# A small-world of weak ties provides optimal global integration of self-similar modules in functional brain networks

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Lazaros K. Gallos, Jul 20, 2015 Available from:-
- "In a series of papers [7] [25] [26] [27] [28], the research group of Makse et al. analyze the fractality of complex networks including box counting technique and algorithms, renormalization and growth approaches, etc. Using their techniques 55 of fractal analysis on networks, they study the biological networks [29] [30] [31], such as function brain networks. "

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**ABSTRACT:**In this paper, we use the Sierpinski gasket to construct evolving networks whose node set is the solid regular triangles in the construction of the Sierpinski gasket up to the stage and any two nodes are neighbors if and only if the corresponding solid triangles are in contact with each other on boundary. Using the encoding method, we show that our evolving networks are scale-free (power-law degree distribution) and have the small-world effect (small average path length and high clustering coefficient).Physica A: Statistical Mechanics and its Applications 05/2015; 436. DOI:10.1016/j.physa.2015.05.048 · 1.72 Impact Factor -
- "For these clusters, we also compute the fractal dimension in terms of eq.(1), where this time r max corresponds to the diameter of the network . This is the same methodology that was implemented in [20]. Note that for this system we need to take a slightly larger maximum distance threshold d = 800m to ensure we are well within the cities definition. "

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**ABSTRACT:**Urban systems present hierarchical structures at many different scales. These are observed as administrative regional delimitations, which are the outcome of geographical, political and historical constraints. Using percolation theory on the street intersections and on the road network of Britain, we obtain hierarchies at different scales that are independent of administrative arrangements. Natural boundaries, such as islands and National Parks, consistently emerge at the largest/regional scales. Cities are devised through recursive percolations on each of the emerging clusters, but the system does not undergo a phase transition at the distance threshold at which cities can be defined. This specific distance is obtained by computing the fractal dimension of the clusters extracted at each distance threshold. We observe that the fractal dimension presents a maximum over all the different distance thresholds. The clusters obtained at this maximum are in very good correspondence to the morphological definition of cities given by satellite images, and by other methods previously developed by the authors (Arcaute et al. 2015). -
- "While it is still unclear how much the physiological or functional impact of a projection changes with its fiber density (e.g., Vanduffel et al. 1997), such graded networks might be more suitably analyzed by approaches that account for the differential weight of pathways (Rubinov and Sporns 2011). The issue is underlined by the fact that examples can be found where brain networks resemble a large-world network when only the stronger connections are taken into account, while incorporating the weakest connections shrinks them to a small-world network (Gallos et al. 2012). Thus, an analysis that takes into account the weight of the connections, especially in very dense networks, and employs weighted versions of network metrics including the small-world index (e.g. "

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**ABSTRACT:**A matter of network topologyIt is commonly assumed that the brain is a small-world network (e.g., Sporns and Honey 2006). Indeed, one of the present authors claimed as much 15 years ago (Hilgetag et al. 2000). The small-worldness is believed to be a crucial aspect of efficient brain organization that confers significant advantages in signal processing (e.g., Lago-Fernández et al. 2000). Correspondingly, the small-world organization is deemed essential for healthy brain function, as alterations of small-world features are observed in patient groups with Alzheimer’s disease (Stam et al. 2007), autism (Barttfeld et al. 2011) or schizophrenia spectrum diseases (Liu et al. 2008; Wang et al. 2012; Zalesky et al. 2011).While the colloquial idea of a small, interconnected world has a long tradition (e.g., Klemperer 1938), the present concept of small-world features of networks is frequently associated with the Milgram experiment (Milgram 1967) that demonstrated surprisingly short paths across ...Brain Structure and Function 04/2015; DOI:10.1007/s00429-015-1035-6 · 4.57 Impact Factor