How random is dice tossing? Nagler, Jan; Richter, Peter
Physical review. E, Statistical, nonlinear, and soft matter physics,
09/2008, Volume:
78, Issue:
3 Pt 2
Journal Article
Tossing the dice is commonly considered a paradigm for chance. But where in the process of throwing a cube does the randomness reside? After all, for all practical purposes the motion is described by ...the laws of deterministic classical mechanics. Therefore the undisputed status of dice as random number generators calls for a careful analysis. This paper is an attempt in that direction. As a simplified model of a dice a barbell with two marked masses at its tips and only two final positions is considered. It is shown how, depending on initial conditions and the degree of dissipation during bounces, the outcome is only more or less unpredictable: the system is not truly random but pseudorandom--even under conditions where it appears to be random.
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Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a ...random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of α, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime αε0.6,0.95 where it is known that only one giant component, denoted C(1) , initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime C(1) stops growing and eventually a second giant component, denoted C(2), emerges in a continuous percolation transition. The delay between the emergence of C(1) and C(2) and their asymptotic sizes both depend on the value of α and we establish by several techniques that there exists a bifurcation point α(c)=0.763±0.002. For αε0.6,α(c)), C(1) stops growing the instant it emerges and the delay between the emergence of C(1) and C(2) decreases with increasing α. For αε(α(c),0.95, in contrast, C(1) continues growing into the supercritical regime and the delay between the emergence of C(1) and C(2) increases with increasing α. As we show, α(c) marks the minimal delay possible between the emergence of C(1) and C(2) (i.e., the smallest edge density for which C(2) can exist). We also establish many features of the continuous percolation of C(2) including scaling exponents and relations.
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Multilevel selection is an important organizing principle that crucially underlies evolutionary processes from the emergence of cells to eusociality and the economics of nations. Previous studies on ...multilevel selection assumed that the effective higher-level selection emerges from lower-level reproduction. This leads to selection among groups, although only individuals reproduce. We introduce selective group extinction, where groups die with a probability inversely proportional to their group fitness. When accounting for this the critical benefit-to-cost ratio is substantially lowered. Because in game theory and evolutionary dynamics the degree of cooperation crucially depends on this ratio above which cooperation emerges previous studies may have substantially underestimated the establishment and maintenance of cooperation.
The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing ...phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of "explosive percolation" was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.
The flea model by Ehrenfest describes the jumps of a fixed number of fleas between two dogs. In each time step a randomly selected flea jumps on the other dog. We study directed and undirected ...multiurn models in a one-dimensional ring. The introduced models represent generalizations of three recently proposed multiurn models which themselves are generalizations of Ehrenfest's model. The models are solved analytically. For the directed case we find oscillations of the average number of balls or fleas in a certain urn before the system reaches its equilibrium state. The discussed models may serve as basic models of dynamics of granular media in connected periodic compartment systems.
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Explosive Percolation describes the abrupt onset of large-scale connectivity that results from a simple random process designed to delay the onset of the transition on an underlying random network or ...lattice. Explosive percolation transitions exhibit an array of novel universality classes and supercritical behaviors including a stochastic sequence of discontinuous transitions, multiple giant components, and lack of self-averaging. Many mechanisms that give rise to explosive percolation have been discovered, including overtaking, correlated percolation, and evolution on hierarchical lattices. Many connections to real-world systems, ranging from social networks to nanotubes, have been identified and explosive percolation is an emerging paradigm for modeling these systems as well as the consequences of small interventions intended to delay phase transitions. This review aims to synthesize existing results on explosive percolation and to identify fruitful directions for future research.
Who votes now? Leighley, Jan E; Leighley, Jan E; Nagler, Jonathan
2013., 20131124, 2013, 2014-01-01
eBook
Who Votes Now?compares the demographic characteristics and political views of voters and nonvoters in American presidential elections since 1972 and examines how electoral reforms and the choices ...offered by candidates influence voter turnout. Drawing on a wealth of data from the U.S. Census Bureau's Current Population Survey and the American National Election Studies, Jan Leighley and Jonathan Nagler demonstrate that the rich have consistently voted more than the poor for the past four decades, and that voters are substantially more conservative in their economic views than nonvoters. They find that women are now more likely to vote than men, that the gap in voting rates between blacks and whites has largely disappeared, and that older Americans continue to vote more than younger Americans. Leighley and Nagler also show how electoral reforms such as Election Day voter registration and absentee voting have boosted voter turnout, and how turnout would also rise if parties offered more distinct choices.
Providing the most systematic analysis available of modern voter turnout,Who Votes Now?reveals that persistent class bias in turnout has enduring political consequences, and that it really does matter who votes and who doesn't.
We study the critical behavior of a general contagion model where nodes are either active (e.g. with opinion A, or functioning) or inactive (e.g. with opinion B, or damaged). The transitions between ...these two states are determined by (i) spontaneous transitions independent of the neighborhood, (ii) transitions induced by neighboring nodes and (iii) spontaneous reverse transitions. The resulting dynamics is extremely rich including limit cycles and random phase switching. We derive a unifying mean-field theory. Specifically, we analytically show that the critical behavior of systems whose dynamics is governed by processes (i-iii) can only exhibit three distinct regimes: (a) uncorrelated spontaneous transition dynamics (b) contact process dynamics and (c) cusp catastrophes. This ends a long-standing debate on the universality classes of complex contagion dynamics in mean-field and substantially deepens its mathematical understanding.
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