326 research outputs found
Stable and unstable attractors in Boolean networks
Boolean networks at the critical point have been a matter of debate for many
years as, e.g., scaling of number of attractor with system size. Recently it
was found that this number scales superpolynomially with system size, contrary
to a common earlier expectation of sublinear scaling. We here point to the fact
that these results are obtained using deterministic parallel update, where a
large fraction of attractors in fact are an artifact of the updating scheme.
This limits the significance of these results for biological systems where
noise is omnipresent. We here take a fresh look at attractors in Boolean
networks with the original motivation of simplified models for biological
systems in mind. We test stability of attractors w.r.t. infinitesimal
deviations from synchronous update and find that most attractors found under
parallel update are artifacts arising from the synchronous clocking mode. The
remaining fraction of attractors are stable against fluctuating response
delays. For this subset of stable attractors we observe sublinear scaling of
the number of attractors with system size.Comment: extended version, additional figur
Does dynamics reflect topology in directed networks?
We present and analyze a topologically induced transition from ordered,
synchronized to disordered dynamics in directed networks of oscillators. The
analysis reveals where in the space of networks this transition occurs and its
underlying mechanisms. If disordered, the dynamics of the units is precisely
determined by the topology of the network and thus characteristic for it. We
develop a method to predict the disordered dynamics from topology. The results
suggest a new route towards understanding how the precise dynamics of the units
of a directed network may encode information about its topology.Comment: 7 pages, 4 figures, Europhysics Letters, accepte
Topology of biological networks and reliability of information processing
Biological systems rely on robust internal information processing: Survival
depends on highly reproducible dynamics of regulatory processes. Biological
information processing elements, however, are intrinsically noisy (genetic
switches, neurons, etc.). Such noise poses severe stability problems to system
behavior as it tends to desynchronize system dynamics (e.g. via fluctuating
response or transmission time of the elements). Synchronicity in parallel
information processing is not readily sustained in the absence of a central
clock. Here we analyze the influence of topology on synchronicity in networks
of autonomous noisy elements. In numerical and analytical studies we find a
clear distinction between non-reliable and reliable dynamical attractors,
depending on the topology of the circuit. In the reliable cases, synchronicity
is sustained, while in the unreliable scenario, fluctuating responses of single
elements can gradually desynchronize the system, leading to non-reproducible
behavior. We find that the fraction of reliable dynamical attractors strongly
correlates with the underlying circuitry. Our model suggests that the observed
motif structure of biological signaling networks is shaped by the biological
requirement for reproducibility of attractors.Comment: 7 pages, 7 figure
Scale-free networks are not robust under neutral evolution
Recently it has been shown that a large variety of different networks have
power-law (scale-free) distributions of connectivities. We investigate the
robustness of such a distribution in discrete threshold networks under neutral
evolution. The guiding principle for this is robustness in the resulting
phenotype. The numerical results show that a power-law distribution is not
stable under such an evolution, and the network approaches a homogeneous form
where the overall distribution of connectivities is given by a Poisson
distribution.Comment: Submitted for publicatio
Self-organized critical neural networks
A mechanism for self-organization of the degree of connectivity in model
neural networks is studied. Network connectivity is regulated locally on the
basis of an order parameter of the global dynamics which is estimated from an
observable at the single synapse level. This principle is studied in a
two-dimensional neural network with randomly wired asymmetric weights. In this
class of networks, network connectivity is closely related to a phase
transition between ordered and disordered dynamics. A slow topology change is
imposed on the network through a local rewiring rule motivated by
activity-dependent synaptic development: Neighbor neurons whose activity is
correlated, on average develop a new connection while uncorrelated neighbors
tend to disconnect. As a result, robust self-organization of the network
towards the order disorder transition occurs. Convergence is independent of
initial conditions, robust against thermal noise, and does not require fine
tuning of parameters.Comment: 5 pages RevTeX, 7 figures PostScrip
Topological Evolution of Dynamical Networks: Global Criticality from Local Dynamics
We evolve network topology of an asymmetrically connected threshold network
by a simple local rewiring rule: quiet nodes grow links, active nodes lose
links. This leads to convergence of the average connectivity of the network
towards the critical value in the limit of large system size . How
this principle could generate self-organization in natural complex systems is
discussed for two examples: neural networks and regulatory networks in the
genome.Comment: 4 pages RevTeX, 4 figures PostScript, revised versio
Characterizing the network topology of the energy landscapes of atomic clusters
By dividing potential energy landscapes into basins of attractions
surrounding minima and linking those basins that are connected by transition
state valleys, a network description of energy landscapes naturally arises.
These networks are characterized in detail for a series of small Lennard-Jones
clusters and show behaviour characteristic of small-world and scale-free
networks. However, unlike many such networks, this topology cannot reflect the
rules governing the dynamics of network growth, because they are static spatial
networks. Instead, the heterogeneity in the networks stems from differences in
the potential energy of the minima, and hence the hyperareas of their
associated basins of attraction. The low-energy minima with large basins of
attraction act as hubs in the network.Comparisons to randomized networks with
the same degree distribution reveals structuring in the networks that reflects
their spatial embedding.Comment: 14 pages, 11 figure
Regulatory networks and connected components of the neutral space
The functioning of a living cell is largely determined by the structure of
its regulatory network, comprising non-linear interactions between regulatory
genes. An important factor for the stability and evolvability of such
regulatory systems is neutrality - typically a large number of alternative
network structures give rise to the necessary dynamics. Here we study the
discretized regulatory dynamics of the yeast cell cycle [Li et al., PNAS, 2004]
and the set of networks capable of reproducing it, which we call functional.
Among these, the empirical yeast wildtype network is close to optimal with
respect to sparse wiring. Under point mutations, which establish or delete
single interactions, the neutral space of functional networks is fragmented
into 4.7 * 10^8 components. One of the smaller ones contains the wildtype
network. On average, functional networks reachable from the wildtype by
mutations are sparser, have higher noise resilience and fewer fixed point
attractors as compared with networks outside of this wildtype component.Comment: 6 pages, 5 figure
Partitioning and modularity of graphs with arbitrary degree distribution
We solve the graph bi-partitioning problem in dense graphs with arbitrary
degree distribution using the replica method. We find the cut-size to scale
universally with . In contrast, earlier results studying the problem in
graphs with a Poissonian degree distribution had found a scaling with ^1/2
[Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also
generalize to the problem of q-partitioning. They can be used to find the
expected modularity Q [Newman and Grivan, Phys. Rev. E, 69, 2004] of random
graphs and allow for the assessment of statistical significance of the output
of community detection algorithms.Comment: Revised version including new plots and improved discussion of some
mathematical detail
Growing Scale-Free Networks with Small World Behavior
In the context of growing networks, we introduce a simple dynamical model
that unifies the generic features of real networks: scale-free distribution of
degree and the small world effect. While the average shortest path length
increases logartihmically as in random networks, the clustering coefficient
assumes a large value independent of system size. We derive expressions for the
clustering coefficient in two limiting cases: random (C ~ (ln N)^2 / N) and
highly clustered (C = 5/6) scale-free networks.Comment: 4 pages, 4 figure
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