"Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites"

One of my paper was published in final form last week. Here are some details.

Title: Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites

Authors: Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter

Abstract: It is important to choose the geographical distributions of public resources in a fair and equitable manner. However, it is complicated to quantify the equity of such a distribution; important factors include distances to resource sites, availability of transportation, and ease of travel. We use persistent homology, which is a tool from topological data analysis, to study the availability and coverage of polling sites. The information from persistent homology allows us to infer holes in a distribution of polling sites. We analyze and compare the coverage of polling sites in Los Angeles County and five cities (Atlanta, Chicago, Jacksonville, New York City, and Salt Lake City), and we conclude that computation of persistent homology appears to be a reasonable approach to analyzing resource coverage.

What Happens in Glasgow Stays in Glasgow

I am off to Glasgow for the 2024 World Science-Fiction Convention (i.e., WorldCon). I am involved in a talk and some panels.

"The Topology of Data"

Our introduction to a topological data analysis (TDA) for a general physics audience was published today in Physics Today. Here are some details.

Title: The Topology of Data

Authors: Mason A. Porter, Michelle Feng, and Eleni Katifori

Lede: Topological data analysis, which allows systematic investigations of the “shape” of data, has yielded fascinating insights into many physical systems.

"Analysis of Spatial and Spatiotemporal Anomalies Using Persistent Homology: Case Studies with COVID-19 Data"

I'm posting about one of my papers that was published in journal form a couple of months ago. I waited for a while because the journal made surprise, unwanted changes after the galley-proof stage — and unsurprisingly I objected very strongly to what they did — and I tried and failed to get those surprise changes addressed. They are very small, but they annoy me (and, as a matter of principle, they should not have made surprise wording changes between the version that we approved and the version that we published). Anyway, here are some details about the article.

Title: Analysis of Spatial and Spatiotemporal Anomalies Using Persistent Homology: Case Studies with COVID-19 Data

Authors: Abigail Hickok, Deanna Needell, and Mason A. Porter

Abstract: We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a data set and quantify the persistence of such voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one- to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (which are an approach for visualizing PH), to track how the locations of the anomalies change with time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code at a single point in time. Second, we study a year-long data set of COVID-19 case rates in neighborhoods of the city of Los Angeles.

"Topological Data Analysis of Spatial Systems"

A book chapter of ours was just published in final form. It is Chapter 16 in this book. Here are some other details about it.

Title: Topological Data Analysis of Spatial Systems

Authors: Michelle Feng, Abigail Hickok, and Mason A. Porter

Abstract: In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review a recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two cases studies: street networks in Shanghai and anomalies in the spread of COVID-19 infections. We then summarize our results and provide an outlook on TDA in spatial systems.

"Connecting the Dots: Discovering the “Shape” of Data"

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

Title: Connecting the Dots: Discovering the “Shape” of Data

Authors: Michelle Feng, Abighail Hickok, Yacoub H. Kureh, Mason A. Porter, and Chad M. Topaz

Abstract: Scientists use a mathematical subject called topology to study the shapes of objects. An important part of topology is counting the number of pieces and the number of holes in an object, and researchers use this information to group objects into different types. For example, a doughnut has the same number of holes and the same number of pieces as a teacup with one handle, but it is different from a ball. In studies that resemble activities like “connect-the-dots,” scientists use ideas from topology to study the “shape” of data. Ideas and methods from topology have been used to study the branching structures of veins in leaves, voting in elections, flight patterns in models of bird flocking, and more.

Here is my tweet, in case you want to share it on social media.

Our introduction to topological data analysis (TDA) for teenagers and preteens is finally out in final form in Frontiers for Young Minds: https://t.co/6KQF2yyJUn@michellehfeng, Abby Hickok, Yacoub Kureh, MAP, & @chadtopaz

(plus special guest appearances by several Pokémon)

— Mason Porter (@masonporter) March 18, 2021

"Persistent Homology of Geospatial Data: A Case Study with Voting"

A new paper of mine is out in final form today. Here are some details.

Title: Persistent Homology of Geospatial Data: A Case Study with Voting

Authors: Michelle Feng and Mason A. Porter

Abstract: A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct Vietoris–Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing simplicial complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexes from two-dimensional geospatial data (such as maps). We apply these methods to a California precinct-level voting data set, and we thereby demonstrate that our new constructions can capture geometric characteristics that are missed by distancebased constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish short-persistence features that occur only for a narrow range of distance scales (e.g., voting patterns in densely populated cities) from short-persistence noise by incorporating information about other spatial relationships between regions.

"Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence"

One of my papers was just published in final form today. Here are some details.

Title: Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Authors: Michelle Feng and Mason A. Porter

Abstract: Spatial networks are ubiquitous in social, geographical, physical, and biological applications. To understand the large-scale structure of networks, it is important to develop methods that allow one to directly probe the effects of space on structure and dynamics. Historically, algebraic topology has provided one framework for rigorously and quantitatively describing the global structure of a space, and recent advances in topological data analysis have given scholars a new lens for analyzing network data. In this paper, we study a variety of spatial networks—including both synthetic and natural ones—using topological methods that we developed recently for analyzing spatial systems. We demonstrate that our methods are able to capture meaningful quantities, with specifics that depend on context, in spatial networks and thereby provide useful insights into the structure of those networks. We illustrate these ideas with examples of synthetic networks and dynamics on them, street networks in cities, snowflakes, and webs that were spun by spiders under the influence of various psychotropic substances.

An Awesome Ph.D. Defense by Dr. Yacoub Kureh!

Congratulations to my Ph.D. student Yacoub Kureh on an awesome defense of his thesis!

CONGRATULATIONS to Dr. Yacoub Kureh (@Yacoub_Kureh) on a successful and very impressive defense of his PhD thesis, called "Nonlinear opinion models and other networked systems"!!! pic.twitter.com/fXDNIE1GRX

— Mason Porter (@masonporter) May 28, 2020

A Spectacular Ph.D. Thesis Defense by Michelle Feng!

Congratulations to my Ph.D. student Michelle Feng on a superb defense of her thesis!

CONGRATULATIONS to Dr. Michelle Feng (@michellehfeng ) on a successful — and, in fact, spectacular — defense of her Ph.D. thesis, called "Topological Tools for Understanding Complex Systems"!!!!

You will be hearing a lot from her in the future. pic.twitter.com/gpjueTA6CW

— Mason Porter (@masonporter) April 28, 2020

"Quantifying “Political Islands” with Persistent Homology"

Here is a new expository article (in SIAM News) by my Ph.D. student Michelle Feng and me about our work on spatial topological data analysis.

I finally get to share my new @TheSIAMNews article (by @michellehfeng and me): "Quantifying “Political Islands” with Persistent Homology"https://t.co/ocpZN43EHg

— Mason Porter (@masonporter) January 28, 2020

You may also be interested in our associated research article and our recent follow-up article.

My TDA (Topological Data Analysis) Origin Story

Based on what I saw as an undergraduate, I thought that algebraic topology was hopelessly abstract, and then I encountered Konstantin Mischaikow's work when I was a postdoc at Georgia Tech. He was using these ideas to analyze experimental data from areas like fluid mechanics. This stuck in my head, but I didn't work on these topics for many years. However, it stuck in the back of my head for about a decade, as this had made an impression on me. (I was aware of work of some others as well, but this is the one that made an impression, because of the close collaboration with experimentalists.) I was spending a bunch of time on granular networks as well as on generalizing network analysis from graphs to various more complicated structures (and I also had the desire to look more at "higher-order" interactions more generally).

During one of my daily arXiv routines, I noticed a paper by Konstantin and collaborators that used topological data analysis (TDA), so I saw that we were looking at the same systems, but in different ways. I contacted him, visited him early in 2013, and we started a joint TDA project --- but it turned out to be on spreading dynamics on networks, rather than on granular networks. Our first paper (which was led by Dane Taylor and coauthored with many other excellent people, including my Oxford colleague Heather Harrington) was published in final form in Nature Communications in 2015. I viewed this as just one paper; I never intended to start a large new direction in my research program. Back at Oxford, one student saw that I was part of that and wanted to work with Heather and me on applications of TDA. Then more students saw the 2015 paper and what this student was doing, and they wanted to work with us on TDA.

After I moved to UCLA, more students (starting with Michelle Feng) saw that I had some papers on TDA and wanted to work with me on those topics, partly because they wanted to do things with applications but also wanted to continue pursuing more theoretical mathematical subjects as well. I also really like the idea of taking "traditionally pure" areas of mathematics and bringing more and more of them into applications. It's a really exciting thing to do. And the work on applications also yields really great insights into the mathematical theory. (Because it does go in both directions, after all.)

Most recently, at least among people who have officially joined my group, Abby Hickok saw the work that Michelle and I have been doing, and she has ideas for building further on that work. And now TDA (along with work involving the intersection of dynamics, networks, and simplicial complexes) has become an important part of my research program,

Anyway, it was an all an accident.

What Happens in Torino Stays in Torino

I am heading to Torino for a workshop on topological data analysis and networks. It's going to be fun!

Awesome Talks by My Students and Postdoc at the Snowbird Conference!

Awesome talks by my students & postdocs at #SIAMDS19!

Heather Brooks (@HZinnbrooks): https://t.co/SDWGU4tTeR

Michelle Feng (@michellehfeng): https://t.co/EpOQTqrsom

Yacoub Kureh (@Yacoub_Kureh): https://t.co/Y31nfpCjm7

Alice Schwarze (@aliceschwarze): https://t.co/k9Ete7BkEv pic.twitter.com/MGojbXkqdH

— Mason Porter (@masonporter) June 10, 2019

Topological Data Analysis of Geographic Distance to Donuts

Clearly, it must be done. It must be done!

Often from a single position you could see two @dunkindonuts #redditDataIsBeautiful pic.twitter.com/SDfHVXmKub

— Suzanne Sindi (@SuzanneSindi) February 16, 2019

"Topological Data Analysis of Continuum Percolation with Disks"

One of my papers came out in final form today. Here are some details and a story.

Title: Topological Data Analysis of Continuum Percolation with Disks

Authors: Leo Speidel, Heather A. Harrington, S. Jonathan Chapman, and Mason A. Porter

Abstract: We study continuum percolation with disks, a variant of continuum percolation in two-dimensional Euclidean space, by applying tools from topological data analysis. We interpret each realization of continuum percolation with disks as a topological subspace of [0,1]^2 and investigate its topological features across many realizations. Specifically, we apply persistent homology to investigate topological changes as we vary the number and radius of disks, and we observe evidence that the longest persisting invariant is born at or near the percolation transition.


And to give a story, or at least the hint of the interesting relationship that I sometimes have with typesetters and editors, here is a note that I received from them while we were working on the galley proofs.

Update (8/05/18): A nice way of phrasing things is that we're in a nonassociative situation, and hyphens are a great tool to indicate exactly (and tersely) where the parentheses should be to group terms in a way that renders their meaning unambiguous. (And, naturally, if somebody makes a change in my text that I don't like, my immediate desire is to change it back.)

Proposal: Use Computational Topology to Study Aversion to Pictures of Holes

Clearly, we need to use topological data analysis (in which one tries to algorithmically compute things like holes and their generalizations) to study aversions to images of clusters of holes.

For science!

"Persistent Homology of Time-Dependent Functional Networks Constructed from Coupled Time Series"

One of my papers came out in final form today. Here are the details.

Title: Persistent Homology of Time-Dependent Functional Networks Constructed from Coupled Time Series

Authors: Bernadette J. Stolz, Heather A. Harrington, and Mason A. Porter

Abstract: We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.