A complete graph is an undirected graph on n nodes such that every node is connected to every other node. Say you remove an edge from a complete graph on n nodes. What is the new clustering coefficient (C)? Assume that n>=3.
I get:
C = 1 - 6/(n*(n-1)) (wrong!)
To find the new clustering coefficient (C) of a complete graph on n nodes after removing an edge, we need to consider the effect of the missing edge on the local clustering of each node.
The clustering coefficient of a node measures the degree to which its neighbors are also connected to each other. In a complete graph, each node is connected to every other node, so the clustering coefficient is initially 1 for all nodes.
When an edge is removed, we need to determine the impact on the clustering coefficient of the nodes that were previously connected by that edge. Let's analyze the effect for an arbitrary node, which we'll call Node A, that was connected to Node B by the removed edge.
Before the edge was removed, Node A had n-1 neighbors (excluding itself) and every pair of those neighbors was connected to each other. After the edge removal, Node A will have n-2 neighbors, as it is no longer directly connected to Node B. The key is that Node A's remaining neighbors are still fully connected to each other, as they were in the complete graph.
The number of edges between Node A's remaining neighbors can be calculated as the number of edges in a complete graph on (n-2) nodes, which is given by (n-2)(n-3)/2. Here, (n-2) is the number of nodes and (n-3)/2 is the number of edges in a complete graph.
Therefore, the updated clustering coefficient for Node A is ((n-2)(n-3)/2) divided by the maximum number of possible edges between Node A's remaining neighbors, which is (n-2)(n-1)/2.
To find the average clustering coefficient for the complete graph after removing an edge, we need to consider all nodes in the graph and compute the average of their updated clustering coefficients.
The correct formula for the new clustering coefficient C is:
C = (Sum of updated clustering coefficients for all nodes) / n
Unfortunately, there isn't a simple closed-form formula to express C in terms of n. The expression you provided, 1 - 6/(n*(n-1)), is incorrect.
To compute the new clustering coefficient, you would need to calculate the updated clustering coefficient for each node and then sum them up, finally dividing by n.