124,129 research outputs found
On local weak limit and subgraph counts for sparse random graphs
We use an inequality of Sidorenko to show a general relation between local
and global subgraph counts and degree moments for locally weakly convergent
sequences of sparse random graphs. This yields an optimal criterion to check
when the asymptotic behaviour of graph statistics such as the clustering
coefficient and assortativity is determined by the local weak limit. As an
application we obtain new facts for several common models of sparse random
intersection graphs where the local weak limit, as we see here, is a simple
random clique tree corresponding to a certain two-type Galton-Watson branching
process
Ranking Intersecting Lorenz Curves
This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations two alternative sequences of nested dominance criteria between Lorenz curves are introduced. At the limit the systems of dominance criteria appear to depend solely on the income share of either the worst-off or the best-off income recipient. This result suggests two alternative strategies for increasing the number of Lorenz curves that can be strictly ordered; one that places more emphasis on changes that occur in the lower part of the income distribution and the other that places more emphasis on changes that occur in the upper part of the income distribution. Both strategies turn out to depart from the Gini coefficient; one requires higher degree of downside and the other higher degree of upside inequality aversion than what is exhibited by the Gini coefficient. Furthermore, it is demonstrated that the sequences of dominance criteria characterize two separate systems of nested subfamilies of inequality measures and thus provide a method for identifying the least restrictive social preferences required to reach an unambiguous ranking of a given set of Lorenz curves. Moreover, it is demonstrated that the introduction of successively more general transfer principles than the Pigou-Dalton principle of transfers forms a helpful basis for judging the normative significance of higher degrees of Lorenz dominance. The dominance results for Lorenz curves do also apply to generalized Lorenz curves and thus provide convenient characterizations of the corresponding social welfare orderings.generalized Gini families of inequality measures, rank-dependent measures of inequality, Gini coefficient, partial orderings, Lorenz dominance, Lorenz curve, general principles of transfers
Ranking intersectiong Lorenz Curves.
This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained witout introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations two alternative sequences of nested dominance criteria between Lorenz curves are introduced. At the limit the systems of dominance criteria appear to depend solely on the income share of either the worst-off or the best-off income recipient. This result suggests two alternative strategies for increasing the number of Lorenz curves that can be stricly ordered; one that focuses on changes that take place in the lower part of the income distribution. Furthermore, it is demonstrated that the sequences of dominance criteria characterize two separate systems of nested subfamilies of inequality measures and thus provide a method for identifying the least restrictive social preferences required to reach an unambiguous ranking of Lorenz curves.Lorenz curve; partial orderings; rank-dependent measures of inequality; generalized Gini families of inequality measures; principles of transfers and mean-spread-preserving transformations
The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices
We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type s for this process is power law with exponent 2+1+δqs+β1-qs/αqs, but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively
Determining the Number of Samples Required to Estimate Entropy in Natural Sequences
Calculating the Shannon entropy for symbolic sequences has been widely
considered in many fields. For descriptive statistical problems such as
estimating the N-gram entropy of English language text, a common approach is to
use as much data as possible to obtain progressively more accurate estimates.
However in some instances, only short sequences may be available. This gives
rise to the question of how many samples are needed to compute entropy. In this
paper, we examine this problem and propose a method for estimating the number
of samples required to compute Shannon entropy for a set of ranked symbolic
natural events. The result is developed using a modified Zipf-Mandelbrot law
and the Dvoretzky-Kiefer-Wolfowitz inequality, and we propose an algorithm
which yields an estimate for the minimum number of samples required to obtain
an estimate of entropy with a given confidence level and degree of accuracy
A Sufficient Condition for Graphic Sequences with Given Largest and Smallest Entries, Length, and Sum
We give a sufficient condition for a degree sequence to be graphic based on
its largest and smallest elements, length, and sum. This bound generalizes a
result of Zverovich and Zverovich
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