Triadic motifs and dyadic self-organization in the World Trade Network

In self-organizing networks, topology and dynamics coevolve in a continuous feedback, without exogenous driving. The World Trade Network (WTN) is one of the few empirically well documented examples of self-organizing networks: its topology strongly depends on the GDP of world countries, which in turn depends on the structure of trade. Therefore, understanding which are the key topological properties of the WTN that deviate from randomness provides direct empirical information about the structural effects of self-organization. Here, using an analytical pattern-detection method that we have recently proposed, we study the occurrence of triadic "motifs" (subgraphs of three vertices) in the WTN between 1950 and 2000. We find that, unlike other properties, motifs are not explained by only the in- and out-degree sequences. By contrast, they are completely explained if also the numbers of reciprocal edges are taken into account. This implies that the self-organization process underlying the evolution of the WTN is almost completely encoded into the dyadic structure, which strongly depends on reciprocity.Comment: 12 pages, 3 figures; Best Paper Award at the 6th International Conference on Self-Organizing Systems, Delft, The Netherlands, 15-16/03/201

Structure and Evolution of the World Trade Network

The \emph{World Trade Web} (WTW), the network defined by the international import/export trade relationships, has been recently shown to display some important topological properties which are tightly related to the Gross Domestic Product of world countries. While our previous analysis focused on the static, undirected version of the WTW, here we address its full evolving, directed description. This is accomplished by exploiting the peculiar reciprocity structure of the WTW to recover the directed nature of international trade channels, and by studying the temporal dependence of the parameters describing the WTW topology.Comment: Proceedings of the "First Bonzenfreies Colloquium on Market Dynamics and Quantitative Economics", Alessandria (ITALY) September 9-10, 2004. One of the three awarded talk

Experimental evidence for the interplay between individual wealth and transaction network

We conduct a market experiment with human agents in order to explore the structure of transaction networks and to study the dynamics of wealth accumulation. The experiment is carried out on our platform for 97 days with 2,095 effective participants and 16,936 times of transactions. From these data, the hybrid distribution (log-normal bulk and power-law tail) in the wealth is observed and we demonstrate that the transaction networks in our market are always scale-free and disassortative even for those with the size of the order of few hundred. We further discover that the individual wealth is correlated with its degree by a power-law function which allows us to relate the exponent of the transaction network degree distribution to the Pareto index in wealth distribution.Comment: 6 pages, 7 figure

The scale-free topology of market investments

We propose a network description of large market investments, where both stocks and shareholders are represented as vertices connected by weighted links corresponding to shareholdings. In this framework, the in-degree ($k_{in}$) and the sum of incoming link weights ($v$) of an investor correspond to the number of assets held (\emph{portfolio diversification}) and to the invested wealth (\emph{portfolio volume}) respectively. An empirical analysis of three different real markets reveals that the distributions of both $k_{in}$ and $v$ display power-law tails with exponents $\gamma$ and $\alpha$. Moreover, we find that $k_{in}$ scales as a power-law function of $v$ with an exponent $\beta$. Remarkably, despite the values of $\alpha$, $\beta$ and $\gamma$ differ across the three markets, they are always governed by the scaling relation $\beta=(1-\alpha)/(1-\gamma)$. We show that these empirical findings can be reproduced by a recent model relating the emergence of scale-free networks to an underlying Paretian distribution of `hidden' vertex properties.Comment: Final version accepted for publication on Physica

An ensemble approach to the analysis of weighted networks

We present a new approach to the calculation of measures in weighted networks, based on the translation of a weighted network into an ensemble of edges. This leads to a straightforward generalization of any measure defined on unweighted networks, such as the average degree of the nearest neighbours, the clustering coefficient, the `betweenness', the distance between two nodes and the diameter of a network. All these measures are well established for unweighted networks but have hitherto proven difficult to define for weighted networks. Further to introducing this approach we demonstrate its advantages by applying the clustering coefficient constructed in this way to two real-world weighted networks.Comment: 4 pages 3 figure

Null Models of Economic Networks: The Case of the World Trade Web

In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the World Trade Web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models.Comment: 39 pages, 46 figures, 2 table

Emergence of weight-topology correlations in complex scale-free networks

Different weighted scale-free networks show weights-topology correlations indicated by the non linear scaling of the node strength with node connectivity. In this paper we show that networks with and without weight-topology correlations can emerge from the same simple growth dynamics of the node connectivities and of the link weights. A weighted fitness network is introduced in which both nodes and links are assigned intrinsic fitness. This model can show a local dependence of the weight-topology correlations and can undergo a phase transition to a state in which the network is dominated by few links which acquire a finite fraction of the total weight of the network.Comment: (4 pages,3 figures

Self-organized network evolution coupled to extremal dynamics

The interplay between topology and dynamics in complex networks is a fundamental but widely unexplored problem. Here, we study this phenomenon on a prototype model in which the network is shaped by a dynamical variable. We couple the dynamics of the Bak-Sneppen evolution model with the rules of the so-called fitness network model for establishing the topology of a network; each vertex is assigned a fitness, and the vertex with minimum fitness and its neighbours are updated in each iteration. At the same time, the links between the updated vertices and all other vertices are drawn anew with a fitness-dependent connection probability. We show analytically and numerically that the system self-organizes to a non-trivial state that differs from what is obtained when the two processes are decoupled. A power-law decay of dynamical and topological quantities above a threshold emerges spontaneously, as well as a feedback between different dynamical regimes and the underlying correlation and percolation properties of the network.Comment: Accepted version. Supplementary information at http://www.nature.com/nphys/journal/v3/n11/suppinfo/nphys729_S1.htm

The International Trade Network

Bilateral trade relationships in the international level between pairs of countries in the world give rise to the notion of the International Trade Network (ITN). This network has attracted the attention of network researchers as it serves as an excellent example of the weighted networks, the link weight being defined as a measure of the volume of trade between two countries. In this paper we analyzed the international trade data for 53 years and studied in detail the variations of different network related quantities associated with the ITN. Our observation is that the ITN has also a scale invariant structure like many other real-world networks.Comment: 9 pages, 7 figure

A complementary view on the growth of directory trees

Trees are a special sub-class of networks with unique properties, such as the level distribution which has often been overlooked. We analyse a general tree growth model proposed by Klemm {\em et. al.} (2005) to explain the growth of user-generated directory structures in computers. The model has a single parameter $q$ which interpolates between preferential attachment and random growth. Our analysis results in three contributions: First, we propose a more efficient estimation method for $q$ based on the degree distribution, which is one specific representation of the model. Next, we introduce the concept of a level distribution and analytically solve the model for this representation. This allows for an alternative and independent measure of $q$. We argue that, to capture real growth processes, the $q$ estimations from the degree and the level distributions should coincide. Thus, we finally apply both representations to validate the model with synthetically generated tree structures, as well as with collected data of user directories. In the case of real directory structures, we show that $q$ measured from the level distribution are incompatible with $q$ measured from the degree distribution. In contrast to this, we find perfect agreement in the case of simulated data. Thus, we conclude that the model is an incomplete description of the growth of real directory structures as it fails to reproduce the level distribution. This insight can be generalised to point out the importance of the level distribution for modeling tree growth.Comment: 16 pages, 7 figure