4,500 research outputs found
Symmetry in Critical Random Boolean Network Dynamics
Using Boolean networks as prototypical examples, the role of symmetry in the
dynamics of heterogeneous complex systems is explored. We show that symmetry of
the dynamics, especially in critical states, is a controlling feature that can
be used both to greatly simplify analysis and to characterize different types
of dynamics. Symmetry in Boolean networks is found by determining the frequency
at which the various Boolean output functions occur. There are classes of
functions that consist of Boolean functions that behave similarly. These
classes are orbits of the controlling symmetry group. We find that the symmetry
that controls the critical random Boolean networks is expressed through the
frequency by which output functions are utilized by nodes that remain active on
dynamical attractors. This symmetry preserves canalization, a form of network
robustness. We compare it to a different symmetry known to control the dynamics
of an evolutionary process that allows Boolean networks to organize into a
critical state. Our results demonstrate the usefulness and power of using the
symmetry of the behavior of the nodes to characterize complex network dynamics,
and introduce a novel approach to the analysis of heterogeneous complex
systems
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
ON THE CAYLEY GRAPHS OF BOOLEAN FUNCTIONS
A Boolean function is a function and we denote the set of all -variable Boolean functions by . For the vector is called the Walsh spectrum of , where , where is the vector space of dimension over the two-element field . In this paper, we shall consider the Cayley graph associated with a Boolean function . We shall also find a complete characterization of the bent Boolean functions of order and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set .
The effect of negative feedback loops on the dynamics of Boolean networks
Feedback loops in a dynamic network play an important role in determining the
dynamics of that network. Through a computational study, in this paper we show
that networks with fewer independent negative feedback loops tend to exhibit
more regular behavior than those with more negative loops. To be precise, we
study the relationship between the number of independent feedback loops and the
number and length of the limit cycles in the phase space of dynamic Boolean
networks. We show that, as the number of independent negative feedback loops
increases, the number (length) of limit cycles tends to decrease (increase).
These conclusions are consistent with the fact, for certain natural biological
networks, that they on the one hand exhibit generally regular behavior and on
the other hand show less negative feedback loops than randomized networks with
the same numbers of nodes and connectivity
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