Speeding up switch Markov chains for sampling bipartite graphs with given degree sequence

We consider the well-studied problem of uniformly sampling (bipartite) graphs with a given degree sequence, or equivalently, the uniform sampling of binary matrices with fixed row and column sums. In particular, we focus on Markov Chain Monte Carlo (MCMC) approaches, which proceed by making small changes that preserve the degree sequence to a given graph. Such Markov chains converge to the uniform distribution, but the challenge is to show that they do so quickly, i.e., that they are rapidly mixing. The standard example of this Markov chain approach for sampling bipartite graphs is the switch algorithm, that proceeds by locally switching two edges while preserving the degree sequence. The Curveball algorithm is a variation on this approach in which essentially multiple switches (trades) are performed simultaneously, with the goal of speeding up switch-based algorithms. Even though the Curveball algorithm is expected to mix faster than switch-based algorithms for many degree sequences, nothing is currently known about its mixing time. On the other hand, the switch algorithm has been proven to be rapidly mixing for several classes of degree sequences. In this work we present the first results regarding the mixing time of the Curveball algorithm. We give a theoretical comparison between the switch and Curveball algorithms in terms of their underlying Markov chains. As our main result, we show that the Curveball chain is rapidly mixing whenever a switch-based chain is rapidly mixing. We do this using a novel state space graph decomposition of the switch chain into Johnson graphs. This decomposition is of independent interest

The intrinsic vulnerability of networks to epidemics

Contact networks are convenient models to investigate epidemics, with nodes and links representing potential hosts and infection pathways, respectively. The outcomes of outbreak simulations on networks are driven both by the underlying epidemic model, and by the networks’ structural properties, so that the same pathogen can generate different epidemic dynamics on different networks. Here we ask whether there are general properties that make a contact network intrinsically vulnerable to epidemics (that is, regardless of specific epidemiological parameters). By conducting simulations on a large set of modelled networks, we show that, when a broad range of network topologies is taken into account, the effect of specific network properties on outbreak magnitude is stronger than that of fundamental pathogen features such as transmission rate, infection duration, and immunization ability. Then, by focusing on a large set of real world networks of the same type (potential contacts between field voles, Microtus agrestis), we showed how network structure can be used to accurately assess the relative, intrinsic vulnerability of networks towards a specific pathogen, even when those have limited topological variability. These results have profound implications for how we prevent disease outbreaks; in many real world situations, the topology of host contact networks can be described and used to infer intrinsic vulnerability. Such an approach can increase preparedness and inform preventive measures against emerging diseases for which limited epidemiological information is available, enabling the identification of priority targets before an epidemic event

Rapid mixing of the switch Markov chain for strongly stable degree sequences

The switch Markov chain has been extensively studied as the most natural Markov chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We show that the switch chain for sampling simple undirected graphs with a given degree sequence is rapidly mixing when the degree sequence is so‐called strongly stable. Strong stability is satisfied by all degree sequences for which the switch chain was known to be rapidly mixing based on Sinclair's multicommodity flow method up until a recent manuscript of Erdős and coworkers in 2019. Our approach relies on an embedding argument, involving a Markov chain defined by Jerrum and Sinclair in 1990. This results in a much shorter proof that unifies (almost) all the rapid mixing results for the switch chain in the literature, and extends them up to sharp characterizations of P‐stable degree sequences. In particular, our work resolves an open problem posed by Greenhill and Sfragara in 2017

Topology of complex networks: Models and analysis

DOI: 10.1017/S000497271600100

A survey of dental practice and equipment characteristics.

The aim of this study was to determine what would be regarded as essential equipment required for setting up a dental practice and the cost thereof. This information could be useful for the newly qualified dentist. Seventy five completed questionnaires returned from a randomly selected group of practising dentists were analysed to obtain this information. Equipment costs were obtained from a dental supply house and are tabulated. Guidelines recommended for setting up a practice include determining the type of service to be rendered, selecting the equipment required for this purpose, seeking expert financial advice and commencing the practice on a conservative basis with only the essential items of equipment.Articl

Evaluation of the high-pressure extrusion technique as a method for sizing plasmid DNA-containing cationic liposomes

Tumorimmunolog