Thierry Lorey
email: thierry.lorey@kedgebs.com

Mazhar Mughal
email: mazhar.mughal@esc-pau.fr

Rashid Javed
email: rachidjaved@gmail.com

Pandemic buying: Covid-19 and purchasing behaviour of French households

Keywords: Covid-19, lockdown, purchasing behaviour, France, Italy, Spain. JEL codes: D14, E00, G51, I10

In this study, we use comprehensive French consumer data from the Nielsen ScanTrack retailer panel to analyse the progression of purchasing trends of French households during the coronavirus outbreak. Our results are threefold: First, we observe three temporal phases, namely the normal (pre-Covid-19) period, the anticipation period, and the lockdown period. During the three weeks of anticipation (24 February -15 March), sales of basic and cleaning products increased by 17%. The first week of the preventive lockdown saw panic buying with sales jumping by 30%, followed by a more subdued growth during the subsequent weeks of the lockdown. We found these time trends to be similar to those observed in Italy and Spain. Second, sales of the hypermarkets fell by 6% during the lockdown, while those of supermarkets and convenience stores rose by 13% and 28% respectively. Online sales through drive stores (80%) and home delivery (93%) showed the highest growth. Covid-19 appears to have accentuated the joint evolution of two major phenomena: strong development of E-commerce, and the desire of the households to visit small shops close to their homes. Third, in terms of product categories, we observe an increase in basic, easy-to-store food products, an increasing consumption of fresh, local & organic products, a significant increase in the purchase of basic hygiene products, and a decline in the purchase of cosmetics. In the latter case, Covid-19 seems to have accentuated a fundamental trend in household purchases in France: "eat local".

Introduction

The Covid-19 pandemic has impacted the economic lives of the people like no other crisis in living memory. From employment and income to household savings and consumption, all aspects of human economic activity have been drastically affected. In this study, we aim to analyse the impact of Covid-19 on the purchasing behaviour of French households. The study has three objectives. First, we use comprehensive French consumer data from the Nielsen ScanTrack retailer panel to analyse the progression of purchasing trends of French households during the coronavirus outbreak. We examine the changes in weekly sales relative to those observed in 2019 and compare them with trends in corresponding Italian and Spanish data. In addition to identifying salient trends in aggregate data, we explore changing trends in the turnover of French retail sector. In addition, we analyze the sales of various product categories during the preventive lockdown.

France is one of Europe's worst hit countries with over a hundred thousand reported Covid-19 fatalities1 . The country reported its first case of the infection on 24 January 2020, and its first confirmed fatality on 14th February 2020. This was the first reported death due to the novel coronavirus outside Asia. Initially, the spread of the virus in France was slow with only 12 positive cases reported within one month of the onset of the pandemic. The spread of the outbreak turned exponential in the month of March, and by the time the French government decided to enter preventive lockdown on 17th March 2020, there were already 6,633 confirmed coronavirus cases and 148 deaths reported in the country. On 11th May when gradual lifting of the first lockdown began, there were a total of 138,854 coronavirus cases and 26,310 related deaths. Figure 1 shows total coronavirus cases in France (panel 1) and total deaths (panel 2) with the red line splitting the pre-lockdown and lockdown periods.

[Figure 1] For our research, we employ weekly mean sales from January 1, 2020, to 26 April 2020.

The variables analyzed correspond to purchases by French households, translated into changes in sales based on a comparison between the year 2020 and 2019. For comparison with Spain and Italy, we obtained data from mid-February up to end April 2020.

Our study makes a number of contributions to the literature: First, unlike most of the literature on the topic which is based on partial or sectorspecific data, we base our analysis on a comprehensive, representative dataset of retail sales which covers nearly all of the 29.2 million French households in the first half of 2020. Data from the Nielsen ScanTrack retailer panel used in the study are drawn from on-and off-line stores resulting from the collaboration with all French retailers. Nielsen receives on a daily basis nearly all the transactions from buying households that take place in hypermarkets, supermarkets, convenience stores, hard-discounters, drive-in collection points and home delivery including Amazon and Cdiscount. This represents approximately 16,000 points of sale in France, i.e. more than 98.5% of the total turnover of the distribution universe in France.

Second, some studies on consumer behaviour during the Covid-19 outbreak describe changes in consumer preferences that have taken place during the lockdowns (see for instance, [START_REF] Andersen | Pandemic, Shutdown and Consumer Spending: Lessons from Scandinavian Policy Responses to COVID-19[END_REF][START_REF] Baker | How Does Household Spending Respond to an Epidemic? Consumption During the 2020 COVID-19 Pandemic[END_REF][START_REF] Fanelli | Changes in the Food-Related Behaviour of Italian Consumers during the COVID-19 Pandemic[END_REF][START_REF] Laguna | The impact of COVID-19 lockdown on food priorities . Results from a preliminary study using social media and an online survey with Spanish consumers[END_REF][START_REF] Martin-Neuninger | What Does Food Retail Research Tell Us About the Implications of Coronavirus (COVID-19) for Grocery Purchasing Habits[END_REF][START_REF] Sorrentino | and lifestyle in italy . A disaster management perspective[END_REF]. However, these studies do not point to the three distinct purchasing patterns that we identify in this analysis.

Third, studies such as [START_REF] Hall | Beyond panic buying: consumption displacement and COVID-19[END_REF], [START_REF] Hashem | Examining the Influence of COVID 19 Pandemic in Changing Customers ' Examining the Influence of COVID 19 Pandemic in Changing Customers ' Orientation towards E-Shopping[END_REF], Martin-Neuninger and Ruby (2020) focus on one or more distribution channels to examine changing consumer behaviour during the Covid-19 outbreak. To our knowledge, these studies have not highlighted the joint evolution of household purchases for the different distribution channels. In this study, we compared the changes in consumer preferences for these channels and highlight the increasing role of online shopping in France. In addition, we point to the gain in significance of the convenience stores.

Finally, although studies such as de Paulo Farias and de Araújo (2020), [START_REF] Fanelli | Changes in the Food-Related Behaviour of Italian Consumers during the COVID-19 Pandemic[END_REF], [START_REF] Laguna | The impact of COVID-19 lockdown on food priorities . Results from a preliminary study using social media and an online survey with Spanish consumers[END_REF], [START_REF] Nicola | The socio-economic implications of the coronavirus pandemic (COVID-19): A review[END_REF] and [START_REF] Yu | The impact of COVID-19 on food prices in China: evidence of four major food products from Beijing, Shandong and Hubei Provinces[END_REF] analyse the demand for food and resulting price changes in various countries, as far as we know, these studies do not analyse and compare the evolution of household food and non-food purchases during the Covid-19 pandemic. Besides comparing changing purchasing patterns of various product categories, we underscore the gains local and organic food products made during the first lockdown.

Brief review of relevant literature

The literature review on the impact of Covid-19 on household purchases can be divided into three parts: time analysis of household purchases, change in preference for distribution channels, and change in goods basket. In the following, we briefly discuss literature on the three aspects.

Time analysis of household purchases

Some studies describe the changes in consumption habits before and during the preventive lockdowns. For example, [START_REF] Andersen | Pandemic, Shutdown and Consumer Spending: Lessons from Scandinavian Policy Responses to COVID-19[END_REF] analyze transaction data from large banks in Scandinavia and show that consumers spending dropped by 29% in Denmark and 25% in Sweden as a result of the lockdown.

Using household financial data, [START_REF] Baker | How Does Household Spending Respond to an Epidemic? Consumption During the 2020 COVID-19 Pandemic[END_REF] explore how households consumption responds to epidemics, and identify two temporal phases. During a first step, households increase their spending for example on food items, before decreasing their overall spending in the second step. In another study, Martin-Neuninger and Ruby (2020) analyse the consequences of the lockdown on consumer grocery purchasing habits in New Zealand. They show that before the lockdown, during what we will call the "anticipation period", many households began to panic buy, and stock-pile products like food and gloves. This panic buying can be expected during periods of fear and uncertainty. It was reinforced by the spread of misinformation on the internet. Panic buying causes shortage of many basic products in mass markets.

Change in Preference for distribution channels (pre-lockdown vs lockdown)

Taking the case of the service sector in New Zealand, [START_REF] Hall | Beyond panic buying: consumption displacement and COVID-19[END_REF] provide evidence for spatial displacement of consumption based on consumer spending patterns. Furthermore, they highlight the link between increased spending in some consumption categories and stockpiling behaviours. [START_REF] Martin-Neuninger | What Does Food Retail Research Tell Us About the Implications of Coronavirus (COVID-19) for Grocery Purchasing Habits[END_REF] show that supermarkets in New Zealand implemented measures during the lockdown that can have an impact on household consumption behaviour. These include: (a) reducing the opening hours to reduce panic buying ; (b) maintaining distance between customers and staff ; (c) allowing the entrance of the supermarket on a " one in, one out" basis; (d) advising their customer to pay on a contactless way; (e) demanding customers to follow floor signage to the supermarket. Some of these measures were also implemented by supermarkets in France, such as the largest supermarket chain in France "Intermarché". These measures have had potential implications on the shopping process, for example less interactions with retail staff and shorter decision-making process due to time constraints. This new purchasing process may have caused changes in the nature and price of the products (basic vs valorized) purchased by French households, and their frequentation of alternative distribution channels. According to [START_REF] Hashem | Examining the Influence of COVID 19 Pandemic in Changing Customers ' Examining the Influence of COVID 19 Pandemic in Changing Customers ' Orientation towards E-Shopping[END_REF], the COVID19 pandemic managed to change customer behaviour towards greater dependence on online shopping and e-payment methods. This tendency of households to use E-commerce in the event of a pandemic crisis is not new. Indeed, the demand for online shopping is reported also to have increased during the SARS epidemic in 2002[START_REF] Kee | Intended usage of Online supermarkets: The Singapore case[END_REF][START_REF] Kee | Intended usage of Online supermarkets: The Singapore case[END_REF].

Change in goods basket and income shock

Extant literature examines factors that could affect consumer preferences for food products. [START_REF] Nicola | The socio-economic implications of the coronavirus pandemic (COVID-19): A review[END_REF], for example, identify panic buying and stockpiling of food to be causing high demand. [START_REF] Yu | The impact of COVID-19 on food prices in China: evidence of four major food products from Beijing, Shandong and Hubei Provinces[END_REF] find that in China, the Covid-19 pandemic had no significant impact on rice and wheat flour prices, i.e. basic necessities. However, de Paulo Farias and de Araújo (2020) show that food prices in Brazil depend on the regions and the period in which the food products were traded.

The pandemic can also represent an income shock for households. Different researchers showed that the Covid-19 pandemic resulted in job losses [START_REF] Mcdowell | Working From Home and Job Loss Due to the COVID-19 Pandemic Are Associated With Greater Time in Sedentary Behaviors[END_REF], parttime work [START_REF] Dias | The Challenges for Labour Market Policy during the COVID-19 Pandemic.pdf[END_REF], and therefore less disposable income for everyday purchases. [START_REF] Griffith | Shopping Around : How Households Adjusted Food Spending Over the Great Recession[END_REF] showed that in response to economic shocks, households can adjust their shopping behavior: they reduce their real food expenditure, but maintain the number of calories that they purchased, and the nutritional quality of these calories.

Specifically, [START_REF] Gicheva | Investigating Income Effects in Scanner Data : Do Gasoline Prices Affect Grocery Purchases[END_REF] investigated Income Effects in Scanner data: they found that the quantity weighted price paid for products decreases when gasoline prices increase: "consumers are able to save money on groceries by shifting purchases toward promotional items". These changes in household consumption patterns can sometimes have beneficial effects on health: [START_REF] Ruhm | ARE RECESSIONS GOOD FOR YOUR HEALTH ?[END_REF] show that health indicators can be better during a period of economic recession than during a period of economic development, which constitutes a paradox.

Analysis of consumption trends

Now we explore the three aspects of changes in French consumer behaviour observed during the lockdown period discussed in the literature review, namely the temporal aspect, the spatial aspect and the product category aspect.

Change in purchasing trends over time

We begin by comparing the purchasing trends of French consumers during 2019 and 2020. Figure 2 shows week-wise percentage change in the amount of total sales. Three periods can be clearly identified:

The first period corresponds to the first eight weeks of the year (1 January -23 February, week 1 to week 8), and is characterized by normal consumption. Although some cases of coronavirus had been reported in France, the situation had yet not taken an alarming turn. During this period, total sales showed a steady progression of 1 to 2 percent.

[Figure 2]

The second period could be termed Anticipation time. It corresponds to three weeks ( 24February -15 March, Week 9 to week 11). By then, Covid-19 had begun to badly affect the country and the number of cases had begun to rise exponentially. In order to reduce the risk of catching or transmitting the virus, consumers increased their purchase of cleaning and hygiene products (e.g. disinfectants, floor cleaners, tissue papers). In view of the lockdown which was seeming increasingly likely, consumers also began to stock food. Compared to the same period last year, the sales of basic food products (e.g. pasta, rice, canned vegetables, milk, eggs, delicatessens) significantly increased.

Together, the sales of these two categories of products (cleaning products and food) increased by 17.6% during this period with 6, 10 and 38% increase witnessed during the first, second and third weeks of the period respectively (Figure 5).

The third period, that began on 16 March (week 12 to week 17), corresponds to the lockdown period. The first week (16 -22 March) saw panic buying with sales jumping by 30%. The rise in consumption slowed in the second (+6%) and third week (+7%) and subsequently plateaued out, averaging 9.4% during the six weeks ending on 26 April. The three countries entered the lockdown phase at about the same time: (with one week of one another). On 9 March 2020, Italy's President of Ministerial Council Giuseppe Conte announced lockdown beginning from the next day, while Pedro Sánchez, the head of the Spanish government, announced the same on 15th March.

[Figure 3] The pattern of three purchasing periods observed for France can also be seen in the case of Italy and Spain (Figure 3). During the initial period (weeks 6 and 7)3 , sales showed a steady increase (3% in Italy, 4% in Spain). This evolution is similar to the one observed in France. The next period (week 8,9,10) saw a double-digit anticipatory increase in sales in Italy. Sales respectively grew by 11, 18 and 15% during the three weeks. This growth in purchases in Italy is somewhat higher than that observed in France, because Italy was the first European country affected by Covid, and the first to take drastic measures. The increase in Spain during this period was a more gradual 8%.

During the first week of lockdown (week 11), sales in both countries exploded (+28% in Italy, +78% in Spain). Afterwards, the rate of increased slowed down but the growth remained in the double-digits. This purchasing growth seen in the two countries during the lockdown is somewhat higher than that observed in France.

Change in preference for distribution channels

France's retail market scene has traditionally been dominated by hypermarkets 4 , supermarkets 5 and hard discount stores6 , respectively corresponding to 40, 33.6 and 12% of market share since 2015. The three collectively account for 86% of the sales turnover. In contrast, convenience stores and e-commerce hold small shares of the retail market, even though that of the latter is increasing7 .

[Figure 4] Three trends could be discerned:

1) The turnover of hypermarkets fell by 6% during the lockdown. Hypermarkets are usually situated in the outskirts of urban centres where hundreds of thousands of consumers shop every day. In order to maintain social distancing, only a limited number of customers were allowed to enter at a time after waiting for their turn outside. The resulting inconvenience accompanied by the desire to avoid physical proximity may have contributed to customers reducing their purchases from hypermarkets.

However, this observation can be challenged on two grounds: (1) The waiting lines and limited entrances existed for both hypermarkets and supermarkets; (2) The travelling and administrative restrictions. Starting at noon on March 17 2020, the 4 A hypermarket is a self-service retail store deriving more than one-third of its sales from food and having a sales floor area of 2,500 m² or more. The average surface area of a hypermarket in France is 5,300 m2, which is three times less than the average surface area of a Wal-Mart Supercenter (18,300m 2) in the U.S.A. Source: https://www.insee.fr/en/metadonnees/definition/c1825 5 A supermarket is an establishment for self-service retail sales with more than two thirds of its turnover in foodstuffs, and a sales area of between 400 and 2,500 m². source: https://www.insee.fr/en/metadonnees/definition/c1449

French government implemented a series of travel restrictions 8 . People were required to remain at home, except to run errands, make business trips, travel for health reasons, for family emergencies, or to perform individual physical activity near home. In addition, for all of these reasons (including "shopping"), individuals who were found away from home must present the law enforcement agents a waiver certification justifying the need to travel. These administrative justification undoubtedly discouraged consumers from making trips to hypermarkets far from home.

2) Conversely, sales of supermarkets and convenience stores increased during the lockdown. The former grew by 13% and the latter a more spectacular 28%. The two types of stores are smaller and usually located within the urban areas. Hard discount stores, which sell a small range of products at lower prices also benefited from the trend with an increase of turnover of 7% during the period. This can also be explained by an anticipation of lower incomes and income shock [START_REF] Griffith | Shopping Around : How Households Adjusted Food Spending Over the Great Recession[END_REF], due to the expected increase in unemployment in the second and third quarter of 2020 9 .

The most notable trend during the lockdown, however, was the increase in online shopping which grew by 81%. Online sales through both drive stores (80%) and home delivery (93%) grew substantially. These Drive stores hold significant weight in the French distribution: there were 4,162 Drive stores in France in 2020 compared to 2,257 Hypermarkets. 87% of Drive stores in France are attached to a physical store 10 . According to the consulting firm Food Service Vision, home meal delivery has grown 47% over 2 years (2018)(2019)(2020), with a turnover of 5 billion euros in 202011 . 8 www.vie-publique.fr, Republique française, 24 mars 2020. Entry into force of the state of health emergency (Entrée en vigueur de l'état d'urgence sanitaire). 9 https://www.insee.fr/fr/statistiques/4930129, In the third quarter of 2020, the unemployment rate rebounded sharply to 9.0% (Au troisième trimester 2020, le taux de chômage rebondit fortement et attaint 9,0%). 10 https://www.lsa-conso.fr,French professional magazine specialized in distribution, Drive expansion in 2020 continues ("La folle année du drive se poursuit »).

Besides, there was a 24% increase in restaurant delivery users compared to prepandemic times12 .

Online shopping was doing well even before that with an increase of 39% observed during the pre-lockdown period. Figure 5 [Figure 5] The respective performance of the five categories of retail stores during the lockdown has helped accelerate the long-term trends observed in France: greater reliance on online shopping and loss of market share by hypermarkets concomitant with increasing share of supermarkets and convenience stores. People are driving less for their grocery and are increasingly buying from nearby stores at the cost of product variety. Indeed, during the lockdown period, there was an increased focus on the purchase of essential products that are easy to store, e.g. frozen, fresh and grocery products (see Section 3.3).

The trends observed in Italy and Spain were somewhat similar. According to European E-commerce report (2021), during the first two months of the pandemic, Italian consumers who bought online grew by 2 million. In Spain, the share of fresh produce grew from a low 2% of e-commerce before the pandemic to 3.5% afterwards.

Change in the goods basket

The basket of goods that French consumers buy also underwent significant change during the lockdown (Figure 6).

1) The sale of food items that could be stored easily increased substantially. The sale of salty and sweet frozen products grew by 34 and 28% respectively, that of salty grocery (e.g. pasta, rice and kitchen items) by 23% and sweets and cake products by 7%. This reflects a generalized trend of stocking food items borne out of uncertainty about the length of lockdown and worries about food shortages. This change in food purchasing was likely also reinforced by the closure of bars and restaurants between March 15 and May 28, 2020 13 . Going to restaurant is deeply rooted in the French culture, with 65% of the population estimated to go to a restaurant at least once a month in 2019 14 . Traditional outof-home purchases such as pizza or sweet and savory pies likely shifted to inhome purchases.

[Figure 6] 2) Another trend that was observed during the period was increasing recourse to local, organic and environmental-friendly products 15 (Nielsen Scantrack, statement for the 16 March -26 April 2020 period). The sale of fresh products, particularly those organic or produced locally, increased, that of dairy products by 19% and non-dairy products by 9%. Likewise, the sale of organic and local vegetables and fruits increased. According to the Nielsen Scantrack report dated 26 th April 2020, consumers, worried about getting infected, consumers considered local fresh products to be less likely to be contaminated with coronavirus given shorter transportation time and fewer steps involved in the handling and packing process. In addition, this increase in organic and local fruit and vegetable sales during the lockdown period is mainly due to the online channel.

[Table 1] 13 https://www.weblex.fr › weblex-actualite › coronavirus-.Legal informations online for Entrepreneurs. " Coronavirus (Covid-19) : réouverture des cafés, bars, restaurants sous quelles conditions ? « 14 Source: statista 2022) (Website: https://www.statista.com/statistics/1241740/frequency-going-tothe-restaurant-france/ 15 A local product is a product whose producer is located within 50 km. An organically grown product is either an agricultural product or a foodstuff. The method of production excludes the use of synthetic chemicals and Genetically modified Organisms (GMOs), and limits the use of inputs (products brought to the land and crops such as fertilizers, insecticides, etc.). An environmental-friendly product is a product of sustainable agriculture. In France, this definition includes products produced according to the high environmental value (HVE) standard. Source: Nielsen Scantrack Panels, week 12 to week 17 2020, lockdown period. Somewhat similar trends were reported from Italy and Spain: According to Fanelli (2021), Italian consumers reduced their purchases and consumption of ready-to-eat foods and consumed more fresh fruits and vegetables. There was a shift towards the purchase of food in small retail stores and an increased frequency of food shopping in Italy during the COVID-19 pandemic, possibly because super and hypermarkets were considered to be too crowded and less safe than smaller stores. In another study from Italy, [START_REF] Sorrentino | and lifestyle in italy . A disaster management perspective[END_REF] identify four categories of behavioural consumer profiles: (1) digital, (2) homescape lovers, (3) responsible and (4) self-care oriented. Based on an online questionnaire of Spanish population, [START_REF] Laguna | The impact of COVID-19 lockdown on food priorities . Results from a preliminary study using social media and an online survey with Spanish consumers[END_REF] show that purchasing frequency increased for products bought for health (pasta and vegetables) or mood improvement (nuts, cheese and chocolates) but decreased for products with a short shelf-life (fish, seafood) and those perceived to be unhealthy or leading to weight gain (sugary bakery goods). There was a reduction of shopping frequency but no changes in shopping location.

3) Consumers also bought more cleaning and hygiene products during the lockdown period: the sale of gloves, soap, disinfectant gels and cleaning products increased by 17% while that of tissue and toilet papers rose by 7%. This trend is borne out of the greater emphasis on cleanliness and personal hygiene in order to reduce the risk of virus infection, and the lockdown itself (people stayed at home and feared of shortage). 4) A final trend worth remarking is the 11% drop in the purchase of beauty and personal care products (e.g. creams, lotions, shaving products) as reduced faceto-face interaction decreased the need to care for physical appearance to the usual extent.

Concluding remarks

In this study, we analysed the changes taking place in the purchasing behaviour of French consumers during the first Covid-19 preventive lockdown. We used panel data on French retail market from Nielsen Scantrack to identify key sales trends in household shopping. Our main contribution is to have highlighted simultaneously three different temporal phases during the Covid-19, the transformation of household purchase in the different distribution channels, and four new trends in the product consumption. To our knowledge, this multi-dimensional revolution in consumption behavior has not been highlighted jointly via panel tools by other researchers in the case of France. Our study highlights the ability of households to adapt their consumption patterns to a new economic situation, and confirms the studies of [START_REF] Griffith | Shopping Around : How Households Adjusted Food Spending Over the Great Recession[END_REF] and [START_REF] Gicheva | Investigating Income Effects in Scanner Data : Do Gasoline Prices Affect Grocery Purchases[END_REF].

First, we highlighted three temporal phases, namely the normal period, the anticipation period, and the lockdown period. We compared these time trends with those prevalent in Italy and Spain, and found them mostly similar.

Second, we highlighted the transformation of household purchases during the covid-19 outbreak by simultaneously analysing the use of different distribution channels. We pointed out that Covid-19 appears to have accentuated the joint evolution of two major phenomena:

(1) strong development of digital commerce, and (2) the desire of the households to frequent small shops close to their homes. The latter helps maintain social distancing (a short-term objective) and at the same time reduces the transport (a long-term ecological objective). This development in favour of small nearby stores is a major emerging trend in household purchases in France [START_REF] Labbé-Pinlon | Impact de la proximité perçue d'un magasin sur la fidélité des clients : le cas des magasins d'enseignes alimentaires de proximité[END_REF].

Finally, we highlighted the joint purchasing preferences of households in the form of four three trends: (a) an increase in local, basic, easy-to-store food products; (b) a significant increase in the purchase of basic hygiene products; (c) a decline in the purchase of cosmetics, which were perceived as less important during a period of social isolation; (d) An increase in fresh dairy and fresh non-dairy products, specifically organic and local vegetables and fruits. This increase was mostly realized in the online channels. This constitutes a paradox for the big tendency call " localism", because internet shopping has a strong environmental cost From a public policy perspective, this highlights the need to redefine what a "local product" stands for in a multi-channel and digital environment (Fuentes-Bautista 2014). Indeed, the French government has yet not clearly defined and regulated "local products". Here again, the Covid-19 crisis has accentuated a social phenomenon which involves the purchase of local, French products in connection with convenience stores, small supermarkets and surprisingly online channels. This trend, popularized by the slogan "eat local" in France, has turned into a worldwide trend [START_REF] Reich | Food as Ideology: Measurement and Validation of Locavorism[END_REF].

  These trends are largely comparable to those seen in Italy and Spain, two of the countries in the European Union that have been most affected by the Covid-19 pandemic. The three neighbours (Italy, France and Spain) suffered high numbers of reported deaths (35,123 fatalities in Italy, 30,223 fatalities in France and 28,436 fatalities in Spain) 2 .

Figure 4

 4 Figure 4 shows percentage change in sales of the five distribution channels before (1 January -15 March 2020) and during the lockdown (16 March -26 April 2020) with respect to sales in the corresponding periods in 2019.

  below highlights two phenomena: (1) The value market share of E-Commerce grew slightly during the years 2017-2019; therefore, this positive development already existed. (2) This value market share increased significantly under the impact of Covid-19, reaching 9.9% of the value market share.

Figure 1 :Figure 2 :

 12 Figure 1: Total coronavirus cases and fatalities in France (2020)

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Evolution of the turnover in Italy and Spain before pre-and lockdown periods

Figure 6 :

 6 Figure 6: Purchases by product families during the lockdown period

Table 1 :

 1 Evolution of the turnover offline vs online of local & organic fruit & vegetables Offline Online Local & organic fruit & vegetables : Nielsen Scantrack, 16 th March to 26 th April 2020

	-10%	+ 116%

Source

European Centre for Disease Prevention and Control updated 9th April

2022, https://www.ecdc.europa.eu/en/geographical-distribution-2019-ncov-cases

European Centre for Disease Prevention and Control updated 29th July 2020, https://www.ecdc.europa.eu/en/geographical-distribution-2019-ncov-cases

Data for the first five weeks are not available.

A hard discount store is an outlet for the sale of foodstuffs on a self-service basis, under a specific ensign, and with a surface area of 400 to 800 m2. The product mix on sale is restricted to commodities. Source : https://www.insee.fr/en/metadonnees/definition/c1699

(Source: Nielsen Scantrack, 26th April 2020).

https://www.latribune.fr/entreprises-finance/services/distribution/livraison-de-repas-la-nouvellefolie-des-citadins-francais-898932.html. La Tribune, French business magazine, Meal delivery: the new craze of French city dwellers.

https://www.statista.com/statistics/1176841/restaurant-delivery-user-growth-duringcovid-in-europe-bycountry/#:~:text=As%20the%20coronavirus%20(COVID%2D19,compared%20to%20pre%2D pandemic%20times.


Luciano Da Fontoura Costa
email: luciano@ifsc.usp.br

A Literal Approach to Network Modularity

A module, or community, of a network consists of one of its subgraphs whose nodes are more interconnected one another than with the remainder of the network. Several real-world and theoretical networks contain modules, which typically play an important role regarding the overall topological and dynamical properties of the network. Detecting modules in a network constitutes an important and challenging problem that is often approached from the perspective of a given criterion quantifying modularity, which can be based on topological or spectral features of the network, among other possibilities. In the present work, we approach a definition of the modularity criterion that is as literally close as possible to the typical textual concept of modularity that commences this abstract. The potential of the obtained literal modularity criterion is illustrate with respect to several model networks, as well as its application in the identification of optimally modular networks derived from datasets by using the coincidence methodology and as basis for an agglomerative clustering approach to community detection in terms of the coincidence similarity.

Introduction

The concept that goes by the names of cluster, group and module recurs frequently in the physical sciences and technology. This concept can be associated to an ability of human intuition to recognize relatively separated concentrations of entities -such as clouds in the sky, bunches on grape trees, and constellations in the firmament. As much important are abstract clusters, such as those related to similar patterns, words, concepts, objects, actions, etc. The identification of clusters is so critical as to have motivated the development of whole areas including pattern recognition and classification (e.g. [START_REF] Duda | Pattern Classification[END_REF][START_REF] Koutrombas | Pattern Recognition[END_REF][START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF]).

Yet, despite so much continued efforts, there is not definitive or consensual definition of what a cluster is. In conceptual terms, given a universe of distinct types of patters, a cluster is often understood as a set of entities that are more similar one another than with the others. This concept, which seems to summarize our own cognitive experience regarding related issues, has motivated a large number of respective more formal definitions involving a wide range of mathematical and computational resources and methods, with emphasis on statistical and physics-based methods. One typical approach is to define an index to measure clustering, and then apply it as a means to identify clusters. For instance, dispersion matrices (e.g. [START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF][START_REF] Duda | Pattern Classification[END_REF]), quantifying data variation in the features of objects in given groups, can be mathemati-cally combined so as to yield a respective measurement of how well the groups are separated. However, each of the many proposed methods have specific features which make them particularly suitable or unsuitable to specific types of data and their interrelationships.

With the substantial increase of interest on graphs and complex networks along the last decades (e.g. [START_REF] Wasserman | Social Network Analysis: Methods and Applications[END_REF][START_REF] Newman | Networks: An introduction[END_REF][START_REF] Barabási | Network Sience[END_REF][START_REF] Da | Analyzing and modeling real-world phenomena with complex networks: a survey of applications[END_REF]), an important subject analogous to the problem of data clustering steadily grown in importance. This issue relates to the definition, quantification, and detection of communities or modules in networks, which has great importance in network science since several real-world and theoretical networks present modular structure. Several definitions of modularity have been proposed (e.g. [START_REF] Newman | Finding community structure in networks using the eigenvectors of matrices[END_REF][START_REF] Newman | Networks: An introduction[END_REF]), based on a wide range of mathematical and computational concepts including spectral properties of matrices, multivariate statistics, and pattern recognition.

The motivation of the present work is to take literally the conceptual definition of modularity often found in the literature, which can be expressed as follows:

A network module, or community, consists of one of its portions that is more interconnected within itself than with the remainder of the network.

We start by motivating, presenting and discussing possible literal approaches to define and quantify the modularity of a subgraph within a given network, and then progress to describe the extension of these individual approaches to quantify the literal modularity of a complete network partitioned into subgraphs. Interestingly, it is also shown that the overall literal modularity has a direct relationship with the Jaccard index. The presented concepts and methods are then illustrated with respect to several randomly uniform networks, followed by the application of these resources to the optimization of the modularity of networks obtained from datasets described by features by using the coincidence methodology [START_REF] Da | Coincidence complex networks[END_REF]. The also important problem of detecting communities in a given network is then addressed by combining the literal modularity with an agglomerative clustering of the network nodes by considering the coincidence similarity while considering the distances from each node to all other nodes as features, with encouraging results.

Literal Modularity of Individual Subgraphs

Given a network, each of its subgraphs can be defined a graph containing a subset of the network nodes as well as the links between these nodes. In the present section we will focus on defining a literal modularity of a given subgraph respectively to the network from which it is derived. For simplicity's sake, we will be initially restricted to undirected networks, but the extension of these concepts to directed and weighted networks will be addressed in Section 9.

Figure 1 presents a diagram of a subgraph S k as well as the links that connect this subgraph to the remainder of the original network. Therefore, the network can be understood as consisting of the subgraph, the remainder network, and their interconnection. The subgraph is understood to contain e i (k) bidirectional (or undirected) edges or links, and the number of bidirectional links between S and the remainder of the network is expressed as e e (k). The literal modularity of the subgraph S can now be compactly defined as:

L(S) = e i (S) e e (S) (1)
with e e (k) = 0. Observe that the literal modularity can take unbound non-negative values starting from 0.

An alternative possible definition that may also be considered consists of making:

L w (S) = e i (S) + e e (S) e e (S) (2)
The latter definition corresponds to the reciprocal of the conductance, or normalized cut (e.g. [START_REF] Chung | Spectral graph theory[END_REF][START_REF] Kannan | On clusterings: Good, bad and spectral[END_REF][START_REF] Leskovec | Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF]) of the subgraph S.

The two definitions above are mostly similar in case e e (S) is relatively small, but diverge when e e (S) is more substantial. To any extent, we have that:

L w (S) ≥ L(S) (3)
For simplicity's sake, except when relating modularity and similarity in Section 6 the present work focuses on the definition of literal modularity as in Equation 3.

In case the number of nodes in the subgraph (n) and in the overall network (N ) are known or should be considered, the above equation can be modified as:

L p (S) = ei(S) n 2 ee(S) 2n(N -n) = p i (S) p e (S) (4)
Thus, p i (k) and p e (k) correspond to the uniform link density or interconnection probability respectively to the subgraph and its interconnection with the remainder of the network, respectively.

In the present work we will be restricted to literal modularity as defined in Equation 3. Also, observe that in case the subset has the same number of nodes as the remainder network, the two literal modularities will result identical values.

Figure 2 illustrates the calculation of the literal modularity of the delimitated subgraph respectively to the remainder of the network. A value of L = 5/8 is obtained, which seems to be conceptually suitable.

Interestingly, nothing is assumed or taken into account regarding the inner structure of either the subgraph or the remainder of the network. All that matters here is the global topology between the subgraph and the remainder of the network. In this way, the issue of modularity can remain specific to the overall, generic interconnection of edges between the subgroup and the remainder of the network, without requiring quantifications or assumptions of other properties of the network including its type, average degree, minimal paths, etc. One immediate issue implied by the incorporation of such additional requirements while defining and quantifying modularity is that it tends to become specific to the features and hypotheses adopted, while the networks of interest may not completely adhere to them. At the same time, if two or more requirements about the modules are imposed from the outset, it becomes more difficult to identify and understand the respective influences of the requirements on the obtained results.

However, because these properties can indeed be of interest, it is always possible to define respective indices to be integrated with the modularity quantification. For instance, in case it is necessary to take into account how much the subgraphs and/or networks of interest adhere to a uniformly random edge distribution, a specific index u(S) can be devised. In case both indices vary in the interval [0, 1], an interesting approach consists of obtaining an index a(S) taking into account these two constraints to correspond to the product between the two indices, i.e.:

a(S) = L(S) u(s) (5)
or to their geometrical average. More generally, several properly normalized indices (e.g. with values in the interval [0, 1]), can be combined into the modularity expression:

a(S) = L(S) [I 1 (s)] α1 [I 2 (s)] α2 . . . [ I R (s)] α R (6)
where α i are weights controlling the contribution of each of the involved requirements. Other types of combinations, including weighted arithmetic average, be equally considered.

In this manner, the effects of the each of the imposed requirements remain disentangled, allowing a direct and simplified quantification and study of the several involved aspects expected from the mixed modularity.

Several other indices reflecting respective requirements or expected characteristics of the modules can be plugged into Equation 6, including indices quantifying the regularity of the subgraphs respectively to topological measurements (e.g. degree, clustering coefficient, average shortest distances), adherence to specific theoretical connectivity models (e.g. uniformly random, scale free, modular, etc.), as well as indices related to non-topological properties of the nodes that happen to be of special interest in each problem.

A generic number of these indices can be therefore combined in several mathematical ways allowing several types of requisites and hypotheses to be taken into account and quantified. This approach consequently decouples the issue of modularity definition and quantification from other networks aspects, allowing attention to be focused on aspects that are exclusively related to the concept of modularity in terms of relative distribution on internal and external connectivity, which corresponds to the approach described in the present work.

Let us now discuss some of the interesting properties of the literal modularity of a subgraph.

First, we need to treat the case e e (S) = 0. We immediately have that:

lim ee(S)→0 L(S) = ∞ (7)
This case indicates that the subgraph is modular to such a high degree that it becomes an isolated component, which does not need to be necessarily connected. Therefore, the infinite obtained value does reflect that this case indeed corresponds to the maximum modularity expected from any subgraph. In practice, this can be indicated by a special label or value, such as:

If e e (S) = 0 =⇒ L(S) = ∞ (8)
Another interesting possibility consists in assigning 1 to the literal modularity in these cases, as this will have absolutely no effect on the respective geometrical average to be used in that case (see Section 3).

The next interesting specific situation concerns the case when e e (S) = 1, which implies:

L(S) = e i (S) (9)
which corresponds to the maximum literal modularity, except for the case e e (S) = 0, that can be obtained. In this case, the literal modularity becomes identical to the number of links in the subgraph.

Yet another situation to be taken into account regards subgraphs corresponding to nodes with degree one, being characterized by e i (S) = 0 and e e (S) = 1. The problem with this type of subgraph is that it will nullify geometrical averages taken for the characterization of the overall literal modularity of a network. An interesting possibility here is to consider such subgraphs as having a self-loop, in which case e i (S) = 1, e e (S) = 1, and M(S) = 1, which does not affect in any way geometrical averages.

Given a subgraph, its minimal literal modularity will be verified whenever e i is maximum. Thus, in general it can be said that the literal modularity of a subgraph of a given network takes values lower bound by 0 while being infinitely upper bound.

Literal Modularity of Sets of Subgraphs and Whole Networks

Having presented the literal modularity and discussed some of its properties, we are now in position to extend this index to a whole set of subgraphs, and not only the isolated subgraph of interest. Given a set os subgraphs S k , k-= 1, 2, . . . , M , derived from a network A, the literal modularity of this set can be defined as the geometrical average between the values L(S k ), i.e.:

L a (A) = M M k=1, ee(S k ) =0 e i (S k ) e e (S k ) = M M k=1, ee(S k ) =0 e i (S k ) M M k=1, ee(S k ) =0
e e (S k ) [START_REF] Chung | Spectral graph theory[END_REF] The subscript 'a' in L a (A) stands for absolute, in contrast to a relative index to be defined later in this section. In principle, there are not restrictions or hypotheses regarding the subgraphs of A to have their conjoint literal modularity quantified.

One of the reasons for adopting the geometrical average instead of the traditional average is that the former implements a more demanding quantification of the modularity, penalizing more intensely configurations departing from the maximum modularity. It is interesting to keep in mind that the geometrical average, which is particularly strict regarding the similarity between the averaged literal modularities, assumes that each of the subgraphs considered as candidates for modules have intrinsic importance, otherwise it should have been weighted with a small value or not included in the calculation of the overall literal modularity.

The weighted geometrical mean of the subgraph literal modularities can be defined as:

L w (A) = W M k=1, ee(S k ) =0 e i (S k ) e e (S k ) w k (11)
where w i are the weights associated to respective individual modularities, and:

W = M i=1 w i (12)
If necessary, the influence of a few little modular subgraphs can be controlled by leaving the outlier cases or implementing some weighting scheme. Other forms of combining the individual literal modularities, including the arithmetic average and weighted averages, can also be adopted. Another interesting approach consists in ordering the obtained individual literal modularity modules and then taking successive groups g elements, from 1 to M , into the overall literal modularity estimation. The value of g for which the largest modularity decrease is observed provides an possible value to be taken as the overall modularity. However, each of the several manners of combining the individual modularities into a single overall value has its intrinsic features, so that the choice should consider the interests specific to each application, as well as on the characteristics of each given dataset.

The literal modularity of a whole network A can now be defined in terms of its possible partitions.

Let the original, complete network A be partitioned into M subgraphs, i.e.:

A = M k=1 S k , with S k ∩ S j = φ for i, j = 1, 2, . . . , M (13)
where φ is the empty set. The links between the subgraphs are understood to be also contained in the subgraphs.

In case all subgraphs are minimally connected to the remainder of the network (i.e. each subgraph is not an isolated component), the maximum overall literal modularity is therefore achieved if and only all individual subgraphs have identical literal modularity values L(S k ) = v. This is illustrated in Figure 3 respectively to a network corresponding to a linear string of 6 nodes which is aimed to be divided into three communities or modules.

It is of particular relevance the fact that both situations [e i (S k ) = 1, e e (S k ) = 1] and [∀e i (S k ), e e (S k ) = 0] therefore yield L(S k ) = 1. Indeed, these two types of interconnections can be understood not to affect the overall literal modularity. In the former case, which corresponds to a single node attached to the remainder of the network through a single edge, this could indeed be expected, as this does not change the overall modularity. In the latter case, the literalmodularity of isolated nodes would be otherwise infinite, strongly affecting the obtained modularity value. This alternative, therefore, do not take isolated connected components of a network into account. Other schemes can be adopted in the case of alternative requirements.

In general, given a network A with a total of E bidirectional edges to be partitioned into M communities, the respective peak literal modularity index can be readily calculated as:

L a,max (A) ≈ E -1 M (14)
Observe that this value is approximated to the closest integer (e.g. rounded) values.

The estimation of the maximum literal modularity for given E and M allows the normalization of the absolute version of the literal modularity given in Equation 10. A possible manner to implement this normalization is as follows:

L r (A) ≈ L a (A) L a,max (A) = L a (A) E M (15)
with 0 ≤ L r (A) ≤ 1.

In the case of isolated subgraphs, these can be treated in several manners, such as assigning a non-infinite value to them, such as twice the number of their respective inner edges. One possibility of particular interest consists in assigning the number of edges as the individual literal modularity of each isolated component, which corresponds to understanding that that component is attached to remainder of the network through a single connection.

Uniformly Random Modules

In order to better illustrate the potential of the literal modularity, we compare it to the classical modularity respectively to modular networks composed by two modules characterized by uniformly random interconnections. The examples in Figure 5 relates to the same type of networks, with constant probabilities of connections within each subgraph (equal to 0.7), but with their relative sizes increasing from left to right, while the interconnecting probability within one of the subgraphs increases from top to down.

The literal modularity presents reasonable quantifications in every case, allowing a greater excursion than the classical modularity

Robustness to False Positive

As with clustering, from which community finding directly relates, it is important to evaluate the literal modularity of networks respectively to false positives, which can be done by considering random partitions in model networks with uniformly random distribution of interconnections.

In order to do so, 1000 networks with 200 nodes and Similar tiny modularity values (compare, for instance, with the literal modularities in Fig. 4) were obtained for other link probabilities, network sizes, number of subgraphs in the partition, and the relative size of the partition subgraphs, indicating remarkable robustness of the literal modularity approach with respect to false positives induced by random fluctuations of the network overall interconnectivity. This can be understood as a direct consequence of the fact that the literal modularity approach does not have a statistical basis, and no hypothesis are made concerning the uniformity of the edges distributions or any other property of the subgraphs and network except for the net number of edges. subraphs of similar sizes. Markedly low literal modularity values have been obtained in all configurations, all them being smaller than 1.5, corroborating the robustness of the literal modularity quantification of network modularity with respect to false positives. This is a direct consequence of the fact that this type of modularity quantification is not statistical and makes no hypothesis whatsoever regarding the structure of properties of the network and subgraph candidates, except for the number of edges.

Relating the Overall Literal Modularity and the Coincidence Similarity

As we will show in this section, the arithmetic-based overall literal modularity and the Jaccard similarity index are intrinsically interrelated, in an opposite sense as could be expected because two highly modular multisets will tend not to be similar. We will approach this interesting relationship in the case of two multiets A and B. Please refer to Figure 7. We have from the definition of individual literal modu-larity of a set that:

L(A) = |A| |A ∩ B| = n A n s (16) L(B) = |B| |A ∩ B| = n B n s (17)
where |A| stands for the cardinality of A, corresponding to the number of elements inside it.

The Jaccard similarity index between these two multisets is defined as:

J (A, B) = |A ∩ B| |A ∪ B| = n s n A + n B -n s = = 1 n A ns + n B ns -1 = 1 L(A) + L(B) -1 (18)
When L(y 1 (x)), L(y 2 (x)) 1, which is often the case, we can write:

J (A, B) ≈ 1 L(A) + L(B) = = 2 1 2 [L({A, B})] = = 2 L a ({A, B}) (19)
Thus, in summary we can write:

J (A, B) ≈ 2 L a ({A, B}) (20)
Which establishes an inverse relationship between the Jaccard index and the overall literal modularity between two sets (or subgraphs).

Application Example: Optimizing Coincidence Networks

Given a dataset with each of its N element described in terms of a set of M features, it is possible to apply the coincidence methodology [START_REF] Da | Coincidence complex networks[END_REF] to translate it into a respective network in which each data element is represented by a node, while the pairwise coincidence [START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF][START_REF] Da | Coincidence complex networks[END_REF] between two data element is taken as the weight of the respective network link. Two parameters are involved in the coincidence approach to translating datasets into networks, namely T and α. The former parameter is an overall threshold applied on the obtained coincidence similarity values. The latter parameter controls the relative contribution of pairwise features that have the same or opposite signs on the overall resulting coincidence value e.g. [START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF][START_REF] Da | Coincidence complex networks[END_REF]).

Given a dataset and an instantiation of the two parameters, the coincidence methodology yields a respective network which can be optimized respectively to some sought property, such as modularity. The classical modularity index has been adopted in previous applications of the coincidence methodology (e.g. [START_REF] Da | Coincidence complex networks[END_REF]), but here we consider also the literal modularity. The basic idea is to vary α and/or T while seeking for the configuration leading to maximum modularity.

In the present work, we shall fix T and vary α along equally spaced values while monitoring the respectively obtained classical and literal modularities. We will apply this methodology on the dataset of handwritten characters, containing 50 samples of each of the characters 'c', 'e', and 'o' (as in [START_REF] Da | Coincidence complex networks[END_REF]), after each of them have been characterized by four real-valued features corresponding to the written area, the width and height of each character, and the arc length of their external contour.

Figure 8 depicts the obtained classical and literal modularity values in terms of the parameter α respectively to T = 0.3 (a) and T = 0.7 (b).

Figure 9 shows scatter-plots obtained by plotting the literal modularity versus the classical modularity respectively to the translation of the handwritten dataset by using the coincidence methodology respectively to T = 0.3 (a) and T = 0.7 (b).

All modularity curves are characterized by a peak taking place at some specific value of α, which is therefore considered optimal for the respective configurations. Interestingly, markedly distinct results have been produced by each of the two modularities for both T = 0.3 and T = 0.7, with the peak of literal modularity taking place for values of the parameter α that are smaller than those obtained by the classical modularity in both these situations.

The optimally modular coincidence networks representations of the handwritten characters by using the two adopted thresholds and modularity approaches are presented in Figure 10. The optimal networks obtained for the handwritten characters data can be considered to have the modules more well separated than those obtained by using the classical modularity. Actually, a substantial separated component corresponding to the handwritten character 'c' has been obtained for both T = 0.3 and T = 0.7 while adopting the literal modularity approach. This result suggests that the coincidence methodology for translating datasets into networks can benefit from the consideration of the literal modularity while identifying optimally modular networks, or in cases in which the adopted optimization criterion involves the modularity (e.g. [START_REF] Reis | Enzyme similarity networks[END_REF]).

Of particular importance is the fact that the above lit- eral modularity optimization was supervised, taking for granted the original handwritten letters category. In this case, by relaxing requirements such as the imposition of uniformly regular interconnections within modules, the literal modularity allows enhanced flexibility while looking for a maximally modular configuration. The relaxation of additional restrictions is only possible provided the supplied original categories of each data element can be taken with certainty.

8 Community Finding by Coincidence-Based Hierarchical Clustering While the previous application example regarding the translation of datasets into maximally modular networks can be understood as being supervised, in the sense that the categories of the nodes are known and supplied a priori, the problem of identifying the modules in a given network, which is often referred to as community finding, represents a substantially more challenging task since it is not supervised because the categories of the nodes are not know a priori. Indeed, even the number of categories may not be known, so that the community finding algorithm also needs to estimate this property.

In this section we describe an approach to community finding that is based on the literal modularity suggested in the present work as well as on the agglomerative clustering (e.g. [START_REF] Da | Real-valued Jaccard and coincidence based hierarchical clustering[END_REF][START_REF] Duda | Pattern Classification[END_REF]) of adjacent network nodes while considering the coincidence similarity [START_REF] Da | Coincidence complex networks[END_REF] between their distances to all other network nodes as feature vector. There are several types of agglomerative clustering available, and here we choose the single-linkage approach, which has been found to be less susceptible to false positive result [START_REF] Tokuda | Revisiting agglomerative clustering[END_REF].

One additional problem typically imposed while maximizing the literal modularity in a non-supervised approach is that the groups are not available and need to be identified. In general, it is not enough to consider only the literal modularity in these cases, because the inner structure of the modules may also matter. However, this can be immediately incorporated into the optimization as describe above, i.e. by using mixed modularity indices where the literal modularity is combined with other requirements, such as uniform or scale-free degree distribution.

Given a network A to have its modules identified, first we calculate the distance from each node to all other nodes in the network. These values are taken as the feature vectors assigned to each node. Initially, each node corresponds to a module. The coincidence similarity is then calculated between each pair of nodes, and the modules belonging to the pair of nodes defining the current maximum coincidence value, and that happen to be originally interconnected (according to the original adjacency matrix), are merged, while bookkeeping their labels and memberships. The processing proceeds until only a single cluster is obtained. The literal modularity is calculated for each possible stage along the agglomerative clustering, with the configuration corresponding to the maximum modularity taken as the most modular solution. The pairwise disconnected modules are assigned respective literal modularity equal to 1 during the estimation of the overall network modularity.

Figure 11(b) illustrates the dendrogram resulting from the application of the above described method to a network obtained from the three groups of points, plus 5 uniformly distributed noise points, in a 2D space as shown in Figure 11(a). The parameterless version of the coincidence similarity (i.e. α = 0.5) has been adopted for simplicity's sake, implying that even more effective results could be probably achieved by optimizing over that parameter.

Given a dataset and an instantiation of the two parameters, the coincidence methodology yields a respective net-work which can be optimized respectively to some sought property, such as modularity. The classical modularity index has been often adopted in previous applications of the coincidence methodology (e.g. [START_REF] Da | Coincidence complex networks[END_REF]), but here we consider the literal modularity. The basic idea is to vary the dendrogram threshold T while seeking for the configuration leading to maximum combined modularity.

Because this example concerns the more challenging unsupervised classification (clustering), we incorporate an additional requirement about the uniformity of the connections within each group that is a candidate for a module. This uniformity index u(S) is calculated so as to reflect the horizontal extension of each group S in the respective dendrogram, and then multiplied by the literal modularity (Eq. [?]), therefore yielding a combined modularity index that demands both high literal modularity and high uniformity as estimated by the respectively adopted index.

The network with the maximum literal modularity obtained by the considered method is shown in Figure 11(b). As could be expected, two large communities (in blue and green) have been properly identified.

It is of particular interest to observe that virtually all groups in the obtained dendrogram are relatively long, which corroborates the potential of the coincidence methodology for strict quantification of the similarity between the compared groups, therefore reducing the chances of false positives in the identification of possible modules. At the same time, it can be observed that the branches corresponding to the similarity between the nodes belonging to the original groups have been well spread along the horizontal axis of the dendrogram, therefore revealing a more detailed characterization of the respectively involved similarity relationships.

Three modules have been properly identified, each corresponding to the original groups.

Directed and Weighted Networks

Though the present work has been so far limited to undirected networks, the extension of all presented concepts and methods to directed and weighted networks is relatively direct, as discussed in the following. Weighted networks are characterized by the links being associated to non-negative numerical weigh values, such as corresponding to pairwise similarity between the properties of nodes. In these cases, the concept of node degree becomes node strength, which corresponds to the sum of the weights of the edges associated to the node of interest. Thus, the concept of subgraph literal modularity can be immediately adapted by replacing the number of inner and interconnecting edges by the sum of the respective weights. However, in order to avoid too large weights associated to some edges to influence strongly the literal modularity, a non-linear transformation can be applied to the pairwise weights (e.g. taking the square root of each weight). In addition, when the sum of the weights of the links interconnecting the subgraph with the remainder of the networks is smaller than 1, too strong modularities can be obtained that will bias the literal modularity. In these cases, it is interesting to take the weight of the external links as corresponding to the maximum between the sum of weights and 1, i.e.:

e e (S) = min W ext , 1 (21)
The case of a subgraph S in directed networks can be approached immediately by considering separately the links outgoing from the subgraphs to the remainder of the network, and vice-versa. The number of these nodes can then be used to define two literal modularities as:

Networks that are both directed and weighted can be addressed by combining the above presented approaches.

Concluding Remarks

Given the importance of communities in several theoretical and real-world interconnected systems, network modularity has motivated a great of interest from the community (e.g. [START_REF] Newman | Finding community structure in networks using the eigenvectors of matrices[END_REF][START_REF] Newman | Networks: An introduction[END_REF]). In the present work, we approached this interesting and relevant problem in terms of a definition of modularity that is as close as possible to the intuitive interpretation often found in the literature. More specifically, this means taking into account only the terms explicitly present in those intuitive definitions. First, we focused on the subgraph of interest, irrespectively to the remainder of the network except for the respective interconnections. Second, only the number of links within the subgraph and the number of interconnecting links are taken into account, irrespectively to the topology of the subgraph or overall network. Then it became possible to define the subgraph modularity in a surprisingly direct and compact manner as the ratio between the number of internal and interconnecting links. In a modified version, the above index corresponds to the conductance or normalized cut (e.g. [START_REF] Chung | Spectral graph theory[END_REF][START_REF] Kannan | On clusterings: Good, bad and spectral[END_REF][START_REF] Leskovec | Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF]) of a subgraph.

After discussing some of the main properties of the literal modularity respective to single subgraphs, we used this concept to derive an index quantifying the overall modularity of the network, partitioned into M subgraphs. This was achieved by taking the geometrical average of the individual subgraph literal modularities. Interestingly, the overall literal modularity (in the case of arithmetic average) was then shown to be directly related to the Jaccard index.

The potential of the suggested literal modularity was then illustrated respectively to the characterization of networks involving two uniformly random subgraphs, with encouraging results. The possibility to adopt the literal modularity as a criterion for selecting the most modular network obtained along an agglomerative clustering while considering coincidence values between the distances from each node to all other nodes was also presented.

All in all, the proposed literal modularity concepts and methods present the following interesting properties: (i) a direct (literal) implementation of the conceptual understanding of modularity; (ii) great conceptual simplicity; (iii) nearly negligible computational cost; (iv) applied to each subgraph independently of the remainder of the network; (v) at least for the data and cases addressed in the present work, capable of providing an effective quantification of the separation between clusters; (vi) does not require analysis of the topological structure of the subgraph to be quantified as a module; (vii) closely related to the concept of reduced networks; (viii) natural and directly extensible to directed and weighted networks; (ix) can be applied to any type of network or subgraph, including disconnected components, trees, and completely regular (but finite) lattices; and (x) cannot be influenced by any property of the subgraph or overall network including number of nodes, average degree, etc.

The simple method to find communities by using literal modularity and agglomerative clustering based on the coincidence similarity between the distance from each node to the others also presents several interesting characteristics, including small computational expenses, in the sense that for a network with N nodes, only N basic steps are required, each of which implementing the merging of a successively less similar node.

As with every scientific and technological applications, those proposed in the present work should be understood as providing tentative and preliminary information about the respectively considered problem and datasets, to be further validated in terms of other approaches and additional data and experiments. In addition, the literal modularity concept is by no means generally optimal, and several other existing indices will be best suited for specific approaches and types of data.

The presented concepts, methods, and results pave the way to a large number of future developments. These include the investigation of the effect of replacing the number of edges by the respective edge densities in the definition of literal modularity of a subgraph. In addition, the concepts and methods can be applied to a large number of different problems and data type, not only for its own sake, but also as a way to further evaluate and understand their properties. Another particularly promising possibility consists in adapting the literal modularity concept and methods to the also important problem of clustering data elements characterized by respective features. This can be achieved immediately by considering the coincidence similarity between the features while adopting the literal modularity as the criterion to be optimized.

Figure 1 :

 1 Figure1: A subgraph S k of a network, interconnected to its remainder. n i (k) is the number of bidirectional links within the subgraph, irrespectively of its topology, and ne(k) is the number of bidirectional links interconnecting S k to the remainder of the network. The literal modularity of the subgraph S k is defined as L(S k ) = n i (k)/ne(k). Observe that the properties of the latter are completely immaterial in our literal approach to modularity.

Figure 2 :

 2 Figure 2: An example of literal modularity calculation. The subgraph of interest, represented encircled by the blue contour, is characterized by e i (S) = 5 and ee(S) = 8 undirected links, from which we readily get L = 5/8 . Observe that the modularity of the subgraph does not depend neither on the remainder of the network to which it is attached nor on the properties of the subgraph other than its number of links.

Figure 3 :

 3 Figure 3: The maximum literal modularity of a string network with 6 nodes is achieved for 3 modules S k of 2 adjacent nodes, with e i (S k ) = 1 inner interconnection and ee(S k ) = 2 external interconnections with the remainder of the network. The overall modularity in this case is L(A) = 0.5.

Figure 4

 4 Figure 4 presents the literal and classic modularity values obtained for networks containing two subgraphs with uniformly random interconnections. The relative size of the two subgraphs increases along the horizontal axis (left to right), while the interconnecting probability increases along the vertical axis (top down).

Figure 4 :

 4 Figure 4: Literal and classical modularities quantified respectively to networks composed of two interconnected subgraphs with inner uniformly random interconnections. The relative size of the subgraphs increases from left to right, while the interconnecting probability increases from top to down.

Figure 5 :

 5 Figure 5: Literal and classical modularities quantified respectively to networks composed of two subgraphs with uniformly random interconnections. The size of one of the subgraphs increases from left to right, while the inner connectivity of one of the subgraphs increases from top to down.

Figure 6 :

 6 Figure 6: Distribution of literal modularity of randomly uniform networks with 200 nodes with p = 0.05 partitioned into M = 2, 10subraphs of similar sizes. Markedly low literal modularity values have been obtained in all configurations, all them being smaller than 1.5, corroborating the robustness of the literal modularity quantification of network modularity with respect to false positives. This is a direct consequence of the fact that this type of modularity quantification is not statistical and makes no hypothesis whatsoever regarding the structure of properties of the network and subgraph candidates, except for the number of edges.

Figure 7 :

 7 Figure 7: Two multisets A and B, each with respective n A and n B elements, presenting a shared intersection s with ns elements.

Figure 8 :

 8 Figure 8: Optimizing coincidence networks: (a) The standard modularity values M(A) and the literal modularity L(A) of coincidencebased networks with T = 0.3 respective to three handwritten characters ('c','e', and'o') in terms of the optimization parameter α. Interestingly, the maximum modularity values are found respectively to two markedly different values of α. (b) the same, but with T = 0.7.

Figure 9 :

 9 Figure 9: Scatterplots illustrating the relationship between the classical and literal modularities applied to the translation of the handwritten dataset by using the coincidence methodology for T = 0.3 (a) and T = 0.7.

Figure 10 :

 10 Figure 10: Optimally modular coincidence networks obtained by: (a) adopting literal modularity with T = 0.3; (b) adopting classical modularity with T = 0.7; (c) adopting literal modularity with T = 0.7; (d) adopting classical modularity with T = 0.7. The optimally modular handwritten characters networks obtained by the literal modularity can be intuitively considered to have subgraphs more intensely separated than those obtained while adopting the classical modularity.

Figure 11 :

 11 Figure 11: Community detection by using agglomerative clustering based on single linkage of the ninth-power of the coincidence similarities between the distances from each node to all other nodes (a). Dendrogram obtained by the application of the coincidence-based agglomerative clustering (b), while taking into account the product of the literal modularity and an index adopted to quantify the uniformity of the dendrogram groups respectively to their extension along the vertical dendrogram axis. The longer the extension of the group, the less uniform it is deemed to be. The values along the y-axis correspond to one minus the ninth-power of the coincidence similarity values. The three original groups have been properly identified in this case.

L

  out (S) = e i (S) e e,out (S) (22) L in (S) = e i (S) e e,in (S) (23) Other approaches are possible, such as combining both types of edges between the subgraph and the remainder of the network: L c (S) = e i (S) e e,out (S) + e e,in (S)

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).

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