text:
\section{Introduction}
\label{sec:intro}

Large self-organizing networks, such as the Internet, the World Wide
Web, social and biological networks, often exhibit power-law degrees. In
simple words, a random variable $X$ has a power-law distribution
with tail exponent $\gamma>0$ if its tail probability ${\mathbb P}(X>x)$ is
roughly proportional to $x^{-\gamma}$, for large enough $x$.
Power-law distributions are {\it heavy tailed} since the tail
probability decreases much more slowly than negative exponential, and
thus one observes extremely large values of $X$ much more frequently
than in the case of light tails. In the network context,
such networks are called \emph{scale free}, and the vertices having
huge degrees are called hubs.
Statistical analysis of complex networks characterized by
power-law degrees has received massive attention in recent literature, see
e.g.~\cite{Chakrabarti2006graph_mining,Mitzenmacher2004,Newman03survey}
for excellent surveys. Nevertheless, there still are many fundamental open problems.
One of them is how to measure \emph{dependencies} between network
parameters.

An important property of networks is the dependence between
the degrees of direct neighbours. Often, this dependence is
characterized by the \emph{assortativity coefficient} of the network,
introduced by Newman in \cite{Newm02a}. The assortativity coefficient
is the correlation coefficient between the vector of degrees on each
side of an edge, as a function of all the edges.
See \cite[Table I]{Newm02a} for a list of assortativity coefficients
for various real-world networks. The empirical data suggest that
social networks tend to be
assortative (i.e., the assortativity coefficient is
positive), while technological and biological networks tend to be
disassortative. In \cite[Table I]{Newm02a}, it is striking that,
typically, larger disassortative networks have an
assortativity coefficient that is closer to 0 and therefore
appear to have approximate {\it uncorrelated} degrees
across edges.  Similar conclusions can be drawn from \cite{Newm03d},
see in particular \cite[Table II]{Newm03d}. In this paper,
we explain this effect mathematically, and conclude that the
assortativity coefficient is \emph{not} the way to describe
dependencies between degrees on edges in the case of
scale-free networks.

Instead, we propose a solution based on the \emph{ranks} of degrees
to deal with degree-degree dependencies in networks. This rank correlation
approach is in fact standard and even classical in the area of multivariate analysis,
falling under the category of `concordance measures' - dependency measures
based on order rather than exact values of two stochastic variables.
The huge advantage of such dependency measures is that they
work well \emph{independently} of the number of finite moments
of the degrees,  while the assortativity coefficient, despite the fact that it is
always in $[-1,1]$, suffers from a strong dependence on the extreme values of
the degrees. This was already noted in the 1936 paper by H. Hotelling and M. R. Pabst~\cite{Hotelling1936}:
{\it `Certainly where there is complete absence of knowledge of the form of the bivariate distribution, and especially if it is believed not to be normal, the rank correlation coefficient is to be strongly recommended as a means of testing the existence of relationship.'}
Among recent applications of rank correlation measures, such as Spearman's rho~\cite{Spearman1904} and the closely related Kendall's tau~\cite{Kendall1938}, is measuring concordance between two rankings for a set of documents in web search. In this application field many other measures for rank distances have been proposed, see e.g.~\cite{Kumar2010rank_correlations} and references therein.

We will show in numerical experiments that statistical estimators for degree-degree dependencies, based on rank correlations, are consistent. That is, for graphs of different sizes but similar structure (e.g.\ Preferential Attachment graphs of 1.000 and 100.000 nodes, respectively), these estimators give consistent values, and the variance of the estimator decreases as the size of the graph grows. We also analytically and numerically show that the assortativity coefficient absolutely does not have this basic property when degree distributions are heavy-tailed!

The paper is organized as follows. We start with formal definitions of the sample correlation coefficient and the sample rank correlation in Section~\ref{sec-corr}. In Section~\ref{sec:linear_model} we study a model with linear dependencies and demonstrate that, when sample size grows to infinity, the sample correlation coefficient (assortativity) does not converge to a constant but rather to a random variable involving stable distributions. We also verify numerically that the rank correlation provides a consistent statistical estimator for this model. Next, in Section~\ref{sec:nonnegative} we prove that if random variables are heavy-tailed and non-negative, then the sample correlation coefficient never converges to a negative value. Thus, such sequence will never be classified as `disassortative'. We illustrate this result by an example of two nonnegative but negatively correlated random variables. This result is extended to random graphs in Section~\ref{sec-RGs}, where also numerical results are provided for assortativity coefficient and rank correlations in three different Configuration Models, and a Preferential Attachment graph.

\section{Correlations between random variables}
\label{sec-corr}
\subsection{Sample correlation coefficient}
\label{sec:corrcoef}

The assortativity coefficient is in fact a statistical estimator of a \emph{correlation coefficient} $\rho$ for the degrees on the two ends of an arbitrary edge in a graph. In this section we formally define such estimator.
The {correlation coefficient} $\rho$ for two random variables
$X$ and $Y$ with ${\rm Var}(X),{\rm Var}(Y)<\infty$ is defined by
    \[    \rho=\frac{{\mathbb E}[XY]-{\mathbb E}[X]{\mathbb E}[Y]}{\sqrt{{\rm Var}(X)}\sqrt{{\rm Var}(Y)}}.
    \]
By Cauchy-Schwarz, $\rho\in [-1,1]$, and $\rho$ measures the \emph{linear
dependence} between the random variables $X$ and $Y$.
We can approximate $\rho$ from a sample by computing the
\emph{sample correlation coefficient}
    \eqn{
    \label{rhon-def}
    \rho_n=\frac{\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X}_n)(Y_i-\bar{Y}_n)}
    {S_n(X)S_n(Y)},
    }
where
    \[
    \bar{X}_n=\frac{1}{n}\sum_{i=1}^n X_i,
    \qquad
    \bar{Y}_n=\frac{1}{n}\sum_{i=1}^n Y_i
    \]
denote the \emph{sample averages} of $(X_i)_{i=1}^n$ and
$(Y_i)_{i=1}^n$, while
    \eqn{
    \label{Sn-def}
    S_n^2(X)=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X}_n)^2,
    \qquad
    S_n^2(Y)=\frac{1}{n-1}\sum_{i=1}^n (Y_i-\bar{Y}_n)^2
    }
denote the \emph{sample variances.}

It is well known that,
again under the assumption that ${\rm Var}(X),{\rm Var}(Y)<\infty$,
the estimator $\rho_n$ of $\rho$ is \emph{consistent,}
i.e.,
\[ \rho_n\convp \rho,\]
where $\convp$ denotes convergence in probability. In practice,
however, we tend not to know whether ${\rm Var}(X),{\rm Var}(Y)<\infty$,
since $S_n^2(X)<\infty$ and $S_n^2(Y)<\infty$ clearly always
hold a.s., for any sample, and, therefore, one might be tempted to
always use the sample correlation coefficient $\rho_n$ in
\eqref{rhon-def}. In this paper we investigate what happens to
$\rho_n$ when ${\rm Var}(X),{\rm Var}(Y)=\infty$, and show that  the use of
$\rho_n$ in random graphs is uninformative, and it leads to deceptive behavior in the context
of a linear dependence, such as in \eqref{eq:X} below.

\subsection{Rank correlations}
\label{sec:spearman}

For two-dimensional data $((X_i, Y_i))_{i=1}^n$, let $r_i^{X}$ and $r_i^{Y}$ be the rank of an observation $X_i$ and $Y_i$, respectively,  when the sample values $(X_i)_{i=1}^n$ and
$(Y_i)_{i=1}^n$ are arranged in a descending order. The idea of rank correlations is in evaluating statistical dependences on the data $((r_i^X,r_i^Y))_{i=1}^n$, rather than on the original data $((X_i, Y_i))_{i=1}^n$.
Rank transformation is convenient, in particular because the two components of the resulting vector $(r^X_i,r_i^Y)$ are realisations of identical uniform distributions, implying many nice mathematical properties.

The statistical correlation coefficient for the rank is known as Spearman's rho~\cite{Spearman1904}:
\begin{equation}
\label{eq:spearman}
\rho_n^{\rm rank}=\frac{\sum_{i=1}^n(r_i^X-(n+1)/2)(r_i^Y-(n+1)/2)}{\sqrt{\sum_{i=1}^n(r_i^X-(n+1)/2)^2\sum_i^n(r_i^Y-(n+1)/2)^2}}=1-\frac{6\sum_{i=1}^nl_i^2}{n^3-n},
\end{equation}
where $l_i=r_i^X-r_i^Y$, see \cite{Hotelling1936}. The mathematical properties of the Spearman's rho have been extensively investigated in the literature. In particular, if $((X_i, Y_i))_{i=1}^n$ consists of independent realizations of $(X,Y)$, and the joint distribution function of $X$ and $Y$ is differentiable, then $\rho_n^{\rm rank}$ is a consistent estimator, and its standard deviation is of the order $1/\sqrt{n}$ (see e.g. \cite{Borkowf2002}, where exact asymptotic expressions are derived).

\section{Linear dependencies}
\label{sec:linear_model}

It is well known that $\rho$ in general measures {\it linear} dependence between two
random variables. Thus, it is natural to check how this measure, and
our proposed alternative measures, perform when the
relation between $X$ and $Y$ are described through the following linear model:
    \begin{eqnarray}
    \label{eq:X} X&=\alpha_1U_1+\cdots+\alpha_mU_m,
	\qquad
    Y=\beta_1U_1+\cdots+\beta_mU_m,
    \end{eqnarray}
where $U_j$, $j=1,\ldots,m$, are independent identically distributed
(i.i.d.) random variables with regularly varying tail, and tail exponent $\gamma$. By definition, the
random variable $U$ is {\it regularly varying} with index
$\gamma>0$, if
\eqn{
    \label{tail-U}
    {\mathbb P}(U>x)={\mathbb P}(V>x)=L(x)x^{-\gamma},
    }
where $L(x)$ is a slowly varying function, that is, for $u>0$,
$L(ux)/L(x)\to 1$ as $x\to\infty$, for instance, $L(u)$ may be
equal to a constant or $\log(u)$.
Note that the random variables $X$ and $Y$ have the same distribution when $(\beta_1,\ldots, \beta_m)$
is a permutation of $(\alpha_1,\ldots, \alpha_m)$. Our main result in this section
is the following theorem:

\begin{Theorem}[Weak convergence of correlation coefficient]
\label{thm-corr-coef}
Let $((X_i,Y_i))_{i=1}^n$ be i.i.d.\ copies of the random variables
$(X,Y)$ in \eqref{eq:X}, and where
$(U_j)_{j=1}^m$ are i.i.d.\ random variables satisfying \eqref{tail-U}
with $\gamma\in (0,2)$, so that ${\rm Var}(U_j)=\infty$.
Then,
    \eqn{
    \label{rhon-asym}
    \rho_n\convd \rho\equiv \frac{\sum_{j=1}^m \alpha_m\beta_m Z_j}{\sqrt{\sum_{j=1}^m \alpha_m^2 Z_j}
    \sqrt{\sum_{j=1}^m \beta_m^2 Z_j}},
    }
where $(Z_j)_{j=1}^m$ are i.i.d.\ random variables having stable distributions with parameter $\gamma/2\in (0,1)$,
and $\convd$ denotes convergence in distribution.
In particular,
$\rho$ has a density on $[-1,1]$, which is strictly positive on $(-1,1)$ if
there exist $k,l$ such that $\alpha_k\beta_k<0<\alpha_l\beta_l$, while the density
is positive on $(a,1)$ when $\alpha_k\beta_k\geq 0$ for every $k$, where
    {\eqn{\label{eq:a}
    a=\min_{S\subset\{1,2,\ldots,m\}, |S|\ge 2}\frac{\sum_{j=1}^m \alpha_j\beta_j {\bf 1}_{\{j\in S\}}}{\sqrt{\sum_{j=1}^m \alpha_j^2 {\bf 1}_{\{j\in S\}}}\sqrt{\sum_{j=1}^m \beta_j^2 {\bf 1}_{\{j\in S\}}}}\in (0,1).
    }}
\end{Theorem}

In order to prove the theorem we need the following technical result:

\begin{Lemma}[Asymptotics of sums in stable domain]
\label{lem-sums-stable}
Let $(U_{i,j})_{i\in [n], j\in[2]}$ be i.i.d. random variables satisfying \eqref{tail-U} for some $\gamma\in (0,2)$. Then
there exists a sequence $a_n$ with $a_n=n^{2/\gamma}\ell(n)$, where $n\mapsto \ell(n)$ is
slowly varying, such that
    \eqn{
    \label{stable-conv}
    \frac{1}{a_n}\sum_{i=1}^n U_{i,1}^2\convd Z_1,
	\qquad
	\frac{1}{a_n}\sum_{i=1}^n U_{i,1}U_{i,2}\convp 0,
    }
where $Z_1$ is stable with parameter $\gamma/2$ and $\convp$ denotes
convergence in probability.
\end{Lemma}

\proof Denote by $F$ the distribution function of $U$.
The proof of the first statement in \eqref{stable-conv} is classical when we note that
the distribution function of $U^2$ equals $u\mapsto F(\sqrt{u})$, which, by
\eqref{tail-U}, is in the domain of attraction of a stable $\gamma/2$
random variable. In particular, we can identify $a_n=[1-F]^{-1}(1/n^2)$.
To prove the second part of \eqref{stable-conv}, we write
    \eqn{
    \lbeq{[1-F]-infvar-ub}
    1-F(x)=
    {\mathbb P}(U>x)\leq c' x^{-\gamma'},\qquad x\ge 0,
    }
which is valid for any $\gamma'\in (1,\gamma)$ by \eqref{tail-U} and Potter's theorem.
We next study the distribution function of $U_1U_2$
which we denote by $H$,
where $U_1$ and $U_2$ are two independent copies of the
random variable $U$. When $F$
satisfies \eqref{[1-F]-infvar-ub}, then it is not hard to see
that there exists a $C>0$ such that
    \eqn{
    \label{1-G-bd}
    1-H(u)\leq C (1+\log{u}) u^{-\gamma'}.
    }
Indeed, assume that $F$ has a density $f(w)=c w^{-(\gamma'+1)}$, for $w\geq 1$.
Then
    $$
    1-H(u)
    =\int_1^{\infty} f(w) [1-F](u/w)dw.
    $$
Clearly, $1-F(w)= c' w^{-\gamma'}$ for $w\geq 1$ and $1-F(w)=1$ otherwise. Substitution of this
yields
    $$
    1-H(u)
    \le cc' \int_1^{u} w^{-(\gamma'+1)} (u/w)^{-\gamma'}dw+c\int_u^{\infty}
    w^{-(\gamma'+1)}\,dw
    \leq C (1+\log{u}) u^{-\gamma'}.
    $$
    When $F$ satisfies \eqref{[1-F]-infvar-ub}, then $U_1$ and $U_2$
are stochastically upper bounded by $U^*_1$ and $U^*_2$ with
distribution function $F^*$ satisfying $1-F^*(w)=c'w^{-\gamma'}\vee 1$,
where $(x\vee y)=\max\{x,y\}$, and the claim in \eqref{1-G-bd} follows from the above computation.

By the bound in \eqref{1-G-bd}, the random variables $U_{i,1}U_{i,2}$
are stochastically bounded from above by random variables $P_i$
that are in the domain of attraction of a stable $\gamma'$
random variable. As a result, there exists $b_n=n^{1/\gamma'}\ell'(n)$, where $n\mapsto \ell'(n)$ is
slowly varying, such that
    \[
    \frac{1}{b_n}\sum_{i=1}^n P_i\convd W,
    \]
where $W$ is stable $\gamma'$. {By choosing $\gamma'>\gamma/2$}, we get $b_n/a_n\to 0$, so we obtain
the second statement in \eqref{stable-conv}
\qed
\bigskip

\proof[Proof of Theorem~\ref{thm-corr-coef}]  We start by noting that
    \eqn{
    \label{rhon-def-rep}
    \rho_n=\frac{\frac{1}{n-1}\sum_{i=1}^n (X_iY_i-\bar{X}_n\bar{Y}_n)}
    {S_n(X)S_n(Y)},
    }
and
    \eqn{
    \label{Sn-def-rep}
    S_n^2(X)=\frac{1}{n-1}\sum_{i=1}^n(X_i^2-\bar{X}_n^2),
    \qquad
    S_n^2(Y)=\frac{1}{n-1}\sum_{i=1}^n (Y_i^2-\bar{Y}_n^2).
    }
We continue to identify the asymptotic behavior of
    \[
    \sum_{i=1}^nX_i^2,
    \qquad
    \sum_{i=1}^nY_i^2,
    \qquad
    \sum_{i=1}^nX_iY_i.
    \]
The distribution of  $((X_i,Y_i))_{i=1}^n$ is described in
terms of an array $(U_{i,j})_{i\in [n], j\in [m]}$, which are i.i.d.\
copies of a random variable $U$. In terms of these random variables, we can identify
    \eqn{
    \label{sum-prods}
    \sum_{i=1}^nX_iY_i
    =\sum_{j=1}^m \alpha_j\beta_j \Big(\sum_{i=1}^n U_{i,j}^2\Big)
    +\sum_{j_1\neq j_2=1}^m \alpha_{{j_1}}\beta_{{j_2}} \Big(\sum_{i=1}^n U_{i,j_1}U_{i,j_2}\Big).
    }
The sums $\sum_{i=1}^n U_{i,j}^2$ are i.i.d.,
and {by Lemma~\ref{lem-sums-stable}, $\sum_{i=1}^n U_{i,j_1}U_{i,j_2}$ is of a smaller
order. Hence, from \eqref{sum-prods} we obtain that}
    \[
    \frac{1}{a_n} \sum_{i=1}^nX_iY_i
    \convd \sum_{j=1}^m \alpha_j\beta_j Z_j.
    \]
Therefore, by
taking $\alpha=\beta$, we also obtain
    \eqn{
    \label{conv-sec-mom}
    \frac{1}{a_n} \sum_{i=1}^nX_i^2
    \convd \sum_{j=1}^m \alpha_j^2 Z_j,
    \qquad
    \frac{1}{a_n} \sum_{i=1}^nY_i^2
    \convd \sum_{j=1}^m \beta_j^2 Z_j,
    }
and these convergence hold \emph{simultaneously}. As a result, \eqref{rhon-asym}
follows. It remains to establish properties of the limiting random variable
$\rho$ in \eqref{rhon-asym}.

The density of $Z_i$ is strictly positive on $(0,\infty)$, so that
the density of $\rho$ is strictly positive on $(-1,1)$ when the sign of
$\alpha_i\beta_i$ is both positive as well as negative. When $\alpha_i\beta_i\geq 0$ for every $i$,
on the other hand, the density of $\rho$ is strictly positive on the support of $\rho$, which is $(a,1)$, where
    \eqn{
    \label{def-interval-a}
    a=\inf_{z_1, \ldots, z_m} \frac{\sum_{j=1}^m \alpha_j\beta_j z_j}{\sqrt{\sum_{j=1}^m \alpha_j^2 z_j}
    {\sqrt{\sum_{j=1}^m \beta_j^2z_j}}}\in (0,1).
    }

{Denote the function that is minimized by $a(z_1, \ldots, z_m)$. Note that rescaling $z_j=cz_j$, $j=1,\ldots,m$, does not change the value of $a(z_1, \ldots, z_m)$. In particular, we can choose $c=(\max\{z_1,z_2,\ldots,z_m\})^{-1}$. Thus, without loss of generality, we can assume that $z_j\in (0,1]$, $j=1,2,\ldots,m$  and $z_k=1$ for at least one $k=1,2,\ldots, m$. In that case, $a$ is a continuous function of $z_j\in [0,1]$, $j\ne k$. Taking a derivative of $a$ with respect to $z_j$ we obtain that the sign of the derivative is defined by the sign of the expression{\sloppy

}
	\eqn{\label{eq:a-sign}
	-a(z_1, \ldots, z_m)(\alpha_j^2+\beta_j^2)+2\alpha_j\beta_j.
	}
Since (\ref{eq:a-sign}) is decreasing in $a$, the derivative of $a(z_1, \ldots, z_m)$ w.r.t.\
$z_j$ cannot equal zero. (Indeed, if (\ref{eq:a-sign}) is zero in some point $z_j^*$, then (\ref{eq:a-sign}) is {\it positive} on $(z_j^*, z_j^*+\varepsilon)$ for some small $\varepsilon$ only if $a$ is {\it decreasing} on $(z_j^*, z_j^*+\varepsilon)$, thus, we obtain a contradiction.) We conclude that $a$ achieves its minimum when all $z_j$'s equal either zero or one, and at least one of the values must equal one. Finally, if only one value $z_j$ is equal to one and the rest are equal to zero, then we obtain $a=1$, which is a maximal possible value of $a$. Thus, at least two values of $z_j$ must equal one.{\sloppy

}
}
\qed

To illustrate the result of Theorem \ref{thm-corr-coef}, consider the example
with $U_j$'s
from a Pareto distribution satisfying ${\mathbb P}(U>x)=1/x^{1.1}$, $x\ge 1$, so $L(x)=1$ and $\gamma=1.1$
in \eqref{tail-U}. The exponent $\gamma=1.1$ is as observed for the World Wide Web~\cite{Broder00}. In \eqref{eq:X}, we choose $m=3$ and
$\alpha_i$, $\beta_i$, $i=1,2,3$, as specified in Table~\ref{tab:linear_model}.
We generate $N$ data samples $((X_i,Y_i))_{i=1}^n$ and compute $\rho_n$ and $\rho_n^{\rm rank}$ for each of the $N$ samples.
Thus, we obtain the vectors $(\rho_{n,j})_{j=1}^N$ and $(\rho^{\rm rank}_{n,j})_{j=1}^N$ of $N$ independent realisations for $\rho_n$ and $\rho_n^{\rm rank}$, respectively, where the sub-index $j=1,\ldots,N$ denotes the $j$th realization of $((X_i,Y_i))_{i=1}^n$. We then compute
	\begin{align}
	\label{eq:EN}
	{\mathbb E}_N(\rho_n)=\frac{1}{N}\sum_{j=1}^N \rho_{n,j},
    	\quad &{\mathbb E}_N(\rho^{\rm rank}_n)=\frac{1}{N}\sum_{j=1}^N \rho^{\rm rank}_{n,j};\\
	\label{eq:sigmaN}
	\sigma_N(\rho_n)=\sqrt{\frac{1}{N-1}\sum_{j=1}^N (\rho_{n,j}-{\mathbb E}_N(\rho_n))^2},\quad
	&\sigma_N(\rho^{\rm rank}_n)=\sqrt{\frac{1}{N-1}\sum_{j=1}^N (\rho^{\rm rank}_{n,j}-
	{\mathbb E}_N(\rho^{\rm rank}_n))^2}.
	\end{align}

The results are presented in Table~\ref{tab:linear_model}. We clearly see that $\rho_n$ has a significant standard deviation, of which estimators are similar for different values of $n$. This means that in the limit as $n\to\infty$, $\rho_n$ is a random variable with a significant spread in its values, as stated in Theorem~\ref{thm-corr-coef}. Thus, by evaluating $\rho_n$ for one sample $((X_i,Y_i))_{i=1}^n$ we will obtain a random number, even when $n$ is huge.
The convergence to a non-trivial distribution is directly
seen in Figure~\ref{fig:rho} because the plots for the two values of $n$ almost coincide. Note
that in all cases, the density is fairly uniform, ensuring a comparable probability
for all feasible values and rendering the value obtained in a specific realization
even more uninformative.
\begin{table}[htb]%
{\small\centerline{
\begin{tabular}{|c|l|c|c|c|c|}
\hline
&$N$&$10^3$&\multicolumn{3}{|c|}{$10^2$}\\
\cline{2-6}
Model parameters&$n$&$10^2$&$10^3$&$10^4$&$10^5$\\
\hline
&${\mathbb E}_N(\rho_n)$&0.4395&0.4365&0.4458&0.4067\\
$\alpha=(1/2,1/2,0)$&$\sigma_N(\rho_n)$&0.3399&0.3143&0.3175&0.3106\\
\cline{2-6}
$\beta=(0,1/2,1/2)$&${\mathbb E}_N(\rho^{\rm rank}_n)$&0.4508&0.4485&0.4504&0.4519\\
&$\sigma_N(\rho^{\rm rank}_n)$&0.0922&0.0293&0.0091&0.0033\\
\hline
&${\mathbb E}_N(\rho_n)$&0.8251&0.7986&0.8289&0.8070\\
$\alpha=(1/2,1/3,1/6)$&$\sigma_N(\rho_n)$&0.1151&0.1125&0.1108&0.1130\\
\cline{2-6}
$\beta=(1/6,1/3,1/2)$&${\mathbb E}_N(\rho^{\rm rank}_n)$&0.8800&0.8850&0.8858&0.8856\\
&$\sigma_N(\rho^{\rm rank}_n)$&0.0248&0.0073&0.0023&0.0007\\
\hline
&${\mathbb E}_N(\rho_n)$&-0.3052&-0.3386&-0.3670&-0.3203\\
$\alpha=(1/2,-1/3,1/6)$&$\sigma_N(\rho_n)$&0.6087&0.5841&0.5592&0.5785\\
\cline{2-6}
$\beta=(1/6,1/2,-1/3)$&${\mathbb E}_N(\rho^{\rm rank}_n)$&-0.3448&-0.3513&-0.3503&-0.3517\\
&$\sigma_N(\rho^{\rm rank}_n)$&0.1202&0.0393&0.0120&0.0034\\
\hline
\end{tabular}}
\caption{\small Estimated mean and standard deviation of $\rho_n$ and $\rho^{\rm rank}_n$ in $N$ samples with linear dependence (\ref{eq:X}).}
\label{tab:linear_model}
}
\end{table}

\begin{figure}[ht]
\centerline{\includegraphics[height=1.2in]{corr1-N.jpg}\hspace{.1in}
\includegraphics[height=1.2in]{corr2-N.jpg}\hspace{.1in}
\includegraphics[height=1.2in]{corr3-N.jpg}
}
\caption{\small The empirical distribution function $F_N(x)={\mathbb P}(\rho_n\le x)$ for the $N=1.000$ observed values of $\rho_n$ ($n=1.000$, $n=10.000$), in the case of linear dependence \eqref{eq:X}.}
\label{fig:rho}
\end{figure}

On the other hand, from Table~\ref{tab:linear_model} we clearly see that the behaviour of rank correlations is exactly as we can expect from a good statistical estimator.  The obtained average values are consistent while the standard deviation of $\rho_n^{\rm rank}$ decreases approximately as $1/\sqrt{n}$ as $n$ grows large.
Therefore, $\rho_n^{\rm rank}$ converges to a deterministic number.

\section{Sample correlation coefficient for non-negative variables}
\label{sec:nonnegative}
In this section, we investigate correlations between non-negative heavy-tailed random variables. Our main result in this section shows that the correlation coefficient is asymptotically non-negative:

\begin{Theorem}[Asymptotic non-negativity of correlation coefficient for positive r.v.'s]
\label{thm-corr-coef-pos}
Let $((X_i,Y_i))_{i=1}^n$ be i.i.d.\ copies of non-negative random variables
$(X,Y)$, where $X$ and $Y$ satisfy
    \eqn{
    \label{tail-XY}
    {\mathbb P}(X>x)=L_{\scriptscriptstyle X}(x)x^{-\gamma_{\scriptscriptstyle X}},
    \qquad
    {\mathbb P}(Y>y)=L_{\scriptscriptstyle Y} (y)y^{-\gamma_{\scriptscriptstyle Y}},\qquad x,y\ge 0,
    }
with $\gamma_{\scriptscriptstyle X}, \gamma_{\scriptscriptstyle Y}\in (0,2)$, so that ${\rm Var}(X)={\rm Var}(Y)=\infty$.
Then, any limit point of the sample correlation coefficient is non-negative.
\end{Theorem}

We illustrate Theorem \ref{thm-corr-coef-pos} with a useful example. Let $(U_i)_{i=1}^n$
be a sequence of i.i.d.\ random variables satisfying \eqref{tail-U} for some $\gamma\in (0,2)$,
and where $U\geq 0$ a.s. Let $(X,Y)=(0,2U)$ with probability $1/2$ and
$(X,Y)=(2U,0)$ with probability $1/2$. Then, $XY=0$ a.s., while ${\mathbb E}[X]={\mathbb E}[Y]={\mathbb E}[U]$
and ${\rm Var}(X)={\rm Var}(Y)=2{\mathbb E}[U^2]-{\mathbb E}[U]^2=2{\rm Var}(U)+{\mathbb E}[U]^2$
Therefore, if ${\rm Var}(U)<\infty$,
    \eqn{
    \label{rhon-pos-fin-var}
    \rho_n\convp \rho=-\frac{{\mathbb E}[U]^2}{2{\rm Var}(U)+{\mathbb E}[U]^2}
    \in (-1,0).
    }
The asymptotics in \eqref{rhon-pos-fin-var} is quite reasonable, since the random
variables $(X,Y)$ are highly negatively dependent: When $X>0$, $Y$ must be equal to 0,
and vice versa. Instead, when $(U_i)_{i=1}^n$
is a sequence of i.i.d.\ random variables satisfying \eqref{tail-U} for some $\gamma\in (0,2)$,
and where $U\geq 0$ a.s., then $\rho_n\convp0$, which is not what we would like.

Table~\ref{tab:non-neg} shows the empirical mean and standard deviation of the estimators $\rho_n$ and $\rho_n^{\rm rank}$. Here ${\mathbb P}(U>x)=x^{-1.1}$, $x\ge 1$, as in Table~\ref{tab:linear_model}.
As predicted by Theorem~\ref{thm-corr-coef-pos}, the sample correlation coefficient (assortativity) converges to zero as $n$ grows large, while $\rho_n^{\rm rank}$ consistently shows a clear negative dependence, and the precision of the estimator improves as $n\to\infty$.
This explains why strong disassortativity is not observed in large samples of power-law data.

\begin{table}%
{
{\small\centerline{
\begin{tabular}{|l|c|c|c|c|c|}
\hline
$N$&\multicolumn{2}{|c|}{$10^3$}&\multicolumn{3}{|c|}{$10^2$}\\
\hline
$n$&$10$&$10^2$&$10^3$&$10^4$&$10^5$\\
\hline
${\mathbb E}_N(\rho_n)$&-0.4833&-0.1363&-0.0342&-0.0077&-0.0015\\
$\sigma_N(\rho_n)$&0.1762&0.0821&0.0245&0.0064&0.0011\\
\hline
${\mathbb E}_N(\rho^{\rm rank}_n)$&-0.6814&-0.4508&-0.4485&-0.4504&-0.4519\\
$\sigma_N(\rho^{\rm rank}_n)$&0.1580&0.0283&0.0082&0.0024&0.0007\\
\hline
\end{tabular}}}
\caption{\small The  mean and standard deviation of $\rho_n$ and $\rho_n^{\rm rank}$ in $N$ simulations of $((X_i,Y_i))_{i=1}^n$, where $X=2UI$, $Y=2U(1-I)$, $I$ is a Bernoulli$(1/2)$ random variable, ${\mathbb P}(U>x)=x^{-1.1}$, $x\ge 1$.}
\label{tab:non-neg}}
\end{table}
\medskip

\noindent
{We next prove Theorem \ref{thm-corr-coef-pos}:\\
{\it Proof of Theorem \ref{thm-corr-coef-pos}.}
Clearly $\sum_{i=1}^nX_iY_i\geq 0$ when $X_i\geq 0, Y_i\geq 0$, so that
    \[
    \rho_n\geq -\frac{\frac{1}{n-1}\sum_{i=1}^n\bar{X}_n\bar{Y}_n}
    {S_n(X)S_n(Y)}
    =-\frac{n}{n-1} \frac{\bar{X}_n}{S_n(X)}\frac{\bar{Y}_n}{S_n(Y)}.
    \]
It remains to show that if ${\rm Var}(X)=\infty$, then $\bar{X}_n/S_n(X)\convp 0$. Indeed, if $\gamma\in (1,2)$ then $\bar{X}_n\convp {\mathbb E}[X]<\infty$ by the strong
law of large numbers, and  from \eqref{tail-XY}, (\ref{Sn-def}) and Lemma~\ref{stable-conv} it follows that
$S_n(X)=n^{2/\gamma_{\scriptscriptstyle X}+o(1)-1}\to\infty$ as $n\to\infty$. When $\gamma\in(0,1]$, instead, $\bar{X}_n=n^{1/\gamma_{\scriptscriptstyle X}-1+o(1)}$,
so that $\bar{X}_n/S_n(X)=n^{-1/\gamma_{\scriptscriptstyle X}+o(1)}\convp 0.$
This proves the claim.}
\qed

\section{Applications to random graph models}
\label{sec-RGs}
The correlation coefficient is particularly important
in the setting of degree-degree correlations in real-world networks. Let $G=(V,E)$
be a graph with vertex set $V$ and edge set $E$.
The assortativity coefficient of $G$ is equal to
(see, e.g., \cite[(4)]{Newm02a})
    \[
    \rho(G)=\frac{\frac{1}{|E|}\sum_{ij\in E} D_iD_j-\Big(\frac{1}{|E|} \sum_{ij\in E} \tfrac12 (D_i+D_j)\Big)^2}
    {\frac{1}{|E|}\sum_{ij\in E} \tfrac12(D_i^2+D_j^2) -\Big(\frac{1}{|E|} \sum_{ij\in E} \tfrac12 (D_i+D_j)\Big)^2},
    \]
where the sum is over directed edges of $G$, and $D_i$ is the degree of vertex $i$, i.e., $ij$ and $ji$ are two distinct edges. The assortativity coefficient is equal to the correlation
coefficient of the sequence of random variables $((D_i, D_j))_{ij\in E}$. Thus, the assortativity
coefficient is the correlation coefficient between two sequences of non-negative
random variables, as studied in Theorem \ref{thm-corr-coef-pos}. We refer to
\cite{Newm10a} for an extensive introduction to networks, their empirical properties and
models for them.

{This section is organized as follows. In Section \ref{sec-no-disass-RG-seq}
we show that all limit points of the assortativity coefficients for sequences of growing scale-free
random graphs with power-law exponent $\gamma<3$ are non-negative,
a result that is similar in spirit to Theorem \ref{thm-corr-coef-pos}.
We highlight this statement by presenting theoretical and numerical results for
several random graph examples where
the assortativity coefficient yields unexpected and unwanted results.
In Section \ref{sec:random_assortativity} we present an example
of a sequence of random graphs where the assortativity coefficient converges to a
proper random variable, as observed in the i.i.d.\ setting in Theorem \ref{thm-corr-coef}.
}

\subsection{No disassortative scale-free random graph sequences}
\label{sec-no-disass-RG-seq}
{We compute that
    \[
    \frac{1}{|E|} \sum_{ij\in E} \tfrac12 (D_i+D_j)
    =\frac{1}{|E|} \sum_{i\in V} D_i^2,
    \qquad
    \frac{1}{|E|}\sum_{ij\in E} \tfrac12(D_i^2+D_j^2)
    =\frac{1}{|E|} \sum_{i\in V} D_i^3.
    \]}
Thus, $\rho(G)$ can be written as
    \eqn{
    \label{eq:rhoG}
    \rho(G)=\frac{\sum_{ij\in E} D_iD_j-\frac{1}{|E|} \Big(\sum_{i\in V} D_i^2\Big)^2}
    {\sum_{i\in V} D_i^3 -\frac{1}{|E|} \Big(\sum_{i\in V} D_i^2\Big)^2}.
    }

Consider a sequence of graphs $(G_n)_{n\geq 1}$, where
$n$ denotes the number of vertices $n=|V|$ in the graph. Since many real-world networks
are quite large, we are interested in the behavior of
$\rho(G_n)$ as $n\rightarrow \infty$. Note that this
discussion applies both to sequences of real-world networks
of increasing size, as well as to graph sequences of random
graphs. We start by generalizing Theorem \ref{thm-corr-coef-pos}
to this setting:

\begin{Theorem}[Asymptotic non-disassortativity of scale-free graphs]
\label{thm-corr-coef-pos-scale-free}
{ Let $(G_n)_{n\geq 1}$  be a sequence of graphs of size $n$
satisfying that there exist $\gamma\in (1,3)$ and $0<c<C<\infty$ such that
$cn\leq |E|\leq Cn$,  $c n^{1/\gamma}\le \max_{i\in [n]} D_i \leq C n^{1/\gamma}$ and $cn^{(2/\gamma)\vee 1}\leq \sum_{i\in [n]} D_i^2 \leq Cn^{(2/\gamma)\vee 1}$.
Then, any limit point of the assortativity coefficients
$\rho(G_n)$ is non-negative.}
\end{Theorem}

\proof
We note that $D_i\geq 0$ for every $i\in V$,
so that, from (\ref{eq:rhoG})
    \[
    \rho(G_n)\geq \underline{\rho}(G_n)\equiv
    -\frac{\frac{1}{|E|}\Big(\sum_{i\in V} D_i^2\Big)^2}
    {\sum_{i\in V} D_i^3-\frac{1}{|E|}\Big(\sum_{i\in V} D_i^2\Big)^2}.
    \]
By assumption, $\sum_{i\in V} D_i^3\geq (\max_{i\in [n]} D_i)^3
\geq c^3 n^{3/\gamma}$, whereas $\frac{1}{|E|}\Big(\sum_{i\in V} D_i^2\Big)^2
\leq (C^2/c) n^{2(2/\gamma\vee 1)-1}=(C^2/c) n^{[(4/\gamma-1)\vee 1]}$. Since $\gamma\in (1,3)$ we have $(4/\gamma-1)\vee 1<3/\gamma$, so that
	\[
	\frac{\sum_{i\in V} D_i^3}{\frac{1}{|E|}\Big(\sum_{i\in V} D_i^2\Big)^2}
	\rightarrow \infty.
	\]
This proves the claim.
\qed
\medskip

In the literature, many examples are reported of real-world networks
where the degree distribution obeys a power law (see \cite{AlbBar02,Newman03survey}
for surveys of real-world networks and their degree properties).
In particular, for scale-free networks, the proportion $p_k$ of vertices of degree
$k$ is close to $p_k\approx ck^{-\gamma-1}$, and most values of $\gamma$ reported in the literature
are in $(1,3)$, see e.g., \cite[Table I]{AlbBar02}
or \cite[Table I]{Newman03survey}. When this is the case, we
can expect that
    \[
    |E|= \sum_{i\in V} D_i\sim \mu n,
    \]
where $\mu={\mathbb E}[D]$, while $\max_{i\in V}D_i\sim n^{1/\gamma}$, and
    \[
    \frac{1}{n} \sum_{i\in V} D_i^p
    \sim
    \begin{cases}
    \mu_p &\text{when }\gamma>p,\\
    c n^{p/\gamma-1} &\text{when }\gamma<p,
    \end{cases}
    \]
where $\mu_p={\mathbb E}[D^p]$.
In particular, the conditions of Theorem \ref{thm-corr-coef-pos-scale-free} hold and
$\underline{\rho}(G_n)\rightarrow 0 \quad \text{when }\gamma<3.$
Thus, the asymptotic degree-degree correlation of the graph sequence
$(G_n)_{n\geq 1}$ is non-negative. As a result,
there exist no disassortative scale-free graph
sequences.

We next consider four random graph models to highlight our result. In the remainder of this section we first describe three models: Configuration model, Configuration model with intermediate vertices, and Preferential Attachment model. Then we present the numerical results for these models in Table~\ref{tab:RG}. As we see from the results, in all these models assortativity converges to zero as $n$ grows.

\paragraph{The configuration model.}
The \emph{configuration model} was invented by
Bollob\'as in \cite{Boll80b}, inspired by
\cite{BenCan78}. Its connectivity structure
was first studied by Molloy and Reed \cite{MolRee95,MolRee98}.
It was popularized by Newman, Srogatz and Watts \cite{NewStrWat00}, who realized
that it is a useful and simple model for real-world networks.
Given a {\it degree sequence}, namely a sequence of $n$
positive integers ${\boldsymbol d} = (d_1,d_2,\ldots, d_n)$
with $\ell_n=\sum_{i\in [n]} d_i$ assumed to be even, the configuration model (CM) on $n$
vertices and degree sequence ${\boldsymbol d}$ is constructed as follows:

Start with $n$ vertices and $d_i$ half-edges adjacent to vertex $i$. The graph is constructed by randomly pairing each half-edge to some other half-edge to form an edge. Number the half-edges from $1$ to
$\ell_n$ in some arbitrary order. Then, at each step,
two half-edges that are not already paired are chosen uniformly at random among
all the unpaired half-edges and are paired to form a single edge
in the graph. These half-edges are removed from the list of
unpaired half-edges. We continue with this procedure of choosing and pairing two
unpaired half-edges
until all the half-edges are paired. Although self-loops may occur,
these become rare as $n\to\infty$ (see e.g.\ \cite{Boll01} or \cite{Jans06b}
for more precise results in this direction). We consider both
the cases where the self-loops are removed and we collapse multiple edges to
a single edge, as well as the setting where we keep the self-loops and
multiple edges. As we will see in the simulations, these two cases are
qualitatively similar.

We investigate the CM where the degrees are i.i.d.\ random variables, and note that
the probability that two vertices are directly connected is close to
$d_id_j/\ell_n$. Since this is of product form in $i$ and $j$, the degrees at either end
of an edge are close to being independent, and in fact are asymptotically independent.
Therefore, one expects
the assortativity coefficient of
the configuration model to converge to 0 in probability, irrespective of
the degree distribution.

\paragraph{Configuration model with intermediate vertices.}
We next adapt the configuration model slightly, by
replacing every edge by two edges that meet at a middle vertex.
Denote this graph by $\bar{G}_n=(\bar{V}_n, \bar{E}_n)$, while the
configuration model is $G_n=(V_n,E_n)$. In this model,
there are $n+\ell_n/2$ vertices and $2\ell_n$ edges (recall that $ij$ and $ji$ are two different edges).
For $st\in \bar{E}_n$, the degree of either vertex $s$ or vertex $t$ equals
2, and the degree of the other vertex in the edge is equal to $D_i$, where $i$ is the unique
vertex in the original configuration model that corresponds to
$s$ or $t$. Therefore,
    \[
    \frac{1}{|\bar{E}_n|}\sum_{st\in \bar{E}_n} \bar{D}_s\bar{D}_t
    =\frac{2}{\ell_n}\sum_{i\in V_n} D_i^2,
    \]
and for $p\ge 2$,
    \[
    \frac{1}{|\bar{E}_n|}\sum_{s\in \bar{V}_n} \bar{D}_s^p
    =\frac{1}{2\ell_n}2^p(\ell_n/2)+\frac{1}{2\ell_n}\sum_{i\in V_n}D_i^{p}
	=2^{p-2}+\frac{1}{2\ell_n}\sum_{i\in V_n}D_i^{p},
    \]
where $\mu_p={\mathbb E}[D^p]$.
As a result, when $\gamma>3$,
    \[
    \rho(\bar{G}_n)\convp \frac{2\mu_2/\mu_1-(1+\mu_2/(2\mu_1))^2}
	{(2+\mu_3/(2\mu_1))-(1+\mu_2/(2\mu_1))^2}
    <0.
    \]
The fact that the degree-degree correlation is negative is quite reasonable,
since in this model, vertices of high degree are only connected
to vertices of degree 2, so that there is negative dependence
between the degrees at either end of an edge.
When $\gamma<3$, on the other hand,  $\mu_3={\mathbb E}[D^3]=\infty$, and thus
    \[
    \rho(\bar{G}_n)\convp 0,
    \]
which is inappropriate, as the negative dependence of
the degrees persists.

\paragraph{Preferential Attachment model.} We consider the basic version of the undirected Preferential Attachment model (PAM), where each new vertex adds only one edge to the network, connecting to the existing nodes with probability proportional to their degrees.  In this case, it is well known that $\gamma=2$
(see e.g., \cite{BarAlb99} or \cite{BolRioSpeTus01}). We see that the assortativity
converges to zero, as indicated in Theorem~\ref{thm-corr-coef-pos-scale-free},
while Spearman's rank correlation indicates that degrees are negatively
dependent. This can be understood by noting that the majority of edges of vertices
with high degrees, which are old vertices,  come from vertices which are added
late in the graph growth process and thus have small degree. On the other hand,
by the growth mechanism of the PAM,  vertices with low degree are
more likely to be connected to vertices having high degree, which indeed suggests
negative degree-degree dependencies.

\paragraph{Numerical results.} To illustrate our results, we have generated the two configuration models and the Preferential Attachment model of different sizes. For fair comparison, we chose $\gamma=3$ for the configuration model: ${\mathbb P}(D\ge x)=x^{-2}$, $x\ge 1$. Since in the configuration model self-loops and multiple edges are possible, we considered two versions: the original model with self-loops and double edges present, and the model where self-loops and double-edges are removed.

The rank correlation coefficient $\rho^{\rm rank}(G)$ is computed using (\ref{eq:spearman}) as follows. We define the random variables $X$ and $Y$ as the degrees on two ends of a random {\it undirected} edge in a graph (that is, when rank correlations are computed, $ij$ and $ji$ represent the same edge). For each edge, when the observed degrees are $a$ and $b$, we assign $[X=a,Y=b]$ or $[X=b,Y=a]$ with probability 1/2. Furthermore, many values of $X$ and $Y$ will be the same because a degree $d$ will appear at the end of an edge $d$ times. We resolve the draws by adding independent random variables, uniformly distributed on $[0,1]$, to each value of $X$ and $Y$. The results are presented in Table~\ref{tab:RG}.
\begin{table}[htb]%
{\small\centerline{
\begin{tabular}{|c|l|c|c|c|c|}
\hline
&$N$&$10^3$&\multicolumn{2}{|c|}{$10^2$}&$10$\\
\cline{2-6}
Model&$n$&$10^2$&$10^3$&$10^4$&$10^5$\\
\hline
&${\mathbb E}_N(\rho(G_n))$&0.0021&-0.0013&0.0001&-0.0003\\
Configuration model&$\sigma_N(\rho(G_n))$&0.0672&0.0212&0.0068&0.0024\\
\cline{2-6}
with self-loops and double edges&${\mathbb E}_N(\rho^{\rm rank}_n)$&0.0012&-0.0010&-0.0002&-0.0002\\
&$\sigma_N(\rho^{\rm rank}(G_n))$&0.0656&0.0202&0.0066&0.0014\\
\hline
&${\mathbb E}_N(\rho(G_n))$&-0.0785&-0.0346&-0.0115&-0.0046\\
Configuration model&$\sigma_N(\rho(G_n))$&0.0686&0.0274&0.0102&0.0039\\
\cline{2-6}
without self-loops and double edges&${\mathbb E}_N(\rho^{\rm rank}(G_n))$&-0.0615&-0.0151&-0.0040&-0.0002\\
&$\sigma_N(\rho^{\rm rank}(G_n))$&0.0836&0.0337&0.0075&0.0024\\
\hline
&${\mathbb E}_N(\rho(\bar{G}_n))$&-0.2589&-0.1243&-0.0587&-0.0303\\
Configuration model&$\sigma_N(\rho(\bar{G}_n))$&0.0872&0.0509&0.0255&0.0189\\
\cline{2-6}
with intermediate vertices&$\displaystyle{{\mathbb E}_N(\rho^{\rm rank}(\bar{G}_n))}$&-0.7482&-0.7499&-0.7498&-0.7501\\
&$\sigma_N(\rho^{\rm rank}(\bar{G}_n))$&0.0121&0.0036&0.0011&0.0006\\
\hline
&${\mathbb E}_N(\rho(G_n))$&-0.2597&-0.1302&-0.0607&-0.0294\\
Preferential attachment&$\sigma_N(\rho(G_n))$&0.0550&0.0261&0.0127&0.0088\\
\cline{2-6}
&${\mathbb E}_N(\rho^{\rm rank}(G_n))$&-0.4167&-0.4151&-0.4166&-0.4158\\
&$\sigma_N(\rho^{\rm rank}(G_n))$&0.0695&0.0202&0.0066&0.0022\\
\hline
\end{tabular}}
\caption{\small Estimated mean and standard deviation of $\rho(G)$ and $\rho^{\rm rank}(G)$ in random graphs.}
\label{tab:RG}
}
\end{table}

Within the same model, the graphs of different sizes are constructed by the same algorithm. Thus, their mixing patterns are exactly the same. As we predicted, the assortativity reduces in absolute value with the graph size, resulting in asymptotically neutral mixing for all models. On the contrary, the rank correlation coefficient consistently shows neutral mixing for the configuration model, moderately disassortative mixing for the Preferential Attachment graph, and strongly disassortative mixing for the configuration model with intermediate edges.

{
\subsection{Random graphs with asymptotically random assortativity}
\label{sec:random_assortativity}

Finally, we discuss the possibility that $\rho(G_n)$ in (\ref{eq:rhoG}) converges to a random
variable when the number of vertices tends to infinity.
Under the assumptions of Theorem~\ref{thm-corr-coef-pos-scale-free}, we have that
	\begin{align}
	\label{eq:crossproducts1}
	\sum_{ij\in E} D_iD_j&
	\le \max_{i\in[n]}D_i\sum_{ij\in E_n} D_i
	=\max_{i\in[n]}D_i\Big(\sum_{i\in V_n} D^2_i\Big)
	\le C^2n^{1/\gamma+(2/\gamma\vee 1)},\\
	\label{eq:crossproducts2}
	\sum_{ij\in E} D_iD_j&\ge \max_{i\in[n]}D_i\ge c n^{1/\gamma},\\
	\label{eq:crossproducts3}
	\sum_{ij\in E} D_iD_j
	&\ge\sum_{i\in V_n} D^2_i\ge  cn^{2/\gamma\vee 1}.
	\end{align}
Further, from the proof of Theorem~\ref{thm-corr-coef-pos-scale-free}, we know that
	\eqan{
	\label{eq:cubes}\sum_{i\in V} D_i^3&\geq (\max_{i\in [n]} D_i)^3
	\geq c^3 n^{3/\gamma},
	}
and
	\eqan{
	\label{eq:squares}
	\frac{1}{|E|}\Big(\sum_{i\in V} D_i^2\Big)^2
	&\leq (C^2/c) n^{(4/\gamma-1)\vee 1},
	}
where we see that (\ref{eq:squares}) is vanishing compared to (\ref{eq:cubes}).
The convergence of (\ref{eq:rhoG}) to a random variable can only take place if
the crossproducts on the left-hand side of (\ref{eq:crossproducts1} -- \ref{eq:crossproducts3})
are of the same order of magnitude as the left-hand side of (\ref{eq:cubes}).
As we see from the above, this is possible for $\gamma\in (1,3)$. However,
the convergence will be slow because an easy calculation shows that the maximal
difference in the order of magnitude between the right-hand sides of
(\ref{eq:crossproducts1}) and (\ref{eq:squares}) is $n^{1/2}$.

Below we present an example where we prove that $\rho(G_n)$ indeed
converges to a random variable, and illustrate numerically how the
distribution of  $\rho(G_n)$ changes as $n$ grows large. However,
due to the slow convergence, a substantially larger computational
capacity is needed in order to (almost) achieve the limiting distribution.

\paragraph{A collection of complete bipartite graphs.}
Take $((X_i,Y_i))_{i=1}^n$ to be an i.i.d.\ sample of random
variables as in \eqref{eq:X}, where $\alpha_1=\alpha_2=\beta_1=b$,
$\beta_2=ab$ for some $b>0$ and $a>1$.
Then, for $i=1, \ldots, n$, we create a complete bipartite graph of
$X_i$ and $Y_i$ vertices, respectively. These $n$ complete bipartite graphs
are not connected to one another. We denote such collection of $n$ bipartite
graphs by $G_n$. The graph $G_n$  has
$|V|=\sum_{i=1}^n (X_i+Y_i)$ vertices and $|E|=2\sum_{i=1}^n X_iY_i$ edges.
Further,
    	\[
    	\sum_{i\in V} D_i^p
    	=\sum_{i=1}^n (X_i^pY_i+Y_i^pX_i),
	\qquad
	\sum_{ij\in E}D_iD_j=2\sum_{i=1}^n (X_iY_i)^2.
    	\]
Assume that the $U_j$ satisfy \eqref{tail-U} with $\gamma\in(3,4)$, so that ${\mathbb E}[U^3]<\infty$,
but ${\mathbb E}[U^4]=\infty$. As a result, $|E|/n \convp 2{\mathbb E}[XY]<\infty$
and $\frac{1}{n} \sum_{i\in V} D_i^2\convp {\mathbb E}[XY(X+Y)]<\infty$. Further,
	\[
	n^{-4/\gamma}b^{-4}\sum_{i=1}^n (X_i^3Y_i+Y_i^3X_i)
	\convd (a^3+a) Z_1+ 2Z_2,
	\qquad
	n^{-4/\gamma}b^{-4}\sum_{i=1}^N (X_iY_i)^2
	\convd a^2 Z_1+Z_2,
	\]
where $Z_1$ and $Z_2$ and two independent stable distributions with parameter $\gamma/4$.
As a result,
	\[
	\rho(G_n)\convd \frac{2a^2Z_1+2Z_2}{(a+a^3)Z_1+2Z_2},\; \mbox{as $n\to\infty$}.
	\]
which is a proper random variable taking values in $(2a/(1+a^2), 1)$.

In Table~\ref{tab:bipartite} we present numerical results for
$\rho(G_n)$ and $\rho^{\rm rank}(G_n)$. Here we choose
$b=1/2$, $a=2$, and $U$ has a generalized Pareto distribution
${\mathbb P}(U>x)=((1.8+x)/2.8)^{-2.8}$, $x> 1$.
	\begin{table}[htb]%
	{\small\centerline{
	\begin{tabular}{|l|c|c|c|c|}
	\hline
	$n$&$10^2$&$10^3$&$10^4$&$10^5$\\
	\hline
	${\mathbb E}_N(\rho(G_n))$&0.6855&0.7293&0.7877&0.8224\\
	$\sigma_N(\rho(G_n))$&0.1389&0.0681&0.0614&0.0629\\
	\hline
	${\mathbb E}_N(\rho^{\rm rank}(G_n))$&0.7556&0.8370&0.8577&0.8641\\
	$\sigma_N(\rho^{\rm rank}(G_n))$&0.0791&0.0379&0.0247&0.0128\\
	\hline
	\end{tabular}}
\caption{\small Estimated mean and standard deviation of $\rho(G_n)$ and $\rho^{\rm rank}(G_n)$ for the collection of $n$ complete bipartite graphs. The number of realizations for
each graph size is $N=100$.}
\label{tab:bipartite}
}
\end{table}

Note that in this model there is a genuine dependence between the correlation
measure and the graph size. Indeed, if $n=1$ then the assortativity coefficient
equals $-1$ because nodes with larger degrees are connected to nodes with
smaller degrees. However, when the graph size grows, the positive correlations
start dominating because of the positive linear dependence between $X$ and $Y$.
We see that again the rank correlation captures the relation faster and gives
consistent results with decreasing dispersion of values. Finally, Figure~\ref{fig:bipartite}
shows the changes in the empirical distribution of $\rho(G_n)$ as $n$ grows.
	\begin{figure}[ht]
	\centerline{\includegraphics[height=3in]{bi-partite2-8.jpg}}
	\caption{\small The empirical distribution function ${\mathbb P}(\rho(G_n)\le x)$ for the
	$N=100$ observed values of $\rho(G_n)$, where $G_n$ is a collection of $n$
	complete bipartite graphs.}
	\label{fig:bipartite}
	\end{figure}
It is clear that a part of the probability mass is spread over the interval $(0.8,1)$. In the limit,
$\rho(G_n)$ has a non-zero density on this interval. The difference between the
crossproducts and the expectation squared in $\rho(G_n)$ is only of the order
$n^{1/\gamma}$, which is $n^{1/2.8}$ in our example, thus, the convergence is
too slow to observe it at $n=100.000$.
}

\section{Discussion}
\label{sec:discussion}
In this paper, we have investigated dependency measures for power-law random variables.
We have argued that the correlation coefficient, despite its appealing feature that it is always in $[-1,1]$, is inappropriate to describe dependencies between heavy-tailed random variables
since it yields insensible results. Indeed, the two main problems with the
sample correlation coefficient are that (a) it can
converge to a proper random variable when the sample size tends to infinity, indicating that
it fluctuates tremendously for different samples, and (b) that it is always asymptotically non-negative when
dealing with non-negative random variables (even when these are obviously negatively
dependent). In the context of random graphs, the first deficiency means that the
assortativity can have a non-vanishing variance even when the size of the graph
is huge, the second means that there do not exist asymptotically disassortative scale-free
graphs. We give proofs for the facts stated above, and illustrate the results using simulations.

Further research is needed to study rank correlations on graphs. Although the numerical results suggest consistency of the Spearman's rho estimator on random graphs, the values of the vectors $(X_i,Y_i)_{ij\in E}$ are in general not independent. Indeed, if one edge emanates from a node with degree 250, the there are 499 other edges ($ij$ and $ji$ being different edges) for which either $X_i$ or $Y_i$ is equal to $250$. Thus, the consistency of the estimator does not generally follow and needs a case by case proof.

Rank correlations are a special case of the broader concept of copulas that are widely used in multivariate analysis, in particular in application in mathematical finance and risk management. There is a heated discussion in this area about the adequacy and informativeness of such measures, see e.g.\ \cite{Mikosch2006copulas} and consequent reactions. There are several points of criticism. In particular, Spearman's rho uses rank transformation, which changes the observed values of the degrees. Then, first of all, what exactly does Spearman's rho tell us about the dependence between the original values? Second of all, no substantial justification exists for the rank transformation, besides its mathematical convenience.
We thus do not claim that Spearman's rho is {\it the} solution for the problem. The main point of
this paper is rather that the assortativity coefficient is {\it not} a solution at all, and that better
solutions must be sought and can be found.

Raising the discussion to a higher level, random variables $X$ and $Y$ are positively
dependent when a large realization of $X$ typically implies a large realization of $Y$.
A strong form of this notion is when ${\mathbb P}(X>x, Y>y)\geq {\mathbb P}(X>x){\mathbb P}(Y>y)$
for every $x,y\in {\mathbb R}$, but for many purposes this notion is too
restrictive.  The covariance for non-negative random variables
is obtained by integrating the above inequality over $x,y\geq 0$, so that
it is true for `typical' values of $x,y$. In many cases, however, we
are particularly interested in certain values of $x,y$.
Another class of methods for measuring rank correlations is
based on the  \emph{angular measure}, a notion originating
in the theory of multivariate extremes, for which the above inequality
is investigated for \emph{large} $x$ and $y$, so that it describes
the {\it tail dependence} for a random vector $(X,Y)$, that is, the dependence
between extremely large values of $X$ and $Y$, see e.g. \cite{Resnick_HT}. Such tail
dependence is characterized by an {\it angular measure} on $[0,1]$. Informally, a
concentration of the angular measure around the points 0 and 1 indicates
independence of large values, while concentration around some other number
$a\in(0,1)$ suggests that a certain fraction of large values of
$Y$ comes together with large values of $X$.
In~\cite{Volkovich08WWW,Volkovich2009complex} a first attempt
was made to compute the angular measure between
in-degree of a node and its importance measured by the
Google PageRank algorithm. Strikingly, completely different dependence structures were
discovered in Wikipedia (independence), Preferential Attachment
networks (complete dependence) and the Web (intermediate case).

\section*{Acknowledgment}

We thank Yana Volkovich for the code generating a Preferential Attachment graph.

\bibliographystyle{plain}

{\small

redpajama_set_name: RedPajamaArXiv


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The Mayor of Cambridgeshire and Peterborough – the choice between more of the same or something different?
Posted by Sebastian Conway-Jarrett | Apr 28, 2021 | Cambridge, Cambridgeshire, Eastern Echo, News, People, Peterborough, Politics, Views | 0 |
The article was written by Oliver Sargent, Account Manager at Dev Comms
November 2016 was when councillors across Cambridgeshire and Peterborough voted to back a devolution deal to form a new Combined Authority.
Since then, the first-ever term of a Cambridgeshire and Peterborough Mayor is coming to an end with James Palmer hoping to repeat what he achieved in 2017 by being re-elected to the highest office in the region.
Whilst it's safe to say the campaign for the next mayor of Cambridgeshire and Peterborough slips under the radar compared to other mayoral battles, this election still matters, with the winning Mayor having powers over areas such as housing, transport and economic development.
For those who missed the recent TV debate between the three main candidates, it was made clear that housing and transport are the big issues locally, which is reflected in the policies that are being proposed.
So who, and what, are each of the mayoral candidate's key policies?
The incumbent is hard to look past as James Palmer will once again be representing the Conservative party, who remains a strong favourite for a second term in office.
Though whether his cavalier attitude towards releasing an official manifesto, which he still has no intention of publishing or even creating, gains him the votes he needs, remains to be seen.
Yet it appears that most of his policies will remain the same, with many being carried over from 2017. His headline policy targets the affordable housing provision on new developments, with units being made available to purchase for £100,000, regardless of their market assessed value and the discount applied.
The scheme has its critics, perhaps justified, with a limited number of these homes having been built or even in the pipeline. This hasn't stopped the current Mayor from heralding the policy a success, and once again campaigning to scale up and create better synergy with authorities to implement the idea into Local Plans.
Another key policy, which will be familiar to anybody who knows the local politics, is Mr Palmer's metro plans (CAM), which he remains committed to delivering. The proposal would see autonomous vehicles, using underground tunnels in central Cambridge, with the entire scheme expected to cover 88 miles.
The idea around the metro is to ensure sustainable development, whilst avoiding inappropriate and poorly planned development, thus resulting in a more even spread across the Combined Authority.
Likely to be James Palmer's closest rival, the Liberal Democrats will be hoping to improve on their runners-up position at the 2017 election. The party have put forward Aidan Van de Weyer who is Deputy Leader of South Cambridgeshire Council.
The Liberal Democrats won the highest share of the vote in Cambridge City and were a close second to the Conservatives in South Cambridgeshire. They will be hoping to do well in these areas again, especially given the candidate's background.
Unlike Palmer, Van de Weyer has released an official manifesto, which he uses to criticise the proposed Metro (CAM). Claiming he would 'rip up' much of the work done by the current Mayor, he is yet to disclose full details of his own plans for transport.
When it comes to housing, Van de Weyer intends to prioritise secure rent over shared ownership, ensure minimum space standards are met and end the current £100K Homes scheme.
A victory for Van de Weyer on May 6 would also mean a more aggressive move towards a pro-green agenda, with an investment strategy for the transition to zero-carbon across the county.
Finally, thought should be given to the Labour candidate, Nik Johnson, who is a Councillor at Huntingdonshire District Council and a doctor at the local hospital.
Although on paper, the party seem to have little chance of success come May, they do however propose a number of notable housing policies.
On affordable housing, Johnson believes the combined authority should follow the Cambridge City Council model of building council houses but accepts the success of this would depend on funding from the central government. He has also campaigned on a minimum of 50 per cent affordable housing to be provided in new developments.
When it comes to transport, we should expect a number of green initiatives and a greater focus on bus franchising.
So who's going to win?
It is hard to predict how the performance of national politics will play on the minds of constituents, however, one thing is for sure – two key issues are at the top of the agenda; housing and transport.
Whether people are postal voting or heading to the polling station on May 6th, the election is a test of Palmer's popularity and performance. But will voters want more of the same or something different?
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Solo Piano with airy melodies recorded on spot in a diary form. Soft, Relaxed & a bit dreamy. The pianist's own instrumental spiritual meditation.
Solo Piano Music by Swedish pianist Christian Lindquist. (a Whisperings.com artist) Soft, calm and airy. Quiet solo piano that is full of warmth and emotion. Influenced by Keith Jarrett and David Foster solo recordings.
Christian is the only Swedish pianist available on http://www.solopianoradio.com - a Whisperings artist.
Born in 1969 on the Ockero islands outside Gothenburg in the west coast of Sweden. His family has lived there for generations. There is proximity to open sea and a stunning nature.
I have always liked the sunrise, the freshness of an early morning. That has also become my time and very often I sit down and play my piano. I push record and play until the inspiration is out.
"Solo Piano Diary" is about my feelings, mood and my hunger for expression. Playing in a diary form is a quest for honesty, where suddenly the tone that was insipid yesterday is inspired today. The piano diary has become a way for me to verify my soul and to whittle truth out of my heart. The melodies are therefore a profound mix of pain and of peace.
Please let me leave my comment here although my English words are not enough. I tried to listen to these beautiful works not only in my own room alone, but also driving in my car looking at the view out of windows. I realized and sensed at sudden that Christian are very closed to me and I was able to feel his kindness, tenderness and sorrow. Because he knows of these essences in our life so well, which is light in our life while shadow in our life.
Every note and every small sound fragment touch my heart. Thank you so much for sharing such a beautiful world and universe with me. And I am very glad to know you such a broad world.
I wish your more creative life in your sounds.
With a gentle touch as a skillful painter Lindquist portrays the beauty of life with the music of his piano.
I play it at work to get som "help" to relax and not get stressed....And I play it for the kids just before bedtime to get them to slow down, and then I let it play when they are falling asleep. I am so happy I bought it!
Excellent cd with many opportunitys for slowing down and meditate and think. I have hes cd at work wich gives me valubile moments to take a breath and slow down. It´s also a very personal record and a great idea to just press "record" and play what comes in to your mind. That is a brave thing to do.


text:
Matter & Energy
Science Bulletin
Home Anthropology It's true: Latinos age slower than other ethnicities
It's true: Latinos age slower than other ethnicities
A UCLA study is the first to show that Latinos age at a slower rate than other ethnic groups. The findings, published in the current issue of Genome Biology, may one day help scientists understand how to slow the aging process for everyone.
"Latinos live longer than Caucasians, despite experiencing higher rates of diabetes and other diseases. Scientists refer to this as the 'Hispanic paradox,'" said lead author Steve Horvath, a professor of human genetics at the David Geffen School of Medicine at UCLA. "Our study helps explain this by demonstrating that Latinos age more slowly at the molecular level."
According to the Centers for Disease Control and Prevention, Latinos in the U.S. live an average of three years longer than Caucasians, with a life expectancy of 82 versus 79. At any age, healthy Latino adults face a 30% lower risk of death than other racial groups, reports a 2013 study in the American Journal of Public Health.
The UCLA team used several biomarkers, including an "epigenetic clock" developed by Horvath in 2013, to track an epigenetic shift linked to aging in the genome. Epigenetics is the study of changes to the DNA molecule that influence which genes are active but don't alter the DNA sequence itself.
Horvath and his colleagues analyzed 18 sets of data on DNA samples from nearly 6,000 people. The participants represented seven different ethnicities: two African groups, African-Americans, Caucasians, East Asians, Latinos and an indigenous people who are genetically related to Latinos. Called the Tsimane, the latter group lives in Bolivia.
When the scientists examined the DNA from blood-which reveals the health of a person's immune system — they were struck by differences linked to ethnicity. In particular, the scientists noticed that, after accounting for differences in cell composition, the blood of Latinos and the Tsimane aged more slowly than other groups.
According to Horvath, the UCLA research points to an epigenetic explanation for Latinos' longer life spans. For example, the biological clock measured Latino women's age as 2.4 years younger than non-Latino women of the same age after menopause.
"We suspect that Latinos' slower aging rate helps neutralize their higher health risks, particularly those related to obesity and inflammation," said Horvath, who is also a professor of biostatistics at the Fielding School of Public Health at UCLA. "Our findings strongly suggest that genetic or environmental factors linked to ethnicity may influence how quickly a person ages and how long they live."
The Tsimane aged even more slowly than Latinos. The biological clock calculated the age of their blood as two years younger than Latinos and four years younger than Caucasians. The finding reflects the group's minimal signs of heart disease, diabetes, hypertension, obesity or clogged arteries.
"Despite frequent infections, the Tsimane people show very little evidence of the chronic diseases that commonly afflict modern society," observed coauthor Michael Gurven, a professor of anthropology at UC Santa Barbara. "Our findings provide an interesting molecular explanation for their robust health."
In another finding, the researchers learned that men's blood and brain tissue ages faster than women's from the same ethnic groups. The discovery could explain why women have a higher life expectancy than men.
Horvath and his colleagues next plan to study the aging rate of other human tissues and to identify the molecular mechanism that protects Latinos from aging.
Source: University of California, Los Angeles (UCLA), Health Sciences
Research Reference:
Steve Horvath, Michael Gurven, Morgan E. Levine, Benjamin C. Trumble, Hillard Kaplan, Hooman Allayee, Beate R. Ritz, Brian Chen, Ake T. Lu, Tammy M. Rickabaugh, Beth D. Jamieson, Dianjianyi Sun, Shengxu Li, Wei Chen, Lluis Quintana-Murci, Maud Fagny, Michael S. Kobor, Philip S. Tsao, Alexander P. Reiner, Kerstin L. Edlefsen, Devin Absher, Themistocles L. Assimes. An epigenetic clock analysis of race/ethnicity, sex, and coronary heart disease. Genome Biology, 2016; 17 (1) DOI: 10.1186/s13059-016-1030-0
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» JustCoding News: Inpatient
JustCoding News: Inpatient, December 7, 2011
Want to receive articles like this one in your inbox? Subscribe to JustCoding News: Inpatient!
As the weather cools, the heat is on coders to properly report the high number of pneumonia cases they tend to see during the winter months.
It's not always easy, considering the changing face of pneumonia testing and treatment and the number of documentation requirements for coding. In particular, cases "without a smoking gun," such as pneumonia without a positive chest x-ray, can be particularly challenging for clinician and coder alike, said Lolita M. Jones, RHIA, CCS, sole principal of Lolita M. Jones Consulting Services in Fort Washington, MD. Jones spoke along with Joy J. King, RHIA, CCS, CCDS, principal of Joy King Consulting, LLC, in Birmingham, AL, during HCPro's September 8 audio conference, "Top ICD-9-CM Trouble Spots: Master Clinical Background and Coding Guidelines for Accurate Coding."
However, a solid understanding of both the clinical aspects and the guidelines for pneumonia coding can help coders correctly report pneumonia during the long winter months and beyond.
Diagnosing pneumonia
"Research is showing that we shouldn't be surprised to see more and more clinical diagnoses in the absence of positive chest x-rays," Jones said. There are a number of reasons why.
For example, Jones noted that even when looking at the same x-rays, radiologists couldn't always agree whether an infiltrate was present, according to an article by Edward Doyle in the February 2006 issue of Today's Hospitalist. Doyle also found that CT scans may actually be a better tool for diagnosing clinical pneumonia.
"A chest x-ray is probably still the first line of defense, the first thing a physician orders to figure out if [pneumonia] is present, but we may be getting to a point when a CT scan of the chest may soon become the prevailing test," Jones said.
That said, the chest x-ray for infiltrates may remain the go-to diagnosis tool simply because of a number of quality care initiatives that require physicians to treat pneumonia patients within a certain number of hours after admission.
Those initiatives "are looking for the physicians to say, for example, 'I treated this particular condition because I found infiltrates on the x-ray so I know I'm treating pneumonia,' " Jones said, explaining that a physician knowing in his or her gut that a patient has pneumonia regardless of a negative chest x-ray doesn't always cut it with external review organizations. "Unfortunately, the quality guidelines and parameters out there haven't caught up with the fact that there are a number of conditions out there that can be treated based on signs and symptoms even if diagnostic tests are negative, and pneumonia is one of them."
Another challenge for pneumonia diagnoses is the rise of drug-resistant organisms. Prescribed antibiotics may not work the first or even second time. Drug resistance was fairly uncommon in the past, according to Jones, but that's no longer the case.
"We need to be sensitive to the fact that you can't say it wasn't pneumonia because the first round of antibiotics didn't work ... the patients may have a clinical pneumonia that is due to a bacteria or virus, for which the patient is drug-resistant," she said.
As the medical industry adapts, coders may see other alternative testing methods documented more frequently in the record. For example, physicians may order C-reactive protein (CRP) tests to test for bacterial pneumonia.
This is a finger stick test that measures the patient's level of CRP, which is stimulated by bacteria and rises in the presence of an infection. A very high level can indicate pneumonia, according to Doyle.
Similarly, physicians can test for rising procalcitonin levels or S-TREM (soluble triggering receptor expressed on myeloid cells), Jones said.
"[Instead of] the tests that we're used to seeing and the treatment protocols that we're used to seeing, I think we really are moving into the next generation. And what we've become used to seeing in the past—what a lot of our queries have been based around, along with tests, documentation training, in-services with physicians—a lot of it needs to begin to move forward according to what's really being seen out there," Jones said. "So much of it is changing. We have to look at what is going on right now and how the practice of medicine is changing."
Pneumonia coding guidelines
Despite the changing clinical preferences for diagnosing pneumonia, the positive chest x-ray is still considered the standard by recovery audit contractors (RAC), the Office of Inspector General, and other auditors, King said. As such, coders need to watch for it in documentation.
Coders should also look for indication that fluids were provided to the patient, particularly prior to a chest x-ray. If a patient presents with dehydration, he or she would usually need fluids before an infiltrate would show up on a chest x-ray regardless of the presence of pneumonia, King explained.
Without a positive chest x-ray, coders should look for other signs and symptoms documented in the record, she said. These could include a heart rate above 100 bpm or a respiratory rate above 25, rales, crackles, rhonchi, a dullness to percussion, or decreased breath sounds.
"Certainly, if you have a physician advisor or champion, or a [clinical documentation improvement] program, communication with the attending physicians about the importance of documenting more about their clinical diagnosis of pneumonia when they don't have that infiltrate is going to be increasingly important," King said.
Another issue coders should watch for is hypoxemia with pneumonia.
Unlike with respiratory failure, hypoxemia is not inherent to pneumonia. Per Coding Clinic, Second Quarter 2006, if a physician documents hypoxemia in the record, coders need to report it separately from the code for the pneumonia diagnosis, according to King.
Finally, there is the ongoing issue of assuming causal organisms based on sputum cultures. As discussed in Coding Clinic, Second Quarter 1998, coders simply cannot do it, King explained.
"Sputum cultures are often misleading or negative," she said. "The physician must actually document a link between the results on the culture and the pneumonia itself in order for coders to link them. This continues to be something that coders struggle with."
ICD-10 coding for pneumonia
Documentation requirements shouldn't change much with the switch to ICD-10-CM/PCS, according to Jones. "We still need a definitive diagnosis of pneumonia, or at the very least a statement on the discharge summary that the pneumonia was not ruled out so that we can know what we're dealing with from a coding standpoint."
The codes, however, will certainly change; there are numerous codes for pneumonia in the new system, including the following:
J18.0 (bronchopneumonia, unspecified organism)
J18.1 (lobar pneumonia, unspecified organism)
J18.2 (hypostatic pneumonia, unspecified organism)
J18.8 (other pneumonia, unspecified organism)
J18.9 (pneumonia, unspecified organism)
"This is just a sampling of the numerous codes out there for ICD-10," Jones explained. "There's a completely different batch of codes for when you have an organism specified."
In particular, coders may want to note ICD-10-CM code J18.9, which is essentially the replacement code for the ICD-9-CM code 486 (pneumonia, unspecified organism), she said.
Additional clinical pneumonia coding guidelines
Need more guidance on coding for clinical pneumonia? Coding Clinic may well have the answers you need; the topic has been addressed in the publication numerous times. King notes the following Coding Clinic issues
Fourth Quarter 2010, p. 135
First Quarter 2010, pp. 3, 12
Third Quarter 2009, p. 16
Fourth Quarter 2008, pp. 69, 140
Second Quarter 2006, pp. 20, 24
Second Quarter 2003, pp. 21–22
Fourth Quarter 1999, p. 6
Third Quarter 1998, p. 7
Second Quarter 1998, pp. 3–5, 7
First Quarter 1998, p. 8
Fourth Quarter 1995, p. 52
First Quarter 1994, pp. 17–18
First Quarter 1991, p. 13
Third Quarter 1988, pp. 11, 13
M-A 1985, p. 6
This list can be a very valuable tool as you develop guidelines for coding and querying for your coding staff, King says.
Editor's note: This article was originally published in the November issue of Medical Records Briefing. E-mail your questions to Senior Managing Editor Andrea Kraynak, CPC, at akraynak@hcpro.com.
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text:
The Treasure of Consecrated Life
By Elizabeth Krump
"What's a Nun Run? Do you dress up as nuns and run?" Last year, new interest was generated in religious life among my single friends when a few of us decided to organize a Nun Run. The day-long event was an opportunity to visit local religious orders in Vancouver to get to know their way of life better. While many of us knew religious Sisters by sight, we had never had the opportunity to have a conversation with one of them.
Incredibly, nearly twenty beautiful young women attended our first Nun Run to visit the Dominican Sisters of St. Cecilia and the cloistered Poor Clare Sisters. We prayed with the Sisters and laughed with them, but the best part of the Nun Run was hearing the stories of these ordinary women who had made the choice to live in an extraordinary way for Christ.
Nuns and sisters are what we call consecrated vocations. Consecrated life takes many forms (priests, nuns, monks, etc), but is ultimately a way of life which strives to imitate Christ through chastity, poverty and obedience. Blessed John Paul II describes consecrated life as "the choice of total self-giving to God in Christ (Vita Consecrata)." It's a radical path to take in the modern world, but contrary to popular belief, it is neither lonely or stifling, but joy filled.
On February 2, the Feast of the Presentation of Jesus, the Church celebrates the consecration of Jesus to God by his earthly parents. It is fitting that the Church also designates this feast as World Day for Consecrated Life. Pope John Paul II established this day in 1997 to recognize consecrated life as a treasure and a gift in the Church.
What is the treasure of consecrated life? Pope John Paul II said, "the consecrated life is not something isolated and marginal, but a reality which affects the whole Church (Vita Consecrata)." Therefore, we should not think of it as separate from the lives of ordinary (lay) Christians. In fact, he goes so far as to say that, "the consecrated life is at the very heart of the Church as a decisive element for her mission, since it 'manifests the inner nature of the Christian calling' and the striving of the whole Church as Bride towards union with her one Spouse (Vita Consecrata)."
What does he mean by the 'inner nature of the Christian calling'? John Paul II is speaking about the fact that consecrated individuals are in a life-giving relationship... with Christ. They are living outwardly what we are all invited to experience inwardly: a relationship with Jesus.
Judith Ameresekere, a fourth year Psychology student at Simon Fraser University is open to the call of consecrated life. She describes nuns as, "A witness: they believe so strongly that they are willing to lay down their lives for [Christ]." The treasure of consecrated life is their example of selfless love. They show us how to love others and they also reflect the total, self-giving love Christ has for us.
On our Nun Run, someone asked one of the Sisters what the best part of her vocation was. This young Dominican, a convert of atheism, still in her twenties and radiating with life, answered, "I get to wear my wedding dress every day." This was a young woman in love: in love with Christ, living as a Bride.
On World Day for Consecrated Life, pray for the men and women who are living this call to selfless, true love. Their radical call deserves our unceasing support in prayer.
Elizabeth Krump is a student at Simon Fraser University in British Columbia, where she has been involved with Catholic Christian Outreach. Elizabeth has written for the B.C. Catholic and with two friends started Live 31 Vancouver, a blog/social group/discussion group for women trying to live their faith in the modern world.
Photos/Elizabeth Krump


text:
Changing Climate a Hot Topic Among Himalayan Villagers, Study Finds
Warmer temperatures, variable monsoons, and other signs of climate change are a hot topic of conversation among many Himalayan villagers, according to scientific sampling of climate change perception among local peoples. "This area is cold and it's often raining. Even during the non-monsoon times there is mist and fog so inevitably conversations here turn to...
Warmer temperatures, variable monsoons, and other signs of climate change are a hot topic of conversation among many Himalayan villagers, according to scientific sampling of climate change perception among local peoples.
"This area is cold and it's often raining. Even during the non-monsoon times there is mist and fog so inevitably conversations here turn to weather," said Kamal Bawa, biologist at the University of Massachusetts, Boston (UMB), and president of Ashoka Trust for Research in Ecology and the Environment (ATREE) in Bangalore, India. "When you stop and have a cup of tea in someone's kitchen, the conversation invariably turns to the weather. But then they soon start talking about how the weather has been changing."
Bawa is also a member of the National Geographic Committee for Research and Exploration.
Bawa didn't set out to study Himalayan perceptions of climate change. But after hearing the same themes repeated again and again during household conversations he decided to investigate. With UMB graduate student Pahupati Chaudhary, he surveyed some 500 homes spread across 18 villages in Darjeeling Hills, West Bengal, India and Nepal's Ilam district. The pair found some surprisingly consistent observations.
Three-fourths of the people surveyed believe that their weather is getting warmer and two-thirds believe that summer and monsoon season have begun earlier over the past ten years. Seventy percent believe that water sources are drying up while forty-six percent said that they think there is less snow on the high mountains.
Many villagers also told Chaudhary they'd noticed shifts in some species ranges and earlier flowering and budding of plants. New pests have also arrived, villagers routinely reported, to plague crops and people—including mosquitoes where none had been before.
Most of these changes were reported by much higher percentages of people living at higher altitudes than by those at lower altitudes. "We've shown in earlier research that people at high altitudes seem to be more sensitive to climate change, and of course it's known that climate change is more severe at higher altitudes, so that's not a surprise," Bawa said.
Many Himalayan peoples live in areas where predicted and observed impacts of climate change, like species migration, are more acute. Many of them also live "close to the land," where agricultural-based livelihoods make them especially attuned to weather patterns.
Listening to Locals Can Help Climate Science
Scientific data on climate change have been hard to come by in the region, Bawa reported. Few weather stations dot remote and high-altitude locales and where they do exist their data are often incomplete.
But where data can be found they seem to corroborate local observations, Bawa said, citing his own research on temperature and rainfall records as well as the work of other scientists listed in recent Biology Letters and Current Science reports of Bawa and Chaudhary's research.
"Governments in the region are now gearing up towards more research," Bawa said. "But it will take time to gather this climate data." That's why local knowledge can be such valuable human intelligence, he added. It can be gathered quickly and widely and used to "jump start" scientific efforts.
"There seems to be quite a bit of knowledge residing with local communities, in the Himalaya and elsewhere, and we can really use that knowledge to formulate scientific questions for further research and make more rapid assessments of the impacts of climate change."
Bawa said it's hard to determine to what extent local peoples are familiar with the global dialogue on climate change, or how much that might have influenced their perceptions. But most of those he spoke with didn't identify a clear cause for the changes they'd observed.
"We're saying that people seem to be aware that the climate is changing, but they may not necessarily be aware of why it's changing. I think when you come to that question people don't have any ideas—or they may have some very different ideas."
Bawa pointed to a recent study of this topic in Tibet, where many respondents believed that humans are causing climate change—but not by producing greenhouse gasses as most climate scientists believe. "They seem to think that the climate is changing because the Gods are not happy and perhaps the people in the younger generations are not praying enough."
This research was supported in part by a Committee for Research and Exploration grant of the National Geographical Society.
Brian Handwerk
Brian Handwerk is a freelance writer based in Amherst, New Hampshire.


Public discussions about how the UK is to exit from the European Union have been too simplified, and have failed to come up with any solution that recognises that only England and Wales in fact voted to leave. Brendan O'Leary, Professor of Political Science at the University of Pennsylvania, outlines a way forward where those nations wanting to remain in the EU might be able to do so.
Brendan O'Leary is Lauder Professor of Political Science at the University of Pennsylvania.
This article first appeared on the LSE EUROPP blog.
Given the arrangements you outline. I would suggest that Wales would also wish to be part of a EU UK.


The unmarked trail to Carlon Falls began in the Stanislaus National Forest on the north bank of the South Fork Tuolumne River. The trail entered Yosemite National Park through a metal maze.
The metal maze you walk through when entering Yosemite National Park from this trail.
I hiked alone through the shady forest on my way to Carlon Falls. Spider webs strewn across the trail tickled my face as I broke the invisible snares.
The sandy trail bordering the river led to a private beach.
It turned away from the water then led steeply uphill along an eroded hillside, down a slippery slope and back towards the water.
A tree that I had to climb over because it was blocking the trail.
The closer I came to the falls, the harder it was to see a clear path. Climbing over boulders and rocks, I forged my own way.
Making my way to the river included walking through this rocky patch.
Scanning the area, I eventually found a barely visible animal trail and followed that up to the next level. The granite near the water's edge was smoothly polished.
A lower cascade of Carlon Falls.
Looking up, I could clearly see the trail that I missed. Returning was much easier when walking on the correct path.
The upper most cascade of Carlon Falls.
This entry was posted in Yosemite Hikes and tagged Carlon Falls, Hiking, South Fork Tuolumne River, Stanislaus National Forest, Yosemite on May 12, 2012 by Janet.
Beautiful photos! I totally agree with your profile statement.


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Home/Insight For Today Devotional/Chuck Swindoll Devotional 5 January 2023 || Insight For Today
Chuck Swindoll Devotional 5 January 2023 || Insight For Today
Welcome to Insight For Today Devotional written by Chuck Swindoll. Charles Swindoll is an evangelical Christian pastor, author, educator, and radio preacher. He founded Insight for Living, headquartered in Frisco, Texas. He is currently senior pastor at Stonebriar Community Church, in Frisco, Texas. Read, meditate and pray as directed by God's spirit.
Today's Topic: God's Forgiveness of Us
Scripture: Psalm 103:1–12
Today's Message 5 January 2023 || Insight For Today
Truth be told, it's God's forgiveness of us that makes possible our forgiving others.
When on the cross Jesus Christ paid in full the penalty of our sin, God's wrath was expressed against Him—the One who took our place. God was therefore satisfied in the epochal sacrifice . . . allowing all who would turn, in faith, to the Son of God to be totally, once-for-all, forgiven. Christ's blood washed away our sin. And from the moment we believe in Him, we stand forgiven, relieved of guilt, before a satisfied God, thereby freeing Him to shower upon us His grace and love.
Remember the verse from that grand old song the church has sung for years?
My sin—O, the bliss of this glorious tho't—
My sin—not in part, but the whole,
Is nailed to the cross, and I bear it no more,
Praise the Lord, praise the Lord, O my soul!1
That says it well but not as beautifully as the song from the oldest of all hymnals—the Psalms:
And all that is within me, bless His holy name.
And forget none of His benefits;
Who pardons all your iniquities,
Who heals all your diseases;
Who crowns you with lovingkindness and compassion;
Who satisfies your years with good things,
So that your youth is renewed like the eagle . . . .
. . . He has not dealt with us according to our sins,
Nor rewarded us according to our iniquities.
For as high as the heavens are above the earth,
So great is His lovingkindness toward those who fear Him.
(Psalm 103:1–5, 10–12)
It's important to understand that we can never forgive others, horizontally, if not for what Christ has already done for us, vertically. Not until we fully accept and appropriate God's infinite and complete forgiveness on our behalf can we carry out the tough assignment of forgiving others.
It isn't long before anyone who gets serious about serving others must come to terms with forgiving others as well.
Yes, must.
January 5, 2023 Your Daily Prayer For Thursday
Dr. David Jeremiah Turning Point 5th January 2023
Chuck Swindoll Devotional 28 January 2023 || Insight For Today


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text:
Union Cabinet reshuffle on cards
Edited By Odishatv Bureau Published By Odishatv Bureau On Jun 14, 2011 - 8:59 AM
New Delhi: An expansion of the Union Cabinet and reshuffle of the ministry and appointment of some Governors and top bureaucrats in some of the key ministries are expected to take place later this month.
Sources said the changes may take place once Prime Minister Manmohan Singh undertakes the process of consultation with Congress President Sonia Gandhi after her return from a holiday this week-end.
The talk in ruling party circles is that the exercise could take place later this month or early July, ahead of of the monsoon session of Parliament.
The possibility of reshuffle has brightened after the victory of Mamata Banerjee, who became the Chief Minister of West Bengal, and resigned as Railway Minister at the Centre.
There is speculation that her party colleague Mukul Roy, who has been given additional charge of Minister of State in the Railway, will be elevated to the Cabinet rank in the Railway ministry.
Ministers like Kapil Sibal are holding more than one portfolio. While there are some vacancies including that created by the resignation of A Raja in the wake of the 2G spectrum scam and Prithviraj Chavan, who became Chief Minister of Maharashtra.
There is also talk of appointment of Governors with some incumbent holding the post in more than one state like Rajasthan.
In Tamil Nadu, S S Barnala has far outlasted his tenure as Governor of the state and the Centre may be looking at at a new candidate.
In the bureaucracy, Home Secretary Gopal K Pillai is retiring this month end while Defence Secretary Pradeep Kumar and Foreign Secretary Nirupama Rao will retire next month.


(For licensing or usage, contact licensing@viralhog.com)As a fire protection contractor, I have access to lots of restaurant kitchens in my city. The following videos are typical of most Chinese restaurants that I visit.
I had to take multipile short videos in order to not be noticed by the staff.
From unrefrigerated chicken and beef pretty much on the floor, to bug traps on top of canned food, this place was disgusting.


37% The original file has 22262 bytes (21.7k) and is available from the project website.
There you can find the official minified version, too, which brings down the size to 8206 bytes (8.0k, 37%).
Most recent activity on February 11, 2019 @ 01:58.