Finding HeavyPaths in Weighted Graphs and a Case-Study on Community Detection

Abstract

A heavy path in a weighted graph represents a notion of connectivity and ordering that goes beyond two nodes. The heaviest path of length in the graph, simply means a sequence of nodes with edges between them, such that the sum of edge weights is maximum among all paths of length . It is trivial to state the heaviest edge in the graph is the heaviest path of length 1, that represents a heavy connection between (any) two existing nodes. This can be generalized in many different ways for more than two nodes, one of which is finding the heavy weight paths in the graph. In an influence network, this represents a highway for spreading information from a node to one of its indirect neighbors at distance . Moreover, a heavy path implies an ordering of nodes. For instance, we can discover which ordering of songs (tourist spots) on a playlist (travel itinerary) is more pleasant to a user or a group of users who enjoy all songs (tourist spots) on the playlist (itinerary). This can also serve as a hard optimization problem, maximizing different types of quantities of a path such as score, flow, probability or surprise, defined as edge weight. Therefore, if one can solve the Heavy Path Problem (HPP) efficiently, they can as well use HPP for modeling and reduce other complex problems to it. More precisely, we aim at finding k heaviest (top-k) paths of a given length . The weight of a path is defined as a monotone aggregation of individual edge weights using functions such as sum or product. We argue finding simple paths is way more practical than finding paths with cycles in applications such as: routing, playlist recommendation, itinerary planning and influence maximization. Avoiding cycles results in lists without repeated items and with better diversity. Simple paths are also expected to have a higher utility in practice. This makes HPP NP-hard and inapproximable. We propose an efficient algorithm that despite its exponential (theoretical) worst case running time, achieves the exact answer of the NP-hard problem in many useful cases and study the problem from different perspectives.