Affirming a conjecture of Erdös and Rényi we prove that for any (real number) c1 > 0 for some c2 > 0, if a graph G has no c1(log n) nodes on which the graph is complete or edgeless (i.e. G exemplifies |G| 9 (c1 log n)22) then G has at least 22 non-isomorphic (induced) subgraphs. Latest Revision 97/Aug/14 I thank Alice Leonhardt for the beautiful typing Typeset by AMS-TEX 1 6 2 7 r e v i s i o n...

The Erdös-Rényi model of a network is simple and possesses many explicit expressions for average and asymptotic properties, but it does not fit well to real-word networks. The vertices of these networks are often structured in prior unknown clusters (functionally related proteins or social communities) with different connectivity properties. We define a generalisation of the Erdös-Rényi model c...

Rényi entropy dimension describes the rate of growth of coding cost in the process of lossy data compression in the case of exponential dependence between the code length and the cost of coding. In this paper we generalize the Csiszár estimation of the Rényi entropy dimension of the mixture of measures for the case of general probability metric space. This result determines the cost of encoding...

Rényi entropy refers to a generalized class of entropies that have been used in several applications. In this work, we derive a non-parametric distance between distributions based on the quadratic Rényi entropy. The distributions are estimated via Parzen density estimates. The quadratic complexity of the distance evaluation is mitigated with GPUbased parallelization. This results in an efficien...

In this paper, we show that 1) additive energy is not appropriate for discussing the validity of Tsallis or Rényi statistics for nonextensive systems at equilibrium; 2) equilibrium N-body systems with nonad-ditive energy or entropy should be described by generalized statistics whose nature is prescribed by the existence of thermodynamic equilibrium. The nonextensive statistics may be Tsallis or...

We associate to the p-th Rényi entropy a definition of entropy power, which is the natural extension of Shannon’s entropy power and exhibits a nice behaviour along solutions to the p-nonlinear heat equation in Rn. We show that the Rényi entropy power of general probability densities solving such equations is always a concave function of time, whereas it has a linear behaviour in correspondence ...

Many axiomatic definitions of entropy, such as the Rényi entropy, of a random variable are closely related to the `α-norm of its probability distribution. This study considers probability distributions on finite sets, and examines the sharp bounds of the `β-norm with a fixed `α-norm, α 6= β, for n-dimensional probability vectors with an integer n ≥ 2. From the results, we derive the sharp bound...

Entropy is well known to be Schur concave on finite alphabets. Recently, the authors have strengthened the result by showing that for any pair of probability distributions P and Q with Q majorized by P , the entropy of Q is larger than the entropy of P by the amount of relative entropy D(P ||Q). This result applies to P and Q defined on countable alphabets. This paper shows the counterpart of t...

In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraphs occur. The mean behaviour of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, what is the probability of an Erdős-Rényi graph containing an excessively large number of a given subgraph? Using the large d...