Hello everyone and sorry for the dumb title. I am going over Hong & Dey's paper "Network Analysis of cosmic Structures"; https://arxiv.org/abs/1504.00006v1.
Specifically I am quite confused with what it is written in sections 2.1.1 and 2.1.2.
From 2.1.1
I do not get how each i-th vertex can be represented as a unit vector. I have constructed a 4 x 4 matrix which satisfies the (1)-(3) properties for simple, undirected networks but what if the network is much more complex? Then each row/column will not a be unit vector (Having constructed a simple 4 x 4 symmetric matrix with main diagonal being all-zero, the resulting graph is just 4 vertices and each pair connected to each other (1-2, 3-4). If I want to connect the 3rd to the 2nd it won't be a unit vector. I also don't get what this "$k$" represents $e_i \equiv \delta_{ik}$ (while the paper then refers to the normal Kronecker $\delta_{ij}$.
Moving on to section 2.1.2
Could you give me links or more detailed analysis of equations 3,4 and 5? (With actual graphs showing example etc so i can get a more solid understanding of the matter?). Lastly about equation 2, say I am interested in vertex 2 then the index "$j$" what values would it take? (An example here would be appreciated)
Thank you
They seem to represent the $i$th vertex simply as the $i$th vector of the standard basis of $\Bbb R^n$, they just use a somewhat convoluted formulation to express this. This fact is unrelated to any properties of the adjacency matrix $A$. This is also unrelated to whether the vertices represent points in $\Bbb R^3$ in a different context.
In a simpler formulation I would consider the set $V$ or vertices, the vector space $\Bbb R^V$, and the endomorphism $\alpha$ of this space induced by $v\mapsto \sum_{vw\in E}w$, which in terms of the standard basis is given by the adjacency matrix $A$.
(3) is just a restatement of $\sum_{vw\in E}w=\alpha v$. $(4)$ is jusr a less legible rewriting of $(5)$, and $(5)$ follows by induction from the recursive definition of a path of length $r+1$ from $v$ to $w$ as a path of length $r$ from $v$ to an arbitrary vertex $u$, followed by an edge from $u$ to $w$.
Lastly, in $(2)$, $j$ runs from $1$ to $n$ (or viewed differently, through the set $V$ of vertices). As the entries in the $i$th column represent to number of edges from the $i$th vertex to the $j$th vertex, $1\le j\le n$, the sum of the entries of this column is the number of edges originating at vertex $i$.