History-dependent percolation on multiplex networks

The structure of interconnected systems and its impact on the system dynamics is a much-studied cross-disciplinary topic. Although various critical phenomena have been found in different models, study of the connections between different percolation transitions is still lacking. Here we propose a unified framework to study the origins of the discontinuous transitions of the percolation process on interacting networks. The model evolves in generations with the result of the present percolation depending on the previous state, and thus is history-dependent. Both theoretical analysis and Monte Carlo simulations reveal that the nature of the transition remains the same at finite generations but exhibits an abrupt change for the infinite generation. We use brain functional correlation and morphological similarity data to show that our model also provides a general method to explore the network structure and can contribute to many practical applications, such as detecting the abnormal structures of human brain networks.

Determining In-plane and Thru-plane Percolation Thresholds for Carbon Nanotube Thin Films Deposited on Paper Substrates Using Impedance Spectroscopy

ABSTRACTConcentration- and layer-dependent percolation thresholds can be determined for carbon nanotube (CNT) films deposited from aqueous dispersions on paper substrates at both the surface of the deposited film (in-plane) and through the thickness of the paper (thru-plane) using impedance spectroscopy. By analyzing the impedance spectra as a function of the number of layers (solution concentration is constant) or the solution concentration (number of layers is constant), the electrical properties and percolation thresholds for CNT-paper composites can be determined. In-plane measurements show that percolation occurs at 4 layers when 1 mg/mL solution concentration is used. In the thru-plane direction, the films are already percolated at 1 mg/mL concentration, which is confirmed by varying the concentration of the solution used to deposit 1 layer films. A second percolation event happens between 8 and 12 layers due to an increased number of interconnections of CNTs within the paper substrate. The lowest sheet resistance achieved was 100 Ω/□.

Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.