Subjective interestingness of subgraph patterns

Abstract

The utility of a dense subgraph in gaining a better understanding of a graph has been formalised in numerous ways, each striking a different balance between approximating actual interestingness and computational efficiency. A difficulty in making this trade-off is that, while computational cost of an algorithm is relatively well-defined, a pattern's interestingness is fundamentally subjective. This means that this latter aspect is often treated only informally or neglected, and instead some form of density is used as a proxy. We resolve this difficulty by formalising what makes a dense subgraph pattern interesting to a given user. Unsurprisingly, the resulting measure is dependent on the prior beliefs of the user about the graph. For concreteness, in this paper we consider two cases: one case where the user only has a belief about the overall density of the graph, and another case where the user has prior beliefs about the degrees of the vertices. Furthermore, we illustrate how the resulting interestingness measure is different from previous proposals. We also propose effective exact and approximate algorithms for mining the most interesting dense subgraph according to the proposed measure. Usefully, the proposed interestingness measure and approach lend themselves well to iterative dense subgraph discovery. Contrary to most existing approaches, Learn (2016) 105:41-75 our method naturally allows subsequently found patterns to be overlapping. The empirical evaluation highlights the properties of the new interestingness measure given different prior belief sets, and our approach's ability to find interesting subgraphs that other methods are unable to find.

Explainable Subgraphs with Surprising Densities: A Subgroup Discovery Approach

“…By De Bie [4], it was argued to choose P as the Iprojection of the previous background distribution onto the set of distributions consistent with the presented pattern. Then Van Leeuwen et al [19] showed that the resulting P is again a product of Bernoulli distribution:…”

“…Given a pattern (W 1 , W 2 , I, k W ), and a background distribution defined by P , the probability of the presence of the pattern is the probability of getting k W or more (for I = 0), or fewer than k W (for I = 1) successes in n W trials with possibly different success probabilities p u,v . While it is impractical to compute these probabilities exactly, using the same approach as Van Leeuwen et al [19] they can be tightly upper bounded using the general Chernoff/Hoeffding bound [9], as follows:…”

Discovering Interesting Cycles in Directed Graphs

“…where w(e) −log(Pr(µ(e) ≥ e )) denotes the information content of the edge e, with Pr(·) denoting the probability under the background distribution P. Note that w(e) ≥ 0 for all e ∈ V × V . e DL can be computed similarly as in the work of Leeuwen et al [17]. To communicate a cycle pa ern C to the user, we need to communicate |C | nodes.…”

Subjectively Interesting Connecting Trees

“…As shown in [6], given prior beliefs on the degrees of the nodes, the maximum entropy distribution factorizes as:…”

“…In general, an edge will have a high probability if the corresponding expected block diagonal sum is high, see Eqs. (3) and (6). Note that the citation network should (in theory) be a directed acyclic graph, since no paper can cite a paper with a higher publication year.…”