Subjectively Interesting Connecting Trees

Abstract

Abstract. Consider a large network, and a user-provided set of query nodes between which the user wishes to explore relations. For example, a researcher may want to connect research papers in a citation network, an analyst may wish to connect organized crime suspects in a communication network, or an internet user may want to organize their bookmarks given their location in the world wide web. A natural way to show how query nodes are related is in the form of a tree in the network that connects them. However, in sufficiently dense networks, most such trees will be large or somehow trivial (e.g. involving high degree nodes) and thus not insightful. In this paper, we define and investigate the new problem of mining subjectively interesting trees connecting a set of query nodes in a network, i.e., trees that are highly surprising to the specific user at hand. Using information theoretic principles, we formalize the notion of interestingness of such trees mathematically, taking in account any prior beliefs the user has specified about the network. We then propose heuristic algorithms to find the best trees efficiently, given a specified prior belief model. Modeling the user's prior belief state is however not necessarily computationally tractable. Yet, we show how a highly generic class of prior beliefs, namely about individual node degrees in combination with the density of particular sub-networks, can be dealt with in a tractable manner. Such types of beliefs can be used to model knowledge of a partial or total order of the network nodes, e.g. where the nodes represent events in time (such as papers in a citation network). An empirical validation of our methods on a large real network evaluates the different heuristics and validates the interestingness of the given trees.

Explainable Subgraphs with Surprising Densities: A Subgroup Discovery Approach

“…If the analyst believes that people in different class years are less likely to connect with each other, the discovered pattern would end up being more informative and useful as it contrasts more with this kind of belief. As shown by Adriaens et al [1], the resulting background distribution is also a product of Bernoulli distributions, one for each of the random variable h u,v ∈ {0, 1}):…”

“…We informally also refer to the extension as the subgroup. Now a description-induced subgraph can be formally defined as: 1 We consider undirected graphs for the sake of presentation and consistency with most literature. However, we note that all our results can be easily extended to directed graphs and graphs with self-edges.…”

“…We also define a pattern syntax informing the analyst about the edge density between two different subgroups. More formally, we define a bi-subgroup pattern as a quadruplet (W 1 , W 2 , I, k W ), where W 1 and W 2 are two descriptions, and I ∈ {0, 1} indicates whether the number of connections between ε(W 1 ) and ε(W 2 ) is upper bounded (1) or lower bounded (0) by the threshold k W . The maximum number of connections between the extensions ε(W 1 ) and ε(W 2 ) is denoted by n W |ε(W 1 )||ε(W 2 )| − 1 2 |ε(W 1 ∧ W 2 )|(|ε(W 1 ∧ W 2 )|+1) for undirected graphs without self-edges.…”

Subjectively interesting connecting trees and forests

“…Relation to conference version This paper is based on the conference paper (Adriaens et al 2017). The following material is completely new:…”

“…-A general method for efficiently fitting background distributions for a large class of prior belief types. Adriaens et al (2017) discussed how to model prior beliefs about vertex degrees and a time-order relation between the vertices in the graph. The current paper extends this to prior knowledge on the density of any part of the adjacency matrix-a generalization of previous work, which also has important applicability in contexts beyond this work (Sects.…”