612 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
Spectral densities of scale-free networks
The spectral densities of the weighted Laplacian, random walk and weighted
adjacency matrices associated with a random complex network are studied using
the replica method. The link weights are parametrized by a weight exponent
. Explicit results are obtained for scale-free networks in the limit of
large mean degree after the thermodynamic limit, for arbitrary degree exponent
and .Comment: 14 pages, two 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
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