There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Why doesn't your home page appear on the first page of search results, even when you query your own name? How do other web pages always appear at the top? What creates these powerful rankings? And how? The first book ever about the science of web page rankings, Google's PageRank and Beyond supplies the answers to these and other questions and more. The book serves two very different audiences: the curious science reader and the technical computational reader. The chapters build in mathematical sophistication, so that the first five are accessible to the general academic reader. While other chapters are much more mathematical in nature, each one contains something for both audiences. For example, the authors include entertaining asides such as how search engines make money and how the Great Firewall of China influences research. The book includes an extensive background chapter designed to help readers learn more about the mathematics of search engines, and it contains several MATLAB codes and links to sample web data sets. The philosophy throughout is to encourage readers to experiment with the ideas and algorithms in the text. Any business seriously interested in improving its rankings in the major search engines can benefit from the clear examples, sample code, and list of resources provided. Many illustrative examples and entertaining asides MATLAB code Accessible and informal style Complete and self-contained section for mathematics review

/pdf/google-s-pagerank-and-beyond-the-science-of-search-engine-33ceqwp28g.pdf

Google's PageRank and Beyond: The Science of Search Engine Rankings

Tables of integrals

We consider the set G consisting of graphs of fixed order and weighted edges. The vertex set of graphs in G will correspond to point masses and the weight for an edge between two vertices is a functional of the distance between them. We pose the problem of finding the best vertex positional configuration in the presence of an additional proximity constraint, in the sense that, the second smallest eigenvalue of the corresponding graph Laplacian is maximized. In many recent applications of algebraic graph theory in systems and control, the second smallest eigenvalue of Laplacian has emerged as a critical parameter that influences the stability and robustness properties of dynamic systems that operate over an information network. Our motivation in the present work is to "assign" this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph. In this venue, one would then be able to "synthesize" information graphs that have desirable system theoretic properties.

On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian

The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem and can be solved efficiently either numerically or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with $n$ nodes, the path has the largest value of optimal total effective resistance and the complete graph has the least.

/pdf/minimizing-effective-resistance-of-a-graph-1rz525m2td.pdf

Minimizing Effective Resistance of a Graph

Complete multipartite graphs and Braess edges

For an irreducible stochastic matrix T, we consider a certain condition
number (T), which measures the sensitivity of the stationary distribution
vector to perturbations in T, and study the extent to which the column sum
vector for T provides information on (T). Specifically, if cT is the column
sum vector for some stochastic matrix of order n, we define the set S(c) =
{A|A is an n × n stochastic matrix with column sum vector cT }. We then
characterise those vectors cT such that (T) is bounded as T ranges over the
irreducible matrices in S(c); for those column sum vectors cT for which  is
bounded, we give an upper bound on  in terms of the entries in cT , and
characterise the equality case.

/pdf/column-sums-and-the-conditioning-of-the-stationary-3qcq5phwiw.pdf

Column sums and the conditioning of the stationary distribution for a stochastic matrix

Traditionally, a parallel application is partitioned, mapped and then routed on a network of compute nodes where the topology of the interconnection network is fixed and known beforehand. Such a topology often comes with redundant links to accommodate the communication patterns of a wide range of applications. With recent advances in technology for optical circuit switches, it is now possible to construct a network with much fewer links, and to make the link endpoints configurable to suit the communication pattern of a given application. While this is economical (saving both links and the power to run them), it raises the difficult problem of how to configure the network and how to reconfigure it quickly when the application's communication pattern changes. In this paper, we propose the Kirchhoff index (KI) of a certain weighted graph related to the interconnection network as a proxy for its communication throughput. Our usage of this metric is based on a theoretical analogy between resistances in an electrical network and communication loads in the interconnection network. We show how mathematical techniques for reducing KI can be used to configure a network in a dramatically shorter time as compared to the current state-of-the-art scheme.

/pdf/a-network-configuration-algorithm-based-on-optimization-of-3yf40zg8nl.pdf

A Network Configuration Algorithm Based on Optimization of Kirchhoff Index

We consider spectral properties of the class T d,l, of (0,1)-matrices M which satisfy where Jk denotes the all ones matrix, and lk the identify matrix, of order k. These matrices generalize tournamet matrices. This paper establishes some of the basic properties for the eigenvalues of matrices in T d,l.

/pdf/on-multipartite-tournament-matrices-with-constant-team-size-3xqukq3br2.pdf

On multipartite tournament matrices with constant team size

https://www.hamilton.ie/skirkland/tnnfinal.pdf

Steve Kirkland

Papers

Complete multipartite graphs and Braess edges

Column sums and the conditioning of the stationary distribution for a stochastic matrix

A Network Configuration Algorithm Based on Optimization of Kirchhoff Index

On multipartite tournament matrices with constant team size

Totally nonnegative (0,1)-matrices