Network probability matrix

The network probability matrix describes the probability structure of a network based on the historical presence or absence of edges in a network. For example, individuals in a social network are not connected to other individuals with uniform random probability. The probability structure is much more complex. Intuitively, there are some people whom a person will communicate with or be connected more closely than others. For this reason, real-world networks tend to have clusters or cliques of nodes that are more closely related than others (Albert and Barabasi, 2002, Carley [year], Newmann 2003). This can be simulated by varying the probabilities that certain nodes will communicate.

The edge probabilities can be derived from empirical data in several ways. Given network data collected over multiple time periods on a group of subjects, the edge probabilities can be estimated by the proportion of edge occurrences, e_ij, for each cell in the adjacency matrix, . In the case of communication networks, statistical distributions can be fitted to the time between messages for each potential edge in the network. For a specified period of time t, the edge probability p for each set of entities i and j can be found. Under certain assumptions, the following is true:

[line missing, possibly as a result of copying and pasting]

In practice, the function must be estimated using techniques such as maximum likelihood estimation; it is the probability density function for the time between communications from node v_i to v_j, and represent the parameters of the density. It may be desirable to construct a network based on a restriction such as, “two emails within a time period demonstrate a relationship, but one does not.” In this case, it is necessary to compose a function of random variables. If [?] represents the probability density function of time between two sets of two emails and [?] represents the probability density function of time between one set of two emails, then the following is true under certain assumptions:

[line missing, possibly as a result of copying and pasting]

It is possible to generalize this idea; if the probability that x or more communications occur within time t, then the following is true:

[line missing, possibly as a result of copying and pasting]

This newly proposed framework for viewing the probability space of a social network preserves the same flexibility for modeling dyadic relationships, however, it provides researchers with a means to understand the probability space of the network and thus devise more robust and appropriate statistical tests for social network analysis.

The network probability matrix was originally proposed by Ian McCulloh.

External links

• Center for Computational Analysis of Social and Organizational Systems (CASOS) at Carnegie Mellon University
• U.S. Military Academy Network Science Center
• The Center for Interdisciplinary Research on Complex Systems at Northeastern University

References

• McCulloh, I., Lospinoso, J. & Carley, K.M. (2007). Probability Mechanics in Communications Networks. In Proceedings of the 12th International Conference on Applied Mathematics of the World Science Engineering Academy and Society, Cairo, Egypt. 30–31 December 2007.
• "Understanding Network Science," http://www.zangani.com/blog/2007-1030-networkingscience
• Linked: The New Science of Networks, A.-L. Barabási (Perseus Publishing, Cambridge (2002).
• Network Science, The National Academies Press (2005)ISBN 0-309-10026-7