Tuesday, December 03, 2024
"Dynamical Importance and Network Perturbations"
Title: Dynamical Importance and Network Perturbations
Authors: Ethan Young and Mason A. Porter
Abstract: The leading eigenvalue λ of the adjacency matrix of a graph exerts much influence on the behavior of dynamical processes on that graph. It is thus relevant to relate notions of importance of network structures to λ and its associated eigenvectors. We study a previously derived measure of edge importance known as “dynamical importance,” which estimates how much λ changes when one removes an edge from a graph or adds an edge to it. We examine the accuracy of this estimate for several undirected network structures and compare it to the relative change in λ after an edge removal or edge addition. We then derive a first-order approximation of the change in the leading eigenvector. We also consider the effects of edge additions on Kuramoto dynamics on networks, and we express the Kuramoto order parameter in terms of dynamical importance. Through our analysis and computational experiments, we find that studying dynamical importance can improve understanding of the relationship between network perturbations and dynamical processes on networks.
Tuesday, December 12, 2023
"Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning"
Title: Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning
Authors: Tachin Ruangkriengsin and Mason A. Porter
Abstract: We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system’s equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system’s possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we examine the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.
Friday, July 28, 2023
What Happens in Sunnyvale Stays in Sunnyvale
Friday, February 10, 2023
"An Adaptive Bounded-Confidence Model of Opinion Dynamics on Networks "
Title: An Adaptive Bounded-Confidence Model of Opinion Dynamics on Networks
Authors: Unchitta Kan, Michelle Feng, and Mason A. Porter
Abstract: Individuals who interact with each other in social networks often exchange ideas and influence each other’s opinions. A popular approach to study the spread of opinions on networks is by examining bounded-confidence models (BCMs), in which the nodes of a network have continuous-valued states that encode their opinions and are receptive to other nodes’ opinions when they lie within some confidence bound of their own opinion. In this article, we extend the Deffuant–Weisbuch (DW) model, which is a well-known BCM, by examining the spread of opinions that coevolve with network structure. We propose an adaptive variant of the DW model in which the nodes of a network can (1) alter their opinions when they interact with neighbouring nodes and (2) break connections with neighbours based on an opinion tolerance threshold and then form new connections following the principle of homophily. This opinion tolerance threshold determines whether or not the opinions of adjacent nodes are sufficiently different to be viewed as ‘discordant’. Using numerical simulations, we find that our adaptive DW model requires a larger confidence bound than a baseline DW model for the nodes of a network to achieve a consensus opinion. In one region of parameter space, we observe ‘pseudo-consensus’ steady states, in which there exist multiple subclusters of an opinion cluster with opinions that differ from each other by a small amount. In our simulations, we also examine the roles of early-time dynamics and nodes with initially moderate opinions for achieving consensus. Additionally, we explore the effects of coevolution on the convergence time of our BCM.
Thursday, December 29, 2022
"Mixed Logit Models and Network Formation"
Title: Mixed Logit Models and Network Formation
Authors: Harsh Gupta and Mason A. Porter
Abstract: The study of network formation is pervasive in economics, sociology, and many other fields. In this article, we model network formation as a ‘choice’ that is made by nodes of a network to connect to other nodes. We study these ‘choices’ using discrete-choice models, in which agents choose between two or more discrete alternatives. We employ the ‘repeated-choice’ (RC) model to study network formation. We argue that the RC model overcomes important limitations of the multinomial logit (MNL) model, which gives one framework for studying network formation, and that it is well-suited to study network formation. We also illustrate how to use the RC model to accurately study network formation using both synthetic and real-world networks. Using edge-independent synthetic networks, we also compare the performance of the MNL model and the RC model. We find that the RC model estimates the data-generation process of our synthetic networks more accurately than the MNL model. Using a patent citation network, which forms sequentially, we present a case study of a qualitatively interesting scenario—the fact that new patents are more likely to cite older, more cited, and similar patents—for which employing the RC model yields interesting insights.
Thursday, December 30, 2021
"Epidemic Thresholds of Infectious Diseases on Tie-Decay Networks"
Title: "Epidemic Thresholds of Infectious Diseases on Tie-Decay Networks"
Authors: Qinyi Chen and Mason A. Porter
Abstract: In the study of infectious diseases on networks, researchers calculate epidemic thresholds to help forecast whether or not a disease will eventually infect a large fraction of a population. Because network structure typically changes with time, which fundamentally influences the dynamics of spreading processes and in turn affects epidemic thresholds for disease propagation, it is important to examine epidemic thresholds in models of disease spread on temporal networks. Most existing studies of epidemic thresholds in temporal networks have focused on models in discrete time, but most real-world networked systems evolve continuously with time. In our work, we encode the continuous time-dependence of networks in the evaluation of the epidemic threshold of a susceptible–infected–susceptible (SIS) process by studying an SIS model on tie-decay networks. We derive the epidemic-threshold condition of this model, and we perform numerical experiments to verify it. We also examine how different factors—the decay coefficients of the tie strengths in a network, the frequency of the interactions between the nodes in the network, and the sparsity of the underlying social network on which interactions occur—lead to decreases or increases of the critical values of the threshold and hence contribute to facilitating or impeding the spread of a disease. We thereby demonstrate how the features of tie-decay networks alter the outcome of disease spread.
Friday, July 16, 2021
My Current Mathematics Genealogy
My current Mathematics Genealogy (with doctoral students only): https://t.co/EAOiY0tasi
— Mason Porter (@masonporter) July 16, 2021
My MGP page (24 doctoral students): https://t.co/p35XlLXjoA
Current doctoral students: Abby Hickok, Grace Li, Jerry Luo, Kaiyan Peng, Mina Shahi
Also: I can't wait to have grandstudents! pic.twitter.com/dOU2wviu1V
Wednesday, June 17, 2020
Congratulations to the 2020 Graduates from Our Research Group! (Humans First; Mathematicians Second)
CONGRATULATIONS to the Class of 2020 from the research group!
And thanks to Prof. Chad Topaz for a wonderful commencement speech (one of the best I ever heard and so meaningful to me). Thanks to the guests who joined us.
The graduating class of 2020:
Postdoc: Heather Zinn Brooks
Undergrads: Qinyi Chen, Unchitta Kanjanasaratool, Tony Liu
PhD students: Michelle Feng, Yacoub Kureh, William Oakley
#truecolors
Here are a couple of pictures.
And here is the text of Chad's speech.
"Are mathematicians human?"
I was very unsure of how to begin this speech, and so I did the natural thing: I Googled “graduation speech opening” and I found a site with the following advice, which I am going to obey.
1. Offer formal words recognizing the honored guests.
I hereby decree that this honored group seems familiar; have we met before?
2. Use humor. If you are confident that your humor will work, making everyone laugh will be a great start.
I am confident. Please go ahead and laugh now.
3. Enthusiastically congratulate graduates on their success.
From my heart, congratulations.
4. State the topic of your speech.
Those of you who know me know that I often go against the crowd. Some graduation speech themes are all too common, so I’ll be avoiding the following messages in this speech:
- Challenges are opportunities
- Be yourself
- The world is your oyster
- Love will triumph over all
- With great power comes great responsibility
So, for this speech, I have chosen the topic “Are mathematicians human?"
I was a postdoctoral fellow in this department from 2003 - 2006. During the last year of my postdoc, and wanting desperately to stay in academia, I applied for tenure track jobs. While I was lucky to get a couple of offers, none of the offers would have resolved my dual career couple issues. It seemed reasonable that I'd have a better chance of finding work in Los Angeles than my husband would in rural Maine. So I made one of the hardest decisions of my life: I stayed in Los Angeles and, in doing so, quit being a mathematician. But not before, in a moment of extreme frustration and angst, I committed the only physically violent act of my adult life by hurling a bowl of oatmeal at the wall of my apartment.
Sometimes we don't realize what part of our identity means until we don't have it anymore. I didn't see this coming, but when I could no longer call myself a mathematician, or at least not professionally, I wasn't sure what I was anymore.
The next year of my life began fairly miserably. I worked as a college administrator on another campus in a unit that was a poor fit for me. I had few personal goals other than feeling sorry myself. Gradually, though, things turned around. I slowly built stronger relationships with friends. I got involved in service work related to issues I cared about. I rediscovered how much I loved playing chamber works with other musicians. I reconnected with all of the parts of myself that I had let shrink.
I stopped identifying principally as a mathematician and started identifying principally as a human.
But what makes us human? Characteristics and abilities once thought to distinguish us from other animals -- the use of tools, the ability to recognize ourselves, the size of our brains, and much more -- turn out to be found in various corners of the animal kingdom. Some evidence from the natural sciences, though, does suggest what makes humans unique.
First, we are human because of the degree to which we are wired to help each other. In psychology experiments, children as young as 14 months will spontaneously help a person who is struggling or who looks worried or who drops an item. At age two years, children will help someone who isn't even aware of their own need for help, say, because they didn't realize they had dropped an item. And rewards don't seem to play a role. In experiments on 20 month old toddlers, those who had previously received rewards for being helpful acted equally helpfully as a control group that hadn't received rewards. Other species certainly have been observed to engage in helping behavior, but within different parameters. For instance, close by on the evolutionary tree, chimpanzees will share food. Chimps, however, appear to be far more selective about their helping behavior, sharing only with close relatives or potential mates. In short, humans appear to be wired to be indiscriminately cooperative.
Second, we are human because of our ability to imagine and know things beyond our senses. Take the famous psychology experiment called the Sally-Anne task. In this task, there are two dolls named Sally and Anne. A young child, who is the subject of the experiment, sees Sally putting a marble in a basket while Anne watches. Then Sally leaves the room. While she's gone, Anne removes the marble from the basket and puts it inside a box. Then Sally comes back into the room. The experimenter asks the child where Sally will look for the marble, most children answer that Sally will look in the basket, where she had originally left it. The child knows that the marble is not there, but understands that Sally will have a different thought. Let me emphasize this: it's not merely that the child knows that the marble moved. The child can put themself in the mindset of another person and imagine what that person thinks. On the other hand, in a version of the experiment designed to assess chimpanzees, the chimps generally failed the test.
So two possible answers to "what makes us human" are our level of radical, selfless cooperation and our capacity to put ourselves in the shoes of another, a quality we sometimes refer to as empathy.
This speech would not be authentic if I did not call out that we are living during challenging times. Now let's be real: epidemic disease has been with humankind for a long time, with the earliest records of an influenza-like epidemic coming from central and Southern Asia around 1200 BC. Racial injustice has been with us in the United States since before we even WERE the United States. Still, we seem to be in an especially challenging moment right now, with bungled public health efforts and the continued killing of Black people by the police. It can be hard to believe that cooperation and empathy are our nature. So I take solace in the aforementioned scientific evidence and I say thank goodness for lab experiments.
It's not just on the national and international stages that the better parts of human nature can be obscured. You will have moments in your professional and personal lives when you will have the opportunity to put your humanity second to more immediate or more tangible or just plan easier ends. My message to you today is simple: always, first and foremost, be a human.
I asked Mason to give me a few very brief words about the human qualities of each of you, graduates. Mason respected my request... except for the brevity part! So please know that when I mention each of you, the brevity is mine, not his.
Heather, you are a valued and highly respected source of wisdom.
Michelle, your creativity and passion are inspirational.
Yacoub, you are driven to help, from each individual student up to saving the world.
Will, your good cheer powers not only your own efforts, but lifts those around you.
Quinyi, you have a rare blend of determination and humility.
Unchitta, you have limitless compassion.
Tony, you fearlessly reach out to those in need.
By the way, while today is mostly about you, graduates, it is also a little bit about Mason. It doesn't escape my notice that your advisor displays stellar human qualities as well, which is perhaps why we are drawn to him.
To the seven honored and accomplished celebrants, I wish you hearty congratulations and all the best for the future. Thanks to your hard work and dedication, you are outstanding students, scholars, teachers, mathematicians, thinkers. But most of all, you are outstanding humans.
Tuesday, June 09, 2020
"Spatial Strength Centrality and the Effect of Spatial Embeddings on Network Architecture"
Title: "Spatial Strength Centrality and the Effect of Spatial Embeddings on Network Architecture"
Authors: Andrew Liu and Mason A. Porter
Abstract: For many networks, it is useful to think of their nodes as being embedded in a latent space, and such embeddings can affect the probabilities for nodes to be adjacent to each other. In this paper, we extend existing models of synthetic networks to spatial network models by first embedding nodes in Euclidean space and then modifying the models so that progressively longer edges occur with progressively smaller probabilities. We start by extending a geographical fitness model by employing Gaussian-distributed fitnesses, and we then develop spatial versions of preferential attachment and configuration models. We define a notion of “spatial strength centrality” to help characterize how strongly a spatial embedding affects network structure, and we examine spatial strength centrality on a variety of real and synthetic networks.
Sunday, June 07, 2020
The COVID-19 Graduating Class
I bought one of these for each graduating member of my research group. pic.twitter.com/H8W7Dxg2aF
— Mason Porter (@masonporter) June 7, 2020
Thursday, June 04, 2020
A Great Ph.D. Defense by Dr. Will Oakley
Congratulations to Dr. Will Oakley for a successful defense of his Ph.D. dissertation, titled "Strong Consensus-Seeking in a Model of Social Consensus Formation"! pic.twitter.com/SxCEOkMpdr
— Mason Porter (@masonporter) June 4, 2020
Thursday, May 28, 2020
An Awesome Ph.D. Defense by Dr. Yacoub Kureh!
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
Thursday, April 30, 2020
"Community Matters"
Title: Community Matters
Authors: Anna C. F. Lewis, Nick S. Jones, Mason A. Porter, and Charlotte M. Deane
Tuesday, April 28, 2020
A Spectacular Ph.D. Thesis Defense by Michelle Feng!
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"!!!!
— Mason Porter (@masonporter) April 28, 2020
You will be hearing a lot from her in the future. pic.twitter.com/gpjueTA6CW
Monday, February 24, 2020
"Automatic Generation of School Bus Routes in Los Angeles"
Title: Automatic Generation of School Bus Routes in Los Angeles
Authors: Mason A. Porter, David J. Spender, and Cu Hauw ("Willy") Hung
Abstract: The goal of our project is to automatically generate school bus routes for the Los Angeles Unified School District (LAUSD). We examined four algorithms, including two from the existing literature and two new ones that we developed. A major focus of our work was the construction of “mixed-load routes,” which transport students from multiple schools. Based on our measurements (whose imperfections we discuss), three of the four algorithms perform at least as well as the existing route plan, and one of those three performs better than the existing route plan. We also delivered a user-friendly routing program to LAUSD that uses one of these algorithms, and we have made our software publicly available. Our insights and results are also applicable to other school districts that permit mixed-load routing.
Monday, January 27, 2020
"Quantifying “Political Islands” with Persistent Homology"
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.
Monday, December 16, 2019
What Happens in Vancouver Stays in Vancouver (2019 Edition)
We'll be binge-gaming, and in particular we are going to get as far as we can through Pandemic Legacy: Season 1, which was a gift from my one of recently-finished doctoral students. (The gift actually helped provide an impetus for this trip to happen, as getting a regular set of players for a 'legacy' or other campaign is not easy for me.)
As I'll be playing a Pandemic game, I guess I won't be escaping from networks entirely. :)
Thursday, November 28, 2019
My TDA (Topological Data Analysis) Origin Story
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.
Monday, November 11, 2019
Tales from the ArXiv: "Tautological Algebra"
Saturday, November 09, 2019
Visualization of the Zachary Karate Club Network Using a Cappuccino Embedding
In case you ever wanted to see the Karate Club network visualized using a capuccino embedding.
— Mason Porter (@masonporter) November 10, 2019
(Thanks to University of Michigan postdoc Sofia Piltz for trying this out for me at BeanBerry Cafe in Ann Arbor. She'll try it with a higher-contrast and more circular picture later.) pic.twitter.com/nqLpcJrfmY
Sofia found this picture in one of Petter Holme's presentations, although it reminds me of one of them from old papers and t-shirt designs.


