The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs
Random graph models with limited choice have been studied extensively with the goal of
understanding the mechanism of the emergence of the giant component. One of the …
understanding the mechanism of the emergence of the giant component. One of the …
Very fast construction of bounded-degree spanning graphs via the semi-random graph process
Semi-random processes involve an adaptive decision-maker, whose goal is to achieve
some predetermined objective in an online randomized environment. They have algorithmic …
some predetermined objective in an online randomized environment. They have algorithmic …
[PDF][PDF] Random graphs: Theory and applications from nature to society to the brain
M Kang, Z Petrasek - Internationale Mathematische Nachrichten, 2014 - math.tugraz.at
The theory of random graphs deals with asymptotic properties of graphs equipped with a
certain probability distribution; for example, it studies how the component structure of a …
certain probability distribution; for example, it studies how the component structure of a …
The evolution of subcritical Achlioptas processes
O Riordan, L Warnke - Random Structures & Algorithms, 2015 - Wiley Online Library
In Achlioptas processes, starting from an empty graph, in each step two potential edges are
chosen uniformly at random, and using some rule one of them is selected and added to the …
chosen uniformly at random, and using some rule one of them is selected and added to the …
The phase transition in bounded-size Achlioptas processes
O Riordan, L Warnke - arXiv preprint arXiv:1704.08714, 2017 - arxiv.org
Perhaps the best understood phase transition is that in the component structure of the
uniform random graph process introduced by Erd\H {o} s and R\'enyi around 1960. Since the …
uniform random graph process introduced by Erd\H {o} s and R\'enyi around 1960. Since the …
Aggregation models with limited choice and the multiplicative coalescent
The percolation phase transition and the mechanism of the emergence of the giant
component through the critical scaling window for random graph models, has been a topic of …
component through the critical scaling window for random graph models, has been a topic of …
Convergence of Achlioptas processes via differential equations with unique solutions
O Riordan, L Warnke - Combinatorics, Probability and Computing, 2016 - cambridge.org
In Achlioptas processes, starting from an empty graph, in each step two potential edges are
chosen uniformly at random, and using some rule one of them is selected and added to the …
chosen uniformly at random, and using some rule one of them is selected and added to the …
[PDF][PDF] Functional Central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs
S Bhamidi, A Budhiraja, A Sakanaveeti - arXiv preprint arXiv:2412.13672, 2024 - arxiv.org
We study inhomogeneous random graphs with a finite type space. For a natural
generalization of the model as a dynamic network-valued process, the paper achieves two …
generalization of the model as a dynamic network-valued process, the paper achieves two …
Bounded-size rules: The barely subcritical regime
Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited
choice along with randomness in the evolution of the system. Typically one starts with the …
choice along with randomness in the evolution of the system. Typically one starts with the …
Random k‐SAT and the power of two choices
W Perkins - Random Structures & Algorithms, 2015 - Wiley Online Library
We study an Achlioptas‐process version of the random k‐SAT process: a bounded number
of k‐clauses are drawn uniformly at random at each step, and exactly one added to the …
of k‐clauses are drawn uniformly at random at each step, and exactly one added to the …