Average luminosity distance in inhomogeneous universes

The paper studies the correction to the distance modulus induced by inhomogeneities and averaged over all directions from a given observer. The inhomogeneities are modeled as mass-compensated voids in random or regular lattices within Swiss-cheese universes. Void radii below 300 Mpc are considered, which are supported by current redshift surveys and limited by the recently observed imprint such voids leave on CMB. The averaging over all directions, performed by numerical ray tracing, is non-perturbative and includes the supernovas inside the voids. Voids aligning along a certain direction produce a cumulative gravitational lensing correction that increases with their number. Such corrections are destroyed by the averaging over all directions, even in non-randomized simple cubic void lattices. At low redshifts, the average correction is not zero but decays with the peculiar velocities and redshift. Its upper bound is provided by the maximal average correction which assumes no random cancelations between different voids. It is described well by a linear perturbation formula and, for the voids considered, is 20% of the correction corresponding to the maximal peculiar velocity. The average correction calculated in random and simple cubic void lattices is severely damped below the predicted maximal one after a single void diameter. That is traced to cancellations between the corrections from the fronts and backs of different voids. All that implies that voids cannot imitate the effect of dark energy unless they have radii and peculiar velocities much larger than the currently observed. The results obtained allow one to readily predict the redshift above which the direction-averaged fluctuation in the Hubble diagram falls below a required precision and suggest a method to extract the background Hubble constant from low redshift data without the need to correct for peculiar velocities.Comment: 34 pages, 21 figures, matches the version accepted in JCA

Average distance in growing trees

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barab\'asi--Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to $m=1$ nodes. Average node-node distance $d$ is calculated numerically in evolving trees as dependent on the number of nodes $N$. The results for $N$ not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance $d$ for large $N$ can be approximated by $d=2\ln(N)+c_1$ for the exponential trees, and $d=\ln(N)+c_2$ for the scale-free trees, where the $c_i$ are constant. We derive also iterative equations for $d$ and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.Comment: 6 pages, 3 figures, Int. J. Mod. Phys. C14 (2003) - in prin

Relations between Average Distance, Heterogeneity and Network Synchronizability

By using the random interchanging algorithm, we investigate the relations between average distance, standard deviation of degree distribution and synchronizability of complex networks. We find that both increasing the average distance and magnifying the degree deviation will make the network synchronize harder. Only the combination of short average distance and small standard deviation of degree distribution that ensures strong synchronizability. Some previous studies assert that the maximal betweenness is a right quantity to estimate network synchronizability: the larger the maximal betweenness, the poorer the network synchronizability. Here we address an interesting case, which strongly suggests that the single quantity, maximal betweenness, may not give a comprehensive description of network synchronizability.Comment: 14 pages, and 7 figures (to be published in Physica A

Average Distance, Diameter, and Clustering in Social Networks with Homophily

I examine a random network model where nodes are categorized by type and linking probabilities can differ across types. I show that as homophily increases (so that the probability to link to other nodes of the same type increases and the probability of linking to nodes of some other types decreases) the average distance and diameter of the network are unchanged, while the average clustering in the network increases

Mutually unbiased bases in dimension six: The four most distant bases

We consider the average distance between four bases in dimension six. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in dimension six. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically-found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.Comment: 10 pages, 2 figures, 1 tabl