Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: From flip processes to dynamical systems on graphons   We introduce a class of random graph processes, which we call \emph{flip
processes}. Each such process is given by a \emph{rule} which is a function
$\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled
$k$-vertex graphs into itself ($k$ is fixed). The process starts with a given
$n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling
$k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced
graph $F:=G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(F)$. This class contains
several previously studied processes including the Erd\H{o}s--R\'enyi random
graph process and the triangle removal process. Actually, our definition of
flip processes is more general, in that $\mathcal{R}(F)$ is a probability
distribution on $\mathcal{H}_k$, thus allowing randomised replacements.
  Given a flip process with a rule $\mathcal{R}$, we construct time-indexed
trajectories $\Phi:\mathcal{W}_0\times [0,\infty)\rightarrow\mathcal{W}_0$ in
the space of graphons. We prove that for any $T > 0$ starting with a large
finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm, with high
probability the flip process will stay in a thin sausage around the trajectory
$(\Phi(W_0,t))_{t=0}^T$ (after rescaling the time by the square of the order of
the graph).
  These graphon trajectories are then studied from the perspective of dynamical
systems. Among others topics, we study continuity properties of these
trajectories with respect to time and initial graphon, existence and stability
of fixed points and speed of convergence (whenever the infinite time limit
exists). We give an example of a flip process with a periodic trajectory.

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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: cs