Curious multisection identities by index factorization

This manuscript introduces a general multisection identity expressed equivalently in terms of infinite double products and/or infinite double series, from which several new product or summation identities involving special functions including Gamma, hyperbolic trigonometric, polygamma, zeta and Jacobi theta functions, are derived. It is shown that a parameterized version of this multisection identity exists, a specialization of which coincides with the standard multisection identity

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

Information Horizons in Networks

We investigate and quantify the interplay between topology and ability to send specific signals in complex networks. We find that in a majority of investigated real-world networks the ability to communicate is favored by the network topology on small distances, but disfavored at larger distances. We further discuss how the ability to locate specific nodes can be improved if information associated to the overall traffic in the network is available.Comment: Submitted top PR

Asymptotic behavior of the Kleinberg model

We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability proportional to r^{-a}) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for ad+1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of a's

Optimization in task--completion networks

We discuss the collective behavior of a network of individuals that receive, process and forward to each other tasks. Given costs they store those tasks in buffers, choosing optimally the frequency at which to check and process the buffer. The individual optimizing strategy of each node determines the aggregate behavior of the network. We find that, under general assumptions, the whole system exhibits coexistence of equilibria and hysteresis.Comment: 18 pages, 3 figures, submitted to JSTA

Link Prediction with Social Vector Clocks

State-of-the-art link prediction utilizes combinations of complex features derived from network panel data. We here show that computationally less expensive features can achieve the same performance in the common scenario in which the data is available as a sequence of interactions. Our features are based on social vector clocks, an adaptation of the vector-clock concept introduced in distributed computing to social interaction networks. In fact, our experiments suggest that by taking into account the order and spacing of interactions, social vector clocks exploit different aspects of link formation so that their combination with previous approaches yields the most accurate predictor to date.Comment: 9 pages, 6 figure

On Compact Routing for the Internet

While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

Mean-field solution of the small-world network model

The small-world network model is a simple model of the structure of social networks, which simultaneously possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either large or small number of shortcuts.Comment: 14 pages, 2 postscript figure