关于网络图论新生节点同组连接数计算逻辑的疑问

Hey Stephen, great question—this is a super common sticking point with mean-field approximations for growing network models, especially when intra-group connections are involved. Let's unpack this step by step.

First, your initial intuition makes perfect sense: if each newborn node allocates a fraction f of its m total links to other newborns, we'd expect each node to have mf connections to its groupmates. So why the m(1+f) expression?

There are two likely explanations for this discrepancy:

1. Possible Mix-Up in Symbol Definitions #

Sometimes, different sources flip the definition of f without explicit clarification. For example, some models define f as the fraction of links going to older nodes, not newborns. If that were the case here, the fraction going to newborns would be 1-f, and a derivation might end up with a term like m(1+f) if there's a double-counting correction (more on that next). But based on your description of the problem, this seems less likely—just something to double-check in your source material.

2. Double-Counting in Mean-Field Degree Calculation #

In continuous time mean-field approximations, we often calculate expected degree by accounting for both outgoing and incoming links. Let's break this down for your model:

• Outgoing links to newborns: Each node explicitly creates mf links to other newborns in its group.
• Incoming links from newborns: Every other newborn in the group (there are n-1 of them) also has a mf/(n-1) chance of linking to our target node (since they're uniformly selecting mf nodes from the n-1 peers). For large n, this simplifies to an expected mf incoming links from groupmates.

Wait, that would give a total expected intra-group degree of mf + mf = 2mf, which still doesn't match m(1+f). So there's probably a different twist here—maybe the problem defines m as the number of links a node receives, not sends? Or perhaps the fraction f is defined relative to the number of intra-group links, not total links?

Alternatively, let's look at the other expression you mentioned: m(1-f)n as the number of links to each older node. That also seems off—typically, preferential attachment means each older node gets new links proportional to its current degree, not a fixed m(1-f)n. This makes me wonder if there's a misinterpretation of the problem's wording: maybe "newborn nodes come in groups of n" means each group collectively creates m total links, not each individual node? That would change the math entirely.

My best advice right now is to go back to the derivation where you found the m(1+f) term and check how the variables are defined at each step. Often, these confusing terms come from a subtle shift in what m or f represents halfway through the math.

备注：内容来源于stack exchange，提问作者Stephen Moreton