Consider a large graph or network, and a user-provided set of query vertices 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 do this is to connect the vertices in the form of a tree structure that is present in the graph. However, in sufficiently dense graphs, most such trees will be large or somehow trivial (e.g. involving high degree vertices) and thus not insightful. Extending previous research, we define and investigate the new problem of mining subjectively interesting trees connecting a set of query vertices in a graph, 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 certain prior beliefs the user has specified about the graph. A remaining problem is efficiently fitting a prior belief model. We show how this can be done for a large class of prior beliefs. Given a specified prior belief model, we then propose heuristic algorithms to find the best trees efficiently. An empirical validation of our methods on a large real graphs evaluates the different heuristics and validates the interestingness of the given trees.

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.

Social networks often provide only a binary perspective on social ties: two individuals are either connected or not. While sometimes external information can be used to infer the strength of social ties, access to such information may be restricted or impractical to obtain. Sintos and Tsaparas (KDD 2014) first suggested to infer the strength of social ties from the topology of the network alone, by leveraging the Strong Triadic Closure (STC) property. The STC property states that if person A has strong social ties with persons B and C, B and C must be connected to each other as well (whether with a weak or strong tie). They exploited this property to formulate the inference of the strength of social ties as a NP-hard maximization problem, and proposed two approximation algorithms. We refine and improve this line of work, by developing a sequence of linear relaxations of the problem, which can be solved exactly in polynomial time. Usefully, these relaxations infer more fine-grained levels of tie strength (beyond strong and weak), which also allows one to avoid making arbitrary strong/weak strength assignments when the network topology provides inconclusive evidence. Moreover, these relaxations allow us to easily change the objective function to more sensible alternatives, instead of simply maximizing the number of strong edges. An extensive theoretical analysis leads to two efficient algorithmic approaches. Finally, our experimental results elucidate the strengths of the proposed approach, while at the same time questioning the validity of leveraging the STC property for edge strength inference in practice. Keywords Strong triadic closure • Strength of social ties • Linear programming • Convex relaxations • Half-integrality

Cycles in graphs o en signify interesting processes. For example, cyclic trading pa erns can indicate ine ciencies or economic dependencies in trade networks, cycles in food webs can identify fragile dependencies in ecosystems, and cycles in nancial transaction networks can be an indication of money laundering. Identifying such interesting cycles, which can also be constrained to contain a given set of query nodes, although not extensively studied, is thus a problem of considerable importance. In this paper, we introduce the problem of discovering interesting cycles in graphs. We rst address the problem of quantifying the extent to which a given cycle is interesting for a particular analyst. We then show that nding cycles according to this interestingness measure is related to the longest cycle and maximum mean-weight cycle problems (in the unconstrained se ing) and to the maximum Steiner cycle and maximum mean Steiner cycle problems (in the constrained se ing). A complexity analysis shows that nding interesting cycles is NPhard, and is NP-hard to approximate within a constant factor in the unconstrained se ing, and within a factor polynomial in the input size for the constrained se ing. e la er inapproximability result implies a similar result for the maximum Steiner cycle and maximum mean Steiner cycle problems. Motivated by these hardness results, we propose a number of e cient heuristic algorithms. We verify the e ectiveness of the proposed methods and demonstrate their practical utility on two real-world use cases: a food web and an international trade-network dataset.

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