Distributed Computation of Top-kk Degrees in Hidden Bipartite Graphs

Hidden graphs are flexible abstractions that are composed of a set of known vertices (nodes), whereas the set of edges are not known in advance. To uncover the set of edges, multiple edge probing queries must be executed by evaluating a function $f(u,v)$ that returns either true or false, if nodes $u$ and $v$ are connected or not respectively. Evidently, the graph can be revealed completely if all possible $n(n-1)/2$ probes are executed for a graph containing $n$ nodes. However, the function $f()$ is usually computationally intensive and therefore executing all possible probing queries result in high execution costs. The target is to provide answers to useful queries by executing as few probing queries as possible. In this work, we study the problem of discovering the top-$k$ nodes of a hidden bipartite graph with the highest degrees, by using distributed algorithms. In particular, we use Apache Spark and provide experimental results showing that significant performance improvements are achieved in comparison to existing centralized approaches

Core Discovery in Hidden Graphs

Massive network exploration is an important research direction with many applications. In such a setting, the network is, usually, modeled as a graph $G$, whereas any structural information of interest is extracted by inspecting the way nodes are connected together. In the case where the adjacency matrix or the adjacency list of $G$ is available, one can directly apply graph mining algorithms to extract useful knowledge. However, there are cases where this is not possible because the graph is \textit{hidden} or \textit{implicit}, meaning that the edges are not recorded explicitly in the form of an adjacency representation. In such a case, the only alternative is to pose a sequence of \textit{edge probing queries} asking for the existence or not of a particular graph edge. However, checking all possible node pairs is costly (quadratic on the number of nodes). Thus, our objective is to pose as few edge probing queries as possible, since each such query is expected to be costly. In this work, we center our focus on the \textit{core decomposition} of a hidden graph. In particular, we provide an efficient algorithm to detect the maximal subgraph of $S_k$ of $G$ where the induced degree of every node $u \in S_k$ is at least $k$. Performance evaluation results demonstrate that significant performance improvements are achieved in comparison to baseline approaches.Comment: 12 pages, 4 figure

Edge Estimation with Independent Set Oracles

We study the task of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an $n$-vertex graph, using (i) $\mathrm{polylog}(n)$ bipartite independent set queries, or (ii) ${n}^{2/3} \cdot\mathrm{polylog}(n)$ independent set queries.Comment: A preliminary version appeared in the proceedings of ITCS 201