2,295 research outputs found

### Integral and Series Representations of Riemann's Zeta function, Dirichelet's Eta Function and a Medley of Related Results

Contour integral representations for Riemann's Zeta function and Dirichelet's
Eta (alternating Zeta) function are presented and investigated. These
representations flow naturally from methods developed in the 1800's, but
somehow they do not appear in the standard reference summaries, textbooks or
literature. Using these representations as a basis, alternate derivations of
known series and integral representations for the Zeta and Eta function are
obtained on a unified basis that differs from the textbook approach, and
results are developed that appear to be new.Comment: 26 page

### Detecting degree symmetries in networks

The surrounding of a vertex in a network can be more or less symmetric. We
derive measures of a specific kind of symmetry of a vertex which we call degree
symmetry -- the property that many paths going out from a vertex have
overlapping degree sequences. These measures are evaluated on artificial and
real networks. Specifically we consider vertices in the human metabolic
network. We also measure the average degree-symmetry coefficient for different
classes of real-world network. We find that most studied examples are weakly
positively degree-symmetric. The exceptions are an airport network (having a
negative degree-symmetry coefficient) and one-mode projections of social
affiliation networks that are rather strongly degree-symmetric

### Organizational structure and communication networks in a university environment

The ``six degrees of separation" between any two individuals on Earth has
become emblematic of the 'small world' theme, even though the information
conveyed via a chain of human encounters decays very rapidly with increasing
chain length, and diffusion of information via this process may be very
inefficient in large human organizations. The information flow on a
communication network in a large organization, the University of Oslo, has been
studied by analyzing e-mail records. The records allow for quantification of
communication intensity across organizational levels and between organizational
units (referred to as ``modules"). We find that the number of e-mails messages
within modules scales with module size to the power of $1.29\pm .06$, and the
frequency of communication between individuals decays exponentially with the
number of links required upwards in the organizational hierarchy before they
are connected. Our data also indicates that the number of messages sent by
administrative units is proportional to the number of individuals at lower
levels in the administrative hierarchy, and the ``divergence of information"
within modules is associated with this linear relationship. The observed
scaling is consistent with a hierarchical system in which individuals far apart
in the organization interact little with each other and receive a
disproportionate number of messages from higher levels in the administrative
hierarchy.Comment: 9 pages, 3 figure

### Navigation in a small world with local information

It is commonly known that there exist short paths between vertices in a
network showing the small-world effect. Yet vertices, for example, the
individuals living in society, usually are not able to find the shortest paths,
due to the very serious limit of information. To theoretically study this
issue, here the navigation process of launching messages toward designated
targets is investigated on a variant of the one-dimensional small-world network
(SWN). In the network structure considered, the probability of a shortcut
falling between a pair of nodes is proportional to $r^{-\alpha}$, where $r$ is
the lattice distance between the nodes. When $\alpha =0$, it reduces to the SWN
model with random shortcuts. The system shows the dynamic small-world (SW)
effect, which is different from the well-studied static SW effect. We study the
effective network diameter, the path length as a function of the lattice
distance, and the dynamics. They are controlled by multiple parameters, and we
use data collapse to show that the parameters are correlated. The central
finding is that, in the one-dimensional network studied, the dynamic SW effect
exists for $0\leq \alpha \leq 2$. For each given value of $\alpha$ in this
region, the point that the dynamic SW effect arises is $ML^{\prime}\sim 1$,
where $M$ is the number of useful shortcuts and $L^{\prime}$ is the average
reduced (effective) length of them.Comment: 10 pages, 5 figures, accepted for publication in Physical Review

### Two-dimensional small-world networks: navigation with local information

Navigation process is studied on a variant of the Watts-Strogatz small world
network model embedded on a square lattice. With probability $p$, each vertex
sends out a long range link, and the probability of the other end of this link
falling on a vertex at lattice distance $r$ away decays as $r^{-\alpha}$.
Vertices on the network have knowledge of only their nearest neighbors. In a
navigation process, messages are forwarded to a designated target. For $\alpha
<3$ and $\alpha \neq 2$, a scaling relation is found between the average actual
path length and $pL$, where $L$ is the average length of the additional long
range links. Given $pL>1$, dynamic small world effect is observed, and the
behavior of the scaling function at large enough $pL$ is obtained. At $\alpha
=2$ and 3, this kind of scaling breaks down, and different functions of the
average actual path length are obtained. For $\alpha >3$, the average actual
path length is nearly linear with network size.Comment: Accepted for publication in Phys. Rev.

### Dynamic rewiring in small world networks

We investigate equilibrium properties of small world networks, in which both
connectivity and spin variables are dynamic, using replicated transfer matrices
within the replica symmetric approximation. Population dynamics techniques
allow us to examine order parameters of our system at total equilibrium,
probing both spin- and graph-statistics. Of these, interestingly, the degree
distribution is found to acquire a Poisson-like form (both within and outside
the ordered phase). Comparison with Glauber simulations confirms our results
satisfactorily.Comment: 21 pages, 5 figure

### A novel approach to study realistic navigations on networks

We consider navigation or search schemes on networks which are realistic in
the sense that not all search chains can be completed. We show that the
quantity $\mu = \rho/s_d$, where $s_d$ is the average dynamic shortest distance
and $\rho$ the success rate of completion of a search, is a consistent measure
for the quality of a search strategy. Taking the example of realistic searches
on scale-free networks, we find that $\mu$ scales with the system size $N$ as
$N^{-\delta}$, where $\delta$ decreases as the searching strategy is improved.
This measure is also shown to be sensitive to the distintinguishing
characteristics of networks. In this new approach, a dynamic small world (DSW)
effect is said to exist when $\delta \approx 0$. We show that such a DSW indeed
exists in social networks in which the linking probability is dependent on
social distances.Comment: Text revised, references added; accepted version in Journal of
Statistical Mechanic

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