Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape

Strategy Logic with Imperfect Information

We introduce an extension of Strategy Logic for the imperfect-information setting, called SLii, and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, the problem turns out to be undecidable. We introduce a syntactical class of "hierarchical instances" for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model. We prove that model-checking SLii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises previous ones, such as decidability of multi-player games with imperfect information and hierarchical observations, and decidability of distributed synthesis for hierarchical systems. To establish the decidability result, we introduce and study QCTL*ii, an extension of QCTL* (itself an extension of CTL* with second-order quantification over atomic propositions) by parameterising its quantifiers with observations. The simple syntax of QCTL* ii allows us to provide a conceptually neat reduction of SLii to QCTL*ii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTL*ii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable. The decidability result for SLii follows since the reduction maps hierarchical instances of SLii to hierarchical formulas of QCTL*ii

Reparameterizing the Birkhoff Polytope for Variational Permutation Inference

Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation, but fully Bayesian inference poses a severe challenge in this high-dimensional, discrete space. To surmount this challenge, we start with the usual step of relaxing a discrete set (here, of permutation matrices) to its convex hull, which here is the Birkhoff polytope: the set of all doubly-stochastic matrices. We then introduce two novel transformations: first, an invertible and differentiable stick-breaking procedure that maps unconstrained space to the Birkhoff polytope; second, a map that rounds points toward the vertices of the polytope. Both transformations include a temperature parameter that, in the limit, concentrates the densities on permutation matrices. We then exploit these transformations and reparameterization gradients to introduce variational inference over permutation matrices, and we demonstrate its utility in a series of experiments

Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points---a classic topic in computer science---in the analysis of probabilistic programs. This leads us to two new martingale-based techniques, too. We give rigorous proofs for their soundness and completeness. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales

Price Competition under Limited Comparability

This paper studies market competition when firms can influence consumers' ability to compare market alternatives, through their choice of price "formats". We introduce random graphs as a tool for modelling limited comparability of formats. Our main results concern the interaction between firms' equilibrium price and format decisions and its implications for industry profits and consumer switching rates. We show that narrow regulatory interventions that aim to facilitate comparisons may have adverse consequences for consumer welfare. Finally, we argue that our limited-comparability approach provides a new perspective into the phenomenon of product differentiation.price competition, industrial organization, limited comparability, bounded rationality, framing, consumer protection, product differentiation, complexity

Mathematical modelling of gas flow networks in pellet induratlon systems

The objective of this research is to develop a simulation software tool, GASFLO, which should evaluate pressure, flow and temperature distributions of process gas in pellet induration system networks. Pellet induration systems are complex industrial systems composed of heterogenous components. The magnitude of gas through leaks i.e. the air entering or leaving the system from the points other than the known exits, is substantial and it adversely effects the performance of induration process. These leaks are very difficult to measure because of the hostile environment in the plant. The modelling of such industrial systems requires a notable amount of experimentation so the tool has been designed to enable the user modeller to change the component models and solution algorithms easily. The conventional methods for flow network simulation are based on process centred approach, mostly composed of homogeneous components. For ease of computation, the non-pipe elements are modelled with an approximate linear or non-linear generic equation, whose coefficients can simulate different states of the element. The resulting set of non-linear equations is linearised and solved simultaneously using some iterative method. By contrast, GASFLO is based on device centred or unit based approach, and uses a two level hierarchical solution algorithm. The pellet induration system network is first idealised into a connected graph of streams (sets of serially connected components) and nodes. At the top or coordination level the flow and pressure distributions satisfying the Kirchhoff's laws are evaluated for the connected graph. At the lower or component level the exact mathematical models of components ale computed, in order of their occurrence in respective streams, using coordination variables as parameters. The converged flows are used for the temperature computation. The solution algorithm requires partitioning of the connected graph into forest and coforest structures, for which secondary algorithms have been developed using specific heuristics relevant to the pellet induration systems. The rigorous application of software engineering techniques for the design and implementation of software, enabled the resolution of the complexity of the modelled system, embedded the characteristics of 'quality software' into the resulting code and benefits from object orientation, even though it is implemented in standard FORTRAN 77. GASFLO predicted results are in a good agreement with the measured results, it has been validated for a real life pellet induration system. It has been applied to simulate several practical scenarios, like addition of extra wind boxes to the zones and to determine how the plant production can be increased by certain ratio, such simulations were not feasible otherwise. GASFLO takes less than a minute to simulate a real-life pellet induration system on a 486 PC. The combined simulation with an other software tool, INDSYS, which evaluates the heat distribution in the solids, is also feasible