Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.

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Random graph dynamics

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

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Random Graphs and Complex Networks

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

Optimal inequality, Power sum, Subadditive, Superadditive, Power mean in- equality. Abstract: Recently Feng Qi has presented a sharp inequality between the sum of squares and the exponential of the sum of a nonnegative sequence. His result has been extended to more general power sums by Huan-Nan Shi, and, independently, by Yu Miao, Li-Min Liu, and Feng Qi. In this note we generalize those inequalitites by introducing weights and permitting more general functions. Inequalities in the opposite direction are also presented.

https://www.maths.tcd.ie/EMIS/journals/JIPAM/images/022_08_JIPAM/022_08.pdf

On an inequality of Feng Qi.

We adapt the concept of evolutionary stability to familial selection when a game theoretic conflicts between siblings determines the survival rate of each sibling in monogamous, exogamous families in a diploid, panmictic population. Similarly to the classical evolutionary game theory, the static condition of evolutionary stability of mixed Nash equilibrium implies the local stability of the genotype dynamics, in spite of that the mating table based genotype dynamics is not a replicator dynamics. We apply our general result to the case where a matrix game determines the survival rate of siblings. In our numerical studies we consider the prisoner’s dilemma between siblings, when the cooperator and defector behaviour are unequally determined by a recessive-dominant allele pair at an autosomal locus. When the prisoner’s dilemma game is strict (cf. iterated one) and the cooperator phenotype is recessive resp. dominant, then the cooperator and defector phenotypes are the unique stable phenotypes, respectively. When the prisoner’s dilemma game is not strict, both phenotypes coexist, independently of the genotype-phenotype mapping. However, the frequencies of the phenotypes are different according to which phenotype is dominant.

Game of full siblings in Mendelian populations

We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefﬁcient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a ﬁxed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a ﬁxed multiple of the average degree, and the maximal degree.

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A random graph of moderate density

We discuss recent developments in the following important areas of Alfred Renyi’s research interest: axiomatization of quantitative dependence measures, qualitative independence in combinatorics, conditional qualitative independence in statistics/data science and in measure theory/probability theory, and finally, prime gaps that are responsible for Renyi’s early career reputation. Most authors of this paper are main contributors to the new developments.

Rényi 100, Quantitative and qualitative (in)dependence

Due to the popularity of randomly evolving graph processes, there exists a randomized version of many recursively defined graph models. This is also the case with the cherry tree, which was introduced by Bukszar and Prekopa to improve Bonferroni type upper bounds on the probability of the union of random events. Here we consider a substantially extended random analogue of that model, embedding it into a general time dependent branching process.

Tamás F. Móri

Papers

On an inequality of Feng Qi.

Game of full siblings in Mendelian populations

A random graph of moderate density

Rényi 100, Quantitative and qualitative (in)dependence

Random cherry graphs