Infinite Schrödinger networks

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and $$p > p_c(G)$$ p > p c ( G ) then there exists a positive constant $$c_p$$ c p such that $$\begin{aligned} \mathbf {P}_p(n \le |K| < \infty ) \le e^{-c_p n} \end{aligned}$$ P p ( n ≤ | K | < ∞ ) ≤ e - c p n for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We deduce the following two corollaries: Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999). For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

Semisupervised Community Detection by Voltage Drops

Many applications show that semisupervised community detection is one of the important topics and has attracted considerable attention in the study of complex network. In this paper, based on notion of voltage drops and discrete potential theory, a simple and fast semisupervised community detection algorithm is proposed. The label propagation through discrete potential transmission is accomplished by using voltage drops. The complexity of the proposal isOV+Efor the sparse network withVvertices andEedges. The obtained voltage value of a vertex can be reflected clearly in the relationship between the vertex and community. The experimental results on four real networks and three benchmarks indicate that the proposed algorithm is effective and flexible. Furthermore, this algorithm is easily applied to graph-based machine learning methods.

Persisting randomness in randomly growing discrete structures: graphs and search trees

The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known. Comment: Official journal file

The Hexagonal Packing Lemma and Discrete Potential Theory

One of the questions concerning the Hexagonal Packing Lemma ([1], [3], [4]) is the rate of convergence of Sn. It was suggested in [3] and [4] that Sn = 0(1/n). In the following we prove this conjecture under the additional condition of some "nice" behaviour of the "circle function".