Conditional Reliability in Uncertain Graphs

TL;DR

This paper introduces the concept of conditional reliability in uncertain graphs, where edge probabilities depend on external conditions, and presents methods to optimize reliability under these conditions, with proven complexity results and empirical validation.

Contribution

It formalizes the problem of conditional reliability, proves its computational hardness, and proposes practical algorithms with empirical evaluation on real-world graphs.

Findings

The problem is NP-hard and does not admit a PTAS.

Proposed methods are effective and scalable on large real-world graphs.

The study extends reliability analysis to condition-dependent edge probabilities.

Abstract

Network reliability is a well-studied problem that requires to measure the probability that a target node is reachable from a source node in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of existence. Many approaches and problem variants have been considered in the literature, all assuming that edge-existence probabilities are fixed. Nevertheless, in real-world graphs, edge probabilities typically depend on external conditions. In metabolic networks a protein can be converted into another protein with some probability depending on the presence of certain enzymes. In social influence networks the probability that a tweet of some user will be re-tweeted by her followers depends on whether the tweet contains specific hashtags. In transportation networks the probability that a network segment will work properly or not might depend on external conditions such as weather or time of the day. In this paper we overcome this limitation and focus on conditional reliability, that is assessing reliability when edge-existence probabilities depend on a set of conditions. In particular, we study the problem of determining the k conditions that maximize the reliability between two nodes. We deeply characterize our problem and show that, even employing polynomial-time reliability-estimation methods, it is NP-hard, does not admit any PTAS, and the underlying objective function is non-submodular. We then devise a practical method that targets both accuracy and efficiency. We also study natural generalizations of the problem with multiple source and target nodes. An extensive empirical evaluation on several large, real-life graphs demonstrates effectiveness and scalability of the proposed methods.