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.
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