Topological Data Analysis of Complex Spatial Systems

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Topological Data Analysis of
Complex Spatial Systems
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA

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(a) (b) (c)
FIG. 8. Visualization of the three different spatial partitions of Peru’s provinces on a map. (a) Broad climate partition
into coast (yellow), mountains (brown), and jungle (green); (b) detailed climate partition, in which we start with the
broad partition and then further divide the coast and mountains into northern coast, central coast, southern coast, northern
mountains, central mountains, and southern mountains; and (c) the administrative partition of Peru. We obtained province
boundaries from [82] and plot the maps in MATLAB.
We use the term “spatial partitions” to describe partitions that have high z-Rand scores in
comparison to the manual climate or administrative partitions. For multilayer networks, we
also compare the algorithmic partitions to partitions that contain a planted temporal change in
community structure. For these comparisons, we group the multilayer nodes into ones that occur
before or after a “critical” time tc (i.e., partitions into a “pre-tc” community and a “post-tc”
community). We test the set t = 1,1+D,1+2D,...,1+D ⇥ bT
D c 1 of times that we
use to create the multilayer network, and we report the time with the highest z-Rand score as the
critical time tc. We also test for pairs of critical times (yielding a partition into three communities)
by examining all possible pairs of critical times tc1 and tc2 in the same manner. We use the term
“temporal partitions” to describe algorithmic partitions of the disease-correlation networks that
yield high z-Rand scores in these comparisons.
4.3.1 Modularity Maximization Using the NG Null Model. We first study the community
structures of the 700 overlapping static networks formed by taking t = {1,2,...,700} and using
D = 80. (There are 779 time points in total.) The community structures that we obtain from max-
imizing modularity have a strong spatial organization, as suggested by the high z-Rand scores
when compared to topographical partitions. As one can see in Fig. 9(a), in which we plot the
z-Rand scores versus the centers of the time windows that correspond to the static networks, the
spatial organization is especially evident starting in the year 2000. In our subsequent figures,
time points that we indicate on the axes also correspond to the centers of the associated time
windows.
As one can see from a plot of number of epidemic cases over time (see Fig. 7), this transition
seems to occur near the time of the largest countrywide epidemic in the data, and the subse-
quent period includes recurring yearly epidemics that were linked to climatic patterns in prior
Spatial Systems
• Space has a major influence on
the structures of networks and
other complex systems
• Useful reference: Marc
Barthelemy, Morphogenesis of
Spatial Networks, 2018

Slime molds and fungal networks
• See, e.g., work by Mark
Fricker and collaborators

Leaf-Venation
Patterns
(Eleni Katifori and
collaborators)

Spiders: Spinning Webs While Under the Influence

The border between
Belgium and the
Netherlands at Baarle-
Nassau/Baarle-Hertog

Topological Data Analysis
• Algorithmic methods to study high-dimensional data in a
quantitative manner
• Data from point clouds, networks, etc.
• Examine the “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” (beyond pairwise) interactions in data

My TDA “Origin Story”
• Available at:
https://twitter.com/masonporter/status/1200127512371556352
• The short short version
• In college (1994–1998), I saw some algebraic topology, but it looked very abstract and it seemed
more on the ‘purer’ (i.e., theoretical) end of mathematics
• As a postdoc (2002–2005) at Georgia Tech, I saw Konstantin Mischaikow using ideas from
computational topology on experimental data (e.g., in fluid mechanics)
• In late 2012, I noticed that Konstantin and I were both working on granular materials (him with
TDA, me with network analysis), so I contacted him and we arranged a visit.
• We did a project on TDA on spreading processes on networks (Taylor et al., Nature Commun., 2015)
• An Oxford doctoral student saw that paper and wanted to work on TDA with me.
• Another Oxford student saw a couple of previous papers of mine with TDA and wanted to work on that with
me.
• Rinse, wash, and repeat — and unintentionally now I do a lot of TDA stuff.

(in practice, usually persistent homology)
• Michelle Feng, Abigail Hickok, Yacoub H. Kureh, Mason A. Porter,
Chad M. Topaz, “Connecting the Dots: Discovering the “Shape” of
Data”, Frontiers for Young Minds, 2021
• Chad Topaz’s introductory article (for a general audience) in DSWeb
• https://dsweb.siam.org/The-Magazine/Article/topological-data-analysis
• Nina Otter, MAP, Ulrike Tillmann, Peter Grindrod, and Heather A.
Harrington [2017], “A Roadmap for the Computation of Persistent
Homology”, European Physical Journal — Data Science, Vol. 6: 17

SIAM News
Jan–Feb 2020 issue

Example: Point-Cloud Data
[Figure from: Michelle Feng, Abigail Hickock, Yacoub H. Kureh, MAP, & Chad M. Topaz,
“Connecting the Dots: Discovering the “Shape” of Data”, Frontiers for Young Minds, 2021]

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of 2D Voting Data
Michelle Feng & MAP [2021], “Persistent Homology of Geospatial Data:
A Case Study with Voting”, SIAM Review, Vol. 63, No. 1: 67–99

Quantifying “Political Islands”
How do we detect red voters in a sea of blue?
(Or light blue voters in a sea of dark blue?)

Precinct-Level Voting Data
• How do people vote?
• How can we identify
geographical or temporal
patterns in voting?
• Our paper: geographical
• Can we automatically
characterize 2D
geographical outliers?
• Voting data (compiled by Los
Angeles Times) for all
California precincts in 2016
election

TDA and Voting Data
• Topological methods allow us to find and
identify holes, if we have a nice enough
space to search for those holes
• They also allow us to relate the presence
of holes to global structure
• Want to find “political islands”
• Red voters in a sea of blue, etc.
• Consider these as “holes” in a manifold in which
all precincts vote similarly
• Maybe we can also say something about
the structure of a county?

Barcodes
• A method of visualizing the PH of a
point cloud
• Each interval represents a feature in
dimension n
• Left endpoint = “birth” of a feature
• Right endpoint = “death” of a feature
• Visually, long features are “more
persistent”

Persistence Diagrams
• Another way of visualizing PH
• Put the filtration on both the
horizontal and vertical axes
• If a feature is born at b and dies at d,
we place a point at (b,d)
• The height above the diagonal
indicates persistence
• Pink circles: H0
• Blue squares: H1

Distance-Based Constructions:
Vietoris–Rips (VR) and Alpha Complexes
•VR complex [Jigglypuff]
• Surround each point in a point cloud with balls of radius !
• For a set of n + 1 points, if the pairwise distance between any two points is
less than !, build an n–simplex. The resulting simplicial complex is X!
•Alpha complex
• Compute the Delaunay triangulation of the point cloud
• X! is the simplicial complex formed by the set of edges and triangles
whose radii are at most !

Topological Data Analysis of Complex Spatial Systems

Summary: Distance-Based Constructions
•Advantages
• Easy to construct
• Fast algorithms, built into many packages
• Easy to interpret
• Embedded in Euclidean space, built-in parameter selection
•Disadvantages
• Which parameter values are appropriate ones?
• Persistence doesn’t always measure what we want it to
• Sensitive to rescaling
• Requires data in point-cloud form

Adjacency Construction
• Use network adjacency to define simplices
• If n + 1 nodes are all pairwise adjacent, define an n–simplex
• Given appropriate node data (or edge data), we construct a
filtration
• Note that filtration is not determined by distance
• In our data, filtration corresponds to strength of precinct
preference for a specific candidate
• For example, we can find light-blue precincts in a sea of dark blue

Summary: Adjacency Construction
• Advantages
• Does not depend on distance scaling
• Suitable for networks that aren’t easy or natural to embed in
Euclidean space
• Disadvantages
• Still only works on discrete data
• Sensitive to choices of construction of the underlying network
• It requires the nodes or edges to have associated data to construct
a meaningful filtration

Level-Set VR Construction
• Use data in surface form
• Take map of all precincts with similar voting patterns, and consider the outer
contour to be the 0 level set of some 3D object
• Evolve the surface outward with forces on a triangular grid according to the
level-set equation
• Take the collection of filled grid cells to be 2-simplices (and take grid lines to
be edges; and take points to be vertices)
• The filtration is given by the time steps of the evolution

Level Sets and PH
• The level-set method is a very fast method for front
propagation
• Persistence corresponds to the size of a feature: larger
holes take longer to fill
• We’re still “thickening” a point cloud (as in VR complexes),
except that we start with a manifold

Summary: Level-Set VR Construction
•Advantages
• We can use the underlying shape of a map
• We maintain some notion of geographic size of holes via the mesh size
• Faster than previous VR method on large data sets
•Disadvantages
• Difficult to associate generators of holes with the original precincts on the map
• Potentially not well-suited to less granular data
• Captures geographic features (e.g., bodies of water) that may not be desirable

A key point from the MF + MAP paper
• Our new constructions allow us to distinguish short-persistence
features that occur only for a narrow range of distance scales (e.g.,
voting behaviors in densely populated cities) from short-persistence
noise by incorporating information about other spatial relationships
between precincts
• Note: “Short persistence” with respect to the usual filtrations that
don’t take geospatial nature of the problem into account

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Persistent Homology on
Other Spatial Data
Michelle Feng & MAP [2020], Physical Review Research, Vol. 2, No. 3: 033426

Spiders Spinning Under the Influence
• The Marshall Space Flight Center studied the webs of spiders that were exposed to
various chemicals. (There is a NASA Tech Brief from 1995.)
• Earlier work, starting in 1948 by Swiss pharmacologist Peter N. Witt
• They concluded that more toxic chemicals resulted in more deformed spiderwebs

PH with Level-Set
Complexes on
Spiderwebs
Pink circles: H0
Blue squares: H1

Los Angeles
(gridlike)
(a) Aleppo and (b) Barcelona
(interrupted grids)
(a) Nanyang and (b) London
(not gridlike)

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Analysis of Spatiotemporal Anomalies Using
Persistent Homology: Case Studies with
COVID-19 Data
Abigail Hickok, Deanna Needell, and MAP, arXiv:2107.09188

COVID-19 Data Sets
• COVID-19 per capita vaccination rates in the different zip
codes of New York City
• Fully vaccinated people on 23 February 2021
• COVID-19 case rates in neighborhoods in the city of Los
Angeles
• Running 14-day mean per capita case rate from 25 April
2020 through 25 April 2021

Constructing a Simplicial Complex
• (1) Construct a 2D simplicial complex
for each region.
• (2) Glue their boundaries together in a
way that respects the geographical
region boundaries.

Filtration Functions (example: sublevel filtration)
Per capita vaccination
rate (NYC) or running
14-day mean of per
capita case rate (LA)

Case Study: Vaccination Rates in NYC
• Each point in the PD
corresponds to a zip code
(which we label by Borough)
that has a higher vaccination
rate than its neighboring zip
codes.

• We use “vineyards” to examine
the birth and death of features
over time.
• A continuous “stack” of PDs through
time. Points in the PD trace out curves
(“vines”) through time.

Conclusions
• Topological data analysis (TDA), such as by computing persistent homology
(PH), can give insights into large-scale structures in networks and other
complex systems
• Important: going beyond pairwise interactions in networks
• Persistent homology of spatial and spatiotemporal data
• By looking at 2D data, we can do systematic comparisons between different types of
constructions (topologically, fewer things can happen)
• Incorporate information from applications of interest into PH approaches
• Short-persistence features versus short-persistence noise: Need to think
carefully about how one constructs simplicial complexes
• Serendipity in research: You can end up writing a lot of papers on a topic
without intending to make it a big part of your research program
• Students and postdocs driving you into new research areas is the best thing ever™

Topological Data Analysis of Complex Spatial Systems

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