Related papers: Heterogeneous-k-core versus Bootstrap Percolation …
Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at…
By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\in\mathbb{N}$…
Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of…
Directed and heterogeneous hypergraphs capture directional higher-order interactions with intrinsically asymmetric functional dependencies among nodes. As a result, damage to certain nodes can suppress entire hyperedges, whereas failure of…
The heterogeneous structure implies that a very few nodes may play the critical role in maintaining structural and functional properties of a large-scale network. Identifying these vital nodes is one of the most important tasks in network…
The study of random graphs has become very popular for real-life network modeling such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice $\mathbb Z^d$, $d\ge1$, is a…
We consider the class of {\em separable} $k$-hypergraphs, which can be viewed as uniform analogs of threshold Boolean functions, and the class of {\em equatable} $k$-hypergraphs. We show that every $k$-hypergraph is either separable or…
Real networks often have severe degree heterogeneity, with the maximum, average, and minimum node degrees differing significantly. This paper examines the impact of degree heterogeneity on statistical limits of network data analysis.…
The percolation behavior of aligned rigid rods of length $k$ ($k$-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The $k$-mers, containing $k$ identical units (each…
Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each…
Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding…
Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a…
Core decomposition is a classic technique for discovering densely connected regions in a graph with large range of applications. Formally, a $k$-core is a maximal subgraph where each vertex has at least $k$ neighbors. A natural extension of…
Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of…
Profiling core-periphery structures in networks has attracted significant attention, leading to the development of various methods. Among these, the rich-core method is distinguished for being entirely parameter-free and scalable to large…
Networks may, or may not, be wired to have a core that is both itself densely connected and central in terms of graph distance. In this study we propose a coefficient to measure if the network has such a clear-cut core-periphery dichotomy.…
Complex networks display various types of percolation transitions. We show that the degree distribution and the degree-degree correlation alone are not sufficient to describe diverse percolation critical phenomena. This suggests that a…
In $d > 2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power-laws, thus it is called mixed-order or hybrid type. It has recently…
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the…
Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed…