On the expected number of equilibria in a multi-player multi-strategy evolutionary game

In this paper, we analyze the mean number $E(n,d)$ of internal equilibria in a general $d$-player $n$-strategy evolutionary game where the agents' payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next we characterize the asymptotic behavior of $E(2,d)$, estimating its lower and upper bounds as $d$ increases. Two important consequences are obtained from this analysis. On the one hand, we show that in both cases the probability of seeing the maximal possible number of equilibria tends to zero when $d$ or $n$ respectively goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger $n$ and $d$, numerical results are provided and discussed.Comment: 26 pages, 1 figure, 1 table. revised versio

The Lovasz number of random graphs

We study the Lovasz number theta along with two further SDP relaxations theta1, theta1/2 of the independence number and the corresponding relaxations of the chromatic number on random graphs G(n,p). We prove that these relaxations are concentrated about their means Moreover, extending a result of Juhasz, we compute the asymptotic value of the relaxations for essentially the entire range of edge probabilities p. As an application, we give an improved algorithm for approximating the independence number in polynomial expected time, thereby extending a result of Krivelevich and Vu. We also improve on the analysis of an algorithm of Krivelevich for deciding whether G(n,p) is k-colorable

On the Ramsey-Tur\'an number with small ss-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur\'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) = \Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta}) = O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.Comment: 25 p

On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

Let \cC be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over \F_q, the finite field of $q$ elements. Let # \cC(\F_{q^n}) be the number of \F_{q^n}-rational points on \cC. Under a certain multiplicative independence condition on the roots of the zeta-function of \cC, we derive an asymptotic formula for the number of $n =1, ..., N$ such that (# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2} belongs to a given interval \cI \subseteq [-1,1]. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve \E is defined over \Q and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$.Comment: 14 page

The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs