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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a seed node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution pi uses O(1/||pi||_2 + d_avg) queries; we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily; the results of Katzir et al. give an upper bound that is worse by a multiplicative factor t_mix(1/n^4).
The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes; in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges; we show that this is tight.

Varun Kanade, Frederik Mallmann-Trenn, and Victor Verdugo. How Large Is Your Graph?. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kanade_et_al:LIPIcs.DISC.2017.34, author = {Kanade, Varun and Mallmann-Trenn, Frederik and Verdugo, Victor}, title = {{How Large Is Your Graph?}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {34:1--34:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.34}, URN = {urn:nbn:de:0030-drops-79767}, doi = {10.4230/LIPIcs.DISC.2017.34}, annote = {Keywords: Estimation, Random Walks, Social Networks} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We consider the problem of stable matching with dynamic preference lists. At each time-step, the preference list of some player may change by swapping random adjacent members. The goal of a central agency (algorithm) is to maintain an approximately stable matching, in terms of number of blocking pairs, at all time-steps. The changes in the preference lists are not reported to the algorithm, but must instead be probed explicitly. We design an algorithm that in expectation and with high probability maintains a matching that has at most O((log n)^2 blocking pairs.

Varun Kanade, Nikos Leonardos, and Frédéric Magniez. Stable Matching with Evolving Preferences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kanade_et_al:LIPIcs.APPROX-RANDOM.2016.36, author = {Kanade, Varun and Leonardos, Nikos and Magniez, Fr\'{e}d\'{e}ric}, title = {{Stable Matching with Evolving Preferences}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.36}, URN = {urn:nbn:de:0030-drops-66597}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.36}, annote = {Keywords: Stable Matching, Dynamic Data} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intra-cluster edge probability p, and inter-cluster edge probability q. We focus on the sparse case, i.e. p, q = O(1/n), which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction
of the conjecture was proved by Mossel, Neeman and Sly (2012), and more recently the positive direction was proven independently by Massoulie and Mossel, Neeman, and Sly.
In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

Varun Kanade, Elchanan Mossel, and Tselil Schramm. Global and Local Information in Clustering Labeled Block Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 779-792, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{kanade_et_al:LIPIcs.APPROX-RANDOM.2014.779, author = {Kanade, Varun and Mossel, Elchanan and Schramm, Tselil}, title = {{Global and Local Information in Clustering Labeled Block Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {779--792}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.779}, URN = {urn:nbn:de:0030-drops-47384}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.779}, annote = {Keywords: stochastic block models, information flow on trees} }