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Learning and Verifying Graphs using Queries with a Focus on Edge Counting
"... Abstract. We consider the problem of learning and verifying hidden graphs and their properties given query access to the graphs. We analyze various queries (edge detection, edge counting, shortest path), but we focus mainly on edge counting queries. We give an algorithm for learning graph partitions ..."
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Abstract. We consider the problem of learning and verifying hidden graphs and their properties given query access to the graphs. We analyze various queries (edge detection, edge counting, shortest path), but we focus mainly on edge counting queries. We give an algorithm for learning graph partitions using O(n log n) edge counting queries. We introduce a problem that has not been considered: verifying graphs with edge counting queries, and give a randomized algorithm with error ǫ for graph verification using O(log(1/ǫ)) edge counting queries. We examine the current state of the art and add some original results for edge detection and shortest path queries to give a more complete picture of the relative power of these queries to learn various graph classes. Finally, we relate our work to Freivalds ’ ‘fingerprinting technique ’ – a probabilistic method for verifying that two matrices are equal by multiplying them by random vectors. 1
Topology Discovery of Sparse Random Graphs With Few Participants
"... We consider the task of topology discovery of sparse random graphs using endtoend random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discover ..."
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We consider the task of topology discovery of sparse random graphs using endtoend random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain endtoend measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario(a), ourproposedalgorithm resultsinasublineareditdistance guarantee using a sublinear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sublinear numberof uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of endtoend information along all the paths between the participants.
Polynomial Time Optimal Query Algorithms for Finding Graphs with Arbitrary Real Weights
"... We consider the problem of finding the edges of a hidden weighted graph and their weights by using a certain type of queries as few times as possible, with focusing on two types of queries with additive property. For a set of vertices, the additive query asks the sum of weights of the edges with bot ..."
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We consider the problem of finding the edges of a hidden weighted graph and their weights by using a certain type of queries as few times as possible, with focusing on two types of queries with additive property. For a set of vertices, the additive query asks the sum of weights of the edges with both ends in the set. For a pair of disjoint sets of vertices, the crossadditive query asks the sum of weights of the edges crossing between the two sets. These queries are related to DNA sequencing and finding Fourier coefficients of pseudoBoolean functions, and have been paid attention to in computational learning. In this paper, we achieve an ultimate goal of recent years for graph finding, by constructing the first polynomial time algorithms with optimal query complexity for the general class of graphs with n vertices and at most m edges in which the weights of edges are arbitrary) real numbers. The algorithms are randomized and their query complexities are O m log n log m which improve the best known bounds by a factor of log m. To build a key component for graph finding, we consider coin weighing with a spring scale which itself has been paid attention to in a long history of combinatorial search. We construct the first polynomial time algorithm with optimal query complexity for the general case in which the weight differences between counterfeit and authentic coins are arbitrary real numbers. We also construct the first polynomial time optimal query algorithm for finding Fourier coefficients of a certain class of pseudoBoolean functions.
Active Learning of Interaction Networks
, 2009
"... From molecular arrangements to biological organisms, our world is composed of systems of small components interacting with and affecting each other. Scientists often learn the structure of such systems by tampering with them and making observations. In this thesis, we develop methods for automating ..."
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From molecular arrangements to biological organisms, our world is composed of systems of small components interacting with and affecting each other. Scientists often learn the structure of such systems by tampering with them and making observations. In this thesis, we develop methods for automating this process from an active learning perspective, a setting where the learner is not restricted to making passive observations, but can choose to query the data. First, we consider the setting of learning hidden graphs with queries. Each query type is motivated by a realworld problem, from genome sequencing to evolutionary tree reconstruction. We give new algorithms for learning graphs and also consider the problem of verifying the results of the learning task. Next, we turn to value injection queries, which model experiments used to identify gene regulatory networks. We analyze the complexity of learning large alphabet and analog circuits with value injection queries. We then apply this model to social networks, allowing the learner to activate and suppress agents in the network, and we give an optimal algorithm and matching lower bound for this problem. Finally, we examine the passive learner, who watches the output of agents in a social network and must deduce the most likely underlying network. Last, we consider a classical problem in query learning: learning finite automata, which themselves are networks of connected states. We introduce label queries as a generalization of the well studied membership queries. We give algorithms for learning automata using label queries and analyze other models for learning automata.