Vinko Zlatić

Wikipedias: collaborative web-based encyclopedias as complex networks

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 the ensemble of Wikipedias in different languages and of different sizes, point to the existence of a unique growth process. We also compare Wikipedias to other previously studied networks.

Random hypergraphs and their applications

In the last few years we have witnessed the emergence, primarily in online communities, of new types of social networks that require for their representation more complex graph structures than have been employed in the past. One example is the folksonomy, a tripartite structure of users, resources, and tags-labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database. Here we propose a mathematical model of such tripartite structures that represents them as random hypergraphs. We show that it is possible to calculate many properties of this model exactly in the limit of large network size and we compare the results against observations of a real folksonomy, that of the online photography website Flickr. We show that in some cases the model matches the properties of the observed network well, while in others there are significant differences, which we find to be attributable to the practice of multiple tagging, i.e., the application by a single user of many tags to one resource or one tag to many resources.

Hypergraph topological quantities for tagged social networks

Recent years have witnessed the emergence of a new class of social networks, which require us to move beyond previously employed representations of complex graph structures. A notable example is that of the folksonomy, an online process where users collaboratively employ tags to resources to impart structure to an otherwise undifferentiated database. In a recent paper, we proposed a mathematical model that represents these structures as tripartite hypergraphs and defined basic topological quantities of interest. In this paper, we extend our model by defining additional quantities such as edge distributions, vertex similarity and correlations as well as clustering. We then empirically measure these quantities on two real life folksonomies, the popular online photo sharing site Flickr and the bookmarking site CiteULike. We find that these systems share similar qualitative features with the majority of complex networks that have been previously studied. We propose that the quantities and methodology described here can be used as a standard tool in measuring the structure of tagged networks.

Robustness and assortativity for diffusion-like processes in scale-free networks

By analysing the diffusive dynamics of epidemics and of distress in complex networks, we study the effect of the assortativity on the robustness of the networks. We first determine by spectral analysis the thresholds above which epidemics/failures can spread; we then calculate the slowest diffusional times. Our results shows that disassortative networks exhibit a higher epidemiological threshold and are therefore easier to immunize, while in assortative networks there is a longer time for intervention before epidemic/failure spreads. Moreover, we study by computer simulations the sandpile cascade model, a diffusive model of distress propagation (financial contagion). We show that, while assortative networks are more prone to the propagation of epidemic/failures, degree-targeted immunization policies increases their resilience to systemic risk.

Reciprocity of Networks with Degree Correlations and Arbitrary Degree Sequences

Although most of the real networks contain a mixture of directed and bidirectional (reciprocal) connections, the reciprocity r has received little attention as a subject of theoretical understanding. We study the expected reciprocity of networks with arbitrary input and output degree sequences and given 2-node degree correlations by means of statistical ensemble approach. We demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to r is revealed. Numerical experiments using network randomization methods show very good agreement to our analytical estimations.

On the rich-club effect in dense and weighted networks

For many complex networks present in nature only a single instance, usually of large size, is available. Any measurement made on this single instance cannot be repeated on different realizations. In order to detect significant patterns in a real-world network it is therefore crucial to compare the measured results with a null model counterpart. Here we focus on dense and weighted networks, proposing a suitable null model and studying the behaviour of the degree correlations as measured by the rich-club coefficient. Our method solves an existing problem with the randomization of dense unweighted graphs, and at the same time represents a generalization of the rich-club coefficient to weighted networks which is complementary to other recently proposed ones.

Topologically biased random walk and community finding in networks

We present an approach of topology biased random walks for undirected networks. We focus on a one-parameter family of biases, and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks. This analogy is extended through the use of parametric equations of motion to study the features of random walks vs parameter values. Furthermore, we show an analysis of the spectral gap maximum associated with the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state. Applications of these studies allow ad hoc algorithms for the exploration of complex networks and their communities.

Model of Wikipedia growth based on information exchange via reciprocal arcs

We show how reciprocal arcs significantly influence the structural organization of Wikipedias, online encyclopedias. It is shown that random addition of reciprocal arcs in the static network cannot explain the observed reciprocity of Wikipedias. A model of Wikipedia growth based on preferential attachment and on information exchange via reciprocal arcs is presented. An excellent agreement between in-degree distributions of our model and real Wikipedia networks is achieved without fitting the distributions, but by merely extracting a small number of model parameters from the measurement of real networks.

Influence of reciprocal edges on degree distribution and degree correlations

Reciprocal edges represent the lowest-order cycle possible to find in directed graphs without self-loops. Representing also a measure of feedback between vertices, it is interesting to understand how reciprocal edges influence other properties of complex networks. In this paper, we focus on the influence of reciprocal edges on vertex degree distribution and degree correlations. We show that there is a fundamental difference between properties observed on the static network compared to the properties of networks, which are obtained by simple evolution mechanism driven by reciprocity. We also present a way to statistically infer the portion of reciprocal edges, which can be explained as a consequence of feedback process on the static network. In the rest of the paper, the influence of reciprocal edges on a model of growing network is also presented. It is shown that our model of growing network nicely interpolates between Barabási-Albert (BA) model for undirected and the BA model for directed networks.

Networks with arbitrary edge multiplicities

One of the main characteristics of real-world networks is their large clustering. Clustering is one aspect of a more general but much less studied structural organization of networks, i.e. edge multiplicity, defined as the number of triangles in which edges, rather than vertices, participate. Here we show that the multiplicity distribution of real networks is in many cases scale-free, and in general very broad. Thus, besides the fact that in real networks the number of edges attached to vertices often has a scale-free distribution, we find that the number of triangles attached to edges can have a scale-free distribution as well. We show that current models, even when they generate clustered networks, systematically fail to reproduce the observed multiplicity distributions. We therefore propose a generalized model that can reproduce networks with arbitrary distributions of vertex degrees and edge multiplicities, and study many of its properties analytically.