19-09-20 Sizemore, Ann E., 2019¶

Sizemore, Ann E., Jennifer E. Phillips-Cremins, Robert Ghrist, and Danielle S. Bassett. "The importance of the whole: topological data analysis for the network neuroscientist." Network Neuroscience 3, no. 3 (2019): 656-673.

Original | [Mendeley]

ToC¶

00. Abstract
01. Intro
- Figure 1
02. When should we use topological data analysis?
03. Pieces and parts of the simplicical complex
- Box 1. From data to simplicial complex
- 03.01. Chain groups and boudaries
- [03.02. Homological algebra][0302]
[04. Homology from complex to complex: persistebt homology][04]
- [04.01. Filtrations][0401]
- [04.02. Persistent homology][0402]
- [04.03. Extracting topological feautres][0403]
[05. Conclusion][05]

00. Abstract¶

01. Intro¶

01 ¶01¶

algebraic topology

01 ¶02¶

neteork topology

Figure 1¶

01 ¶03¶

persistent homology

01 ¶04¶

01 ¶05¶

02. When should we use topological data analysis?¶

02 ¶01¶

02 ¶02¶

03. Pieces and parts of the simplicical complex¶

03 ¶01¶

simplicial complex

unit of $k + 1$ nodes is $k$ -simplex

$k$ -simplex
nodes $\{v_0, \cdots, v_k\}$

rule:
if $s$ is simplex in $K$ and $s' \subseteq s$ , then $s' \in K$

$k$ -skeleton ( $X^k$ ): collection of all cimplices w dim at most $k$

$k$ -cycle: looped patterns of $k$ -simplices

03 ¶02¶

simplex dist

03 ¶03¶

face: subset of a simplex
e.g., if $\{v_1, v_2, v_3\}$ is 2-simplex, it contains 1-simplex $\{v_1, v_2\}$ ,
where $\{v_i, v_j\}$ is face of $\{v_i, v_j, v_k\}$

Figure 2¶

03.01. Chain groups and boudaries¶

03.01 ¶01¶

cavities w/i simplicial complexes as akin to bubbles under water

Box 1. From data to simplicial complex¶

box1 ¶01¶

clique complex (flag complex) :
assign $k$ -simplex to each $(k+1)$ -clique

box1 ¶02¶

nerve complex
$\text{#simplices} \times \text{#vertices}$ mat
$\{v_i, v_j, v_k\}$

box1 ¶03¶

Vietoris-Rips complex
$\text{pairwise distance} < \epsilon$

Figure 3¶

03.01 ¶02¶

$\text{#nodes} \times \text{#edges}$ binary mat ( $\partial_1$ )

first / zeroth chain groups ( $C_1$ , $C_0$ ), 1- / 0-chains

03.01 ¶03¶

boudary operator

$\ker{(L)} = \{ v \in V | L(v) = 0 \}$

$\partial_1(C_1) \subseteq C_0$

boundary

03.01 ¶04¶

let $C_k$ (the $k^{\text{th}}$ )

boundary map $\partial_k$
$\text{dim}(C_{k-1}) \times \text{dim}(C_k)$ binary mat

boudary of $1$ -chain: sum of vertices
boudary of $2$ -chain: sum of edges
boudary of $3$ -chain: shell-forming sum of $2$ -simplices

chain group $C_k$ , boudary operator $\partial_k$ ,
$\partial_k: C_k \rightarrow C_{k-1}$

03.01 ¶05¶

$\cdots \overset{\partial_{k+3}}{\longrightarrow} C_{k+2} \overset{\partial_{k+2}}{\longrightarrow} C_{k+1} \overset{\partial_{k+1}}{\longrightarrow} C_{k} \overset{\partial_{k}}{\longrightarrow} C_{k-1} \overset{\partial_{k-1}}{\longrightarrow} C_{k-2} \overset{\partial_{k-2}}{\longrightarrow} \cdots$

Figure 4¶

03.02. Homological algebra¶

03.02 ¶01¶

$k$ -cycle subspace:
$Z_k = \ker{\partial_k}$

$k$ -boudary subspace: $B_k$
subspace of $Z_k$