80 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Generating Polynomials of Exponential Random Graphs
The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).
In the past, determining when a probability distribution has strong negative dependence has proven to be difficult ([Pem00, BBL09]). The negative dependence of a probability distribution is characterized by properties of its corresponding generating polynomial ([BBL09]). This thesis bridges the theory of exponential random graphs with the geometry of their generating polynomials, namely, when and how they satisfy the stable or Lorentzian properties ([Wag09, BBL09, BH20, AGV21]). We provide necessary and sufficient conditions as well as full characterizations of the parameter space for when this model has a stable or Lorentzian generating polynomial. This is done using a well-developed dictionary between probability distributions and their corresponding multiaffine generating polynomials.
In particular, we characterize when the generating polynomial of a random graph model with a large symmetry group is irreducible. We assert that the edge parameter of the exponential random graph model does not affect stability and that the triangle and k-star parameters are necessarily related if the model is stable or Lorentzian. We also provide full Lorentzian and stable characterizations for the model on K3 and a Lorentzian characterization for specializations of the model on K4
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Large deviations for the isoperimetric constant in 2D percolation
Isoperimetric profile describes the minimal boundary size of a set with a
prescribed volume. Itai Benjamini conjectured that the isoperimetric profile of
the giant component in supercritical percolation experiences an averaging
effect and satisfies the law of large numbers. This conjecture was settled by
Biskup-Louidor-Procaccia-Rosenthal for 2D percolation, and later resolved by
Gold for higher-dimensional lattices.
However, more refined properties of the isoperimetric profile, such as
fluctuations and large deviations, remain unknown. In this paper, we determine
the large deviation probabilities of the isoperimetric constant in 2D
supercritical percolation, answering the question by
Biskup-Louidor-Procaccia-Rosenthal. Interestingly, while the large deviation
probability is of surface order in the entire upper tail regime, a phase
transition occurs in the lower tail regime, exhibiting both surface and volume
order large deviations.Comment: 35 pages, 2 figure
Bootstrap Percolation, Connectivity, and Graph Distance
Bootstrap Percolation is a process defined on a graph which begins with an
initial set of infected vertices. In each subsequent round, an uninfected
vertex becomes infected if it is adjacent to at least previously infected
vertices. If an initially infected set of vertices, , begins a process in
which every vertex of the graph eventually becomes infected, then we say that
percolates. In this paper we investigate bootstrap percolation as it
relates to graph distance and connectivity. We find a sufficient condition for
the existence of cardinality 2 percolating sets in diameter 2 graphs when . We also investigate connections between connectivity and bootstrap
percolation and lower and upper bounds on the number of rounds to percolation
in terms of invariants related to graph distance.Comment: 18 pages, 11 figure
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 277, GIScience 2023, Complete Volume
LIPIcs, Volume 277, GIScience 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
12th International Conference on Geographic Information Science: GIScience 2023, September 12–15, 2023, Leeds, UK
No abstract available
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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