Rapid mixing of the switch Markov chain for strongly stable degree sequences and 2-class joint degree matrices

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 use comparison arguments with other, less natural but simpler to analyze, Markov chains, to show that the switch chain mixes rapidly in two different settings. We first study the classic problem of uniformly sampling simple undirected, as well as bipartite, graphs with a given degree sequence. We apply an embedding argument, involving a Markov chain defined by Jerrum and Sinclair (TCS, 1990) for sampling graphs that almost have a given degree sequence, to show rapid mixing for degree sequences satisfying strong stability, a notion closely related to P-stability. This results in a much shorter proof that unifies the currently known rapid mixing results of the switch chain and extends them up to sharp characterizations of P-stability. In particular, our work resolves an open problem posed by Greenhill (SODA, 2015).Secondly, in order to illustrate the power of our approach, we study the problem of uniformly sampling graphs for which, in addition to the degree sequence, a joint degree distribution is given. Although the problem was formalized over a decade ago, and despite its practical significance in generating synthetic network topologies, small progress has been made on the random sampling of such graphs. The case of a single degree class reduces to sampling of regular graphs, but beyond this almost nothing is known. We fully resolve the case of two degree classes, by showing that the switch Markov chain is always rapidly mixing. Again, we first analyze an auxiliary chain for strongly stable instances on an augmented state space and then use an embedding argument.</p

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

Sedimentology of a Mid-Late Ordovician carbonate mud-mound complex from the Katmandu nappe in central Nepal

peer reviewedaudience: researcher, professional, studentThis sedimentological study of the Godavari quarry is the first relating to the Palaeozoic Tethyan sedimentary rocks of the Katmandu nappe (Central Nepal). Sedimentological analyses led to the identification of six microfacies belonging to a large carbonate mud-mound complex, which can be divided into mound, flank and off-mound main depositional settings. Identification of two dasycladaceans (Dasyporella cf. silurica (Stolley, 1893) and Vermiporella sp.) in the mound facies gives a Mid-Late Ordovician age to this newly discovered Godavari carbonate mud-mound, which makes this mound one of the oldest ever described in the Asian continent. The mound microfacies are characterized by a high micritic content, the presence of stromatactis and the prevalence of red coloured sediments (the red pigmentation probably being related to organic precipitation of iron). The flank microfacies are characterized by a higher crinoid and argillaceous content and the presence of bio- and lithoclasts concentrated in argillaceous lenses. Finally, the off-mound microfacies show very few bioclasts and a high argillaceous content. Palaeoenvionmental interpretation of microfacies, in terms of bathymetry, leads us to infer that the Godavari mud-mound started to grow in a deep environment setting below the photic and wave action zones and that it evolved to occupy a location below the fair weather wave base. Cementation of cavities within the mound facies underlines a typical transition from a marine to a burial diagenetic environment characterized by: (1) a radiaxial non luminescent feroan calcite cement (marine) showing a bright orange luminescent band in its middle part; (2) a bright zoned orange fringe of automorphic feroan calcite (meteoric phreatic); (3) a dull orange xenomorphic feroan calcite cement in the centre of cavities (burial) and (4) a saddle dolomite within the centre of larger cavities. The faunal assemblage (diversity and relative proportion) of the Godavari mound facies is dominated by crinoids and ostracods, which makes this carbonate mud-mound comparable to the Meiklejohn Peak mounds (Nevada)