170 research outputs found
Strong-majority bootstrap percolation on regular graphs with low dissemination threshold
International audienceConsider the following model of strong-majority bootstrap percolation on a graph. Let r ≥ 1 be some integer, and p ∈ [0, 1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v) + r)/2 of its neighbours are active. Given any arbitrarily small p > 0 and any integer r, we construct a family of d = d(p, r)-regular graphs such that with high probability all vertices become active in the end. In particular, the case r = 1 answers a question and disproves a conjecture of Rapaport, Suchan, Todinca and Verstraëte [38]
Aspects of random graphs
The present report aims at giving a survey of my work since the end of my PhD thesis "Spectral Methods for Reconstruction Problems". Since then I focussed on the analysis of properties of different models of random graphs as well as their connection to real-world networks. This report's goal is to capture these problems in a common framework. The very last chapter of this thesis about results in bootstrap percolation is different in the sense that the given graph is deterministic and only the decision of being active for each vertex is probabilistic; since the proof techniques resemble very much results on random graphs, we decided to include them as well. We start with an overview of the five random graph models, and with the description of bootstrap percolation corresponding to the last chapter. Some properties of these models are then analyzed in the different parts of this thesis
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Hipsters on Networks: How a Small Group of Individuals Can Lead to an Anti-Establishment Majority
The spread of opinions, memes, diseases, and "alternative facts" in a
population depends both on the details of the spreading process and on the
structure of the social and communication networks on which they spread. In
this paper, we explore how \textit{anti-establishment} nodes (e.g.,
\textit{hipsters}) influence the spreading dynamics of two competing products.
We consider a model in which spreading follows a deterministic rule for
updating node states (which describe which product has been adopted) in which
an adjustable fraction of the nodes in a network are hipsters,
who choose to adopt the product that they believe is the less popular of the
two. The remaining nodes are conformists, who choose which product to adopt by
considering which products their immediate neighbors have adopted. We simulate
our model on both synthetic and real networks, and we show that the hipsters
have a major effect on the final fraction of people who adopt each product:
even when only one of the two products exists at the beginning of the
simulations, a very small fraction of hipsters in a network can still cause the
other product to eventually become the more popular one. To account for this
behavior, we construct an approximation for the steady-state adoption fraction
on -regular trees in the limit of few hipsters. Additionally, our
simulations demonstrate that a time delay in the knowledge of the
product distribution in a population, as compared to immediate knowledge of
product adoption among nearest neighbors, can have a large effect on the final
distribution of product adoptions. Our simple model and analysis may help shed
light on the road to success for anti-establishment choices in elections, as
such success can arise rather generically in our model from a small number of
anti-establishment individuals and ordinary processes of social influence on
normal individuals.Comment: Extensively revised, with much new analysis and numerics The abstract
on arXiv is a shortened version of the full abstract because of space limit
Dynamical Phase Transitions in Graph Cellular Automata
Discrete dynamical systems can exhibit complex behaviour from the iterative
application of straightforward local rules. A famous example are cellular
automata whose global dynamics are notoriously challenging to analyze. To
address this, we relax the regular connectivity grid of cellular automata to a
random graph, which gives the class of graph cellular automata. Using the
dynamical cavity method (DCM) and its backtracking version (BDCM), we show that
this relaxation allows us to derive asymptotically exact analytical results on
the global dynamics of these systems on sparse random graphs. Concretely, we
showcase the results on a specific subclass of graph cellular automata with
``conforming non-conformist'' update rules, which exhibit dynamics akin to
opinion formation. Such rules update a node's state according to the majority
within their own neighbourhood. In cases where the majority leads only by a
small margin over the minority, nodes may exhibit non-conformist behaviour.
Instead of following the majority, they either maintain their own state, switch
it, or follow the minority. For configurations with different initial biases
towards one state we identify sharp dynamical phase transitions in terms of the
convergence speed and attractor types. From the perspective of opinion dynamics
this answers when consensus will emerge and when two opinions coexist almost
indefinitely.Comment: 15 page
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