Network Topology of an Experimental Futures Exchange

Many systems of different nature exhibit scale free behaviors. Economic systems with power law distribution in the wealth is one of the examples. To better understand the working behind the complexity, we undertook an empirical study measuring the interactions between market participants. A Web server was setup to administer the exchange of futures contracts whose liquidation prices were coupled to event outcomes. After free registration, participants started trading to compete for the money prizes upon maturity of the futures contracts at the end of the experiment. The evolving `cash' flow network was reconstructed from the transactions between players. We show that the network topology is hierarchical, disassortative and scale-free with a power law exponent of 1.02+-0.09 in the degree distribution. The small-world property emerged early in the experiment while the number of participants was still small. We also show power law distributions of the net incomes and inter-transaction time intervals. Big winners and losers are associated with high degree, high betweenness centrality, low clustering coefficient and low degree-correlation. We identify communities in the network as groups of the like-minded. The distribution of the community sizes is shown to be power-law distributed with an exponent of 1.19+-0.16.Comment: 6 pages, 12 figure

The effect of aging on network structure

In network evolution, the effect of aging is universal: in scientific collaboration network, scientists have a finite time span of being active; in movie actors network, once popular stars are retiring from stage; devices on the Internet may become outmoded with techniques developing so rapidly. Here we find in citation networks that this effect can be represented by an exponential decay factor, $e^{-\beta \tau}$, where $\tau$ is the node age, while other evolving networks (the Internet for instance) may have different types of aging, for example, a power-law decay factor, which is also studied and compared. It has been found that as soon as such a factor is introduced to the Barabasi-Albert Scale-Free model, the network will be significantly transformed. The network will be clustered even with infinitely large size, and the clustering coefficient varies greatly with the intensity of the aging effect, i.e. it increases linearly with $\beta$ for small values of $\beta$ and decays exponentially for large values of $\beta$. At the same time, the aging effect may also result in a hierarchical structure and a disassortative degree-degree correlation. Generally the aging effect will increase the average distance between nodes, but the result depends on the type of the decay factor. The network appears like a one-dimensional chain when exponential decay is chosen, but with power-law decay, a transformation process is observed, i.e., from a small-world network to a hypercubic lattice, and to a one-dimensional chain finally. The disparities observed for different choices of the decay factor, in clustering, average node distance and probably other aspects not yet identified, are believed to bear significant meaning on empirical data acquisition.Comment: 8 pages, 9 figures,V2, accepted for publication in Phys. Rev.

Mean-field theory for clustering coefficients in Barabasi-Albert networks

We applied a mean field approach to study clustering coefficients in Barabasi-Albert networks. We found that the local clustering in BA networks depends on the node degree. Analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees. Clustering coefficient of a whole network calculated from our approach perfectly fits numerical data.Comment: 8 pages, 3 figure

Graph theoretic analysis of protein interaction networks of eukaryotes

Thanks to recent progress in high-throughput experimental techniques, the datasets of large-scale protein interactions of prototypical multicellular species, the nematode worm Caenorhabditis elegans and the fruit fly Drosophila melanogaster, have been assayed. The datasets are obtained mainly by using the yeast hybrid method, which contains false-positive and false-negative simultaneously. Accordingly, while it is desirable to test such datasets through further wet experiments, here we invoke recent developed network theory to test such high throughput datasets in a simple way. Based on the fact that the key biological processes indispensable to maintaining life are universal across eukaryotic species, and the comparison of structural properties of the protein interaction networks (PINs) of the two species with those of the yeast PIN, we find that while the worm and the yeast PIN datasets exhibit similar structural properties, the current fly dataset, though most comprehensively screened ever, does not reflect generic structural properties correctly as it is. The modularity is suppressed and the connectivity correlation is lacking. Addition of interlogs to the current fly dataset increases the modularity and enhances the occurrence of triangular motifs as well. The connectivity correlation function of the fly, however, remains distinct under such interlogs addition, for which we present a possible scenario through an in silico modeling.Comment: 7 pages, 6 figures, 2 table

Exact scaling properties of a hierarchical network model

We report on exact results for the degree $K$, the diameter $D$, the clustering coefficient $C$, and the betweenness centrality $B$ of a hierarchical network model with a replication factor $M$. Such quantities are calculated exactly with the help of recursion relations. Using the results, we show that (i) the degree distribution follows a power law $P_K \sim K^{-\gamma}$ with $\gamma = 1+\ln M /\ln (M-1)$, (ii) the diameter grows logarithmically as $D \sim \ln N$ with the number of nodes $N$, (iii) the clustering coefficient of each node is inversely proportional to its degree, $C \propto 1/K$, and the average clustering coefficient is nonzero in the infinite $N$ limit, and (iv) the betweenness centrality distribution follows a power law $P_B \sim B^{-2}$. We discuss a classification scheme of scale-free networks into the universality class with the clustering property and the betweenness centrality distribution.Comment: 4 page

Effect of correlations on network controllability

A dynamical system is controllable if by imposing appropriate external signals on a subset of its nodes, it can be driven from any initial state to any desired state in finite time. Here we study the impact of various network characteristics on the minimal number of driver nodes required to control a network. We find that clustering and modularity have no discernible impact, but the symmetries of the underlying matching problem can produce linear, quadratic or no dependence on degree correlation coefficients, depending on the nature of the underlying correlations. The results are supported by numerical simulations and help narrow the observed gap between the predicted and the observed number of driver nodes in real networks

Modularity map of the network of human cell differentiation

Cell differentiation in multicellular organisms is a complex process whose mechanism can be understood by a reductionist approach, in which the individual processes that control the generation of different cell types are identified. Alternatively, a large scale approach in search of different organizational features of the growth stages promises to reveal its modular global structure with the goal of discovering previously unknown relations between cell types. Here we sort and analyze a large set of scattered data to construct the network of human cell differentiation (NHCD) based on cell types (nodes) and differentiation steps (links) from the fertilized egg to a crying baby. We discover a dynamical law of critical branching, which reveals a fractal regularity in the modular organization of the network, and allows us to observe the network at different scales. The emerging picture clearly identifies clusters of cell types following a hierarchical organization, ranging from sub-modules to super-modules of specialized tissues and organs on varying scales. This discovery will allow one to treat the development of a particular cell function in the context of the complex network of human development as a whole. Our results point to an integrated large-scale view of the network of cell types systematically revealing ties between previously unrelated domains in organ functions.Comment: 32 pages, 7 figure

Spatial stochastic resonance in 1D Ising systems

The 1D Ising model is analytically studied in a spatially periodic and oscillatory external magnetic field using the transfer-matrix method. For low enough magnetic field intensities the correlation between the external magnetic field and the response in magnetization presents a maximum for a given temperature. The phenomenon can be interpreted as a resonance phenomenon induced by the stochastic heatbath. This novel "spatial stochastic resonance" has a different origin from the classical stochastic resonance phenomenon.Comment: REVTex, 5 pages, 3 figure