Today’s massive and dynamic data sets have motivated many computer scientists and mathematicians to study classical problems in combinatorics and graph theory in various settings. In certain settings the algorithms’ access to the data is restricted while in others the resources are limited. In this talk we explore such settings in three directions: ranking of objects, property testing in social networks and finding communities in dynamic graphs.
In the first part, we discuss algorithms on permutations as well as prefixes of permutations also known as top-k lists. The study of later particularly arises in big data scenarios when analysis of the entire permutation is costly or not interesting. We start by discussing random walks on the set of full rankings or permutations of n objects, a set whose size is n!. Since 1990s to today, various versions of this problem have been studied, each for different motivation.
The second part of the talk is devoted to property testing in social networks: given a graph, an algorithm seeks to approximate several parameters of a graph just by accessing the graph by means of oracles and while querying these oracles as few times as possible. We introduce a new oracle access model which is applicable to social networks, and assess the complexity of estimating parameters such as number of users (vertices) in this model.
In the third part of the talk, we introduce a new dynamic graph model which is based on triadic closure: a friendship is more likely to be formed between a pair of users with a larger number of mutual friends. We find upper bounds for the rate of graph evolution in this model and using such bounds we devise algorithms discovering communities. I will finish this talk by proposing new directions and presenting related open problems.