Wiki Collaboration Graph

A graph is a set of vertices (represented by points) and edges connecting pairs of vertices. We have seen several applications of graph theory, such as representing highway systems. Graphs are also useful for representing and analyzing social networks. In a social network, each person is represented by a vertex and two people are connected by an edge if they are friends.

The study of social networks and relationships allows us to understand phenomena such as how ideas and news spread among groups of people, and how interconnected we are. The social networks of Facebook, Twitter, and LinkedIn (to name a few) are continuously and extensively studied. We will study the collaboration graph of contributors to the Brilliant wiki project.

The Brilliant Wiki Collaboration graph is the graph with vertices representing wiki authors and edges between pairs of authors who have collaborated on a Brilliant wiki page. We have removed any authors who have no collaborators, as well as all staff members. Here are a few statistics on the wiki collaboration graph (as of March 2015):

\[ \begin{align} \mbox{Number of pages: } & 535\\ \mbox{Number of vertices/contributors: } & 328\\ \mbox{Number of edges/collaborations: } & 1480\\ \end{align}\]

The interactive Wiki Collaboration Graph is posted here. Hover over any vertex to highlight the neighbors in the graph. (Note the resemblance to the Brilliant logo!)

You may notice that the edges seem bundled. This is because some authors have worked on mostly algebra pages, while other authors have worked on mostly geometry pages, and similar subjects are grouped together.

Another property you may notice is that the wiki collaboration graph is not connected -- there are several groups of authors who are connected to each other but are not connected to anyone else. A connected component in an undirected graph is a subset of vertices such that every vertex can be reached by every other vertex through a sequences of edges (which forms a path). There is one giant component in the wiki collaboration graph with 212 vertices, and the remaining connected components are quite small. There are two connected components with 4 vertices, one connected component with 3 vertices, eight connected components with 2 vertices, and the remaining connected components have 1 vertex each. These isolated components may also occur in social graphs, such groups of people living on an island who have no contact with the outside world.

The giant component, which is a connected component containing a significant number of the vertices in the network, is also observed in many other large-scale networks, such as the internet graph. Why is there only one giant component in the wiki collaboration network? Imagine what would happen if there are two giant components. Then because there are a large number of vertices in the both giant components, there is a good chance that at some point, one person from the first giant component eventually collaborates on a wiki page with another person from the second giant component. This will merge the two giant components into a single component!

The degree of a vertex \(v\) in a graph is the number of edges including vertex \(v\). In the wiki collaboration graph, the degree of a vertex is the number of collaborators she or he has. Here is the distribution of degrees in the wiki collaboration graph:

The interactive version of this histogram is here.

The contributors with the greatest number of collaborators are:

\[ \begin{array}{ccccc} \mbox{Anuj: } 46 & & \mbox{Nihar: } 39 & & \mbox{Yash: } 38\\ \mbox{Hemang: } 36 & & \mbox{Azhaghu: } 31 & & \mbox{Mursalin: } 31\\ \mbox{Ashley: } 26 & & \mbox{Sharky: } 25 & & \mbox{Prasun: } 22\\ \mbox{A: } 20 & & \mbox{Ishan: } 20 & & \mbox{Julian: } 20\\ \mbox{Krishna: } 19 & & \mbox{Agnishom: } 18 & & \mbox{Christopher: } 18\\ \mbox{Trevor: } 18 & & \mbox{Omkar: } 17 & & \mbox{Satvik: } 16\\ \mbox{Ritu: } 15 & & \mbox{Hamza: } 15 & & \mbox{Archit: } 14\\ \mbox{Curtis: } 12 & & \mbox{GAUTAM: } 12 & & \mbox{Jeremy: }12\\ \mbox{Daniel: } 11 & & \mbox{megh: } 11 & & \mbox{Mohnish: } 11\\ \mbox{Anant: } 11 & & & & \\ \end{array}\]

Similarly, we can consider the number of wiki pages with 1 author, 2 authors, 3 authors, etc.

The interactive version of this histogram is here.

The pages with the highest number of contributions are:

Fermat's Little Theorem: 10 contributors
Chinese Remainder Theorem: 8 contributors
Prime Factorization: 7 contributors
Divisibility Rules: 7 contributors
Pythagorean Theorem: 7 contributors
Fractions: 6 contributors
Extended Euclidean Algorithm: 6 contributors
Applying the Arithmetic Mean-Geometric Mean: 6 contributors
Cyclic Quadrilaterals: 6 contributors

Do the distributions in the above images remind you of any distributions you are familiar with? We will return to this topic when we study the power-law phenomenon for degree distributions of networks.

Similar graphs have been created for collaborations among researchers. Paul Erdős was a prolific mathematician who wrote roughly 1,500 mathematical articles in his lifetime, in collaboration with 511 co-authors. A researcher's Erdős number is the distance in the collaboration graph from the researcher to Paul Erdős. For example, all of the 511 co-authors on papers with Paul Erdős have Erdős number 1, the set of people who are co-authors with people of Erdős number 1 (but do not have Erdős number 1 themselves) have Erdős number 2, etc.

We have seen many great contributions from the wiki writing project from all over the world and would like to thank all of the wiki writers for helping to make Brilliant wiki into the very best math resource online. Thank you on behalf of the many people who are benefiting from your experience and knowledge!!