Diameter Minimization by Shortcutting with Degree Constraints

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.Comment: A shorter version of this work has been accepted at the IEEE ICDM 2022 conferenc

Minimizing Hitting Time between Disparate Groups with Shortcut Edges

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. 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. In this paper we address the problem of adding a given number of shortcut edges in the network so as to directly minimize the average hitting time and the maximum hitting time between two disparate groups. Our algorithm for minimizing average hitting time is a greedy bicriteria that relies on supermodularity. In contrast, maximum hitting time is not supermodular. Despite, we develop an approximation algorithm for that objective as well, by leveraging connections with average hitting time and the asymmetric k-center problem.Comment: To appear in KDD 202

Subjectively interesting connecting trees and forests

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

Block-Approximated Exponential Random Graphs

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 both 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 probabalistic model for this task.Comment: Accepted for DSAA 2020 conferenc

Application of cerium chloride to improve the acid resistance of dentine

OBJECTIVE: To investigate the effect of cerium chloride, cerium chloride/fluoride and fluoride application on calcium release during erosion of treated dentine. METHODS: Forty dentine samples were prepared from human premolars and randomly assigned to four groups (1-4). Samples were treated twice a day for 5 days, 30s each, with the following solutions: group 1 placebo, group 2 fluoride (Elmex fluid), group 3 cerium chloride and group 4 combined fluoride and cerium chloride. For the determination of acid resistance, the samples were consecutively eroded six times for 5 min with lactic acid (pH 3.0) and the calcium release in the acid was determined. Furthermore, six additional samples per group were prepared and used for EDS analysis. SEM pictures of these samples of each group were also captured. RESULTS: Samples of group 1 presented the highest calcium release when compared with the samples of groups 2-4. The highest acid resistance was observed for group 2. Calcium release in group 3 was similar to that of group 4 for the first two erosive attacks, after which calcium release in group 4 was lower than that of group 3. Generally, the SEM pictures showed a surface coating for groups 2-4. No deposits were observed in group 1. CONCLUSION: Although fluoride showed the best protective effect, cerium chloride was also able to reduce the acid susceptibility of dentine significantly, which merits further investigation

Testing Cluster Properties of Signed Graphs

Publisher Copyright: © 2023 Owner/Author.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.Peer reviewe