Ising model on a random network with annealed or quenched disorder

We study the equilibrium properties of an Ising model on a disordered random network where the disorder can be quenched or annealed. The network consists of fourfold coordinated sites connected via variable length one-dimensional chains. Our emphasis is on nonuniversal properties and we consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one-dimensional fluctuations arising from the chains. We use analytic methods in the annealed case, and a Monte Carlo simulation for the quenched disorder. Our objective is to study the difference between quenched and annealed results with a broad random distribution of interaction parameters. The former represents a situation where the time scale associated with the randomness is very long and the corresponding degrees of freedom can be viewed as frozen, while the annealed case models the situation where this is not so. We find that the transition temperature and the entropy associated with one-dimensional fluctuations are always higher for quenched disorder than in the annealed case. These differences increase with the strength of the disorder up to a saturating value. We discuss our results in connection to physical systems where a broad distribution of interaction strengths is present.

On the Maximum Number of Linearly Independent Cycles and Paths in a Network

Central to network tomography is the problem of identifiability, the ability to identify internal network characteristics uniquely from end-to-end measurements. This problem is often underconstrained even when internal network characteristics such as link delays are modeled as additive constants. While it is known that the network topology can play a role in determining the extent of identifiability, there is a lack in the fundamental understanding of being able to quantify it for a given network. In this paper, we consider the problem of identifying additive link metrics in an arbitrary undirected network using measurement nodes and establishing paths/cycles between them. For a given placement of measurement nodes, we define and derive the ``link rank'' of the network-the maximum number of linearly independent cycles/paths that may be established between the measurement nodes. We achieve this in linear time. The link rank helps quantify the exact extent of identifiability in a network. We also develop a quadratic time algorithm to compute a set of cycles/paths that achieves the maximum rank.

A Few Case Studies on the Correlation of Particle Network and Its Stability on the Ionic Conductivity of Solid-Liquid Composite Electrolytes

We discuss here the crucial role of the particle network and its stability on the long-range ion transport in solid liquid composite electrolytes. The solid liquid composite electrolytes chosen for the study here comprise nanometer sized silica (SiO2) particles having various surface chemical functionalities dispersed in nonaqueous lithium salt solutions, viz, lithium perchlorate (LiClO4) in two different polyethylene glycol based solvents. These systems constitute representative examples of an independent class of soft matter electrolytes known as ``soggy sand'' electrolytes, which have tremendous potential in diverse electrochemical devices. The oxide additive acts as a heterogeneous dopant creating free charge carriers and enhancing the local ion transport. For long-range transport, however, a stable spanning particle network is needed. Systematic experimental investigations here reveal that the spatial and time dependent characteristics of the particle network in the liquid solution are nontrivial. The network characteristics are predominantly determined by the chemical makeup of the electrolyte components and the chemical interactions between them. It is noteworthy that in this study the steady state macroscopic ionic conductivity and viscosity of the solid liquid composite electrolyte are observed to be greatly determined by the additive oxide surface chemical functionality, solvent chemical composition, and solvent dielectric constant.