Related papers: Optimal Path and Minimal Spanning Trees in Random …
In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the…
Consider~\(n\) nodes distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and~\(MSTC_n\) denotes…
We study the current flow paths between two edges in a random resistor network on a $L\times L$ square lattice. Each resistor has resistance $e^{ax}$, where $x$ is a uniformly-distributed random variable and $a$ controls the broadness of…
The minimum spanning tree of a graph is a well-studied structure that is the basis of countless graph theoretic and optimization problem. We study the minimum spanning tree (MST) perturbation problem where the goal is to spend a fixed…
Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less…
Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdos and Renyi (ER), evolving networks introduced by…
Quantitative descriptions of network structure in big data can provide fundamental insights into the function of interconnected complex systems. Small-world structure, commonly diagnosed by high local clustering yet short average path…
A long series of recent results and breakthroughs have led to faster and better distributed approximation algorithms for single source shortest paths (SSSP) and related problems in the CONGEST model. The runtime of all these algorithms,…
We study the distribution $P(\sigma)$ of the equivalent conductance $\sigma$ for Erd\H{o}s-R\'enyi (ER) and scale-free (SF) weighted resistor networks with $N$ nodes. Each link has conductance $g\equiv e^{-ax}$, where $x$ is a random number…
Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph, and use it to develop a continuum perturbation expansion for the MST on…
We consider two variational models for transport networks, an urban planning and a branched transport model, in which the degree of network complexity and ramification is governed by a small parameter $\varepsilon>0$. Smaller $\varepsilon$…
Let $G=(V,E)$ be an $n$-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a {\em single-source shortest-path tree} (SPT) of $G$ with a \emph{sparse} set of \emph{auxiliary} edges selected from $E$, in…
Minimum spanning trees (MSTs) provide a convenient representation of datasets in numerous pattern recognition activities. Moreover, they are relatively fast to compute. In this paper, we quantify the extent to which they are meaningful in…
Many real world networks (graphs) are observed to be 'small worlds', i.e., the average path length among nodes is small. On the other hand, it is somewhat unclear what other average path length values networks can produce. In particular, it…
Average trapping time (ATT) is central in the trapping problem since it is a key indicator characterizing the efficiency of the problem. Previous research has provided the scaling of a lower bound of the ATT for random walks in general…
The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in scaled coordinates that respect these…
Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees…
The aim of the study was to compare the epidemic spread on static and dynamic small-world networks. The network was constructed as a 2-dimensional Watts-Strogatz model (500x500 square lattice with additional shortcuts), and the dynamics…
In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square,all edges must respect the ``coordinatewise'' partial order and there must be a directed path from each…
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…