See Models of network formation

• Edges (like hyperlinks) may also disappear. They may also appear at times after the nodes are added.
• Nodes may also disappear (like websites).
• Preferential attachment could be non-linear on degree, or it could depend on other network property of the node.
• Nodes may have some intrinsic fitness too.

Models with extra edge addition

Model can consist of the BA model, but with an extra process carried out at each step. A given number of edges $w$ is added to the network between two nodes with a probability proportional to their degree. One can again construct a master equation, and get a power law degree distribution.

Similar models exist that generalize the Price's model instead of BA.

Edge removal

Simple model: at each update step we remove $v$ edges chosen uniformly at random from set of all edges. The probability that node $i$ looses an edge connected to it, for each of these removals, is $2k_i/\sum_i k_i$. This is because randomly choosing an edge means randomly choosing a pair of stubs, and $i$ will loose an edge when either of these randomly chosen stubs coincides with one of the $k_i$ stubs incident to $i$. The probability of this happening for each of the randomly chosen stubs is $k_i/\sum_i k_i$, and the probability that either stubs is from $i$ is $2k_i/\sum_i k_i-\text{probability both ends on same edge}$. However the $\text{probability both ends on same edge}$ is $0$ because the BA network formation model doesn't allow self-edges to form. Therefore we are left with $2k_i/\sum_i k_i$, as in Newman's book.

Models with edge addition and removal

One can also combine the two models above. The master equation in this case, becomes more complicated, because $p_k$ now depends on both $p_{k+1}$ and $p_{k-1}$. Generating function methods need then to be used. Se Newman section 14.4.2 or the paper Exact solutions for models of evolving networks with addition and deletion of nodes for detailed calculation, and a power law degree distribution is still obtained (though with different exponent of course), as long as edge removal rate is not too high.

One can also do analogous for removal and addition of nodes.

Non-linear preferential attachment

Attachment probability may depend nonlinearly on degree, i.e. we have a nonlinear attachment kernel.

One can still derive an asymptotic form of the degree distribution for the case $a_k \propto k^\gamma$, of interest because empirical networks have shown this form of preferential attachment. For $1/2 <\gamma <1$, the degree distribution is no longer a power law, but an "stretched exponential" of the form:

$p_k \sim k^{-\gamma} \exp{(-\frac{\mu k^{1-\gamma}}{c(1-\gamma)})}$

This function decays slower than exponential because $1-\gamma <1$. There are also similar but more complicated expressions for other $\gamma$ in the range $(0,1)$.

One can also calculate the case for superlinear preferential attachment with $\gamma >1$. In this case it turns out that the behaviour is that a "leader" emerges in the network, gaining a non-zero fraction of all edges, asymptotically, with the rest having degree less than some fixed constant. See here for more.

Nodes with inherent fitness

Inherent fitness aka attractiveness may vary across nodes in the network.

See Bose-Einstein condensation in complex networks and Competition and multiscaling in evolving networks for a model. In it a fitness value, $\eta_i$, is assigned to each node (sampled from a given distribution $\rho(\eta)$), and is unchanged thereafter. Now, the attachment kernel depends on $\eta$ as well: $a_k(\eta)$. The same method used as for the section Degree distribution as a function of time of creation above can be used (with $\eta$ instead of $t$), and a solution can be analytically obtained for the case $a_k(\eta) = \eta k$, and a power law distribution is obtained for each $\eta$, but not overall, as it depends on what $\rho(\eta)$ is.

In Bose-Einstein condensation in complex networks, they show an interesting effects that happens for some choices of $\rho(\eta)$, where a few nodes (a constant number of them, so as a fraction, they go as $1/n$ and go to 0 as $n \rightarrow \infty$ and so don't appear in $p_k$) have a degree proportional to $n$, and so do contribute to quantities like $\langle k \rangle$. This is analogous to Bose condensation. However, it is not known which $\rho(\eta)$ will produce condensation, and computer simulations suggest, that whether condensation occurs or not may depend on the fluctuations and thus not be deterministic (see Polya's run; is this at all related to Ross–Littlewood paradox?

There are also interesting work on the statistics of the node with maximum fitness (which changes more and more rarely as a higher value of $\eta$ is sampled at some updates). These follow so-called record dynamics Slow dynamics from noise adaptation.

More relevant review articles:

Statistical mechanics of complex networks

Complex networks: Structure and dynamics

Evolution of networks