Network equations according to Grok

Common mathematical equations describing networks depend on the context, such as graph theory, network analysis, or specific applications like social networks or communication systems. Here are some key equations:

1. Degree of a Node:
   ( d(v) = \sum_{u \in V} A(u, v) )
   Where ( d(v) ) is the degree of node ( v ), and ( A(u, v) ) is the adjacency matrix element indicating a connection between nodes ( u ) and ( v ).
2. Clustering Coefficient:
   ( C_i = \frac{2T_i}{d_i (d_i – 1)} )
   Where ( C_i ) is the clustering coefficient for node ( i ), ( T_i ) is the number of triangles through node ( i ), and ( d_i ) is the degree of node ( i ).
3. Average Path Length:
   ( L = \frac{1}{n(n-1)} \sum_{i \neq j} d(i, j) )
   Where ( L ) is the average path length, ( n ) is the number of nodes, and ( d(i, j) ) is the shortest path length between nodes ( i ) and ( j ).
4. Betweenness Centrality:
   ( C_B(v) = \sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}} )
   Where ( C_B(v) ) is the betweenness centrality of node ( v ), ( \sigma_{st} ) is the total number of shortest paths from node ( s ) to ( t ), and ( \sigma_{st}(v) ) is the number of those paths passing through ( v ).
5. Eigenvector Centrality:
   ( x_v = \frac{1}{\lambda} \sum_{u \in M(v)} x_u )
   Where ( x_v ) is the centrality score of node ( v ), ( M(v) ) is the set of neighbors of ( v ), and ( \lambda ) is a constant (the largest eigenvalue of the adjacency matrix).

These equations are foundational in network science, used to analyze connectivity, influence, and structure. For more specific applications, additional models like the Erdős-Rényi random graph probability (( p )) or the Barabási-Albert scale-free model might apply.

See what ChatGPT said about the same question here.