Mallows permutations as stable matchings

We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

Transitions across agriculture-savannah edges: Why did the rodent cross the edge?

Human-induced landscape alteration by agriculture is ubiquitous and impacts ecological processes and ecosystem services. The link between altered and native vegetation (hereafter edges) can create anthropogenic barriers for wildlife movement. Movement response to edges is dependent on landscape and functional group characteristics that shape spatial behavior in individuals; and affect population dynamics, dispersal, diversity, gene flow, and nutrient distribution. Few studies exist on small-mammal functional group movement response to edges, despite its integral role in behavioral and ecological processes. I analyzed the effect of altered land use on movement response in African savannah generalist Mastomys natalensis and specialist Lemniscomys rosalia. Each species was live-trapped for 2 months at 8 sites, across 2 boundary classes—homestead agriculture vs. savannah and commercial agricultural vs. savannah. Twelve individuals of each species were tracked and translocated at each transect with alternating treatments. Movement characteristics by M. natalensis showed a preference for altered land-use areas, possibly due to resource utilization. Conversely, L. rosalia showed a definitive trend (80%) to avoid edge crossing, regardless of abutting vegetative community, possibly due to their inability to cross and utilize altered landscapes. These results support the a priori hypothesis that altered land-use areas greatly impact specialists moving across the edge but only minimally affect generalist movements. Altered landscapes can affect small mammal movements and have cascading effects that alter vegetative and faunal communities.

Improvement of layered directed acyclic graph layout in CourseViewer

CourseViewer is a software that uses interactive diagrams to assist students, teachers, and course coordinators in analyzing information related to academic transcripts and course curriculum, which are represented as layered directed acyclic graphs of subjects and prerequisites. Recent improvements in the layout of these graphs included edge crossing minimization and better horizontal positioning of nodes. This work continues this list of improvements by means of researching edge bundling techniques that group edges of layered directed acyclic graphs, in order to simplify graph understanding. We selected and implemented an edge bundling technique in CourseViewer. We also exemplify course curricula in which we applied this technique.

A Bottleneck Matching Problem with Edge-Crossing Constraints

Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given [Formula: see text] red points and [Formula: see text] blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless [Formula: see text]. We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points.

Overview with details for exploring geo-located graphs on maps

Geo-located graph drawings often suffer from map visualization problems, such as overplotting of nodes as well as edges and location of parts of the graph being outside of the screen. One cause of these problems is often an irregular distribution of nodes on the map. Zooming and panning do not solve the problems, as they either only show the overview of the whole graph or only the details of a part of the graph. We present an interactive graph drawing technique that overcomes these problems without affecting the overall geographical structure of the graph. First, we introduce a method that uses insets to visualize details of small or remote areas. Second, to prevent the subgraphs within insets from overplotting and edge crossing, we introduce a local area re-arrangement. Moreover, insets are automatically drawn/hidden and repositioned in accordance with the user’s navigation. We test our technique on real-world geo-located graph data and show the effectiveness of our approach for showing overview and details at the same time. Additionally, we report on expert feedback concerning our approach.