Network Capacity Bound for Personalized PageRank in Multimodal Networks

In a former paper the concept of Bipartite PageRank was introduced and a theorem on the limit of authority flowing between nodes for personalized PageRank has been generalized. In this paper we want to extend those results to multimodal networks. In particular we introduce a hypergraph type that may be used for describing multimodal network where a hyperlink connects nodes from each of the modalities. We introduce a generalisation of PageRank for such graphs and define the respective random walk model that can be used for computations. we finally state and prove theorems on the limit of outflow of authority for cases where individual modalities have identical and distinct damping factors.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1702.0373

Glassy behavior induced by geometrical frustration in a hard-core lattice gas model

We introduce a hard-core lattice-gas model on generalized Bethe lattices and investigate analytically and numerically its compaction behavior. If compactified slowly, the system undergoes a first-order crystallization transition. If compactified much faster, the system stays in a meta-stable liquid state and undergoes a glass transition under further compaction. We show that this behavior is induced by geometrical frustration which appears due to the existence of short loops in the generalized Bethe lattices. We also compare our results to numerical simulations of a three-dimensional analog of the model.Comment: 7 pages, 4 figures, revised versio

Normalisation Control in Deep Inference via Atomic Flows

We introduce `atomic flows': they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax

Dynamical transitions and quantum quenches in mean-field models

We develop a generic method to compute the dynamics induced by quenches in completely connected quantum systems. These models are expected to provide a mean-field description at least of the short time dynamics of finite dimensional system. We apply our method to the Bose-Hubbard model, to a generalized Jaynes-Cummings model, and to the Ising model in a transverse field. We find that the quantum evolution can be mapped onto a classical effective dynamics, which involves only a few intensive observables. For some special parameters of the quench, peculiar dynamical transitions occur. They result from singularities of the classical effective dynamics and are reminiscent of the transition recently found in the fermionic Hubbard model. Finally, we discuss the generality of our results and possible extensions

Vertex Clustering in Random Graphs via Reversible Jump Markov Chain Monte Carlo

Networks are a natural and effective tool to study relational data, in which observations are collected on pairs of units. The units are represented by nodes and their relations by edges. In biology, for example, proteins and their interactions, and, in social science, people and inter-personal relations may be the nodes and the edges of the network. In this paper we address the question of clustering vertices in networks, as a way to uncover homogeneity patterns in data that enjoy a network representation. We use a mixture model for random graphs and propose a reversible jump Markov chain Monte Carlo algorithm to infer its parameters. Applications of the algorithm to one simulated data set and three real data sets, which describe friendships among members of a University karate club, social interactions of dolphins, and gap junctions in the C. Elegans, are given

Hitting time for the continuous quantum walk

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.Comment: 12 pages, 1figur