Search

This paper considers inhomogeneous Erdős–Rényi random graphs 𝔾 N on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices { i , j } is retained with probability 𝜀 N f (i N , j N) , 1 ≤ i ≠ j ≤ N , independently of other edges, where f : [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , ∞) is a continuous function such that f (x , y) = f (y , x) for all x , y ∈ [ 0 , 1 ]. We study the empirical distribution of both the adjacency matrix A N and the Laplacian matrix Δ N associated with 𝔾 N , in the limit as N → ∞ when lim N → ∞ 𝜀 N = 0 and lim N → ∞ N 𝜀 N = ∞. In particular, we show that the empirical spectral distributions of A N and Δ N , after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where f (x , y) = r (x) r (y) with r : [ 0 , 1 ] → [ 0 , ∞) a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks. [ABSTRACT FROM AUTHOR]

In this article, we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit description of the moments of the limiting measure. We also show that in some special cases the Gaussian assumption can be relaxed. The description of the limiting measure can also be made via its Stieltjes transform which is characterized as the solution of a functional equation. In two special cases, we get a description of the limiting measure -- one as a free product convolution of two distributions, and the other one as a dilation of the Wigner semicircular law. [ABSTRACT FROM AUTHOR]

Iodine deficiency disorders (IDD) constitute the single most important preventable cause of mental handicap at global level. Recognizing the importance of coordination and synergy of the activities of wide range of universal salt iodisation (USI) stakeholders, WHO/ Unicef'/ ICCIDD has prescribed a national multi-sectoral coalition as one of the ten indicators essential for attaining sustainable elimination of IDD at national level. Challenge for coordination among different stakeholders of IDD/USI is even greater in democratic and diverse country like India. In the present article we present successful experience from India regarding formation of a national coalition and contributions made by the coalition towards promoting USI in India. The activities of the national coalition in India are classified into three phases; 1) Phase 1- year 2006 to 2009- the inception; 2) Phase 2- year 2009 to 2012- consolidation; 3) Phase 3- year 2013 and ongoing- expansion. The National coalition for Sustained Optimal Iodine Intake (NSOI) has been instrumental in ensuring greater coordination and synergy amongst IDD and USI stakeholders in India and partially responsible for the current 71 percentage household level coverage of adequately iodised salt. The most significant contribution of the national coalition has been to act as a high level advocacy channel and provide a platform for regular dialogue for all partners of the coalition. With "mission" approach and allocation of optimal resource, India can achieve and should achieve USI by 2015, an apt culmination of a decade of existence of the national coalition. [ABSTRACT FROM AUTHOR]