245,412 research outputs found
The Degree Distribution of Random k-Trees
A power law degree distribution is established for a graph evolution model
based on the graph class of k-trees. This k-tree-based graph process can be
viewed as an idealized model that captures some characteristics of the
preferential attachment and copying mechanisms that existing evolving graph
processes fail to model due to technical obstacles. The result also serves as a
further cautionary note reinforcing the point of view that a power law degree
distribution should not be regarded as the only important characteristic of a
complex network, as has been previously argued
Degree-dependent intervertex separation in complex networks
We study the mean length of the shortest paths between a vertex of
degree and other vertices in growing networks, where correlations are
essential. In a number of deterministic scale-free networks we observe a
power-law correction to a logarithmic dependence, in a wide range of network
sizes. Here is the number of vertices in the network, is the
degree distribution exponent, and the coefficients and depend on a
network. We compare this law with a corresponding dependence obtained
for random scale-free networks growing through the preferential attachment
mechanism. In stochastic and deterministic growing trees with an exponential
degree distribution, we observe a linear dependence on degree, . We compare our findings for growing networks with those for
uncorrelated graphs.Comment: 8 pages, 3 figure
Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks
Various real-life networks exhibit degree correlations and heterogeneous
structure, with the latter being characterized by power-law degree distribution
, where the degree exponent describes the extent
of heterogeneity. In this paper, we study analytically the average path length
(APL) of and random walks (RWs) on a family of deterministic networks,
recursive scale-free trees (RSFTs), with negative degree correlations and
various , with an aim to explore the
impacts of structure heterogeneity on APL and RWs. We show that the degree
exponent has no effect on APL of RSFTs: In the full range of
, behaves as a logarithmic scaling with the number of network nodes
(i.e. ), which is in sharp contrast to the well-known double
logarithmic scaling () previously obtained for uncorrelated
scale-free networks with . In addition, we present that some
scaling efficiency exponents of random walks are reliant on degree exponent
.Comment: The definitive verion published in Journal of Mathematical Physic
Scaling limits of slim and fat trees
We consider Galton--Watson trees conditioned on both the total number of
vertices and the number of leaves . The focus is on the case in which
both and grow to infinity and , with . Assuming the exponential decay of the offspring distribution, we show
that the rescaled random tree converges in distribution to Aldous' Continuum
Random Tree with respect to the Gromov--Hausdorff topology. The scaling depends
on a parameter which we calculate explicitly. Additionally, we
compute the limit for the degree sequences of these random trees.Comment: 37 pages, 2 figure
Random Sequential Renormalization of Networks I: Application to Critical Trees
We introduce the concept of Random Sequential Renormalization (RSR) for
arbitrary networks. RSR is a graph renormalization procedure that locally
aggregates nodes to produce a coarse grained network. It is analogous to the
(quasi-)parallel renormalization schemes introduced by C. Song {\it et al.}
(Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it et
al.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easier
to implement. In this first paper we apply RSR to critical trees and derive
analytical results consistent with numerical simulations. Critical trees
exhibit three regimes in their evolution under RSR: (i) An initial regime
, where is the number of nodes at some step in the
renormalization and is the initial size. RSR in this regime is described
by a mean field theory and fluctuations from one realization to another are
small. The exponent is derived using random walk arguments. The
degree distribution becomes broader under successive renormalization --
reaching a power law, with and a variance
that diverges as at the end of this regime. Both of these results
are derived based on a scaling theory. (ii) An intermediate regime for
, in which hubs develop, and
fluctuations between different realizations of the RSR are large. Crossover
functions exhibiting finite size scaling, in the critical region , connect the behaviors in the first two regimes. (iii)
The last regime, for , is characterized by the
appearance of star configurations with a central hub surrounded by many leaves.
The distribution of sizes where stars first form is found numerically to be a
power law up to a cutoff that scales as with
A Plotless Density Estimator based on the Asymptotic Limit of Ordered Distance Estimation Values
Estimation of tree density from point-tree distances is an attractive option for quick inventory of new sites, but estimators that are unbiased in clustered and dispersed situations have not been found. Noting that bias of an estimator derived from distances to the kth nearest neighbor from a random point tends to decrease with increasing k, a method is proposed for estimating the limit of an asymptotic function through a set of ordered distance estimators. A standard asymptotic model is derived from the limiting case of a clustered distribution. The proposed estimator is evaluated against 13 types of simulated generating processes, including random, clustered, dispersed and mixed. Performance is compared with ordered distance estimation of the same rank, and with fixed-area sampling with the same number of trees tallied. The proposed estimator consistently performs better than ordered distance estimation, and nearly as well as fixed area sampling in all but the most clustered situations. The estimator also provides information regarding the degree of clustering or dispersion
Assessment of Three Mitochondrial Genes (16S, Cytb, CO1) for Identifying Species in the Praomyini Tribe (Rodentia: Muridae)
The Praomyini tribe is one of the most diverse and abundant groups of Old World rodents. Several species are known to be involved in crop damage and in the epidemiology of several human and cattle diseases. Due to the existence of sibling species their identification is often problematic. Thus an easy, fast and accurate species identification tool is needed for non-systematicians to correctly identify Praomyini species. In this study we compare the usefulness of three genes (16S, Cytb, CO1) for identifying species of this tribe. A total of 426 specimens representing 40 species (sampled across their geographical range) were sequenced for the three genes. Nearly all of the species included in our study are monophyletic in the neighbour joining trees. The degree of intra-specific variability tends to be lower than the divergence between species, but no barcoding gap is detected. The success rate of the statistical methods of species identification is excellent (up to 99% or 100% for statistical supervised classification methods as the k-Nearest Neighbour or Random Forest). The 16S gene is 2.5 less variable than the Cytb and CO1 genes. As a result its discriminatory power is smaller. To sum up, our results suggest that using DNA markers for identifying species in the Praomyini tribe is a largely valid approach, and that the CO1 and Cytb genes are better DNA markers than the 16S gene. Our results confirm the usefulness of statistical methods such as the Random Forest and the 1-NN methods to assign a sequence to a species, even when the number of species is relatively large. Based on our NJ trees and the distribution of all intraspecific and interspecific pairwise nucleotide distances, we highlight the presence of several potentially new species within the Praomyini tribe that should be subject to corroboration assessments
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