	meteorological data intend to reflect the land endowment of every municipality and thus their
	historical path of inequality (i.e. structural inequality). It is worth noting that relying on
	meteorological variables to instrument inequality has already been done in previous studies
	(e.g. Nepal et al., 2010; Ramcharan, 2010).
	, "one confusion in the
	theoretical and empirical analysis of inequality is between what we could call structural
	inequality and market inequality". The current high levels of inequality observed in Latin
	American countries are primarily linked to structural inequality with historical foundations
	(colonization, slavery and land distribution). The combination of agricultural and

meteorological data is therefore well-suited to account for the structural component of inequality. Unfortunately, data on land suitability are not available at the scale of Mexican J o u r n a l P r e -p r o o f municipalities

Table 1 : BMI-based categories by gender

 1

	BMI categories (%)	Men	Women	Whole sample
	Underweight	1.14	0.94	1.00
	Normal weight	28.93	21.24	23.88
	Overweight	41.26	36.89	38.39
	Obesity	28.67	40.93	36.73
	Total	100	100	100
	Source: Authors' calculations.

Table 2 : IV multilevel estimations of the impact of inequalities on BMI-based indicators

 2 Bootstrapped standard errors (500 replications) in parentheses. Levels of statistical significance: *** p<0.01, ** p<0.05, * p<0.1. All control variables are included. Sargan-Hansen overidentification tests are performed using two-stage least square (2SLS) estimator.

	Dep. var.:	Body mass index (kg/m²)		Underweight (<18.5kg/m²)	Overweight (>25kg/m²)	Obesity (>30kg/m²)
		Whole Sample	Women	Men	Women	Men	Women	Men	Women	Men
	Gini Index	-0.4332***	-0.5168***	-0.2678	-0.0007	-0.0017	-0.0177*	-0.0082	-0.0317**	-0.0178
		(0.1315)	(0.1682)	(0.2558)	(0.0017)	(0.0025)	(0.0094)	(0.0142)	(0.0126)	(0.0193)
	Fitted residuals from stage 1	0.4100*** (0.1372)	0.4895*** (0.1727)	0.2573 (0.2752)	0.0018 (0.0021)	0.0007 (0.0025)	0.0131 (0.0103)	0.0086 (0.0152)	0.0268* (0.0137)	0.0136 (0.0212)
	Observations	6,106	3,983	2,123	3,983	2,123	3,983	2,123	3,983	2,123
	Number of municipalities	214	214	214	214	214	214	214	214	214
	Overidentification test (J stat)	1.766	0.755	3.305	1.531	0.867	0.010	2.333	1.983	4.896
	P-value	0.414	0.686	0.192	0.465	0.648	0.995	0.311	0.371	0.087
	Notes:

Source: Authors' calculations.

Table 3 : 3SLS estimations of the relationship between income inequality, potential pathways and BMI

 3 Standard errors in parentheses. Levels of statistical significance: *** p<0.01, ** p<0.05, * p<0.1. All control variables are included. Source: Authors' calculations.

	Whole Sample	Women	Men


Hussein Chouman

Alahdab Luay

Rafael Colares

Annie Gravey

Philippe Gravey

Hervé Kerivin

Michel Morvan

Annegret K Wagler

Assessing the Health of Flexgrid Optical Networks

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Assessing the Health of Flexgrid Optical Networks Hussein Chouman (1) , Luay Alahdab (1) , Rafael Colares (2) , Annie Gravey (1) , Philippe Gravey (1) , Hervé Kerivin (2) , Michel Morvan (1) , Annegret Wagler (2) (1) Institut Mines Télécom, IMT Atlantique, Brest, France (2) Laboratory LIMOS, CNRS UMR 6158, Clermont-Ferrand, France Email: hussein.chouman@imt-atlantique.fr

Abstract-This paper focuses on quantifying the efficiency of an off-line dimensioning process for flexgrid optical networks. We assume that a network operator attempts to optimize the mapping of a set of demands on a given topology and wishes to assess the remaining capacity of its network once all demands are satisfied. Minimizing the amount of blocked traffic is not an appropriate objective function since all demands should be satisfied. Two other well-known objective functions are thus considered: one is related to path lengths and the other aims to curtail spectrum fragmentation; an alternate objective function, also focusing on spectrum fragmentation, is introduced. Several network level metrics are then proposed, some of them extending existing link or path level metrics; these new metrics either characterize the global fragmentation within the network or the network's remaining capacity (i.e. the network's capability to host future demands). We try to reach a better understanding of the pros and cons of the chosen objective functions in order to provide insight on how flexgrid network dimensioning should be performed. This is done by applying off-line dimensioning to some instances (flexgrid network and set of demands) and assessing the impact of the selected objective function on the network level metrics.

Index Terms-RSA, Optical network, flexgrid, optimization, fragmentation

I. INTRODUCTION

In a flexgrid based optical network, the spectrum is split into frequency slices; an optical channel is characterized by one or several contiguous slices dedicated to this specific channel on all links of the path it is routed on. On the one hand, this allows a network operator to support different rates and thus to more efficiently use the available spectrum; on the other hand, the contiguity (on each link) and continuity (over all the links of a path) constraints respectively yield the well known spectrum "horizontal" and "vertical" fragmentation issues, both implying that some spectrum slices cannot be used over some links [START_REF] Amar | How problematic is spectrum fragmentation in operator's gridless network?[END_REF].

Although a well-known phenomenon, it is neither easy to quantify spectrum fragmentation, nor is it easy to assess its impact on resource allocation efficiency in a flexgrid network. The paper thus focuses on deriving quantitative "network health" indicators and highlighting the relationship between spectrum fragmentation and these indicators. We address a network operator's requirement: assuming that a given set of demands has to be mapped on a certain network, is it possible to assess how much spectrum resources are wasted and to gain insight on how much additional traffic could be supported once the initial set of demands is mapped? In the following, we first discuss state-of-the-art results regarding network dimensioning and fragmentation metrics. The paper's contributions are then highlighted.

A. Dimensioning a flexgrid network

In flexgrid optical networks, the problem of establishing lightpaths (optical paths) that compete for spectrum resources is called Routing and Spectrum Assignment (RSA). Offline RSA considers a static set of demands, whereas online RSA deals with dynamic demands. The off-line RSA problem is formulated as an Integer Linear Programming (ILP) optimization problem. Thus, mapping demands on a certain network will depend on the selected objective function. There is no clear indications in the literature which objective function yields the "best" network state, as the latter is not specified. In [START_REF] Amar | How problematic is spectrum fragmentation in operator's gridless network?[END_REF], the total number of transponders is minimized in order to reduce the cost necessary to satisfy all demands. In [START_REF] Ruiz | Column generation algorithm for RSA problems in flexgrid optical networks[END_REF], [START_REF] Klinkowski | Valid inequalities for the routing and spectrum allocation problem in elastic optical networks[END_REF], the volume of blocked traffic is minimized. In [START_REF] Żotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF], the sum of lengths of the paths carrying the demands is minimized to limit the total length of selected paths and thus reduce the global spectrum occupancy and delay. Furthermore, slices are assumed to be indexed between 1 and the total number of slices on a link. Other works aim to limit fragmentation by minimizing the max slice-index in the network [START_REF] Walkowiak | Optimization of multicast traffic in elastic optical networks with distanceadaptive transmission[END_REF], or the summation, over links, of the max slice-index of each link in the network [START_REF] Ghallaj | Optimizing spectrum allocation in flex-grid optical networks[END_REF].

B. Fragmentation Metrics

External Fragmentation (EF ) [START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF] is defined for each link as the ratio of the maximum number of free contiguous slices to the total number of free slices. Shannon's Entropy (SE) [START_REF] Wright | Simulation results of Shannon entropy based flexgrid routing and spectrum assignment on a real network topology[END_REF] considers how slices are distributed on a link as follows:

SE = - N i=1 D i /D log(D i /D)
where D is the total number of frequency slices in the spectrum, D i is the number of slices of the ith block of either contiguous free or contiguous used slices and N is the number of such blocks. Access Blocking Probability (ABP ) [START_REF] Amar | Traffic forecast impact on spectrum fragmentation in gridless optical networks[END_REF] computes the ratio between the number of possible demands that can be placed on sets of contiguous free slices on a certain link and the number that could be placed if all free slices were contiguous. As the above are either link level or path level metrics, it is not straightforward to derive from them relevant network level indicators. Moreover, as the usability of a given set of contiguous free slices depends on the "granularity" of the transponder (i.e., the number of slices n required by the transponder), a fragmentation metric should ideally be expressed as a function of n. However, this is the case neither for EF nor for SE and although ABP does take into account transponders' granularity, it only provides an average measure of fragmentation over available transponders granularities. Wasted-Unusable-Free Ratio (W U F R) [START_REF] Pederzolli | Path-based fragmentation metric and RSA algorithms for elastic optical networks[END_REF] is a path level metric that also depends on transponder granularity: it is defined as the ratio of the number of wasted slices (those that are only free on some, but not all, of the links of a path) and unusable slices (those free on all links of a path but belonging to a block of free slices that is smaller than the number of slices requested by a transponder) to the number of free slices. Lastly, note that none of the above metrics are translated into network level metrics, i.e. where all paths between all origin and destination (o-d) pairs are taken into account, and therefore do not provide any quantitative insight on the network's remaining capacity (i.e. the network's capability to host future demands while taking into account the granularity of transponders and network's wasted resources).

C. Contributions

This work focus on the off-line RSA problem. We consider three different metrics to be minimized: two are wellknown [START_REF] Żotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF], [START_REF] Walkowiak | Optimization of multicast traffic in elastic optical networks with distanceadaptive transmission[END_REF], whereas the paper introduces a new one that aims to curtail spectrum fragmentation. We then introduce new network level metrics, as natural extensions of existing link/path level metrics; these new metrics aim to characterize either the global fragmentation within the network or the network's remaining capacity. Finally, we try to reach a better understanding of the pros and cons of the chosen objective functions, following an approach applied in [START_REF] Balon | How well do traffic engineering objective functions meet traffic engineering requirements?[END_REF] in the context of the off-line design of IP networks. The comparison between objective functions is performed using our new proposed network level metrics.

The rest of the paper is organized as follows: Sec. II presents a recent ILP formulation used to solve the off-line RSA problem [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF]. In Sec. III, several network level metrics are proposed while Sec. IV applies optimization algorithms to a practical dimensioning problem, highlighting the relationship between spectrum fragmentation and network health indicators. Sec. V concludes the paper. II. OFF-LINE DIMENSIONING Fig. 1 shows the different components of the process modelled in this work: the inputs (topology and set of demands), how to solve this problem (RSA algorithm) for a given objective function q, and the output which is denoted as mapping q .

A. Set of demands

A given demand to an optical transport network is expressed as a requested bit rate to be carried between an o-d pair. Since we put the emphasis in this study on the RSA problem, the modulation format used by the transponder(s) serving a given demand has to be defined prior addressing the RSA problem. Moreover, we do not allow for an intermediate regeneration of the lightpaths. Thus each lightpath is associated to a pair of transponders, selected among a set of possible ones characterized by their bit rate, the number of spectrum slices they occupy and a transmission reach. Therefore, between each o-d pair, there should exist at least one path shorter than the maximum reach allowed by the transponder set. In these conditions, for each demand we select the smallest transponder bit rate that is able to carry it. If none of the transponders is able to carry the demand while satisfying the transmission reach limit (or if the requested capacity is higher than the highest transponder bit rate), the demand is split into smaller ones till we find suitable transponders. Once this preliminary step has been performed, the set of traffic demands addressed to the RSA module is thus expressed in terms of number of slices, submitted to a maximum distance condition.

B. Defining the RSA problem

An input of the RSA problem is composed of

• an optical-network topology represented as an undirected, loopless, and connected graph G = (V, E), where V corresponds to the set of network nodes and E to the set of network links, • an edge-length (in kms) vector ∈ R E + , • an optical spectrum (available slices) defined by the set of slice indexes S = {1, . . . , s}, and • a multiset K of traffic demands where each k ∈ K is specified by an origin node o k ∈ V , a destination node d k ∈ V \ {o k }, a requested number w k ∈ N + of slices, and a transmission reach k ∈ R + (in kms). Both w k and k characterize a type of requested transponder.

The off-line RSA problem aims to determine for each traffic demand k ∈ K, a lightpath composed of a route (i.e., an (o k , d k )-path P k in G) whose length does not exceed the transmission reach k and a channel (i.e., a subset S k ⊆ S of slices) such that:

• the w k slices allocated to traffic demand k (i.e., slices in S k ) are the same all along the route P k (spectrum continuity), • the channel allocated to traffic demand k is composed of

w k contiguous slices (i.e., S k = {i, i + 1, . . . , i + w k -1} where 1 ≤ i ≤ s -w k + 1 (spectrum contiguity),
• a same slice cannot be allocated to two different traffic demands on the same link (non-overlapping spectrum), • the value taken by a given metric is optimized.

C. A new ILP for solving the RSA problem

An ILP model for the above off-line RSA problem was introduced in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF]. This model has a polynomial number of variables as it is based on the so-called edge-node formulation for the route determination and a non-polynomial number of constraints that are handled in a polynomial time through the well-known cutting-plane approach [START_REF] Gomory | Outline of an algorithm for integer solutions to linear programs[END_REF]. This tractable model is the first ILP model for the off-line RSA problem that considers the entire set of feasible solutions and whose validity is independent of the considered objective function.

D. Objective functions

The optimization objective functions used in this paper are:

E. Mapping demands: optimality gap

The ILP model with either objective functions is solved through a branch-and-cut framework, that is, a cutting-plane algorithm to handle the non-polynomial numbers of constraints embedded in a branch-and-bound approach that partitions the whole problem into smaller ones [START_REF] Wolsey | Integer Programming[END_REF]. The efficiency of a branch-and-cut approach significantly depends on the strength of both the lower bounds (i.e., best possible objective values that the linear relaxations provide) and the upper bounds (i.e., best found objective values given by feasible solutions encountered during the whole solution process). To improve the latter bounds, we incorporate a greedy-based heuristic that is frequently called during the course of the branching tree. Such an approach may fail to find an optimal solution within a time limit but it usually provides a feasible solution whose quality can be assessed when put in perspective with the lower bound. This is known as the (relative objective) gap, defined as: best found objectivebest possible objective best found objective , and the smaller this gap is, the better the feasible solution is proven to be.

F. Reducing the size of the ILP

To run simulations that use the above optimization model, the number s of available slices on the links needs to be specified. For example, a fiber with a 4-THz spectrum has a nominal number of slices that equals 320 (i.e., 12.5 GHz per slice). However, it would be better to decrease this number in order to decrease both the numbers of variables and constraints in the ILP model and consequently, the time necessary to obtain an optimal solution. Let an admissible link configuration be a number s of available slices on each link that accommodates all demands and such that the proportion of used slices on all links is lower than a given threshold T (e.g., T = 80%). Before running our simulations, we first perform an initial process that aims at finding an admissible link configuration. Starting with s = 320, this initial process can be described as follows:

1) minimize the number of blocked demands N blocked , i.e.,

min k∈K (1 - s∈S z k s )
2) and if N blocked = 0 and the maximum spectrum occupation observed over all the links does not exceed T , reduce s by 40 and go back to the first step to recompute N blocked .

At the end of the initial process we have reached a number s of available slices per link that satisfies both the non-blocking and spectrum utilization conditions.

III. NETWORK LEVEL METRICS

Selecting s as specified in Sec. II-F ensures that there always is a feasible solution (i.e., without blocking), regardless of the objective function; however, the mapping depends on the selected objective function. How can differences between obtained mappings be characterized? This section proposes network level metrics to characterize the "network health", and classifies these metrics into "network state" and "remaining capacity" metrics.

A. Network State Metrics

1) Total number of holes: This metric is the sum of the number of holes on all the links (a spectrum hole on a link is a set of contiguous free slices).

2) Network level fragmentation metrics: For a given mapping, having computed F M e , the value of a link fragmentation metric for each link e, the network level fragmentation F M mapping of this mapping is defined as the average over the set of links:

F M mapping = e∈E F M e | E | (1)
3) Optimization relative metrics: Let us define metric p, q as the value taken by metric p in mapping q (obtained with objective function q). If mapping p is optimal (zero relative gap), the best possible value for metric p, q is obtained when p = q; this may not be the case otherwise. For each metric p, the difference percentage with reference to metric p, p is: M etric p, q = (metric p, q -metric p, p )/metric p, p .

(

A negative value indicates that mapping p is sub-optimal, and that mapping q yields a smaller value for metric p. A positive value indicates that, although mapping p may still be suboptimal, it nevertheless yields a smaller value for metric p than mapping q .

B. Remaining Capacity Metrics

We build remaining capacity metrics by extending an approach described in [START_REF] Pederzolli | Path-based fragmentation metric and RSA algorithms for elastic optical networks[END_REF].

1) Wasted and accessible slices: Let p m,i be the i th path between the o-d pair m. On each link of p m,i , there are used slices and free slices, the latter being partitioned into wasted and accessible slices as explained in Sec. I-B; let us define W m,i (respectively A m,i ) as the number of wasted (respectively accessible) slices on path p m,i . Metric W m,i (respectively A m,i ) is W m,i (respectively A m,i ) divided by the number of links of the path p m,i . Dimensionless metrics are obtained by dividing the above metrics by s.

2) Number of supplementary transponders: On a given path, accessible slices are grouped into "path-holes", that is, sets of continuous and contiguous accessible slices separated by either wasted or used slices. If p m,i satisfies transponder's reach and has at least one path-hole j of size larger than n, then it is defined as a n-feasible path. The number of supplementary transponders with granularity n that could be accommodated on p m,i is:

X m,i (n) = J j=1 A m,i,j n (3)
where x/n denotes the integer division of x by n and J denotes the total number of path-holes in a n-feasible path p m,i . Fig. 2 illustrates the above concepts by computing the remaining capacity metrics W BE , A BE and X BE (n) on path B-C-D-E between nodes B and E with 3-slice and 6-slice transponders. There are 18 used slices and 36 free slices, which are partitioned into W BE = 9 wasted slices (in blue) and A BE = 27 accessible slices (in orange). As there are 3 links on the path and s = 18, W BE = 3 and A BE = 9. Dimensionless metrics (respectively 16.7% and 50 %) are obtained if we divide by s= 18. The accessible slices of the path are grouped into two path-holes: path-hole 1 , A (BE, 1) = 3, can accommodate one 3-slice transponder, while path-hole 2 ,A (BE, 2) = 6, can accommodate either two 3-slice transponders or a single 6-slice transponder. For each path i of o-d pair m and remaining capacity metric RC m,i (either W m,i , A m,i , or X m,i (n)), we compute network level metrics by first averaging RC over the number of feasible paths I (which can be I m or I m (n) depending on the considered RC metric), and then averaging over the number M of o-d pairs. A network level remaining capacity metric is thus defined as:

RC = M m=1 I i=1 RC m,i I M. (4)
IV. ASSESSING THE OFF-LINE DIMENSIONING PROCESS We now illustrate how the choice of the objective function impacts the mappings obtained when performing off-line RSA. Due to space constraints, most of the results presented in this paper correspond to a single instance (i.e., a set of demands to be mapped on a given network); however, it has been checked that consistent results are obtained when different instances are considered.

Fig. 3 illustrates how the network level metrics are computed on each of the obtained mappings.

A. Problem Specification

The network, the set of demands to be mapped on it and how the RSA optimizer is implemented are now described.

1) Network and transponders: We have considered a small network, namely the 9-Node M-NSF, which is based on the 9-Node NSF topology with 13 links [START_REF] Pavon-Marino | Offline impairment aware RWA algorithms for crosslayer planning of optical networks[END_REF]. The modification on the original NSF topology consists in dividing link lengths by 2 to consider transponders with suitable transmission reach. The considered transponders require 3, 5 or 6 slices and respectively support 100 Gb/s (3000 km reach), 200 Gb/s (1500 km reach), or 400 Gb/s (600 km reach).

2) Set of demands and admissible link configuration: The total bit-rate traffic to be mapped on the M-NSF network is 6.2 Tb/s. The resulting set of demands are derived as explained in Sec. II-A and correspond to 59 demands, totalling 246 slices. Running the process described in Sec. II-F yields s = 120.

3) Running the RSA: The model is implemented using Cplex 12.10 and runs on a high performance computing cluster with a 32 Gigabytes memory. The program stops and provides either an optimal solution or a feasible solution with a positive relative gap when a time limit is reached. B. Qualitative assessment of spectrum occupancy Fig. 4 shows link occupancy for all 3 mappings of the demands. Links and slices are represented as a two-dimension array, where each row represents a link composed of used (black) and free (white) slices.

In mapping 1 and mapping 2 , most of the used slices are squeezed on the left-hand side of the spectrum, whereas in Fig. 4. Slice distribution in the M-NSF topology for mapping 1 (left), mapping 2 (middle) and mapping 3 (right). Black: used slice, white: free slice mapping 3 , used slices are distributed all over the spectrum. This was to be expected as both objective functions 1 and 2 are used to limit link fragmentation by focusing on the largest used slice per demands, whereas objective function 3 focuses on path lengths, irrespective of the location of the used slices.

Furthermore, we see that in mapping 2 , the max slice indexes are similar on most of the links, whereas they significantly differ in mapping 1 , showing more compact sets of used slices. Note that the achieved value of metric 2 in mapping 1 is only slightly larger than the optimum achieved in mapping 2 ; therefore, compacting the used slices by using objective function 1 does not significantly increase the value of the largest used slice over all edges.

In the following, we assess how this qualitative assessment translates into quantitative network level metrics.

C. Network state metrics

With a simulation-time limit set to 28 hours, an optimal mapping 3 is obtained in less than 40 minutes, while an optimal mapping 2 is obtained in 27 hours. On the other hand, a suboptimal mapping 1 (with a 23.9% relative gap) is provided when the program stops due to the time limit.

Table I lists the network level state metrics computed for the three mappings of the M-NSF instance. 1) Total number of holes: The respective numbers of holes obtained for the 3 mappings confirm the previous observations. First, the numbers of holes in mapping 1 and mapping 2 are close; a more compact spectrum obtained in mapping 1 translates in a lower achieved total number of holes. On the other hand, roughly twice more holes are present in mapping 3 .

2) Network level fragmentation metrics: For all 3 network level fragmentation metrics, the hierarchy of mappings, from the least fragmented to the most fragmented, is mapping 1 , mapping 2 , mapping 3 . The latter, as expected, presents really larger values than the former two; also, mapping 1 , al-though sub-optimal, is still significantly less fragmented than mapping 2 according to EF , SE, and ABP .

3) Relative Optimization Metrics: Considering Metric 1,2 , we see that mapping 2 reaches a value for metric 1 which is significantly larger than the value reached by mapping 1 , although the latter is sub-optimal. On the other hand, as Metric 2,1 equals 5%, we see that mapping 1 achieves a value quite close to the optimum for metric 2. As expected, mapping 3 fails to provide values for metrics 1 and 2 close to those obtained by mapping 1 and mapping 2 . Conversely, mapping 2 and even more mapping 1 , provide values for metric 3 rather close to the optimum achieved in mapping 3 ; this means that reducing spectrum fragmentation (using objective functions 1 or 2) is not made at the cost of increasing significantly path lengths.

D. Remaining capacity metrics

We now quantify the impact of fragmentation on network's remaining capacity.

1) Wasted and accessible slices: Fig. 5 presents the numbers of wasted and accessible slices of the obtained mappings as defined in Sec. III-B. The proportion of wasted slices is much larger for mapping 3 than for the two other mappings; obviously, it also means that the proportion of accessible slices for mapping 3 is significantly lower than the ones than for the two other mappings. The wasted and accessible numbers for mapping 1 and mapping 2 are quite similar, with more than 40% of accessible slices, the former providing a slightly better performance than the latter.

2) Number of supplementary transponders: For each considered transponder's granularity n in {3,5,6}, Fig. 6 plots X(n). X(n) is twice as large (for n = 3) or even three times larger (for n = 5 or 6) for mapping 1 and mapping 2 than for mapping 3 . The gain is larger for a larger n; indeed, some available slices cannot all be used since a given transponder Fig. 6. No. of supplementary transponders versus n for M-NSF topology requires a set of several contiguous available slices. In the M-NSF network, we see in Fig. 6 that similar values for X(n) are obtained for mapping 1 and mapping 2 , the former providing a slightly better performance.

We have also studied a smaller instance, namely the 5-Node Spain network with 7 links [START_REF] Mariño | Net2plan: a publicly-available open-source network planner[END_REF], s = 40, and a set of 15 demands (totalling 68 slices). Optimal mappings were obtained for the 3 objective functions. Fig. 7 shows that, for this instance, the performance in terms of X(n) is significantly better for objective function 1 than for both objective functions 2 and 3. This paper has highlighted the major impact of the selected objective function on both fragmentation metrics and remaining capacity metrics. We observed consistent results on all studied instances (only part of them were presented here).

Using the path length related objective function (objective function 3) yields a very quick optimization process but provides a mapping which presents a large global spectrum fragmentation level, and a very low remaining capacity.

Using a spectrum usage related objective function (objective functions 1 and 2) yields harder optimization task. For the M-NSF instance, the required computation time to provide an optimal solution with objective function 2 and only a sub-optimal solution for objective function 1 is 50 times larger. However, both mappings present global fragmentation indicators that are half of those obtained for mapping 3 and X(n) values that are twice or three times larger than those for mapping 3 . The path length indicator (metric 3) values obtained for mapping 1 and mapping 2 are only slightly larger than the optimum reached for mapping 3 .

At the cost of a harder optimization task, objective function 1 provides better results than objective function 2, even when a sub-optimal mapping 1 compares with an optimal mapping 2 .

The on-going work includes improving the ILP formulation in order to tackle larger instances of off-line RSA and identifying how to best use some of the newly defined network health metrics to decide when the network need to be re-dimensioned in order to avoid any blocking. Indeed, network level metrics can be used in other frameworks, including off-line RAS based on heuristics and on-line RSA used for mapping new demands on an operational network.

This work was carried out within the framework of the French research project FLEXOPTIM (ANR-17-CE25-0006).

Fig. 1 .

 1 Fig. 1. RSA Optimization for objective function q

Fig. 2 .

 2 Fig. 2. Remaining capacity metrics for Path BE .

3 )

 3 Network remaining capacity: For o-d pair m, let I m denote the total number of paths that can be used for m and I m (n) denote the number of n-feasible paths (I m (n) ≤ I m ).

Fig. 3 .

 3 Fig. 3. Comparison block.

Fig. 5 .

 5 Fig. 5. Network's average wasted and accessible slices of the M-NSF topology

Fig. 7 .

 7 Fig. 7. No. of supplementary transponders versus n for Spain topology

  minimizing metric 1, the sum of the demand's maximum slice used over all the demands, if and only if the route P k uses edge e, and • minimizing metric 3, the sum of the route lengths over all the demands[START_REF] Żotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF],

	min	sz k s ,	(Objective function 1)
	k∈K s∈S
	where z k s is the binary variable that equals 1 if and only
	if slice s is the last slice assigned to demand k,
	• minimizing metric 2, the largest index of an allocated
	slice over all the edges [5],
	min max e∈E	p e ,	(Objective function 2)
	where for a given edge e in E, p e = k∈K x k e and binary variable x k e equals 1 min e x k e . (Objective function 3) s∈S sz k s
	k∈K e∈E

•

TABLE I NETWORK

 I STATE METRICS FOR M-NSF INSTANCE.

	Metric	Mapping 1	Mapping 2	Mapping 3
	Total number holes	52	66	99
	EF	0.070	0.080	0.204
	SE	0.389	0.482	0.800
	ABP	0.011	0.017	0.030
	Metric 1, q	0 %	38 %	116 %
	Metric 2, q	5 %	0 %	64 %
	Metric 3, q	11 %	22 %	0 %