Farrukh Jamal

M Elgarhy

Arslan Nasir

Gamze Ozel

Mamode Naushad

Khan

M Arslan Nasir

Naushad Mamode

Generalized Inverted Kumaraswamy Generated Family of Distributions: Theory and Applications

Keywords: Kumaraswamy distribution, quantile function, Simulation, maximum likelihood estimation. Mathematics Subject Classification. 60E05, 62E15, 62F10

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Introduction

In the recent times, there has been an increased interest in applying some inverted distributions to data applications in the areas of medical, economic and engineering sciences (See Calabria and 1 [START_REF] Calabria | On the maximum likelihood and least squares estimation in the inverse Weibull distribution[END_REF], Al-Dayian (1999),Abd El-Kader et al. (2003), [START_REF] Prakash | Inverted exponential distribution under a Bayesian view point[END_REF]). Such distributions include the inverse Weibull, inverted Burr Type XII, the Pareto Type I and the exponentiated inverted Weibull models. However, to further improve the goodness of fit especially in exploring tail properties, researchers have also considered to derive new generators for univariate continuous families of distributions by introducing one or more additional shape parameter(s) to the baseline distribution.

In this context, some well-known generators are the beta-G by [START_REF] Eugene | Beta-normal distribution and its applications[END_REF] and [START_REF] Jones | Families of distributions arising from the distributions of order statistics[END_REF], Kumaraswamy-G (Kw-G) by [START_REF] Cordeiro | A new family of generalized distributions[END_REF] , McDonald-G (Mc-G) by [START_REF] Alexander | Generalized beta generated distributions[END_REF], gamma-G (type 1) by [START_REF] Zografos | On families of beta-and generalized gamma-generated distributions and associated inference[END_REF], gamma-G (type 2) by [START_REF] Ristic | The gamma-exponentiated exponential distribution[END_REF], gamma-G (type 3) by [START_REF] Torabi | The gamma-uniform distribution and its application[END_REF], logistic-G by [START_REF] Torabi | The logistic-uniform distribution and its application[END_REF], exponentiated generalized-G by Cordeiro The probability density function (pdf) corresponding to (1.1) is

π(x; α, β) = α β (1 + x) -α-1 1 -(1 + x) -α β-1 . (1.
2)

The curves of the pdf show that the IKw distribution exhibits a long right tail and as illustrated in the paper by Al-Fattah et al. (2017), IKw produces optimistic predictions of rare events occurring in the right tail of the distribution as compared with other distributions. However, in many applied areas such as lifetime analysis, finance and insurance, the data are seen to exhibit heavy-tail while in practice, there is also a strong need to generate distributions with symmetric, left-skewed, rightskewed and reverse J-shaped and hence to provide consistently better fits than other generated models under the same underlying distribution. The basic motivations for obtaining GIKw-G family in practice are: (i) to make the kurtosis more flexible as compared to the baseline model; (ii) to produce skewness for symmetrical distributions; (iii) to construct heavy-tailed distributions that are not longer-tailed for modeling real data; (iv) to generate distributions with symmetric, left-skewed, right-skewed and reversed-J shape; (v) to provide consistently better fits than other generated models under the same underlying distribution.

In this context, this paper proposes a new generalized Inverted Kumaraswamy family of distributions (GIKw-G) using the generator G γ 1-G γ . For the conditions on the baseline distribution see [START_REF] Alzaatreh | A new method for generating families of continuous distributions[END_REF].

The cdf of the new family, GIKw-G, is given by

F (y) = α β G γ (x,ξ) 1-G γ (x,ξ) 0 (1 + y) -α-1 1 -(1 + y) -α β-1 dy = [1 -(1 -G γ (x, ξ)) α ] β . (1.3)
where the corresponding density function to (3) is given by

f (x) = α β γ g(x, ξ) G γ-1 (x, ξ) (1 -G γ (x, ξ)) α-1 [1 -(1 -G γ (x, ξ)) α ] β-1 , α, β, γ > 0. (1.4)
where ξ is the parametric space of the baseline distribution. Now onward G(x, ξ) = G(x) and g(x, ξ) = g(x). The survival, hazard rate and reversed hazard rate functions of the GIKw-G family are, respectively, given by

S(x) = 1 -[1 -(1 -G γ (x)) α ] β , h(x) = α β γ g(x) G γ-1 (x) (1 -G γ (x)) α-1 [1 -(1 -G γ (x)) α ] β-1 1 -[1 -(1 -G γ (x)) α ] β , H(x) = α β γ G γ-1 (x) (1 -G γ (x)) α-1 1 -(1 -G γ (x)) α .
Note that when γ = 1 we obtain the exponentiated generalized family of distributions that was introduced by [START_REF] Cordeiro | The exponentiated generalized class of distribution[END_REF]. The GIKw-G family is easily simulated by inverting (1.3) as follows: if u has a uniform U (0, 1) distribution, then the solution of the nonlinear equation is given by

Q x (u) = G -1 1 -1 -u 1 β 1 α 1 γ
.

The rest of the paper is organized as follows: In Section 2, four special models of the new family are given with their plots of density and hazard rate functions. In Section 3 , Structural Properties are discussed while in Section 4, parameter estimation and a Monte Carlo simulation study are presented. In Section 5, the proposed GIKw-G is applied to some real-life data and also compared with other well-known distributions.

F (x) = 1 -1 - x θ γ α β
The plots of density and hazard rate functions of the GIKw-U are presented in Figure 1. As seen in 1(a), the density function has reversed J and upside-down bathtub shapes.

.3 θ = 1 α = 2 β = 1 γ = 3 θ = 1 α = 1.5 β = 1.5 γ = 1 θ = 1 α = 2 β = 1.2 γ = 1 θ = 1 α = 1 β = 1 γ = 1 θ = 1 α = 0.8 β = 2 γ = 5 θ = 1 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 x hrf α = 2 β = 0.5 γ = 0.3 θ = 1 α = 2 β = 1 γ = 3 θ = 1 α = 1.5 β = 1.5 γ = 1 θ = 1 α = 2 β = 1.2 γ = 1 θ = 1 α = 1 β = 0.5 γ = 0.2 θ = 1 α = 0.1 β = 0.1 γ = 0.3 θ = 1

Generalized inverted Kumaraswamy-Weibull (GIKw-W) distribution

We consider the Weibull distribution with scale and shape parameters a, b > 0. The pdf and cdf are g(x) = a b x b-1 e -a x b and G(x) = 1 -e -a x b , respectively. Then, the pdf of the GIKw-W is given by

f (x) = a b α β γ x b-1 e -a x b 1 -e -a x b γ-1 1 + 1 -e -a x b γ α-1 1 -1 + 1 -e -a x b γ α β-1 ,
The corresponding cdf takes the following form

F (x) = 1 -1 + 1 -e -a x b γ α β
Figure 2 shows the plots of density and hazard functions of GIKw-W distribution. The density function in Figure 2(a) has reversed J, left, right skewed and symmetrical shapes. The hrf in Figure 2(b) gives bathtub, upside-down bathtub and increasing shapes.

Generalized inverted Kumaraswamy-Log logistic (GIKw-LL) distribution

We consider the log-logistic distribution with scale parameter θ > 0. The pdf and cdf are given by g(x) = θ x θ-1 (1+x θ ) 2 and G(x) = x θ 1+x θ , respectively. Then, the cdf of the GIKw-LL distribution is given

= 2.1 β = 0.2 γ = 2 a = 1 b = 0.6 α = 2 β = 1 γ = 9 a = 0.8 b = 1.2 α = 1.5 β = 1.5 γ = 1.2 a = 1 b = 1 α = 1.2 β = 2 γ = 2 a = 3 b = 0.6 α = 1 β = 0.5 γ = 0.6 a = 1 b = 2 α = 1 β = 1 γ = 1 a = 1 b = 1
f (x) = α β γ θ x θ-1 (1 + x θ ) 2 x θ 1 + x θ γ-1 1 - x θ 1 + x θ γ α-1 1 -1 - x θ 1 + x θ γ α β-1
The corresponding cdf takes the following form

F (x) = 1 -1 - x θ 1 + x θ γ α β
The plots of density and hazard rate functions of GIKw-LL are given in Figure 3. The density in Figure 3(a) gives reversed J, nearly symmetrical and positively skewed shapes. As seen in Figure 3(b), the hazard rate function has the bathtub, upside-down bathtub, increasing and decreasing shapes.

Generalized inverted Kumaraswamy-Lomax (GIKw-Lx) distribution

We consider the Lomax distribution with scale parameter a, b > 0. The pdf and cdf are g

(x) = a b 1 + x b a-1 and g(x) = 1 -1 + x b a
, respectively. Then, the pdf of GIKw-L is given by

f (x) = α β γ a b 1 + x b a-1 1 -1 + x b a γ-1 1 -1 -1 + x b a γ α-1 1 -1 -1 -1 + x b a γ α β-1
The corresponding cdf takes the following form The plots of density and hazard rate functions of the GIKw-L are given in Figure 4. Figure 4(a)

F (x) = 1 -1 -1 -1 + x b a γ α β (a) (b) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 x pff α = 1.2 β = 0.5 γ = 0.2 θ = 8 α = 2 β = 1 γ = 2 θ = 3 α = 2 β = 1.4 γ = 1 θ = 1 α = 3 β = 0.7 γ = 5 θ = 2.6 α = 1 β = 0.5 γ = 2 θ = 0.6 α = 0.1 β = 0.1 γ = 0.3 θ = 1 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x hrf α = 2 β = 0.5 γ = 0.3 θ = 1 α = 2 β = 1 γ = 3 θ = 1 α = 1.5 β = 1.5 γ = 1 θ = 1 α = 2 β = 1.2 γ = 1 θ = 1 α = 1 β = 0.5 γ = 0.2 θ = 1 α = 0.1 β = 0.1 γ = 0.3 θ = 1
shows that reversed J and positively skewed shapes are possible for the density function.

α = 2 β = 1 γ = 0.5 δ = 0.8 µ = 1 α = 1.5 β = 1.5 γ = 1.2 δ = 1 µ = 1 α = 1.5 β = 1.5 γ = 1.2 δ = 3 µ = 1.5 α = 5 β = 1.5 γ = 1.2 δ = 3 µ = 1.5 α = 2 β = 2 γ = 1.2 δ = 2 µ = 2 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 x hrf α = 1 β = 1 γ = 1 δ = 1 µ = 1 α = 2 β = 1 γ = 9 δ = 0.8 µ = 1.2 α = 1.5 β = 1.5 γ = 1.2 δ = 1 µ = 1 α = 2 β = 1 γ = 1 δ = 8 µ = 1 α = 2 β = 0.2 γ = 0.2 δ = 0.2 µ = 0.3

Structural Properties

The several types of moments of a random variable are important especially in applied work. Some of the most important features and characteristics of a distribution can be studied through moments, e.g., tendency, dispersion, skewness and kurtosis, mean deviations, Bonferroni and Lorenz curves, etc. In this section, we will provide the expansions of GIKw-G densities, rth moment, sth incomplete moment, moment generating function and mean deviations.

Useful expansions

Here, the infinite mixture representations for the cdf and pdf of the GIKw-G family are given in terms of baseline densities. Consider following series expansion

(1 -z) b = ∞ j=0 b j (-1) j z j . (3.1)
Using the series expansion in (3.1), the equation (1.3) becomes

F (x) = ∞ j=0 β j (-1) j ∞ i=0 α j i (-1) i G γ i
where

G γ i = ∞ q=0 ∞ q=k γ i k k q (-1
) q+k G q (x). Now the above expression can be written as

F (x) = ∞ j=0 ∞ i=0 β j α j i (-1) i+j ∞ q=0 ∞ q=k γ i k k q (-1) q+k G q (x)
. Rewriting the above expression, we obtain an expansion for the cdf of the GIKw-G family

F (x) = ∞ q=0 a q H q (x), (3.2)
where

a q = ∞ j=0 ∞ i=0 β j α j i (-1) i+j ∞ q=k γ i k k q (-1
) q+k and H q (x) = G q (x). Similarly, we have an expansion for the pdf of the GIKw-G family as

f (x) = ∞ q=0 a q h q-1 (x) (3.3)
where h q-1 (x) = q × g(x) G q-1 (x).

Moments and moment generating function

Moments are necessary and important in any statistical analysis, especially in applications. It can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis). The rth moment of GIKw-G family is given by

µ r = ∞ 0 x r f (x) dx
. Using the infinite mixture representation of the pdf in equation (3.3), we have

µ r = ∞ q=0 a q ∆ r where ∆ r = ∞ 0 x r h q-1 (x) dx
Similarly, the sth incomplete moment of GIKw-G family is defined as

µ s = x 0 x s f (x) dx
From the infinite mixture representation of pdf in equation ( 3.3), we get

µ s = ∞ q=0 a q T s (x)
where

T s (x) = x 0 x s h q-1 (x) dx.
Now, we define the moment generating function of GIKw-G family as

M 0 (t) = ∞ 0 e t x f (x) dx
From the infinite mixture representation of pdf in equation (3.3), we obtain

M 0 (t) = ∞ q=0 a q M q-1 (t)
where M q-1 (t) =

x 0 e t x h q-1 (x) dx.

Note that the integrals ∆ r , T s (x) and M q-1 (t) depend only for any choice of baseline distribution.

The mean deviations of GIKw-G family is obtained by

δ 1 = 2 µ F (µ) -2 J(µ), δ 2 = µ -2 J( x)
where µ = mean = E(X), x = median = Q x (0.5) and j(c) =

j(c) = ∞ q=0 a q S q-1 (c)
where S q-1 (c) = c 0

x h q-1 (x) dx the integral depends on G(x) and g(x).

Entropies

The entropy of a random variable X is a measure of variation of uncertainty and has been used in many fields such as physics, engineering, and economics. In this section, we will consider two well known entropy measures, Reǹyi and Shannon entropies. According to [START_REF] Reǹyi | On measures of entropy and information[END_REF], the Reǹyi entropy is defined by

I R = 1 1 -R log ∞ 0 f R (x) dx (3.4) where f R (x) = α β γ g(x) G γ-1 (x) (1 -G γ (x)) α-1 [1 -(1 -G γ (x)) α ] β-1 R
. Using series expansion in equation (3.1), we have

f R (x) = ∞ i=0 w i (j) g γ(i+R)-R (x)
Here, we define

w i (j) = (α β γ) R ∞ j=0 R(β-1) j α(j+R)-R i (-1) i+j γ(i+R)-R+1 and g γ(i+R)-R (x) = [γ(i + R) -R + 1] g R (x) G γ(i+R)-R (x). Now equation (3.5) becomes I R = 1 1 -R log   ∞ i=0 w i (j) ∞ 0 g γ(i+R)-R (x) dx   (3.5)
The above integral depends on G(x) and g(x).

The Shannon entropy of a random variable X is defined by [START_REF] Shannon | A mathematical theory of communication[END_REF] as

η x = -E [log f (x)]
From equation (1.4), we have

η x = -E log γ + log g(x) + (γ -1) log (1 -G γ (x)) + log r G γ (x) 1 -G γ (x) (3.6) Let T = G γ (x) 1-G γ (x) then G(x) = T 1+T 1 γ and 1 -G γ (x) = 1 1+T . Now equation (3.6) becomes η x = -E log γ + log g(x) + 1 - 1 γ log T + 1 + 1 γ log(1 + T ) + log r(T )
After some algebra we have

η x = η g + η T -log γ -1 - 1 γ ∞ j=0 (-1) j j! B 1 -j α , β Γ(1 -j) Γ (1) -ψ(1 -j) + 1 α 1 + 1 γ Γ (1) -ψ(β + 1) (3.7)
where η T is the Shannon entropy of IKw distribution and η g is the Shannon entropy of baseline distribution. where ψ(a) = Γ (a) Γ(a) and Γ (1) is Euler's constant.

Order Statistics

Order statistics have been extensively applied in many fields of statistics, such as reliability and life testing. They enter in the problems of estimation and hypothesis tests in a variety of ways. We now discuss some properties of the order statistics for the GIKw-G family. Let X i:n denote the ith order statistic from a random sample X 1 , . . . , X n from the GIKw-G family. Then, the density function of ith order statistic is given by

f i:n (x) = n! (i -1)!(n -i)! f (x) F i-1 (x) [1 -F (x)] n-i
Using series expansion in equation (3.1), we have

f i:n (x) = n! (i -1)!(n -i)! n-i j=0 n -i j (-1) j f (x) F j+i-1 (x) (3.8)
Consider f (x) F j+i-1 (x), using infinite mixture representation in equations (3.2) and (3.3), we obtain 

 ∞ q=0 a q h q-1 (x)   × ∞ m=0 a m H m (x) j+i-1
From power series raised to a positive power expansion

∞ k=0 a k G k (x) r = ∞ k=0 d k:r G k (x) we have   ∞ q=0 a q q g(x) G q (x)   × ∞ m=0 d m:j+i-1 G m (x)
After simplification, we have

  ∞ q=0 ∞ m=0 a q d m:j+i-1 q g(x) G q+r (x)   Now equation (3.8) becomes f i:n (x) = n! (i -1)!(n -i)! n-i j=0 n -i j (-1) j ∞ q=0 ∞ m=0 a q d m:j+i-1 q g(x) G q+r (x)
Rewriting the equation, the density function of ith order statistic is given by

f i:n (x) = n-i j=0 ∞ q=0 ∞ m=0 V j (q, m) h q+m-1 (x)
, where

V j (q, m) = n! (i-1)!(n-i)! n-i j (-1) j a q d m:j+i-1 q q+m and h q+m-1 (x) = (q + m) g(x) G q+m-1 .
4 Estimation and Simulation

Estimation

Here, we consider the estimation of the unknown parameters of GIKw-G family by the maximum likelihood method for the complete samples. The maximum likelihood estimates (MLEs) enjoy desirable properties that can be used when constructing confidence intervals and deliver simple approximations that work well in finite samples. The resulting approximation for the MLEs in distribution theory is easily handled either analytically or numerically. Let x 1 , ..., x n be a sample of size n from GIKw-G family given in (1.3). The log-likelihood function of GIKw-G family for the vector of parameter Θ = (α, β, γ , ξ) T is given by

(Θ) = n log(α β γ) + n i=1 log g(x i , ξ) + (γ -1) n i=1 log G(x i , ξ) + (α -1) n i=1 log [1 -G γ (x i , ξ)] +(β -1) n i=1 log [1 -{1 -G γ (x i , ξ)} α ]
The components of score vector U = (U β , U α , U γ , U ξ ) T are given by

U β = n β + n i=1 log [1 -{1 -G γ (x i , ξ)} α ] U α = n α + n i=1 log [1 -G γ (x i , ξ)] + (β -1) n i=1 {1 -G γ (x i , ξ)} α log {1 -G γ (x i , ξ)} 1 -{1 -G γ (x i , ξ)} α U γ = n γ + n i=1 log G(x i , ξ) + (α -1) n i=1 G γ (x i , ξ) log G(x i , ξ) 1 -G γ (x i , ξ) +(β -1)α n i=1 {1 -G γ (x i , ξ)} α-1 G γ (x i , ξ) log G(x i , ξ) 1 -{1 -G γ (x i , ξ)} α U ξ = n i=1 g ξ (x i , ξ) g(x i , ξ) + (γ -1) n i=1 G ξ (x i , ξ) G(x i , ξ) -γ (α -1) n i=1 G γ-1 (x i , ξ) G ξ (x i , ξ) 1 -G γ (x i , ξ) -α γ (β -1) n i=1 {1 -G γ (x i , ξ)} α-1 G γ-1 (x i , ξ) G ξ (x i , ξ) 1 -{1 -G γ (x i , ξ)} α
Setting U β , U α , U γ and U ξ equal to zero and solving these equations simultaneously yields the the MLEs. These equations cannot be solved analytically, and analytical softwares are required to solve them numerically.

Simulation study

In this section, an extensive numerical investigation will be carried out to evaluate the performance of the MLEs for the GIKw-W model. Performance of estimators is evaluated through their biases, and mean square errors (MSEs) for different sample sizes. A numerical study is performed using Mathematica ( 9) software. Different sample sizes (n = 30, 50, 75, 100, 200 and 300) and different values of parameters are considered. The experiment is repeated 3000 times. In each experiment, the estimates of the parameters are obtained by maximum likelihood method. The means, MSEs and biases for the different estimators are reported from these experiments. The values in Table 1 indicate that the estimates are quite stable and, more importantly, are close to the true values for the these sample sizes. From Table 1, it is observed that in general the standard deviation decreases as n increases. The simulation study shows that the maximum likelihood method is appropriate for estimating the GIKw-W parameters. In fact, the MSEs of the parameters tend to be closer to the true parameter values when n increases. This fact supports that the asymptotic normal distribution provides an adequate approximation to the finite sample distribution of the MLEs. The normal approximation can be often times improved by using bias adjustments to these estimators.

Conclusion

This paper introduces a new of models called the "GIKw-G" family distributions which can generate all classical continuous distributions while is proved suitable for data of heavy-tails and exhibiting different features of skewness and reverse J-shapes. For any parent continuous distribution G, we define corresponding GKIw-G distribution. We study some of statistical and mathematical properties of the new generator, such as ordinary moments, quantile functions, Shannon and Reǹyi, entropies and order statistics. We discuss maximum likelihood estimates of the model parameters. Four applications of the new family and a simulation study demonstrate its usefulness and potentiality to analysis of real data. This new family and its generated models is commendably useful in several areas such as engineering, survival and lifetime data, hydrology, economics, among others.

  et al. (2013), transformedtransformer (T-X) by Alzaatreh et al. (2013), Weibull-G by Bourguignon et al. (2014), Lomax-G by Cordeiro et al. (2014b), and logistic-X by Tahir et al. (2014). Other recent family of distributions are Garhy-G by Elgarhy et al. (2016), Kumaraswamy Weibull-G by Hassan and Elgarhy (2016a), exponentiated Weibull-G by Hassan and Elgarhy (2016b), additive Weibull-G by Hassan and Hemeda (2016), exponentiated extended-G by Elgarhy et al. (2017), and type II half logistic-G by Hassan et al. (2017). Most recently, Al-Fattah et al. (2017) introduced the inverted Kumaraswamy (IKw) distribution with two positive shape parameters α > 0 and β > 0, where the cumulative distribution function (cdf) is given by Π(x; α, β) = 1 -(1 + x) -α β , x > 0 (1.1)

Figure 1 (

 1 b) shows that the hazard rate function gives increasing and bathtub shapes. β = 0.5 γ = 0

Figure 1 :

 1 Figure 1: Plots of pdf and hrf for the GIKw-U distribution with several values of parameters

  β = 0.5 γ = 1 a = 1 b = 1 α = 2 β = 1 γ = 12 a = 10.8 b = 1.2 α = 1.5 β = 1.5 γ = 1.2 a = 1 b = 1 α = 1 β = 0.8 γ = 12 a = 0.8 b = 2 α = 3 β = 3 γ = 2 a = 0.1 b

Figure 2 :

 2 Figure 2: Plots of pdf and hrf for the GIKw-W distribution with several values of parameters

Figure 3 :

 3 Figure 3: Plots of pdf and hrf for the GIKw-LL distribution with several values of parameters

Figure 4 (

 4 b) also shows that the hazard rate function has upside-down bathtub, increasing and decreasing shapes.

Figure 4 :

 4 Figure 4: Plots of pdf and hrf for the GIKw-L distribution with several values of parameters

  x) dx. Using the infinite mixture representation of pdf in equation (3.3), we get

Figure 5 :Figure 6 :

 56 Figure 5: Estimated pdf in (a) and cdf in (b) for the first data set and estimated pdf in (c) and cdf in (d) for the second data set, respectively.

Table 1 :

 1 The parameter estimations from the GIKw-W distribution using the maximum likelihood method

	n	Par Init MLE	Bias	MSE	Init MLE	Bias	MSE
		γ	2	2.0915 0.0915 0.2296	2	2.0905 0.0905 0.2085
		α	2	2.0580 0.0580 0.1427	2	2.0488 0.0488 0.1308
	50	β	1.5 1.5316 0.0316 0.0522 1.5 1.5330 0.0330 0.0485
		a	2	2.0752 0.0752 11.7134	2	2.1234 0.1234 0.4287
		b	0.5 0.5293 0.0293 0.0200	2	2.1119 0.1119 0.2870
		γ	2	2.0679 0.0679 0.1361	2	2.0559 0.0559 0.1356
		α	2	2.0311 0.0311 0.0882	2	2.0430 0.0430 0.0897
	75	β	1.5 1.5255 0.0255 0.0335 1.5 1.5196 0.0196 0.0332
		a	2	2.0775 0.0775 0.2373	2	2.0952 0.0952 0.2366
		b	0.5 0.5204 0.0204 0.0114	2	2.0679 0.0679 0.1647
		γ	2	2.0434 0.0434 0.0940	2	2.0433 0.0433 0.0945
		α	2	2.0290 0.0290 0.0625	2	2.0259 0.0259 0.0610
	100	β	1.5 1.5161 0.0161 0.0239 1.5 1.5155 0.0155 0.0241
		a	2	2.0630 0.0630 0.1512	2	2.0598 0.0598 0.1523
		b	0.5 0.5136 0.0136 0.0074	2	2.0526 0.0526 0.1197
		γ	2	2.0190 0.0190 0.0425	2	2.0227 0.0227 0.0426
		α	2	2.0164 0.0164 0.0302	2	2.0122 0.0122 0.0293
	200	β	1.5 1.5071 0.0071 0.0115 1.5 1.5089 0.0089 0.0115
		a	2	2.0335 0.0335 0.0679	2	2.0274 0.0274 0.0663
		b	0.5 0.5063 0.0063 0.0034	2	2.0273 0.0273 0.0547
		γ	2	2.0128 0.0128 0.0272	2	2.0155 0.0155 0.0282
		α	2	2.0070 0.0070 0.0194	2	2.0104 0.0104 0.0200
	300	β	1.5 1.5045 0.0045 0.0074 1.5 1.5066 0.0066 0.0077
		a	2	2.0173 0.0173 0.0426	2	2.0211 0.0211 0.0433
		b	0.5 0.5031 0.0031 0.0021	2	2.0199 0.0199 0.0351

Table 2 :

 2 The MLEs and their standard errors (in parentheses) for the first data set.

	Distribution	α	β	γ	a	b
	GIKw-W	3.8371	9.2165	0.1699	0.0120	1.5582
		(6.2626) (25.2396) (0.4264) (0.0279) ( 1.3545)
	GIKw	2.0265	0.8848	2.64300	-	-
		( 1.8893) ( 0.6646) ( 3.8870)	-	-
	IKw	1.7408	2.1058	-	-	-
		(0.3237) ( 0.5373)	-	-	-
	W	-	-	-	0.5262	1.0102
		-	-	-	(0.1176) (0.1326)

Table 3 :

 3 The goodness-of-fit statistics for the first data set

	Distribution	A *	W *	ˆ	K-S	P-value
	GIKw-W	0.1832 0.0270 54.7047 0.0770 0.9899
	GIKw	0.2805 0.0379 55.7572 0.0955 0.9153
	IKw	0.2764 0.0378 55.7706 0.0965 0.9090
	Weibull	0.3000 0.0462 55.4496 0.0918 0.9366

Table 4 :

 4 The MLEs and their standard errors (in parentheses) for the second data set.

	Distribution	α	β	γ	a	b
	GIKw-W	0.1199	1.1017	3.3626	6.0015	1.1272
		(0.0277) ( 0.2661) ( 0.0524) ( 0.0066) ( 0.0066)
	GIKw	1.1631	1.847	1.4473	-	-
		(1.022) ( 1.1798) (1.3924)	-	-
	IKw	2.4601	4.1715	-	-	-
		(0.4213) ( 1.2783)	-	-	-
	W	-	-	-	0.4560	1.4633
		-	-	-	(0.1141) (0.2029)

Table 5 :

 5 The goodness-of-fit statistics for the second data set

	Distribution	A *	W *	ˆ	K-S	P-value
	GIKw-W	0.1813 0.0231 39.1069 0.0748 0.9989
	GIKw	0.2653 0.0409 40.8189 0.1092 0.8665
	IKw	0.3323 0.04948 41.2380 0.1111 0.8523
	Weibull	0.1921 0.0299 39.9909 0.0788 0.9900

Table 6 :

 6 The MLEs and their standard errors (in parentheses) for the third data set.

	Distribution	α	β	γ	a	b
	GIKw-W	0.1308	1.2393	2.7063	4.7484	1.2909
		(0.0297) (0.3783) ( 1.6289) (0.0038) (0.0037)
	GIKw	1.9540	1.4209	3.9458	-	-
		(1.8055) ( 1.0186) ( 5.6701)	-	-
	IKw	2.9876	8.5933	-	-	-
		(0.4731) (3.1240)	-	-	-
	W	-	-	-	0.3154	1.8088
		-	-	-	(0.0906) (0.2491)

Table 7 :

 7 The goodness-of-fit statistics for the third data set

	Distribution	A *	W *	ˆ	K-S	P-value
	GIKw-W	0.1038 0.0144 37.9810 0.0547 0.9988
	GIKw	0.3226 0.0514 30.3017 0.1097 0.8627
	IKw	0.3505 0.0561 39.4255 0.1143 0.8280
	Weibull	0.1693 0.0219 38.6432 .0689	0.9901

Table 8 :

 8 The MLEs and their standard errors (in parentheses) for the fourth data set.

	Distribution	α	β	γ	a	b
	GIKw-W	0.1705	4.5176	5.4997	1.9419	0.5709
		(0.0212) ( 1.0214)	( 0.0069) ( 0.0029) ( 0.0028)
	GIKw	0.3096	4.2274	45.4405	-	-
		(1.1821) ( 16.1314) ( 16.8344)	-	-
	IKw	1.3768	60.9128	-	-	-
		(0.1377) ( 23.9816)	-	-	-
	W	-	-	-	0.0085	1.2685
		-	-	-	(0.0039) (0.1089)

Table 9 :

 9 The goodness-of-fit statistics for the fourth data set

	Distribution	A *	W *	K-S	P-value
	GIKw-W	0.7649 0.1039 293.2774 0.0981 0.6691
	GIKw	1.2613 0.1848 295.4949 0.1018 0.5197
	IKw	1.2044 0.1743 295.4044 0.0954 0.5047
	Weibull	1.0078 0.1470 296.9085 0.1060 0.4681

The histogram of the all data sets and the estimated pdf's and cdf's of the GIKw-W model and the other competitive models are displayed in Figures5 and 6. It is clear from Figures5 and 6that the GIKw-W model provide better fits to these four data sets.

Applications

In this section, real data modeling performance of GIKw-W distribution is compared with several well-known distributions. We provide four applications to real data sets to illustrate the applicability of the GIKw-G family. We focus on the GIKw-W distribution introduced in Section 2. We compute the log-likelihood function evaluated at the MLEs ( ˆ ) using a limited-memory quasi-Newton code for bound-constrained optimization (L-BFGS-B). The goodness-of-fit measures, Anderson-Darling (A * ), Cramèr-von Mises (W * ) are computed for all models and used to decide the best model.The lower values of these statistics and higher p-values of K-S indicate good fits. The value for the Kolmogorov Smirnov (KS) statistic and its p-value are also provided. We compare the GIKw-W distribution with those of the Weibull (W), generalized inverted Kumaraswamy (GIKw) and Kumaraswamy (IKw).

The first data is corresponding to the vinyl chloride data obtained from clean upgrading, monitoring wells in mg/L [START_REF] Bhaumik | Testing parameters of a gamma distribution for small samples[END_REF]. The data set are given as follows: 5.1, 1.2, 1.3, 0.6, 0.5, 2.4, 0.5, 1.1, 8.0, 0.8, 0.4, 0.6, 0.9, 0.4, 2.0, 0.5, 5.3,3.2, 2.7, 2.9, 2.5, 2.3, 1.0, 0.2, 0.1, 0.1, 1.8, 0.9, 2.0, 4.0, 6.8, 1.2, 0.4, 0.2.

The second data set is obtained from [START_REF] Murthy | Weibull Models, Wiley Series in Probability and Statistics[END_REF]. The data refers to the time between failures for repairable items. The data set are given as follows: 1.43, 0. The fourth data is based on the waiting times between 65 consecutive eruptions of Blowhole (1998). The data set are given as follows: 83, 51, 87, 60, 28, 95, 8, 27, 15, 10, 18, 16, 29, 54, 91, 8, 17, 55, 10, 35,47, 77, 36, 17, 21, 36, 18, 40, 10, 7, 34, 27, 28, 56, 8, 25, 68, 146, 89, 18, 73, 69, 9, 37, 10, 82, 29, 8, 60, 61, 61, 18, 169, 25, 8, 26, 11, 83, 11, 42, 17, 14, 9, 12. The MLEs and the goodness of fit statistics of the models for all data sets are presented in Tables 2 to 9, respectively. As seen in Table 3, 5, 7, and 9 the GIKw-W model gives the lowest values. Therefore, it is clear that GIKw-W distribution provides the overall best fit and therefore could be chosen as the more adequate model for explaining data sets.


Pascal Casari

Denis Cartié

Peter Davies

CHARACTERISATION OF NOVEL K-COR SANDWICH STRUCTURES

Keywords: K-Cor™, pin reinforced foam cores, fracture, damage tolerance. 1

Sandwich beams reinforced with K-Cor™ are characterized in terms of stiffness, strength and fracture by means of static tests, and compared to traditional sandwich materials. Stiffness is evaluated with a series of three flexure tests applied to each sample. Strength tests show different failure mechanisms for the two architectures. Mode I fracture properties are measured and a significant increase in the core toughness of reinforced samples is revealed.

INTRODUCTION

Polymer foam core sandwich construction is becoming more popular for cost and weight efficiency reasons. However, the lack of mechanical properties can sometimes be a problem. The present paper considers a solution where the polymer foam is locally reinforced by rigid carbon fibre rods. This 'pin' reinforcement technology has been developed recently 1 and is known under the commercial name of X-Cor™ and K-Cor™.

X-Cor™ and K-Cor™ sandwich panels are structures composed of skins separated with a polymeric foam in which cured carbon fibre rods have been inserted in a pre-determined truss configuration. During the manufacturing process, the pins extend beyond each foam surface. For the X-Cor™ panels, uncured face sheets are pressed on each side of the core. During cure, the pins enter the face sheets. The K-Cor™ construction varies by the fact that the length of pins which extends outside the polymeric foam core is pressed flat on the surface of the foam by a hot-press process. The skins (of any material) are then adhesively bonded onto the surface of the core. For both type of cores, the truss transfers some of the through-thickness and shear loads, and enhances the properties of the core [START_REF] Cartié | The effect of pin reinforcement upon the throughthickness compressive strength of foam cored sandwich panels[END_REF][START_REF] Marasco | Mechanical properties balance in novel Z-pinned sandwich panels: Out of plane properties[END_REF][START_REF] Partridge | Manufacture and performance of Zpinned composites[END_REF]. Due to the fact that there is no need for an adhesive film in the X-cor™ configuration, the aerospace industry developed this configuration to replace Nomex® honeycomb cores in weight saving exercises. The truss was designed to reproduce the general mechanical properties of the cores already in use. For a specific application, (Sikorsky Comanche access doors) a weight saving of 10 to 15% was achieved by using X-cor™ as opposed to Nomex honeycomb [START_REF] Cartensen | X-cor advanced sandwich core material[END_REF]. The transition between solid laminates and X-cor™ truss has been studied by O'Brien et al. [START_REF] O'brien | Exploratory investigation of failure mechanisms in transition regions between solid laminates and X-cor® truss sandwich[END_REF]. They showed that the failure modes are very specific to the loading conditions applied locally on the truss. Because of the different mechanical links between the pins and the skins in K-cor™, there is no knowledge of the failure modes in the solid laminate to sandwich transition area.

MANUFACTURING

The sandwich panels were constructed with woven carbon fibre face sheets and Rohacell® polymethacrymalimide foam core. Some of the foam cores were reinforced by pultruded carbon fibre / epoxy rods, creating a 2 dimensional truss. The details are as follows.

Two 290mm x 210mm x 12mm blocks of Rohacell® 51IG (density 51kg/m 3 ) were chamfered along two edges, creating the ramps. One foam block was reinforced by 0.5mm diameter pultruded T650 carbon fibre / epoxy rods. The rods were inserted using a 6-Axis Kawasaki robot arm (see Figure 1). The pins were inserted in two directions at nominal angles of ±22°, creating a truss in the direction 1-3 of the samples (see Figure 2.). The pin truss extended beyond each surface of the laminate for a free length of approximately 3mm. The pins were folded back, flush with the foam surface using a hot press process. The beams have been specifically designed to study the stress transfer in the transition area between sandwich and solid laminates. The skins are relatively thick in order to study a design of sandwich structure commonly used in the marine industry. Specimen Geometry.

The face sheets were constructed using 8 plies of 977-6 resin / IM7 woven carbon fibre pre-preg. An AF163-2 adhesive film was used between the foam core and the surface plies. The 8 plies of the top face sheet were laid up and debulked every ply under vacuum to assure good shape conformity with the core. The panels were cured in an autoclave at 135°C at a pressure of 0.2 MPa (2 bars) for 3 hours. 40 mm wide samples were cut using a diamond coated circular saw. The nominal geometry of the samples is given in Figure 2.

STIFFNESS

For evaluating the effect of pins on the stiffness, a set of three flexure tests is applied to each sample. This ensures that the beams are loaded with various bending moments and shear forces and allows a more accurate characterization of shear stiffnesses. The three tests are shown in (Table 1) : Three point bending. The bending moment is maximum in the center whereas the shear force alternates from a constant positive value on one side of the loading point to an opposite one on the other side. Four point bending test. The aim of this test is to ensure a central part with no shear force and a constant bending moment Four point bending test with an asymmetric arrangement. This produces a bending moment diagram with a zero value in the centre, and a shear force which is maximum at this point.

Table 1. Tests and corresponding internal loads on the sandwich samples.

Each test has been conducted on 3 samples from classical and reinforced sandwich beams. The samples have been loaded between 0.5 and 1.5 kN in order to produce a deflection around 1 mm.

FRACTURE

Damage development in flexure

Beam specimens with and without pins were loaded to failure in three point flexure. Two test configurations were used, with the loading on alternate facings of the beams. The same fixture as above was employed. The loading rate was 5mm/min.

Configuration 1

This is the more severe configuration as the joint between the sandwich and the beam end is subjected to both shear and tensile loads. For the sandwich without pins the first damage appears at the end of the foam, Figure 3a, as a shear crack. Further loading results in compression failure below the upper load point, and continuing the test leads to core crushing and a final tensile failure in the lower facing, Figure 3b.

The inclusion of pins delays the appearance of first damage. Some cracks still appear in the core foam but the pins maintain the overall integrity of the sandwich up to compressive failure below the upper loading point, Figure Figure 3c. They also delay the crushing of the core so that final tensile failure in the lower facing appears later. The displacement-load curve shows these different steps after an initial linear part. Load-displacement, configuration 1.

Configuration 2

Turning the specimen and loading from the other side results in compression rather than tension at the foam ends. Figure 5 shows examples of load-displacement plots. The loads are slightly higher for both materials. For the classical beam (Figure 6a) the failure mechanisms are mostly crushing of the core and indentation of the skin which lead to a yield point on the load-displacement curve following the elastic region. The beam reinforced with K-cor™ (Figure 6b) shows a part cracked in shear on the right side. This series of cracks corresponds to the first peak on the loaddisplacement curve. Failure mechanisms of sandwich beams in bending in configuration 2.

Crack propagation in the core

The inclusion of pins clearly has a beneficial effect on the flexural response of these sandwich structures. An additional benefit to be expected is improved damage tolerance. Extensive work on delamination of pinned laminates has clearly shown that pins play a major role in slowing crack propagation or even stopping it completely [START_REF] Partridge | Delamination resistant laminates by Z-Fiber® pinning. Part I Manufacture and fracture performance[END_REF]. It is therefore of interest to examine the role of through-core pinning on crack behaviour in sandwich materials. Some preliminary crack propagation tests have been performed using a Double Cantilever Beam (DCB) specimen, Figure 7. The starter crack was introduced at core mid-thickness with a saw, then sharpened by a razor blade. Notches in both specimens were the same length, 40 mm from the load point. Load was introduced by bonding aluminium blocks to the facings. Loading is initially mode I. In the foam without pins the crack propagated at mid-thickness in a symmetrical manner as shown in Figure 7. In the K-Cor™ specimen the starter crack did not propagate, a damage zone developed near the lower core/facing interface and propagated there as shown in Figure 7. Two types of DCB specimen -sandwich and K-cor™.

Figure 8 shows examples of load-displacement plots. It is apparent that considerably higher energy is necessary to propagate a crack in the pinned foam. The approximate fracture energies (G c ) are 460 J/m² for the foam and 1690 J/m² for the K-Cor™, at least three times higher. The through thickness pinning provides an effective barrier to crack propagation as shown in Figure 9. These pins must be pulled out or broken before the crack can advance. It should be emphasized that these are preliminary results for one K-Cor™ configuration. An advantage of these materials is that the pinning parameters can be tailored for a particular application; optimization of pin material, density, angle and sandwich manufacturing conditions may result in further significant improvements in damage tolerance.

CONCLUSION

This paper presents results from a preliminary study of the mechanical behaviour of a pin reinforced foam core sandwich material. Stiffness, fracture strength and damage tolerance have been examined, and it is apparent that for a similar weight all these properties can be significantly improved.

Figure 1 . 6 -

 16 Figure 1. 6-Axis Robot in use at Cranfield University and Close Up on the Insertion head.

Figure 2 .

 2 Figure 2.Specimen Geometry.

Figure 3 .

 3 Figure 3.Failure mechanisms of sandwich beams in bending in configuration 1.

Figure 4 .

 4 Figure 4.Load-displacement, configuration 1.

Figure 5 .

 5 Figure 5.

  Figure 6.Failure mechanisms of sandwich beams in bending in configuration 2.

  Figure 7.Two types of DCB specimen -sandwich and K-cor™.

Figure 8 .

 8 Figure 8. Load-displacement plots, DCB tests, corresponding to 55 mm crack propagation in both materials.

Figure 9 .

 9 Figure 9. K-Cor™ DCB specimen, details of crack initiation region.

Table 2

 2 directly reports the bending stiffness results. Obviously, the stiffness of samples reinforced with K-Cor™ is much higher than the classical sandwich value.

	Material	Sample	3 points	4 pts symmetrical 4 pts antisymmetrical
		1	0.28	0.52	0.58
	No pins	2	0.27	0.4	0.43
		3	0 .25	0.57	0.40
		1	0.80	1.39	2.12
	With pins	2	0.82	1.20	1.77
		3	0.80	1.05	1.97

Table 2 .

 2 Bending stiffness (kN/mm).

Aztex Inc, 303 Bear Hill Road, Whaltham, MA, 02154-1196, USA. 1


Samy Skander Bahoura

A COMPACTNESS RESULT FOR AN ELLIPTIC EQUATION IN DIMENSION 2

Keywords: Mathematics Subject Classification: 35J60 35B45 35B50 blow-up, boundary, elliptic equation, a priori estimate, Lipschitz condition, starshaped domains

We give a blow-up behavior for the solutions of an elliptic equation under some conditions. We also derive a compactness creterion for this equation.

INTRODUCTION AND MAIN RESULTS

Let us consider the following operator:

L ǫ := ∆ + ǫ(x 1 ∂ 1 + x 2 ∂ 2 ) = div[a ǫ (x)∇] a ǫ (x) with a ǫ (x) = e ǫ|x| 2
2 .

We consider the following equation:

(P ǫ ) -∆u -ǫ(x 1 ∂ 1 u + x 2 ∂ 2 u) = -L ǫ u = V e u in Ω ⊂ R 2 , u = 0 in ∂Ω.
Here, we assume that: Ω starshaped, and, u ∈ W 1,1 0 (Ω), e u ∈ L 1 (Ω), 0 ≤ V ≤ b, 1 ≥ ǫ ≥ 0.

When ǫ = 0 the previous equation was studied by many authors with or without the boundary condition, also for Riemann surfaces see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF][START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF][START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF][START_REF] Chen | Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces[END_REF][START_REF] Chang | Scalar curvature equation on 2and 3-spheres[END_REF][START_REF] De Figueiredo | A priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF][START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF][START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF][START_REF] Nagasaki | Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities[END_REF][START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF] where one can find some existence and compactness results. Also we have a nice formulation in the sens of the distributions of this Problem in [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF].

Among other results, we can see in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] the following important Theorem, Theorem A (Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) If (u i ) i and (V i ) i are two sequences of functions relative to the problem (P 0 ) with ǫ = 0 and,

0 < a ≤ V i ≤ b < +∞ then it holds, sup K u i ≤ c,
with c depending on a, b, K and Ω.

We can find in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] an interior estimate if we assume a = 0 but we need an assumption on the integral of e u i , namely:

Theorem B(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to the problem (P 0 ) with, 0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, then it holds; sup

K u i ≤ c,
with c depending on b, C, K and Ω.

The condition Ω e u i dy ≤ C is a necessary condition in the Problem (P ǫ ) as showed by the following counterexample for ǫ = 0:

Theorem C (Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).There are two sequences (u i ) i and (V i ) i of the problem (P 0 ) with;

0 ≤ V i ≤ b < +∞, Ω e u i dy ≤ C, such that, sup Ω u i → +∞.
To obtain the two first previous results (Theorems A and B) Brezis and Merle used an inequality (Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) obtained by an approximation argument and they used Fatou's lemma and applied the maximum principle in W 1,1 0 (Ω) which arises from Kato's inequality. Also this weak form of the maximum principle is used to prove the local uniform boundedness result by comparing a certain function and the Newtonian potential. We refer to [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF] for a topic about the weak form of the maximum principle.

Note that for the problem (P 0 ), by using the Pohozaev identity, we can prove that Ω e u i is uniformly bounded when 0 < a ≤ V i ≤ b < +∞ and ||∇V i || L ∞ ≤ A and Ω starshaped, when a = 0 and ∇ log V i is uniformly bounded, we can bound uniformly Ω V i e u i . In [START_REF] Ma | Convergence for a Liouville equation[END_REF] Ma-Wei have proved that those results stay true for all open sets not necessarily starshaped in the case a > 0.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] Chen-Li have proved that if a = 0 and Ω e u i is uniformly bounded and ∇ log V i is uniformly bounded then (u i ) i is bounded near the boundary and we have directly the compactness result for the problem (P 0 ). Ma-Wei in [START_REF] Ma | Convergence for a Liouville equation[END_REF] extend this result in the case where a > 0.

When ǫ = 0 and if we assume V more regular we can have another type of estimates called sup + inf type inequalities. It was proved by Shafrir see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF] that, if (u i ) i , (V i ) i are two sequences of functions solutions of the Problem (P 0 ) without assumption on the boundary and 0 < a ≤ V i ≤ b < +∞ then it holds:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
We can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF] an explicit value of C a b = a b . In his proof, Shafrir has used the blow-up function, the Stokes formula and an isoperimetric inequality see [START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF]. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Now, if we suppose (V i ) i uniformly Lipschitzian with A its Lipschitz constant then C(a/b) = 1 and c = c(a, b, A, K, Ω) see Brezis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also have in [START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF], an extension of the Brezis-Li-Shafrir result to compact Riemannian surfaces without boundary. One can see in [START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses when the solutions blow-up. Here the notion of isolated blow-up point is used. Also one can see in [START_REF] Chen | Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces[END_REF] refined estimates near the isolated blow-up points and the bubbling behavior of the blow-up sequences.

Here we give the behavior of the blow-up points on the boundary and a proof of a compactness result with Lipschitz condition. Note that our problem is an extension of the Brezis-Merle Problem.

The Brezis-Merle Problem (see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) is:

Problem. Suppose that V i → V in C 0 ( Ω) with 0 ≤ V i . Also, we consider a sequence of solutions (u i ) of (P 0 ) relative to (V i ) such that, Ω e u i dx ≤ C, is it possible to have: ||u i || L ∞ ≤ C = C(b, C, V, Ω)?
Here we give blow-up analysis on the boundary when V (similar to the prescribed curvature when ǫ = 0) are nonegative and bounded, and on the other hand, if we add the assumption that these functions (similar to the prescribed cruvature) are uniformly Lipschitzian, we have a compactness of the solutions of the problem (P ǫ ) for ǫ small enough. (In particular we can take a sequence of ǫ i tending to 0):

For the behavior of the blow-up points on the boundary, the following condition is sufficient,

0 ≤ V i ≤ b,
The condition V i → V in C 0 ( Ω) is not necessary. But for the compactness of the solutions we add the following condition:

||∇V i || L ∞ ≤ A.
Our main results are:

Theorem 1.1. Assume that max Ω u i → +∞, where (u i ) are solutions of the probleme (P ǫ i ) with:

0 ≤ V i ≤ b,
and

Ω e u i dx ≤ C, ǫ i → 0,
then, after passing to a subsequence, there is a finction u, there is a number N ∈ N and N points x 1 , . . . , x N ∈ ∂Ω, such that,

∂ ν u i → ∂ ν u + N j=1 α j δ x j , α j ≥ 4π, in the sens of measures on ∂Ω. u i → u in C 1 loc ( Ω -{x 1 , . . . , x N }). Theorem 1.2.
Assume that (u i ) are solutions of (P ǫ i ) relative to (V i ) with the following conditions:

0 ≤ V i ≤ b, ||∇V i || L ∞ ≤ A and Ω e u i ≤ C, ǫ i → 0.
Then we have: First remark that:

||u i || L ∞ ≤ c(b, A, C, Ω),
-∆u i = ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ) + V i e u i ∈ L 1 (Ω) in Ω ⊂ R 2 , u i = 0 in ∂Ω.
and,

u i ∈ W 1,1 0 (Ω).
By the corollary 1 of Brezis-Merle see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] we have e u i ∈ L k (Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] imply that:

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
Also remark that, we have for two positive constants C q = C(q, Ω) and C 1 = C 1 (Ω) (see [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF]) :

||∇u i || L q ≤ C q ||∆u i || L 1 ≤ (C ′ q + ǫC 1 ||∇u i || L 1 ), ∀ i and 1 < q < 2.
Thus, if ǫ > 0 is small enough and by the Holder inequality, we have the following estimate:

||∇u i || L q ≤ C ′′ q , ∀ i and 1 < q < 2.
Step 1: interior estimate First remark that, if we consider the following equation:

-∆w i = ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ) ∈ L q , 1 < q < 2 in Ω ⊂ R 2 , w i = 0 in ∂Ω. If we consider v i the Newtonnian potential of ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ),
we have:

v i ∈ C 0 ( Ω), ∆(w i -v i ) = 0.
By the maximum principle w i -v i ∈ C 0 ( Ω) and thus w i ∈ C 0 ( Ω).

Also we have by the elliptic estimates that w i ∈ W 2,1+ǫ ⊂ L ∞ , and we can write the equation of the Problem as:

-∆(u i -w i ) = Ṽi e u i -w i in Ω ⊂ R 2 , u i -w i = 0 in ∂Ω.
with,

0 ≤ Ṽi = V i e w i ≤ b, Ω e u i -w i ≤ C.
We apply the Brezis-Merle theorem to u i -w i to have:

u i -w i ∈ L ∞ loc (Ω),
and, thus:

u i ∈ L ∞ loc (Ω).

Step2: boundary estimate

Set ∂ ν u i the inner derivative of u i . By the maximum principle ∂ ν u i ≥ 0.

We have:

∂Ω ∂ ν u i dσ ≤ C.
We have the existence of a nonnegative Radon measure µ such that,

∂Ω ∂ ν u i ϕdσ → µ(ϕ), ∀ ϕ ∈ C 0 (∂Ω).
We take an x 0 ∈ ∂Ω such that, µ(x 0 ) < 4π. Set B(x 0 , ǫ) ∩ ∂Ω := I ǫ . We choose a function η ǫ such that,

           η ǫ ≡ 1, on I ǫ , 0 < ǫ < δ/2, η ǫ ≡ 0, outside I 2ǫ , 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I 2ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
We take a ηǫ such that,

-∆η ǫ = 0 in Ω ⊂ R 2 , ηǫ = η ǫ in ∂Ω.
Remark: We use the following steps in the construction of ηǫ :

We take a cutoff function η 0 in B(0, 2) or B(x 0 , 2):

1-We set η ǫ (x) = η 0 (|x -x 0 |/ǫ) in the case of the unit disk it is sufficient.

2-Or, in the general case: we use a chart (f, Ω) with f (0) = x 0 and we take µ ǫ (x) = η 0 (f (|x|/ǫ)) to have connected sets I ǫ and we take η ǫ (y

) = µ ǫ (f -1 (y)). Because f, f -1 are Lipschitz, |f (x) -x 0 | ≤ k 2 |x| ≤ 1 for |x| ≤ 1/k 2 and |f (x) -x 0 | ≥ k 1 |x| ≥ 2 for |x| ≥ 2/k 1 > 1/k 2 , the support of η is in I (2/k 1 )ǫ .            η ǫ ≡ 1, on f (I (1/k 2 )ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside f (I (2/k 1 )ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I (2/k 1 )ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
3-Also, we can take: µ ǫ (x) = η 0 (|x|/ǫ) and η ǫ (y) = µ ǫ (f -1 (y)), we extend it by 0 outside f (B 1 (0)). We have

f (B 1 (0)) = D 1 (x 0 ), f (B ǫ (0)) = D ǫ (x 0 ) and f (B + ǫ ) = D + ǫ (x 0 ) with f and f -1 smooth diffeo- morphism.            η ǫ ≡ 1, on a the connected set J ǫ = f (I ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside J ′ ǫ = f (I 2ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (J ′ ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
And,

H 1 (J ′ ǫ ) ≤ C 1 H 1 (I 2ǫ ) = C 1 4ǫ, because f is Lipschitz.
Here H 1 is the Hausdorff measure.

We solve the Dirichlet Problem:

∆η ǫ = ∆η ǫ in Ω ⊂ R 2 , ηǫ = 0 in ∂Ω.
and finaly we set ηǫ = -η ǫ + η ǫ . Also, by the maximum principle and the elliptic estimates we have :

||∇η ǫ || L ∞ ≤ C(||η ǫ || L ∞ + ||∇η ǫ || L ∞ + ||∆η ǫ || L ∞ ) ≤ C 1 ǫ 2 ,
with C 1 depends on Ω.

As we said in the beguening, see also [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF], we have:

||∇u i || L q ≤ C q , ∀ i and 1 < q < 2.
We deduce from the last estimate that, (u i ) converge weakly in W 1,q 0 (Ω), almost everywhere to a function u ≥ 0 and Ω e u < +∞ (by Fatou lemma). Also, V i weakly converge to a nonnegative function V in L ∞ . The function u is in W 1,q 0 (Ω) solution of :

-∆u = V e u ∈ L 1 (Ω) in Ω ⊂ R 2 , u = 0 in ∂Ω.
According to the corollary 1 of Brezis-Merle result, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], we have e ku ∈ L 1 (Ω), k > 1. By the elliptic estimates, we have u ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).

We denote by f • g the inner product of any two vectors f and g of R 2 .

We can write,

-∆((u i -u)η ǫ ) = (V i e u i -V e u )η ǫ -2∇(u i -u) • ∇η ǫ + ǫ i (∇u i • x)η ǫ . (1)
We use the interior esimate of Brezis-Merle, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF],

Step 1: Estimate of the integral of the first term of the right hand side of (1).

We use the Green formula between ηǫ and u, we obtain,

Ω V e u ηǫ dx = ∂Ω ∂ ν uη ǫ ≤ Cǫ = O(ǫ) (2)
We have,

-∆u i -ǫ i ∇u i • x = V i e u i in Ω ⊂ R 2 , u = 0 in ∂Ω.
We use the Green formula between u i and ηǫ to have:

Ω V i e u i ηǫ dx = ∂Ω ∂ ν u i η ǫ dσ -ǫ i Ω (∇u i • x)η ǫ = = ∂Ω ∂ ν u i η ǫ dσ + o(1) → µ(η ǫ ) ≤ µ(J ′ ǫ ) ≤ 4π -ǫ 0 , ǫ 0 > 0 (3)
From ( 2) and (3) we have for all ǫ > 0 there is i 0 such that, for i ≥ i 0 ,

Ω |(V i e u i -V e u )η ǫ |dx ≤ 4π -ǫ 0 + Cǫ (4)
Step 2.1: Estimate of integral of the second term of the right hand side of (1).

Let Σ ǫ = {x ∈ Ω, d(x, ∂Ω) = ǫ 3 } and Ω ǫ 3 = {x ∈ Ω, d(x, ∂Ω) ≥ ǫ 3 }, ǫ > 0. Then, for ǫ small enough, Σ ǫ is an hypersurface. The measure of Ω -Ω ǫ 3 is k 2 ǫ 3 ≤ meas(Ω -Ω ǫ 3 ) = µ L (Ω -Ω ǫ 3 ) ≤ k 1 ǫ 3 .
Remark: for the unit ball B(0, 1), our new manifold is B(0, 1 -ǫ 3 ).

(Proof of this fact; let's consider d(x, ∂Ω) = d(x, z 0 ), z 0 ∈ ∂Ω, this imply that (d(x, z 0 )) 2 ≤ (d(x, z)) 2 for all z ∈ ∂Ω which it is equivalent to (z -z 0 ) • (2x -z -z 0 ) ≤ 0 for all z ∈ ∂Ω, let's consider a chart around z 0 and γ(t) a curve in ∂Ω, we have;

(γ(t) -γ(t 0 ) • (2x -γ(t) -γ(t 0 )) ≤ 0 if
we divide by (t -t 0 ) (with the sign and tend t to t 0 ), we have γ ′ (t 0 ) • (x -γ(t 0 )) = 0, this imply that x = z 0 -sν 0 where ν 0 is the outward normal of ∂Ω at z 0 ))

With this fact, we can say that S = {x, d(x, ∂Ω)

≤ ǫ} = {x = z 0 -sν z 0 , z 0 ∈ ∂Ω, -ǫ ≤ s ≤ ǫ}. It is sufficient to work on ∂Ω. Let's consider a charts (z, D = B(z, 4ǫ z ), γ z ) with z ∈ ∂Ω such that ∪ z B(z, ǫ z )
is cover of ∂Ω . One can extract a finite cover (B(z k , ǫ k )), k = 1, ..., m, by the area formula the measure of S ∩ B(z k , ǫ k ) is less than a kǫ (a ǫ-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary).

We write,

Ω |∇(u i -u) • ∇η ǫ |dx = Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx + Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx. (5)
Step 2.1.1:

Estimate of Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
First, we know from the elliptic estimates that

||∇η ǫ || L ∞ ≤ C 1 /ǫ 2 , C 1 depends on Ω
We know that (|∇u i |) i is bounded in L q , 1 < q < 2, we can extract from this sequence a subsequence which converge weakly to h ∈ L q . But, we know that we have locally the uniform convergence to |∇u| (by the Brezis-Merle's theorem), then, h = |∇u| a.e. Let q ′ be the conjugate of q.

We have, ∀f ∈ L q ′ (Ω)

Ω |∇u i |f dx → Ω |∇u|f dx
If we take f = 1 Ω-Ω ǫ 3 , we have:

for ǫ > 0 ∃ i 1 = i 1 (ǫ) ∈ N, i ≥ i 1 , Ω-Ω ǫ 3 |∇u i | ≤ Ω-Ω ǫ 3 |∇u| + ǫ 3 . Then, for i ≥ i 1 (ǫ), Ω-Ω ǫ 3 |∇u i | ≤ meas(Ω -Ω ǫ 3 )||∇u|| L ∞ + ǫ 3 = ǫ 3 (k 1 ||∇u|| L ∞ + 1) = O(ǫ 3 ).
Thus, we obtain,

Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 1) = O(ǫ) (6)
The constant C 1 does not depend on ǫ but on Ω.

Step 2.1.2: Estimate of

Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
We know that, Ω ǫ ⊂⊂ Ω, and ( because of Brezis-Merle's interior estimates)

u i → u in C 1 (Ω ǫ 3 ). We have, ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ≤ ǫ 3 , for i ≥ i 3 .
We write,

Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx ≤ ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ||∇η ǫ || L ∞ = C 1 ǫ = O(ǫ) for i ≥ i 3 , For ǫ > 0, we have for i ∈ N, i ≥ i ′ , Ω |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 2) = O(ǫ) (7)
From (4) and (7), we have, for ǫ > 0, there is i ′′ such that, i ≥ i ′′ ,

Ω |∆[(u i -u)η ǫ ]|dx ≤ 4π -ǫ 0 + ǫ2C 1 (2k 1 ||∇u|| L ∞ + 2 + C) = 4π -ǫ 0 + O(ǫ) (8)
We choose ǫ > 0 small enough to have a good estimate of (1).

Indeed, we have:

-∆[(u i -u)η ǫ ] = g i,ǫ in Ω ⊂ R 2 , (u i -u)η ǫ = 0 in ∂Ω. with ||g i,ǫ || L 1 (Ω) ≤ 4π -ǫ 0 /2.
We can use Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] to conclude that there are q ≥ q > 1 such that:

Vǫ(x 0 ) e q|u i -u| dx ≤ Ω e q|u i -u|ηǫ dx ≤ C(ǫ, Ω).
where, V ǫ (x 0 ) is a neighberhooh of x 0 in Ω. Here we have used that in a neighborhood of x 0 by the elliptic estimates, 1 -Cǫ ≤ ηǫ ≤ 1.

Thus, for each x 0 ∈ ∂Ω -{x 1 , . . . , xm } there is ǫ 0 > 0, q 0 > 1 such that:

B(x 0 ,ǫ 0 ) e q 0 u i dx ≤ C, ∀ i.
By the elliptic estimate see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF] we have:

||u i || C 1,θ [B(x 0 ,ǫ)] ≤ c 3 ∀ i.
We have proved that, there is a finite number of points x1 , . . . , xm such that the squence (u i ) i is locally uniformly bounded in C 1,θ , (θ > 0) on Ω -{x 1 , . . . , xm }.

Proof of theorem 1.2:

The Pohozaev identity gives :

∂Ω 1 2 (x • ν)(∂ ν u i ) 2 dσ + ǫ Ω (x • ∇u i ) 2 dx + ∂Ω (x • ν)V i e u i dσ = Ω (x • ∇V i + 2V i )e u i dx
We use the boundary condition and the fact that Ω is starshaped and the fact that ǫ > 0 to have that:

∂Ω (∂ ν u i ) 2 dx ≤ c 0 (b, A, C, Ω). (9)
Thus we can use the weak convergence in L 2 (∂Ω) to have a subsequence ∂ ν u i , such that:

∂Ω ∂ ν u i ϕdx → ∂Ω ∂ ν uϕdx, ∀ ϕ ∈ L 2 (∂Ω),
Thus, α j = 0, j = 1, . . . , N and (u i ) is uniformly bounded.

Remark 1: Note that if we assume the open set bounded starshaped and V i uniformly Lipschitzian and between two positive constants we can bound, by using the inner normal derivative Ω e u i .

Remark 2: If we assume the open set bounded starshaped and ∇ log V i uniformly bounded, by the previous Pohozaev identity (we consider the inner normal derivative) one can bound Ω V i e u i uniformly.

Remark 3: One can consider the problem on the unit ball and an ellipse. These two problems are differents, because:

1) if we use a linear transformation, (y 1 , y 2 ) = (x 1 /a, x 2 /b), the Laplcian is not invariant under this map.

2) If we use a conformal transformation, by a Riemann theorem, the quantity x • ∇u is not invariant under this map.

We can not use, after using those transofmations, the Pohozaev identity.

A COUNTEREXAMPLE

We start with the notation of the counterexample of Brezis and Merle.

The domain Ω is the unit ball centered in (1, 0).

Lets consider z i (obtained by the variational method), such that:

-∆z i -ǫ i x • ∇z i = -L ǫ i (z i ) = f ǫ i .
With Dirichlet condition. By the regularity theorem we have z i ∈ C 1 ( Ω).

We have:

||f ǫ i || 1 = 4πA.
Thus by the duality theorem of Stampacchia or Brezis-Strauss, we have:

||∇z i || q ≤ C q , 1 ≤ q < 2.
We solve:

-∆w i = ǫ i x • ∇z i , With Dirichlet condition.
By the elliptic estimates, w i ∈ C 1 ( Ω) and w i ∈ C 0 ( Ω) uniformly.

By the maximum principle we have:

z i -w i ≡ u i .
Where u i is the function of the counterexemple of Brezis Merle. We write:

-∆z i -ǫ i x • ∇z i = f ǫ i = V i e z i .
Thus, we have: Ω e z i ≤ C 1 , and 0 ≤ V i ≤ C 2 , and,

z i (a i ) ≥ u i (a i ) -C 3 → +∞, a i → O.

3 2 .

 2 PROOF OF THE THEOREMSProof of theorem 1.1:


Anna Maria Vegni
email: annamaria.vegni@uniroma3.it

Valeria Loscrí
email: valeria.loscri@inria.fr

Pietro Manzoni
email: pmanzoni@disca.upv.es

Analysis of Small-World Features in Vehicular Social Networks

 proves that the time-evolving dynamics of the vehicular social network graph presents a small-world structure, where nodes tend to connect through clusters whose average distance is low on average. This result highlights how VSNs are a specific class of vehicular ad hoc networks with peculiar features that distinguish them from classical online social networks.

I. INTRODUCTION

Social networks offer the opportunity to distribute information, direct traffic, create a dialogue with specific groups, and deploy applications that allow users to participate directly in a coordinated way. However, when we change the context, and pass from online social networks (OSNs) [START_REF] Staab | Social networks applied[END_REF] to mobile/vehicular social networks (VSNs) [START_REF] Vegni | A survey on vehicular social networks[END_REF], the quest for an efficient and effective specific solution to distribute information is clearly a crucial issue.

In this paper, we analyze SCARF from a network graph evolution perspective and observe that the obtained time-evolving VSN follows a small-world behavior [START_REF] Barabási | Network Science[END_REF]. According to Watts and Strogatz [START_REF] Watts | Collective dynamics of "small-world" networks[END_REF], small-world networks are a class of networks that are "highly clustered, yet have small characteristic path lengths, like random graphs". These characteristics result in networks with unique properties of regional specialization with efficient information transfer. Determining whether a network is a "small-world" is important in understanding the system's behavior, and can provide useful data, like the diameter, or path length, of a network. Moreover, knowing the network's general topology allows to better understand the overall characteristics and behavior of such systems. In our context, spotting a congestion in specific links allows to forecasts possible critical situations if those links fail and prepare the network for failure tolerance.

This paper is organized as follows. Section II describes how the small-world model can be applied into VSNs. We start from the Barabási-Albert model, which generates scalefree networks based on the growth and preferential attachment features, and we inquire if this model can also be suitable for VSNs. Through an analytical analysis, in Section III we demonstrate that the time-evolving graph theoretic model with SCARF carries to a small-world structure, then stating SCARF is more suitable for VSNs, instead of the Barabási-Albert model. Finally, conclusions are drawn at the end of the paper.

II. SMALL-WORLD MODEL FOR VSNS

Let a VSN be modeled through the Barabási-Albert (BA) model, thus generating a scale-free vehicular social network. The VSN is initially made of m 0 vehicles. The BA model states that a vehicle attempting to connect will be likely linked to existing vehicles already connected to the VSN.

The probability that the l-th link (with l = 1, 2, . . . , m 0 ) of the new vehicle connects to the i-th vehicle in the VSN depends on the degree k i as [START_REF] Barabási | Network Science[END_REF]:

(k i ) = k i j k j , with j = i. (1)
Eq. ( 1) can be translated into the preferential attachment feature of the BA model, so that a new node tends to link to the "most" connected nodes (i.e., the nodes with higher degree). In this way, there is a node evolution towards hubs. Hubs represent the nodes with highest degree, and are able to attract new nodes accessing the network. This reflects the richgets-richer phenomenon, so that new nodes are more likely to connect to the "most" connected nodes than to the "least" connected nodes. After t time steps, the BA model generates a network with N = t + m 0 nodes and (m 0 + mt) links. Considering the time evolution of the VSN, the probability that a node v will be connected to the i-th node will be:

p vi = k i 2t -1 , with v = i (2)
which represents the linearized chord diagram (LCD) [START_REF] Bollobás | The degree sequence of a scale-free random graph process[END_REF], where we did not consider the existence of self-loops.

In this context, we can apply SCARF as a message rebroadcasting technique towards selected vehicles showing a social degree that makes them "social vehicles". SCARF distinguishes different types of vehicles, based on their social features, such as relationships (i.e., a friend vehicle or a family vehicle), common interests (i.e., a classmate vehicle or a commuter vehicle) and social activity (i.e., a hub vehicle) [START_REF] Vegni | SCARF: A SoCial-Aware Reliable Forwarding Technique for Vehicular Communications[END_REF].

III. ANALYTICAL PERFORMANCE

In this section, we present the analytical results obtained in a vehicular social network, following the time-evolving BA model. Our aim is to demonstrate that SCARF approach is well-suitable for data dissemination in VSNs, as it takes into account not only vehicular features, but also social parameters, and thus it allows the VSN to evolve in a small-world structure. Compared to typical approaches for small-world networks, SCARF shows that the network evolution follows a smallworld nature, i.e., the distance between two randomly chosen nodes in the network is short, and the nodes tend to form more clusters.

Let us consider a network graph comprised of 50 nodes. Then, five new nodes will enter the VSN, trying to connect to existing nodes. In this scenario, we compare SCARF technique to the BA model. Analytical results are expressed in terms of average (i) distance, and (ii) clustering coefficient, for different network sizes (i.e., from 50 to 55 nodes). These metrics were chosen in order to assess how many vehicles are connected, how their links evolve, how central and important a node is. This analysis aims to determine if SCARF produces a VSN with a small-world structure that is when nodes have a higher degree, form clusters, and present a lower distance on average, as stated by the following definition.

Definition 1 (Small-world structure): A network graph has a small-world structure if nodes tend forming clusters and the average distance is low. Similarly, a network graph with nodes having low betweenness and high centrality presents a small-world structure.

In SCARF, each vehicle entering the VSN tries to connect to high social degree nodes, which also have high network degree. It follows that the average distance with SCARF for different vehicles entering the VSN is the lowest with respect to the BA approach. In Fig. 1, the BA model shows highest average distance. This means that SCARF highlights a smallworld behavior in VSNs, so that nodes are most connected to each others. Also, considering the average diameter D (i.e., the maximum average distance) in the network with 55 nodes, we obtain D S = 42.20 and D B = 44.23, using SCARF and BA techniques, respectively.

The clustering coefficient captures the degree to which the neighbors of a given node link to each other. The local clustering coefficient gives information if two neighbors of a node link to each other, and it is one of the most important metrics for the social network analysis. Fig. 2 depicts the average clustering coefficient for different network scenarios. This is a measure of the degree to which nodes tend to cluster together. Highest performance is obtained with SCARF approach, which provides a clustering coefficient of ≈ 0.8, value that keeps increasing for higher network sizes, and represents that the neighbors of the i-th node link to each other. In contrast, lower values are experienced for the BA model, meaning that neighboring vehicles limit the clustering and to connect to each others.

IV. CONCLUSIONS

In this paper, we focused on the key role played by an effective dissemination scheme in a VSN context.

We first analyzed SCARF from a graph evolution perspective and we observed that the obtained time-evolving VSN follows a small-world behavior [START_REF] Barabási | Network Science[END_REF]. Then, we compared the performance of SCARF to the BA model. Best performance is with the SCARF technique, since the time-evolving dynamics of the vehicular social network graph allow forming a smallworld structure, where nodes tend to connect through clusters and the average distance is lower.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Average distance in different network scenarios.