Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks

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.

Power-law graphs with small diameter: Framework, structural properties, and average trapping time

Here, we propose a simple algorithmic framework for creating power-law graphs with small diameters and then study structural properties, for instance, average degree, on graphs built. The results show that our graphs have not only some commonly seen properties including scale-free feature, small-world property, and disassortative structure, but also many rarely found characteristics, such as the density feature due to power-law exponents equal to 2 and the diameter equivalent to 2, compared to most previous scale-free models. In addition, we also consider the trapping problem on the proposed graphs and then find that they have more optimal trapping efficiency by means of their own average trapping times achieving the theoretical lower bound, a phenomenon that is seldom observed in existing scale-free models. We conduct extensive simulations, and the results show that empirical simulations are consistent with theoretical analysis.

Dynamics of networks in a viscoelastic and active environment

We investigate the dynamics of fractals and other networks in a viscoelastic and active environment. The viscoelastic dynamics is modeled based on the generalized Langevin equation, where the activity is introduced to it by means of the exponentially correlated noise. The intramolecular interactions are taken into account by the bead-spring picture. The microscopic connectivity (studied in the form of Vicsek fractals, of dual Sierpiński gaskets, of NT_D trees, and of a family of deterministic small-world networks) reveals itself in the multiscale monomeric dynamics, which shows vastly different behaviors in the active and passive baths. In particular, the dynamics under active forces leads to a swelling that is characterized through power laws which are not present in the passive case. In all cases, the dynamics reflects the broad scaling behavior of the density of states and not necessarily the maximal relaxation time of the structures in a passive bath, as it is exemplified on the NT_D trees.

Mean first-passage time for random walks on generalized deterministic recursive trees

We describe a technique that allows the exact analytical computation of the mean first passage time (MFPT) for infinite families of trees using their recursive properties. The method is based in the relationship between the MFPT and the eigenvalues of the Laplacian matrix of the trees but avoids their explicit computation. We apply this technique to find the MFPT for a family of generalized deterministic recursive trees. The method, however, can be adapted to other self-similar tree families.

Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees

The determination of mean first-passage time (MFPT) for random walks in networks is a theoretical challenge, and is a topic of considerable recent interest within the physics community. In this paper, according to the known connections between MFPT, effective resistance, and the eigenvalues of graph Laplacian, we first study analytically the MFPT between all node pairs of a class of growing treelike networks, which we term deterministic uniform recursive trees (DURTs), since one of its particular cases is a deterministic version of the famous uniform recursive tree. The interesting quantity is determined exactly through the recursive relation of the Laplacian spectra obtained from the special construction of DURTs. The analytical result shows that the MFPT between all couples of nodes in DURTs varies as N ln N for large networks with node number N. Second, we study trapping on a particular network of DURTs, focusing on a special case with the immobile trap positioned at a node having largest degree. We determine exactly the average trapping time (ATT) that is defined as the average of FPT from all nodes to the trap. In contrast to the scaling of the MFPT, the leading behavior of ATT is a linear function of N. Interestingly, we show that the behavior for ATT of the trapping problem is related to the trapping location, which is in comparison with the phenomenon of trapping on fractal T-graph although both networks exhibit tree structure. Finally, we believe that the methods could open the way to exactly calculate the MFPT and ATT in a wide range of deterministic media.