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The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the component size are derived for an arbitrary given distribution over small degrees $dleq L$ and given expected degree, and numerical implementations show that these bounds are close already for small values of $L$. On the other hand, examples illustrate that, for a fixed degree tail, the component size can vary substantially depending on the distribution over small degrees.
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In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. Our contribution is three-fold: (1) We prove that, if $mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.
We consider the sums $S_n=xi_1+cdots+xi_n$ of independent identically distributed random variables. We do not assume that the $xi$s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${bf P}{M>x}$ as $xtoinfty$, provided that $M=sup{S_n, nge1}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.
To determine the size of an extensive air shower it is not necessary to have knowledge of the function that describes the fall-off of signal size from the shower core (the lateral distribution function). In this paper an analysis with a simple Monte Carlo model is used to show that an optimum ground parameter can be identified for each individual shower. At this optimal core distance, $r_mathrm{opt}$, the fluctuations in the expected signal, $S(r_mathrm{opt})$, due to a lack of knowledge of the lateral distribution function are minimised. Furthermore it is shown that the optimum ground parameter is determined primarily by the array geometry, with little dependence on the energy or zenith angle of the shower or choice of lateral distribution function. For an array such as the Pierre Auger Southern Observatory, with detectors separated by 1500 m in a triangular configuration, the optimum distance at which to measure this characteristic signal is close to 1000 m.
It is likely that the holographic principle will be a consequence of the would be theory of quantum gravity. Thus, it is interesting to try to go in the opposite direction: can the holographic principle fix the gravitational interaction? It is shown that the classical gravitational interaction is well inside the set of potentials allowed by the holographic principle. Computations clarify which role such a principle could have in lowering the value of the cosmological constant computed in QFT to the observed one.
We test the hypothesis that the initial mass function (IMF) is determined by the density probability distribution function (PDF) produced by supersonic turbulence. We compare 14 simulations of star cluster formation in 50 solar mass molecular cloud cores where the initial turbulence contains either purely solenoidal or purely compressive modes, in each case resolving fragmentation to the opacity limit to determine the resultant IMF. We find statistically indistinguishable IMFs between the two sets of calculations, despite a factor of two difference in the star formation rate and in the standard deviation of $log(rho)$. This suggests that the density PDF, while determining the star formation rate, is not the primary driver of the IMF.
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