Random copying in space

Random copying is a simple model for population dynamics in the absence of selection, and has been applied to both biological and cultural evolution. In this work, we investigate the effect that spatial structure has on the dynamics. We focus in particular on how a measure of the diversity in the population changes over time. We show that even when the vast majority of a population's history may be well-described by a spatially-unstructured model, spatial structure may nevertheless affect the expected level of diversity seen at a local scale. We demonstrate this phenomenon explicitly by examining the random copying process on small-world networks, and use our results to comment on the use of simple random-copying models in an empirical context.Comment: 26 pages, 11 figures. Based on invited talk at AHRC CECD Conference on "Cultural Evolution in Spatially Structured Populations" at UCL, September 2010. To appear in ACS - Advances in Complex System

Spatially self-organized resilient networks by a distributed cooperative mechanism

The robustness of connectivity and the efficiency of paths are incompatible in many real networks. We propose a self-organization mechanism for incrementally generating onion-like networks with positive degree-degree correlations whose robustness is nearly optimal. As a spatial extension of the generation model based on cooperative copying and adding shortcut, we show that the growing networks become more robust and efficient through enhancing the onion-like topological structure on a space. The reasonable constraint for locating nodes on the perimeter in typical surface growth as a self-propagation does not affect these properties of the tolerance and the path length. Moreover, the robustness can be recovered in the random growth damaged by insistent sequential attacks even without any remedial measures.Comment: 34 pages, 12 figures, 2 table

Randomness and Complexity in Networks

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.Comment: Survey paper based on talk given at the workshop on ``Stochastic Networks and Internet Technology'', Centro di Ricerca Matematica Ennio De Giorgi, Matematica nelle Scienze Naturali e Sociali, Pisa, 17th - 21st September 2007. To appear in proceeding

Quasispecies distribution of Eigen model

We study sharp peak landscapes (SPL) of Eigen model from a new perspective about how the quasispecies distribute in the sequence space. To analyze the distribution more carefully, we bring forth two tools. One tool is the variance of Hamming distance of the sequences at a given generation. It not only offers us a different avenue for accurately locating the error threshold and illustrates how the configuration of the distribution varies with copying fidelity $q$ in the sequence space, but also divides the copying fidelity into three distinct regimes. The other tool is the similarity network of a certain Hamming distance $d_{0}$, by which we can get a visual and in-depth result about how the sequences distribute. We find that there are several local optima around the center (global optimum) in the distribution of the sequences reproduced near the threshold. Furthermore, it is interesting that the distribution of clustering coefficient $C(k)$ follows lognormal distribution and the curve of clustering coefficient $C$ of the network versus $d_{0}$ appears as linear behavior near the threshold.Comment: 13 pages, 6 figure

Adiabatic quantum algorithm for search engine ranking

We propose an adiabatic quantum algorithm for generating a quantum pure state encoding of the PageRank vector, the most widely used tool in ranking the relative importance of internet pages. We present extensive numerical simulations which provide evidence that this algorithm can prepare the quantum PageRank state in a time which, on average, scales polylogarithmically in the number of webpages. We argue that the main topological feature of the underlying web graph allowing for such a scaling is the out-degree distribution. The top ranked $\log(n)$ entries of the quantum PageRank state can then be estimated with a polynomial quantum speedup. Moreover, the quantum PageRank state can be used in "q-sampling" protocols for testing properties of distributions, which require exponentially fewer measurements than all classical schemes designed for the same task. This can be used to decide whether to run a classical update of the PageRank.Comment: 7 pages, 5 figures; closer to published versio

The Degree Distribution of Random k-Trees

A power law degree distribution is established for a graph evolution model based on the graph class of k-trees. This k-tree-based graph process can be viewed as an idealized model that captures some characteristics of the preferential attachment and copying mechanisms that existing evolving graph processes fail to model due to technical obstacles. The result also serves as a further cautionary note reinforcing the point of view that a power law degree distribution should not be regarded as the only important characteristic of a complex network, as has been previously argued

Quantum copying can increase the practically available information

While it is known that copying a quantum system does not increase the amount of information obtainable about the originals, it may increase the amount available in practice, when one is restricted to imperfect measurements. We present a detection scheme which using imperfect detectors, and possibly noisy quantum copying machines (that entangle the copies), allows one to extract more information from an incoming signal, than with the imperfect detectors alone. The case of single-photon detection with noisy, inefficient detectors and copiers (single controlled-NOT gates in this case) is investigated in detail. The improvement in distinguishability between a photon and vacuum is found to occur for a wide range of parameters, and to be quite robust to random noise. The properties that a quantum copying device must have to be useful in this scheme are investigated.Comment: 10 pages, 6 figures, accepted PR