Model reduction and aggregation are of key importance for simulation and analysis of large-scale systems, such as molecular dynamics, large swarms of robotic vehicles, and animal aggregations. We study a nonlinear network which exhibits areas of internally dense and externally sparse interconnections. The densely connected nodes in these areas synchronize in the fast time-scale, and behave as aggregate nodes that dominate the slow dynamics of the network. We first derive a singular perturbation model which makes this time-scale separation explicit and, next, prove the validity of the reduced-model approximation on the infinite time interval.
"We develop a novel aggregation approach by exploiting sparsity properties that the web inherently has, as stated by (ii) above. The particular approach has a close relation to the singular perturbation analysis for large-scale systems with network structures    . The method there however requires a stronger sparsity notion on the underlying graph, which seems difficult to expect in the web. "
[Show abstract][Hide abstract] ABSTRACT: The PageRank algorithm employed at Google assigns a measure of importance to
each web page for rankings in search results. In our recent papers, we have
proposed a distributed randomized approach for this algorithm, where web pages
are treated as agents computing their own PageRank by communicating with linked
pages. This paper builds upon this approach to reduce the computation and
communication loads for the algorithms. In particular, we develop a method to
systematically aggregate the web pages into groups by exploiting the sparsity
inherent in the web. For each group, an aggregated PageRank value is computed,
which can then be distributed among the group members. We provide a distributed
update scheme for the aggregated PageRank along with an analysis on its
convergence properties. The method is especially motivated by results on
singular perturbation techniques for large-scale Markov chains and multi-agent
IEEE Transactions on Automatic Control 03/2012; 57(11). DOI:10.1109/TAC.2012.2190161 · 2.78 Impact Factor
"where the matrix I − (1 − m) A 22 is nonsingular. This expression is motivated by the time-scale separation in singular perturbation based approaches of , , . Substituting this into the recursion ( "
[Show abstract][Hide abstract] ABSTRACT: At Google, the PageRank algorithm helps rankings in search results by providing measures of web page importance. This paper builds upon the distributed randomized approach for this algorithm proposed in our recent works. To reduce computation and communication, we develop a method to systematically aggregate web pages into groups by exploiting the sparsity inherent in the web. Each group computes an aggregated PageRank, which can be distributed among group members. We provide a decentralized scheme for its computation and analyze convergence properties.
[Show abstract][Hide abstract] ABSTRACT: We study stability properties of interconnected hybrid systems with application to large-scale logistics networks. Hybrid systems are dynamical systems that combine two types of dynamics: continuous and discrete. Such behaviour occurs in wide range of applications. Logistics networks are one of such applications, where the continuous dynamics occurs in the production and processing of material and the discrete one in the picking up and delivering of material. Stability of logistics networks characterizes their robustness to the changes occurring in the network. However, the hybrid dynamics and the large size of the network lead to complexity of the stability analysis. In this thesis we show how the behaviour of a logistics networks can be described by interconnected hybrid systems. Then we recall the small gain conditions used in the stability analysis of continuous and discrete systems and extend them to establish input-to-state stability (ISS) of interconnected hybrid systems. We give the mixed small gain condition in a matrix form, where one matrix describes the interconnection structure of the system and the other diagonal matrix takes into account whether ISS condition for a subsystem is formulated in the maximization or the summation sense. The small gain condition is sufficient for ISS of an interconnected hybrid system and can be applied to an interconnection of an arbitrary finite number of ISS subsystems. We also show an application of this condition to particular subclasses of hybrid systems: impulsive systems, comparison systems and the systems with stability of only a part of the state. Furthermore, we introduce an approach for structure-preserving model reduction for large-scale logistics networks. This approach supposes to aggregate typical interconnection patterns (motifs) of the network graph. Such reduction allows to decrease the number of computations needed to verify the small gain condition.
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