Social Networks, Poisson, And ARMA

[John Creighto]

Social Properties and First order Links
I wasn't sure to put this in the math or sociology form but I already have two Social Networks topics posted in the Math forum and I think I would like to devote more specific topics to the math forum.


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Social Networks

I've being thinking recently about the relationship between social networks and dynamic models and well I have many questions it is clear that the underlying social network specifies how dependent people are with respect to common properties, such as age, location, ethnicity, wealth, professions, hobbies, etc...

People who are simmilar with respect to these dimensions will have a greater probability of forming a social tie. With regards to modeling it is easier to divide these dimensions into discrete bins or groups both for computational reasons and ease of gathering information.

Similarly data is usually collected in discrete time steps and usually discrete time models are computationally easier. With a given time period \Delta T there is a probability of two individuals forming some kind of social link. For instance they can exchange information, spread a disease, form a friendship, write a paper together, etc...

If any point in time a link is equally likely then Poisson statistics are appropriate. Thus within any bin or between any two bins we can assign a parameter lambda which is the average number of links formed within the bin in one time step. Unfortunately with Poisson statistics the variance is equal to the mean so there will be large uncertainty as to the number of links formed.

The links within a period of time can represent the exchange of some quantity (information, disease, ideas, etc...). Let:

F_A be the fraction of people in bin A that have this quanity.
F_B be the fraction of people in bin B that have this quantity.
Let N^1_{A,B} Be the number of first order links between bin A and bin B formed in one time step.

Then the amount of this quantity transferred from bin A to Bin B due to first order links is given by:

F_AN^1_{A,B}(1-F_B)

Similarly the amount of this quantity transferred from Bin B to Bin A due to first order links is given by:

F_BN^1_{A,B}(1-F_A)

 

[John Creighto]

Second Order Links

In a non directional network one might suspect second order links to follow poison statistics. I will verify this another time. For now I'm interested in direction networks, which are relevant to the spread of information and disease.

If \lambda is the probability of N occurrences in a time interval T, then the proability of an occurence within an infentesimal unit of time is

{\lambda dt \over T}

If a first order link occurs in time t. Then the fraction of these links which will be second order links is given by:

{(T-t) \over T} \lambda

Therefore the total number of second order links is given by:

\int_{t=0}^T{\lambda dt \over T}{(T-t) \over T} \lambda={\lambda^2 \over 2}

For large lambda higher order links dominate. Therefore, one suitable strategy would be to choose the bin size/time step small enough so that lambada is about 1/10. That way one only needs to consider first order links.

 

[changeseeker]

There is a large community of people working on these sorts of problems -- Wasserman and Faust come to mind immediately. So does Brian Uzzi. I would sign up on SOCNET and post your question there. I would also check the Pajek community and the SunBelt conference.