609 research outputs found
Multiple scale theory of topology driven pattern on directed networks
Dynamical processes on networks are currently being considered in different
domains of cross-disciplinary interest. Reaction-diffusion systems hosted on
directed graphs are in particular relevant for their widespread applications,
from neuroscience, to computer networks and traffic systems. Due to the
peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points
can turn unstable, on a directed support, because of the topology of the
network, a phenomenon which cannot be induced on undirected graphs. A linear
analysis can be performed to single out the conditions that underly the
instability. The complete characterization of the patterns, which are
eventually attained beyond the linear regime of exponential growth, calls
instead for a full non linear treatment. By performing a multiple time scale
perturbative calculation, we here derive an effective equation for the non
linear evolution of the amplitude of the most unstable mode, close to the
threshold of criticality. This is a Stuart-Landau equation whose complex
coefficients appear to depend on the topological features of the embedding
directed graph. The theory proves adequate versus simulations, as confirmed by
operating with a paradigmatic reaction-diffusion model
Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs
A normal form approximation for the evolution of a reaction-diffusion system
hosted on a directed graph is derived, in the vicinity of a supercritical Hopf
bifurcation. Weak diffusive couplings are assumed to hold between adjacent
nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE)
is obtained, whose coefficients depend on the parameters of the model and the
topological characteristics of the underlying network. The CGLE enables one to
probe the stability of the synchronous oscillating solution, as displayed by
the reaction-diffusion system above Hopf bifurcation. More specifically,
conditions can be worked out for the onset of the symmetry breaking instability
that eventually destroys the uniform oscillatory state. Numerical tests
performed for the Brusselator model confirm the validity of the proposed
theoretical scheme. Patterns recorded for the CGLE resemble closely those
recovered upon integration of the original Brussellator dynamics
The second will be first: competition on directed networks
Multiple sinks competition is investigated for a walker diffusing on directed
complex networks. The asymmetry of the imposed spatial support makes the system
non transitive. As a consequence, it is always possible to identify a suitable
location for the second absorbing sink that screens at most the flux of agents
directed against the first trap, whose position has been preliminarily
assigned. The degree of mutual competition between pairs of nodes is
analytically quantified through apt indicators that build on the topological
characteristics of the hosting graph. Moreover, the positioning of the second
trap can be chosen so as to minimize, at the same time the probability of being
in turn shaded by a thirdly added trap. Supervised placing of absorbing traps
on a asymmetric disordered and complex graph is hence possible, as follows a
robust optimization protocol. This latter is here discussed and successfully
tested against synthetic data
Hopping in the crowd to unveil network topology
We introduce a nonlinear operator to model diffusion on a complex undirected
network under crowded conditions. We show that the asymptotic distribution of
diffusing agents is a nonlinear function of the nodes' degree and saturates to
a constant value for sufficiently large connectivities, at variance with
standard diffusion in the absence of excluded-volume effects. Building on this
observation, we define and solve an inverse problem, aimed at reconstructing
the a priori unknown connectivity distribution. The method gathers all the
necessary information by repeating a limited number of independent measurements
of the asymptotic density at a single node that can be chosen randomly. The
technique is successfully tested against both synthetic and real data, and
shown to estimate with great accuracy also the total number of nodes
Global topological control for synchronized dynamics on networks
A general scheme is proposed and tested to control the symmetry breaking
instability of a homogeneous solution of a spatially extended multispecies
model, defined on a network. The inherent discreteness of the space makes it
possible to act on the topology of the inter-nodes contacts to achieve the
desired degree of stabilization, without altering the dynamical parameters of
the model. Both symmetric and asymmetric couplings are considered. In this
latter setting the web of contacts is assumed to be balanced, for the
homogeneous equilibrium to exist. The performance of the proposed method are
assessed, assuming the Complex Ginzburg-Landau equation as a reference model.
In this case, the implemented control allows one to stabilize the synchronous
limit cycle, hence time-dependent, uniform solution. A system of coupled real
Ginzburg-Landau equations is also investigated to obtain the topological
stabilization of a homogeneous and constant fixed point
Stochastic Turing Patterns on a Network
The process of stochastic Turing instability on a network is discussed for a
specific case study, the stochastic Brusselator model. The system is shown to
spontaneously differentiate into activator-rich and activator-poor nodes,
outside the region of parameters classically deputed to the deterministic
Turing instability. This phenomenon, as revealed by direct stochastic
simulations, is explained analytically, and eventually traced back to the
finite size corrections stemming from the inherent graininess of the
scrutinized medium.Comment: The movies referred to in the paper are provided upon request. Please
send your requests to Duccio Fanelli ([email protected]) or Francesca
Di Patti ([email protected]
Optimal search strategies on complex multi-linked networks
In this paper we consider the problem of optimal search strategies on multi-linked networks, i.e. graphs whose nodes are endowed with several independent sets of links. We focus preliminarily on agents randomly hopping along the links of a graph, with the additional possibility of performing non-local hops to randomly chosen nodes with a given probability. We show that an optimal combination of the two jump rules exists that maximises the efficiency of target search, the optimum reflecting the topology of the network. We then generalize our results to multi-linked networks with an arbitrary number of mutually interfering link sets
Can a microscopic stochastic model explain the emergence of pain cycles in patients?
A stochastic model is here introduced to investigate the molecular mechanisms
which trigger the perception of pain. The action of analgesic drug compounds is
discussed in a dynamical context, where the competition with inactive species
is explicitly accounted for. Finite size effects inevitably perturb the
mean-field dynamics: Oscillations in the amount of bound receptors
spontaneously manifest, driven by the noise which is intrinsic to the system
under scrutiny. These effects are investigated both numerically, via stochastic
simulations and analytically, through a large-size expansion. The claim that
our findings could provide a consistent interpretative framework to explain the
emergence of cyclic behaviors in response to analgesic treatments, is
substantiated.Comment: J. Stat. Mech. (Proceedings UPON2008
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