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We give a characterization of equilibrium measures for $p$-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For $p=2$, this provides, in the special case of trees, a converse to a theorem of Benjamini and…

This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on X_1,...,X_{N_n} where X_1,X_2,... are i.i.d. in \Re^2 with distribution F and N_n is independent of the X_i and…

We study the local profiles of trees. We show that, in contrast with the situation for general graphs, the limit set of k-profiles of trees is convex. We initiate a study of the defining inequalities of this convex set. Many challenging…

Let n points be taken at random on a circle of unit circumference and clockwise ordered. Uniform spacings are defined as the clockwise arc-lengths between the successive points from this sample. We are interested in the asymptotic behavior…

We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism.

Symmetry -- invariance to certain operators -- is a fundamental concept in many branches of physics. We propose ways to measure symmetric properties of vertices, and their surroundings, in networks. To be stable to the randomness inherent…

Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between…

Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by…

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with…

A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and…

The graph edit distance is used for comparing graphs in various domains. Due to its high computational complexity it is primarily approximated. Widely-used heuristics search for an optimal assignment of vertices based on the distance…

We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…

Centrality indices are used to rank the nodes of a graph by importance: this is a common need in many concrete situations (social networks, citation networks, web graphs, for instance) and it was discussed many times in sociology,…

We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…

We obtain general inequalities constraining the difference between the average of an arbitrary function of a phenotypic trait, which includes the fitness landscape of the trait itself, in the presence or in the absence of natural selection.…

In this paper we study a new variant of graph arboricity, which requires all the forests to have the same number of edges (up to a difference of 1). We prove that the new variant, which we call equitable arboricity, is equivalent to…

We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…

The comprehensive characterization of the structure of complex networks is essential to understand the dynamical processes which guide their evolution. The discovery of the scale-free distribution and the small world property of real…

An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. However, when k = 1 the complexity of the problem is polynomial. In this paper, some tools for…

We generalize finite-sample bounds for convex clustering to the setting where affinity weights appearing in the objective correspond to a general connected graph. These bounds and their analysis lead to a better understanding of clustering…