Barabasi-Albert model
* This is just the undirected version of the Price model. The new vertex comes
with c links and joins the old vertices preferentially.
* Notice that in this case we don't have to have the parameter 'a' because the
graph is undirected, and hence there is no problem in having the connection
probability exactly proportional to the degree.
* It is easy to write the degree distribution of the BA model using the
solution for the Price graph: one just needs to substitute a = c.
The reasoning for this can be understood by the following argument:
- Suppose the graph is actually directed, and the new vertex joins the
network only with out-links.
- Since in the BA model, the probability is proportional to the total
degree (which is equal to the out-degree plus in-degree), the constant a
is simply equal to the out-degree of each vertex.
* It can be shown that asymptotically, the degree distribution p_k ~ k^(-3)
Extensions of preferential attachment
The original models of Price and Barabasi-Albert discard many realistic
processes that are present in the real-world networks. A large number of
variants of these models have been proposed over the years to explain other
important characteristics like high clustering, assortativity etc. Below we
briefly mention some of these mechanisms:
(1) Addition of extra edges
- In real networks, the edges are not formed only by the new nodes
that join the network. In fact, a large number of edges are added much
after the correspoding nodes have joined the network. For example, it is
common for two webpages to form a hyperlink between them much after they
are created. The major exception to this rule, and the one that Price
was interested in, is the citation network.
(2) Removal of edges
- It is common for real-networks to lose links: old friendships are
forgotten, Internet connections are removed, people are unfollowed on
social media, roads are closed and so on. Several models on network
formation incorporate these features. Again, the citation network is an
exception.
(3) Non-linear preferential attachment
- In the Price model and the BA model, the connection probability is
precisely proportional to the degree k. However, it is certainly possible
that at least in some networks the probability has a more general form
k^{gamma}, where gamma is a constant. Models involing such 'non-linear
preferential attachment' show that in most cases, the distribution
deviates from a pure-power law.
(4) Fitness models
- The idea of this class of models is to make the connection probability
proportinal not only to the degree of an old vertex, but also to its
intrinsic quality. For example, a new webpage may not want to link to an
old webpage just because that old page has a high degree, but also because
it is better than others independent of the degree values. Thus, in these
models, it becomes possible for a vertex that joins a network late to gain
substantial amount of links because of its intrinsic fitness. This is not
possible in the original preferential attachment models, except by a rare
chance. A famous model in this class is the Bianconi-Barabasi model.
* Other models of network formation
Preferential attachment is not the only mechanism that is widely employed
to explain the structure of the real-world networks. There exist other
models which to a varying extent try to explain other empirical features
(many times also the degree distributions). Some of these classes are
given below:
1. Vertex coping models
2. Network optimization models
3. Triadic closure models
4. Small-world models