  In this article we presented a brief study of the main network models with
growth and preferential attachment. Such models are interesting because they
present several characteristics of real systems. We started with the classical
model proposed by Barabasi and Albert: nodes are added to the network
connecting preferably to other nodes that are more connected. We also presented
models that consider more representative elements from social perspectives,
such as the homophily between the vertices or the fitness that each node has to
build connections. Furthermore, we showed a version of these models including
the Euclidean distance between the nodes as a preferential attachment rule. Our
objective is to investigate the basic properties of these networks as
distribution of connectivity, degree correlation, shortest path, cluster
coefficient and how these characteristics are affected by the preferential
attachment rules. Finally, we also provided a comparison of these synthetic
networks with real ones. We found that characteristics as homophily, fitness
and geographic distance are significant preferential attachment rules to
modeling real networks. These rules can change the degree distribution form of
these synthetic network models and make them more suitable to model real
networks.
