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Centrality Analysis

Centrality Analysis

The Centrality Analysis feature allows you to calculate network-based metrics on your model. This helps you find the most important reactions or metabolites by quantifying their connectivity and influence in the network.

All Centrality metrics are calculated using the Python NetworkX package.

Centrality Analysis Workflow

Steps

  1. Click the Centrality Analysis button.
  2. Select the desired metrics to be calculated and click "Calculate and download results".
  3. A zip file will be downloaded containing the selected metrics in .csv format.

Degree Centrality

Degree centrality is defined as the number of direct edges a node has in a given network. For directed metabolic pathways, degree centrality is split into in-degree (number of incoming edges) and out-degree (number of outgoing edges).

Mathematical Formulae:

For a directed graph \(G = (V,E)\) and node \(v\):

  • In-degree centrality:
\[ C_{D}^{in}(v) = \frac{\deg^{in}(v)}{|V|-1} \]
  • Out-degree centrality:
\[ C_{D}^{out}(v) = \frac{\deg^{out}(v)}{|V|-1} \]

where:

  • \(\deg^{in}(v)\) = number of incoming edges to \(v\)
  • \(\deg^{out}(v)\) = number of outgoing edges from \(v\)

Both values are normalized by \(|V|-1\), so that:

\[ 0 \leq C_D(v) \leq 1 \]

Betweenness Centrality

Betweenness centrality measures how frequently a node lies on shortest paths between other nodes in the network. This helps identify bottleneck reactions that control the flow of metabolites.

Mathematical Formula:

For a graph \(G = (V,E)\) and node \(v\):

\[ C_B(v) = \sum_{s \neq v \neq t \in V} \frac{\sigma_{st}(v)}{\sigma_{st}} \]

where:

  • \(\sigma_{st}\) = number of shortest paths between nodes \(s\) and \(t\)
  • \(\sigma_{st}(v)\) = number of those paths that pass through node \(v\)

Normalization:

To scale values between 0 and 1:

  • Undirected graphs:
\[ C_B^{norm}(v) = \frac{C_B(v)}{(n-1)(n-2)/2} \]
  • Directed graphs:
\[ C_B^{norm}(v) = \frac{C_B(v)}{(n-1)(n-2)} \]

where \(n = |V|\) is the total number of nodes.

Closeness Centrality

Closeness centrality explains how close a node is to all other nodes in the network based on the average length of shortest paths.

Mathematical Formula:

For a connected graph \(G = (V,E)\) and node \(v\):

\[ C_C(v) = \frac{n-1}{\sum_{t \in V \setminus \{v\}} d(v,t)} \]

where:

  • \(n = |V|\) is the total number of nodes
  • \(d(v,t)\) is the shortest path distance between \(v\) and \(t\)

The denominator is the farness (total distance from \(v\) to all other nodes), and closeness is its reciprocal.

Eigenvector Centrality

Eigenvector centrality measures a node's importance based on its connections and the importance of the nodes it connects to.

Mathematical Formula:

For a node \(v\):

\[ x_v = \frac{1}{\lambda} \sum_{t \in V} A_{vt} \, x_t \]

where:

  • \(A\) is the adjacency matrix of the graph
  • \(x_v\) is the centrality score of node \(v\)
  • \(\lambda\) is the largest eigenvalue of \(A\)

In vector form:

\[ Ax = \lambda x \]

The components of \(x\) give the eigenvector centrality scores of the nodes.

PageRank Centrality

PageRank is a variant of eigenvector centrality that measures a node's importance based on both the number of incoming links and the importance of the linking nodes, while also incorporating a probability of randomly jumping to another node.

Mathematical Formula:

For a directed graph \(G = (V,E)\) and node \(v\):

\[ PR(v) = \frac{1-d}{N} + d \sum_{u \in M(v)} \frac{PR(u)}{\deg^{out}(u)} \]

where:

  • \(N = |V|\) is the total number of nodes
  • \(d\) is the damping factor (usually 0.85)
  • \(M(v)\) is the set of nodes linking to \(v\)
  • \(\deg^{out}(u)\) is the out-degree of node \(u\)

The algorithm is solved iteratively until values converge, giving a normalized importance score for each node.