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

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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

Abstract We consider Laplacian fractional revival between two vertices of a graph X. Assume that it occurs at time τ between vertices 1 and 2. We prove that for the spectral decomposition L=∑r=0qθrEr L = \sum\nolimits_{r = 0}^q {{\theta _r}{E_r}} of the Laplacian matrix L of X, for each r = 0, 1, . . . , q, either Ere1 = Ere2, or Ere1 = −Ere2, depending on whether eiτθr equals to 1 or not. That is to say, vertices 1 and 2 are strongly cospectral with respect to L. We give a characterization of the parameters of threshold graphs that allow for Laplacian fractional revival between two vertices; those graphs can be used to generate more graphs with Laplacian fractional revival. We also characterize threshold graphs that admit Laplacian fractional revival within a subset of more than two vertices. Throughout we rely on techniques from spectral graph theory.

/pdf/fractional-revival-of-threshold-graphs-under-laplacian-3mc8zng6mi.pdf

Fractional Revival of Threshold Graphs Under Laplacian Dynamics

http://web.mat.bham.ac.uk/P.Butkovic/Grant%202/GursoyKirklandMasonSergeev2014.pdf

The Markov Chain Tree Theorem in commutative semirings and the State Reduction Algorithm in commutative semifields

A conjecture of Grone and Merris states that for any graph $G$, its Laplacian spectrum, $\Lambda(G)$, is majorized by its conjugate degree sequence, $D^*(G)$. That conjecture prompts an investigation of the relationship between $\Lambda(G)$ and $D^*(G),$ and Merris has characterized the graphs $G$ for which the multisets $\Lambda(G)$ and $D^*(G)$ are equal. In this paper, we provide a constructive characterization of the graphs $G$ for which $\Lambda(G)$ and $D^*(G)$ share all but two elements.

/pdf/near-threshold-graphs-wxbu2m42t2.pdf

Near Threshold Graphs

On the sequence of powers of a stochastic matrix with large exponent

Traditionally, a parallel application is partitioned, mapped and then routed on a network of compute nodes where the topology of the interconnection network is known beforehand and is homogeneous. However, such homogeneity in interconnects is rarely required or needed for several important classes of applications. Nevertheless such interconnects are designed this way, i.e., with redundant links, to accommodate the communication patterns of a wide range of applications. However, with recent advances in technology for optical circuit switches, it is now possible to construct 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. Since the space of all configurable topologies is large and determining the quality of a topology is a time-consuming process, it is not feasible to explore the entire space. One way of dealing with this limitation is to start the search from a ``good'' initial topology and then conduct a restricted search around it. The success of such a strategy crucially depends on the choice of the initial or seed topology. In the past, such an initial topology was computed by mimicking the communication requirements of the application. In this paper, we propose a different approach by showing that interconnect topologies such as chordal rings(circulant graphs) chosen based on metrics such as bisection width and average shortest path length can provide a better starting point. The topology obtained by searching around such an initial topology provides almost as good a performance as an application-specific initial topology, and the search time is significantly reduced.

/pdf/seeds-for-a-heterogeneous-interconnect-51a5pn6teo.pdf

Steve Kirkland

Papers

Fractional Revival of Threshold Graphs Under Laplacian Dynamics

The Markov Chain Tree Theorem in commutative semirings and the State Reduction Algorithm in commutative semifields

Near Threshold Graphs

On the sequence of powers of a stochastic matrix with large exponent

Seeds for a Heterogeneous Interconnect