Stochastic Actor-Based Models for Network Local-Dynamics

Social networks analysis is a statistical technique to analyze phenomena in which is present some type of relation between some type of social-actors (people, web pages, country, corporations and so on). SNA breaks with standard statistical methods, it synthesizes the phenomen object of analysis in form of graphical structure and thanks to geometric-descriptive approach and statistical-probabilistic analysis it allows to find patterns and regularity among the structure related to the phenomenon. SNA could be used to give answer in many scientific fields according to the aims and objectives of researchers. For example in marketing and politics it is important to understand who are the most important and/or the most influence actors in the network, in literature these people are called opinion − leader, furthermore in criminal network intelligence SNA allows to understand how to break and/or to isolate groups of actors for example the connection between cocaine providers and distributors minimyzing a predefined cost function.
Network Analysis allows to identify roles and positions of social actors, veryfing statistical hypothesis about relations, associations among relations, patterns and regularities. In some case is also possible to compare results that carry out from different networks. Moreover in the last twenty years Scientists have developed models to deal with the future structure of the networks, these methods are based on panel network data and they take into account the behavior of actors.
In this chapter, the main notions and results of network analysis will be shown.

Network Data
Remember that a network is a structure that comes out by the union of graphs and data. There are two types of variables in a network data set: structural variables and composition variables. Structural variables are measured on pairs of actors and represent the relation among them. In specific, the kinds of relations include:
• Friendship, Respect.
• Transictions or transfers of material and non material resources such as money or communications.
• Movement, formal roles and kinship.
Composition variables are the classical attributes that are measured on the actors e.g. salary, age, sex ecc. These type of variables is not necessary to study network while structural variables are essential.
For a social network study may be very difficult to understand which actors to include for the analysis namely it is hard to determinate a reference population. Generally reserchers use two type of approaches to specify a population. The first is the realist approach that focuses on actor set boundaries and membership as preceived by the actor themselves. For example a baby-gang is acknowledged as a social entity by its members and these members know the other members that belong to the gang. The other approach is the nominalist, namely researcher construct the list of actors for its analytical purposes. Network data are mostly gathered by questionnaries, interview, archivial records or experiments. However there are a lot of iusses about network data sampling that won’t treat in this work.
For a network the term mode is refferred to a distinct set of entities on which variables are measured. One − mode network is a network in which actors comes from one set of actors namley actors are of the same types. In Two − mode network actors belong to two different sets e.g. affiliation network are network in which there are a set of actors and a set of clubs to which the actors belong. In these type of networks is possible to study relation among actors of the same set and among actors of different sets. This thesis focuses essentially on one-mode network.
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