Rate of Convergence of Increasing Path-Vector Routing Protocols

A good measure of the rate of convergence of path-vector protocols is the number of synchronous iterations required for convergence in the worst case. From an algebraic perspective, the rate of convergence depends on the expressive power of the routing algebra associated with the protocol. For example in a network of $n$ nodes, shortest-path protocols are guaranteed to converge in $O(n)$ iterations. In contrast the algebra underlying the Border Gateway Protocol (BGP) is in some sense too expressive and the protocol is not guaranteed to converge. There is significant interest in finding well-behaved algebras that still have enough expressive power to satisfy network operators. Recent theoretical results have shown that by constraining routing algebras to those that are ``strictly increasing'' we can guarantee the convergence of path-vector protocols. Currently the best theoretical worst-case upper bound for the convergence of such algebras is $O(n!)$ iterations. However in practice it is difficult to find examples that do not converge in $n$ iterations. In this paper we close this gap. We first present a family of network configurations that converges in $\Theta(n^2)$ iterations, demonstrating that the worst case is $\Omega(n^2)$ iterations. We then prove that path-vector protocols with a strictly increasing algebra are guaranteed to converge in $O(n^2)$ iterations. Together these results establish a tight $\Theta(n^2)$ bound. This is another piece of the puzzle in showing that ``strictly increasing" is, at least on a technical level, a reasonable constraint for practical policy-rich protocols. {In memory of Abha Ahuja

An objective method for the production of isopach maps and implications for the estimation of tephra deposit volumes and their uncertainties

Characterization of explosive volcanic eruptive processes from interpretation of deposits is a key for assessing volcanic hazard and risk, particularly for infrequent large explosive eruptions and those whose deposits are transient in the geological record. While eruption size—determined by measurement and interpretation of tephra fall deposits—is of particular importance, uncertainties for such measurements and volume estimates are rarely presented. Here, tephra volume estimates are derived from isopach maps produced by modeling raw thickness data as cubic B-spline curves under tension. Isopachs are objectively determined in relation to original data and enable limitations in volume estimates from published maps to be investigated. The eruption volumes derived using spline isopachs differ from selected published estimates by 15–40 %, reflecting uncertainties in the volume estimation process. The formalized analysis enables identification of sources of uncertainty; eruptive volume uncertainties (>30 %) are much greater than thickness measurement uncertainties (~10 %). The number of measurements is a key factor in volume estimate uncertainty, regardless of method utilized for isopach production. Deposits processed using the cubic B-spline method are well described by 60 measurements distributed across each deposit; however, this figure is deposit and distribution dependent, increasing for geometrically complex deposits, such as those exhibiting bilobate dispersion. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s00445-015-0942-y) contains supplementary material, which is available to authorized users

An Agda Formalization of Üresin and Dubois’ Asynchronous Fixed-Point Theory

In this paper we describe an Agda-based formalization of results from \"{U}resin \& Dubois' ``Parallel Asynchronous Algorithms for Discrete Data.'' That paper investigates a large class of iterative algorithms that can be transformed into asynchronous processes. In their model each node asynchronously performs partial computations and communicates results to other nodes using unreliable channels. \"{U}resin \& Dubois provide sufficient conditions on iterative algorithms that guarantee convergence to unique fixed points for the associated asynchronous iterations. Proving such sufficient conditions for an iterative algorithm is often dramatically simpler than reasoning directly about an asynchronous implementation. These results are used extensively in the literature of distributed computation, making formal verification worthwhile. Our Agda library provides users with a collection of sufficient conditions, some of which mildly relax assumptions made in the original paper. Our primary application has been in reasoning about the correctness of network routing protocols. To do so we have derived a new sufficient condition based on the ultrametric theory of Alexander Gurney. This was needed to model the complex policy-rich routing protocol that maintains global connectivity in the internet

Tracking urban activity growth globally with big location data.

In recent decades, the world has experienced rates of urban growth unparalleled in any other period of history and this growth is shaping the environment in which an increasing proportion of us live. In this paper, we use a longitudinal dataset from Foursquare, a location-based social network, to analyse urban growth across 100 major cities worldwide. Initially, we explore how urban growth differs in cities across the world. We show that there exists a strong spatial correlation, with nearby pairs of cities more likely to share similar growth profiles than remote pairs of cities. Subsequently, we investigate how growth varies inside cities and demonstrate that, given the existing local density of places, higher-than-expected growth is highly localized while lower-than-expected growth is more diffuse. Finally, we attempt to use the dataset to characterize competition between new and existing venues. By defining a measure based on the change in throughput of a venue before and after the opening of a new nearby venue, we demonstrate which venue types have a positive effect on venues of the same type and which have a negative effect. For example, our analysis confirms the hypothesis that there is large degree of competition between bookstores, in the sense that existing bookstores normally experience a notable drop in footfall after a new bookstore opens nearby. Other place types, such as museums, are shown to have a cooperative effect and their presence fosters higher traffic volumes to nearby places of the same type