The scaling of the minimum sum of edge lengths in uniformly random trees

[Abstract] The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.Ministerio de Economía, Industria y Competitividad; TIN2013-48031- C4-1-PXunta de Galicia; R2014/034Agència de Gestió d'Ajuts Universitaris i de Recerca; 2014SGR 890Ministerio de Economía, Industria y Competitividad; TIN2014-57226-PMinisterio de Economía, Industria y Competitividad; FFI2014-51978-C2-2-

The sum of edge lengths in random linear arrangements

Spatial networks are networks where nodes are located in a space equipped with a metric. Typically, the space is two-dimensional and until recently and traditionally, the metric that was usually considered was the Euclidean distance. In spatial networks, the cost of a link depends on the edge length, i.e. the distance between the nodes that define the edge. Hypothesizing that there is pressure to reduce the length of the edges of a network requires a null model, e.g., a random layout of the vertices of the network. Here we investigate the properties of the distribution of the sum of edge lengths in random linear arrangement of vertices, that has many applications in different fields. A random linear arrangement consists of an ordering of the elements of the nodes of a network being all possible orderings equally likely. The distance between two vertices is one plus the number of intermediate vertices in the ordering. Compact formulae for the 1st and 2nd moments about zero as well as the variance of the sum of edge lengths are obtained for arbitrary graphs and trees. We also analyze the evolution of that variance in Erdos-Renyi graphs and its scaling in uniformly random trees. Various developments and applications for future research are suggested

Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models

We study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion $\delta$ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as $1 + \Theta(\delta^2)$. We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.Comment: 24 pages, 3 figure

Random Planar Lattices and Integrated SuperBrownian Excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R-L of the support of the one-dimensional ISE. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat's construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and http://www.iecn.u-nancy.fr/~chassain

Optimal Path and Minimal Spanning Trees in Random Weighted Networks

We review results on the scaling of the optimal path length in random networks with weighted links or nodes. In strong disorder we find that the length of the optimal path increases dramatically compared to the known small world result for the minimum distance. For Erd\H{o}s-R\'enyi (ER) and scale free networks (SF), with parameter $\lambda$ ($\lambda >3$), we find that the small-world nature is destroyed. We also find numerically that for weak disorder the length of the optimal path scales logaritmically with the size of the networks studied. We also review the transition between the strong and weak disorder regimes in the scaling properties of the length of the optimal path for ER and SF networks and for a general distribution of weights, and suggest that for any distribution of weigths, the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter $Z=\ell_{\infty}/A$ where $A$ plays the role of the disorder strength, and $\ell_{\infty}$ is the length of the optimal path in strong disorder. The relation for $A$ is derived analytically and supported by numerical simulations. We then study the minimum spanning tree (MST) and show that it is composed of percolation clusters, which we regard as "super-nodes", connected by a scale-free tree. We furthermore show that the MST can be partitioned into two distinct components. One component the {\it superhighways}, for which the nodes with high centrality dominate, corresponds to the largest cluster at the percolation threshold which is a subset of the MST. In the other component, {\it roads}, low centrality nodes dominate. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network.Comment: review, accepted at IJB

Critical random forests

Let $F(N,m)$ denote a random forest on a set of $N$ vertices, chosen uniformly from all forests with $m$ edges. Let $F(N,p)$ denote the forest obtained by conditioning the Erdos-Renyi graph $G(N,p)$ to be acyclic. We describe scaling limits for the largest components of $F(N,p)$ and $F(N,m)$, in the critical window $p=N^{-1}+O(N^{-4/3})$ or $m=N/2+O(N^{2/3})$. Aldous described a scaling limit for the largest components of $G(N,p)$ within the critical window in terms of the excursion lengths of a reflected Brownian motion with time-dependent drift. Our scaling limit for critical random forests is of a similar nature, but now based on a reflected diffusion whose drift depends on space as well as on time

Critical random graphs: limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure