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Maximizing the quality index modularity has become one of the primary methods for identifying the clustering structure within a graph. As contemporary networks are not static but evolve over time, traditional static approaches can be inappropriate for specific tasks. In this work we pioneer the NP-hard problem of online dynamic modularity maximization. We… (More)

Maximizing the quality index <i>modularity</i> has become one of the primary methods for identifying the clustering structure within a graph. Since many contemporary networks are not static but evolve over time, traditional static approaches can be inappropriate for specific tasks. In this work, we pioneer the NP-hard problem of online dynamic modularity… (More)

J o u r n a l o f P r o b a b i l i t y Electron. Abstract We consider a system of N particles on the real line that evolves through iteration of the following steps: 1) every particle splits into two, 2) each particle jumps according to a prescribed displacement distribution supported on the positive reals and 3) only the N right-most particles are… (More)

Consider a d-ary rooted tree (d ≥ 3) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S * n of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to… (More)

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