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**1 - 2**of**2**### Algorithms for Offline Tracking of Connected Components in Large Evolving Networks

"... Given a large evolving network with time information on its edges, we are interested in how and when its connected components are formed through time. Such information is useful while analyzing the characteristics of the network’s snapshots taken at different time points. This analysis can be used t ..."

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Given a large evolving network with time information on its edges, we are interested in how and when its connected components are formed through time. Such information is useful while analyzing the characteristics of the network’s snapshots taken at different time points. This analysis can be used to answer various queries such as what is the time point where two people are first connected in a professional network, how the scientific communities merged over time in a citation graph, or how conversations are formed and attracted new users in a forum discussion. The sensitivity of such an analysis increases with the number of time points and the cost of the analysis increase along with. We propose efficient algorithms and a compact representation of component structures evolving through time for both directed and undirected networks. For an undirected network with m edges, the time complexity of the algorithm is almost linear with m. For the directed case, the time complexity is O(mlog τ) where τ is the number of snapshots.

### Journal of Machine Learning Research (2011) Submitted 07/2011; Published Adaptive Content Search Through Comparisons

"... We study the problem of navigating through a database of similar objects using comparisons. This problem is known to be strongly related to the small-world network design problem. However, contrary to prior work, which focuses on cases where objects in the database are equally popular, we consider h ..."

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We study the problem of navigating through a database of similar objects using comparisons. This problem is known to be strongly related to the small-world network design problem. However, contrary to prior work, which focuses on cases where objects in the database are equally popular, we consider here the case where the demand for objects may be heterogeneous. We show that, under heterogeneous demand, the small-world network design problem is NP-hard. Given the above negative result, we propose a novel mechanism for smallworld design and provide an upper bound on its performance under heterogeneous demand. The above mechanism has a natural equivalent in the context of content search through comparisons, and we establish both an upper bound and a lower bound for the performance of this mechanism. These bounds are intuitively appealing, as they depend on the entropy of the demand as well as its doubling constant, a quantity capturing the topology of the set of target objects. Finally, based on these results, we propose an adaptive learning algorithm for content search that meets the performance guarantees achieved by the above mechanisms. 1.