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

Levich Institute and Physics Department, City College of New York, New York, NY 10031, USA.
Proceedings of the National Academy of Sciences (Impact Factor: 9.67). 02/2012; 109(8):2825-30. DOI: 10.1073/pnas.1106612109
Source: PubMed


The human brain is organized in functional modules. Such an organization presents a basic conundrum: Modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer. It is commonly accepted that small-world architecture of short paths and large local clustering may solve this problem. However, there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity, a global property unrelated to local clustering. Here, we present a possible solution to this puzzle. We first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks. These modules are "large-world" self-similar structures and, therefore, are far from being small-world. However, incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well-defined modules. Remarkably, weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost. This trade-off architecture is reminiscent of the "strength of weak ties" crucial concept of social networks. Such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain.

Download full-text


Available from: Lazaros K. Gallos,
  • Source
    • "Indirect paths indicate the presence of weaker links in the functional connectivity architecture (Goni et al., 2014). While most previous investigations of the connectome used thresholding procedures to discarded weaker links from analysis, the importance of weak links is being increasingly appreciated in recent times (Gallos et al., 2012; Schwarz and "
    [Show abstract] [Hide abstract]
    ABSTRACT: Spatial variation in connectivity is an integral aspect of the brain's architecture. In the absence of this variability, the brain may act as a single homogenous entity without regional specialization. In this study, we investigate the variability in functional links categorized on the basis of the presence of direct structural paths (primary) or indirect paths mediated by one (secondary) or more (tertiary) brain regions ascertained by diffusion tensor imaging. We quantified the variability in functional connectivity using an unbiased estimate of unpredictability (functional connectivity entropy) in a neuropsychiatric disorder where structure-function relationship is considered to be abnormal; 34 patients with schizophrenia and 32 healthy controls underwent DTI and resting state functional MRI scans. Less than one-third (27.4% in patients, 27.85% in controls) of functional links between brain regions were regarded as direct primary links on the basis of DTI tractography, while the rest were secondary or tertiary. The most significant changes in the distribution of functional connectivity in schizophrenia occur in indirect tertiary paths with no direct axonal linkage in both early (P = 0.0002, d = 1.46) and late (P = 1 × 10(-17) , d = 4.66) stages of schizophrenia, and are not altered by the severity of symptoms, suggesting that this is an invariant feature of this illness. Unlike those with early stage illness, patients with chronic illness show some additional reduction in the distribution of connectivity among functional links that have direct structural paths (P = 0.08, d = 0.44). Our findings address a critical gap in the literature linking structure and function in schizophrenia, and demonstrate for the first time that the abnormal state of functional connectivity preferentially affects structurally unconstrained links in schizophrenia. It also raises the question of a continuum of dysconnectivity ranging from less direct (structurally unconstrained) to more direct (structurally constrained) brain pathways underlying the progressive clinical staging and persistence of schizophrenia. Hum Brain Mapp, 2015. © 2015 The Authors. Human Brain Mapping Published by Wiley Periodicals, Inc. © 2015 The Authors. Human Brain Mapping Published by Wiley Periodicals, Inc.
    Human Brain Mapping 08/2015; DOI:10.1002/hbm.22932 · 5.97 Impact Factor
  • Source
    • "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. "
    [Show abstract] [Hide abstract]
    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 · 1.73 Impact Factor
  • Source
    • "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. "
    [Show abstract] [Hide abstract]
    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).
Show more