Implementation and introduction to extended mapper algorithm from topological data analysis. Connections to graph theory and network analysis.

1. Multiscale Mapper Networks
By Colleen M. Farrelly

2. Problem
 Data contains many
underlying structures
and relationships.
 Current methods (such
as k-means
clustering):
◦ Don’t capture all of these
structures
◦ Struggle with certain
data properties
(dimensionality)
◦ Provide little information
about connectedness
between
clusters/individuals
◦ Instability

3. Recent Solutions
 Nonlinear distance
metrics
◦ Random forest-based
◦ Manifold learning-based
 Hierarchical clustering
◦ Nested clustering
approach
 Multiscale K-Nearest
Neighbors
◦ Adjust number of
neighbors to slice data
 Still don’t provide a
comprehensive view of
data structure

4. Topology Overview
 Branch of mainly pure
mathematics
 Study of changes in
function behavior on
different shapes
(called manifolds)
 Can examine locally-
variant and globally-
invariant properties
 Classify
similarities/differences
between shapes
based on these
characteristics
Algebra can be used to build
more complex structures
from basic building blocks

5. Topology and Data
 Data clouds can be turned into discrete shapes
combinations (simplices)
 Identify key topological features across different slices of
the data (circles, holes…)
◦ Classified by Betti numbers (dimension plus feature type)
 Find connected components of similar topological
structure
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6. Mapper Algorithm
 Topological clustering
◦ Define distance metric
 Linear or nonlinear
◦ Define filtration function
 Linear, density-based…
◦ Slice multidimensional
dataset with Morse function
 Type of function associated
with gradient flow and critical
point identification on smooth
manifolds
◦ Examine function behavior
across slice (level set)
◦ Cluster function behavior
◦ Graph cluster connections
 Type of extended Reeb Graph
Response
gradations
Outliers

7. Multiscale Extension of
Mapper
 Instability of single-
scale mapper algorithm
◦ Clusters may change with
scale
◦ Connections may change
with scale
 Filtrations at multiple
resolution settings
 Connections change as
lens zooms in or out
◦ Contains information
about underlying data
structure and
relationships
◦ Hierarchy of Reeb graphs
◦ Topological summary

8. Graph Theory Extensions of
Mapper
 Cluster relationships from
Mapper give an adjacency
matrix and distance metric
◦ Clusters as vertices
◦ Nested hierarchy as edges
◦ Connected/unconnected
components
◦ Centrality of certain points
◦ Bridges linking disparate
clusters
◦ Path lengths between
clusters
 Can apply network
analytics to assess cluster
relationships and
individual connections
across clusters
This is a weighted,
undirected graph!

9. Network Extensions of Multiscale
Mapper
 Graph theory
algorithms applied to
Mapper results dig
deeper into:
◦ Data topology/structure
◦ Nature of individuals’
similarities across
multivariate distribution
 Examine across
different lenses
◦ Hierarchy of networks
connected through
individuals common to
multiple networks
◦ Analyze across slices to
gain deeper insight into
network and underlying
data structures

10. Information from Each
Network Hubs
◦ Direct connection
to many other
clusters
 Betweenness
◦ Non-extremity
measure
 Diversity
◦ Information
contained
 Bridges
◦ Connection
between less-
related
components
 Graph Laplacian
◦ Eigenvectors with
connection/bridge
weights
 Centrality
◦ Weight direct
connections and
bridges for
importance to
network
 Vertices
◦ Clusters at a
particular resolution
 Edges
◦ Connections between
clusters
◦ Individuals common
between clusters
 Levels
◦ Level sets (height
slices) containing one
or more vertices
◦ Individuals bridging
levels

11. Combined Insight of
Extensions
 Multiple resolutions
◦ Cluster hierarchy
 Evolving cluster structure
 More complete picture of
individual classifications
◦ Network hierarchy
 Evolving network structure
 More complete picture of
cluster relationships and
structure
 More complete picture of
individual connections

12. Example Demonstration
 Demo dataset of 7th grade SAT
scores
 Group-level data mining of results
Transition
Transition
Emergence of
subgroups
Split into
two distinct
groups

13. Individual Mining Results
 Map back to individuals
◦ Bridging individuals
 Transition between clusters
 Multivariate cut-off scores
determination
◦ Isolated individuals
 Outliers and outlier groups
 Unique response or predictors
subsets
◦ Consistently clustered
individuals
 Cohesive subgroups in data
 Underlying similarity of
predictors or response

14. Conclusion
New method ameliorates some of the issues
with clustering methods
◦ Robust
◦ Works in high dimensions
◦ Captures connectedness
◦ Stable
◦ Provides hierarchy
◦ Quantify relationships