In a network, we define shell $ell$ as the set of nodes at distance $ell$ with respect to a given node and define $r_ell$ as the fraction of nodes outside shell $ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $ell$ as a function of $r_ell$. Further, we find that $r_ell$ follows an iterative functional form $r_ell=phi(r_{ell-1})$, where $phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_ell$ found in shells with $ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real world networks the theoretical prediction of $r_ell$ deviates from the empirical $r_ell$. We introduce a network correlation function $c(r_ell)equiv r_{ell+1}/phi(r_ell)$ to characterize the correlations in the network, where $r_{ell+1}$ is the empirical value and $phi(r_ell)$ is the theoretical prediction. $c(r_ell)=1$ indicates perfect agreement between empirical results and theory. We apply $c(r_ell)$ to several model and real world networks. We find that the networks fall into two distinct classes: (i) a class of {it poorly-connected} networks with $c(r_ell)>1$, which have larger average distances compared with randomly connected networks with the same degree distributions; and (ii) a class of {it well-connected} networks with $c(r_ell)<1$.
In order to investigate whether government regulations against corruption can affect the economic growth of a country, we analyze the dependence between Gross Domestic Product (GDP) per capita growth rates and changes in the Corruption Perceptions Index (CPI). For the period 1999-2004 on average for all countries in the world, we find that an increase of CPI by one unit leads to an increase of the annual GDP per capita by 1.7 %. By regressing only European transition countries, we find that $Delta$CPI = 1 generates increase of the annual GDP per capita by 2.4 %. We also analyze the relation between foreign direct investments received by different countries and CPI, and we find a statistically significant power-law functional dependence between foreign direct investment per capita and the country corruption level measured by the CPI. We introduce a new measure to quantify the relative corruption between countries based on their respective wealth as measured by GDP per capita.
