Themis Kokolakakis
email: t.kokolakakis@shu.ac.uk

Fernando Lera-López

Analysis of the determinants of sports participation in Spain and England

 (European Commission, 2004). The methodology used was that of the Standard Eurobarometer polls. In the aforementioned poll, the definition of sport is not explicitly stated and it is dependent on the understanding of the interviewee.

Table 1 presents some results of the latest Eurobarometer survey classifying participation in 'at least once a week' and 'at least three times a week' categories. In 2004, 38% of the European citizens (24 member states) participated in sport at least once a week. Forty per cent of the EU citizens answered that they never play sport. Participation rates at least once a week fluctuate in a range of 22% (Portugal) to 76% (Finland). The Scandinavian countries are the most sporting countries, with participation rates in excess of 70% at least once per week. On the other hand, sports participation tends to be lower in some southern countries (Portugal 22%, Greece 26%, and Italy 27%) and new member states (Hungary 20%, Slovakia 24%). The participation rates (at least once per week) of the UK and Spain are 45% and 37% correspondingly.

If the definition changes from participating 'at least once a week' to 'at least three times a week', the sports participation level of the 24 EU states reduces by more than half (from 38% to 17%). At this level of frequency, the lowest participation rate is recorded in Portugal (8%), followed by Italy (9%) and Austria (12%). The highest participation rates are still recorded in the Scandinavian countries with Finland leading the way at 45%, followed by Sweden at 40%. The 2004 participation rates (at least three times a week) for Spain and England are 21% and 23% correspondingly.

It must be pointed out that the year 2004 was rich in sports events (prominent among them the Athens Olympics) which helped to boost sports participation among some European countries. In terms of the socio-demographic characteristics, the poll of the EU citizens verifies results that are in agreement with our present analysis of the English and Spanish sports participation levels. Specifically, there is:

• A significant gender inequality in EU sports participation, with 41% of men participating at least once per week compared to 35% of women.

• A negative relationship between age and sports participation, meaning that as age increases, the percentage of citizens participating at least once per week decreases.

• A positive relation between sports participation and education; the participation rate increases significantly, as we switch from people that finished education at 15 years of age or earlier (20%), to people that finished education at the age of 20 or older (50%).

It is important to note that the poll shows the existence of a positive relationship between availability of free time and sports participation. In fact, the lack of free time is identified as the most significant factor for absence of sport activities by 34% of interviewees. This was followed by personal dislike of sport (25%). It is extremely significant that 'supply of sport facilities' was not identified as a major factor preventing citizens to participate. Only 3% of the interviewees claimed this as a reason. The time effect hides a plethora of income effects that often contradict each other. Although this is not investigated in the Eurobarometer, the implication is that citizens with a lot of free time, such as students, part-time workers, and short-term unemployed are going to have a time advantage over the rest. Similarly, citizens of high incomes can potentially put themselves in a position of indirectly purchasing free time through buying services from gardeners, house cleaners, tutors for children etc. Time disadvantaged groups include the selfemployed and citizens taking care of large families. The free-time significance brings about a further dimension in sports participation. Sports participation is in direct conflict with products of mass culture indoor games. A lot of time is spent on indoor electronic entertainment e.g. Television, DVDs, Internet, video games. In some age groups, this time is so extensive, that it is reasonable to say that indoor entertainment is a major factor that checks the growth of sports participation. In the poll, 86% of the interviewees agreed to the statement that sport is a genuine alternative to divert from interior activities, such as television, video games, and the Internet.

Outside the Eurobarometer publication there have been three efforts promoting harmonization in the measurement of sport participation rates: COMPASS, HETUS and IPAQ.

COMPASS was an initiative of the Italian National Olympic Committee, UK Sport and Sport England. The objective was to 'examine existing systems for the collection and analysis of sports participation data in European countries with a view to identifying ways in which harmonisation may be achieved, so that greater comparability of data from different European countries will become possible' [START_REF] Compass | Sports Participation in Europe[END_REF]. Table 2 presents indicative statistics for seven EU countries using 1999 COMPASS data. COMPASS statistics, although not comparable with the polls, have been adjusted on the basis of participating once per week 1 . The percentage rates between 1999 and 2004 divert more in the cases of Italy and Spain than in countries with high sports participation, such as Finland and Sweden. This reflects the definition of sport under the COMPASS methodology (in particular the treatment of walking) rather than changes in actual participation rates. In the COMPASS study, people that took part in walking and no other activity were classified as non-participants. This definition was not a problem in the case of Finland, yet it excluded a larger proportion of people in countries with small sports participation rates, such as Italy.

The Harmonised European Time Use Survey (HETUS) provides an alternative way of examining European sports participation. Since the 1990s, this was supported by Eurostat, in order 1 The COMPASS participation rates were adjusted proportionately to convert the class of participating at least 60 times per year to participating at least 52 times per year. The numbers here are indicative of the evolution of sport participation over the examined period. Walking/hiking and sport in general. In the case of Spain, the time spent on walking and hiking is very considerable, at 36 minutes per day.

Finally, IPAQ stands for: International Physical Activity Questionnaire. Its purpose is to derive a set of instruments that can help us estimate international indices for Physical Activity.

The case of the UK and Spain

In Great Britain, historically, sports participation figures were derived from the General Household Survey. The latter surveys over 20,000 people aged 16+. For the period 2005/6, in England, according to the Active People Survey, 21% of adults participated on average at least three times a week and 40% on average at least once a week. These figures are close to the 2004 Eurobarometer statistics of 23% and 45% respectively (European Commission, 2004). Table 4 Nevertheless, from 1975, the Centre for Sociological Research (CIS) has applied a questionnaire to analyse the Spanish sports behaviour. CIS is an independent entity established to study Spanish society, mainly through public opinion polls. CIS, together with the National Sports Council (CSD), has given special attention to sports participation in Spain developing a national survey every five years. The last survey was made in 2005 with a sample of 8,170 adults (aged 15+) [START_REF] García Ferrando | Posmodernidad y deporte: Entre la individualización y la masificación[END_REF]. It is a nationally representative individual survey, the only one of its kind carried out in Spain, and contains economic and socio-demographic information.

Table 5 presents the evolution of sports participation in Spain during the period 1975-2005. According to this evolution, there has been a stagnation point in sports participation in Spain in 1995. From 1975to 1985 sports participation increased significantly. In the case of 'at least one activity', sports participation grew constantly until 1995, although participation in more than one sporting activities was in the doldrums since 1990. In this section, we present an outline of the theoretical models that underpin current research on sport demand and empirical research about factors affecting the level of sports participation. According to [START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF], economic decision-making theories in relation to sports can be broken down into two main types: neoclassical and heterodox approaches.

Neoclassical approaches employ a rational-choice framework to model individual sports participation. The idea is to maximise subjective utility subject to certain constraints, mainly relating to budget and time. Implicitly, sport uses "non-economic" and "non-obligated" time, the demand for which reflects the trade-off between the utility derived from consumption of goods and the opportunity cost of an hour of sport. More recent variants, following [START_REF] Becker | A Theory of the Allocation of Time[END_REF] theories, have integrated time allocation into the consumption decisions faced by households.

Consequently, individual time allocation can be modelled in terms of the allocation of activities between household members, revealing the influence of household member preferences [START_REF] Downward | On leisure demand: a Post Keynesian critique of neoclassical theory[END_REF]. Overall, the neoclassical demand model would consider the price of a sport activity, income, and prices of other goods. In addition, a time element is also considered, as sport involves consumption of time. In fact, most often (51% participate outside official channels), it involves only consumption of time, rather than money (European Commission, 2004). [START_REF] Gratton | The Economics of Sport and Recreation[END_REF] point out that the neoclassical analysis has become too restrictive for an analysis of sport participation. That leads towards multidisciplinary heterodox methodologies and a more complex perception of sports motivations.

Heterodox economic theories consider a wider set of methodological and theoretical principles than neoclassical theory [START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF]. These theories involve economic, sociological, and psychological approaches. The Post-Keynesian approach emphasises that individual behaviour is linked to broader aspects of social behaviour such as the importance of 10 social values, while the consumption of sport involves learning-by-doing and spillover effects [START_REF] Lavoie | Post Keynesian consumer theory: Potential synergies with consumer research and economic psychology[END_REF]. The sociological analysis of sports focuses on explaining sporting activities in terms of concrete social situations and the construction of identities by individuals in choice situations.

According to this theory, sporting styles and individual preferences are linked not only to individual feelings, but also to social pressure and the influence of habits [START_REF] Bourdieu | Distinction: a social critique of the judgement of taste[END_REF].

Bourdieu presented two different factors to explain divergent tastes in sports: economic capital (income) and cultural capital (education). Finally, the psychological approach argues that the individual's preferences and tastes are not given [START_REF] Scitovsky | The joyless economy[END_REF]; they evolve and change over the life-span. Sensation-seeking, awakening, concern, pleasure or anxiety can be potential sources of demand for sport and leisure activities. Additionally Lipsombe (1999) discussed the influence of 'peak experience' within the psychological framework.

Despite a lack of consensus in the definition of sport, it is possible to make general assessments of the roles played by economic, individual and social variables. The empirical evidence has shown that sports participation decreases with age (Farrel and Shields, 2002;European Commission, 2004;[START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF][START_REF] Downward | Social interactions and the demand for sport: An economic analysis[END_REF][START_REF] Breuer | Demographic and economic factors concerning the inclusion in the German sport system. A micro-analysis of the years 1985 to 2005[END_REF][START_REF] Wicker | Promoting sport for all age-specific target groups. The impact of sport infrastructure[END_REF]. Nevertheless, longitudinal analyses show that the assumption of decreasing sports activity with increasing age is only correct for men [START_REF] Breuer | Decreasing sports activity with increasing age? Findings from a 20year longitudinal and cohort sequence analysis[END_REF].

Differences in sports participation can be attributed to biological and physical limitations and to changes in activity preferences as people grow older [START_REF] Barber | Canadian participation rates in ten sport and fitness activities[END_REF]. Against this mainstream view, Rogers (1977) argued that it is habit rather than age on its own that adversely affects sports participation.

Gender roles have been found to be a major form of social pressure and, correspondingly, a source of constraint in leisure participation [START_REF] Culp | Adolescent girls and outdoor recreation: a case study examining constraints and effective programming[END_REF]. There is a consensus about the fact that men, in general, participate more in sport [START_REF] Gratton | The Economics of Sport and Recreation[END_REF][START_REF] Wilson | The paradox of social class and sports involvement. The roles of cultural and economic capital[END_REF] [START_REF] García Ferrando | Posmodernidad y deporte: Entre la individualización y la masificación[END_REF][START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF]Lera-López and Rapún-Garate, 2007;[START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF][START_REF] Downward | Sports Economics: Theory, Evidence and Policy[END_REF]. Gender differences can be attributed to biological factors, and cultural and social influences, reflecting differences in family responsibilities as well as differences regarding behaviour, social expectations and work.

Traditionally, the economic perspective emphasises the relevance of economic variables in sports participation. Variables used to measure financial status include income level, occupation, and professional status. The literature provides evidence that lower income may act as a barrier to sports participation [START_REF] Wilson | The paradox of social class and sports involvement. The roles of cultural and economic capital[END_REF][START_REF] Stempel | Adult participation sports as cultural capital: A test of Bourdieu's theory of the field of sports[END_REF][START_REF] Humphreys | Participation in physical activity and government spending on parks and recreation[END_REF][START_REF] Lera-López | The demand for sport: Sport consumption and participation models[END_REF][START_REF] Breuer | Demographic and economic factors concerning the inclusion in the German sport system. A micro-analysis of the years 1985 to 2005[END_REF][START_REF] Wicker | Promoting sport for all age-specific target groups. The impact of sport infrastructure[END_REF]. In general, smaller sports participation rates are present among occupational segments in the lower socio-economic groups [START_REF] García Ferrando | Posmodernidad y deporte: Entre la individualización y la masificación[END_REF][START_REF] Lera-López | The demand for sport: Sport consumption and participation models[END_REF]. Other studies have looked at worktime, which is negatively associated with sports participation [START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF][START_REF] Breuer | Demographic and economic factors concerning the inclusion in the German sport system. A micro-analysis of the years 1985 to 2005[END_REF]. The factor of free time is rarely addressed directly in sport surveys. Since time is finite, any increase in the time devoted to sport will always be constrained by competing demands for time from leisure, work and other uses. Time-related constraints have been perceived as one of the most relevant barriers to physical activity and sports participation (Alexandris and Carrol, 1999). [START_REF] Webber | Student Participation in Sporting Activities[END_REF], examining the University student participation patterns concluded that there is an inverse relationship between participation and hours of work, and a positive one between participation and sport literacy (familiarity) or sport investment.

The sociological perspective highlights the role of education in explaining sports participation.

A higher level of education might lead to a greater awareness of the benefits and importance of sport. Education also implies habits acquired in the student environment, where access to facilities is easy and relatively inexpensive. In turn, a sports culture brings about higher number of sports related roles in the social structure, increasing its direct economic importance. Thus, education is expected to be positively related to sports participation. This is supported by empirical evidence [START_REF] Desbordes | Marketing du Sport[END_REF][START_REF] Stempel | Adult participation sports as cultural capital: A test of Bourdieu's theory of the field of sports[END_REF][START_REF] Downward | Exploring the economic choice to participate in sport: Results from the 2002 General Household Survey[END_REF][START_REF] Humphreys | Participation in physical activity and government spending on parks and recreation[END_REF][START_REF] Breuer | Demographic and economic factors concerning the inclusion in the German sport system. A micro-analysis of the years 1985 to 2005[END_REF][START_REF] García | Estimation of a structural model of the determinants of the time spent on physical activity and sport: Evidence for Spain, Economic Discussion Papers 02-2009[END_REF]. In a similar way, ethnic cultures may have a significant effect on sports participation [START_REF] Thompson | Leisure Time in Midlife: What are the Odds?[END_REF].

Definition of sports participation, description of data and methodology

Definition of sports participation used

In this paper, participation in sports is broadly defined by two criteria: a) it must involve physical activity lasting at least 30 minutes; b) it must be practiced for recreational purposes and/or as a competitive activity. It includes forty different activities considered as sport in Spain and England, following the Council of Europe's definition (Council of Europe, 1992). Table 6 lists the number of sports included in the analysis. Note that walking is omitted, because of a definitional incompatibility in this activity between the two countries.

The Spanish data were collected in 2005 based on a face-to-face questionnaire survey designed on a stratified random sample of people 16 to 74 years old, in the 17 regions of Spain.

The sample (7,078 valid respondents) is stratified by gender and age. Table 7 presents The selection of the independent variables was based on existing consistencies in the variables of the two datasets, excluding factors that are treated inconsistently between the two surveys. Table 7 provides a description of the socio-democratic variables used and the number of respondents within each specified category. Note that although the variables of the two models are constructed in a comparable way, they may not represent identical categories. For example the English variable "A levels/below degree average" is not identical to the Spanish "medium education".

Sports participation comparisons

Table 8 presents the percentage participation rates associated with each independent variable. Note that in the case of England, the dataset is weighted. The main result is that the sport participation rate in England is significantly higher than that of Spain (47.7% against 36.7%).

Moreover, the gender difference in participation in England is much lower than that in Spain (11.2% against 16.2% respectively). Sports participation, in both England and Spain, declines by approximately forty percentage points, as we move from the youngest to the oldest age category. However, the participation rates of the youngest groups in England and Spain differ by 10% (66.2% and 56.1% respectively). In general (except in the aforementioned case), transition to older groups does not bring analogous decreases in participation in England and Spain. The most noteworthy case is the fall in participation from the group 25-34 to 35-44 where in England the participation falls by 4.5% points (from 56.5% to 52.0%), while in Spain it falls by 8.7% points (from 46.2% to 37.5%). Interestingly, the inverse is true in the transition from 35-44 to 45-54 age group, where the English dataset 14 reports a decrease of 11.2% (from 52.0% to 40.8%) unlike the Spanish dataset which reports a fall of only 8% (from 37.5% to 29.5%).

In both countries, people with higher education tend to participate more. The Spanish dataset presents greater discrepancies in participation among the various education levels than in England. While the difference in sports participation between people of higher and medium education in both countries is very small (2.2% in the UK and 5.4% in Spain), the gap in participation between people of lower and no qualifications grows to 18.3% (45.6% versus 27.3%) in the UK and 15% (27.3% versus 12.3%) in Spain.

Professional and occupational status appears to have, generally, the same effect in terms of participation in both societies. Between the two samples, many categories such as 'Skilled nonmanual', 'Skilled-manual' and 'Unskilled-manual', present analogous percentage differences to the overall participations (47.7% and 36.7% in the UK and Spain respectively). Nevertheless, the groups with the highest participation rates, such as 'Professional employees' and 'students', present a small difference in participation between the two countries, while on the other end 'Unemployed' vary only by 1%. England and Spain differ mostly across the groups of 'Housekeeping' (36.7% against 19.0%); 'Retired' (29.4% against 20.2%); and 'Sick' (17.6% against 34.9%). The fact that sport participation in the latter category is almost double in Spain compared to the UK, may be due to the small size of the group (43 observations).

Logit modelling

The aforementioned descriptive statistical analysis based on Tables 7 and8 allowed for the presentation of some interesting 'facts', but it is not appropriate to capture the 'effects' of the variables examined. The latter can be captured through econometric analysis, examining the variables as a whole, and accounting for factor interactions such as: 'people on lower education 15 levels tend to be older'; 'people with professional occupation tend to have higher education'; 'students generally belong to low age levels'; 'housekeepers are usually females' etc.

Note that the constructed econometric models account (to some extent) for biases such as the ones explained above but do not explore these issues in great depth; the reason being that they do not include any interaction dummy variables designed for this purpose.

The modelling exercise involves a logistic regression of the observed participation rates on the reported economic and demographic factors described in Tables 7 and8. The regression model which is calculated using Binary Logistic Regression in SPSS has the form:

ε β + =         - X p p 1 ln (1)
where p is the probability of participation while β and X are vectors with 18 elements (including the constant). All the explanatory variables of the model are binary capturing the factors: Gender, Age, Education, and Occupation (see Table 9). In the model, the base category is defined when all explanatory variables are equal to zero; in this case, the right hand side of the regression equation is equal to the constant. In the constructed models, the base category corresponds to the following socio-demographic profile: 1. Female; 2. Age: 16-24; 3. Education: High; 4. Occupation: Professional/Managerial. The advantage in using a logistic non-linear model rather than a linear OLS one is that the expected participation rates generated from the former are designed to have a minimum value of zero and a maximum of one. This makes it ideal for binary variables such as participation (0 stands for non-participation and 1 for participation). OLS generated expected values outside this range would be meaningless (i.e. OLS may return results outside the 0-1 domain).

Note that a non-linear logistic model does not make the unrealistic 'constant returns' assumption embodied in the coefficient values of the linear ones. Therefore, we intentionally do 9 reports them together with other statistics including the 95% confidence interval of the odds ratios.

For instance, the expected participation rates of an English or Spanish respondent with a profile matching that of the base category are 72.3% and 60.8% respectively. The expected participation rates are calculated by solving the estimated regression equation in terms of p . In the case of base category:

c p p =         - 1 ln (2)
where c is the constant, solving for p gives:

Exp( c ) = Exp(c) 1 Exp(c) 1 + = ⇒ - p p p (3)
In the English model c equals 0.96 while Exp( c ) is equal to 2.61.

In order to assess the effect of being 'Male' (against 'Female') ceteris paribus, instead of comparing just 'male' against 'female' as in descriptive statistics, a comparison such as that between 'Male; Age: 16-24; Education: Higher; Occupation: Professional' against 'Female; Age: 16-24; Education: Higher; Occupation: Professional' is drawn. To estimate the new expected probability ( ' p ) the following calculation is required:

Exp(c+B)= Exp(B)] Exp(c) [ 1 Exp(B)] [Exp(c) B) Exp(c 1 B) Exp(c ' ' 1 ' × + × = + + + = ⇒ - p p p (4)
where B = 0.34 (in the English model).

Note that p'-p compares the effect of being 'male' against 'female' (given the rest of the base category). However, the change in magnitude from p to p' depends on the value of p. In other words, if the remaining characteristics of the base category were defining another social group change p' ap a . Hence, there is no one to one relationship between the value of a coefficient and the effect on sports participation; the definition of the base category is of vital importance, defining the position we operate on the logistic curve. However, as mentioned above, in the next section we choose to report only coefficients ( B ) and odds ratios Exp( B ). Note that the latter is the exponential value of the former. Therefore, according to the exponential function:

• When 0 ≤ B , 1
Exp(B) ≤ (i.e. negative or neutral effects)

• When 0 > B , 1 ) ( > B Exp (i.e.

positive effects)

Note that, an odds ratio of 0.5 indicates a lesser negative effect than of 0.2.

Econometric results

Table 9 summarises the regression results from the English and Spanish models, which are based on sample sizes of 319,131 and 7,078 respondents respectively (see Table 7). In the former, all variables except 'Occupation: Student' are significant at 1% level. In the Spanish model, all variables except 'Occupation: Sick' are significant at 5% level. Note that the two aforementioned variables were also insignificant at 90% confidence level. The former is due to the small coefficient value which implied an almost equivalent effect with the base case 'Occupation: Professional'; the latter is possibly justified on grounds of small group size (i.e. only 43 respondents reported 'Sick').

The odds-ratios of the constants in the English and Spanish models are 2.16 and 1.55. As discussed above, this implies that the expected participation levels of the respondents who belong to the base category are 72.3% and 60.8% respectively. Recall that the base category refers to 'Female'; 'Higher Education';and 'Professional Occupation'. All the coefficients of the two models agree in sign. Both models support the positive effects of being male, young, highly educated, and having high occupational status on sports F o r P e e r R e v i e w 18 participation. Specifically, all the signs, apart from 'Gender: Males ', the constants and 'Occupation: Student' in the Spanish model are negative. The 'switch on' of any dummy explanatory variable would result to a deviation from the specification of the base category, and consequently, to a change in the expected participation rate. However, these changes usually appear to be disproportionate.

Between the English and the Spanish models, the odds ratios for being male differ significantly (1.40 against 2.04). In other words, the gender coefficient in the Spanish model (0.71) is more than twice as big as the English one (0.34). This implies that gender in Spain is a more important factor than in England with men participating much more than women.

Compared to the age group '16-24', the older age groups '55-64' and '65-74' appear to have a more negative effect in England (0.29 and 0.23) than in Spain (0.40 and 0.31). Another interesting finding is that a switch from 'Age: 16-24' to 'Age: 25-34' causes a greater fall in expected participation level in England (0.64) than in Spain (0.69). In turn, switching from 'Age: 25-34' to 'Age: 35-44' yields a greater fall in Spain (from 0.69 to 0.53) than in England (from 0.64 to 0.58).

The age variable relates to factors such as free time (in retirement), health and income.

The age coefficients of the English model are distributed across a wider value range than In Spain. Consequently, unlike gender, age is a more important factor in England than in Spain. In contrast, education appears to be more important in Spain. Even after accounting for definitional differences, it is evident that the Spanish coefficients for education are subjected to a greater value range than the English counterparts. In both countries, having a medium education level, as opposed to high, results to a relatively small drop in participation (slightly greater in Spain). The most interesting finding though is that a sub-sequent comparison between 'Education: Low' with 'Education: Medium' yields a much greater negative effect in Spain (from 0.76 to 0.39) than in Compared to 'Occupation: Professional', any other occupation results to a lower expected participation rate (odds ratios less than unity) except in the case of being student in Spain (1.32).

This may be due to the established student culture in the UK of working alongside full time studies, restricting significantly free time. Skilled occupations in England (0.82 and 0.77) appear to have similar in magnitude effects compared to Spain (0.78 and 0.77). On the contrary, unskilled labour yields a greater negative effect in England (0.66 against 0.77).

Being Unemployed in England (0.50) has a greater negative effect in participation than in Spain (0.73). The definition of unemployment includes both short-term (less than 12 months) and long-term (more than 12 months). It should be noted that unemployment, relates to both income and free time effects. Specifically, unemployed are likely to have more free time but less income to spend than professionals. The same applies in the case of students. Furthermore, 'Occupation: Housekeeping' appears to have a more negative effect in England (0.62) than in Spain (0.73). It is noteworthy that the vast majority of respondents who claimed this occupation are women (e.g. 96.3% in the English dataset), hence there is an explicit relationship with the gender variable.

'Occupation: Sick' yields the most negative effect compared to the base case 'Occupation: Professional'. The small size of Spanish respondents in this category (43 respondents) resulted to the exclusion of the variable due to insignificance (p > 0.1).

Finally, retirement has a negative effect in participation in both England and Spain (0.84 and 0.68) compared to the base category. The participation rate of people in retirement is affected positively through more free time and negatively through less income and old age. Given that the age variables account for the age effect, if the free time effect is significantly great we expect a F o r P e e r R e v i e w 20 positive effect on retirement and vice versa. In the case of England, retirement despite having a negative coefficient has the greatest odds ratio within the occupation group (0.84). This means that the sports participation of people in retirement is only disadvantaged compared to the base category; it is better positioned than any other occupational group in England. Had we considered any other base category in occupation, the effect of retirement in England would appear to be positive. The aforementioned English position is in total contrast to the sports participation of the people in retirement in Spain; this is by far the less active occupational category, as indicated by the odds ratio of 0.52. We expect that the people in retirement will fare much better if we include walking in the sports participation definition. However, the comparison between the two nations clearly illustrates that if sports policies in Spain targeted the group of people at the age of retirement, a lot of potential could be realised. analysing sports participation in both countries. This is also in accordance to the shared experience of the EU countries as outlined in the Eurobarobeter publication (European Commission, 2004), confirming the empirical evidence in other countries.

Conclusions and implications

Higher education level, professional occupation, younger age and being male are factors associated with more sports participation. Although there is no difference in the direction of the factor effects on participation between England and Spain, there is considerable variation in their relative strength. For example, the age effect is more profound in England, while the gender and education effects are more important in Spain. In particular, the Spanish population group with 'low' educational level suffers an unusual fall in sport participation compared to the base categories. This illustrates the importance of factors related to active citizenship on sports participation with implications on sports policy.

The results reveal that in the context of a general European support for sports participation, European countries should develop specific sport promotion policies. For example in both countries, it was verified that as age increases, sports participation declines. From the odds ratios we can see that there is no reversal of this pattern. However, what is interesting is not so much the statement of the direction of the relationship, but the degree of the participation fall among the considered age groups. In the case of England, an unusual fall occurs when we switch from the 35-44 age group to 45-54 (odds ratios decline from 0.58 to 0.39). A similar exaggerated effected can be detected in Spain when switching from the 25-34 age group to 35-44. In the case of Spain this may be accounted by the pressure of marriage and having children. In the case of the UK, there is a direct relationship with employment status. People aged 45-55 in the UK, are usually moving to the top jobs especially in the AB occupational categories, often changing residency and as a result, they experience a breakdown in their personal sport networks and less In Spain among the elderly, women and less educated groups, sports participation rates are particularly low compared to the English situation. Hence, special emphasis needs to be placed on the health and well-being effects of sports practice among these social groups. Promotion of the benefits of sports practice through conferences and social activities in, for example, pensioners clubs should be encouraged. In addition, particular promotions of some specific sporting activities such as swimming and gym for the less educated groups, the elderly and women, should be promoted. By comparing the Spanish and English experience, it can be concluded that the safest way to increase sports participation in Spain is through taking advantage of the free time capital owned by pensioners. This has clearly been achieved in the UK, indicating the feasibility of such policy; in Spain this occupation group is the most disadvantaged in terms of sports participation.

In England overall, sports policy should be focused on preventing the unusual (according to Spanish experience) dips in sports participation in the age groups, 25-34 and especially 45-54. Of even greater importance is the class divide that appears in the UK but not in Spain. For example in the case of unemployed the odds ratios in England and Spain are 0.5 and 0.73 respectively. A similar divergence can be observed in the group of partly skilled/unskilled people between the two countries. We suspect that this observation reflects a failure of civic engagement at large in the aforementioned social groups in the UK. Although we do not endeavour to address this problem, it is clear that the UK has a lot to learn in this matter from the Spanish experience.

  's sports participation statistics was in the Eurobarometer 2004 publication 'The Citizens of the European Union and Sport'

  Time Use Surveys. Most European countries that conducted Time Use Surveys since the late 1990s, have considered the Eurostat guidelines.

  It has been carried out annually since 1971 by the Office of Population, Censuses and Surveys and provided sports participation estimates up to the year 2002. In 2005, Sport England started commissioning the Active People Survey, a major annual survey of the sports behaviour of the English population. It was conducted by Ipsos MORI, and its sample size for the first year was 363,724 adults (aged 16+).

  review about determinants of sports participation

  the percentage distribution of respondents by socio-demographic characteristic. The sampling error based on a 95% confidence level and a 50/50 split ( p / 1p ) is ± 1.11%. The English dataset comes from the Active People 2005/2006 Survey. The survey was conducted through a telephone questionnaire. Approximately 1000 random interviews were conducted per Local Authority, across a twelve-month period starting in October 2005. The validsample size used in this analysis is 319,131. This emanates from filtering out people of age 75 or higher (like in the Spanish dataset) and respondents who refused to answer or stated 'don't know' interest. The sampling error, based on a 95% confidence level and a 50/50 split ( p / 1p ), is ± 0.16%.

  effects of variables through changes in expected participation rates compared to the base category, but we draw comparisons only based upon their coefficient values (B) or their odds ratios Exp(B). Table

  rate p a , the same c and B , would result to a different (in magnitude)

  88 to 0.72). A further movement from 'Education: Low' to 'Education: No qualifications' yields an almost equal drop in expected participation rates.

  Over the past ten years, sports participation appears to have reached a stagnation point in many European countries. Consequently, a serious concern among governments has been the development of adequate sports policies to promote sports participation and increase individuals' wellbeing and social integration. In this context, a strong increase in academic interest in sports participation research has emerged. This paper investigates the role of socio-demographic characteristics, educational and economic variables on sports participation in a comparative way in two European countries: a Southern country (Spain) and a Northern country (England). It is evident that sports participation behaves in a similar fashion in England and in Spain albeit at different levels of sports engagement.Despite having a higher level of participation in England, the factors of influence are similar. It has been demonstrated that gender, age, occupation, and education level are all significant factors in

Table 3

 3 illustrates some 2007 results in terms of Participation and Time Used in

  shows the development of sports participation in England during the period 1987 to 2006. Figures are consistent with General Household Survey definitions.

	Survey. The former is an estimate based on the actual 2006 figures and previous trends. In 2006
	we have a reversal of the ten year decline, with sports participation rising to 48% (excluding
	walking) and 68% (in general).
	Switching from the General Household questionnaire to the Active People Survey despite the
	consistency of the definition, may have contributed to the sudden rise in sports participation. It is
	possible that the Active People Survey questionnaire in its detail helps people recall more
	information than GHS.
	In Spain, there are no time series data about sports participation developed by the Spanish F o Office for National Statistics (INE). The only information is provided by the Spanish Time-Use
	r
	Survey conducted by the INE in 2002-03 collecting information about 176 different activities, P r e e including sporting activities and walking only for the 2002 year.
	R e v i e w
	According to Table 4, there has been a decline throughout the 1990s. In the case of 'at least
	one activity' sports participation declined from 64.5% in 1990 to 58.5% in 2002. Similarly in the
	more restrictive definition (excluding walking), sports participation declined from 47.8% in 1990 to
	43.2% in 2002. The last two observations in 2005 and 2006 are based on the Active Peoples
	8

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  This analysis suggests that England and Spain should focus policy on the 45-54 and 35-44 age groups respectively.

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Table 1 .

 1 Sports participation in the European Union, selected countries, 2004

	Country	Participating "at least once a week"	Participating "at least three times a week"
	Portugal	22%	8%
	Greece	26%	16%
	Italy	27%	9%
	Austria	34%	12%
	Spain	37%	21%
	UK	45%	23%
	Netherlands	52%	17%
	Ireland	53%	28%
	Sweden	72%	40%
	Finland	76%	45%
	European Union	38%	17%
	Source: European Commission (2004)

Table 2 .

 2 Compass sports participation "at least once per week",%

	Country	1999
	Finland	73%
	Sweden	61%
	Ireland	31%
	Netherlands	36%
	UK	25%
	Spain	17%
	Italy	11%
	Source: Estimates are based on COMPASS statistics.

Table 8 .

 8 Percentage participation* rates by socio-demographic characteristics Participation is defined as taking part actively in any of the forty sporting activities of Table6, at least once over four weeks and for at least 30 minutes. †Note that the sport participation rates based on Active People Survey are produced after the application of National weights which, according to mid-2006 population estimates and 2001 census data, account for the factors: 1.Gender, 2. Age, 3.Ethnicity, 4.Household size, 5.Working status and 6. NS-SEC categories. ‡The Spanish dataset is random and stratified by gender, age and area given its population size.

					Page
	England (Weighted data †), N = 308,386	Spain (Stratified sample ‡), N = 7,078
	Gender			Gender
	Overall	308,386	47.7% Overall	7,078 36.7%
	Male	152,227	53.4% Male		3,541 44.8%
	Female	156,160	42.2% Female	3,537 28.6%
	Age			Age
	16-24	51,019	66.2% 16-24		1,111 56.1%
	25-34	56,844	56.5% 25-34		1,546 46.2%
	35-44	68,177	52.0% 35-44		1,479 37.5%
	45-54	54,416	40.8% 45-54		1,179 29.5%
	55-64	48,313	32.4% 55-64		929 24.0%
	65-74	29,617	27.1% 65-74		834 16.1%
	Education			Education
	Higher (degree equivalent)	88,672	55.8% Higher (university degree)	1,264 53.6%
	A levels/below degree
	equivalent	80,430	53.1% Medium (secondary education) 1,988 47.7%
	GCSE/trade apprentices/other	92,969	45.6% Lower (elementary education)	3,346 27.3%
	No qualifications	46,317	27.3% No qualifications	480 12.3%
	Professional & Occupational Status		Professional & Occupational Status
	Professional/Managerial	77,245	58.6% Professional	929 53.2%
	Skilled non-manual	45,223	49.8% Medium	1,155 38.3%
	Skilled manual	43,570	49.2% Skilled manual	820 40.2%
	Partly skilled/Unskilled	30,911	43.1% Unskilled Manual	786 34.2%
	Unemployed	20,406	37.2% Unemployed	589 36.2%
	Student	24,956	68.1% Student	610 63.0%
	Housekeeping	14,758	36.7% Housekeeping	956 19.0%
	Sick	7,792	17.6% Sick		43 34.9%
	Retired/Not classified	43,527	29.4% Retired/Not classified	1,190 20.2%

*

30 of 32 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript F o r P e e r R e v i e w 31Table 9 .

 9 Models for sports participation in England and Spain Insignificant variables: 'Occupation: Student' for the English model (p > 0.1); 'Occupation: Sick' for the Spanish model (p > 0.1) All variables in the English model are significant at α = 0.01 level of significance All variables in the Spanish model are significant at α = 0.05 level of significance Note that the variable labels express the comparative categories of the two models

				English model				Spanish model
				95% CI for Odds Ratio			95% CI for Odds Ratio
	Variables	B	SE	Lower Exp(B)	Upper	B	SE	Lower Exp(B)	Upper
	Constant	0.96	0.02		2.61		0.44	0.11		1.55
	Gender: Male	0.34	0.01	1.38	1.40	1.42	0.71	0.06	1.81	2.04	2.29
	Age: 25-34	-0.46	0.02	0.62	0.64	0.65	-0.37	0.10	0.57	0.69	0.84
	Age: 35-44	-0.55	0.02	0.56	0.58	0.60	-0.64	0.10	0.44	0.53	0.65
	Age: 45-54	-0.95	0.02	0.38	0.39	0.40	-0.84	0.11	0.35	0.43	0.53
	Age: 55-64	-1.24	0.02	0.28	0.29	0.30	-0.93	0.13	0.31	0.40	0.50
	Age: 65-74	-1.46	0.02	0.22	0.23	0.24	-1.17	0.17	0.22	0.31	0.43
	Education: Medium	-0.13	0.01	0.86	0.88	0.90	-0.28	0.08	0.65	0.76	0.89
	Education: Lower	-0.32	0.01	0.71	0.72	0.74	-0.93	0.08	0.34	0.39	0.46
	Education: No Qualif. / Not Classif.	-0.78	0.01	0.45	0.46	0.47	-1.44	0.17	0.17	0.24	0.33
	Occupation: Skilled non-manual	-0.20	0.01	0.80	0.82	0.84	-0.25	0.09	0.65	0.78	0.93
	Occupation: Skilled manual	-0.27	0.01	0.75	0.77	0.78	-0.26	0.11	0.62	0.77	0.95
	Occupation: Partly Skilled/Unskilled	-0.42	0.02	0.64	0.66	0.68	-0.27	0.11	0.62	0.77	0.95
	Occupation: Unemployed	-0.70	0.02	0.48	0.50	0.52	-0.32	0.11	0.58	0.73	0.91
	Occupation: Student*						0.28	0.13	1.02	1.32	1.71
	Occupation: Housekeeping	-0.48	0.02	0.60	0.62	0.64	-0.31	0.12	0.58	0.73	0.93
	Occupation: Sick*	-1.39	0.03	0.24	0.25	0.27
	Occupation: Retired / Not Classif.	-0.18	0.02	0.81	0.84	0.87	-0.39	0.14	0.52	0.68	0.88
		N = 319,131				N = 7,078
		R 2 = 0.094 (Cox & Snell), 0.126 (Nagelkerke)		R 2 = 0.134 (Cox & Snell), 0.183 (Nagelkerke)
		Model χ 2 (16) = 31,524.3, (p < 0.01)		Model χ 2 (16) = 1,016.9, (p < 0.01)

*

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

 England 1987[START_REF] García Ferrando | Posmodernidad y deporte: Entre la individualización y la masificación[END_REF]. In % 19871990199319961999[START_REF] Wilson | The paradox of social class and sports involvement. The roles of cultural and economic capital[END_REF][START_REF] Stempel | Adult participation sports as cultural capital: A test of Bourdieu's theory of the field of sports[END_REF]


Regino Criado
email: regino.criado@urjc.es

Julio Flores
email: julio.flores@urjc.es

Alejandro Jose Gacia
email: alejandro.garciadelamo@urjc.es

Del Amo

Jesus Gómez

Universidad Rey

Juan Carlos

Alejandro García

Jesús Gómez-Gardeñes
email: jesus.gomez.gardenes@urjc.es

Miguel Romance
email: miguel.romance@urjc.es

A mathematical model for networks with structures in the mesoscale

Keywords: multilevel networks, clustering, efficiency, mesoscale, random models

  L'archive ouverte pluridisciplinaire

Introduction

During the last years the scientific community has shown that many relevant properties of communication systems, social networks and other biological and technological systems may be described in terms of complex network properties, including structural and dynamical properties and the interplay between them (see, for example, [START_REF] Albert | Statistical mechanics of complex networks[END_REF], [START_REF] Barabási | Linked: The new science of networks[END_REF], [START_REF] Boccaletti | Complex Networks: Structure and dynamics[END_REF], [START_REF] Criado | Choosing a leader on a complex network[END_REF], [START_REF] Criado | Improvements in performance and security for complex networks[END_REF], [START_REF] Newman | The structure and function of complex networks[END_REF] or [START_REF] Strogatz | Exploring complex networks[END_REF] and the references therein). The emergence of these appealing results has forged the Complex Networks Analysis as an attractive and multidisciplinary branch of research that includes topics of sociology (social networks), biological sciences (including metabolic and protein networks, genetic regulatory network and many others), neuro-sciences (neural interaction networks, cortical networks) engineering (phone call networks, computers in telecommunication networks, Internet, the World Wide Web), theoretical physics and applied mathematics.

The working objects of the Complex Networks Analysis are graphs (which are called complex networks) which occasionally prove unable to capture the details present in some real-life problems. One example of this situation can be found in sociology. Social networks analysis is used in the social and behavioral sciences, as well as in economics, marketing, and industrial engineering ( [START_REF] Wasserman | Social Networks Analysis[END_REF]), but some questions related to the structure of social networks have been not properly understood. From the fact that a social network can be understood as a set of people or groups of people with some pattern of contacts or interactions between them ( [START_REF] Scott | Social Networks Analysis: A Handbook[END_REF], [START_REF] Wasserman | Social Networks Analysis[END_REF]) -think of the Facebook network -it might seem that all the connections or social relationships between the members of that network take place at the same level. But the real situation is far from this. The real relationships amongst the members of a social network take place (mostly) inside of different groups (levels) and therefore they cannot be properly modeled if only the natural local-scale point of view used in classic complex network models is taken into account. This phenomenon is related to the fact that several real-life complex systems have a community structure [START_REF] Girvan | Community structure in social and biological networks[END_REF] that takes place in a scale that lies between the local and the global one and that gives information about the structure-functionality relationship. The problem of finding and analyzing the community structure of a complex network is one of the hot spots of the complex network analysis, since these meso-scale structures reveal valuable information about roles of groups of nodes [START_REF] Newman | Detecting community structure in networks[END_REF].

The main goal of this paper is to present the concept of multilevel network as a new model useful to analyze with detail complex systems that have some structures in the mesoscale level. As we pointed out before, this kind of networks are present in many real-life situations, not only in social networks, but also in communication networks and all the other examples that exhibit a community structure. The introduction of a sharp mathematical model that fits this new structures may prove crucial to properly analyze the dynamics that take place in these complex systems which are far from being completely understood. An example of these phenomena is the dissemination of culture in social networks by using the Axelrod Model that has been recently studied by using meso-scale networks [START_REF] Guerra | Dynamical organization towards consensus in Axelrod model on complex networks[END_REF]. Yet the study of other meso-scaled complex networks and their dynamics is one of the scientific challenges of the nowadays complex network analysis. There are other mathematical models that consider non-local structures such as the hypergraphs (or hypernetworks) [START_REF] Berge | Hyper-graphs[END_REF] or the hyper-structures [START_REF] Criado | Hyperstructures, a new approach to complex systems[END_REF] but they are not able to combine the local scale with the global and the mesoscale structures of the system. For example, if we want to model how a rumor is spread within a social network, it is necessary to have in mind not only that different groups are linked only through some of their members but also that two people who know the same person don't have necessarily to know each other, and, in fact, these two people may belong to different groups or levels. If we try to model this situation with hyper-networks, then we will only be taking into account the social groups and not the actual relationships between their members; in contrast if we use the hyper-structure model, then we won't possibly know to which social group each contact between two nodes belongs.

After multilevel networks are defined, we present a structural analysis of these objects in terms of the clustering coefficient and the metric structure which must be properly defined in this setting. We will compare these parameters with the corresponding ones for the underlying projection networks and we will present the analytical relationships between them. Despite the fact that there are sharp connections between the structural properties of the multilevel network and the properties of their projection and layer (or slice) networks, we will present some examples that exhibit the deep differences between the classic complex network approach and this new point of view.

Finally, in section 3, we will give several growing models to produce multilevel random networks, inspired by the Barabási-Albert preferential attachment model of complex networks [START_REF] Barabási | Emergence of scaling in random networks[END_REF]. These methods are inspired by the bipartite networks models introduced by Ramasco et al. [START_REF] Ramasco | Selforganization of collaboration networks[END_REF] and by M. Peltomäki and M. Alava [START_REF] Peltomäki | Correlations in bipartite collaboration networks[END_REF], but in this case we have to take into account the meso-scale nature of the multilevel structure. The use of growing random models is natural since many of the examples in real life that can be modeled by multilevel networks, such as social or transportation networks, are dynamic models whose structure grows in time by addition of new nodes, links and layers. By using these models we perform several random testings that show that we can get very different multilevel networks (i.e. with very different meso-scale structure) which still have similar local and global structures when considered as classic complex network. For example, we exhibit a multilevel network that at the meso-scale is like an Erdős-Rényi random network, but it has a scale-free structure if we consider it as a complex network. These random testings reinforce the idea that in many cases we must take into account the meso-scale structure in order to get deep understanding of the structure and dynamics of many complex systems.

Multilevel Networks: A Structural Analysis

As we pointed out in the previous section, a sort of naive approach to complex systems with structures in the mesoscale could suggest that hyper-networks and hyper-structures fit perfectly to these real-life systems. The key-point that makes these mathematical model not to be the best solution for systems with structures in the mesoscale has to do with the fact that both hyper-networks and hyper-structures are node-based models, while many real systems combine a node-based point of view with a link-based perspective. For example, if we have a look again at a social network with structures in the mesoscale (such as several social groups within its structure), when we consider a relationship between two members of one social group(or several), we have to take into account not only the social groups that hold the members, but also the social groups that hold the relationship itself, i.e., if, for example, there is a relationship between two people that share the same working group and the same sport group, we have to highlight if the relationship is due to sharing the same group of work or it has sport nature. This fact is not particular of social networks with structures in the mesoscale and a similar situation occurs, for example, in public transportation systems, where a link between two stations belonging to several transport lines can occur as a part of different lines.

In order to avoid this node-based nature of hyper-networks and hyper-structures, we propose to introduce the following concept that combines the node-based with the link-based perspective:

Definition 2.1 Let G = (X, E) be a network. A multilevel network is a triple M = (X, E, S), where S = {S 1 , . . . , S p } is a family of subgraphs S q = (X q , E q ) of G such that (i) X = p j=1 X j , (ii) E = p j=1 E j .
The network G is the projection network of M and each subgraph S j ∈ S is called a slice of the multilevel network M. This mathematical model perfectly suits social systems (as well as other complex systems) with structures in the mesoscale, since each social group can be understood as a slice graph in a multilevel network; thus we are simultaneously taking into account the nature of the links (i.e. relationships) and the nodes involved.

It is easy to check that this new mathematical object extends both the classic complex network model and also the hyper-network model [START_REF] Berge | Hyper-graphs[END_REF]. Let us point it out very briefly. On the one hand, if G = (X, E) is a network, then it can be easily seen as a multilevel network by considering M = (X, E, S), where S = {G}.

On the other hand if H = (X, H) is an hyper-network (i.e. X is a non-empty set of nodes and H = {H 1 , . . . , H p } is a family of non-empty subsets of X, each of them called and hyper-link of H), then it can be seen as the multilevel set M = (X, E H , S H ), given by

S H = {K H 1 , . . . , K Hp },
where K H j = (H j , E j ) is the complete network obtained by linking every pair of 4

E H = p r=1 E r .
By using similar arguments we can show that every hyper-structure [START_REF] Criado | Hyperstructures, a new approach to complex systems[END_REF] can be understood as a particular multilevel network, by considering one slice network for each hyper-link in the hyper-structure and each slice graph being a set of isolated nodes.

Once we have introduced this new and novel mathematical object, we have to give suitable structural parameters to analyze it. We can give natural extensions of many of the usual tools of the complex networks' analysis, such as the clustering coefficient, an adjacency matrix/tensor, a natural network representation as a tripartite network or a geodesic structure, among many others.

Clustering of multilevel networks

In this subsection we extend the graph clustering coefficient introduced by Watts and Strogatz in [START_REF] Watts | Collective dynamics of small-world networks[END_REF] to multilevel networks and we establish some relationships between the clustering coefficient of a multilevel network, the clustering coefficient of its slices and the clustering coefficient of its projection network. Recall that given a network G = (X, E) the clustering coefficient of a given node i is defined as

c G (i) =
number of links between the neighbors of i largest possible number of links between the neighbors of i .

We take a/0 to be 0 for every number a. Thus, if we think of three people i,j and k with mutual relations between i and j as well as between i and k, the clustering coefficient of i is supposed to represent the likeliness that j and k are also related. The clustering coefficient of G is usually defined as the average of the clustering coefficients of all nodes.

In order to give a definition of the clustering coefficient of a node in a multilevel network M = (X, E, S), where S = {S 1 , . . . , S p } and S q = (X q , E q ) for every 1 ≤ q ≤ p, we need to introduce some notation. For every node i ∈ X call N (i) the set of all neighbors of the node i in the projection graph G. For every q = 1, . . . , p we will denote N q (i) = N (i) ∩ X q and G q (i) the subgraph of the slice S q generated by N q (i), i.e. G q (i) = (N q (i), E q (i)), where

E q (i) = {k, j} ∈ E q | k, j ∈ N q (i) .
Similarly we will define G(i) as the subgraph of the projection network G generated by N (i), i.e. G(i) = (N (i), E(i)), where In addition the complete graph generated by N q (i) will be denoted as usual by K Nq(i) . Note that the largest possible number of links between the nodes of N q (i) is precisely the number of links in K Nq(i) , i.e.

E(i) = {k, j} ∈ E| k, j ∈ N (i) .
|E q (i)| ≤ |N q (i)|(|N q (i)| -1) 2 ,
where |N q (i)| stands as above for the cardinality of N q (i). With the previous notation we give the following Definition 2.2 Let M = (X, E, S) be a multilevel network. The clustering coefficient of a given node i in M is defined as

C M (i) = 2 p q=1 |E q (i)| p q=1 |N q (i)|(|N q (i)| -1)
.

The clustering coefficient of M is the average of all C M (i) over the set of nodes.

Notice that the clustering coefficient might have been defined differently. For instance we might have considered the clustering coefficient of each S q , say c q (i), and then take the average over the set {S q } p q=1 . However we have opted for a definition that embodies the following idea: it is possible for a given node i to have two neighbors k, j with {i, k} ∈ E q , {i, j} ∈ E q with q = q and such that there is a link {j, k} ∈ E q with q = q, q . This is a natural situation when we think of social networks; indeed, one person i knows j from the aerobic class, i also knows k from her reading club while j and k know each other from the supermarket. This sort of situation produces more links in the subgraphs G q (i) defined above than those already present in the slice S q . Taking the alternative definition based on the clustering coefficients of the slices would not help to discriminate such situations; in contrast the definition proposed does. The following example should clarify this situation and justifies the restrictions of theorem 2.4.

Example 2.3 If we consider the multilevel network M = (X, E, S) where X = {1, 2, 3, 4}, S = {S 1 , S 2 , S 3 } and G is as in the next figure, it is easy to check that c G (i) = 1 for all i ∈ X but however c Sq (i) = 0 for all i ∈ X and q = 1, 2, 3.

As we have seen in the last example, the clustering coefficient of the projected graph G and the clustering coefficient of each slice of M may be very different. However we are interested in establishing relations between the clustering coefficient of the multilevel network, C M (i), and the clustering coefficient in the projected network, c G (i), if possible. That is the target of the next theorem:

(i) = {q ∈ {1, . . . , p} | |N q (i)| < 2} and we denote θ(i) = |A(i)|, then 1 p -θ(i) c G (i) ≤ C M (i) ≤ p c G (i) 1 + (p -1) 4 + θ(i) p -θ(i) . (2.1)
Furthermore, if M is homogeneous (i.e. X q = X for every S q = (X q , E q ) ∈ S), then

1 p -θ(i) c G (i) ≤ C M (i) ≤ c G (i). ( 2

.2)

Proof:

Let us start with the left-hand inequality in (2.1). Using that N q (i) ⊆ N (i) for every 1 ≤ q ≤ p and

E(i) = ∪ p q=1 E q (i), we get that c G (i) = 2|E(i)| |N (i)|(|N (i)| -1) = p q=1 |N q (i)|(|N q (i)| -1) |N (i)|(|N (i)| -1) • 2|E(i)| p q=1 |N q (i)|(|N q (i)| -1) ≤ (p -θ(i)) 2|E(i)| p q=1 |N q (i)|(|N q (i)| -1) ≤ (p -θ(i)) 2 p q=1 |E q (i)| p q=1 |N q (i)|(|N q (i)| -1) = (p -θ(i)) C M (i) , since p q=1 |N q (i)|(|N q (i)| -1) = q / ∈A(i) |N q (i)|(|N q (i)| -1) ≤ (p -θ(i))(|N (i)|(|N (i)| -1)) .
In order to prove the remaining inequality notice that the number of links in the complete graph built up on the set of nodes N (i) never exceeds the sum over q = 1, . . . , p of the numbers of links of the complete graphs built up on the sets of nodes N q (i) plus all the links possibly obtained by joining nodes of N k (i) to nodes of N j (i) where k and j are chosen among all possible pairs (k, j) with 1 ≤ k < j ≤ p, i.e.

|N (i)|(|N (i)| -1) 2 ≤ 1 2 p q=1 |N q (i)|(|N q (i)| -1) + k<j |N k (i)||N j (i)|, (2.3)
where the last sum has p 2 terms. Notice also that we can always assume, by rearranging if necessary, that |N q (i)| ≤ |N q+1 (i)| for 1 ≤ q ≤ p -1. In addition to this, observe finally that

k<j |N j (i)| (|N k (i)| -1) ≤ p q=1 (|N q (i)| -1)|N q (i)|(q -1) ≤ (p -1) p q=1 (|N q (i)| -1)|N q (i)|.
(2.4) Thus, since E q (i) ⊆ E(i) for every 1 ≤ q ≤ p and by using (2.3)

C M (i) = 2 p q=1 |E q (i)| p q=1 |N q (i)|(|N q (i)| -1) ≤ 2p|E(i)| p q=1 |N q (i)|(|N q (i)| -1) = p c G (i) |N (i)|(|N (i)| -1) p q=1 |N q (i)|(|N q (i)| -1) = p c G (i) 1 + 2 k<j |N k (i)||N j (i)| p q=1 |N q (i)|(|N q (i)| -1)
.

(2.5)

Notice that if 1 ≤ k < j ≤ p, then |N k (i)||N j (i)| = (|N k (i)| -1 + 1) |N j (i)| ≤ (|N k (i)| -1) |N j (i)| + |N j (i)|.
Now if we combine the last formula with (2.5), we get that

C M (i) ≤ p c G (i) 1 + 2 k<j |N k (i)||N j (i)| p q=1 |N q (i)|(|N q (i)| -1) ≤ p c G (i) 1 + 2(p -1) + 2 p q=1 (q -1)|N q (i)| p q=1 |N q (i)| (|N q (i)| -1) ≤ p c G (i) 1 + 2(p -1) + 2 (p -1) p q=1 |N q (i)| p q=1 |N q (i)| (|N q (i)| -1)
.

It is easy to check that if q ∈ A(i), then |N q (i)| (|N q (i)| -1) = 0, which makes that

p q=1 |N q (i)| (|N q (i)| -1) = q / ∈A(i) |N q (i)| (|N q (i)| -1) ≥ q / ∈A(i) |N q (i)|.
8

F
(i)| p q=1 |N q (i)| (|N q (i)| -1) ≤ q / ∈A(i) |N q (i)| + q∈A(i) |N q (i)| q / ∈A(i) |N q (i)| ≤ 1 + θ(i) 2(p -θ(i))
,

since if q ∈ A(i), then |N q (i)| ≤ 1, while if q / ∈ A(i), then |N q (i)| ≥ 2.
Finally, if we use the last upper bound, we obtain that

C M (i) ≤ p c G (i) 1 + 2(p -1) + 2 (p -1) p q=1 |N q (i)| p q=1 |N q (i)| (|N q (i)| -1) ≤ p c G (i) 1 + (p -1) 4 + θ(i) p -θ(i)
.

On the other hand, if M is homogeneous, then the same techniques as before show that c

G (i) ≤ (p -θ(i))C M (i). Since M is homogeneous, then N q (i) = N (i) and E q (i) ⊆ E(i) for every 1 ≤ q ≤ p. Therefore we conclude that C M (i) = 2 p q=1 |E q (i)| p q=1 |N q (i)|(|N q (i)| -1) ≤ 2p|E(i)| p q=1 |N (i)|(|N (i)| -1) = 2|E(i)| |N (i)|(|N (i)| -1) = c G (i).
Remark 2.5 The definition of θ(i) shows that the estimation in the previous theorem gets coarser as the number of slices for which i has strictly less than two neighbors (possibly zero) increases; in a "normal" scenario this situation will not occur and the estimation will involve a polynomial of degree two in p. Anyway, since 0 ≤ θ(i) ≤ p and the function f

(x) = x/(p -x) is increasing in [0, p), then the last result ensures that C M (i) ≤ g(p) c G (i),
where g(p) is of order p α with α ∈ [2, 3], but still it is not obvious whether the exponent 2 can be lowered.

As it was said above there are other possible definitions for the clustering of a node in a multilevel network, based on the clusterings of that node in the slices, and then it is natural to have then related by means of estimations. However a remark is in order. Notice that it is possible to construct a network whose slices have clustering coefficients zero for every node while the clustering coefficient of the projection network is 1 for every node. Indeed, if we consider the example 2.3 as our multilevel network M, then it is straightforward to check that for a given node i, its clustering coefficient in each of the slices, c q (i), is zero, and hence every convex combination of them, while the clustering coefficient C M (i) is 1. This means that we cannot expect to estimate C M (i) from convex combinations of {c q (i)} p q=1 . Still it makes sense to give a "slicewise" definition for the clustering coefficient of the node i in a multilevel network M and find estimations by means of the clustering already defined. Again some notation is required. Call

N * q (i) = {j ∈ X | j is a neighbor of i in S q }
and G q (i) = (N * q (i), E q (i)), where

E q (i) = {k, j} ∈ E q | k, j ∈ N * q (i)
Note that G q (i) is the subgraph of the slice S q generated by N * q (i). Thus the following definition is proposed. Definition 2.6 Let M = (X, E, S) be a multilevel network and i be a given node of M. The slice clustering coefficient of i is defined as

C sl G (i) = 2 p q=1 |E q (i)| p q=1 |N * q (i)|(|N * q (i)| -1)
.

Notice that G q (i) is a subgraph of G q (i) as was illustrated in the example above. Accordingly N * q (i) ⊆ N q (i) and hence the largest possible number of links between neighbours of i in the q-slice, cannot exceed the corresponding largest possible number of links between neighbours of i in N q (i). As above we have the relation

|N (i)|(|N (i)| -1) 2 ≤ 1 2 p q=1 |N * q (i)|(|N * q (i)| -1) + k<j |N * k (i)||N * j (i)|,
where the last sum has p 2 terms. Also we can assume, after rearrangement, that

|N * q (i)| ≤ |N * q+1 (i)| for all 1 ≤ q ≤ p.
Thus by mimicking the proof of Theorem 2.4, we can get a relationship between the clustering coefficient in the projected network c G (i) and the slice clustering coefficient C sl G (i).

Theorem 2.7 Let M = (X, E, S) be a multilevel network and i be a given node of Notice that there are other possible variations for the definition of the slice clustering coefficient of a node i. For example, the average over the clustering coefficients c q (i) of the slides:

M. Call A(i) = {q ∈ {1, . . . , p} | |N * q (i)| < 2} and θ(i) = |A(i)|. Then C sl G (i) ≤ p c G (i) 1 + (p -1) 4 + θ(i) p -θ(i) .
C sl G (i) = 1 p p q=1 c q (i) .
The relation between this definition and definition 2.6

p min 1≤q≤p |N * q (i)|(|N * q (i)| -1) 2 C sl G (i) ≤ C sl G (i) ≤ p max 1≤q≤p |N * q (i)|(|N * q (i)| -1) 2 C sl G (i)
is easily derived from the following lemma.

Lemma 2.8 If a 1 , a 2 , . . . , a n ∈ [0, ∞) and b 1 , b 2 , . . . , b n ∈ (0, ∞), then n min 1≤i≤n b i n i=1 b i 1 n n i=1 a i b i ≤ n i=1 a i n i=1 b i ≤ n max 1≤i≤n b i n i=1 b i 1 n n i=1 a i b i . Proof: It is straightforward since n i=1 a i n i=1 b i = n i=1 b i n j=1 b j a i b i .
The following examples illustrate the behavior of the different concepts of clustering previously introduced. As it can be seen, if each one is separately evaluated one gets extremely different values.

Example 2.9 Consider M = (X, E, S) with X = {1, 2, . . . , n} and the slice set S = { § 1 , S 2 }, where

S 1 = ({1, 2, 3}, {{1, 2}, {1, 3}, {2, 3}}) S 2 = ({1, 2, . . . , n}, {{1, 2}, {1, 3}, . . . , {1, n}, {2, 3}}).
In this case

C sl G (1) = 1 2 1 1 + 1 (n -1)(n -2)/2 ∼ 1 2 while c sl G (1) = 1 + 1 1 + (n -1)(n -2)/2 = O 1 n 2 .
We see that the two local clustering coefficients dramatically differ for n 1.

11 On the other hand, if we take M = (X, E, S) with X = {1, 2, . . . , n 2 } and the slice set is given by S = {S 1 , S 2 , . . . , S n }, where S j = (X, E j ) and

E 1 = {i, j} : 1 ≤ i < j ≤ n 2 , E j = {{1, 2}, . . . {1, n}, {j, j + 1}} if 2 ≤ j ≤ n -1, E n = {{1, 2}, . . . {1, n}, {2, n}} , then C sl G (1) = 1 n (n 2 -1)(n 2 -2)/2 (n 2 -1)(n 2 -2)/2 + 1 (n -1)(n -2)/2 + • • • + 1 (n -1)(n -2)/2 = O 1 n 2 , while C sl G (1) = (n 2 -1)(n 2 -2)/2 + 1 + . . . + 1 (n 2 -1)(n 2 -2)/2 + (n -1)(n -2)/2 + . . . + (n -1)(n -2)/2 ∼ 1 .
Also in this case the two local clustering coefficients are very different for n 1.

Example 2.10 Consider M = (X, E, S), where the slice S j = (X j , E j ) is defined by X j = {1, j + 1, j + 2} and E j = {{1, j + 1}, {1, j + 2}, {j + 1, j + 2}} for

1 ≤ j ≤ n -2, X n-1 = {1, 2, n}, E n-1 = {{1, 2}, {1, n}, {2, n}}. Then C sl G (1) = 1 while c G (1) = O(1/n).
Example 2.11 Now consider M = (X, E, S) with X j = X = {1, 2, . . . , n} and

E j = {{1, k} : 2 ≤ k ≤ n} ∪ {{j + 1, k} : 2 ≤ k ≤ n, k = j + 1} for 1 ≤ j ≤ n -1. Then c G (1) = 1 while C sl G (1) = O(1/n).
Example 2.12 If we take M = (X, E, S) with X j = X = {1, 2, . . . , n} and E j = {{1, j +1}, {1, j +2}, {j +1, j +2}} for 1

≤ j ≤ n-2, E n-1 = {{1, 2}, {1, n}, {2, n}}. Then C sl G (1) = 1 while C M (1) = O(1/n 2 ). Example 2.13 Consider M = (X, E, S) with X j = X = {1, 2, . . . , 2n}, A = {{1, k} : 2 ≤ k ≤ n} ∪ {{k, l} : n + 1 ≤ k < l ≤ 2n}, E j = A ∪ {{j + 1, j + 2}} for 1 ≤ j ≤ n-2, E n-1 = A∪{{n, 2}}, E n = {{1, k} : 2 ≤ k ≤ 2n}. Then C M (1) ∼ 1/4 while C sl G (1) = O(1/n 2 ).
Example 2.14 Consider M = (X, E, S) with X i,j = X = {1, 2, . . . , n} and E i,j = {{i, j}} for 1

≤ i < j ≤ n. Then C sl G (1) = 0, C M (1) = O(1/n 2 ) and c G (1) = 1.
Example 2.15 Consider M = (X, E, S) with X = {1, 2, . . . , n}, X 1 = {1, 2, 3}, with E 1 = {{1, 2}, {1, 3}, {2, 3}} and X j = {1, j + 2} with E j = {{1, j + 2}} for j = 2, . . . , n -2.

c G (1) = 1 (n -1)(n -2) = O(1/n 2 ) while C M (1) = 1 + 0 + • • • + 0 1 + 0 + • • • + 0 = 1.
We conclude this section with a final observation. All what has been said refers to local clustering coefficients. The aggregation of the local quantities into a global one, or clustering of M, can naturally be considered. We leave the details to the reader.

Metric Structure: Distances and Efficiency

If we want to introduce metric tools in a multilevel network M = (X, E, S), we first have to give the notion of path and length.

A path ω in M = (X, E, S) is a set of the form ω = {( 1 , • • • , q ), (S 1 , • • • , S q )} such that (i) ( 1 , • • • , q ) is a path in (X, E), (ii) (S 1 , • • • , S q ) is a sequence of slice graphs S 1 , • • • , S q ∈ S,
(iii) For every 1 ≤ j ≤ q, we have that j is an edge in the slice graph S j .

By using this concept we can introduce a metric structure in a multilevel graph M = (X, E, S) as follows.

Definition 2.16 Let M = (X, E, S) be a multilevel network, β ≥ 0 fixed and

ω = {( 1 , • • • , q ), (S 1 , • • • , S q )} be a path in M. The length of ω is the nonnegative value (ω) = q + β q j=2 ∆(j), where ∆(j) = 1 if S j = S j-1 , 0 otherwhise.
The distance in M between two nodes i, j ∈ X is the minimal length among all possible paths in M from i to j.

If we take β = 0, the previous definition gives the natural metric in the projection graph, while if β > 0, we introduce new metrics that take into account, not only the global structure of the projection network, but also the interplay between the slice networks, that help to model the multi-scale nature of real-life social networks. Remark 2.17 We could introduce a more general definition of the length of a path ω in a multilevel network M = (X, E, S) of slices S 1 , . . . , S p by replacing the jumping weight β by a p×p non-negative matrix Θ = (β(S i , S j )) in order to take into account that some slices could be closer to an other and therefore we should take different jumping weights depending on the starting and target slices where the slice jump occurred. Following this idea, the length of a path ω = {( 1 , • • • , q ), (S σ(1) , • • • , S σ(q) )} (where σ : {1, . . . q} -→ {1, . . . , p}) could be defined as

˜ (ω) = q + q j=2 ∆(j),
where ∆(j) = β(S σ(j-1) , S σ(j) ) if S σ(j) = S σ(j-1) , 0 otherwhise.

We could also go further and consider that the jumping weights depend not only on the two slices involved in the slice jump, but also on the actual node in which this jump take place, but in any case, it is straightforward to check that the results, methods and algorithms introduced in this section can be easily mimicked to these more general metric structures.

One way to calculate the distance matrix, that is a matrix Λ = (λ ij ) that contains in each position λ ij the distance in M between vertices i and j in X, is to consider an auxiliary graph in the following way. Every vertex of the multilevel network M is represented by a vertex in the auxiliary graph and, if a vertex in M belongs to two or more slice graphs in M, then we duplicated it as many times as the number of slide graphs it belongs to. Every edge of E is an edge in the auxiliary graph and there is one more (weighted) edge for each vertex duplication between the duplicated vertex and the original one. The distance between a duplicated vertex j and another vertex k is d jk = min{d j k , d j k }, where j and j are the duplications of vertex j in the auxiliary graph.

Once we have defined a distance in a multilevel network M = (X, E, S), it is easy to generalize the concept of efficiency given in [START_REF] Criado | Hyperstructures, a new approach to complex systems[END_REF] to this structure:

Definition 2.18 Let M = (X, E, S) a multilevel network. The efficiency of M is E(M) = 1 n(n -1) i,j∈X,i =j 1 d M (i, j) ,
where n is the number of vertices in X and d M (i, j) is the distance in M between vertices i and j.

It is quite natural to try to establish comparisons between the efficiency E(M) of a multilevel graph M = (X, E, S) as it was just defined before and the efficiency E(G) of the underlying graph as a classic complex network, as it was introduced in [START_REF] Latora | Efficient Behavior of Small-World Networks[END_REF] (see also [START_REF] Criado | New Results on computable efficiency and its stability for complex networks[END_REF]). The actual analytical result that connects these two parameters are given in the following theorem.

1 β + 1 E(G) ≤ E(M) ≤ E(G), (2.6)
where E(G) is the efficiency of the projection network G and E(M) is the efficiency of the multilevel network M.

Proof:

Let i, j ∈ X and ω be a path in M from i to j. We denote by d M (i, j) the distance between i, j in the multilevel network M and by d G (i, j) the distance between i, j in the network G. In a similar way, we denote by M (ω) its length in the multilevel network M and by G (ω) its length in the network G.

It is clear that d M (i, j) ≥ d G (i, j). On the other hand, since ∆(•) ≤ 1 then

d M (i, j) ≤ M (ω) = G (ω) + β G (ω) j=2 ∆(j) ≤ G (ω) + β( G (ω) -1) = G (ω)(β + 1) -β ≤ G (ω)(β + 1), Hence, since it is clear that d G (i, j) = min{ G (ω)|ω is a path in M from i to j}, we get that 1 β + 1 d M (i, j) ≤ d G (i, j)
and therefore, by combining the last expressions with the reverse bound obtained before we obtain that

1 β + 1 1 d G (i, j) ≤ 1 d M (i, j) ≤ 1 d G (i, j) ,
which gives the result.

Remark 2.20

The last bounds are sharp simply by making β -→ 0 + .

We are equally interested in obtaining estimations of the efficiency of a multilevel graph in terms of the efficiencies of its slices. Precisely we obtain the following result.

Theorem 2.21 If M = (X, E, S) is a multilevel network such that S = {S 1 , . . . , S p } and for every 1 ≤ q ≤ p we denote S q = (X q , E q ), then

|X q |(|X q | -1) |X|(|X| -1) E(S q ) ≤ E(M), (2.7)
where |X q | is the number of nodes of S q and |X| is the number of nodes of M . Proof: If we take 1 ≤ q ≤ p, since X q ⊆ X and E q ⊆ E, then for every i, j ∈ X q and every path ω in S q from i to j in S q , we get that

Sq (ω) = M (ω) ≥ d M (i, j). Then d Sq (i, j) ≥ d M (i, j). Therefore i,j∈Xq,i =j 1 d Sq (i, j) ≤ i,j∈Xq,i =j 1 d M (i, j) ≤ i,j∈X,i =j 1 d M (i, j) ,
which makes that

E(S q ) = 1 |X q |(|X q | -1) i,j∈Xq,i =j 1 d Sq (i, j) ≤ 1 |X q |(|X q | -1) i,j∈X,i =j 1 d M (i, j) = |X|(|X| -1) |X q |(|X q | -1) E(M).
Remark 2.22 There is a reverse inequality that can be established when M = (X, E, S) is an homogeneous multilevel network (i.e., a multilevel network such that X q = X for every slice S q = (X q , E q ) ∈ S). Indeed, in this case the following relation is easily obtained

E(S q ) ≤ E(M) ≤ E(G) ≤ (|X| -1)E(S q ). (2.8)
3 Some models for multilevel random networks

In this final section we present some models to produce multilevel random networks, inspired by the Barabási-Albert preferential attachment model of complex networks [START_REF] Barabási | Emergence of scaling in random networks[END_REF] and several bipartite networks models such as the collaboration network model proposed by Ramasco et al. [START_REF] Ramasco | Selforganization of collaboration networks[END_REF] or the sublineal preferential attachment bipartite model introduced by M. Peltomäki and M. Alava [START_REF] Peltomäki | Correlations in bipartite collaboration networks[END_REF]. We propose growing random models since many of the real-life examples that can be modeled by multilevel networks, such as social or transportation networks, are dynamic models and their structure grows in time by addition of new nodes, links and layers. The presented randomized models are determined by the following rules:

(i) Model parameters. Our models have three main parameters: N, m and p new . We set N ∈ N as the number of nodes in the multilevel network while 2 ≤ m ≤ N account of the number of nodes in each layer (i.e. if we take m = 2, we recover the Barabási-Albert model [START_REF] Barabási | Emergence of scaling in random networks[END_REF]).

In our model m will be fixed, but it can also be replaced by any other nonnegative integer random variable if we want to produce more general models. Finally, we set p new ∈ (0, 1] as the probability of joining a new node to the multilevel network. (ii) Initial conditions. We start with a seed multilevel network made of one single layer S 0 of m nodes that are linked all to all, (i.e. S 0 = K m ). We can replace the all-to-all structure by any other structure (such as a scale free or a Erdős-Rényi network), but the results obtained are statistically equivalent.

(iii) Layer composition. At each time step t, a new layer S t of m nodes is added to the multilevel network. In order to determine S t we have to give its nodes and links. We start by choosing randomly an existing node of the multilevel network proportionally to its degree (preferential election). Therefore if at step t -1, the set of nodes of the multilevel network is {v 1 , . . . , v n }, and k i denotes the degree of node v i at time t -1 in the projection network, then we choose the node v i randomly and independently with probability

p i = k i n j=1 k j .
The chosen node will be the first node of the new layer S t and we call it coordinator of the layer. As an alternative, we could choose the coordinator proportionally to the number of layers that it belongs to but, in this case, the model will be similar to the collaboration network model proposed by Ramasco em et al. [START_REF] Ramasco | Selforganization of collaboration networks[END_REF]. Each of the remaining m -1 nodes of S t will be a new node with probability p new and an existing node with probability (1p new ). If we have to add an already existing node, we will uniformly and independently choose it at random. Note that we can replace the uniform random selection by other random procedure (such as preferential selection), but the random tests done suggest that the multilevel network obtained have statistically the same structural properties when N is large enough (N > 10 3 ). Anyway, we should have chosen m nodes ṽ1 , . . . , ṽm (ṽ 1 is the coordinator node) that will belong to the new layer S t .

(iv) Layer inner-structure. Once we have fixed the nodes ṽ1 , . . . , ṽm of the new layer S t , we have to give its links. First, we link all the nodes to the coordinator node in order to warrant that the new layer is connected. In addition to this, we set new links between each pair of nodes, say v i and v j (with 1 < i = j ≤ m) by using a random linking probability p ij that we will present later. At the end of this step we have defined completely the new layer S t .

(v) Finally, we repeat steps (iii) and (iv) until the number of nodes of the multilevel network is at least N .

These rules define a family of growing models for multilevel random networks that cover a wide range of different networks with different meso-scalestructure, simply by changing the linking strategy, p ij , at each step. The simplest choice for this strategy is p ij = 1, i.e. an all-to all strategy, thus linking all the nodes in the new slice. In this way, we recover a model similar to the Ramasco et al. model [START_REF] Ramasco | Selforganization of collaboration networks[END_REF], that can be used to produce hyper-networks with a meso-scale structure similar to that found in scientific collaboration systems.

In this paper we will focus on the following three different linking strategies:

1. Erdős-Rényi type strategy (Model I). We fix a value p ij = p link ∈ [0, 1] so that we add each link {ṽ i , ṽj } randomly and independently with probability p link , for every 2 ≤ i = j ≤ m (in what follows we will use a value p link = 0.31).

Note that this strategy does not take into account the meso-scale structure of the multilevel network, since the existence of the link {ṽ i , ṽj } is independent of the nodes ṽi , ṽj in other slices of the multilevel network. Clearly, this is a toy model inspired by the classic Erdős-Rényi model.

2. Assortative linking strategy (Model II). For every 2 ≤ i = j ≤ m, we add randomly the link {ṽ i , ṽj } proportionally to the number of common slices that hold simultaneously ṽi and ṽj . Hence if we denote by Q ij the number of slices that hold simultaneously ṽi and ṽj at time step t (including S t ) and by q i the number of slices that hold ṽi at time step t (also including S t ), thus the probability of linking node ṽi with node ṽi is given by

p ij = 2Q ij q i + q j ,
for every 2 ≤ i = j ≤ m. The heuristic behind this strategy comes from social networks, since in this type of networks the relationships in a new social group are correlated with the previous relationships between the actors in other social groups [START_REF] Wasserman | Social Networks Analysis[END_REF]. Hence, on the one hand, if two actors that belong to the new social group coincide in many (previous) social groups, then the probability of linking in this new social group is big and ,on the other hand, when two new actors join their first social group, the probability of establishing a relationship between them it is also high.

3. Disassortative linking strategy (Model III). If we take 2 ≤ i = j ≤ m, the probability of linking ṽi with ṽj in the slice S t is inversely proportional to the number of slices that hold simultaneously ṽi and ṽj , i.e. if we denote by p ij such probability, then

p ij = 1 - 2Q ij q i + q j ,
by using the same notation as in the assortative strategy (Model II). Now, the heuristic inspiring this strategy lies in some transportation networks, such as the airline networks, where the links in a new line try to connect nodes that are not connected in previous lines. By using the last expression, the newcomer nodes of the slice will be linked to old nodes that belong to many slices with high probability, while the newcomers will not be linked between them also with high probability. This strategy prevents the multiple reiteration of the In all cases, the probability of joining a new node to the network is fixed to p new = 0.202 as in the empirical data obtained in [START_REF] Peltomäki | Correlations in bipartite collaboration networks[END_REF]. The distribution of each type of network and size of slice is computed for 100 random networks of 5000 nodes each one.

same link in different slices and stresses the dissasortativity between newcomer nodes and nodes that belong to many slices.

Once we have introduced the three linking strategies, we now present some numerical results to shows that, despite the fact that the above tree rules produce hyper-graphs with similar projection networks, the structures found in the mesoscale can be quite different.

Let us start by showing in figure 2 the degree distribution, P (k), of the projection network. It is clear that the degree distribution of the projection network obtained in three models display heavy-tailed profiles, and this fact is more relevant if the number of nodes of each slice is low, since in this case the Big Number Law's make that the degree distribution behaves as a power-law, P (k) ∼ k -γ with 2 ≤ γ ≤ 3. Note that a discretizacion effect appears in all the distribution presented in figure 2 that makes that the degree distribution are oscillatory. This is due to the fact that each slice has the same number of nodes, and all of them are linked to the coordinator

(q) q M-I M-II M-III x -3.2 e -0.1x -2 A B C F E D
Figure 3: Random testing of the hyper-degree distribution, P (q), of the multilevel models (in log scale) for different number of nodes m in each slice and for two different values of p new , combined with some numerical approximation in each case. The cases m = 5 are on the left, m = 20 are centered and those for m = 50 are on the right. Besides, the panels on the top show the case p new = 0.202 while those on the bottom are for p new = 0.9. Each distribution P (q) is averaged over 100 random networks of 5000 nodes each one. node, which makes that the degree of the nodes are concentrated in some values modulus m. In order to avoid this effect, we could modify step (i) replacing m by a non-negative integer random variable that prevents that the degree of each node increases modulo m. This discretization effect is present in the three models and increases when the size of the slices is large, but it is more persistent in Model III, since in this case the dissasortative character of the model stresses this phenomenon.

In addition to the degree distribution of the projection network, in figure 3 we show the distribution P (q) of the number of slices that hold each node (we call it hyper-degree distribution for consistency with the hyper-graph theory notation). Obviously, the hyper-degree of a node, q i , depends on the number of nodes m of each slide and the probability p new of adding a new node for constructing each slide (so that the number of new nodes in each slide is on average mp new ). Therefore, the distribustion P (q) is not correlated with the linking model used to construct the network. In figure 3 we show that the behavior of P (q) depends quite strongly on the particular value of m and p new . In particular, in figures 3.A, 3.B and 3.C we show respectively the cases m = 5, m = 20 and m = 50 when p new is small (p new = 0.202 as in figure 2). For m = 5, P (q) can be approximated by a stretched exponential (figure 3.A), while P (q) presents an exponential decay in the cases m = 20 and m = 50 (figures 3.B and 3.C). When p new is large (p new = 0.9 in figures 3.D, 3.E and 3.F) the hyper-degree distributions become more heterogeneous as they are better 20 approximated by truncanted power-laws. Remarkably, for m = 5 (figure 3.D) we obtain the most heterogeneous distribution since P (q) ∼ q -3 for nearly all the range of q values. The heterogeneity of P (q) decreases with m as the plots for m = 20 and m = 50 (figures 3.E and 3.F) show. In these latter cases the power law behavior spans the small hyper-degree classes while, for large q values, a cut-off of the form exp(-αq 2 ) applies.

Figures 2 and3 have shown that either structure of the projection network and hyper-degree distribution are quite similar for the three random models presented. However, they are very different if we analyze their mesoscale structure, as figure 4 shows. This figure shows, for m = 50 and m = 100, the probability of finding a node connected to hk nodes within a randomly chosen slice of the network. It is straightforward to check that since the Erdős-Rényi type strategy is used in Model I, then the resulting distribution P (hk) is a binomial distribution with parameters m and p link (asymptotically a Poisson distribution). However, note that in figure 2 we found that the degree distribution of the projection network is scale-free. Correspondingly, Models II and III display totally different mesoscale properties as shown by their corresponding distributions P (hk). On one hand in Model II we observe that P (hk) tend to accumulate around the maximum possible value of hk, m, due to the assortative character of the linking strategy used. Conversely, in Model III the association between the nodes belonging to the same slice is much lower due to the disassortativity of the model. Again, while both Models II and III gives two different bimodal averaged distributions P (hk), the corresponding distribution for their projection networks are also of scale-free type. Therefore, these results illustrate that many different multilevel networks (with different mesoscale structures) can produce similar projection networks and therefore that we should take into account

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Figure 1 :

 1 Figure 1: An example of the extreme difference between clustering coefficients

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 14 Theorem 2.19 Let M = (X, E, S) a multilevel network, and G = (X, E). Then

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 182 Figure2: Random testing of the degree distribution of the projection graph in the randomized multilevel models. The model I is on the left, the model II is centered and the Model III is on the right, each of them for m = 5, 20, 50 nodes per slice. In all cases, the probability of joining a new node to the network is fixed to p new = 0.202 as in the empirical data obtained in[START_REF] Peltomäki | Correlations in bipartite collaboration networks[END_REF]. The distribution of each type of network and size of slice is computed for 100 random networks of 5000 nodes each one.

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 4 Figure 4: Random testing of the average degree distribution of each slice for 100 random networks of 5000 nodes each one in the randomized multilevel models I, II and III. The case m = 50 is in panel A (on the left), while the case m = 100 is in panel B (on the right), both for models I, II and III.

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Acknowledgements

The authors of this work have been partially supported by the Spanish Government Project MTM2009-13848.