Synchrony on Networks

Below are my initial impressions about the project. These are just a sketch, and open to change!

Basic (Toy) Model

Throughout, I'll use the word 'spike' interchangeably with 'tweet.'

The basic skeleton is based off of the paper by Aram et al. We consider a network of users which we'll model as a graph with a vertex for each user and an edge denoting a connection between each user. For Twitter data, these edges might indicate follower/followee relationships. Directedness of these edges is an open question. This gives us the basic structural model.

The dynamics occurring on top of this network will take the form of coupled inhomogeneous Poisson processes. That is, each vertex will have a Poisson process on top of it, and the rate of the process will depend on the behavior of the other nodes connected to it. An inhomogeneous Poisson process model is a common (though very rough) approximation of neural behavior. It captures the notion of 'completely random.'

Aram's model sets the instantaneous rate of each user to be a (constant) base rate, plus terms that depend on the behavior of the neighbors of a given user (if a neighbor has recently tweeted, and the user is dynamically connected to that user, they are more likely to tweet). The constant base rate is open to change.

Community Detection

A standard approach to finding communities within a network is to look at the network structure alone. We attempt to partition the nodes in the network such that there are many more edges within a community than between communities. A popular method for doing this is the fast-greedy algorithm developed by Clauset, Newman, and Moore.

One simple extension of this approach would be

Toy Network

One of our first tasks is to design a reasonable (small) network to run the toy model on top of. This network should presumably have apparent community structure, and then we can design on top of this a weighted network that incorporates the notion of dynamical community. These weights would essentially correspond to the weights in Equation 4 of Aram et al.' paper. One option for designing realistic network structure is to use a small network within the Twitter network I have available.