Do the rich get richer? An empirical analysis of the BitCoin transaction network

The possibility to analyze everyday monetary transactions is limited by the scarcity of available data, as this kind of information is usually considered highly sensitive. Present econophysics models are usually employed on presumed random networks of interacting agents, and only macroscopic properties (e.g. the resulting wealth distribution) are compared to real-world data. In this paper, we analyze BitCoin, which is a novel digital currency system, where the complete list of transactions is publicly available. Using this dataset, we reconstruct the network of transactions, and extract the time and amount of each payment. We analyze the structure of the transaction network by measuring network characteristics over time, such as the degree distribution, degree correlations and clustering. We find that linear preferential attachment drives the growth of the network. We also study the dynamics taking place on the transaction network, i.e. the flow of money. We measure temporal patterns and the wealth accumulation. Investigating the microscopic statistics of money movement, we find that sublinear preferential attachment governs the evolution of the wealth distribution. We report a scaling relation between the degree and wealth associated to individual nodes.Comment: Project website: http://www.vo.elte.hu/bitcoin/; updated after publicatio

An extended formalism for preferential attachment in heterogeneous complex networks

In this paper we present a framework for the extension of the preferential attachment (PA) model to heterogeneous complex networks. We define a class of heterogeneous PA models, where node properties are described by fixed states in an arbitrary metric space, and introduce an affinity function that biases the attachment probabilities of links. We perform an analytical study of the stationary degree distributions in heterogeneous PA networks. We show that their degree densities exhibit a richer scaling behavior than their homogeneous counterparts, and that the power law scaling in the degree distribution is robust in presence of heterogeneity

Assortativity and leadership emergence from anti-preferential attachment in heterogeneous networks

Many real-world networks exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Particularly in social networks, the contribution to the total assortativity varies with degree, featuring a distinctive peak slightly past the average degree. The way traditional models imprint assortativity on top of pre-defined topologies is via degree-preserving link permutations, which however destroy the particular graph's hierarchical traits of clustering. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties and tunable realistic assortativity. In our approach, two distinct populations of nodes are added to an initial network seed: one (the followers) that abides by usual preferential rules, and one (the potential leaders) connecting via anti-preferential attachments, i.e. selecting lower degree nodes for their initial links. The latter nodes come to develop a higher average degree, and convert eventually into the final hubs. Examining the evolution of links in Facebook, we present empirical validation for the connection between the initial anti-preferential attachment and long term high degree. Thus, our work sheds new light on the structure and evolution of social networks

Making new connections towards cooperation in the prisoner's dilemma game

Evolution of cooperation in the prisoner's dilemma game is studied where initially all players are linked via a regular graph, having four neighbors each. Simultaneously with the strategy evolution, players are allowed to make new connections and thus permanently extend their neighborhoods, provided they have been successful in passing their strategy to the opponents. We show that this simple coevolutionary rule shifts the survival barrier of cooperators towards high temptations to defect and results in highly heterogeneous interaction networks with an exponential fit best characterizing their degree distributions. In particular, there exist an optimal maximal degree for the promotion of cooperation, warranting the best exchange of information between influential players.Comment: 6 two-column pages, 7 figures; accepted for publication in Europhysics Letter

A Scale-Free Topology Construction Model for Wireless Sensor Networks

A local-area and energy-efficient (LAEE) evolution model for wireless sensor networks is proposed. The process of topology evolution is divided into two phases. In the first phase, nodes are distributed randomly in a fixed region. In the second phase, according to the spatial structure of wireless sensor networks, topology evolution starts from the sink, grows with an energy-efficient preferential attachment rule in the new node's local-area, and stops until all nodes are connected into network. Both analysis and simulation results show that the degree distribution of LAEE follows the power law. This topology construction model has better tolerance against energy depletion or random failure than other non-scale-free WSN topologies.Comment: 13pages, 3 figure

Sustainable growth in complex networks

Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considers both: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, $k_0$. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: $P(k_{0}) \propto k_{0}^{1-\alpha}$, and $P(k)\propto k^{1-\gamma}$, respectively. For the total number of links as a function of the network size, we find empirically $K(N)\propto N^{\beta}$, where $\beta$ is (at the beginning of the network evolution) between 1.25 and 2, while converging to $\sim 1$ for large $N$. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents $\alpha$, $\beta$, $\gamma$, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks.Comment: 5 pages, 2 figures, 1 tabl