My first investigation involves attempting to detect key content or users in a given community. I begin by establishing the model and various mathematical preliminaries.

Mathematical Preliminaries

Basic Definitions

Networks can be modelled by nodes that are connected together by edges. The mathematics of Algebraic Graph Theory gives us the tools we need for this section. In this case, we introduce the language used for directed graphs.

Definition: A graph \(G = (V, E)\) is a pair consisting of a set of nodes \(V\) and a set of edges \(E\) that describe which nodes lead to which others. Specifically, if there is an edge from node \(n_1 \in V\) to \(n_2 \in V\) then \((n_1, n_2) \in E.\)

Another representation of the edges, which is vastly more useful, is the adjacency matrix. The adjacency matrix not only permits us to describe the links between one node to another, but also allows us to weight each one with any real number.

Definition: Say a graph \(G=(V,E)\) has \(n\) nodes. An adjacency matrix \(A\in \mathbb{R}^{n\times n}\) for graph \(G\) is such that \[(A)_{i,j} = \begin{cases} w_{i,j} & \text{if } (i,j) \in E \\ 0 & \text{otherwise} \end{cases},\] where \(w_{i,j} \neq 0.\) We take, for simplicity’s sake, that \(w_{i,j} > 0.\) That is, the matrix \(A\) is non-negative.

The adjacency matrix provides a lens into the nature of the underlying graph it represents. For example, if you would like to know whether a path of length 2 exists from node 3 to node 5, you simply could evaluate \((A^2)\_{3,5}\) and see if it is non-zero. How can we see this? Observe, \[(A^2)_{3,5} = \sum_{l = 1}^n A_{3,l} A_{l, 5}.\] Assuming all elements of \(A\) are non-negative, we can conclude that \(A^2\) is non-zero at position \((3,5)\) only if node 3 leads to some node \(l\) — the intermediate node — which then directly leads to node 5. In other words, a path of length 2 from 3 to 5.

This can be easily generalized to existence of paths of any length by summing all the powers of \(A.\) Note how the following proposition leverages the fact that we assumed \(A\) is non-negative.

Proposition: Let \(M_k(A) = \sum_{l=1}^k A^l.\) A path from node \(i\) to node \(j\) exists if and only if there exists a \(k \in \mathbb{N}\) so that \((M_k(A))_{i,j} > 0.\)

Centrality Measures

Centrality measures are a way to assess the degree to which any given node is "central" to the graph. The definition of “central” here depends on the measure used. They all make an attempt to capture the idea of a node that is highly connected to tons of other nodes in the graph; the social butterfly of the graph, if you will. The centrality measure I choose to use is the Katz Centrality measure.

Definition: Let \(A\) be an adjacency matrix for some graph. Let \(\alpha > 0\) be a sufficiently small constant. The Katz Centrality measure for node \(i\) is, \[c_i = \sum_{k=1}^{\infty}{ \sum_{j=1}^{n}{ \alpha^k (A^k)_{j,i} } }\]

This definition doesn’t have any obvious relationship to centrality. To make the connection clear, let us define another matrix \(\bar{A} = \alpha A.\) and rearrange the measure to get, \[c_i = \sum_{j=1}^{n}{ \left( \sum_{k=1}^{\infty} \bar{A}^k \right)_{j,i} } = \sum_{j=1}^{n}{ \left( \lim_{k\to \infty} M_k(\bar{A}) \right)_{j,i}}\] Side-stepping the question of whether the limit converges (and for what choice of \(\alpha\) it does so), this formulation provides an intuition for the measure. Recalling that \(M_k\) is non-zero at elements that indicate path existence, this measure is attempting to sum up the strength of all possible paths to the node \(i\) — note that it attenuates the longer paths by the constant factor \(\alpha.\)

Of course computing this measure requires evaluating a limit, which isn’t ideal. The next proposition states an established technique used to calculate the measure.

Proposition: The iteration, \(c(k+1) = \alpha A^T(1 + c(k))\) converges to the Katz Centrality measure vector.

We are now ready to dive into the actual model.