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DATA FUNDAMENTALS (Matrices (Eigenproblems (Eigenspectrum (List of…
DATA FUNDAMENTALS
Matrices
Graphs as matrices
Adjacency matrix
row is the "from" of an edge
column is the "to" of an edge
Allows us to analyse a discrete problem as continous
Flow analysis
Problem of flow over a graph or network
Initial distribution
set of weighted edges
examples
packages moving between depots
users moving between webpages
$$ X_{t=1} = AX_{t=0} $$
What about $$ t = \frac{1}{24} $$ or an hour's time?
What about $$ t = \infty $$
Does the network reach an equilibrium or does it oscillate forever?
Stochastic Matrices
Right
all rows add to 1
Left
all columns add to 1
Double
both rows and columns all add to 1
Eigenproblems
Eigenvectors
Represent the "directions" in which the linear map extends
Eigenvalues
represent the scaling of the eigenvectors to achieve the linear map
Eigenspectrum
List of eigenvalues ordered by decreasing absolute value
Principal Component Analysis
Power Iteration
$$ A^{n} x $$
This would expand or collapse very quickly
Fix by normalising by deviding the result at each step by the l infinty norm