Construction and Analysis of Random Networks with Explosive Percolation

The existence of explosive phase transitions in random (Erdös Rényi-type) networks has been recently documented by Achlioptas, D’Souza, and Spencer via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented

Hierarchical networks, power laws, and neuronal avalanches

We show that in networks with a hierarchical architecture, critical dynamical behaviors can emerge even when the underlying dynamical processes are not critical. This finding provides explicit insight into current studies of the brain\u27s neuronal network showing power-lawavalanches in neural recordings, and provides a theoretical justification of recent numerical findings. Our analysis shows how the hierarchical organization of a network can itself lead to power-law distributions of avalanche sizes and durations, scaling laws between anomalous exponents, and universal functions—even in the absence of self-organized criticality or critical points. This hierarchy-induced phenomenon is independent of, though can potentially operate in conjunction with, standard dynamical mechanisms for generating power laws

Nonlinear Dynamics in Combinatorial Games: Renormalizing Chomp

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of unsolved combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that grows (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to transform a combinatorial game like Chomp into a type of dynamical system. Not only does this provide powerful insights into the game of Chomp (yielding a complete probabilistic description of optimal play in Chomp and an answer to a longstanding question about the nature of the winning opening move), but more generally, it offers a mathematical framework for exploring this unexpected relationship between combinatorial games and modern dynamical systems theory

Dynamical Effects of Partial Orderings in Physical Systems

We demonstrate that many physical systems possess an often overlooked property known as a partial-ordering structure. The detection and analysis of this special geometric property can be crucial for understanding a system\u27s dynamical behavior. We review here the fundamental dynamical features common to all such systems, and describe how the partial ordering imposes interesting restrictions on their possible behavior. We show, for instance, that though such systems are capable of displaying highly complex and even chaotic behaviors, most of their experimentally observable behaviors will be simple. Partial orderings are illustrated with examples drawn from many branches of physics, including solid state physics, fluids, and chemical systems. We also describe the consequences of partial orderings on some simple nonlinear models, and prove, for example, that for general two-dimensional mappings with the partial-ordering property, period 3 implies chaos, in analogy with the well-known result of Li and York [Am. Math. Mon. 82, 985 (1975)] for (ordinary) one-dimensional mappings

Combinatorial Games with a Pass: A dynamical systems approach

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.Comment: 39 pages, 13 figures, published versio

The Behavior of Coupled Automata

We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of “synchronization” between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles

Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim

Given an impartial combinatorial game G, we create a class of related games (CISG) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such modifications amount to taking cofinite induced subgraphs (CIS) of the original game graph. Some recent numerical/heuristic work has suggested that the underlying structure and behavior of such “CIS-games” can shed new light on, and bears interesting relationships with, the original games from which they are derived. In this paper we present an analytical treatment of the cofinite induced subgraphs associated with the game of (three-heap) Nim. This constitutes one of the simplest nontrivial cases of a CIS game. Our main finding is that although the structure of the winning strategies in games of CIS-Nim can differ greatly from that of Nim, CIS-Nim games inherit a type of period-two scale invariance from the original game of Nim

33731 Use of digital resource centers for atopic dermatitis patients, caregivers, and health care professionals to improve shared decision-making and proactive management

Overview: To close gaps in atopic dermatitis (AD) care, we developed and analyzed aligned resource centers for patients/caregivers and health care professionals (HCPs). Methods: The patient resource center was designed to support patients/caregivers in being more proactive in their AD care. The HCP-targeted resource center aimed to increase their awareness of AD patient perspectives and improve communication. Surveys were completed by users of both resource centers. Results: Of the 1014 HCPs, 22% were physicians (30% specialists), 19% were nurse practitioners/physician assistants, 47% were nurses, and 12% were pharmacists. Approximately one-half of 801 patients (98% adults) reported that they only treat their AD when experiencing a flare, and only 30% were very satisfied with their care. Only 56% indicated that they make all decisions with their HCP, and only 22% of HCPs reported that they always involve their AD patients/caregivers in shared decision-making. Only 21% of patients always share preferences, goals, or concerns about AD with their HCP, and only 6% of HCPs rated their ability to ask about and understand the impact of AD on patients’ quality of life as “very good.” Actions that patients planned to take included proactive skin care, asking their HCP about additional treatment options, and telling their HCP about the impact of AD on their quality of life. Nearly 60% of 764 HCPs planned to educate AD patients/caregivers about treatment options and expectations. Conclusions: These results highlight communication gaps between AD patients/caregivers and HCPs. Insights from these data can be used to improve shared decision-making