February notes
05 Mar 2019Notes for February 2019.
Minorities in network
I attended a talk about minorities in network, by Claudia Wagner at the Complex network seminar of Sorbonnes University. There was a lot of content, here are a few things I noted.

Contrary to what I thought, a $k$core is not another name for a $k$clique. A $k$core of a graph is a maximal connected subgraph where all the nodes have degree at least $k$. It can have much more nodes than just $k$ (a $k$regular graph is its own $k$core). Such subgraphs can be considered as coherent communities in social networks.

A part of the talk was about minorities in wikipedia. One would like to consider statements such as “women are more linked to other women than to men”. This is not true in general because there are more men pages, thus links have more chance to point to men, but could still be true, proportionally. But it’s a bit too rough to just look at proportions because the graph may be complicated and there might be many correlations going on. One way to deal with this is following: consider the graph of wikipedia bibliographies, note the global gender proportions, then erase the gender of the nodes, and reassign them at random, keeping the right proportions. Now you can compare the neighbourhoods in this new graph and in the original graph, and try to understand what’s going on. One paper on the topic is the following: It’s a Man’s Wikipedia? Assessing Gender Inequality in an Online Encyclopedia

A notion known as Burt efficiency, or brokerage, or structural hole, is more or less the following. A node that belongs to (or is close to) two clusters in a network, can have an advantage over the other nodes, because it can enjoy the information gathered by both communities, and can choose to transfer or not such information. One can define coefficients to measure if a node is or not in such a position.
Multistage optimization
For dynamic algorithms, one is usually concerned with having a good solution at any time, but these solutions do not need to be related. Multistage optimization, introduced in this paper, considers the cases where one should not change the solution too much between the two steps. In other words, in this framework one maximizes the quality of the solution, while minimizing the churn.
[I stumble on the notion in this preprint.]
123 Conjecture
Another edition of the Complex Network seminar by Mohammed Senhaji (that I couldn’t attend) was about the 123 conjecture, which is the following.
In any graph, one can label the edges with label 1, 2, or 3, such that, when each node computes the sum of the labels of its adjacent labels, not two neighbours have the same sum.
A survey about the conjecture is here.
Lovász’s new book
László Lovász wrote a new book: Graphs and geometry.
Vandermonde identity
Vandermonde’s identity is the following:
\[\binom{m+n}{r} = \sum_{k=0}^{r}\binom{m}{k} \binom{n}{rk}.\]I thought it was only a bachelor exercise, until it naturally popped up in the calculation in a recent paper.