Related papers: Large-scale structures in random graphs
A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for…
We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in…
Hierarchical structure and repetition are prevalent in graphs originating from nature or engineering. These patterns can be represented by a class of parametric-structure graphs, which are defined by templates that generate structure by way…
We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the…
The obstruction set for graphs with knotless embeddings is not known, but a recent paper of Goldberg, Mattman, and Naimi indicates that it is quite large. Almost all known obstructions fall into four Triangle-Y families and they ask if…
Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the…
We prove a theorem computing the number of solutions to a system of equations which is generic subject to the sparsity conditions embodied in a graph. We apply this theorem to games obeying graphical models and to extensive-form games. We…
It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to…
Identifying the hidden organizational principles and relevant structures of networks representing complex physical systems is fundamental to understand their properties. To this aim, uncovering the structures involving a network's prominent…
We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then…
Finding groups of connected individuals in large graphs with tens of thousands or more nodes has received considerable attention in academic research. In this paper, we analyze three main issues with respect to the recent influx of papers…
Graph rewrite formalisms are a powerful approach to modeling complex molecular systems. They capture the intrinsic concurrency of molecular interactions, thereby enabling a formal notion of mechanism (a partially ordered set of events) that…
Temporal graphs provide a useful model for many real-world networks. Unfortunately the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which…
Graph vertices are often organized into groups that seem to live fairly independently of the rest of the graph, with which they share but a few edges, whereas the relationships between group members are stronger, as shown by the large…
Deep learning models have been shown to be vulnerable to adversarial attacks. This perception led to analyzing deep learning models not only from the perspective of their performance measures but also their robustness to certain types of…
We consider the problem of enumerating all instances of a given pattern graph in a large data graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges in the data graph, $k=O(1)$…
In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a…
The presence of hierarchy in many real-world networks is not yet fully explained. Complex interaction networks are often coarse-grain models of vast modular networks, where tightly connected subgraphs are agglomerated into nodes for…
A Random Graph is a random object which take its values in the space of graphs. We take advantage of the expressibility of graphs in order to model the uncertainty about the existence of causal relationships within a given set of variables.…
The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs…