Relating the microscopic rules in coalescence-fragmentation models to the cluster-size distribution

Abstract

Coalescence-fragmentation problems are now of great interest across the physical, biological, and social sciences. They are typically studied from the perspective of rate equations, at the heart of which are the rules used for coalescence and fragmentation. Here we discuss how changes in these microscopic rules affect the macroscopic cluster-size distribution which emerges from the solution to the rate equation. Our analysis elucidates the crucial role that the fragmentation rule can play in such dynamical grouping models. We focus our discussion on two well-known models whose fragmentation rules lie at opposite

Modeling Human Conflict and Terrorism Across Geographic Scales

“…The resulting model yields an exponentially cutoff 2.5-exponent power-law for the distribution of cell sizes. We note that generalizations of this model have appeared in the literature-in particular, [35] contains a number of relevant generalizations, including a variable number of agents in time N.t/. A later paper [61] reached similar conclusions to our earlier publication [35] concerning the remarkable robustness of the 2.5 exponent to variations in the model mechanisms.…”

“…We note that generalizations of this model have appeared in the literature-in particular, [35] contains a number of relevant generalizations, including a variable number of agents in time N.t/. A later paper [61] reached similar conclusions to our earlier publication [35] concerning the remarkable robustness of the 2.5 exponent to variations in the model mechanisms. Analysis of a simple version of this model was completed earlier by d'Hulst and Rodgers [8], and real-world applications have focused on financial markets-however the derivation below features general values frag and coal .…”

“…Fourth, the distinction between an insurgent or terrorist (i.e., Red) and the background civilian population (i.e., Green) can be blurred and itself highly fluid. It is no longer the case that a civilian population can be considered some inert background which simply soaks up the violent events as they play out.Our approach to coping with this complexity considers the underlying ecology as interacting populations of heterogenous agents who operate with covert but dynamically evolving communication networks, and who adapt their strategies in response to external events and news, as well as counteradaptation by the relevant state authorities [30,31,[34][35][36]38]. In so doing, we incorporate the combined effects of intra-population grouping dynamics and inter-population attrition dynamics [7,24] thereby generating an intriguing non-equilibrium many body problem.…”

“…This model describes the severity of fatal events. The simplicity of our approach allows a range of analytic mathematical analysis to be performed for both the severity [35] and timing of fatal events [29]. Finally we comment on the comparison to cyber-gangs and street gangs.…”

Equivalent dynamical complexity in a many-body quantum and collective human system

“…coalescence of multiple groups, fragmentation into groups larger than one, a slowly time-varying particle number N, and it holds for a wide range of ν frag and ν coal values. [21][22][23][24] It does not depend on the size of the individual fragmented parts, as long as they are all small, since the size of the largest pieces just dictates the value of s above which the 5/2 power-law result kicks in. Figure 1(b) illustrates the model's predicted analytic form, for the simple demonstrative case of ν frag = ν = 1 − ν coal .…”