Hyun Keun Lee

Epidemic threshold of the susceptible-infected-susceptible model on complex networks.

We demonstrate that the susceptible-infected-susceptible (SIS) model on complex networks can have an inactive Griffiths phase characterized by a slow relaxation dynamics. It contrasts with the mean-field theoretical prediction that the SIS model on complex networks is active at any nonzero infection rate. The dynamic fluctuation of infected nodes, ignored in the mean field approach, is responsible for the inactive phase. It is proposed that the question whether the epidemic threshold of the SIS model on complex networks is zero or not can be resolved by the percolation threshold in a model where nodes are occupied in degree-descending order. Our arguments are supported by the numerical studies on scale-free network models.

Fluctuation theorems and entropy production with odd-parity variables.

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess contribution that can appear only transiently and is called nonadiabatic. Another one is attributed solely to the breakage of detailed balance. The last part that does not satisfy the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables that is activated in a nontransient manner. The latter two parts combine together as the housekeeping (adiabatic) contribution, whose positivity is not guaranteed except when the excess contribution completely vanishes. Our finding reveals that the equilibrium requires the steady-state distribution symmetry for odd-parity variables independently, in addition to the usual detailed balance.

Nonequilibrium steady states in Langevin thermal systems.

Equilibrium is characterized by its fundamental properties, such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relation mutually exclude each other, and no equivalence relation is possible between any two of the three properties. This implies that a steady state of Langevin systems with velocity-dependent forces may maintain some equilibrium properties but not all of them. Our results are illustrated with a few example systems.

Scaling of cluster heterogeneity in percolation transitions.

We investigate a critical scaling law for the cluster heterogeneity H in site and bond percolations in d-dimensional lattices with d = 2,...,6. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability p increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that H diverges algebraically, approaching the percolation critical point p(c) as H |p-p(c)|(-1/σ) with the critical exponent σ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent ν H = 1+d (f)/(d)ν, where d(f) is the fractal dimension of the critical percolating cluster and ν is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.

Degree-ordered percolation on a hierarchical scale-free network.

We investigate the critical phenomena of the degree-ordered percolation (DOP) model on the hierarchical (u,v) flower network with u ≤ v. Highest degree nodes are linked directly without intermediate nodes for u=1, while this is not the case for u ≠ 1. Using the renormalization-group-like procedure, we derive the recursion relations for the percolating probability and the percolation order parameter, from which the percolation threshold and the critical exponents are obtained. When u ≠ 1, the DOP critical behavior turns out to be identical to that of the bond percolation with a shifted nonzero percolation threshold. When u=1, the DOP and the bond percolation have the same vanishing percolation threshold but the critical behaviors are different. Implication to an epidemic spreading phenomenon is discussed.

Collective Helping and Bystander Effects in Coevolving Helping Networks

We study collective helping behavior and bystander effects in a coevolving helping network model. A node and a link of the network represents an agent who renders or receives help and a friendly relation between agents, respectively. A helping trial of an agent depends on relations with other involved agents and its result (success or failure) updates the relation between the helper and the recipient. We study the network link dynamics and its steady states analytically and numerically. The full phase diagram is presented with various kinds of active and inactive phases and the nature of phase transitions are explored. We find various interesting bystander effects, consistent with the field study results, of which the underlying mechanism is proposed.

Percolation transitions with nonlocal constraint.

We investigate percolation transitions in a nonlocal network model numerically. In this model, each node has an exclusive partner and a link is forbidden between two nodes whose r-neighbors share any exclusive pair. The r-neighbor of a node x is defined as a set of at most N(r) neighbors of x, where N is the total number of nodes. The parameter r controls the strength of a nonlocal effect. The system is found to undergo a percolation transition belonging to the mean-field universality class for r<1/2. On the other hand, for r>1/2, the system undergoes a peculiar phase transition from a nonpercolating phase to a quasicritical phase where the largest cluster size G scales as G~N(α) with α=0.74(1). In the marginal case with r=1/2, the model displays a percolation transition that does not belong to the mean-field universality class.

A Baker–Campbell–Hausdorff solution by differential equation

We propose a procedure to figure out the Baker–Campbell–Hausdorff (BCH) solution, ln eX eY, when the exponent is a linear combination of the spin operator along a direction and its ladder operators. The procedure converts the manipulation of the BCH formula into that of a differential equation. It is shown that the fixed point of the differential equation leads to the solution we are looking for. We also remark that the validity of the present method is restricted to the case when the solution branch can be determined in the complex plane.