Doochul Kim

Classification of scale-free networks

While the emergence of a power-law degree distribution in complex networks is intriguing, the degree exponent is not universal. Here we show that the betweenness centrality displays a power-law distribution with an exponent η, which is robust, and use it to classify the scale-free networks. We have observed two universality classes with η ≈ 2.2(1) and 2.0, respectively. Real-world networks for the former are the protein-interaction networks, the metabolic networks for eukaryotes and bacteria, and the coauthorship network, and those for the latter one are the Internet, the World Wide Web, and the metabolic networks for Archaea. Distinct features of the mass-distance relation, generic topology of geodesics, and resilience under attack of the two classes are identified. Various model networks also belong to either of the two classes, while their degree exponents are tunable.

Mechanical restriction versus human overreaction triggering congested traffic states.

A new cellular automaton traffic model is presented. The focus is on mechanical restrictions of vehicles realized by limited acceleration and deceleration capabilities. These features are incorporated into the model in order to construct the condition of collision-free movement. The strict collision-free criterion imposed by the mechanical restrictions is softened in certain traffic situations, reflecting human overreaction. It is shown that the present model reliably reproduces most empirical findings including synchronized flow, the so-called pinch effect, and the time-headway distribution of free flow. The findings suggest that many free flow phenomena can be attributed to the platoon formation of vehicles (platoon effect).

Power laws, scale-free networks and genome biology

Power Laws in Biological Networks.- Graphical Analysis of Biocomplex Networks and Transport Phenomena.- Large-Scale Topological Properties of Molecular Networks.- The Connectivity of Large Genetic Networks.- The Drosophila Protein Interaction Network May Be neither Power-Law nor Scale-Free.- Birth and Death Models of Genome Evolution.- Scale-Free Evolution.- Gene Regulatory Networks.- Power Law Correlations in DNA Sequences.- Analytical Evolutionary Model for Protein Fold Occurrence in Genomes, Accounting for the Effects of Gene Duplication, Deletion, Acquisition and Selective Pressure.- The Protein Universes.- The Role of Computation in Complex Regulatory Networks.- Neutrality and Selection in the Evolution of Gene Families.- Scaling Laws in the Functional Content of Genomes.

Traffic states of a model highway with on-ramp

Several distinct traffic states are identified from real highway traffic data (Kerner and Rehborn, Phys. Rev. Lett. 79 (1997) 4030; Kerner and Rehborn, Phys. Rev. E 53 (1996) R4275; Kerner, Phys. Rev. Lett. 81 (1998) 3797; Kerner and Rehborn, Phys. Rev. E 53 (1996) R1297). Influence of on-ramp flux is important in generating the stop-and-go traffic flow near the ramps (Lee et al., Phys. Rev. Lett. 81 (1998) 1130; Lee et al., Phys. Rev. E 59 (1999) 5101). In this work, we study the phase diagram of the continuum traffic flow model equation in the presence of an on-ramp. Using an open boundary condition, traffic states and metastabilities are investigated for several representative values of the upstream boundary flux and for a range of the on-ramp flux. We find several traffic states such as the pinned localized cluster (PLC) state, the oscillating pinned localized cluster (OPLC) or the recurring hump (RH) state, the oscillating congested traffic (OCT) state and the homogeneous congested traffic (HCT) state. The latter two are traffic jam states. The free flow, the OPLC state and the jam can coexist in a certain metastable region where the free flow can undergo phase transitions to either of the two states under fluctuations. Some of these states are related to observed traffic states.

Phase diagram of a disordered boson Hubbard model in two dimensions.

We study the zero-temperature phase transition of a two-dimensional disordered boson Hubbard model. The phase diagram is constructed in terms of the disorder strength and the chemical potential. Via Monte Carlo simulations, we find a multicritical line separating the weak-disorder regime, where the Mott-insulator-to-superfluid transition occurs, from the strong-disorder regime, where the Bose-glass-to-superfluid transition occurs. On the multicritical line, the insulator-to-superfluid transition has the dynamical critical exponent z = 1.35+/-0.05 and the correlation length critical exponent nu = 0.67+/-0.03. We suggest that the proliferation of the particle-hole pairs screens out the weak-disorder effects.

Asymmetric XXZ chain at the antiferromagnetic transition: spectra and partition functions

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as and their amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. A similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as . The partition functions for this case are found to be that of a non-relativistic free fermion system. From the symmetry of the lattice model under rotation, several identities between the partition functions are found. The scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky - Talapov transition with the space and time coordinates interchanged.

Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω,ϒ)=(±1,∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω,ϒ)=(±1,∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.

Scaling dimensions and conformal anomaly in anisotropic lattice spin models

The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(ay/ax) sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

FINITE-SIZE SCALING AND THE TOROIDAL PARTITION FUNCTION OF THE CRITICAL ASYMMETRIC SIX-VERTEX MODEL

Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The non-universal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant