Wednesday, September 19, 2018

Tuesday, September 18, 2018

Trying to Divide by 0 on a Mechanical Calculator


Monday, September 17, 2018

"Inferring Parameters of Prey Switching in a 1 Predator–2 Prey Plankton System with a Linear Preference Tradeoff"

Another of my papers came out in final published form today.

Title: Inferring Parameters of Prey Switching in a 1 Predator–2 Prey Plankton System with a Linear Preference Tradeoff

Authors: Sofia H. Piltz, Lauri Harhanen, Mason A. Porter, and Philip K. Maini

Abstract: We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models’ ability to fit data on freshwater plankton. We model the predator’s switch from one prey to the other in two different ways: (i) smooth switching using a hyperbolic tangent function; and (ii) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model results quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German–Swiss–Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey-switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate–algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.

Note: This paper is actually the third in a series of papers that arose from Sofia's doctoral thesis. In all three, we studied prey switching in plankton as a dynamical system. However, although we were concerned in all three papers with the same ecological situation, we modeled it in three different mathematical ways: using piecewise-smooth dynamical systems (in paper 1), using fast–slow dynamical systems (in paper 2), and using smooth dynamical systems (in this paper). It is really important to model the same phenomenon in different ways and to compare the qualitative features of the different models against each other as well as to empirical data. I am really pleased with this effort, which Sofia did a superb job of leading.

"Frequency-Based Brain Networks: From a Multiplex Framework to a Full Multilayer Description"

One of my papers just came out in final form. Here are some details.

Title: Frequency-Based Brain Networks: From a Multiplex Framework to a Full Multilayer Description

Authors: Javier M. Buldú and Mason A. Porter

Abstract: We explore how to study dynamical interactions between brain regions by using functional multilayer networks whose layers represent different frequency bands at which a brain operates. Specifically, we investigate the consequences of considering the brain as (i) a multilayer network, in which all brain regions can interact with each other at different frequency bands; and as (ii) a multiplex network, in which interactions between different frequency bands are allowed only within each brain region and not between them. We study the second-smallest eigenvalue λ2 of the combinatorial supra-Laplacian matrix of both the multiplex and multilayer networks, as λ2 has been used previously as an indicator of network synchronizability and as a biomarker for several brain diseases. We show that the heterogeneity of interlayer edge weights and, especially, the fraction of missing edges crucially modify the value of λ2, and we illustrate our results with both synthetic network models and real data obtained from resting-state magnetoencephalography. Our work highlights the differences between using a multiplex approach and a full multilayer approach when studying frequency-based multilayer brain networks.

Bonus: This paper has an easter egg. Can you find it? (Hint: This is Spinal Tap.)

Tuesday, September 11, 2018

A Very Exciting (and Dangerous?) Mathematics Conference

P.S. I wonder what they actually have in mind with that session? I'm at a loss. I suppose it's clearer to the people who are actually attending the conference?

Tuesday, September 04, 2018

Referee Reports for the Watts–Strogatz 'Small World' Paper

This is a great document!

Steve Strogatz posted his referee reports (with annotations for their revision plans) about his genre-defining 1998 ‘small world’ paper with Duncan Watts.

It just goes to show that misguided, skeptical reports (see Referee 2 in the upper right) happen to the best of us.

Referee 2 was confused about the implications of the work to systems other than those studied (and possibly even to those applications)…

(The source is Steve's tweet. I posted the picture separately to orient it correctly.)

Monday, September 03, 2018

Googly Eyes, and Freshness of Fish

Well, this story is fishy. Wow. (And oy vey.)

The headline pretty much sums it up: Store shut down after owners caught sticking googly eyes on fish to look fresher

P.S. There are also other uses for googly eyes.

(Tip of the cap to Renée O'Rear Handley.)

Hindsight's 2020 Presidential Campaign

I don't know about you, but I'll be voting for the team of Hindsight/Forethought in 2020.

Saturday, September 01, 2018

What Happens in Sydney Stays in Sydney

I am at LAX for my trip Down Under!

The only previous time I was in Australia was in 2005 for my interview at University of Sydney. (I am visiting a collaborator at Macquarie University, and I'll be giving one teach each there and at University of Sydney.)

"Isomorphisms in Multilayer Networks"

This paper first appeared on the arXiv in 2015 and was published in advanced access about a year ago. It finally has its final volume and page coordinates, so I am finally writing a blog entry about. At this stage, the paper certainly doesn't feel "new" anymore, but it has some useful ideas in it, and it's also my first paper in an IEEE journal. This paper is also rather unusual for me, in that my coauthor Mikko Kivelä and I decided that the clearest way to present things would be to write the paper in definition–theorem–proof format. I almost never write papers that way. Anyway, here are a few details.

Title: Isomorphisms in Multilayer Networks

Authors: Mikko Kivelä and Mason A. Porter

Abstract: We extend the concept of graph isomorphisms to multilayer networks with any number of “aspects” (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomorphisms for multiplex networks, temporal networks, networks with both of these features, and more. We reduce each of the multilayer network isomorphism problems to a graph isomorphism problem, where the size of the graph isomorphism problem grows linearly with the size of the multilayer network isomorphism problem. One can thus use software that has been developed to solve graph isomorphism problems as a practical means for solving multilayer network isomorphism problems. Our theory lays a foundation for extending many network analysis methods—including motifs, graphlets, structural roles, and network alignment—to any multilayer network.