1,005 research outputs found

### Metastable state en route to traveling-wave synchronization state

The Kuramoto model with mixed signs of couplings is known to produce a
traveling-wave synchronized state. Here, we consider an abrupt synchronization
transition from the incoherent state to the traveling-wave state through a
long-lasting metastable state with large fluctuations. Our explanation of the
metastability is that the dynamic flow remains within a limited region of phase
space and circulates through a few active states bounded by saddle and stable
fixed points. This complex flow generates a long-lasting critical behavior, a
signature of a hybrid phase transition. We show that the long-lasting period
can be controlled by varying the density of inhibitory/excitatory interactions.
We discuss a potential application of this transition behavior to the recovery
process of human consciousness

### Nonequilibrium phase transition by directed Potts particles

We introduce an interface model with q-fold symmetry to study the
nonequilibrium phase transition (NPT) from an active to an inactive state at
the bottom layer. In the model, q different species of particles are deposited
or are evaporated according to a dynamic rule, which includes the interaction
between neighboring particles within the same layer. The NPT is classified
according to the number of species q. For q=1 and 2, the NPT is characterized
by directed percolation, and the directed Ising class, respectively. For $q \ge
3$, the NPT occurs at finite critical probability p_c, and appears to be
independent of q; the $q=\infty$ case is related to the Edwards-Wilkinson
interface dynamics.Comment: 4 pages, latex, 5 PS figure file

### Two Types of Discontinuous Percolation Transitions in Cluster Merging Processes

Percolation is a paradigmatic model in disordered systems and has been
applied to various natural phenomena. The percolation transition is known as
one of the most robust continuous transitions. However, recent extensive
studies have revealed that a few models exhibit a discontinuous percolation
transition (DPT) in cluster merging processes. Unlike the case of continuous
transitions, understanding the nature of discontinuous phase transitions
requires a detailed study of the system at hand, which has not been undertaken
yet for DPTs. Here we examine the cluster size distribution immediately before
an abrupt increase in the order parameter of DPT models and find that DPTs
induced by cluster merging kinetics can be classified into two types. Moreover,
the type of DPT can be determined by the key characteristic of whether the
cluster kinetic rule is homogeneous with respect to the cluster sizes. We also
establish the necessary conditions for each type of DPT, which can be used
effectively when the discontinuity of the order parameter is ambiguous, as in
the explosive percolation model.Comment: 9 pages, 6 figure

### Robustness of the in-degree exponent for the world-wide web

We consider a stochastic model for directed scale-free networks following
power-laws in the degree distributions in both incoming and outgoing
directions. In our model, the number of vertices grow geometrically with time
with growth rate p. At each time step, (i) each newly introduced vertex is
connected to a constant number of already existing vertices with the
probability linearly proportional to the in-degree of a selected vertex, and
(ii) each existing vertex updates its outgoing edges through a stochastic
multiplicative process with mean growth rate of outgoing edges g and variance
$\sigma^2$. Using both analytic treatment and numerical simulations, we show
that while the out-degree exponent $\gamma_{\rm out}$ depends on the
parameters, the in-degree exponent $\gamma_{\rm in}$ has two distinct values,
$\gamma_{\rm in}=2$ for $p > g$ and 1 for $p < g$, independent of different
parameters values. The latter case has logarithmic correction to the power-law.
Since the vertex growth rate p is larger than the degree growth rate g for the
world-wide web (www) nowadays, the in-degree exponent appears robust as
$\gamma_{\rm in}=2$ for the www

### Identification of essential and functionally moduled genes through the microarray assay

Identification of essential genes is one of the ultimate goals of drug
designs. Here we introduce an {\it in silico} method to select essential genes
through the microarray assay. We construct a graph of genes, called the gene
transcription network, based on the Pearson correlation coefficient of the
microarray expression level. Links are connected between genes following the
order of the pair-wise correlation coefficients. We find that there exist two
meaningful fractions of links connected, $p_m$ and $p_s$, where the number of
clusters becomes maximum and the connectivity distribution follows a power law,
respectively. Interestingly, one of clusters at $p_m$ contains a high density
of essential genes having almost the same functionality. Thus the deletion of
all genes belonging to that cluster can lead to lethal inviable mutant
efficiently. Such an essential cluster can be identified in a self-organized
way. Once we measure the connectivity of each gene at $p_s$. Then using the
property that the essential genes are likely to have more connectivity, we can
identify the essential cluster by finding the one having the largest mean
connectivity per gene at $p_m$.Comment: 21 pages, 8 figures, 1 table, LaTe

### Critical behavior of a two-step contagion model with multiple seeds

A two-step contagion model with a single seed serves as a cornerstone for
understanding the critical behaviors and underlying mechanism of discontinuous
percolation transitions induced by cascade dynamics. When the contagion spreads
from a single seed, a cluster of infected and recovered nodes grows without any
cluster merging process. However, when the contagion starts from multiple seeds
of $O(N)$ where $N$ is the system size, a node weakened by a seed can be
infected more easily when it is in contact with another node infected by a
different pathogen seed. This contagion process can be viewed as a cluster
merging process in a percolation model. Here, we show analytically and
numerically that when the density of infectious seeds is relatively small but
$O(1)$, the epidemic transition is hybrid, exhibiting both continuous and
discontinuous behavior, whereas when it is sufficiently large and reaches a
critical point, the transition becomes continuous. We determine the full set of
critical exponents describing the hybrid and the continuous transitions. Their
critical behaviors differ from those in the single-seed case.Comment: 10 pages, 15 figure

### Avalanche size distribution in the Toom interface

We present numerical data of the height-height correlation function and of
the avalanche size distribution for the three dimensional Toom interface. The
height-height correlation function behaves samely as the interfacial
fluctuation width, which diverges logarithmically with space and time for both
unbiased and biased cases. The avalanche size defined by the number of changing
sites caused by a single noise process, exhibits an exponentially decaying
distribution, which is in contrast to power-law distributions appearing in
typical self-organized critical phenomena. We also generalize the Toom model
into arbitrary dimensions.Comment: 16pages, latex, SNUTP 93-7

### Hysteresis and criticality in hybrid percolation transitions

Phase transitions (PTs) are generally classified into second-order and
first-order transitions, each exhibiting different intrinsic properties. For
instance, a first-order transition exhibits latent heat and hysteresis when a
control parameter is increased and then decreased across a transition point,
whereas a second-order transition does not. Recently, hybrid percolation
transitions (HPTs) are issued in diverse complex systems, in which the features
of first-order and second-order PTs occur at the same transition point. Thus,
the question whether hysteresis appears in an HPT arises. Herein, we
investigate this fundamental question with a so-called restricted
Erd\H{o}s--R\'enyi random network model, in which a cluster fragmentation
process is additionally proposed. The hysteresis curve of the order parameter
was obtained. Depending on when the reverse process is initiated, the shapes of
hysteresis curves change, and the critical behavior of the HPT is conserved
throughout the forward and reverse processes

### Avalanche dynamics driven by adaptive rewirings in complex networks

We introduce a toy model displaying the avalanche dynamics of failure in
scale-free networks. In the model, the network growth is based on the
Barab\'asi and Albert model and each node is assigned a capacity or tolerance,
which is constant irrespective of node index. The degree of each node increases
over time. When the degree of a node exceeds its capacity, it fails and each
link connected to it is is rewired to other unconnected nodes by following the
preferential attachment rule. Such a rewiring edge may trigger another failure.
This dynamic process can occur successively, and it exhibits a self-organized
critical behavior in which the avalanche size distribution follows a power law.
The associated exponent is $\tau \approx 2.6(1)$. The entire system breaks down
when any rewired edges cannot locate target nodes: the time at which this
occurs is referred to as the breaking time. We obtain the breaking time as a
function of the capacity. Moreover, using extreme value statistics, we
determine the distribution function of the breaking time.Comment: 4 pages, 5 figure

### Fast Algorithm for Relaxation Processes in Big-data Systems

Relaxation processes driven by a Laplacian matrix can be found in many
real-world big-data systems, for example, in search engines on the
World-Wide-Web and the dynamic load balancing protocols in mesh networks. To
numerically implement such processes, a fast-running algorithm for the
calculation of the pseudo inverse of the Laplacian matrix is essential. Here we
propose an algorithm which computes fast and efficiently the pseudo inverse of
Markov chain generator matrices satisfying the detailed-balance condition, a
general class of matrices including the Laplacian. The algorithm utilizes the
renormalization of the Gaussian integral. In addition to its applicability to a
wide range of problems, the algorithm outperforms other algorithms in its
ability to compute within a manageable computing time arbitrary elements of the
pseudo inverse of a matrix of size millions by millions. Therefore our
algorithm can be used very widely in analyzing the relaxation processes
occurring on large-scale networked systems.Comment: 11 pages, 3 figure

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