A hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O (md log n) where d is the depth of the dendrogram describing the community structure.Expand

This work presents a general technique for inferring hierarchical structure from network data and shows that the existence of hierarchy can simultaneously explain and quantitatively reproduce many commonly observed topological properties of networks.Expand

This paper uses the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram of the stochastic block model, a commonly used generative model for social and biological networks, and develops a belief propagation algorithm for inferring functional groups or communities from the topology of the network.Expand

This work proposes quantum versions of finite-state and push-down automata, and regular and context-free grammars, and finds analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form.Expand

A way of encoding sparse data using a “nonbacktracking” matrix, and it is shown that the corresponding spectral algorithm performs optimally for some popular generative models, including the stochastic block model.Expand

A mean-field solution for the average path length and for the distribution of path lengths in the small-world network model is presented, which is exact in the limit of large system size and either a large or small number of shortcuts.Expand

Two quantum walks on the n-dimensional hypercube are studied, one in discrete time and one in continuous time, showing that the instantaneous mixing time is (π/4)n steps, faster than the Θ(n log n) steps required by the classical walk.Expand

We define a case of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous… Expand

The resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network, and are given exact solutions for the position of this threshold in a variety of cases.Expand

The authors explain why the P vs. NP problem is so fundamental, and why it is so hard to resolve, and lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing.Expand