Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation
  Processes   Network dynamics with point-process-based interactions are of paramount
modeling interest. Unfortunately, most relevant dynamics involve complex graphs
of interactions for which an exact computational treatment is impossible. To
circumvent this difficulty, the replica-mean-field approach focuses on randomly
interacting replicas of the networks of interest. In the limit of an infinite
number of replicas, these networks become analytically tractable under the
so-called "Poisson Hypothesis". However, in most applications, this hypothesis
is only conjectured. Here, we establish the Poisson Hypothesis for a general
class of discrete-time, point-process-based dynamics, that we propose to call
fragmentation-interaction-aggregation processes, and which are introduced in
the present paper. These processes feature a network of nodes, each endowed
with a state governing their random activation. Each activation triggers the
fragmentation of the activated node state and the transmission of interaction
signals to downstream nodes. In turn, the signals received by nodes are
aggregated to their state. Our main contribution is a proof of the Poisson
Hypothesis for the replica-mean-field version of any network in this class. The
proof is obtained by establishing the propagation of asymptotic independence
for state variables in the limit of an infinite number of replicas. Discrete
time Galves-L\"{o}cherbach neural networks are used as a basic instance and
illustration of our analysis.

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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg