Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications

Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between the Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundreds of shells. We also show that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.

Properties of Hydrogen-Bonded Networks in Ethanol-Water Liquid Mixtures as a Function of Temperature: Diffraction Experiments and Computer Simulations

New X-ray and neutron diffraction experiments have been performed on ethanol–water mixtures as a function of decreasing temperature, so that such diffraction data are now available over the entire composition range. Extensive molecular dynamics simulations show that the all-atom interatomic potentials applied are adequate for gaining insight into the hydrogen-bonded network structure, as well as into its changes on cooling. Various tools have been exploited for revealing details concerning hydrogen bonding, as a function of decreasing temperature and ethanol concentration, like determining the H-bond acceptor and donor sites, calculating the cluster-size distributions and cluster topologies, and computing the Laplace spectra and fractal dimensions of the networks. It is found that 5-membered hydrogen-bonded cycles are dominant up to an ethanol mole fraction x_eth = 0.7 at room temperature, above which the concentrated ring structures nearly disappear. Percolation has been given special attention, so that it could be shown that at low temperatures, close to the freezing point, even the mixture with 90% ethanol (x_eth = 0.9) possesses a three-dimensional (3D) percolating network. Moreover, the water subnetwork also percolates even at room temperature, with a percolation transition occurring around x_eth = 0.5.

Spectral analysis for weighted iterated q-triangulation networks

Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated q-triangulation networks, which are generated by iteration operation F(⋅). We add q(q∈N₊) new nodes on each old edge and connect them with two endpoints of the old edge. At the same time, the newly linked edges are given weight factor r(0<r≤1). From the construction of the network, we obtain all the eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated q-triangulation network. Further, as applications of spectra of the normalized Laplacian matrix, we study the Kemeny constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for weighted iterated q-triangulation networks.

Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules

We report on spontaneous rotational symmetry breaking in a minimal model of complex macromolecules with branches and cycles. The transition takes place as the strength of the self-repulsion is increased. At the transition point, the density distribution transforms from isotropic to anisotropic. We analyze this transition using a variational mean-field theory that combines the Gibbs-Bogolyubov-Feynman inequality with the concept of the Laplacian matrix. The density distribution of the broken symmetry state is shown to be determined by the eigenvalues and eigenvectors of this Laplacian matrix. Physically, this reflects the increasing role of the underlying topological structure in determining the density of the macromolecule when repulsive interactions generate internal tension. Eventually, the variational free energy landscape develops a complex structure with multiple competing minima.

The trapping problem of the weighted scale-free treelike networks for two kinds of biased walks

It has been recently reported that trapping problem can characterize various dynamical processes taking place on complex networks. However, most works focused on the case of binary networks, and dynamical processes on weighted networks are poorly understood. In this paper, we study two kinds of biased walks including standard weight-dependent walk and mixed weight-dependent walk on the weighted scale-free treelike networks with a trap at the central node. Mixed weight-dependent walk including non-nearest neighbor jump appears in many real situations, but related studies are much less. By the construction of studied networks in this paper, we determine all the eigenvalues of the fundamental matrix for two kinds of biased walks and show that the largest eigenvalue has an identical dominant scaling as that of the average trapping time (ATT). Thus, we can obtain the leading scaling of ATT by a more convenient method and avoid the tedious calculation. The obtained results show that the weight factor has a significant effect on the ATT, and the smaller the value of the weight factor, the more efficient the trapping process is. Comparing the standard weight-dependent walk with mixed weight-dependent walk, although next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT.

Heterogeneous continuous-time random walks

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

Coherence analysis of a class of weighted networks

This paper investigates consensus dynamics in a dynamical system with additive stochastic disturbances that is characterized as network coherence by using the Laplacian spectrum. We introduce a class of weighted networks based on a complete graph and investigate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. First, the recursive relationship of its eigenvalues at two successive generations of Laplacian matrix is deduced. Then, we compute the sum and square sum of reciprocal of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first- and second-order coherence with network size obey four and five laws, respectively, along with the range of the weight factor. Finally, it indicates that the scalings of our studied networks are smaller than other studied networks when 1d<r≤1.

Spectra of weighted scale-free networks

Much information about the structure and dynamics of a network is encoded in the eigenvalues of its transition matrix. In this paper, we present a first study on the transition matrix of a family of weight driven networks, whose degree, strength, and edge weight obey power-law distributions, as observed in diverse real networks. We analytically obtain all the eigenvalues, as well as their multiplicities. We then apply the obtained eigenvalues to derive a closed-form expression for the random target access time for biased random walks occurring on the studied weighted networks. Moreover, using the connection between the eigenvalues of the transition matrix of a network and its weighted spanning trees, we validate the obtained eigenvalues and their multiplicities. We show that the power-law weight distribution has a strong effect on the behavior of random walks.

Anomalous behavior of trapping in extended dendrimers with a perfect trap

Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generation-dependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers.

Spectrum of walk matrix for Koch network and its application

Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

The eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network. Nevertheless, it is still not well understood how the eigenvalues behave in small-world and scale-free networks, which describe a large variety of real systems. In this paper, we study the eigenvalues for the transition matrix of a network that is simultaneously scale-free, small-world, and clustered. We derive explicit simple expressions for all eigenvalues and their multiplicities, with the spectral density exhibiting a power-law form. We then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks, both of which exhibit unusual behaviors compared with those for other networks, signaling discernible effects of topologies on spectral features. Finally, we use the eigenvalues to count spanning trees in the network.

Equivalence between a generalized dendritic network and a set of one-dimensional networks as a ground of linear dynamics

It has been shown by some existing studies that some linear dynamical systems defined on a dendritic network are equivalent to those defined on a set of one-dimensional networks in special cases and this transformation to the simple picture, which we call linear chain (LC) decomposition, has a significant advantage in understanding properties of dendrimers. In this paper, we expand the class of LC decomposable system with some generalizations. In addition, we propose two general sufficient conditions for LC decomposability with a procedure to systematically realize the LC decomposition. Some examples of LC decomposable linear dynamical systems are also presented with their graphs. The generalization of the LC decomposition is implemented in the following three aspects: (i) the type of linear operators; (ii) the shape of dendritic networks on which linear operators are defined; and (iii) the type of symmetry operations representing the symmetry of the systems. In the generalization (iii), symmetry groups that represent the symmetry of dendritic systems are defined. The LC decomposition is realized by changing the basis of a linear operator defined on a dendritic network into bases of irreducible representations of the symmetry group. The achievement of this paper makes it easier to utilize the LC decomposition in various cases. This may lead to a further understanding of the relation between structure and functions of dendrimers in future studies.

A class of scale-free networks with fractal structure based on subshift of finite type

In this paper, given a time series generated by a certain dynamical system, we construct a new class of scale-free networks with fractal structure based on the subshift of finite type and base graphs. To simplify our model, we suppose the base graphs are bipartite graphs and the subshift has the special form. When embedding our growing network into the plane, we find its image is a graph-directed self-affine fractal, whose Hausdorff dimension is related to the power law exponent of cumulative degree distribution. It is known that a large spectral gap in terms of normalized Laplacian is usually associated with small mixing time, which makes facilitated synchronization and rapid convergence possible. Through an elaborate analysis of our network, we can estimate its Cheeger constant, which controls the spectral gap by Cheeger inequality. As a result of this estimation, when the bipartite base graph is complete, we give a sharp condition to ensure that our networks are well-connected with rapid mixing property.

Full eigenvalues of the Markov matrix for scale-free polymer networks

Much important information about the structural and dynamical properties of complex systems can be extracted from the eigenvalues and eigenvectors of a Markov matrix associated with random walks performed on these systems, and spectral methods have become an indispensable tool in the complex system analysis. In this paper, we study the Markov matrix of a class of scale-free polymer networks. We present an exact analytical expression for all the eigenvalues and determine explicitly their multiplicities. We then use the obtained eigenvalues to derive an explicit formula for the random target access time for random walks on the studied networks. Furthermore, based on the link between the eigenvalues of the Markov matrix and the number of spanning trees, we confirm the validity of the obtained eigenvalues and their corresponding degeneracies.

Two universality classes for random hyperbranched polymers

We grow AB2 random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a "slow growth" (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster-cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a "quick growth" (QG) process which allows for cluster-cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic-power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.