In this paper, a directed network model for world-wide web is presented. The out-degree of the added nodes are supposed to be scale-free and its mean value is $m$. This model exhibits small-world effect, which means the corresponding networks are of very short average distance and highly large clustering coefficient. More interesting, the in-degree distribution obeys the power-law form with the exponent $γ=2+1/m$, depending on the average out-degree. This finding is supported by the empirical data, which has not been emphasized by the previous studies on directed networks.
The analysis of small recurrent substructures, so called network motifs, has become a standard tool of complex network science to unveil the design principles underlying the structure of empirical networks. In many natural systems network nodes are associated with an intrinsic property according to which they can be ordered and compared against each other. Here, we expand standard motif analysis to be able to capture the hierarchical structure in such ordered networks. Our new approach is based on the identification of all ordered 3-node substructures and the visualization of their significanc…
A search technique locating network modules, i.e., internally densely connected groups of nodes in directed networks is introduced by extending the Clique Percolation Method originally proposed for undirected networks. After giving a suitable definition for directed modules we investigate their percolation transition in the Erdos-Renyi graph both analytically and numerically. We also analyse four real-world directed networks, including Google's own webpages, an email network, a word association graph and the transcriptional regulatory network of the yeast Saccharomyces cerevisiae. The obtained…
PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q -> 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q -> 1 it can always be derived from the in-degree distrib…
We reveal a hierarchical, multilayer organization of finite components -- i.e., tendrils and tubes -- around the giant connected components in directed networks and propose efficient algorithms allowing one to uncover the entire organization of key real-world directed networks, such as the World Wide Web, the neural network of \emph{Caenorhabditis elegans}, and others. With increasing damage, the giant components decrease in size while the number and size of tendril layers increase, enhancing the susceptibility of the networks to damage.
Wikipedia is a popular web-based encyclopedia edited freely and collaboratively by its users. In this paper we present an analysis of Wikipedias in several languages as complex networks. The hyperlinks pointing from one Wikipedia article to another are treated as directed links while the articles represent the nodes of the network. We show that many network characteristics are common to different language versions of Wikipedia, such as their degree distributions, growth, topology, reciprocity, clustering, assortativity, path lengths and triad significance profiles. These regularities, found in…
In past ten years, modern societies developed enormous communication and social networks. Their classification and information retrieval processing become a formidable task for the society. Due to the rapid growth of World Wide Web, social and communication networks, new mathematical methods have been invented to characterize the properties of these networks on a more detailed and precise level. Various search engines are essentially using such methods. It is highly important to develop new tools to classify and rank enormous amount of network information in a way adapted to internal network s…
A model for directed networks is proposed and power laws for their in-degree and/or out-degree distributions are derived from the model. It is based on the Barabasi-Albert model and contains two parameters. The parameters serve as regulatory factors for the contributions of new nodes through in- and out-degrees. The model allows a newly added link to connect directly two nodes already present in the network. Such a link is called an inner link, while a link attached to a new node is called an outer link. Using relationships between inner and outer links, we investigate power laws for in- and o…
In this note we study the bilateral merchandise trade flows between 186 countries over the 1948-2005 period using data from the International Monetary Fund. We use Pajek to identify network structure and behavior across thresholds and over time. In particular, we focus on the evolution of trade "islands" in the a world trade network in which countries are linked with directed edges weighted according to fraction of total dollars sent from one country to another. We find mixed evidence for globalization.
The arrangement of network nodes in hyperbolic spaces has become a widely studied problem, motivated by numerous results suggesting the existence of hidden metric spaces behind the structure of complex networks. Although several methods have already been developed for the hyperbolic embedding of undirected networks, approaches able to deal with directed networks are still in their infancy. Here, we propose a framework based on the dimension reduction of proximity matrices reflecting the network topology, coupled with a general conversion method transforming Euclidean node coordinates into hype…
Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and Email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing …
We study random graph models for directed acyclic graphs, an important class of networks that includes citation networks, food webs, and feed-forward neural networks among others. We propose two specific models, roughly analogous to the fixed edge number and fixed edge probability variants of traditional undirected random graphs. We calculate a number of properties of these models, including particularly the probability of connection between a given pair of vertices, and compare the results with real-world acyclic network data finding that theory and measurements agree surprisingly well -- far…
The study of triangles in graphs is a standard tool in network analysis, leading to measures such as the \emph{transitivity}, i.e., the fraction of paths of length $2$ that participate in triangles. Real-world networks are often directed, and it can be difficult to "measure" this network structure meaningfully. We propose a collection of \emph{directed closure values} for measuring triangles in directed graphs in a way that is analogous to transitivity in an undirected graph. Our study of these values reveals much information about directed triadic closure. For instance, we immediately see tha…
Directed networks are ubiquitous and are necessary to represent complex systems with asymmetric interactions---from food webs to the World Wide Web. Despite the importance of edge direction for detecting local and community structure, it has been disregarded in studying a basic type of global diversity in networks: the tendency of nodes with similar numbers of edges to connect. This tendency, called assortativity, affects crucial structural and dynamic properties of real-world networks, such as error tolerance or epidemic spreading. Here we demonstrate that edge direction has profound effects …
We study a two states opinion formation model driven by PageRank node influence and report an extensive numerical study on how PageRank affects collective opinion formations in large-scale empirical directed networks. In our model the opinion of a node can be updated by the sum of its neighbor nodes' opinions weighted by the node influence of the neighbor nodes at each step. We consider PageRank probability and its sublinear power as node influence measures and investigate evolution of opinion under various conditions. First, we observe that all networks reach steady state opinion after a cert…
Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi l…
The availability of data from many different sources and fields of science has made it possible to map out an increasing number of networks of contacts and interactions. However, quantifying how reliable these data are remains an open problem. From Biology to Sociology and Economy, the identification of false and missing positives has become a problem that calls for a solution. In this work we extend one of newest, best performing models -due to Guimera and Sales-Pardo in 2009- to directed networks. The new methodology is able to identify missing and spurious directed interactions, which rende…
Dynamic social opinion models have been widely studied on undirected networks, and most of them are based on spin interaction models that produce a consensus. In reality, however, many networks such as Twitter and the World Wide Web are directed and are composed of both unidirectional and bidirectional links. Moreover, from choosing a coffee brand to deciding who to vote for in an election, two or more competing opinions often coexist. In response to this ubiquity of directed networks and the coexistence of two or more opinions in decision-making situations, we study a non-consensus opinion mo…
Determining the relative importance of nodes in directed networks is important in, for example, ranking websites, publications, and sports teams, and for understanding signal flows in systems biology. A prevailing centrality measure in this respect is the PageRank. In this work, we focus on another class of centrality derived from the Laplacian of the network. We extend the Laplacian-based centrality, which has mainly been applied to strongly connected networks, to the case of general directed networks such that we can quantitatively compare arbitrary nodes. Toward this end, we adopt the idea …
Preferential attachment is the most popular explanation for the emergence of scaling behavior in the World Wide Web, but this explanation has been challenged by the global information hypothesis, the existence of linear preference and the emergence of new big internet companies in the real world. We notice that most websites have an obvious feature that their pages are organized as a tree (namely hidden tree) and hence propose a new model that introduces a hidden tree structure into the Erdős-Ré}yi model by adding a new rule: when one node connects to another, it should also connect to all nod…
As part of their efforts to consistently and reliably identify and preserve the archival records of their organizations, archivists are trying to address the issue of appraising and collecting World Wide Web documents. This paper discusses the approach taken at the Stanford Linear Accelerator Center to collect and preserve documentation of the first World Wide Web site in the United States.
When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analyzing directed networks. Although many notions of a directed module have been proposed, no consensus has been reached. This lack of consensus results partly because there might exist distinct types of modules in a single directed network, whereas most previous studies focused on an independent criterion for modules. To address this issue, we propose a generic notion of the so-called truss s…
We performed a large-scale crawl of the World Wide Web, covering 6.9 Million domains and 57 Million subdomains, including all high-traffic sites of the Internet. We present a study of the correlations found between quantities measuring the structural relevance of each node in the network (the in- and out-degree, the local clustering coefficient, the first-neighbor in-degree and the Alexa rank). We find that some of these properties show strong correlation effects and that the dependencies occurring out of these correlations follow power laws not only for the averages, but also for the boundari…
Directed acyclic graphs are a fundamental class of networks that includes citation networks, food webs, and family trees, among others. Here we define a random graph model for directed acyclic graphs and give solutions for a number of the model's properties, including connection probabilities and component sizes, as well as a fast algorithm for simulating the model on a computer. We compare the predictions of the model to a real-world network of citations between physics papers and find surprisingly good agreement, suggesting that the structure of the real network may be quite well described b…
For many real-world networks only a small "sampled" version of the original network may be investigated; those results are then used to draw conclusions about the actual system. Variants of breadth-first search (BFS) sampling, which are based on epidemic processes, are widely used. Although it is well established that BFS sampling fails, in most cases, to capture the IN-component(s) of directed networks, a description of the effects of BFS sampling on other topological properties are all but absent from the literature. To systematically study the effects of sampling biases on directed networks…
We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysing a set of year-by-year data covering the time interval 1950-2000, we show that the dynamics of all GDP values and the evolution of the WTW (trade flow and topology) are tightly coupled. The probability that two countries are connected depends on their GDP values, supporting rece…
Countries globally trade with tons of waste materials every year, some of which are highly hazardous. This trade admits a network representation of the world-wide waste web, with countries as vertices and flows as directed weighted edges. Here we investigate the main properties of this network by tracking 108 categories of wastes interchanged in the period 2001-2019. Although, most of the hazardous waste was traded between developed nations, a disproportionate asymmetry existed in the flow from developed to developing countries. Using a dynamical model, we simulate how waste stress propagates …
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…
The feed-forward relationship naturally observed in time-dependent processes and in a diverse number of real systems -such as some food-webs and electronic and neural wiring- can be described in terms of so-called directed acyclic graphs (DAGs). An important ingredient of the analysis of such networks is a proper comparison of their observed architecture against an ensemble of randomized graphs, thereby quantifying the {\em randomness} of the real systems with respect to suitable null models. This approximation is particularly relevant when the finite size and/or large connectivity of real sys…
We study the game of go from a complex network perspective. We construct a directed network using a suitable definition of tactical moves including local patterns, and study this network for different datasets of professional tournaments and amateur games. The move distribution follows Zipf's law and the network is scale free, with statistical peculiarities different from other real directed networks, such as e. g. the World Wide Web. These specificities reflect in the outcome of ranking algorithms applied to it. The fine study of the eigenvalues and eigenvectors of matrices used by the rankin…
