15 research outputs found
Properties of a random attachment growing network
In this study we introduce and analyze the statistical structural properties
of a model of growing networks which may be relevant to social networks. At
each step a new node is added which selects 'k' possible partners from the
existing network and joins them with probability delta by undirected edges. The
'activity' of the node ends here; it will get new partners only if it is
selected by a newcomer. The model produces an infinite-order phase transition
when a giant component appears at a specific value of delta, which depends on
k. The average component size is discontinuous at the transition. In contrast,
the network behaves significantly different for k=1. There is no giant
component formed for any delta and thus in this sense there is no phase
transition. However, the average component size diverges for delta greater or
equal than one half.Comment: LaTeX, 19 pages, 6 figures. Discussion section, comments, a new
figure and a new reference are added. Equations simplifie
Subgraphs in random networks
Understanding the subgraph distribution in random networks is important for
modelling complex systems. In classic Erdos networks, which exhibit a
Poissonian degree distribution, the number of appearances of a subgraph G with
n nodes and g edges scales with network size as \mean{G} ~ N^{n-g}. However,
many natural networks have a non-Poissonian degree distribution. Here we
present approximate equations for the average number of subgraphs in an
ensemble of random sparse directed networks, characterized by an arbitrary
degree sequence. We find new scaling rules for the commonly occurring case of
directed scale-free networks, in which the outgoing degree distribution scales
as P(k) ~ k^{-\gamma}. Considering the power exponent of the degree
distribution, \gamma, as a control parameter, we show that random networks
exhibit transitions between three regimes. In each regime the subgraph number
of appearances follows a different scaling law, \mean{G} ~ N^{\alpha}, where
\alpha=n-g+s-1 for \gamma<2, \alpha=n-g+s+1-\gamma for 2<\gamma<\gamma_c, and
\alpha=n-g for \gamma>\gamma_c, s is the maximal outdegree in the subgraph, and
\gamma_c=s+1. We find that certain subgraphs appear much more frequently than
in Erdos networks. These results are in very good agreement with numerical
simulations. This has implications for detecting network motifs, subgraphs that
occur in natural networks significantly more than in their randomized
counterparts.Comment: 8 pages, 5 figure
On the topological classification of binary trees using the Horton-Strahler index
The Horton-Strahler (HS) index has been shown to
be relevant to a number of physical (such at diffusion limited aggregation)
geological (river networks), biological (pulmonary arteries, blood vessels,
various species of trees) and computational (use of registers) applications.
Here we revisit the enumeration problem of the HS index on the rooted,
unlabeled, plane binary set of trees, and enumerate the same index on the
ambilateral set of rooted, plane binary set of trees of leaves. The
ambilateral set is a set of trees whose elements cannot be obtained from each
other via an arbitrary number of reflections with respect to vertical axes
passing through any of the nodes on the tree. For the unlabeled set we give an
alternate derivation to the existing exact solution. Extending this technique
for the ambilateral set, which is described by an infinite series of non-linear
functional equations, we are able to give a double-exponentially converging
approximant to the generating functions in a neighborhood of their convergence
circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos
correcte
Generating functions for multi-labeled trees
Multi-labeled trees are a generalization of phylogenetic trees that are used, for example, in the study of gene versus species evolution and as the basis for phylogenetic network construction. Unlike phylogenetic trees, in a leaf-multi-labeled tree it is possible to label more than one leaf by the same element of the underlying label set. In this paper we derive formulae for generating functions of leaf-multi-labeled trees and use these to derive recursions for counting such trees. In particular,weprove results which generalize previous theorems by Harding on so-called tree-shapes, and by Otter on relating the number of rooted and unrooted phylogenetic trees
On weighted multiway cuts in trees
A min-max theorem is developed for the multiway cut problem of edge-weighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynamic programming type algorithm for this more general problem from biology and show that our min-max theorem does not apply to it. © 1994 The Mathematical Programming Society, Inc
A Fourier inversion formula for evolutionary trees
Abstract-We establish a pair of identities, which will provide a useful tool in the reconstruction of evolutionary trees in Kimura’s 3-parameter model. The starting point of this paper was an attempt for a better understanding and generalization of an Hadamard inverse pair of formulae, which was used in statistics by Cooper [l], in image processing by Andrews [2, Chapters 6,7], and in information theory by Whelchel and Guinn [3]. Recently, Hendy and Penny [4] applied this technique to the spectral analysis of phylogenetic data, for two-state character sequences, and they asked for generalization to four-state character sequences, which is the form of a nucleotide sequence. The most invaluable tool for our work was [5], where discrete Fourier analysis is applied to somewhat similar problems. Let us start with the original theorem of Hendy and Penny [6]. Suppose we are given a tree T with leaf set L, out of which one is a root called R. Set ILI = n. Toss a coin independently for every edge e of the tree with probability and probability 1- pe for a tail. Let u denote a subset of L \ {R}. Let fc denote pe for a head the probabilit