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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us to compare the dynamics of very different networks, in terms of scale and evolution speed., Comment: This paper is withdrawn because it should have replaced 1506.01565 instead of creating a duplicate. This version will now appear as a new version of 1506.01565

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., Comment: 18 pages, 8 figures

Recent evidence shows that in many societies worldwide the relative sizes of the economic and social elites are continuously shrinking. Is this a natural social phenomenon? What are the forces that shape this process? We try to address these questions by studying a Core-Periphery social structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society, the periphery. Herein, we present a novel axiom-based model for the forces governing the mutual influences between the elite and the periphery. Assuming a simple set of axioms, capturing the elite's dominance, robustness, compactness and density, we are able to draw strong conclusions about the elite-periphery structure. In particular, we show that a balance of powers between elite and periphery and an elite size that is sub-linear in the network size are universal properties of elites in social networks that satisfy our axioms. We note that the latter is in controversy to the common belief that the elite size converges to a linear fraction of society (most recently claimed to be 1%). We accompany these findings with a large scale empirical study on about 100 real-world networks, which supports our results.

In many societies there is an elite, a relatively small group of powerful individuals that is well-connected and highly influential. Since the ancient days of Julius Caesar's senate of Rome to the recent days of celebrities on Twitter, the size of the elite is a result of conflicting social forces competing to increase or decrease it. The main contribution of this paper is the answer to the question how large the elite is at equilibrium. We take an axiomatic approach to solve this: assuming that an elite exists and it is influential, stable and either minimal or dense, we prove that its size must be $\Theta(\sqrt{m})$ (where $m$ is the number of edges in the network). As an approximation for the elite, we then present an empirical study on nine large real-world networks of the subgraph formed by the highest degree nodes, also known as the rich-club. Our findings indicate that elite properties such as disproportionate influence, stability and density of $\Theta(\sqrt{m})$-rich-clubs are universal properties and should join a growing list of common phenomena shared by social networks and complex systems such as "small world," power law degree distributions, high clustering, etc., Comment: under submission