Stability in CAN-free graphs

AbstractA class F of graphs characterized by three forbidden subgraphs C, A, N is considered; C is the claw (the unique graph with degree sequence (1, 1, 1, 3)), A is the antenna (a graph with degree sequence (1, 2, 2, 3, 3, 3) which does not contain C), and N is the net (the unique graph with degree sequence (1, 1, 1, 3, 3, 3)). These graphs are called CAN-free. A construction is described which associates with every CAN-free graph G another CAN-free graph G′ with strictly fewer nodes than G and with stbility number α(G′) = α(G) − 1. This gives a good algorithm for determining the stability number of CAN-free graphs

Minimum degree stability of H-free graphs

Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 − 1/r + o(1))( n 2 ) edges, and any H-free graph with that many edges can be made r-partite by deleting o(n 2 ) edges. Here we consider a natural variant of this – the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n 2 ) edges? We determine this least value for all 3- chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the Andrásfai-Erdős-Sós theorem and work of Alon and Sudako

The hidden matching-structure of the composition of strips: a polyhedral perspective

Stable set problems subsume matching problems since a matching is a stable set in a so- called line graph but stable set problems are hard in general while matching can be solved efficiently [11]. However, there are some classes of graphs where the stable set problem can be solved efficiently. A famous class is that of claw-free graphs; in fact, in 1980 Minty [19, 20] gave the first polynomial time algorithm for finding a maximum weighted stable set (mwss) in a claw-free graph. One of the reasons why stable set in claw-free graphs can be solved efficiently is because the so called augmenting path theorem [4] for matching generalizes to claw-free graphs [5] (this is what Minty is using). We believe that another core reason is structural and that there is a intrinsic matching structure in claw-free graphs. Indeed, recently Chudnovsky and Seymour [8] shed some light on this by proposing a decomposition theorem for claw-free graphs where they describe how to compose all claw-free graphs from building blocks. Interestingly the composition operation they defined seems to have nice consequences for the stable set problem that go much beyond claw-free graphs. Actually in a recent paper [21] Oriolo, Pietropaoli and Stauffer have revealed how one can use the structure of this composition to solve the stable set problem for composed graphs in polynomial time by reduction to matching. In this paper we are now going to reveal the nice polyhedral counterpart of this composition procedure, i.e. how one can use the structure of this composition to describe the stable set polytope from the matching one and, more importantly, how one can use it to separate over the stable set polytope in polynomial time. We will then apply those general results back to where they originated from: stable set in claw-free graphs, to show that the stable set polytope can be reduced to understanding the polytope in very basic structures (for most of which it is already known). In particular for a general claw-free graph G, we show two integral extended formulation for STAB(G) and a procedure to separate in polynomial time over STAB(G); moreover, we provide a complete characterization of STAB(G) when G is any claw-free graph with stability number at least 4 having neither homogeneous pairs nor 1-joins. We believe that the missing bricks towards the characterization of the stable set polytope of claw-free graphs are more technical than fundamentals; in particular, we have a characterization for most of the building bricks of the Chudnovsky-Seymour decomposition result and we are therefore very confident it is only a question of time before we solve the remaining case

Numerical equilibrium analysis for structured consumer resource models

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs

A Hebbian approach to complex network generation

Through a redefinition of patterns in an Hopfield-like model, we introduce and develop an approach to model discrete systems made up of many, interacting components with inner degrees of freedom. Our approach clarifies the intrinsic connection between the kind of interactions among components and the emergent topology describing the system itself; also, it allows to effectively address the statistical mechanics on the resulting networks. Indeed, a wide class of analytically treatable, weighted random graphs with a tunable level of correlation can be recovered and controlled. We especially focus on the case of imitative couplings among components endowed with similar patterns (i.e. attributes), which, as we show, naturally and without any a-priori assumption, gives rise to small-world effects. We also solve the thermodynamics (at a replica symmetric level) by extending the double stochastic stability technique: free energy, self consistency relations and fluctuation analysis for a picture of criticality are obtained

Self-Assembly of Geometric Space from Random Graphs

We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice

Handelman 's hierarchy for the maximum stable set problem.

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman's hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies

NIR-spectroscopy for bioprocess monitoring & control

The Ingold port adaption of a free beam NIR spectrometer is tailored for optimal bioprocess monitoring and control. The device shows an excellent signal to noise ratio dedicated to a large free aperture and therefore a large sample volume. This can be seen particularly in the batch trajectories which show a high reproducibility. The robust and compact design withstands rough process environments as well as SIP/CIP cycles. Robust free beam NIR process analyzers are indispensable tools within the PAT/QbD framework for realtime process monitoring and control. They enable multiparametric, non-invasive measurements of analyte concentrations and process trajectories. Free beam NIR spectrometers are an ideal tool to define golden batches and process borders in the sense of QbD. Moreover, sophisticated data analysis both quantitative and MSPC yields directly to a far better process understanding. Information can be provided online in easy to interpret graphs which allow the operator to make fast and knowledge-based decisions. This finally leads to higher stability in process operation, better performance and less failed batches