Very large graphs

TL;DR

This paper discusses the mathematical challenges and properties of very large networks, highlighting differences between dense and sparse networks and reviewing related theoretical and probabilistic approaches.

Contribution

It provides a survey of the mathematical and probabilistic methods used to analyze large-scale networks, emphasizing the unique challenges posed by their size and incomplete information.

Findings

Sparse networks are more practically important.

Dense networks have more complete theoretical results.

Large networks require indirect data collection methods.

Abstract

In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them. These huge networks pose exciting challenges for the mathematician. Graph Theory (the mathematical theory of networks) faces novel, unconventional problems: these very large networks (like the Internet) are never completely known, in most cases they are not even well defined. Data about them can be collected only by indirect means like random local sampling. Dense networks (in which a node is adjacent to a positive percent of others nodes) and sparse networks (in which a node has a bounded number of neighbors) show very different behavior. From a practical point of view, sparse networks are more important, but at present we have more complete theoretical results for dense networks. The paper surveys relations with probability, algebra, extrema graph theory, and analysis.