Local cluster aggregation models of explosive percolation

We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step of the process one edge is selected from a small set of potential edges sharing common vertices and added to the graph. We show that the evolution can be accurately described by a system of differential equations and that such models exhibit the discontinuous emergence of the giant component. Yet, they also obey scaling behaviors characteristic of continuous transitions, with scaling exponents that differ from the classic Erdos-Renyi model.Comment: Final version as appearing in PR

Strongly discontinuous explosive percolation with multiple giant components

We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are considered one at a time and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function asymptotically approaching the value alpha = 1/2. We show that multiple giant components appear simultaneously in a strongly discontinuous percolation transition and remain distinct. Furthermore, tuning the value of alpha determines the number of such components with smaller alpha leading to an increasingly delayed and more explosive transition. The location of the critical point and strongly discontinuous nature are not affected if only edges which span components are sampled.Comment: Final version appearing in PR

Network Growth with Feedback

Existing models of network growth typically have one or two parameters or strategies which are fixed for all times. We introduce a general framework where feedback on the current state of a network is used to dynamically alter the values of such parameters. A specific model is analyzed where limited resources are shared amongst arriving nodes, all vying to connect close to the root. We show that tunable feedback leads to growth of larger, more efficient networks. Exact results show that linear scaling of resources with system size yields crossover to a trivial condensed state, which can be considerably delayed with sublinear scaling.Comment: 4 pages, 4 figure

Resilience and rewiring of the passenger airline networks in the United States

The air transportation network, a fundamental component of critical infrastructure, is formed from a collection of individual air carriers, each one with a methodically designed and engineered network structure. We analyze the individual structures of the seven largest passenger carriers in the USA and find that networks with dense interconnectivity, as quantified by large k-cores for high values of k, are extremely resilient to both targeted removal of airports (nodes) and random removal of flight paths paths (edges). Such networks stay connected and incur minimal increase in an heuristic travel time despite removal of a majority of nodes or edges. Similar results are obtained for targeted removal based on either node degree or centrality. We introduce network rewiring schemes that boost resilience to different levels of perturbation while preserving total number of flight and gate requirements. Recent studies have focused on the asymptotic optimality of hub-and-spoke spatial networks under normal operating conditions, yet our results indicate that point-to-point architectures can be much more resilient to perturbations.Comment: 11 pages, 8 figures, replaced by the version to appear in Physical Review

BML revisited: Statistical physics, computer simulation and probability

Statistical physics, computer simulation and discrete mathematics are intimately related through the study of shared lattice models. These models lie at the foundation of all three fields, are studied extensively, and can be highly influential. Yet new computational and mathematical tools may challenge even well established beliefs. Consider the BML model, which is a paradigm for modeling self-organized patterns of traffic flow and first-order jamming transitions. Recent findings, on the existence of intermediate states, bring into question the standard understanding of the jamming transition. We review the results and show that the onset of full-jamming can be considerably delayed based on the geometry of the system. We also introduce an asynchronous version of BML, which lacks the self-organizing properties of BML, has none of the puzzling intermediate states, but has a sharp, discontinuous, transition to full jamming. We believe this asynchronous version will be more amenable to rigorous mathematical analysis than standard BML. We discuss additional models, such as bootstrap percolation, the honey-comb dimer model and the rotor-router, all of which exemplify the interplay between the three fields, while also providing cautionary tales. Finally, we synthesize implications for how results from one field may relate to the other, and also implications specific to computer implementations.