This year marks the publication of the 60th volume of the venerable SIAM Review. As has become traditional when journals mark anniversaries, the editors of SIREV have compiled a list of the journal’s 10 most read articles. These lists are always interesting, both for what shows up and for what is missing (from my purely subjective point of view).

Number 1 on the list is a modern classic, The Structure and Function of Complex Networks by Mark Newman. At the time this paper appeared in 2003, network science was just getting hot. Newman’s review, which laid out all of the foundational ideas of the field in a very clear way, quickly became the standard reference for definitions and basic results about various kinds of networks. It didn’t hurt that Newman had recently made a splash in the scientific community by analyzing scientific collaboration networks: given that everyone’s favorite topic is themselves, scientists were naturally intrigued by a quantitative study of their own behavior. All kidding aside, Newman’s SIAM Review article has been hugely influential. All kinds of networks have been analyzed using these methods, ranging from social networks to protein interaction networks. As if having the number 1 paper in this list wasn’t enough, Newman is also a coauthor of the 2009 paper Power-Law Distributions in Empirical Data, which is number 6 on the list. The latter paper deals with statistical methods for determining whether or not a data set fits a power-law distribution.

Desmond Higham has the singular distinction of having two singly authored papers on this list, both of them from the Education section of SIAM Review, but both wonderful introductions to their topics for young scientists, or for old scientists who need to learn new tricks. At number 3 on the list, we have An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, which presents the simplest introduction to stochastic differential equations I have ever had the pleasure to read. Then at number 4, we have Modeling and Simulating Chemical Reactions. In the latter, Higham walks us through three levels of description of chemical equations, as Markov chains in the space of species populations, then using the chemical Langevin equation, and finally in the bulk mass-action limit. He derives each method from the preceding one, essentially by focusing on computational methods for simulating them, and then showing that these methods simplify as various assumptions are introduced. I think that these two papers of Higham’s have been successful not only because of his exceptionally clear writing, but because he also provided Matlab code for all his examples. The interested reader can therefore go from reading these papers to doing their own calculations rather quickly. I learned a lot from these papers myself, and I’ve used both of them in a graduate course on stochastic processes. They’re just fantastic resources.

One paper that didn’t appear, and that I had guessed would be there before I looked at the list, is the classic 1978 paper Nineteen Dubious Ways to Compute the Exponential of a Matrix by Cleve Moler (original developer of Matlab, and founder of MathWorks, the company that sells Matlab) and Charles Van Loan (author with the late Gene Golub of the book Matrix Computations, known by people in numerical analysis simply as “Golub and Van Loan”). It’s possible that it didn’t make the list because an expanded version of the original was published in the SIAM Review in 2003, and that this paper’s reads are therefore split between the two versions. However, it’s still a surprise. This is one of those papers that is often mentioned, in part I’m sure because of its mischievous (if accurate) title, but also because it discusses an important problem—matrix exponentials show up all over the place—and does so with exceptional clarity.

There have been lots of papers on singular perturbation theory and the related boundary-layer problems in the SIAM Review over the years, which is perhaps not surprising given how central these methods are to a lot of applied mathematics. In fact, in 1994, the SIAM Review published an issue that contained a collection of papers on singular perturbation methods. I would have thought that at least one paper on this topic would have made the list. My all-time favorite SIREV paper is in fact Lee Segel’s Simplification and Scaling, which I routinely assign as reading to graduate students who need an introduction to the basic ideas of singular perturbation theory, followed closely by Lee Segel and Marshall Slemrod’s The Quasi-Steady-State Assumption: A Case Study In Perturbation, which derives the steady-state approximation for the Michaelis-Menten mechanism using the machinery of singular perturbation theory. The full power of these methods is made evident when they derive a more general condition for the validity of the steady-state approximation than had previously been obtained. The late Lee Segel was one of the great pioneers of mathematical biology. He worked on every important problem in the field, from oscillators to pattern formation, and left us some beautiful applied mathematics. He also left us an absolutely wonderful book, Mathematics Applied to Deterministic Problems in the Natural Sciences, coauthored with Chia-Chiao Lin, who has sadly also left us. Marshall Slemrod is, fortunately, still very much alive. Marshall is probably best known for his elegant work in fluid dynamics, but he has worked on quite a variety of problems in applied mathematics over his long and distinguished career.

It’s interesting to compare SIAM list of “most read” papers to the most cited papers from SIREV (Web of Science search, Oct. 8, 2018). Here they are:

1. Mark Newman’s The Structure and Function of Complex Networks, cited 8333 times, more than twice as often as any other paper published in SIREV. No great surprise there.
2. Fractional Brownian Motions, Fractional Noises and Applications by Benoit Mandelbrot and John van Ness (3554 citations). Perhaps this one should have been on my radar, although I’ll admit that I have never read it. I’ll put it on my reading list now.
3. Power-Law Distributions in Empirical Data, Newman’s other entry on the most-read list, which interestingly comes out much higher in the most-cited ranking that in the most-read list, where it occupies the number 6 spot, with 2885 citations.
4. Semidefinite Programming by Lieven Vandenberghe and Stephen Boyd (2086 citations)
5. Tensor Decompositions and Applications by Tamara G. Kolda and Brett W. Bader (2042 citations, number 2 on the most-read list)
6. Analysis of Discrete Ill-Posed Problems by Means of the L-Curve by Per Christian Hansen (1870 citations)
7. The Mathematics of Infectious Diseases by Herbert W. Hethcote (1813 citations)
8. Atomic Decomposition by Basis Pursuit by Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders (1647 citations)
9. On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws by Amiram Harten, Peter D. Lax, and Bram van Leer (1493 citations). This is the kind of paper we often see on most-cited lists because it discusses practical issues in the numerical solution of PDEs.
10. Mixture Densities, Maximum Likelihood and the EM Algorithm by Richard A. Redner and Homer F. Walker (1256 citations)

It’s interesting, and perhaps a little surprising, how little overlap there is between the most-read and most-cited lists. Just three papers show up on both lists! This is another manifestation of the well-known problem of trying to use any single metric to determine the influence of a paper.

There are other SIREV papers that I really love, even though I wouldn’t have expected them to make this list, sometimes because of their real-world applications, and sometimes just because they describe very clearly some beautiful applied mathematics.

Bryan and Leise’s The $25,000,000,000 Eigenvector: The Linear Algebra behind Google, explains the mathematics behind the Google search engine. It’s both a great educational article on large, sparse matrix eigenvector calculations, and an interesting peek into the workings of one of the most important technologies of our time.

James Keener’s article on The Perron-Frobenius Theorem and the Ranking of Football Teams is a great read, and a fun way to introduce students to the powerful Perron-Frobenius theorem. James Keener has been one of the leading figures in mathematical biology over the last several decades, and is the author, with James Sneyd, of the highly regarded textbook Mathematical Physiology.

I also really enjoyed Diaconis and Freedman’s Iterated Random Functions, which describes some lovely mathematics that connects together Markov chains and fractals, among other things. Persi Diaconis is perhaps best known for his analysis of card shuffling and other games of chance. In fact, another paper of his in the SIAM Review (with Susan Holmes and Richard Montgomery) on Dynamical Bias in the Coin Toss is also a fantastic read.

I could go on, but I think I’ll stop here.

You may have noticed some recurring themes in this post. One is that there is some great writing in the SIAM Review. In fact, I would say that this is a hallmark of SIREV. Regardless of the author or topic, the final published paper always seems to be a great piece of scientific literature. Of course, I might be a little bit biased, having published a Classroom Note in the SIAM Review myself. Another theme of this post is the number of outstanding scientists who have written for SIREV. SIREV makes room for up and comers, but it also regularly gives us the benefit of reading papers by people who have spent decades deepening their knowledge of their respective areas.

So happy birthday, SIAM Review, and many happy returns!