The number of spanning trees of an infinite family of outerplanar,small-world and self-similar graphs

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.     

Boundary Quantum Entanglement of the XXZ Spin Chain with Boundary Impurities

The boundary quantum entanglement for the s = 1/2 X X Z spin chain with boundary impurities is studied via the density matrix renormalization group （DMRG） method. It is shown that the entanglement entropy of the boundary bond （the impurity and the chain spin next to it） behaves differently in different phases. The relationship between the singular points of the boundary entropy and boundary quantum critical points is discussed.     

Stability analysis of coupled map lattices at locally unstable fixed points

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for exploring fundamental features of CMLs, such as their stability properties. Since CMLs can be considered as graphs, we apply methods of spectral graph theory to analyze their stability at locally unstable fixed points for different updating rules, different coupling scenarios, and different types of neighborhoods. Numerical studies are found to be in excellent agreement with our theoretical results.     

Renormalization and destruction of 1/gamma2 tori in the standard nontwist map

Extending the work of del-Castillo-Negrete, Greene, and Morrison [Physica D 91, 1 (1996); 100, 311 (1997)] on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.     

Visibility graph approach to exchange rate series

By means of a visibility graph, we investigate six important exchange rate series. It is found that the series convert into scale-free and hierarchically structured networks. The relationship between the scaling exponents of the degree distributions and the Hurst exponents obeys the analytical prediction for fractal Brownian motions. The visibility graph can be used to obtain reliable values of Hurst exponents of the series. The characteristics are explained by using the multifractal structures of the series. The exchange rate of EURO to Japanese Yen is widely used to evaluate risk and to estimate trends in speculative investments. Interestingly, the hierarchies of the visibility graphs for the exchange rate series of these two currencies are significantly weak compared with that of the other series.     

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent α > 1 of certain monotone combinatorial types. We prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated two-dimensional strong unstable manifold. For monotone families of Lorenz maps we prove that each infinitely renormalizable combinatorial type has a unique representative within the family. We also prove that each infinitely renormalizable map has no wandering intervals, is ergodic, and has a uniquely ergodic minimal Cantor attractor of measure zero.     

An exact,finite, gauge-invariant,non-perturbative approach to QCD renormalization

A particular choice of renormalization, within the simplifications provided by the non-perturbative property of Effective Locality, leads to a completely finite, non-perturbative approach to renormalized QCD, in which all correlation functions can, in principle, be defined and calculated. In this Model of renormalization, only the Bundle chain-Graphs of the cluster expansion are non-zero. All Bundle graphs connecting to closed quark loops of whatever complexity, and attached to a single quark line, provided no ‘self-energy’ to that quark line, and hence no effective renormalization. However, the exchange of momentum between one quark line and another, involves only the cluster-expansion’s chain graphs, and yields a set of contributions which can be summed and provide a finite color-charge renormalization that can be incorporated into all other QCD processes. An application to High Energy elastic pp scattering is now underway.     

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model, belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or W-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit.     

Boundary entropy of one-dimensional quantum systems at low temperature

The boundary beta function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta function, expressing it as the gradient of the boundary entropy s at fixed nonzero temperature. The gradient formula implies that s decreases under renormalization, except at critical points (where it stays constant). At a critical point, the number exp((s) is the "ground-state degeneracy," g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature, except at critical points, where it is independent of temperature. It remains open whether the boundary entropy is always bounded below.     

Search algorithm on strongly regular graphs based on scattering quantum walks

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs(SRGs) with parameters(N, k, λ, μ) achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm's performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs.The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as√N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O(√N) time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.     

Complex networks renormalization: flows and fixed points

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.     

Structural Entropy of the Stochastic Block Models

With the rapid expansion of graphs and networks and the growing magnitude of data from all areas of science, effective treatment and compression schemes of context-dependent data is extremely desirable. A particularly interesting direction is to compress the data while keeping the “structural information” only and ignoring the concrete labelings. Under this direction, Choi and Szpankowski introduced the structures (unlabeled graphs) which allowed them to compute the structural entropy of the Erdős–Rényi random graph model. Moreover, they also provided an asymptotically optimal compression algorithm that (asymptotically) achieves this entropy limit and runs in expectation in linear time. In this paper, we consider the stochastic block models with an arbitrary number of parts. Indeed, we define a partitioned structural entropy for stochastic block models, which generalizes the structural entropy for unlabeled graphs and encodes the partition information as well. We then compute the partitioned structural entropy of the stochastic block models, and provide a compression scheme that asymptotically achieves this entropy limit.     

Phase transitions for information diffusion in random clustered networks

We study the conditions for the phase transitions of information diffusion in complexnetworks. Using the random clustered network model, a generalisation of the Chung-Lurandom network model incorporating clustering, we examine the effect of clustering underthe Susceptible-Infected-Recovered (SIR) epidemic diffusion model with heterogeneouscontact rates. For this purpose, we exploit the branching process to analyse informationdiffusion in random unclustered networks with arbitrary contact rates, and provide noveliterative algorithms for estimating the conditions and sizes of global cascades,respectively. Showing that a random clustered network can be mapped into a factor graph,which is a locally tree-like structure, we successfully extend our analysis to randomclustered networks with heterogeneous contact rates. We then identify the conditions forphase transitions of information diffusion using our method. Interestingly, for variouscontact rates, we prove that random clustered networks with higher clustering coefficientshave strictly lower phase transition points for any given degree sequence. Finally, weconfirm our analytical results with numerical simulations of both synthetically-generatedand real-world networks.     

Components in time-varying graphs

Real complex systems are inherently time-varying. Thanks to new communication systems and novel technologies, today it is possible to produce and analyze social and biological networks with detailed information on the time of occurrence and duration of each link. However, standard graph metrics introduced so far in complex network theory are mainly suited for static graphs, i.e., graphs in which the links do not change over time, or graphs built from time-varying systems by aggregating all the links as if they were concurrent in time. In this paper, we extend the notion of connectedness, and the definitions of node and graph components, to the case of time-varying graphs, which are represented as time-ordered sequences of graphs defined over a fixed set of nodes. We show that the problem of finding strongly connected components in a time-varying graph can be mapped into the problem of discovering the maximal-cliques in an opportunely constructed static graph, which we name the affine graph. It is, therefore, an NP-complete problem. As a practical example, we have performed a temporal component analysis of time-varying graphs constructed from three data sets of human interactions. The results show that taking time into account in the definition of graph components allows to capture important features of real systems. In particular, we observe a large variability in the size of node temporal in- and out-components. This is due to intrinsic fluctuations in the activity patterns of individuals, which cannot be detected by static graph analysis.     

Renormalization for breakup of invariant tori

We present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. Coordinate transformations on the space of maps relate these fixed points, and also induce conjugacies between the corresponding operators.     

Information theory and renormalization group flows

We present a possible approach to the study of the renormalization group (RG) flow based entirely on the information theory. The average information loss under a single step of Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables, which are integrated out, when the slow ones are held fixed. Its positivity results in the monotonic decrease of the informational entropy under renormalization. This, however, does not necessarily imply the irreversibility of the RG flow, because entropy is an extensive quantity and explicitly depends on the total number of degrees of freedom, which is reduced. Only some size-independent additive part of the entropy could possibly provide the required Lyapunov function. We also introduce a mutual information of fast and slow variables as probably a more adequate quantity to represent the changes in the system under renormalization and evaluate it for some simple systems. It is shown that for certain real space decimation transformations the positivity of the mutual information directly leads to the monotonic growth of the entropy per lattice site along the RG flow and hence to its irreversibility.     

Ising deconfinement transition between Feshbach-resonant superfluids

We investigate the phase diagram of bosons interacting via Feshbach-resonant pairing interactions in a one-dimensional lattice. Using large scale density matrix renormalization group and field theory techniques we explore the atomic and molecular correlations in this low-dimensional setting. We provide compelling evidence for an Ising deconfinement transition occurring between distinct superfluids and extract the Ising order parameter and correlation length of this unusual superfluid transition. This is supported by results for the entanglement entropy which reveal both the location of the transition and critical Ising degrees of freedom on the phase boundary.     

Renormalization group study of random quantum magnets

We have developed a very efficient numerical algorithm of the strong disorder renormalization group method to study the critical behaviour of the random transverse field Ising model, which is a prototype of random quantum magnets. With this algorithm we can renormalize an N-site cluster within a time NlogN, independently of the topology of the graph, and we went up to N ～ 4 × 10(6). We have studied regular lattices with dimension D ≤ 4 as well as Erd?s-Rényi random graphs, which are infinite dimensional objects. In all cases the quantum critical behaviour is found to be controlled by an infinite disorder fixed point, in which disorder plays a dominant role over quantum fluctuations. As a consequence the renormalization procedure as well as the obtained critical properties are asymptotically exact for large systems. We have also studied Griffiths singularities in the paramagnetic and ferromagnetic phases and generalized the numerical algorithm for other random quantum systems.     

Threshold effects on renormalization group running of neutrino parameters in the low-scale seesaw model

We show that, in the low-scale type-I seesaw model, renormalization group running of neutrino parameters may lead to significant modifications of the leptonic mixing angles in view of so-called seesaw threshold effects. Especially, we derive analytical formulas for radiative corrections to neutrino parameters in crossing the different seesaw thresholds, and show that there may exist enhancement factors efficiently boosting the renormalization group running of the leptonic mixing angles. We find that, as a result of the seesaw threshold corrections to the leptonic mixing angles, various flavor symmetric mixing patterns (e.g., bi-maximal and tri-bimaximal mixing patterns) can be easily accommodated at relatively low energy scales, which is well within the reach of running and forthcoming experiments (e.g., the LHC).