Cator and Van Mieghem [Phys. Rev. E 89, 052802 (2014)PLEEE81539-375510.1103/PhysRevE.89.052802] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian susceptible-infected-susceptible (SIS) and susceptible-infected-removed (SIR) epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the Fortuin, Kasteleyn, and Ginibre (FKG) inequality. In this Comment, we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is crucial for invoking the FKG inequality. Moreover, we provide an example of a simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.

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In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any k consecutive vertices in the path contain at least one vertex of the subsequence. An n-ary tree, with height h, is a tree whose vertices at distance at most h − 1 to the root have n children. For the case of n-ary trees, we prove that, as h tends to infinity the probability of having such subsequence: tends to 1, if n grows significantly faster than k h/(ek) ; and tends to 0, if n grows significantly slower than k h/(ek) . Date: June 28, 2018. 2010 Mathematics Subject Classification. 60K35, 60C05, 92D15.

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly increasing through it. In the rough Mount Fuji (RMF) model, the fitness function is defined on the graph as ω ( v ) = η ( v ) + θ ⋅ d ( v ) , where θ is a positive number called the drift, d is the distance to the source of the graph and η ( v ) are i.i.d. random variables. In this paper, we determine values of θ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices L 2 and L a l t 2 .

We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, and ending at the origin.