Maximum flow and topological structure of complex networks

The problem of sending the maximum amount of flow $q$ between two arbitrary nodes $s$ and $t$ of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between $s$ and $t$. The average of $q$ over all node pairs with smaller degree $k_{\rm min}$ is $_{k_{\rm min}} \simeq c k_{\rm min}$ for large $k_{\rm min}$ with $c$ a constant implying that the statistics of $q$ is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and $q$ can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient $c$. The applicability of our results to real world networks is tested for the Internet at the autonomous system level.Comment: 7 pages, 4 figure

The complexity of finding two disjoint paths with min-max objective function

AbstractGiven a network G = (V,E) and two vertices s and t, we consider the problem of finding two disjoint paths from s to t such that the length of the longer path is minimized. The problem has several variants: The paths may be vertex-disjoint or arc-disjoint and the network may be directed or undirected. We show that all four versions as well as some related problems are strongly NP-complete. We also give a pseudo-polynomial-time algorithm for the acyclic directed case

An Algorithm for Finding Two Node-Disjoint Paths in Arbitrary Graphs

Given two distinct vertices (nodes) source s and target t of a graph G = (V, E), the two node-disjoint paths problem is to identify two node-disjoint paths between s ∈ V and t ∈ V. Two paths are node-disjoint if they have no common intermediate vertices. In this paper, we present an algorithm with O(m)-time complexity for finding two node-disjoint paths between s and t in arbitrary graphs where m is the number of edges. The proposed algorithm has a wide range of applications in ensuring reliability and security of sensor, mobile and fixed communication networks

Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$, where $\mathcal{T}$ is the rooted real tree naturally associated with a Brownian excursion and φ is a random continuous function from $\mathcal{T}$ into ℝd such that, conditional on $\mathcal{T}$, φ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in ℝd run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric $d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$. Applications of this result include the recovery of the spatial tree $(\mathcal{T},\phi)$ from the set $\mathcal{S}$ alone, which implies in turn that a Dawson–Watanabe super-process can be recovered from its range. Furthermore, $d_{\mathcal{S}}$ can be used to construct a Brownian motion on $\mathcal{S}$, which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained