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

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.

Functional brain networks: great expectations, hard times and the big leap forward

“…Various studies highlighted that the brain shows prominent hierarchical structure, with modules themselves containing other modules [26]. Zooming in and out of brain functional activity reveals a complex fractal structure, showing both self-similarity [27] and selfdissimilarity [28]. Interestingly, these global properties are associated with some mesoscale properties such as assortativity, with hubs in fractal and non-fractal structures, respectively, repelling or attracting each other [29].…”

Robustness and closeness centrality for self-organized and planned cities

“…Between others, an interesting example is the brain cortex which shares many similarities with street networks, as high modularity, fractal structure, etc. [2]. Written language networks in the same way could be described by the interplay between syntactic (embedded in the text contiguity representation [19]) and semantic layer (which relates each content word to other texts) [20].…”

“…[1][2][3][4]. Within transportation systems, street networks possibly represent the largest known infrastructural one [5].…”

Interdisciplinary and physics challenges of network theory

“…Moreover, it is believed that this theory could be fundamental for proposing new routing strategies for packets in the Internet, solving in this way a problem of scalability of the presently used technology [26][27][28][29]. Additionally characterizing brain geometry and topology will contribute to a deeper understanding between brain structure, dynamics and function [21,[30][31][32][33][34]. Finally, geometrical network models [35][36][37][38][39] have been able to generate networks sharing the phenomenology of most complex networks and therefore provide the best way of understanding how all the universal properties of complex networks might emerge at the same time.…”

“…Moreover, the study of the interplay between structural and functional brain networks [14], is recently attracting large attention. It is believed that modularity [32][33][34] plays a role of special importance, together with the small world property, in generating a dynamical phase of frustrated synchronization, where synchronization is sustained but not stationary. It is possible that in the future, advances in the understanding of the geometrical organization of the brain networks will allow to fully identify the structural properties that favour brain dynamics.…”