Tunable Exponent Research Articles

Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states

We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as $1/{r}^{\ensuremath{\alpha}}$, where $r$ denotes the distance between two spins and $\ensuremath{\alpha}$ is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of $\ensuremath{\alpha}$. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}<1$ we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime $\ensuremath{\alpha}\ensuremath{\ge}1$, and qualitative agreement with DMRG in the strong-long range regime $\ensuremath{\alpha}\ensuremath{\le}1$. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.

Universality Class of Ising Critical States with Long-Range Losses.

We show that spatial resolved dissipation can act on d-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as ∝q^{α}, with α a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, 1/r^{(α+d)}. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the nonunitary counterpart of the universality class of long-range interacting Ising models. For α<1 we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial (∼∂_{t}^{2}) and frictional (∼∂_{t}) kinetic coefficients, and driven by a gapless Markovian noise with variance ∝q^{α} at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.

Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

Do Beach Profiles Under Nonbreaking Waves Minimize Energy Dissipation?

AbstractThe hypothesis that equilibrium beach profiles under nonbreaking waves minimize wave energy dissipation was considered by Larson et al. (1999 Coast. Eng. 36 59‐85). Larson et al. approached the hypothesis as a variational problem, assuming a priori that the solution (the extremal profile) followed a power law with a freely tunable exponent, which was varied so as to extremize the relevant functional. Here, we revisit this hypothesis and solve the associated variational problem approximately via analytical means, without a priori assumptions on the mathematical structure of the solution. We remark that for the solution to be realistic, the problem formulation must consider additional constraints; for example, the bed slope angle must not exceed the sediment's angle of repose. Incidentally, the solution we derive recovers the power law prescribed by Larson et al., which is in turn backed by a large body of empirical evidence. However, the exponent of the power in our solution is not an arbitrarily free parameter; it depends on the parametrization of the bed shear stress (the main mechanism by which nonbreaking waves dissipate energy), and predicted values of the exponent are supported by previous research. The power law curve derived here agrees well with empirical data from field and laboratory, suggesting that a principle of energy economy may indeed underpin the particular shape adopted by beach profiles under nonbreaking waves. This theoretical study aims at promoting and aiding further tests of this hypothesis.

Inequality in resource allocation and population dynamics models.

The Hassell model has been widely used as a general discrete-time population dynamics model that describes both contest and scramble intraspecific competition through a tunable exponent. Since the two types of competition generally lead to different degrees of inequality in the resource distribution among individuals, the exponent is expected to be related to this inequality. However, among various first-principles derivations of this model, none is consistent with this expectation. This paper explores whether a Hassell model with an exponent related to inequality in resource allocation can be derived from first principles. Indeed, such a Hassell model can be derived by assuming random competition for resources among the individuals wherein each individual can obtain only a fixed amount of resources at a time. Changing the size of the resource unit alters the degree of inequality, and the exponent changes accordingly. As expected, the Beverton–Holt and Ricker models can be regarded as the highest and lowest inequality cases of the derived Hassell model, respectively. Two additional Hassell models are derived under some modified assumptions.

Dense power-law networks and simplicial complexes.

There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e., their degree distributions follow P(k)∝k^{-γ} with γ∈(1,2]. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time step is constant. Here we present a modeling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent γ=2 or directed networks with power-law out-degree distribution with tunable exponent γ∈(1,2). We also extend the model to that of directed two-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.

A Chemical Reaction Network to Generate Random, Power-Law-Distributed Time Intervals.

In Lévy walks (LWs), particles move with a fixed speed along straight line segments and turn in new directions after random time intervals that are distributed according to a power law. Such LWs are thought to be an advantageous foraging and search strategy for organisms. While complex nervous systems are certainly capable of producing such behavior, it is not clear at present how single-cell organisms can generate the long-term correlated control signals required for a LW. Here, we construct a biochemical reaction system that generates long-time correlated concentration fluctuations of a signaling substance, with a tunable fractional exponent of the autocorrelation function. The network is based on well-known modules, and its basic function is highly robust with respect to the parameter settings.

Diffusion-like recommendation with enhanced similarity of objects

In last decades, diversity and accuracy have been regarded as two important measures in evaluating a recommendation model. However, a clear concern is that a model focusing excessively on one measure will put the other one at risk, thus it is not easy to greatly improve diversity and accuracy simultaneously. In this paper, we propose to enhance the Resource-Allocation (RA) similarity in resource transfer equations of diffusion-like models, by giving a tunable exponent to the RA similarity, and traversing the value of the exponent to achieve the optimal recommendation results. In this way, we can increase the recommendation scores (allocated resource) of many unpopular objects. Experiments on three benchmark data sets, MovieLens, Netflix, and RateYourMusic show that the modified models can yield remarkable performance improvement compared with the original ones.

Bootstrap percolation on spatial networks.

Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of long-range links’ lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameter-dependent critical value for the power-law exponent, above which there is a double phase transition, mixed of a second-order phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a second-order phase transition. We further find a parameter-independent critical value around −1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value −1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the self-organization of spatial structure of online social networks, in terms of the effective function for information spreading.

Enhanced localization, energy anomalous diffusion and resonant mode in harmonic chains with correlated mass-spring disorder

In this work, we study the vibrational modes and energy spreading in a harmonic chain model with diluted second-neighbors couplings and correlated mass-spring disorder. While all nearest neighbor masses are coupled by an elastic spring, second neighbors springs are introduced with a probability pD. The masses are randomly distributed according to the site connectivity , where ni is the connectivity of the site i and α is a tunable exponent. We show that maximum localization of the vibrational modes is achieved for α ≃ 3/4. The time-evolution of the energy wave-packet is followed after an initial localized excitation. While the participation number remains finite, the energy spread is shown to be sub-diffusive after a displacement and super-diffusive after an impulse excitation. These features are related to the development of a power-law tail in the wave-packet distribution. Further, we unveil that the spring dilution leads to the emergence of a resonant localized state which is signaled by a van Hove singularity in the density of states.

Outbreak statistics and scaling laws for externally driven epidemics.

Power-law scalings are ubiquitous to physical phenomena undergoing a continuous phase transition. The classic susceptible-infectious-recovered (SIR) model of epidemics is one such example where the scaling behavior near a critical point has been studied extensively. In this system the distribution of outbreak sizes scales as P(n)∼n-3/2 at the critical point as the system size N becomes infinite. The finite-size scaling laws for the outbreak size and duration are also well understood and characterized. In this work, we report scaling laws for a model with SIR structure coupled with a constant force of infection per susceptible, akin to a "reservoir forcing". We find that the statistics of outbreaks in this system fundamentally differ from those in a simple SIR model. Instead of fixed exponents, all scaling laws exhibit tunable exponents parameterized by the dimensionless rate of external forcing. As the external driving rate approaches a critical value, the scale of the average outbreak size converges to that of the maximal size, and above the critical point, the scaling laws bifurcate into two regimes. Whereas a simple SIR process can only exhibit outbreaks of size O(N1/3) and O(N) depending on whether the system is at or above the epidemic threshold, a driven SIR process can exhibit a richer spectrum of outbreak sizes that scale as O(Nξ), where ξ∈(0,1]∖{2/3} and O((N/lnN)2/3) at the multicritical point.

Structural features beneath neuronal avalanches

Recently, Friedman et al performed high-resolution measurements [1] that strongly supported an universal critical character (in the sense of statistical physics) of neuronal avalanches, despite the abnormally high frequency of large events (bumps in the distributions of size and lifetime of avalanches) deforming the pure power-law behavior expected from analogies to equilibrium critical phenomena that became manifest only for smaller events. Based on simulations of the Kinouchi-Copelli (KC) model, we have shown [2] that such bumps may not be experimental artifacts and really be typical at criticality when the topology of the neural network is the Barabasi-Albert (BA) model, leading to a scale-free degree distribution of exponent -3.0. On the other hand, those simulations could not reproduce the exponents of the power-law region of the avalanche distributions in [1] (namely, -1.7 for the size distribution and -1.9 for the lifetime distribution). Besides that, the KC dynamics on BA topology revealed that the information capacity (entropy of avalanche size distribution) did not exhibit critical optimization, in contrast with an earlier experiment [3]. In this study we investigate the KC dynamics on the Uncorrelated Configuration Model (UCM) [4]. The UCM is a kind of wiring procedure of a neural topology that can lead to scale-free degree distributions with tunable exponents and can be matched to BA model. However, even so their avalanches are not identical. While the UCM also allows the appearance of bumps on the avalanche distributions, it both shows that the information capacity may be critically optimal and exhibits quantitatively accurate values of the critical exponents for small avalanches, suggesting the UCM may be descriptive of some structural features in the systems claimed to exhibit critical dynamics.

Building complex networks with Platonic solids

We propose a unified model to build planar graphs with diverse topological characteristics which are of relevance in real applications. Here convex regular polyhedra (Platonic solids) are used as the building blocks for the construction of a variety of complex planar networks. These networks are obtained by merging polyhedra face by face on a tree-structure leading to planar graphs. We investigate two different constructions: (1) a fully deterministic construction where a self-similar fractal structure is built by using a single kind of polyhedron which is iteratively attached to every face and (2) a stochastic construction where at each step a polyhedron is attached to a randomly chosen face. These networks are scale-free, small-world, clustered, and sparse, sharing several characteristics of real-world complex networks. We derive analytical expressions for the degree distribution, the clustering coefficient, and the mean degree of nearest neighbors showing that these networks have power-law degree distributions with tunable exponents associated with the building polyhedron, and they possess a hierarchical organization that is determined by planarity.

Packing of Soft Asymmetric Dumbbells

We use numerical simulations to study the phase behavior of a system of purely repulsive soft dumbbells as a function of size ratio of the two components and their relative degree of deformability. We find a plethora of different phases, which includes most of the mesophases observed in self-assembly of block copolymers but also crystalline structures formed by asymmetric, hard binary mixtures. Our results detail the phenomenological behavior of these systems when softness is introduced in terms of two different classes of interparticle interactions: (a) the elastic Hertz potential, which has a finite energy cost for complete overlap of any two components, and (b) a generic power-law repulsion with tunable exponent. We discuss how simple geometric arguments can be used to account for the large structural variety observed in these systems and detail the similarities and differences in the phase behavior for the two classes of potentials under consideration.

Noise and critical phenomena in biochemical signaling cycles at small molecule numbers

Biochemical reaction networks in living cells usually involve reversible covalent modification of signaling molecules, such as protein phosphorylation. Under conditions of small molecule numbers, as is frequently the case in living cells, mass-action theory fails to describe the dynamics of such systems. Instead, the biochemical reactions must be treated as stochastic processes that intrinsically generate concentration fluctuations of the chemicals. We investigate the stochastic reaction kinetics of covalent modification cycles (CMCs) by analytical modeling and numerically exact Monte Carlo simulation of the temporally fluctuating concentration. Depending on the parameter regime, we find for the probability density of the concentration qualitatively distinct classes of distribution functions including power-law distributions with a fractional and tunable exponent. These findings challenge the traditional view of biochemical control networks as deterministic computational systems and suggest that CMCs in cells can function as versatile and tunable noise generators.

Weighted Evolving Networks with Self-organized Communities

In order to describe the self-organization of communities in the evolution of weighted networks, we propose a new evolving model for weighted community-structured networks with the preferential mechanisms functioned in different levels according to community sizes and node strengths, respectively. Theoretical analyses and numerical simulations show that our model captures power-law distributions of community sizes, node strengths, and link weights, with tunable exponents of ν ⩾ 1, γ > 2, and α > 2, respectively, sharing large clustering coefficients and scaling clustering spectra, and covering the range from disassortative networks to assortative networks. Finally, we apply our new model to the scientific co-authorship networks with both their weighted and unweighted datasets to verify its effectiveness.

Universal robustness characteristic of weighted networks against cascading failure

We investigate the cascading failure on weighted complex networks by adopting a local weighted flow redistribution rule, where the weight of an edge is (k(i)k(j))theta with k(i) and k(j) being the degrees of the nodes connected by the edge. Assume that a failed edge leads only to a redistribution of the flow passing through it to its neighboring edges. We found that the weighted complex network reaches the strongest robustness level when the weight parameter theta=1, where the robustness is quantified by a transition from normal state to collapse. We determined that this is a universal phenomenon for all typical network models, such as small-world and scale-free networks. We then confirm by theoretical predictions this universal robustness characteristic observed in simulations. We furthermore explore the statistical characteristics of the avalanche size of a network, thus obtaining a power-law avalanche size distribution together with a tunable exponent by varying theta. Our findings have great generality for characterizing cascading-failure-induced disasters in nature.

Power-law strength-degree correlation from resource-allocation dynamics on weighted networks

Many weighted scale-free networks are known to have a power-law correlation between strength and degree of nodes, which, however, has not been well explained. We investigate the dynamic behavior of resource-traffic flow on scale-free networks. The dynamical system will evolve into a kinetic equilibrium state, where the strength, defined by the amount of resource or traffic load, is correlated with the degree in a power-law form with tunable exponent. The analytical results agree well with simulations.

Weighted assortative and disassortative networks model

Real-world networks process structured connections since they have non-trivial vertex degree correlation and clustering. Here we propose a toy model of structure formation in real-world weighted network. In our model, a network evolves by topological growth as well as by weight change. In addition, we introduce the weighted assortativity coefficient, which generalizes the assortativity coefficient of a topological network, to measure the tendency of having a high-weighted link between two vertices of similar degrees. Network generated by our model exhibits scale-free behavior with a tunable exponent. Besides, a few non-trivial features found in real-world networks are reproduced by varying the parameter ruling the speed of weight evolution. Most importantly, by studying the weighted assortativity coefficient, we found that both topologically assortative and disassortative networks generated by our model are in fact weighted assortative.

Efficiency dynamics on scale-free networks with tunable degree exponent

A model, introduced earlier for the dynamics of a generic efficiency measure in a population of agents by Majumdar and Krapivsky (Phys. Rev. E 63, 054101 (2001)), is investigated on scale-free networks whose degree distribution follows a power law with the tunable exponent γ. The model shows a delocalization transition from a stagnant phase to a growing one when decreasing the degree exponent γ of scale-free networks. By taking into account the specific dynamical properties of the model and the geometrical properties of scale-free networks, we predict the appearance of this critical transition. This work is useful for understanding these kinds of transitions occurring in many dynamical processes on scale-free networks.