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Graph valued function, mapping from reals to graph (maybe about realization of functors)?

1

$begingroup$

Are there mappings from vectors/matrices of reals (or vectors/matrices of real-valued functions) to the graph? Motivating example: network of synapses and neurons as the domain and knowledge graph as the codomain. Is there mathematics that researches such objects?

I have received quite fraudulent answer, that is correct (this is big failure of the contemporary math education that focuses not on the ideas and reasoning but on the search for tricky, fraudulent, most easy answers that are eligible to miss the essence of the problem), but that misses the essence of my question.

The essence of my question is this: from the one side there is category of real-valued tensors (simply n-dimensional vectors or vector-valued functions, nothing more fancy from differential geometry) and from the other side there is category of graphs. Functor is mapping between those categories and in essence my question is two questions:

1. What mathematical constructions can realize the functor from the category of n-dimensional vecotrs to the category of graphs;


2. What metrics can be assigned to the realization of those functors and how we can find the optimal realization according to some metric.

Is there math that tries to find answers on any of those two questions?

real-analysis functions graph-theory category-theory neural-networks

share|cite|improve this question

edited Mar 23 at 21:47

TomR

asked Mar 23 at 14:24

TomRTomR

284312

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  An adjacency matrix with nodes and edges with or without values represents a graph and vice versa. For my understanding: What are the elements of your domain and what are the elements of your codomain?
  $endgroup$
  – IV_
  Mar 23 at 18:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Knowledge representation between features and values which are of very different kind can be made by decision trees.
  $endgroup$
  – IV_
  Mar 23 at 21:43
  

  
  
  
  

add a comment | 

1

$begingroup$

Are there mappings from vectors/matrices of reals (or vectors/matrices of real-valued functions) to the graph? Motivating example: network of synapses and neurons as the domain and knowledge graph as the codomain. Is there mathematics that researches such objects?

I have received quite fraudulent answer, that is correct (this is big failure of the contemporary math education that focuses not on the ideas and reasoning but on the search for tricky, fraudulent, most easy answers that are eligible to miss the essence of the problem), but that misses the essence of my question.

The essence of my question is this: from the one side there is category of real-valued tensors (simply n-dimensional vectors or vector-valued functions, nothing more fancy from differential geometry) and from the other side there is category of graphs. Functor is mapping between those categories and in essence my question is two questions:

1. What mathematical constructions can realize the functor from the category of n-dimensional vecotrs to the category of graphs;


2. What metrics can be assigned to the realization of those functors and how we can find the optimal realization according to some metric.

Is there math that tries to find answers on any of those two questions?

real-analysis functions graph-theory category-theory neural-networks

share|cite|improve this question

edited Mar 23 at 21:47

TomR

asked Mar 23 at 14:24

TomRTomR

284312

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  An adjacency matrix with nodes and edges with or without values represents a graph and vice versa. For my understanding: What are the elements of your domain and what are the elements of your codomain?
  $endgroup$
  – IV_
  Mar 23 at 18:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Knowledge representation between features and values which are of very different kind can be made by decision trees.
  $endgroup$
  – IV_
  Mar 23 at 21:43
  

  
  
  
  

add a comment | 

$begingroup$

Are there mappings from vectors/matrices of reals (or vectors/matrices of real-valued functions) to the graph? Motivating example: network of synapses and neurons as the domain and knowledge graph as the codomain. Is there mathematics that researches such objects?

I have received quite fraudulent answer, that is correct (this is big failure of the contemporary math education that focuses not on the ideas and reasoning but on the search for tricky, fraudulent, most easy answers that are eligible to miss the essence of the problem), but that misses the essence of my question.

The essence of my question is this: from the one side there is category of real-valued tensors (simply n-dimensional vectors or vector-valued functions, nothing more fancy from differential geometry) and from the other side there is category of graphs. Functor is mapping between those categories and in essence my question is two questions:

1. What mathematical constructions can realize the functor from the category of n-dimensional vecotrs to the category of graphs;


2. What metrics can be assigned to the realization of those functors and how we can find the optimal realization according to some metric.

Is there math that tries to find answers on any of those two questions?

real-analysis functions graph-theory category-theory neural-networks

share|cite|improve this question

edited Mar 23 at 21:47

TomR

asked Mar 23 at 14:24

TomRTomR

284312

$endgroup$

share|cite|improve this question

edited Mar 23 at 21:47

TomR

asked Mar 23 at 14:24

TomRTomR

284312

share|cite|improve this question

edited Mar 23 at 21:47

TomR

asked Mar 23 at 14:24

TomRTomR

284312

asked Mar 23 at 14:24

TomRTomR

284312

asked Mar 23 at 14:24

TomRTomR

284312

1 Answer
1

active

oldest

votes

2

$begingroup$

Of course, there are plenty of them.

For example, you can assignt to every graph an adjacency matrix, that is a square matrix wit entires 0 and 1: https://en.wikipedia.org/wiki/Adjacency_matrix

or incidence matrix https://en.wikipedia.org/wiki/Incidence_matrix

Of coures you can go vice versa also. Haveing such a matrix you can make a graph.

share|cite|improve this answer

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Starting from your suggestion: is there theory that researches arbitrary valued real matrices and how they can be mapped to adjacency matrices with smaller dimension?
  $endgroup$
  – TomR
  Mar 23 at 14:48
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @TomR Any square matrix can be interpreted as an adjacency matrix in at least two different ways: One, as the adjacency matrix of some (edge) weighted graph, where the vertices are $1, ldots, n$ ($n$ the order of the matrix) and the weight on the edge $(i, j)$ is the $i$-$j$ entry of the matrix. This is especially done if it's a non-negative matrix (no entry is negative). Another way is to look only at the zero-nonzero pattern of the matrix and have edges corresponding to nonzero entries — equivalently, replace all the nonzero entries of the matrix by $1$ to get a usual adjacency matrix.
  $endgroup$
  – M. Vinay
  Mar 23 at 15:33
  

  
  
  
  

add a comment | 

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2

$begingroup$

Of course, there are plenty of them.

For example, you can assignt to every graph an adjacency matrix, that is a square matrix wit entires 0 and 1: https://en.wikipedia.org/wiki/Adjacency_matrix

or incidence matrix https://en.wikipedia.org/wiki/Incidence_matrix

Of coures you can go vice versa also. Haveing such a matrix you can make a graph.

share|cite|improve this answer

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Starting from your suggestion: is there theory that researches arbitrary valued real matrices and how they can be mapped to adjacency matrices with smaller dimension?
  $endgroup$
  – TomR
  Mar 23 at 14:48
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @TomR Any square matrix can be interpreted as an adjacency matrix in at least two different ways: One, as the adjacency matrix of some (edge) weighted graph, where the vertices are $1, ldots, n$ ($n$ the order of the matrix) and the weight on the edge $(i, j)$ is the $i$-$j$ entry of the matrix. This is especially done if it's a non-negative matrix (no entry is negative). Another way is to look only at the zero-nonzero pattern of the matrix and have edges corresponding to nonzero entries — equivalently, replace all the nonzero entries of the matrix by $1$ to get a usual adjacency matrix.
  $endgroup$
  – M. Vinay
  Mar 23 at 15:33
  

  
  
  
  

add a comment | 

2

$begingroup$

Of course, there are plenty of them.

For example, you can assignt to every graph an adjacency matrix, that is a square matrix wit entires 0 and 1: https://en.wikipedia.org/wiki/Adjacency_matrix

or incidence matrix https://en.wikipedia.org/wiki/Incidence_matrix

Of coures you can go vice versa also. Haveing such a matrix you can make a graph.

share|cite|improve this answer

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Starting from your suggestion: is there theory that researches arbitrary valued real matrices and how they can be mapped to adjacency matrices with smaller dimension?
  $endgroup$
  – TomR
  Mar 23 at 14:48
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @TomR Any square matrix can be interpreted as an adjacency matrix in at least two different ways: One, as the adjacency matrix of some (edge) weighted graph, where the vertices are $1, ldots, n$ ($n$ the order of the matrix) and the weight on the edge $(i, j)$ is the $i$-$j$ entry of the matrix. This is especially done if it's a non-negative matrix (no entry is negative). Another way is to look only at the zero-nonzero pattern of the matrix and have edges corresponding to nonzero entries — equivalently, replace all the nonzero entries of the matrix by $1$ to get a usual adjacency matrix.
  $endgroup$
  – M. Vinay
  Mar 23 at 15:33
  

  
  
  
  

add a comment | 

$begingroup$

Of course, there are plenty of them.

For example, you can assignt to every graph an adjacency matrix, that is a square matrix wit entires 0 and 1: https://en.wikipedia.org/wiki/Adjacency_matrix

or incidence matrix https://en.wikipedia.org/wiki/Incidence_matrix

Of coures you can go vice versa also. Haveing such a matrix you can make a graph.

share|cite|improve this answer

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

$endgroup$

share|cite|improve this answer

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125

answered Mar 23 at 14:32

Maria MazurMaria Mazur

50.1k1361125