All homework is to be completed without assistance from any other student or faculty member. If you need
assistance, please contact Prof. McBride. You must use Mathematica to complete the exploration of the problem
computationally. The write up is to be done in Mathematica as well. Homeworks submitted late incur a 10
percentage point penalty per 24 hours late. Thus, home works submitted after 9:ooam Wednesday and by 9:ooam
Thursday will be penalized 10 percentage points; from 9:01am Thursday to 9:ooam Friday, 20 percentage points,
etc.
Last Modified: 2015.09.15
Economic Context
Use of meta-data to analyze relationships is a currently, sometimes hotly, debated topic. Using network analysis of meta-data about relation-
ships between people, interesting patterns can be inferred. In a fun blog post Healy (2013), based in part on work by Han (2009), uses simple
meta-data on membership in organizations in the Boston area in 1775 can identify Paul Revere as a key central figure. Both Han (2009) and
Healy (2013) refer to measures of the centrality of a node in the network. One of the measures they use is the betweenness centrality which
measures the proportion of shortest paths between any two given nodes that a third node lies within.
A separate literature has arisen that uses these same techniques to understand the relationships between co-authors, (e.g. Leskovec, et. al.
(2007)). In the context of co-authors, the question becomes who are the central figures (nodes) in a literature? The pattern of who co-
authored with whom reveals information on the centrality of different authors doing research in the literature.
The assignment is going to use data on astrophysics co-authorships from 1995 to 1999 to undertake an analysis of the network relationships
which differs from the typical centrality measures. Instead of betweenness centrality, we’re going to determine hub and authority scores for
each node in the network. Hubs and authorities are generally used as the basis for ranking nodes related to searching for particular nodes and
were proposed by Kleinberg as alternative to Google’s PageRank.
What does all of this have to do with mathematical economics? A graph, or network, is defined by its adjacency matrix. The adjacency
matrix m is a square matrix of dimension n, where n is the number of nodes. The (i, j)th element of the m is equal to one if a link exists from
node i TO node j. Thus, the adjacency matrix need not be symmetric (links between nodes can be directed or undirected). The algorithm for
computing hub and authorities scores for each node is a set of iterated matrix calculations that will converge.
Model Exploratlon
I Research steps to undertake
To undertake the hub and authority analysis, complete the following steps to determine who the most important “hubs” and “authorities” are
in the astrophysics literature:
1. Carefully read sections 14.2 and 14.6.A of Easley and Kleinberg (2010). Read the other parts of the chapter 14 only if you’re interested.
2. Get the ”citations-starting.nb” Mathematica notebook. Setup the Mathematica notebook to pull in the data and create the adjacency
matrix. Instructions are provided in the notebook.
3. Eco414 students: Complete a 4 step (k = 4) hub and authority algorithm using the data provided. Which astrophysicists are the
important hubs or authorities? Interpret what that means.
4. MA. and 3+1 students: Complete a 4 step (k = 4) hub and authority algorithm using the data provided. This will help you understand
the algorithm. Next, undertake the analysis in the limit as k -> 00. Which astrophysicists are the important hubs or authorities? Interpret
what that means.

All homework is to be completed without assistance from any other student or faculty member. If you need
assistance, please contact Prof. McBride. You must use Mathematica to complete the exploration of the problem
computationally. The write up is to be done in Mathematica as well. Homeworks submitted late incur a 10
percentage point penalty per 24 hours late. Thus, home works submitted after 9:ooam Wednesday and by 9:ooam
Thursday will be penalized 10 percentage points; from 9:01am Thursday to 9:ooam Friday, 20 percentage points,
etc.
Last Modified: 2015.09.15
Economic Context
Use of meta-data to analyze relationships is a currently, sometimes hotly, debated topic. Using network analysis of meta-data about relation-
ships between people, interesting patterns can be inferred. In a fun blog post Healy (2013), based in part on work by Han (2009), uses simple
meta-data on membership in organizations in the Boston area in 1775 can identify Paul Revere as a key central figure. Both Han (2009) and
Healy (2013) refer to measures of the centrality of a node in the network. One of the measures they use is the betweenness centrality which
measures the proportion of shortest paths between any two given nodes that a third node lies within.
A separate literature has arisen that uses these same techniques to understand the relationships between co-authors, (e.g. Leskovec, et. al.
(2007)). In the context of co-authors, the question becomes who are the central figures (nodes) in a literature? The pattern of who co-
authored with whom reveals information on the centrality of different authors doing research in the literature.
The assignment is going to use data on astrophysics co-authorships from 1995 to 1999 to undertake an analysis of the network relationships
which differs from the typical centrality measures. Instead of betweenness centrality, we’re going to determine hub and authority scores for
each node in the network. Hubs and authorities are generally used as the basis for ranking nodes related to searching for particular nodes and
were proposed by Kleinberg as alternative to Google’s PageRank.
What does all of this have to do with mathematical economics? A graph, or network, is defined by its adjacency matrix. The adjacency
matrix m is a square matrix of dimension n, where n is the number of nodes. The (i, j)th element of the m is equal to one if a link exists from
node i TO node j. Thus, the adjacency matrix need not be symmetric (links between nodes can be directed or undirected). The algorithm for
computing hub and authorities scores for each node is a set of iterated matrix calculations that will converge.
Model Exploratlon
I Research steps to undertake
To undertake the hub and authority analysis, complete the following steps to determine who the most important “hubs” and “authorities” are
in the astrophysics literature:
1. Carefully read sections 14.2 and 14.6.A of Easley and Kleinberg (2010). Read the other parts of the chapter 14 only if you’re interested.
2. Get the ”citations-starting.nb” Mathematica notebook. Setup the Mathematica notebook to pull in the data and create the adjacency
matrix. Instructions are provided in the notebook.
3. Eco414 students: Complete a 4 step (k = 4) hub and authority algorithm using the data provided. Which astrophysicists are the
important hubs or authorities? Interpret what that means.
4. MA. and 3+1 students: Complete a 4 step (k = 4) hub and authority algorithm using the data provided. This will help you understand
the algorithm. Next, undertake the analysis in the limit as k -> 00. Which astrophysicists are the important hubs or authorities? Interpret
what that means.