Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamics of Nonlinear Random Walks on Complex Networks   In this paper we study the dynamics of nonlinear random walks. While typical
random walks on networks consist of standard Markov chains whose static
transition probabilities dictate the flow of random walkers through the
network, nonlinear random walks consist of nonlinear Markov chains whose
transition probabilities change in time depending on the current state of the
system. This framework allows us to model more complex flows through networks
that may depend on the current system state. For instance, under humanitarian
or capitalistic direction, resource flow between institutions may be diverted
preferentially to poorer or wealthier institutions, respectively. Importantly,
the nonlinearity in this framework gives rise to richer dynamical behavior than
occurs in linear random walks. Here we study these dynamics that arise in
weakly and strongly nonlinear regimes in a family of nonlinear random walks
where random walkers are biased either towards (positive bias) or away from
(negative bias) nodes that currently have more random walkers. In the weakly
nonlinear regime we prove the existence and uniqueness of a stable stationary
state fixed point provided that the network structure is primitive that is
analogous to the stationary distribution of a typical (linear) random walk. We
also present an asymptotic analysis that allows us to approximate the
stationary state fixed point in the weakly nonlinear regime. We then turn our
attention to the strongly nonlinear regime. For negative bias we characterize a
period-doubling bifurcation where the stationary state fixed point loses
stability and gives rise to a periodic orbit below a critical value. For
positive bias we investigate the emergence of multistability of several stable
stationary state fixed points.

6
The categories of this paper are: astro-ph
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The categories of this paper are: eess
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph