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Given a 3-rank tensor with dimensions $x,y,z$.

Where:

  • $x$: number of graphs (number of samples)
  • $y$: number of nodes/vectors/features (let's say $5$: $a, b, c, d,$ and $e$)
  • $z$: embedding dimension (e.g. $2$ for Cartesian space that consists of horizontal axis and vertical axis)

Assume that a node represents a vector. The problem is to find the relationship between the nodes whether directed (dependent), undirected (interdependent/bidirectional), or unconnected (independent).

So, initially you are given unconnected nodes for all graph samples. How can you find the insight that the nodes are connected or not in a whole graph based on all samples of graphs?

This dataset can be reproduced with Random Number Generator (RNGs).

Let generate random dummy vector values for 100 samples in 2D vector:

a := RNG(dims=(100,2))
b := RNG(dims=(100,2))
c := RNG(dims=(100,2))
d := # initial d value doesn't matter
e := RNG(dims=(100,2))

Let generate dummy relationships between nodes, let's say like simple element-wise addition/multiplication or whatever as long as the operation involving dependencies of another node for un/directed relationships while no other nodes must be involved for unconnected nodes.

  • $a ← a + b + 1$
  • $b ← b × a + 2$
  • $c ← c + b × 3$
  • $d ← c^{b} + 4$
  • $e ← e + 5$

That's why initial $d$ doesn't matter because there is no self dependent in its assignment.

And here is in general:

  • $ a ← A(b) $
  • $ b ← B(a) $
  • $ c ← C(b) $
  • $ d ← D(b,c) $
  • $ e ← E() $

The problems are finding how many and what parameters are needed for each function $A, B, C, D$, and $E$.

In this case, the model inference is resulting:

  • Function $A$ needs one parameter, node $b$.
  • Function $B$ needs one parameter, node $a$, therefore $a$ and $b$ is having undirected relationship.
  • Function $C$ needs one parameter, node $b$. (directed relationship).
  • Function $D$ needs two parameters, node $b$ and node $c$.
  • Function $E$ does not need parameters so it's safely to be said as unconnected node.

yeah

So, to summary. If I have a 3-rank tensor like $G_{x,y,z}$ and a blackbox model $F$. Performing $F(G_{x,y,z})$ will resulting insights like above, where the number of insights is depending on the number of vectors which is $y$ dimension that has length $5$.

What is the $F$? Is it a neural network? If so, how it defined?

Technically, the output of $F$ must be continuous probabilistic with the range of $[0,1]$ for each node's edge to represent dependency strongness. E.g.:

a: {b:0.8, c:0.3, d:0.2, e:0.1}
b: {a:0.9, c:0.1, d:0.2, e:0.1}
c: {a:0.1, b:0.9, d:0.1, e:0.0}
d: {a:0.1, b:0.9, c:0.8, e:0.1}
e: {a:0.1, b:0.0, c:0.0, e:0.0}

Or in table:

From / To a b c d e
a - 0.8 0.3 0.2 0.1
b 0.9 - 0.1 0.2 0.1
c 0.1 0.9 - 0.1 0.0
d 0.1 0.9 0.8 - 0.1
e 0.1 0.0 0.0 0.0 -

It's easy to solved it with neural network if there is ground truth. Where the ground truth is adjacency matrix. But, unfortunately no ground truth.

Muhammad Ikhwan Perwira
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