Complex networks from such different fields as biology, technology or sociology share similar organization principles. The possibility of a unique growth mechanism for a variety of complex networks in different fields such as biology, technology or sociology is of interest, as it promises to uncover the universal origins of collective behavior. The emergence of self-similarity in complex networks raises the fundamental question of the growth process according to which these structures evolve. Here, we use the concept of renormalization as a mechanism for the growth of fractal and non-fractal modular networks. We show that the key principle that gives rise to the fractal architecture of networks is a strong effective "repulsion" (disassortativity) between the most connected nodes (hubs) on all length scales, rendering them very dispersed. More importantly, we show that a robust network comprised of functional modules, such as a cellular network, necessitates a fractal topology, suggestive of an evolutionary drive for their existence. An important result in statistical physics was the generation of fractal geometries by Mandelbrot [1, 2] , the structures of which look the same on all length scales. Their importance stem from the fact that these structures were recognized in numerous examples in Nature, from snowflakes and trees to phase transitions in critical phenomena [2, 3] . While these fascinating patterns are only geometric, new forms of topological fractality have been observed in complex networks [4] where the links rely on interactions between the participants [5, 6] . Examples of topological fractal networks include the hyperlinks in the WWW, physical interactions in protein interaction networks or biochemical reactions in metabolism [4, 7] . Other complex networks such as the Internet do not share the topological fractal property. These fractal complex networks are characterized by the small-world property (as given by the logarithmic dependence of the average distance with the number of nodes) brought about by the "short-cuts" in the network [8], a very wide (power-law or "scale-free" [9]) distribution of connections, and a modular hierarchical structure [10, 11, 12, 13] . However, the fractal sets of Mandelbrot do not exhibit these features. While in our previous work [4], we discovered the fractal nature of organization in many real networks the question remained how these networks have evolved in time. We therefore launch a study of growth mechanisms to understand the simultaneous emergence of fractality, modularity, as well as the small world effect, and the scale-free property in real world complex networks. Our results have important evolutionary implications. They highlight an evolutionary drive towards fractality, inspired by an increase in network robustness. Thus, a robust modular network requires a fractal topology. Furthermore, our analysis indicates that the fractal clusters can be identified with the functional modules in the case of the metabolic network of E. coli. The "democratic" rule of the seminal Erdös-Rényi model [14] (where the nodes in the network are connected at random) was first invoked to explain the small world effect. It was then replaced by the "rich-get-richer" principle of preferential attachment [9] to explain the scale-free property; a discovery carrying important implications on network vulnerability [15, 16] . However these rules do not capture the fractal topologies found in diverse complex networks. We find that models of scale-free networks are not fractals (see Supplementary Materials Section I). Here, we demonstrate a new view of network dynamics where the growth takes place multiplicatively in a correlated self-similar modular fashion, in contrast to the uncorrelated growth of models of preferential attachment [5, 9] . We formalize these ideas by borrowing the concept of "length-scale renormalization" from critical phenomena [3] . In this paper, we will show that the emergence of self-similar fractal networks, such as cellular ones, is due to the strong repulsion (disassortativity [17]) between the hubs at all length scales. In other words, the hubs prefer to grow by connections to lessconnected nodes rather than to other hubs, an effect that can be viewed as an effective hub repulsion. In this new paradigm, the "rich" still get richer, although at the expense of the "poor". In other words, the hubs grow by preferentially linking with less-connected nodes to generate a more robust fractal topology. In contrast, weakly anticorrelated or uncorrelated growth leads to non-fractal topologies such as the Internet. Growth mechanism.-The renormalization scheme tiles a network of N nodes with N B (ℓ B ) boxes using the box-covering algorithm [4] , as shown in Fig. 1a . The boxes contain nodes separated by a distance ℓ B , measured as the length of the shortest path between nodes. Each box is subsequently replaced by a node, and the process is repeated until the whole network is reduced to a single node. The way to distinguish between fractal and non-fractal networks is represented in their scaling properties as seen in Fig. 2a and 2b . Fractal networks can be characterized by the following scaling relations ( Fig. 2a) : where k hub and k B (ℓ B ) are the degree of the most connected node inside each box and that of each box respectively (Fig. 1a ). Although both of them are partial variables, the ratio between them is a global quantity, depending only on the length scale ℓ B , as we showed in [4] . The two exponents d B and d k are the fractal dimension and the degree exponent of the boxes, respectively. While the term "fractal dimension" is usually reserved for geometrical self-similarity, here we relax the usage to include the topological self-similarity as well. For a non-fractal network like the Internet (Fig. 2b) , we have d B → ∞ and d k → ∞; the scaling laws in Eq. (1) are replaced by exponential functions. Based on the results leading to Eq. (1), we propose a network growth dynamics as the inverse of the renormalization procedure. Thus, the coarse-grained networks of smaller size are network structures appearing earlier in time, as exemplified in Fig. 1a . A present time network withÑ (t) nodes is tiled with N B (ℓ B ) boxes of size ℓ B . Each box represents a node in a previous time step, so thatÑ (t − 1) = N B (ℓ B ). The maximum degree of the nodes
Moreover, the importance of modular structures and the improved robustness from the fractality property are discussed in REF .
Origins of fractality in the growth of complex networks
Nature Physics
Nature Physics
Computer Science Physics