Article

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

ABSTRACT

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

Full-text

Available from: Lazaros K. Gallos
  • Source
    • "This is in agreement with the conjecture that finite dimensionality is required for the existence of GPs [3]. Moreover, many real networks, as for instance brain connectomes, have a modular organization, where modules are finite, heterogeneous and weakly connected [39]. Slow dynamics has been reported in models defined on hierarchical modular networks [4] [14] [40]. "
    [Show description] [Hide description]
    DESCRIPTION: We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-sized random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects of outliers both in case of highly fluctuating (natural cutoff) and non-fluctuating (rigid cutoff) most connected verticeLogarithmic and power-law decays in time were found for natural and rigid cutoffs, respectively. This happens in extended regions of the control parameter space $\lambda_1<\lambda<\lambda_2$, suggesting Griffiths effects, induced by the topological inhomogeneities. Optimal fluctuation theory considering sample-to-sample fluctuations of the pseudo thresholds is presented to explain the observed slow dynamics. A quasistationary analysis shows that response functions remain bounded at $\lambda_2$. We argue these to be signals of a smeared transition, caused by rare region effects of the star subgraphs. However, in the thermodynamic limit the Griffiths effects loose their relevancy and a cross over to a conventional critical point at $\lambda_c=0$ branching rate.
    Full-text · Research · Dec 2015
  • Source
    • "In some cases it is even possible to infer these processes from the final structure of the system [5] [6] [7] [8]. Percolation [9] is a natural process that happens in different types of systems and it is used in fields as varied as oil extraction [10], the study of the electrical conductivity of materials [11], polymerization processes [12], epidemic studies [13] and fire spreading [14] or even the modular structure of brain networks [15] or the spread of obesity [16]. Percolation processes are a branch from the field of complex systems that aims to study how geometrical microscopical properties affect the macroscopic configuration of the ensemble. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Road networks are characterised by several structural and geometric properties. Their topological structure determines partially its hierarchical arrangement, but since these are networks that are spatially situated and, therefore, spatially constrained, to fully understand the role that each road plays in the system it is fundamental to characterize the influence that geometrical properties have over the network's behaviour. In this work, we percolate the UK's road network using the relative angle between street segments as the occupation probability. We argue that road networks undergo a non-equilibrium first-order phase transition at the moment the main roads start to interconnect forming the spanning percolation cluster. The percolation process uncovers the hierarchical structure of the roads in the network, and as such, its classification. Furthermore, this technique serves to extract the set of most important roads of the network and to create a hierarchical index for each road in the system.
    Full-text · Article · Dec 2015
  • Source
    • "This topology is efficient for signal processing [5], but doubts have remained if the small-world assumption is generally true for the brain [6]. Despite some evidence that networks obtained from spatially coarse-grained parcellations of the brain are small worlds [7], at a cellular level the brain may be a large-world network after all [8]. Like the small-world property, the hypothesis that the brain has a scale-free degree distribution became popular around the turn of the millennium. "
    [Show abstract] [Hide abstract]
    ABSTRACT: The structural human connectome (i.e.\ the network of fiber connections in the brain) can be analyzed at ever finer spatial resolution thanks to advances in neuroimaging. Here we analyze several large data sets for the human brain network made available by the Open Connectome Project. We apply statistical model selection to characterize the degree distributions of graphs containing up to ~10^6 nodes and ~10^8 edges. The model that in general describes the observed degrees best is a three-parameter generalized Weibull (also known as a stretched exponential) distribution. Thus the degree distribution is heavy-tailed, but not scale-free. We also calculate the topological (graph) dimension D and the small-world coefficient \sigma of these networks. While \sigma suggests a small-world topology, we found that D < 4 showing that long-distance connections provide only a small correction to the topology of the embedding three-dimensional space.
    Full-text · Article · Dec 2015
Show more