# Presentation: 912290

## Ageing in Citation Networks

- How Network Ageing
- Can Affect the Ranking Information

**Dylan Walker**
- Brookhaven National Lab
- Stony Brook University

## Summary

- The Problem: Ranking Citation Networks
- Why the old way is bad
- How we can learn from Google
- CiteRank Model
- Performance on Real Citation Networks
- Optimal Parameters
- Why it works – physical interpretation

## The old way of ranking publications

- Current method of ranking citation networks:
- 		       kin ~ the number of citations received
- But this is unfair:
- New papers have not been around long enough to accrue citations
- All citations are not equal
- A new citation should count more than an old one
- Citations from popular papers should count more
- Google PageRank does this...

**Notes:**
The conventional method of ranking publications, based on the number of citations received, is not very good. It has a number of problems:  For one thing, old papers are preferred, because K_in rises monotonically with time.  A new paper is not highly ranked simply because it has not been around long enough to accrue citations. In reality, all citations are not equal.  Newer citations indicate relevancy in active lines of research. Additionally, citations from popular papers should count more than citations from less popular ones.  This latter point is a key feature addressed by a successful ranking scheme, known as the Google PageRank.  

## Google Predicts Traffic

- Why is Google’s PageRank so successful?
- How do we know it *is* successful?
- PageRank is a model of traffic: The PageRank of a page can be interpreted as the predicted traffic for that page.
- 1010 heads are better than 1:
- An ensemble of random surfers walk on the network.
- Predictions of traffic to a given site are determined from the average visitation.
- Random surfers aren’t smart, but the network is.
- Walking on a network accounts for the self-consistence of popularity.
- So why can’t we use Google on citation network?

**Notes:**
Why is Google so successful?  How do we know that a given ranking is a good ranking? This sounds a lot like a question for engineers.  Let’s turn it into a question for scientists.  Suppose we interpret the PageRank of a webpage as a prediction of future traffic for that page.  Then PageRank is a model of traffic.  It’s a good model if it yields an accurate prediction of traffic.  Now we can ask a more meaningful question: Why is PageRank a good model of traffic?  The answer is that 10^10 heads are better than one.  Google simulates random walkers on the network.  A single walker doesn’t really see that much.  But an army of them see virtually everything.  This means that they can collectively tap the information that’s stored in the network.  In fact, this is why PageRank accounts for popularity in a self consistent way.  A surfer is more likely to arrive at a popular paper and hence more likely to follow a link from that paper.  This is one of the features we are looking for.  So why not use PageRank on a citation network?

## Google and Citation Networks

- Citation networks are fundamentally different from the web
- Citation networks are acyclic and have an intrinsic time-arrow:
- The links on a webpage can be updated at any moment. It is their own responsibility to maintain relevancy.
-  The citations in a publication remain fixed.
- What does this mean for ranking?
- Given enough time, random researchers (surfers) would pile-up at the old edge of the network.
- Aging effects cannot be ignored.
- Can we still model traffic on Citation Networks with random researchers?

**Notes:**
Citation networks are fundamentally different from the web.  For one thing, they are acyclic and have an intrinsic time arrow.  PageRank assumes that web pages are kept up-to-date: the onus of current relevancy is theirs.  But Citations, by definition, are never updated.  What does this mean in terms of our army of random surfers?  It means that, given enough time, they’ll all pile up on the old edge of the citation network.  So can we still use the concept of random researchers to model traffic?  Only if we account for aging explicitly.

## The CiteRank Model of Traffic

- The CiteRank prediction of traffic has two parameters:
- With a fixed probability, each researcher will follow a citation to an adjacent publication
- Probability to follow a link ~
- Distribute random researchers on a citation network according to an initial distribution:
-               , where         ~ characteristic decay time
- The CiteRank algorithm is given by:

**Notes:**
We introduce the CiteRank model of traffic.  It has two parameters.  The first one, alpha is similar to the PageRank parameter. It’s related to the probability of following a citation.  The new parameter, tau_d is related to the aging.  Here’s how it works.  We distribute random researchers on a citation network with an initial distribution rho.  Tau_d is the characteristic time constant of this distribution.  Explicitly, the CiteRank traffic is given by the following series.  By the way, W here is the transfer matrix W= {(1/k_out^j if j cites i ; 0 otherwise}.  The first term of this series is the traffic that comes from initially selecting a paper.  The second term is the traffic that comes from following a single citation... and so on.  We take this series until it converges sufficiently, and we have our predicted traffic.

## Two Real Citation Networks

- To select the best parameters and see if CiteRank is a viable ranking scheme, we evaluate two real citation networks:
- High Energy Physics Theory ArXiv (hep-th):
- 	A snapshot of the high energy physics theory area of arxiv.org from April 2003 (citations ranging from 1992-2003)
- 	2800 papers ; 350,000 citations
- 	no form of peer review
- Physical Review (physrev):
- 	Citation data from all Physical Review journals (citations ranging from 1913-2005)
- 	380,000 papers ; 3,100,000 citations

## CiteRank Optimal Parameters

- The CiteRank predicts traffic
- Ideally, we would like to select parameters that best correlate Ti with real traffic, Tireal.
- However, traffic data is not readily available.
- Can estimate Tireal with the recently accrued citations, Dki .
- Relationship between Tireal and Dki is unclear:
- Assume linearity and test the correlation over range of the model parameters.

**Notes:**
So how can we test CiteRank on these real networks?  Since CiteRank is predicts traffic, we might try to select the parameters that best correlate predicted traffic with actual traffic.  The problem with this is that actual traffic data for the publications in these networks is not readily available.  In lieu of this, we can estimate the actual traffic to a paper with the number of citations accrued over some recent period of time.  We don’t know the exact relationship between real traffic and recent citations.  We start by assuming a linear relationship and examine the correlation over the range of model parameters.  

## Linear Correlation of Ti with Dki

- Linear Correlation of Ti with Dki

- physrev

**Notes:**
So here are the results of that correlation for the physrev network.  The correlation is shown as contours over the parameter plane.  The features are pretty similar for the hep-th network (which I’ll show you in a moment). There’s a fairly sharp peak in the correlation here that is localized around a particular value of alpha.  It’s a little wider over tau_d.  The maximal correlation is achieved for alpha=0.5, tau_d=2.6 yrs.  In case you are interested, the value of the maximum correlation coeff=0.6543.

## Linear Correlation of Ti with Dki

- Linear Correlation of Ti with Dki

- hep-th

**Notes:**
Here’s the same thing for the hep-th network. You’ll notice that the time scale of tau_d is in days here.  We just don’t have nearly as much data here as we did in the physrev case. But if you compare the regions on a similar scale, you see that the features of the correlation plot are remarkably similar.  Keep in mind that these two citation networks are, in fact, very different.  Physrev data is from peer-reviewed publications acquired over a hundred year period.  Hep-th is made up of electronically submitted preprint articles with no associated form of peer-review. 
Here the alpha=0.48  tau_d=363 days.  In case you’re interested the maximal correlation coeff=0.7275 .  

## What if Ti isn’t linear with Dki ?

- The previous correlation contour plots rely on the assumption of linearity between real traffic and recent citations.
- Can we relax this assumption to something more reasonable?
- Assume monotonic relationship only
- There is a correlation measure adapted for such a situation:  Spearman Rank Correlation
- Changes in Dki that do not lead to rank changes will not affect the correlation.
- We should expect peaks that are broadened due to this decrease in sensitivity.

**Notes:**
Of course, you could easily argue that the both of these plots rests on the assumption of linearity between traffic and recent citations.  You’d be perfectly correct.  We can relax this assumption and see what happens.  We might not expect the relation to be linear, but it is certainly reasonable to expect that it is monotonic.  There is a correlation measure adapted for such a situation; it is the Spearman Rank Correlation.  In rank correlation, only the order of the T_i values matters.  That is, a change in recent citations of a publication that does not move that publication up in rank will not affect this correlation.  We should expect some broadening in the peaks of the correlation contour plot, due to this decrease in sensitivity.

## Rank Correlation of Ti with Dki

- Rank Correlation of Ti with Dki

- physrev

**Notes:**
In case you’re interested, the maximal correlation coeff=0.6436

## Rank Correlation of Ti with Dki

- Rank Correlation of Ti with Dki

- hep-th

**Notes:**
In case you’re interested, the maximal correlation coeff=0.5978 .

## Correlation from Age Distribution

- Why is the peak correlation attained at those values of the parameters?
- In what way is traffic prediction getting better?
- Look at linear correlation for physrev
- Take the slice td = 2.6 yrs (optimal) and look at effect of varying a.
- Examine the average age distribution:
- Real citations , Dki
- Predicted traffic , Ti

**Notes:**
Suppose we want to get a better idea of what’s going on in the correlation contour plot.  For example, you might ask: why is the peak in correlation attained?  In what way is the traffic prediction getting better?  Let’s take a look at the linear correlation case for the physrev network and take a slice tau_d=2.6 yrs (the optimal value) and examine the effect of varying alpha.  We can look at how this traffic is distributed over age for both the real citations and the predicted traffic.  How does this shape change with alpha?

## Age Distribution

**Notes:**
Here are the age distribution curves for physrev, tau_d=2.6 yrs and alpha as shown.  The blue curve is the age distribution for recent real citations.  The curves have been artificially separated, for easier comparison.  (Actually the normalization constant has no effect on the correlation).  The red curve matches the shape of real citation data best.  The purple curve below it comes close, but initially is too steep.  That is low values of alpha, predict a traffic age distribution that is too uniform.  Higher values of alpha predict traffic that is weighted too much in favor of recent publications.  Alpha=0.5 is about the “sweet spot”.  

## Concluding Remarks

- Good agreement in estimation of a over networks:
- On average, the typical researcher follows citation chain of length ~ 2
- Future explorations
- Precise relation between Dki and Tireal
- Sampling of actual traffic

## Acknowledgements

- Support:
- Brookhaven National Lab, Division of Material Science, U.S. Department of Energy
- Collaborators:
- S. Maslov, S. Redner, H. Xie, Y. Koon-Kiu, P. Chen
- Thanks to:
- 	Mark Doyle, Marty Blume, Paul Dlug of the Physical Review Editorial Office

## Slide 18

## Citing Age Distribution

- T(t) ~ traffic from CiteRank model as a function of age
- T(t) is comprised of two varieties of traffic:
- Direct traffic ~ Td(t) – arrive at paper via initial selection
- Indirect traffic ~ Ti(t) – arrive at paper via citation
- Pc(t,t’) ~ fraction papers of age t’  papers age t
-                          To good approximation:
- or in fourier space:
- Then, for the tail of T(t), an exp. fit can be made with:
- 					 so, insisting this tail fit real traffic: