Topological Aspects of Critical Systems and Networks

PREFACE.

CONTENTS.

International Symposium on Topological Aspects of Critical Systems and Networks.

Group Photo.

In a scale-free network, there is a small subset of nodes with linkages far heavier than those of others in the network. Intentional attack on one or a few nodes in this group can trigger a cascade of node failures, leading to potential breakdown of the network on a large scale. This typically occurs for situations where the capacities of nodes in the network are small and overload results in node failure. Breakdown of the network can be understood as a phase transition that occurs when the node capacity parameter is decreased through a critical point. However, when the node capacities are sufficiently large, the network can be robust against cascading breakdown. A physical theory to account for these phenomena is reviewed. In situations where overload does not cause node failure but generate traffic congestion, attack can induce persistent oscillations of the network in the sense that its characterizing quantities, such as the diameter, oscillate in time and are never able to restore to their original values. This remarkable phenomenon of network oscillation is also discussed.

In this paper, after reviewing some known results, we extend the analysis of epidemic thresholds for complex networks to more general epidemic dynamics with more than two states. The fact that the epidemic threshold is extinguished for highly heterogeneous networks is transferred to most of these generalized models. However, in some types of models, additional constraints associated with pathogen mutation are needed for the epidemic threshold to disappear.

In this paper, the notion of homotopy reductions is presented to derive a skeleton structure focusing on loops in complex networks. The main result is to show scale-free networks as skeleton structures of random networks under some rules. In addition, homotopy reductions are applied to scale-free real networks(WWW and protein interaction networks¹). The potential of homotopy reduction techniques is also discussed in detail.

We investigate the susceptible-infected-susceptible(SIS) model on complex network. By counting the number of paths from one node to another, we show that the epidemic threshold K_c must satisfy the relation K_c≥ ln(1 + 1/λ₀), where λ₀ represents the largest eigenvalue of the adjacent matrix of network. We also propose the new strategy for the immunization on complex networks.

The intellectual property rights and their effect to the field where they are applied is an important issue especially in the software industry. In this paper we inspect a simple dynamic model of innovation in a lattice of dependencies of a pair of agents that can either collaborate or do not. We use a graph where we add some short– and long–range connections. We inspect a quantity we call mixing, which refers to the proportion of the histories of sites first visited by agents overlap. We show that the mixing of the progression of non–cooperative agents makes a rapid transition from zero when we add connections. Also when the number of long–range connections increases the non–cooperative agents initially mix faster than non–cooperative agents.

Long-tailed distributions in biological systems have been studied. First, we found that lognormal distributions show excellent fit with various data for the duration distribution of disability in aged people, irrespective of their severity and gender. The robust lognormal distribution of disability implies that the incidence of diseases can be completed by many independent subprocesses in succession. Next, we studied food fragmentation by human mastication. A lognormal distribution well fits to the entire region for masticated food fragments for a small number of chewing strokes. Furthermore, the tail of the fragment-size distribution changes from the lognormal distribution to a power-law one as the chewing stroke number increases. The good data fitting by the lognormal and power-law distribution implies that two functions of mastication, a sequential fragmentation with cascade and randomness and a lower threshold for fragment size, may affect the size distribution of masticated food fragments.

Personal income distribution in the USA has a well-defined two-class structure. The majority of population (97–99 %) belongs to the lower class characterized by the exponential Boltzmann-Gibbs ("thermal") distribution, whereas the upper class (1–3 % of population) has a Pareto power-law ("superthermal") distribution. By analyzing income data for 1983–2001, we show that the "thermal" part is stationary in time, save for a gradual increase of the effective temperature, whereas the "superthermal" tail swells and shrinks following the stock market. We discuss the concept of equilibrium inequality in a society, based on the principle of maximal entropy, and quantitatively show that it applies to the majority of population.

The purpose of this paper is to highlight certain newly discovered social phenomena that accord with Zipf's law, in addition to the famous natural and social phenomena including word frequencies, earthquake magnitude, city size, income¹ etc. that are already known to follow it. These phenomena have recently been discovered within the transaction amount (payments or receipts) distributions within two different Community Currencies (CC) that had been initiated as social experiments. One is a local CC circulating in a specific geographical area, such as a town. The other is a virtual CC used among members who belong to a certain community of interest (COI) on the Internet. We conducted two empirical studies to estimate the economic vitalization effects they had on their respective local economies. The results we found were that the amount of transactions (payments and receipts) of the two CCs was distributed according to a power-law distribution with a unity rank exponent. In addition, we found differences between the two CCs with regard to the shapes of their distribution over a low-transaction range. The result may originate from the difference in methods of issuing CCs or in the magnitudes of the minimum-value unit; however, this result calls for further investigation.

Experimental observations of oscillations and chaotic dynamics in the CO catalytic combustion on supported catalysts were reported more than fifteen years ago. Recently, a detailed reaction mechanism including over 20 reaction steps has been proposed for the catalytic CO oxidation, NO_x reduction and hydrocarbons oxidation taking place in a three-way catalytic converter (TWC), the most common reactor for detoxification of automobile exhaust gases. For an unforced lumped model, we report results of stoichiometric network analysis of several reaction subnetworks determining feedback loops, which cause oscillations within certain ranges of parameters.

We found that epithelial cells (Madin-Darby canine kidney cells) migrated collectively on a collagen gel substrate. Time-lapse images of MDCK cells cultured on type-I collagen gels and glass substrates were captured by phase contrast microscopy equipped with an incubation system. On the gel substrate, the directions of cell movement gradually converged on one direction as the number of cells increased, whereas the cells moved randomly on the glass substrate. We also observed "leader" cells, which extended large lamellae and were accompanied by many "follower" cells, migrating in the direction of oriented collagen fibers. Moreover, we found that epithelial cells exhibit wave-like collective movement during formation of an epithelial sheet on a collagen gel substrate. We name this "cellular wave". The cellular wave is categorized as a longitudinal wave, since the cells elongated their anterior edges and contracted their posterior parts alternately in a portion of the epithelial sheet. The morphological changes propagated to the neighboring cells through the cell-cell adhesions. The cellular wave always appeared when the cell density was getting saturated. Therefore, appearance of the cellular wave depends on both the cellular density and the substrate stiffness. We found new phenomena indicating how epithelial sheets organize a cyst structure when embedded in a collagen gel. Turnup of the periphery of the cellular sheet appeared at first, and then collective cell migration occurred until the cyst was formed.

We report here a new kind of behavior that seems to be 'indecisive' in an amoeboid organism, the Physarum plasmodium of true slime mold. The plasmodium migrating in a narrow lane stops moving for a period of time (several hours but the duration differs for each plasmodium) when it encounters the presence of a chemical repellent, quinine. After stopping period, the organism suddenly begins to move again in one of three different ways as the concentration of repellent increases: going through the repulsive place (penetration), splitting into two fronts of going throught it and turning (splitting) and turning back (rebound). In relation to the physiological mechanism for tip migration in the plasmodium, we found that the frontal tip is capable of moving further although the tip is divided from a main body of organism. This means that a motive force of front locomotion is produced by a local process at the tip. Based on this finding, a mathematical model for front locomotion is considered in order to understand the dynamics for both the long period of stopping and three kinds of behavior. A model based on reaction-diffusion equations succeeds to reproduce the experimental observation. The origin of long-time stopping and three different outputs may be reduced to the hidden instabilities of internal dynamics of the pulse, which may be a skeleton structure extracted from much more complex dynamics imbedded in the Physarum plasmodium.

We studied the effect of the size of food sources (FSs) presented to the true slime mould Physarum polycephalum on the tubular networks formed by the organism to absorb nutrient. The amount of plasmodium gathering at an FS was shown to be proportional to both the concentration of nutrient and the surface area of the FS. We presented two FSs to test which connection the organism selected in response to varying amounts of food and derived a simple rule for connection persistence: the longer connection collapses earlier. A mathematical model for tube selection in response to amount of food was derived and predicted our experimental findings regarding the choice of connection. When three FSs were presented to the organism, the longer tubes were also the first to collapse, explained by the relative probability of disconnection. The size of the FS is thus a key parameter determining network formation.

The dynamical property of thermal fluctuation in the double network gels was studied by frequency resolved light scattering method and time resolved impulsive stimulated light scattering method. Both methods give complimentarily wide range dynamical property over than twelve decades between 10 - 10¹³ Hz. The dynamic property in the double network gels shows a hierarchy over five levels.

Recently, the important role of the choroidal circulation has been recognized in various fundus diseases, such as punctate inner choroidopathy (PIC), Vogt-Koyanagi-Harada (VKH) disease, and age-related macular degeneration (AMD). An apparatus based on the laser speckle phenomenon, which we call laser speckle flowgraphy (LSFG) has been used to measure the blood flow velocity in the retina and choroid with the diode laser. PIC presents in myopic women who complain of decreased visual acuity. VKH disease is a bilateral, granulomatous chorioretinitis. The composite map of the LSFG represent that blood flow velocity in the choroid was decreased in PIC and VKH. After the treatment, blood flow velocity in this area was increased in both disease and visual acuity recovered. LSFG appears to be a safe and sensitive means to evaluate these disease progression and response to therapy. A multifractal analysis has been performed for blood flow data in the composite map of the LSFG. In consequence, multifractality becomes clearly worse for the AMD patient compared to normal choroidal blood flows. The multifractal analysis of LSFG data represents an additional useful method to detect the early stage of AMD.

The aim of study was to assess whether any network index of placental surface arteries was associated with neonatal birth weight. Twenty-six placentas were randomly selected between 34 and 41 weeks of gestational ages. Placental weights ranged 385 to 770 g; and neonatal weights ranged 1960 to 3680 g. After visualization of placental surface arteries by a milk injection method, network indices including the number of nodes, network density, network diameter, average distance of nodes, and the degree centralization were determined. These network indices and placental weights were compared with neonatal birth weights. The Number of nodes, network density, network diameter, average distance of nodes, and the degree centralization were found to be as follows (Mean ± SD); 84.7 ± 29.3, 0.0262 ± 0.0088, 15.8 ± 2.77, 7.83 ± 1.13, 0.0263 ± 0.0091, respectively. We found that neonatal birth weights correlate with the number of nodes of placental surface arteries (correlation coefficient R=0.40) and placental weights (R=0.52) both. However, the number of nodes of placental surface arteries was not associated with the placental weights or the gestational age. We for the first time found that a topological factor, i.e., the number of nodes of placental surface arteries correlated with neonatal growth. There was no correlation between numbers of nodes and placental weights. This suggests that the number of nodes affects fetal growth independent of placental weights. A topological factor of placental vasculization might significantly affect fetal growth in utero and determine risks of vascular diseases in their future lives.

We present a field theory for a structurally disordered magnetic system coupled to a metallic environment near a quantum critical point. We show that close to the magnetic quantum critical point droplets are formed due to the disorder and undergo dissipative quantum dynamics. We argue that the problem has a characteristic energy scale, the droplet Kondo temperature, that determines the crossover energy scale from weak to strong coupling. Our results have direct significance for the Griffiths-McCoy singularities of itinerant magnets.

Nuclear Magnetic Resonance (NMR) and Muon Spin Rotation (μSR) experiments are effective for the detection of inhomogeneous statics and/or dynamics in the spin systems housed by non-Fermi liquid (NFL) materials. This paper describes the properties that can be probed using these techniques and the techniques' sensitivity to magnetic disorder. The paper also reviews NMR and μSR work in some heavy fermion, NFL, materials that lie near a quantum critical point. NFL behavior is found at low temperature in a number of materials that can be tuned towards quantum criticality by the suppression of antiferromagnetism. The NFL regime is signaled by unusual behavior of thermodynamic and transport properties. The study of NMR and μSR parameters in some of these systems reveals strongly inhomogeneous magnetic susceptibilities due to structural disorder, which in turn is found to be an important factor in determining their NFL character. At the lowest temperatures, the spin dynamics in such systems are due to cooperative fluctuations of inhomogeneous type, similar to the spin dynamics of spin glasses, but with zero spin-freezing temperature. The spectroscopic data permit the comparison between electronic disorder and slow spin fluctuations, which turn out to be correlated in NFL systems.

For heavy fermion materials, moderate crystallographic disorder can induce new magnetic ground states such as spin glass phases. Here, I will review the experimental situation for one specific case, URh₂Ge₂. I will demonstrate that there is a competition between the spin glass phase and antiferromagnetism and discuss implications of this observation.

The intricate relationship between quantum criticality and phase formation in correlated materials is one of challenging issues to be understood in current condensed matter physics. Herein, we summarize our recent findings in the heavy fermion compounds U(Ru_1-xRh_x)₂Si₂ (x=0, 2, 2.5, 3, and 4 %) that can enlighten our understanding on the open issue. Careful electrical transport and thermodynamic measurements in the high magnetic field B region up to 45 tesla uncover the formation of multiple phases via avoiding the quasiparticle divergence near their putative metamagnetic quantum critical points (QCPs) at low temperature (T) region. From the established B vs. T phase diagram in the series of samples, we find a clear link between the B-dependence of the phases and that of the putative QCP. This observation constitutes an archetypal case that the field-tuned metamagnetic quantum criticality is intimately related to the formation of the thermodynamic phases. In addition, using the Hall effect studies in the x=0 and 4 % samples, we find discontinuous changes of the Hall coefficient at the phase boundaries between the field-induced phases and neighboring Fermi liquid states, indicating the Fermi surface is sharply reconstructed in the phase formation process near the putative QCP.

New magnetotransport measurements of the heavy fermion compound CeRu₂Si₂ suggest that the metamagnetic transition (MMT) that occurs in an applied magnetic field of 7.8 T is continuous down to <10 mK. We propose a new model of the MMT that does not involve localization of the f-electron and the discontinuities in transport properties that such localization would imply.

We have presented recent results of neutron scattering experiments for the pseudo-binary alloy system Ce(Ru_0.9Rh_0.1)₂(Si_1-yGe_y)₂. The base material Ce(Ru_0.9Rh_0.1)₂Si₂ is an itinerant antiferromagnetic(AFM) heavy fermion. By substituting Ge for Si, we effectively apply negative pressure to the itinerant AFM phase through lattice expansion and the magnetic order is shifted to a localized AFM order. A drastic change of the magnetic wave vector has been observed around y ~ 0.26, which may corresponds to a 1st order phase transition between the itinerant- and the localized-AFM phases.

Neutron powder diffraction experiments were carried out on the insulating phase of the mixed compounds SrRu_1-xMn_xO₃ for 0.4 ≤ x ≤ 0.6. We have found that the ferromagnetic phase observed at x ~ 0 changes into the C-type antiferromagnetic phase above x ≥ 0.4. In addition, this transition accompanies the structural variation from orthorhombic phase (x ~ 0) to tetragonal phase (x ≥ 0.4). With increasing x, the AF moment is strongly enhanced from 1.1 μB (x = 0.4) to 2.6 μB (x = 0.6), which is tightly coupled with the elongation of the tetragonal c/a ratio. We suggest from the experimental results that the evolution of the AF order in the insulating phase is generated by the superexchange interaction originating in the strong electronic correlation between the d electrons on Ru and Mn ions.

Static magnetization measurements of Sr₂RuO₄ have been performed down to 0.1 K. For H ∥ [100] an anomalous increase of magnetization is observed at H₂ which is located just below H_c2. Besides the above anomaly, a small kink is found in the magnetization curves at 8.9 kOe, suggesting an additional superconducting transition due to the lifting of the degenerate order parameter. For H ∥ [001] anomalous successive flux-jumps are observed around zero field in the magnetization curves. Possible origins of such anomalous magnetization are discussed from the viewpoints of the pairing symmetry and the topological change in the vortex lattice configuration.

We investigated thermal, magnetic and transport properties for the dilute system Th_1-xPr_xRu₂Si₂ (x = 0.05 and 0.1). The 4f electronic specific heat C_4f/T shows a Schottky anomaly at ~ 10 K, and the 4f contribution to magnetic susceptibility χ_4f shows a saturation tendency below this temperature. Both C_4f/T and χ_4f are well scaled by the doped Pr concentration, and fitted by crystalline-electric-field (CEF) calculations based on the singlet ground state. The electrical resistivity ρ shows normal metallic behavior. The Kondo screening seems not to occur in this dilute system.

⁵¹V nuclear magnetic resonance (NMR) measurements below 100 mK using a ³He-⁴He dilution refrigerator have been carried out to investigate the magnetic states of V ions in the V15 cluster. It is revealed experimentally that the V⁴⁺ (s=1/2) spins on the outer hexagons are in a singlet state and V ions on the inner triangle has s=1/2 spin moments at low temperature.

Fractal body, a porous silica with cross-sectional fractal dimension D_cs = 1.87 was created by a sol-gel reaction of tetramethyl orthosilicate (TMOS) using unique template particles. D_cs was maintained over ca. three decades in pore size from 0.05 - 30 μm and its density ρ = 0.35 g·cm^-3. Based on the obtained D_cs, pore size distribution and ρ, it was concluded that its fractal geometry was closer to Menger sponge (fractal dimension D = 2.73) at the 7th generation, a mathematical model of fractal body. Our experimental strategy would allow us to design fractality of porous materials in real space.

In order to reveal the lattice relaxation processes of photoexcited states in dimetal hlaid chains, we employ a one-dimensional extended Peierls-Hubbard Hamiltonian and simulate the time evolution of photoexcited system within time-dependent Hartree-Fock approximation. We demonstrate that charged solitons, neutral solitons, and polarons are simultaneously photogenerated. This situation strikingly differs from one in conventional monometal-halid chains.

We propose a new analytical approach to discuss the existence of the spin glass (SG) phase. We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards-Anderson model. Our RG equations under the replica symmetric (RS) ansatz show the flow diagram that indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that the latter is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.

Network models are both simple and general. In solid states physics, quantum network models that take into account the wave nature of the electron have been the subject of intense research. One example is the Chalker–Coddington model that describes the quantum Hall effect (J. T. Chalker, P. D. Coddington, J. Phys., C 21 2665 (1988)). In this paper we focus on the transfer matrices that describe these models. We emphasize the symmetry properties of these matrices and the consequences for eigenvalue and transport properties.

The self-averaging nature of statistical quantities at criticality has been studied by examining infinite clusters in two-dimensional bond percolation systems at the percolation threshold p_c. We calculate fractal dimensions Δ_f for individual clusters and obtain the distribution function of Δ_f It has been clarified that the system size dependence of the variance var(Δ_f) can be explained by the scaling theory if eliminating non-fractal critical clusters (NFCC) from the original ensemble of infinite clusters. This implies the absence of the self-averaging nature at the critical point of percolation systems because of the existence of NFCC at p_c even in the thermodynamic limit.

Critical behaviour of the two-dimensional Ising model defined on a pseudo-sphere is numerically studied. Employing the finite-size scaling analysis, we observe a quantitative deviation of static critical exponents from the case of the Ising lattice model on a flat plane. This finding indicates the occurrence of a novel universality class of the Ising model on negatively curved surfaces.

We have investigated electron states confined in a cylindrical surface whose radius is smoothly varied within a limited region. The quantum confinement normal to the surface results in an effective potential energy that is a function of the local surface curvature. This curvature-induced potential strongly affects the electronic structures of the lowest-lying states, and further yields bounded states at the deformed region. These suggest the possibility to control electric conductivity of nanostructures by inducing a local deformation.

Studying quantum pumping in a one-dimensional electron system with a time-varying double δ potential, it has been clarified that nonadiabatic pumping currents are qualitatively different from adiabatic ones. We also elucidate that a finite pure spin current can be expected in a nonadiabatically driven system under a magnetic field, in which no chaotic character is exhibited.

The charge density wave state in a ring-shaped crystal is studied by mapping to the three dimensional uniformly frustrated XY model with quasi-one-dimensional anisotropy in connectivity. Cylindrical bending results frustration between intra- and inter-chain couplings then phase vortex lines are generated even in the ground state. As temperature becomes lower, the system takes a phase transition to the ordered state, which is characterized by the vortex lattice.

Spatiotemporal acoustic field imaging on the (100) plane of the cubic crystal LiF and on a square array of gold pyramids on a glass substrate is performed with an ultrafast optical technique. In both cases wave fronts exhibiting fourfold symmetry are obtained. The results of these two experiments on naturally and artificially anisotropic materials are compared.

By improving the spatial-light-modulator pattern, a clean optical vortex beam for large topological charge was obtained without mixing of nondesired modes or forming an anamorphic circle.

An influence of cosmic topology on a black hole solution is discussed. The solution of a spherically symmetric black hole located at the center of a topological universe is obtained as a toy model. Trajectories of a massive particle around the black hole is found to deviate remarkably from those for the usual Schwarzschild solution at a larger distance from the black hole.

AUTHOR INDEX.