My talk at the 2017 SIAM "Snowbird" conference on applications of dynamical systems (#SIAMDS17).
I spoke in a session on topological data analysis (TDA). My talk concerned persistent homology and its application to Brexit data (including voting data) and "functional networks" from coupled time series from both experiments and output of dynamical systems.
Eventually, a version of these slides that is synchronized with the audio of my talk is supposed to be posted online.
These are the slides for a tutorial talk about "multilayer networks" that I gave at NetSci 2014.
I walk people through a review article that I wrote with my PLEXMATH collaborators: http://comnet.oxfordjournals.org/content/2/3/203
This is a presentation I gave in a workshop on "Language, concepts, history" organized by historian Joanna Innes. It took place on Friday 4/22/16 in Somerville College, Oxford.
I was one of the only people present who was not from the humanities, so it was a rather different-than-usual audience and set of participants for me.
I drew some of these slides from other presentations to rather different audiences. I emphasized rather different parts of some of those slides, so I am not sure if the slides on their own give an accurate reflection of the difference between this presentation and some of my other ones.
I thought the presentation went rather well.
My slides from my 3-hour tutorial on mesoscale structures in networks from the 2016 Lake Como School on Complex Networks (http://ntmb.lakecomoschool.org/).
After my talk, Tiago Peixoto gave a talk on statistical inference of large-scale mesoscale structures in networks. His presentation, which takes a complementary perspective from mine, is available at the following website: https://speakerdeck.com/count0/statisical-inference-of-generative-network-models
Centrality in Time- Dependent NetworksMason Porter
My slides for my keynote talk at the NetSci 2018 (#NetSci2018) conference in Paris, France (June 2018). This talk will take place on Thursday 13 June in the morning.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
This is a colloquium that I presented on 4/22/21: Stockholm University, Nordic Institute for Theoretical Physics (NORDITA), WINQ–AlbaNova Colloquium
Here is a video of my talk: http://video.albanova.se/ALBANOVA20210422/video.mp4
Paper Writing in Applied Mathematics (slightly updated slides)Mason Porter
Here are my slides (which I have updated very slightly) in writing papers in applied mathematics.
There will be an accompanying oral presentation and discussion on Friday 20 April. I am recording the video for that and plan to post it along with these (or a further updated version of these) slides.
These are the slides for a tutorial talk about "multilayer networks" that I gave at NetSci 2014.
I walk people through a review article that I wrote with my PLEXMATH collaborators: http://comnet.oxfordjournals.org/content/2/3/203
This is a presentation I gave in a workshop on "Language, concepts, history" organized by historian Joanna Innes. It took place on Friday 4/22/16 in Somerville College, Oxford.
I was one of the only people present who was not from the humanities, so it was a rather different-than-usual audience and set of participants for me.
I drew some of these slides from other presentations to rather different audiences. I emphasized rather different parts of some of those slides, so I am not sure if the slides on their own give an accurate reflection of the difference between this presentation and some of my other ones.
I thought the presentation went rather well.
My slides from my 3-hour tutorial on mesoscale structures in networks from the 2016 Lake Como School on Complex Networks (http://ntmb.lakecomoschool.org/).
After my talk, Tiago Peixoto gave a talk on statistical inference of large-scale mesoscale structures in networks. His presentation, which takes a complementary perspective from mine, is available at the following website: https://speakerdeck.com/count0/statisical-inference-of-generative-network-models
Centrality in Time- Dependent NetworksMason Porter
My slides for my keynote talk at the NetSci 2018 (#NetSci2018) conference in Paris, France (June 2018). This talk will take place on Thursday 13 June in the morning.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
This is a colloquium that I presented on 4/22/21: Stockholm University, Nordic Institute for Theoretical Physics (NORDITA), WINQ–AlbaNova Colloquium
Here is a video of my talk: http://video.albanova.se/ALBANOVA20210422/video.mp4
Paper Writing in Applied Mathematics (slightly updated slides)Mason Porter
Here are my slides (which I have updated very slightly) in writing papers in applied mathematics.
There will be an accompanying oral presentation and discussion on Friday 20 April. I am recording the video for that and plan to post it along with these (or a further updated version of these) slides.
Mathematical Models of the Spread of Diseases, Opinions, Information, and Mis...Mason Porter
This is my general-audience talk at DiscCon III (2021 WorldCon).
My talk overlapped with the Hugo Award ceremony, but the video will be posted later on the DisCon website for attendees who want to see it.
1. Basics of Social Networks
2. Real-world problem
3. How to construct graph from real-world problem?
4. What graph theory problem getting from real-world problem?
5. Graph type of Social Networks
6. Special properties in social graph
7. How to find communities and groups in social networks? (Algorithms)
8. How to interpret graph solution back to real-world problem?
Community detection from research papers (AAN dataset) using the algorithms:
K-Means
Louvain
Newman-Girvan
github link to code: https://goo.gl/CXej44
github link to project web page: http://goo.gl/7OOkhI
youtube link to video:https://goo.gl/SCpamf
dropbox link to ppt report video: https://goo.gl/cgACzU
Mathematical Models of the Spread of Diseases, Opinions, Information, and Mis...Mason Porter
This is my general-audience talk at DiscCon III (2021 WorldCon).
My talk overlapped with the Hugo Award ceremony, but the video will be posted later on the DisCon website for attendees who want to see it.
1. Basics of Social Networks
2. Real-world problem
3. How to construct graph from real-world problem?
4. What graph theory problem getting from real-world problem?
5. Graph type of Social Networks
6. Special properties in social graph
7. How to find communities and groups in social networks? (Algorithms)
8. How to interpret graph solution back to real-world problem?
Community detection from research papers (AAN dataset) using the algorithms:
K-Means
Louvain
Newman-Girvan
github link to code: https://goo.gl/CXej44
github link to project web page: http://goo.gl/7OOkhI
youtube link to video:https://goo.gl/SCpamf
dropbox link to ppt report video: https://goo.gl/cgACzU
Spectral clustering with motifs and higher-order structuresDavid Gleich
I presented these slides at the #strathna meeting in Glasgow in June 2017. They are an updated and enhanced version of the earlier talks on the subject.
Paper Explained: Understanding the wiring evolution in differentiable neural ...Devansh16
Read my Explanation of the Paper here: https://medium.com/@devanshverma425/why-and-how-is-neural-architecture-search-is-biased-778763d03f38?sk=e16a3e54d6c26090a6b28f7420d3f6f7
Abstract: Controversy exists on whether differentiable neural architecture search methods discover wiring topology effectively. To understand how wiring topology evolves, we study the underlying mechanism of several existing differentiable NAS frameworks. Our investigation is motivated by three observed searching patterns of differentiable NAS: 1) they search by growing instead of pruning; 2) wider networks are more preferred than deeper ones; 3) no edges are selected in bi-level optimization. To anatomize these phenomena, we propose a unified view on searching algorithms of existing frameworks, transferring the global optimization to local cost minimization. Based on this reformulation, we conduct empirical and theoretical analyses, revealing implicit inductive biases in the cost's assignment mechanism and evolution dynamics that cause the observed phenomena. These biases indicate strong discrimination towards certain topologies. To this end, we pose questions that future differentiable methods for neural wiring discovery need to confront, hoping to evoke a discussion and rethinking on how much bias has been enforced implicitly in existing NAS methods.
Topological Data Analysis of Complex Spatial SystemsMason Porter
These are slides from a seminar I gave in "Cardiff" (for the mathematics department at University of Cardiff) on 4/15/20.
You can also find a recording of a similar talk that I gave in March 2020 for MBI (Mathematical Biosciences Institute): https://mbi.osu.edu/events/online-colloquium-mason-porter-spatial-systems-and-topological-data-analysis
To have the ability to “think outside the box” is generally regarded as something positive. At a moment in time when resources are scarce, and the problems facing us are many, innovation and professional excellence becomes a requirement, rather than a matter of choice. At the core of our attempts to come up with new, and better solutions are the digital technologies. Within the structural engineering context, the different types of off-the-shelf packages for finite element analysis play a central role. These “black-box” types of software packages exemplify how user-friendliness may have harmful consequences within a field where knowledge and the successful mastery of relevant skills is key, and consequently- ignorance may lead to fatal results. These tools make any effort “venturing outside” difficult to achieve. A technical paradigm shift is called for- that places learning and creative, informed exploration at the heart of the user experience. Presented during the Knowledge Based Engineering session of the 19th IABSE congress entitled "Challenges in Design and Construction of an Innovative and Sustainable Built Environment" held in Stockholm, September 21-23, 2016.
To have the ability to “think outside the box” is generally regarded as something positive. At a moment in time when resources are scarce, and the problems facing us are many, innovation and professional excellence becomes a requirement, rather than a matter of choice. At the core of our attempts to come up with new, and better solutions are the digital technologies. Within the structural engineering context, the different types of off-the-shelf packages for finite element analysis play a central role. These “black-box” types of software packages exemplify how user-friendliness may have harmful consequences within a field where knowledge and the successful mastery of relevant skills is key, and consequently- ignorance may lead to fatal results. These tools make any effort “venturing outside” difficult to achieve. A technical paradigm shift is called for- that places learning and creative, informed exploration at the heart of the user experience. Presented during the Knowledge Based Engineering session of the 19th IABSE congress held in Stockholm, September 21-23, 2016.
Complex systems,
Software systems,
Database systems,
Operating systems,
Bioinformatics systems,
Social Systems,
Service Oriented Systems,
Cloud Systems,
Ubiquitous systems,
Distributed Version Control Systems (GitHub), and
Software Container Systems (DockerHub and Google App Engine).
Community search is the problem of finding a good community for a given set of query vertices.
In this work we propose a novel method that it is in general more efficient and effective than state-of-the art, it can handle multiple query vertices, it yields optimal communities, and it is parameter free.
Networks in Space: Granular Force Networks and BeyondMason Porter
This is my talk for the Network Geometry Workshop (http://ginestra-bianconi-6flt.squarespace.com) at QMUL on 16 July 2015.
(A few of the slides are adapted from slides by my coauthors Dani Bassett and Karen Daniels.)
Opinion Dynamics on Generalized NetworksMason Porter
This is a talk on opinion dynamics (especially bounded-confidence models) on generalized networks.
It is part of the MIX-NEXT III (Multiscale & Integrative compleX Networks: EXperiments & Theories) satellite at NetSci 2022.
(Thursday 14 July 2022)
Here are my slides (though the animated gifs on a couple of them are stills in this version) of my talk on an introduction to the science of "chaos" at WorldCon 77 in Dublin, Ireland.
This is my attempt to give a gentle introduction to the notion of chaos to a science-fiction audience.
Tutorial on Paper-Writing in Applied Mathematics (Preliminary Draft of Slides)Mason Porter
These are preliminary slides for a tutorial and discussion on "Writing Papers in Applied Mathematics" that I'll be giving at UCLA, first for a few of my own PhD students on 4/6 and later (on 4/20 ?) in a recorded session to a larger UCLA group.
Several people have expressed interest, so I will post the recorded session online and circulate it.
These slides are for my talk for the Somerville College Mathematics Reunion ("Somerville Maths Reunion", 6/24/17): http://www.some.ox.ac.uk/event/somerville-maths-reunion/
This is my attempt at an introduction to data ethics for mathematicians. Mathematicians increasingly need to deal with these kinds of issues, but we don't have the tradition of ethics training from other disciplines.
I welcome comments on how to improve these slides. Did I miss any salient points? Do you want to offer a different perspective on any of these? Do you want to offer any counterpoints? (Please e-mail me directly with comments and suggestions.)
Eventually, I hope to develop these slides further into an article for a venue aimed at mathematical scientists, and of course I would love to have knowledgeable coauthors who can offer a different perspective from mine.
These are slides for my tutorial talk on network dynamics. (The colors are fine in the downloaded version, though there seem to be color issues if you view the slides directly in slideshare.)
Slides from my talk on a systems-level investigation of long-term human migration in Korea. Our paper is available at the following page: http://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.041009
I adapted these slides from the ones created by my coauthor Sang Hoon Lee.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
1. The Topological “Shape” of
Brexit and Functional Networks
Mason A. Porter
Department of Mathematics
UCLA
MS 91, Tues. 5/23, 2:15 PM
(@masonporter)
2. Outline
Introduction
“Brexit” Examples
“Functional Networks” in Neuroscience
Conclusions
Note: Various slides adapted or taken from slides of Bernadette J. Stolz (the first author on most of “my” papers I’ll discuss).
Most of the original research I am presenting is her work.
4. A Few Introductory Resources for
Topological Data Analysis
Chad Topaz’s awesome introductory article in DSWeb
https://dsweb.siam.org/The-Magazine/Article/topological-data-analysis
The most-read DSWeb article of all time
Book: Robert Ghrist, Elementary Applied Topology
https://www.math.upenn.edu/~ghrist/notes.html
Nina Otter, MAP, Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington, “A Roadmap for
the Computation of Persistent Homology”, submitted, arXiv:1506.08903
Bernadette Stolz [2014], Masters Thesis, University of Oxford, Computational Topology in
Neuroscience
http://www.math.ucla.edu/~mason/research/Dissertation-stolz2014-Corr.pdf
Chad Giusti, Robert Ghrist, & Danielle S. Bassett [2016], “Two’s Company, Three (or More) is a
Simplex”, Journal of Computational Neuroscience, Vol. 41, No. 1: 1–14
Links to various resources on my Quora answer on TDA
https://www.quora.com/Applied-Mathematics-What-is-the-background-required-to-study-and-understand-topological-
data-analysis
5. Introduction and Motivation
Algorithmic methods to study high-dimensional data (from point
clouds, networks, etc.) in a quantitative manner
Examine “shape” of data
Persistent homology
Mathematical formalism for studying topological invariants
Fast algorithms
Persistent structures: a way to cope with noise in data
Allows examination of “higher-order” interactions (beyond pairwise) in
data
A major reason for my interest in these methods (e.g., for networks)
6. Topological Data Analysis and
Networks
Typically for weighted networks
In real-world networks, it is hard to extra significant structures (signal) from
insignificant ones (noise).
Sometimes convenient to threshold weights, binarize remaining values, and study
the resulting graph
Loss of important properties of original graph
Study global network characteristics
Large-scale network structure, but of a different type from common ones like community structure
Useful: compare results of TDA approaches to “traditional” network approaches
7. Persistent Homology: Underlying Idea
Idea: Consider a filtration
For example: filter by the threshold for going from a weighted network to a binary network, and
only keep (binarized) edges of at least that threshold.
Study changes in topological structure along filtration by calculating topological
invariants such as Betti numbers
8. Persistent Homology: Underlying Idea
1. Construct a sequence of embedded graphs from a weighted network.
2. Define simplicial complexes.
9. Persistent Homology: Weight Rank
Clique Filtration (WRCF)
(e.g. G. Petri et al., PLOS ONE, 2013)
1. Construct a sequence of embedded graphs from a weighted network.
2. Define k-simplices to be the k-cliques present in the graph.
12. Part I: The Topological “Shape” of
Brexit
B. J. Stolz, H. A. Harrington, & MAP [2016], “The
Topological “Shape” of Brexit”, arXiv:1610.00752
13. Warmup: Network of EU Countries
Connect two countries with an edge if they are
considered neighbors via a border (either in
Europe or abroad), a bridge or a tunnel.
The edge weight is the later of the two years
that the two countries joined the EU.
Consider WRCF
14. Example 2: Referendum Voting Data
Construct 2 point clouds
‘Remain’ point cloud: coordinates of cities in
voting districts that voted to remain in the EU
Gibraltar omitted for simplicity
‘Leave’ point cloud: coordinates of cities in voting
districts that voted to leave the EU
Construct a Vietoris–Rips filtration
Choose a sequence {r1, … , rn} of increasing
distances
In the ith filtration step, we have k-simplices from
unordered (k+1)-tuples with pairwise distance at
most ri
15.
16. Part II: Functional Networks
B. J. Stolz [2014], Masters Thesis, University of Oxford, Computational Topology in
Neuroscience
http://www.math.ucla.edu/~mason/research/Dissertation-stolz2014-Corr.pdf
B. J. Stolz, H. A. Harrington, & MAP [2017], “Persistent Homology of Time-
Dependent Functional Networks Constructed from Coupled Time Series”, Chaos,
Vol. 27, No. 4: 047410
17. Pipeline
E.g. Coupled Kuramoto oscillators (you thought that you could finally avoid them in this session, didn’t you?)
In contrast to using “traditional”
methods for studying weighted
networks
18. What is a Functional Network?
Functional versus Structural Networks
Example from neuroscience:
Structural network: nodes = neurons, edges = synapses
Functional network: nodes = cortical areas, edges = behavioral similarity (quantified by similarity of time series)
Example from ordinary differential equations:
Structural network: nodes = oscillators, edges = coupling between oscillators
Functional network: nodes = oscillators, edges = behavioral similarity (quantified by similarity of time series)
Functional networks are weighted and fully connected (or almost fully connected)
We can study them using persistent homology
Can compare results on large-scale structure to other approaches, such as community structure
21. Kuramoto Data versus Null Models
Simple null model:
Independently reassign
the order of each
oscillator’s time series
according to a uniform
distribution (i.e., scramble
time independently for
each oscillator)
Fourier null model:
Generate surrogate data
by scrambling phases in
Fourier space
22. Example: fMRI Data
Data from D. S. Bassett, N. F. Wymbs, MAP, P. J.
Mucha, J. M. Carlson, & S. T. Grafton [2011],
“Dynamic Reconfiguration of Human Brain
Networks During Learning”, PNAS, Vol. 118, No. 18:
7641–7646
Weighted networks from time-series similarity (wavelet
coherence) of neuronal activity of brain regions during
performance of simple motor task
In the above paper and follow-ups, we studied things like
community structure and core–periphery structure.
Using persistent homology gives another way to examine
large-scale (“mesoscale”) network structures
These data also used in D. S. Bassett, MAP, N. F. Wymbs, S. T.
Grafton, J. M. Carlson, & P. J. Mucha [2013], “Robust
Detection of Dynamic Communities in Networks”, Chaos, Vol.
23, No. 1: 013141
23. Differences in Different Days?
Experimental observations on 3 different days
(20 participants)
Right plot: Average persistence landscapes
Landscape peak shifts to the left in later days
I.e. they are formed by edges with higher weights,
indicating that there is stronger synchronization
between the associated brain regions
24. Conclusions
Computing persistent homology can give insights into large-scale structure of networks
Complements network clustering methods, such as detection of mesoscale features like community
structure and core–periphery structure
Important: going beyond pairwise interactions in networks
Observation: Sometimes relatively short features (e.g. as visualized in short barcodes)
represent meaningful features. (We saw this in both Kuramoto and fMRI data.)
E.g. strongly synchronized Kuramoto oscillators within the same community of a structure network
Contrasts with conventional wisdom: longer (i.e. more persistent) features are supposed to be the signals,
and shorter features are usually construed as noise
Our Brexit example was a toy, but it’s worth looking at that kind of data more seriously
using TDA approaches.
Reminder: If you want to get started on PH, look at our “roadmap” paper: Nina Otter, MAP,
Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington, “A Roadmap for the
Computation of Persistent Homology”, submitted, arXiv:1506.08903