426 research outputs found
Disassortativity of random critical branching trees
Random critical branching trees (CBTs) are generated by the multiplicative
branching process, where the branching number is determined stochastically,
independent of the degree of their ancestor. Here we show analytically that
despite this stochastic independence, there exists the degree-degree
correlation (DDC) in the CBT and it is disassortative. Moreover, the skeletons
of fractal networks, the maximum spanning trees formed by the edge betweenness
centrality, behave similarly to the CBT in the DDC. This analytic solution and
observation support the argument that the fractal scaling in complex networks
originates from the disassortativity in the DDC.Comment: 3 pages, 2 figure
Origin of the mixed-order transition in multiplex networks: the Ashkin-Teller model
Recently, diverse phase transition (PT) types have been obtained in multiplex
networks, such as discontinuous, continuous, and mixed-order PTs. However, they
emerge from individual systems, and there is no theoretical understanding of
such PTs in a single framework. Here, we study a spin model called the
Ashkin-Teller (AT) model in a mono-layer scale-free network; this can be
regarded as a model of two species of Ising spin placed on each layer of a
double-layer network. The four-spin interaction in the AT model represents the
inter-layer interaction in the multiplex network. Diverse PTs emerge depending
on the inter-layer coupling strength and network structure. Especially, we find
that mixed-order PTs occur at the critical end points. The origin of such
behavior is explained in the framework of Landau-Ginzburg theory.Comment: 10 pages, 5 figure
Genuine Non-Self-Averaging and Ultra-Slow Convergence in Gelation
In irreversible aggregation processes droplets or polymers of microscopic
size successively coalesce until a large cluster of macroscopic scale forms.
This gelation transition is widely believed to be self-averaging, meaning that
the order parameter (the relative size of the largest connected cluster)
attains well-defined values upon ensemble averaging with no sample-to-sample
fluctuations in the thermodynamic limit. Here, we report on anomalous gelation
transition types. Depending on the growth rate of the largest clusters, the
gelation transition can show very diverse patterns as a function of the control
parameter, which includes multiple stochastic discontinuous transitions,
genuine non-self-averaging and ultra-slow convergence of the transition point.
Our framework may be helpful in understanding and controlling gelation.Comment: 8 pages, 10 figure
Intrinsic degree-correlations in static model of scale-free networks
We calculate the mean neighboring degree function and
the mean clustering function of vertices with degree as a function
of in finite scale-free random networks through the static model. While
both are independent of when the degree exponent , they show
the crossover behavior for from -independent behavior for
small to -dependent behavior for large . The -dependent behavior
is analytically derived. Such a behavior arises from the prevention of
self-loops and multiple edges between each pair of vertices. The analytic
results are confirmed by numerical simulations. We also compare our results
with those obtained from a growing network model, finding that they behave
differently from each other.Comment: 8 page
Percolation Transitions in Scale-Free Networks under Achlioptas Process
It has been recently shown that the percolation transition is discontinuous
in Erd\H{o}s-R\'enyi networks and square lattices in two dimensions under the
Achlioptas Process (AP). Here, we show that when the structure is highly
heterogeneous as in scale-free networks, a discontinuous transition does not
always occur: a continuous transition is also possible depending on the degree
distribution of the scale-free network. This originates from the competition
between the AP that discourages the formation of a giant component and the
existence of hubs that encourages it. We also estimate the value of the
characteristic degree exponent that separates the two transition types.Comment: 4 pages, 6 figure
Ground state energy of -state Potts model: the minimum modularity
A wide range of interacting systems can be described by complex networks. A
common feature of such networks is that they consist of several communities or
modules, the degree of which may quantified as the \emph{modularity}. However,
even a random uncorrelated network, which has no obvious modular structure, has
a finite modularity due to the quenched disorder. For this reason, the
modularity of a given network is meaningful only when it is compared with that
of a randomized network with the same degree distribution. In this context, it
is important to calculate the modularity of a random uncorrelated network with
an arbitrary degree distribution. The modularity of a random network has been
calculated [Phys. Rev. E \textbf{76}, 015102 (2007)]; however, this was limited
to the case whereby the network was assumed to have only two communities, and
it is evident that the modularity should be calculated in general with communities. Here, we calculate the modularity for communities by
evaluating the ground state energy of the -state Potts Hamiltonian, based on
replica symmetric solutions assuming that the mean degree is large. We found
that the modularity is proportional to regardless of and that only the coefficient depends on . In
particular, when the degree distribution follows a power law, the modularity is
proportional to . Our analytical results are
confirmed by comparison with numerical simulations. Therefore, our results can
be used as reference values for real-world networks.Comment: 14 pages, 4 figure
On continuum modeling of sputter erosion under normal incidence: interplay between nonlocality and nonlinearity
Under specific experimental circumstances, sputter erosion on semiconductor
materials exhibits highly ordered hexagonal dot-like nanostructures. In a
recent attempt to theoretically understand this pattern forming process, Facsko
et al. [Phys. Rev. B 69, 153412 (2004)] suggested a nonlocal, damped
Kuramoto-Sivashinsky equation as a potential candidate for an adequate
continuum model of this self-organizing process. In this study we theoretically
investigate this proposal by (i) formally deriving such a nonlocal equation as
minimal model from balance considerations, (ii) showing that it can be exactly
mapped to a local, damped Kuramoto-Sivashinsky equation, and (iii) inspecting
the consequences of the resulting non-stationary erosion dynamics.Comment: 7 pages, 2 Postscript figures, accepted by Phys. Rev. B corrected
typos, few minor change
Finite-size scaling theory for explosive percolation transitions
The finite-size scaling (FSS) theory for continuous phase transitions has
been useful in determining the critical behavior from the size dependent
behaviors of thermodynamic quantities. When the phase transition is
discontinuous, however, FSS approach has not been well established yet. Here,
we develop a FSS theory for the explosive percolation transition arising in the
Erd\H{o}s and R\'enyi model under the Achlioptas process. A scaling function is
derived based on the observed fact that the derivative of the curve of the
order parameter at the critical point diverges with system size in a
power-law manner, which is different from the conventional one based on the
divergence of the correlation length at . We show that the susceptibility
is also described in the same scaling form. Numerical simulation data for
different system sizes are well collapsed on the respective scaling functions.Comment: 5 pages, 5 figure
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