1,724 research outputs found
Negative circuits and sustained oscillations in asynchronous automata networks
The biologist Ren\'e Thomas conjectured, twenty years ago, that the presence
of a negative feedback circuit in the interaction graph of a dynamical system
is a necessary condition for this system to produce sustained oscillations. In
this paper, we state and prove this conjecture for asynchronous automata
networks, a class of discrete dynamical systems extensively used to model the
behaviors of gene networks. As a corollary, we obtain the following fixed point
theorem: given a product of finite intervals of integers, and a map
from to itself, if the interaction graph associated with has no
negative circuit, then has at least one fixed point
Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks
We are interested in fixed points in Boolean networks, {\em i.e.} functions
from to itself. We define the subnetworks of as the
restrictions of to the subcubes of , and we characterizes a
class of Boolean networks satisfying the following property:
Every subnetwork of has a unique fixed point if and only if has no
subnetwork in . This characterization generalizes the fixed point
theorem of Shih and Dong, which asserts that if for every in
there is no directed cycle in the directed graph whose the adjacency matrix is
the discrete Jacobian matrix of evaluated at point , then has a
unique fixed point. Then, denoting by (resp. )
the networks whose the interaction graph is a positive (resp. negative) cycle,
we show that the non-expansive networks of are exactly the
networks of ; and for the class of
non-expansive networks we get a "dichotomization" of the previous forbidden
subnetwork theorem: Every subnetwork of has at most (resp. at least) one
fixed point if and only if has no subnetworks in (resp.
) subnetwork. Finally, we prove that if is a conjunctive
network then every subnetwork of has at most one fixed point if and only if
has no subnetwork in .Comment: 40 page
Random curves on surfaces induced from the Laplacian determinant
We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
We show that, for a sequence of graphs conformally approximating the
surface, the measures on CRSFs of converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
We set the framework for the study of these probability measures and their
scaling limits and state some of their properties
Random two-component spanning forests
We study random two-component spanning forests (SFs) of finite graphs,
giving formulas for the first and second moments of the sizes of the
components, vertex-inclusion probabilities for one or two vertices, and the
probability that an edge separates the components. We compute the limit of
these quantities when the graph tends to an infinite periodic graph in
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