Applications of Sampling and Estimation on Networks

Networks or graphs are fundamental abstractions that allow us to study many important real systems, such as the Web, social networks and scientific collaboration. It is impossible to completely understand these systems and answer fundamental questions related to them without considering the way their components are connected, i.e., their topology. However, topology is not the only relevant aspect of networks. Nodes often have information associated with them, which can be regarded as node attributes or labels. An important problem is then how to characterize a network w.r.t. topology and node label distributions. Another important problem is how to design efficient algorithms to accomplish tasks on networks. Since nodes often have attributes, an interesting avenue for investigation consists in learning and exploiting existing correlations between node and neighbor attributes for accomplishing a task more efficiently. One of the challenges faced when studying networks in the wild is the fact that in general their topology and information associated with its nodes cannot be directly obtained. Thus, one must resort to collecting the data, but when obtaining the entire network is infeasible, sampling and estimation are the best option. This dissertation investigates the use of sampling and estimation to characterize networks and to accomplish a particular task. More precisely, we study (i) the problem of characterizing directed and undirected networks through random walk-based sampling, (ii) the problem of estimating the set-size distribution from an information-theoretic standpoint, which has application to characterizing the in-degree distribution in large graphs, and (iii) the problem of searching networks to find nodes that exhibit a specific trait while subject to a sampling budget by learning a model from node attributes and structural properties, which has application to recruiting in social networks

Analysis of Relaxation Time in Random Walk with Jumps

We study the relaxation time in the random walk with jumps. The random walk with jumps combines random walk based sampling with uniform node sampling and improves the performance of network analysis and learning tasks. We derive various conditions under which the relaxation time decreases with the introduction of jumps.Comment: 13 page

Sampling-based estimation of in-degree distribution in directed networks

The focus of this thesis is on the estimation of the in-degree distribution in directed networks from sampling network nodes or edges. A number of sampling schemes are considered, including random sampling with and without replacement, and several approaches based on random walks with possible jumps. When sampling nodes, it is assumed that only the out-edges of that node are visible, that is, the in-degree of that node is not observed. The suggested estimation of the in-degree distribution is based on two approaches. The inversion approach exploits the relation between the original and sample in-degree distributions, and can estimate the bulk of the in-degree distribution, but not the tail of the distribution. The tail of the in-degree distribution is estimated through an asymptotic approach, which itself has two versions: one assuming a power-law tail and the other for a tail of general form. The two estimation approaches are examined on synthetic and real networks, with good performance results, especially striking for the asymptotic approach.Bachelor of Scienc

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Analysis of Relaxation Time in Random Walk with Jumps

International audienceWe study the relaxation time in the random walk with jumps. The random walk with jumps combines random walk based sampling with uniform node sampling and improves the performance of network analysis and learning tasks. We derive various conditions under which the relaxation time decreases with the introduction of jumps