The topological structure of complex networks has fascinated researchers for several decades, resulting in the discovery of many universal properties and reoccurring characteristics of different kinds of networks. However, much less is known today about the network dynamics: indeed, complex networks in reality are not static, but rather dynamically evolve over time. Our paper is motivated by the empirical observation that network evolution patterns seem far from random, but exhibit structure. Moreover, the specific patterns appear to depend on the network type, contradicting the existence of a " one fits it all " model. However, we still lack observables to quantify these intuitions, as well as metrics to compare graph evolutions. Such observables and metrics are needed for extrapolating or predicting evolutions, as well as for interpolating graph evolutions. To explore the many faces of graph dynamics and to quantify temporal changes, this paper suggests to build upon the concept of centrality, a measure of node importance in a network. In particular, we introduce the notion of centrality distance, a natural similarity measure for two graphs which depends on a given centrality, characterizing the graph type. Intuitively, centrality distances reflect the extent to which (non-anonymous) node roles are different or, in case of dynamic graphs, have changed over time, between two graphs. We evaluate the centrality distance approach for five evolutionary models and seven real-world social and physical networks. Our results empirically show the usefulness of centrality distances for characterizing graph dynamics compared to a null-model of random evolution, and highlight the differences between the considered scenarios. Interestingly, our approach allows us to compare the dynamics of very different networks, in terms of scale and evolution speed.
Graph structures are often characterized by the frequency of small patterns called @cite_17 @cite_0 @cite_13 @cite_37 , also known as @cite_24 , or @cite_10 . Another important graph characterization, which is studied in this paper, are @cite_25 . Dozens of different centrality indices have been defined over the last years, and their study is still ongoing, with no unified theory yet. We believe that our centrality distance framework can provide new inputs for this discussion.
@cite_37 @cite_10 @cite_0 @cite_24 @cite_13 @cite_25 @cite_17
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Motifs in a given network are small connected subnetworks that occur in significantly higher frequencies than would be expected in random networks. They have recently gathered much attention as a concept to uncover structural design principles of complex networks. [Bioinformatics, 2004] proposed a sampling algorithm for performing the computationally challenging task of detecting network motifs. However, among other drawbacks, this algorithm suffers from a sampling bias and scales poorly with increasing subgraph size. Based on a detailed analysis of the previous algorithm, we present a new algorithm for network motif detection which overcomes these drawbacks. Furthermore, we present an efficient new approach for estimating the frequency of subgraphs in random networks that, in contrast to previous approaches, does not require the explicit generation of random networks. Experiments on a testbed of biological networks show our new algorithms to be orders of magnitude faster than previous approaches, allowing for the detection of larger motifs in bigger networks than previously possible and thus facilitating deeper insight into the field. Network forms of organization, unlike hierarchies or marketplaces, are agile and are constantly adapting as new links are added and dysfunctional ones dropped. We review some of the theoretical and methodological accomplishments and challenges of contemporary research on organizational networks. We then offer an analytic framework that can be used to specify and statistically test simultaneously multilevel, multitheoretical hypotheses about the structural tendencies of organizational networks. We conclude with an empirical study illustrating some of the capabilities of this framework. Complex networks are studied across many fields of science. To uncover their structural design principles, we defined “network motifs,” patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks. We found such motifs in networks from biochemistry, neurobiology, ecology, and engineering. The motifs shared by ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomyces cerevisiae or from those found in the World Wide Web. Similar motifs were found in networks that perform information processing, even though they describe elements as different as biomolecules within a cell and synaptic connections between neurons in Caenorhabditis elegans. Motifs may thus define universal classes of networks. This Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu Network motifs, patterns of local interconnections with potential functional properties, are important for the analysis of biological networks. To analyse motifs in networks the first step is to find patterns of interest. This paper presents 1) three different concepts for the determination of pattern frequency and 2) an algorithm to compute these frequencies. The different concepts of pattern frequency depend on the reuse of network elements. The presented algorithm finds all or highly frequent patterns under consideration of these concepts. The utility of this method is demonstrated by applying it to biological data. Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved.