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The use of machine learning techniques in classical and quantum systems has led to novel techniques to classify ordered and disordered phases, as well as uncover transition points in critical phenomena. Efforts to extend these methods to…
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large…
It has been recently understood (arXiv:1212.2885, arXiv:1310.4764, arXiv:1410.0605) that for a general class of percolation models on $\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes i.a.\ Bernoulli percolation,…
Percolation refers to the emergence of a giant connected cluster in a disordered system when the number of connections between nodes exceeds a critical value. The percolation phase transitions were believed to be continuous until recently…
In this paper we study the macroscopic conduction properties of large but finite binary networks with conducting bonds. By taking a combination of a spectral and an averaging based approach we derive asymptotic formulae for the conduction…
Highly nonconvex granular particles, such as staples and metal shavings, can form solid-like cohesive structures through geometric entanglement (interlocking). The network structure formed by this entanglement, however, remains largely…
We analyze the stability of the network's giant connected component under impact of adverse events, which we model through the link percolation. Specifically, we quantify the extent to which the largest connected component of a network…
We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove…
A simple but powerful network model with $n$ nodes and $m$ partly overlapping layers is generated as an overlay of independent random graphs $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint…
Hyperuniform many-particle systems, which encompass crystals, quasicrystals and certain exotic disordered systems, exhibit an anomalous suppression of density fluctuations on macroscopic length scales relative to those of conventional…
Recently much attention has been paid to the study of the robustness of interdependent and multiplex networks and, in particular, networks of networks. The robustness of interdependent networks can be evaluated by the size of a mutually…
We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest…
A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a `slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation…
We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying…
We introduce a novel percolation model that generalizes the classical Random Connection Model (RCM) to a random simplicial complex, allowing for a more refined understanding of connectivity and emergence of large-scale structures in random…
We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering,…
Can there be a `Litmus test' for determining the nature of transition in models of percolation? In this paper we argue that the answer is in the affirmative. All one needs to do is to measure the `growth exponent' $\chi$ of the largest…
We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size),…
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a…
We report on the exact treatment of a random-matrix representation of bond percolation model on a square lattice in two dimensions with occupation probability $p$. The percolation problem is mapped onto a random complex matrix composed of…