Anders Martinsson

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the n-dimensional binary hypercube, for some n, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concernis with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node v^0 to the all-ones node v^1 tends respectively to 0, 1 and 1, as n tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of v^0 is set to some α = α_n ∈ [0, 1]. We prove that there is a very sharp threshold at α_n = (ln n)/n for the existence of accessible paths from v^0 to v^1 . As a corollary we prove significant concentration, for α below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from v^0 to v^1 existing tends to 1 provided the drift parameter θ = θ_n satisfies n(θ_n) → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.

An improved energy argument for the Hegselmann–Krause model

We show that the freezing time of the d-dimensional Hegselmann–Krause model is where n is the number of agents. This improves the best known upper bound whenever .

The Hegselmann-Krause dynamics on the circle converge

We consider the Hegselmann-Krause dynamics on a one-dimensional torus and provide the first proof of convergence of this system. The proof requires only fairly minor modifications of existing methods for proving convergence in Euclidean space.

Even Flying Cops Should Think Ahead

We study the entanglement game, which is a version of cops and robbers, on sparse graphs. While the minimum degree of a graph G is a lower bound for the number of cops needed to catch a robber in G, we show that the required number of cops can be much larger, even for graphs with small maximum degree. In particular, we show that there are 3-regular graphs where a linear number of cops are needed.

When can multi-agent rendezvous be executed in time linear in the diameter of a plane configuration?

In multi-agent rendezvous it is naturally assumed that agents have a maximum speed of movement. In the absence of any distributed control issues, this imposes a lower bound on the time to rendezvous, for idealised point agents, proportional to the diameter of a configuration. Assuming bounded visibility, we consider Ω(n2 log n) points distributed independently and uniformly at random in a disc of radius n, so that the visibility graph is asymptotically almost surely (a.a.s.) connected. We allow three types of possible interaction between neighbors, which we term signalling, sweeping and tracking. Assuming any such interaction can be executed without significant delay, and assuming each point can generate random bits and has unlimited memory, we describe a randomized algorithm which a.a.s. runs in time O(n), hence in time proportional to the diameter, provided the number of points is o(n3). Several questions are posed for future work.