Related papers: Electric currents in infinite networks
Entanglement, one of the clearest manifestations of non-classical physics, holds significant promise for technological applications such as more secure communications and faster computations. In this paper we explore the use of…
The bounds on the statuses of the nodes in a finite graph established by Entringer, Jackson, and Snyder are extended herein so that they apply to the nodes in a transfinite graph of a certain kind.
In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects,…
The function of a real network depends not only on the reliability of its own components, but is affected also by the simultaneous operation of other real networks coupled with it. Robustness of systems composed of interdependent network…
We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph…
Many physical, biological, and social phenomena can be described by cascades taking place on a network. Often, the activity can be empirically observed, but not the underlying network of interactions. In this paper we offer three…
Modeling power transmission networks is an important area of research with applications such as vulnerability analysis, study of cascading failures, and location of measurement devices. Graph-theoretic approaches have been widely used to…
This article is a mini-review about electrical current flows in networks from the perspective of statistical physics. We briefly discuss analytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic…
Many situations in physics, biology, and engineering consist of the transport of some physical quantity through a network of narrow channels. The ability of a network to transport such a quantity in every direction can be described by the…
Power grids are undergoing major changes from a few large producers to smart grids build upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture, when dynamic nodes can achieve synchronization to a…
Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi_e$ indexed by all the edges in the tree. We assign the resistance $m^d \xi_e$ to each edge $e$ at…
The infinite AC ladder network can exhibit unexpected behavior. Entangling the topology brings even more surprises, found by direct numerical investigation. We consider a simple modification of the ladder topology and explain the numerical…
This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator,…
Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree…
We develop a theory of holomorphic differentials on a certain class of non-compact Riemann surfaces obtained by opening infinitely many nodes.
Distributions of the resilience of transport networks are studied numerically, in particular the large-deviation tails. Thus, not only typical quantities like average or variance but the distributions over the (almost) full support can be…
In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids.…
Given a resistive electrical network, we would like to determine whether all the resistances (edges) in the network are working, and if not, identify which edge (or edges) are faulty. To make this determination, we are allowed to measure…
We give identities for the voltage and resistance functions on a metrized graph to show how these functions behave under any edge deletion/contraction and the identification of any two vertices. This leads to explicit versions of Rayleigh's…
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…