Huiseung Chae

Spatial evolutionary public goods game on complete graph and dense complex networks

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the “tragedy of the commons” and “an anomalous state without any active participants” occurs in real-life situations. When r is low (), the state with only loners is stable, and the state with only defectors is stable when r is high (). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how “tragedy of the commons” disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

Explosive percolation on the Bethe lattice.

Based on self-consistent equations of the order parameter P∞ and the mean cluster size S, we develop a self-consistent simulation method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying the self-consistent simulation to well-known percolation models, random bond percolation, and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing key functions obtained from self-consistent simulations for Achlioptas models with a product rule and a sum rule to the prototype functions, we show that the percolation transition of Achlioptas models on the Bethe lattice is continuous regardless of details of growth rules.

Discontinuous phase transition in a core contact process on complex networks

To understand the effect of generalized infection processes, we suggest and study the core contact process (CCP) on complex networks. In CCP an uninfected node is infected when at least k different infected neighbors of the node select the node for the infection. The healing process is the same as that of the normal CP. It is analytically and numerically shown that discontinuous transitions occur in CCP on random networks and scale-free networks depending on infection rate and initial density of infected nodes. The discontinuous transitions include hybrid transitions with β = 1/2 and β = 1. The asymptotic behavior of the phase boundary related to the initial density is found analytically and numerically. The mapping between CCP with k and static (k+1)-core percolation is supposed from the (k+1)-core structure in the active phase and the hybrid transition with β = 1/2. From these properties of CCP one can see that CCP is one of the dynamical processes for the k-core structure on real networks.

Phase transitions in paradigm shift models.

Two general models for paradigm shifts, deterministic propagation model (DM) and stochastic propagation model (SM), are proposed to describe paradigm shifts and the adoption of new technological levels. By defining the order parameter based on the diversity of ideas, , it is studied when and how the phase transition or the disappearance of a dominant paradigm occurs as a cost in DM or an innovation probability in SM increases. In addition, we also investigate how the propagation processes affect the transition nature. From analytical calculations and numerical simulations is shown to satisfy the scaling relation for DM with the number of agents . In contrast, in SM scales as .

Agglomerative percolation on the Bethe lattice and the triangular cactus

Agglomerative percolation (AP) on the Bethe lattice and the triangular cactus is studied to establish the exact mean-field theory for AP. Using the self-consistent simulation method based on the exact self-consistent equations, the order parameter P? and the average cluster size S are measured. From the measured P? and S, the critical exponents ?k and ?k for k = 2 and 3 are evaluated. Here, ?k and ?k are the critical exponents for P? and S when the growth of clusters spontaneously breaks the Zk symmetry of the k-partite graph. The obtained values are ?2 = 1.79(3), ?2 = 0.88(1), ?3 = 1.35(5) and ?3 = 0.94(2). By comparing these exponents with those for ordinary percolation (?? = 1 and ?? = 1), we also find ?? ?3 > ?2. These results quantitatively verify the conjecture that the AP model belongs to a new universality class if the Zk symmetry is broken spontaneously, and the new universality class depends on k.