Percolation in an ultrametric space

We study percolation on the hierarchical lattice of order $N$ where the probability of connection between two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)},\; \delta >-1$. Since the distance is an ultrametric, there are significant differences with percolation on the Euclidean lattice. There are two non-critical regimes: $\delta <1$, where percolation occurs, and $\delta >1$, where it does not occur. In the critical case, $\delta =1$, we use an approach in the spirit of the renormalization group method of statistical physics and connectivity results of Erd\H{o}s-Renyi random graphs play a key role. We find sufficient conditions on $c_k$ such that percolation occurs, or that it does not occur. An intermediate situation called pre-percolation is also considered. In the cases of percolation we prove uniqueness of the constructed percolation clusters. In a previous paper \cite{DG1} we studied percolation in the $N\to\infty$ limit (mean field percolation) which provided a simplification that allowed finding a necessary and sufficient condition for percolation. For fixed $N$ there are open questions, in particular regarding the existence of a critical value of a parameter in the definition of $c_k$, and if it exists, what would be the behaviour at the critical point

Evolution of cooperation on dynamical graphs

There are two key characteristic of animal and human societies: (1) degree heterogeneity, meaning that not all individual have the same number of associates; and (2) the interaction topology is not static, i.e. either individuals interact with different set of individuals at different times of their life, or at least they have different associations than their parents. Earlier works have shown that population structure is one of the mechanisms promoting cooperation. However, most studies had assumed that the interaction network can be described by a regular graph (homogeneous degree distribution). Recently there are an increasing number of studies employing degree heterogeneous graphs to model interaction topology. But mostly the interaction topology was assumed to be static. Here we investigate the fixation probability of the cooperator strategy in the prisoner’s dilemma, when interaction network is a random regular graph, a random graph or a scale-free graph and the interaction network is allowed to change. We show that the fixation probability of the cooperator strategy is lower when the interaction topology is described by a dynamical graph compared to a static graph. Even a limited network dynamics significantly decreases the fixation probability of cooperation, an effect that is mitigated stronger by degree heterogeneous networks topology than by a degree homogeneous one. We have also found that from the considered graph topologies the decrease of fixation probabilities due to graph dynamics is the lowest on scale-free graphs

Dynamic Graphs on the GPU

We present a fast dynamic graph data structure for the GPU. Our dynamic graph structure uses one hash table per vertex to store adjacency lists and achieves 3.4–14.8x faster insertion rates over the state of the art across a diverse set of large datasets, as well as deletion speedups up to 7.8x. The data structure supports queries and dynamic updates through both edge and vertex insertion and deletion. In addition, we define a comprehensive evaluation strategy based on operations, workloads, and applications that we believe better characterize and evaluate dynamic graph data structures