scholarly journals Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

Author(s):  
Anirban Basak ◽  
Mark Rudelson
2016 ◽  
Vol 05 (01) ◽  
pp. 1650002
Author(s):  
Paulo Manrique ◽  
Victor Pérez-Abreu ◽  
Rahul Roy

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.


Author(s):  
V. F. Kolchin
Keyword(s):  

1968 ◽  
Vol 20 (03/04) ◽  
pp. 548-554
Author(s):  
J Gajewski ◽  
G Markus

SummaryA method for the standardization of human plasminogen is proposed, based on the stoichiometric interaction between plasminogen and streptokinase, resulting in inhibition of proteolytic activity. Activation of a constant amount of plasminogen with increasing amounts of streptokinase yields linearly decreasing activities, as a function of streptokinase, with a sharp transition to a constant residual level. The point of transition corresponds to complete saturation of plasmin with streptokinase in a 1:1 molar ratio, and is therefore a measure of the amount of plasminogen present initially, in terms of streptokinase equivalents. The equivalence point is independent of the kind of protein substrate used, buffer, pH, length of digestion and, within limits, temperature. The method, therefore, is not subject to the variations commonly encountered in the usual determination based on specific activity measurements.


Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.


Author(s):  
Mark Newman

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.


2021 ◽  
Vol 20 (3) ◽  
Author(s):  
Sho Kubota ◽  
Etsuo Segawa ◽  
Tetsuji Taniguchi

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