The graph bottleneck identity

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$. Five types of matrices that determine transitional measures for a digraph are considered, namely, the matrices of path weights, connection reliabilities, route weights, and the weights of in-forests and out-forests. The results obtained have undirected counterparts. In [P. Chebotarev, A class of graph-geodetic distances generalizing the shortest-path and the resistance distances, Discrete Appl. Math., URL http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic

Sizing the length of complex networks

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short

Incidences between points and generalized spheres over finite fields and related problems

Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in \mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in \mathbb{F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace x\in \mathbb{F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{F}_q^d, r\in \mathbb{F}_q$. We prove bounds on the number of incidences between a point set $\mathcal{P}$ and a $Q$-sphere set $\mathcal{S}$, denoted by $I(\mathcal{P},\mathcal{S})$, as the following. $$| I(\mathcal{P},\mathcal{S})-\frac{|\mathcal{P}||\mathcal{S}|}{q}|\le q^{d/2}\sqrt{|\mathcal{P}||\mathcal{S}|}.$$ We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings $\mathbb{Z}_q$ where $q$ is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the number of incidences between a random point set and a random $Q$-sphere set in $\mathbb{F}_q^d$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.Comment: to appear in Forum Mat

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

Matrices of forests, analysis of networks, and ranking problems

The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the stochastic matrix of out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171. Published in Proceedings of the First International Conference on Information Technology and Quantitative Management (ITQM 2013). This version contains some corrections and addition