Diversity maximization problem is a well-studied problem where the goal is to find k diverse items. Fair diversity maximization aims to select a diverse subset of k items from a large dataset, while requiring that each group of items be well represented in the output. More formally, given a set of items with labels, our goal is to find k items that maximize the minimum pairwise distance in the set, while maintaining that each label is represented within some budget. In many cases, one is only interested in selecting a handful (say a constant) number of items from each group. In such scenario we show that a randomized algorithm based on padded decompositions improves the state-of-the-art approximation ratio to √log(m) /(3m), where m is the number of labels. The algorithms work in several stages: (i) a preprocessing pruning which ensures that points with the same label are far away from each other(ii) a decomposition phase, where points are randomly placed in clusters such that there is a feasible solution with maximum one point per cluster and that any feasible solution will be diverse(iii) assignment phase, where clusters are assigned to labels, and a representative point with the corresponding label is selected from each cluster. We experimentally verify the effectiveness of our algorithm on large datasets.

One of the most well-known and simplest models for diversity maximization is the Max-Min Diversification (MMD) model, which has been extensively studied in the data mining and database literature. In this paper, we initiate the study of the Asymmetric Max-Min Diversification (AMMD) problem. The input is a positive integer k and a complete digraph over n vertices, together with a nonnegative distance function over the edges obeying the directed triangle inequality. The objective is to select a set of k vertices, which maximizes the smallest pairwise distance between them. AMMD reduces to the well-studied MMD problem in case the distances are symmetric, and has natural applications to query result diversification, web search, and facility location problems. Although the MMD problem admits a simple 1/2-approximation by greedily selecting the next-furthest point, this strategy fails for AMMD and it remained unclear how to design good approximation algorithms for AMMD.

Structural bias or segregation of networks refers to situations where two or more disparate groups are present in the network, so that the groups are highly connected internally, but loosely connected to each other. Examples include polarized communities in social networks, antagonistic content in video-sharing or news-feed platforms, etc. In many cases it is of interest to increase the connectivity of disparate groups so as to, e.g., minimize social friction, or expose individuals to diverse viewpoints. A commonly-used mechanism for increasing the network connectivity is to add edge shortcuts between pairs of nodes. In many applications of interest, edge shortcuts typically translate to recommendations, e.g., what video to watch, or what news article to read next. The problem of reducing structural bias or segregation via edge shortcuts has recently been studied in the literature, and random walks have been an essential tool for modeling navigation and connectivity in the underlying networks. Existing methods, however, either do not offer approximation guarantees, or engineer the objective so that it satisfies certain desirable properties that simplify the optimization task.

This work initiates the study of property testing in signed graphs, where every edge has either a positive or a negative sign. We show that there exist sublinear query and time algorithms for testing three key properties of signed graphs: balance (or 2-clusterability), clusterability and signed triangle freeness. We consider both the dense graph model, where one queries the adjacency matrix entries of a signed graph, and the bounded-degree model, where one queries for the neighbors of a node and the sign of the connecting edge. Our algorithms use a variety of tools from unsigned graph property testing, as well as reductions from one setting to the other. Our main technical contribution is a sublinear algorithm for testing clusterability in the bounded-degree model. This contrasts with the property of k-clusterability in unsigned graphs, which is not testable with a sublinear number of queries in the bounded-degree model. We experimentally evaluate the complexity and usefulness of several of our testers on real-life and synthetic datasets.

We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer. The problem is motivated by network-design applications, where we want to minimize the worst case communication in the network without excessively increasing the degree of any single vertex, so as to avoid additional overload. We present three algorithms for this task, each with their own merits. The special case of a matching augmentation -when every vertex can be incident to at most one new edge-is of particular interest, for which we show an inapproximability result, and provide bounds on the smallest achievable diameter when these edges are added to a path. Finally, we empirically evaluate and compare our algorithms on several real-life networks of varying types.

An important challenge in the field of exponential random graphs (ERGs) is the fitting of non-trivial ERGs on large graphs. By utilizing fast matrix block-approximation techniques, we propose an approximative framework to such non-trivial ERGs that result in dyadic independence (i.e., edge independent) distributions, while being able to meaningfully model local information of the graph (e.g., degrees) as well as global information (e.g., clustering coefficient, assortativity, etc.) if desired. This allows one to efficiently generate random networks with similar properties as an observed network, and the models can be used for several downstream tasks such as link prediction. Our methods are scalable to sparse graphs consisting of millions of nodes. Empirical evaluation demonstrates competitiveness in terms of both speed and accuracy with state-of-the-art methods-which are typically based on embedding the graph into some low-dimensional space- for link prediction, showcasing the potential of a more direct and interpretable probablistic model for this task.

Abstract 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 thestrengthof 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 theStrong Triadic Closure(STC) property. The STC property states that if personAhas strong social ties with personsBandC,BandCmust 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.

Cycles in graphs often signify interesting processes. For example, cyclic trading patterns can indicate inefficiencies or economic dependencies in trade networks, cycles in food webs can identify fragile dependencies in ecosystems, and cycles in financial 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 first address the problem of quantifying the extent to which a given cycle is interesting for a particular analyst. We then show that finding cycles according to this interestingness measure is related to the longest cycle and maximum mean-weight cycle problems (in the unconstrained setting) and to the maximum Steiner cycle and maximum mean Steiner cycle problems (in the constrained setting). A complexity analysis shows that finding interesting cycles is NP-hard, and is NP-hard to approximate within a constant factor in the unconstrained setting, and within a factor polynomial in the input size for the constrained setting. The latter 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 efficient heuristic algorithms. We verify the effectiveness of the proposed methods and demonstrate their practical utility on two real-world use cases: a food web and an international trade-network dataset.

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.