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