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Citations, Sequence Alignments, Contagion, and Semantics: On Acyclic Structures and their Randomness

Datasets from several domains, such as life-sciences, semantic web, machine learning, natural language processing, etc. are naturally structured as acyclic graphs. These datasets, particularly those in bio-informatics and computational epidemiology, have grown tremendously over the last decade or so. Increasingly, as a consequence, there is a need to build and evaluate various strategies for processing acyclic structured graphs. Most of the proposed research models the real world acyclic structures as random graphs, i.e., they are generated by randomly selecting a subset of edges from all possible edges. Unfortunately the graphs thus generated have predictable and degenerate structures, i.e., the resulting graphs will always have almost the same degree distribution and very short paths. Specifically, we show that if $O(n \log n \log n)$ edges are added to a binary tree of $n$ nodes then with probability more than $O(1/(\log n)^{1/n})$ the depth of all but $O({\log \log n} ^{\log \log n})$ vertices of the dag collapses to 1. Experiments show that irregularity, as measured by distribution of length of random walks from root to leaves, is also predictable and small. The degree distribution and random walk length properties of real world graphs from these domains are significantly different from random graphs of similar vertex and edge size.