Update #3: Any Press is Good Press

The most recent step in my research, defining preferential attachment and deletion for a signed network, has by far been the most exciting and rewarding part thus far.  For the first time in my project this summer I had absolute freedom to put forth my own original ideas and it was thrilling to see how they led to useful results, both expected and unexpected.  To give a glimpse of what I was working on I will quickly summarize my five methods for preferential attachment.  These methods are used to decide which other nodes a new node is likely to connect to when it enters the network.  In other words, when a new user makes an account, with whom will it likely make connections or friendships with.  Each method takes into account a different psychological expectation of user behavior in a social network.

1) Neutral Preferential Attachment – Connections are most likely to be made with the node with the highest total degree (most links) regardless of whether those links are positive or negative.  This can be summarized by the phrase “Any press is good press” because a node is considered popular based on the quantity of connections is has, not taking into account whether those were positive or negative connections.

2) Separate Preferential Attachment – Positive connections are most likely to be made with the node with the highest positive degree (most positive links) and negative connections are most likely to be made with the node with the highest negative degree (most negative links).  While this seems like a very simple idea, it had some surprising effects on the network structure.  This method suggests a network where nodes are inherently good or bad, such that a good node will accumulate more and more good attention and a bad node will accumulate more and more bad attention.

3) Scaled Product Preferential Attachment – Connections are most likely to be made with the node with the highest product of degree*positive degree.  This method was meant to combine a preference for nodes with a high magnitude of connections with a preference for nodes with mostly positive links.  This method gave the most surprising results and had a degree distribution that unfortunately looked nothing like a degree distribution we would see in a real-world network.  However, this new result gave us better intuition as to what to expect for the next two methods.

4) Unscaled Ratio Preferential Attachment – Connections are most likely to be made with the node with the highest ratio of positive degree to negative degree. This method correctly chooses the node with a lot of positive connections and few negative connections.  I was hesitant about this model at first because it neglects to incorporate the overall magnitude of a node, but it ended up being the most useful model and the one I will likely study in more depth.

5) Scaled Ratio Preferential Attachment – Connections are most likely to be made with the node with the highest ratio of positive degree^2 to negative degree.  This method has the advantages of Unscaled Ratio Preferential Attachment but also gives preference to nodes with an overall high magnitude.  Before running my simulation, I had expected this method to be the most useful and most applicable to a real-world network, however it resulted in the same skewed and unusable results as Scaled Product Preferential Attachment.

As you can see, the results of these five methods were far from what I had expected, but this surprise was by far the most rewarding part of my research thus far. With these methods I created an entirely new addition to study of networks and the fact that the results were surprising only solidifies that there is something very significant here that needs to be studied and understood better.  The rest of my research will be to really delve into each network type to hopefully develop an intuition about how each method affects its network structure.