Strange behavior in nx.effective_size for the single node graph

[joyemang33]

Hello! It seems I've created quite a few issue reports, and I apologize for any inconvenience. Please feel free to review them at your own pace, or you can assign them to me if I can assist in creating PR to fix them.

According to the formula

The effective_size should have a well-defined value of 0 for a graph with a single node.
However, if we use Borgatti's method to calculate this graph, the value will lose its definition, and NetworkX will return some strange behavior:

For instance:

import networkx as nx

G = nx.Graph()
G.add_node(0)
print(nx.effective_size(G))

G = nx.Graph([(0, 0)])
print(nx.effective_size(G))

Result:

{0: nan}
ZeroDivisionError

I am not sure whether such behaviors are expected. So it would be highly appreciated if you could further confirm and investigate it.
If they are unexpected, I will create a PR to fix it by returning a single 0 if the graph contains only one node.

Best regards,
Joye

Environment

NetworkX 3.1
Python 3.10

[MridulS]

If they are unexpected, I will create a PR to fix it by returning a single 0 if the graph contains only one node.

Yeah we should be catching this corner case. Thanks for your work on this!! Please do send a PR :)

[MridulS]

Looking at this one again a code comment explicitly states that for the corner case of isolated nodes while using Borgatti's formula it should be undefined (nan)

networkx/networkx/algorithms/structuralholes.py

Lines 147 to 149 in f35e034

	# Effective size is not defined for isolated nodes
	if len(G[v]) == 0:
	effective_size[v] = float("nan")

But the other case is where I guess we need to make a choice. I think for consistency we put in undefined behavior for isolated nodes but according to the reference it should be zero.

networkx/networkx/algorithms/structuralholes.py

Lines 155 to 157 in f35e034

	# Effective size is not defined for isolated nodes
	if len(G[v]) == 0:
	effective_size[v] = float("nan")

Also the ZeroDivisionError is popping up because of the self loop which makes the node 0 not isolated and hence the zero division error isn't caught early on, well I guess the question is does a node with a self loop and no other connection an isolated node?

[joyemang33]

Hi @MridulS. Thanks for your kind help in this issue.
In my opinion, a node with a self-loop and no other connection may need to be regarded as an isolated node, and we need to give a zero value in this case. This is because its effective_size computed by the above equation will still be zero (N(u) \ u = empty set).

That's only my personal opinion, so please feel free to tell me if I misunderstand the question.

Best regards,
Joye

[rossbar]

IIUC, it seems that the returning nan for isolated nodes is agreed upon as the correct, or at least sensible, behavior. So the remaining question is isolated nodes with self-loops. I'm no expert, but looking at Borgatti's formula in the docstring, it seems that an isolated node with a self-edge would still result in 0 - 0/0, and therefore should also result in nan. Absent any explicit information on how self-loops are treated from the Borgatti/Burt papers, this is what I think makes the most sense.