Routes Obey Hierarchy in Complex Networks.

The last two decades of network science have discovered stunning similarities in the topological characteristics of real life networks (many biological, social, transportation and organizational networks) on a strong empirical basis. However our knowledge about the operational paths used in these networks is very limited, which prohibits the proper understanding of the principles of their functioning. Today, the most widely adopted hypothesis about the structure of the operational paths is the shortest path assumption. Here we present a striking result that the paths in various networks are significantly stretched compared to their shortest counterparts. Stretch distributions are also found to be extremely similar. This phenomenon is empirically confirmed on four networks from diverse areas of life. We also identify the high-level path selection rules nature seems to use when picking its paths

Using Pareto optimality to explore the topology and dynamics of the human connectome.

Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an 'economical' small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains

Generative models of the human connectome

The human connectome represents a network map of the brain's wiring diagram and the pattern into which its connections are organized is thought to play an important role in cognitive function. The generative rules that shape the topology of the human connectome remain incompletely understood. Earlier work in model organisms has suggested that wiring rules based on geometric relationships (distance) can account for many but likely not all topological features. Here we systematically explore a family of generative models of the human connectome that yield synthetic networks designed according to different wiring rules combining geometric and a broad range of topological factors. We find that a combination of geometric constraints with a homophilic attachment mechanism can create synthetic networks that closely match many topological characteristics of individual human connectomes, including features that were not included in the optimization of the generative model itself. We use these models to investigate a lifespan dataset and show that, with age, the model parameters undergo progressive changes, suggesting a rebalancing of the generative factors underlying the connectome across the lifespan.Comment: 38 pages, 5 figures + 19 supplemental figures, 1 tabl

Robust estimation of fractal measures for characterizing the structural complexity of the human brain: optimization and reproducibility

High-resolution isotropic three-dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape, volume and topological complexity. In particular, methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions. The usefulness of such measures for characterizing individual differences in brain structure critically depends on their within-subject reproducibility in order to allow the robust detection of between-subject differences. This study analyzes key analytic parameters of three fractal-based methods that rely on the box-counting algorithm with the aim to maximize within-subject reproducibility of the fractal characterizations of different brain objects, including the pial surface, the cortical ribbon volume, the white matter volume and the grey matter/white matter boundary. Two separate datasets originating from different imaging centers were analyzed, comprising, 50 subjects with three and 24 subjects with four successive scanning sessions per subject, respectively. The reproducibility of fractal measures was statistically assessed by computing their intra-class correlations. Results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility. Highest reproducibility with intra-class correlations in the range of 0.9–0.95 is achieved with the correlation dimension. Further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region-specific patterns of individual variability. These results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure, for instance in studies of neurological diseases that do not allow repeated measurements or for disease-course longitudinal studies

Micro-connectomics: probing the organization of neuronal networks at the cellular scale.

Defining the organizational principles of neuronal networks at the cellular scale, or micro-connectomics, is a key challenge of modern neuroscience. In this Review, we focus on graph theoretical parameters of micro-connectome topology, often informed by economical principles that conceptually originated with Ramón y Cajal's conservation laws. First, we summarize results from studies in intact small organisms and in samples from larger nervous systems. We then evaluate the evidence for an economical trade-off between biological cost and functional value in the organization of neuronal networks. Various results suggest that many aspects of neuronal network organization are indeed the outcome of competition between these two fundamental selection pressures.This work was supported by the National Institute of Health Research (NIHR) Cambridge Biomedical Research Centre.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by the Nature Publishing Group

Path ensembles and a tradeoff between communication efficiency and resilience in the human connectome.

Computational analysis of communication efficiency of brain networks often relies on graph-theoretic measures based on the shortest paths between network nodes. Here, we explore a communication scheme that relaxes the assumption that information travels exclusively through optimally short paths. The scheme assumes that communication between a pair of brain regions may take place through a path ensemble comprising the k-shortest paths between those regions. To explore this approach, we map path ensembles in a set of anatomical brain networks derived from diffusion imaging and tractography. We show that while considering optimally short paths excludes a significant fraction of network connections from participating in communication, considering k-shortest path ensembles allows all connections in the network to contribute. Path ensembles enable us to assess the resilience of communication pathways between brain regions, by measuring the number of alternative, disjoint paths within the ensemble, and to compare generalized measures of path length and betweenness centrality to those that result when considering only the single shortest path between node pairs. Furthermore, we find a significant correlation, indicative of a trade-off, between communication efficiency and resilience of communication pathways in structural brain networks. Finally, we use k-shortest path ensembles to demonstrate hemispherical lateralization of efficiency and resilience

Resting state network modularity along the prodromal late onset Alzheimer's disease continuum

Alzheimer's disease is considered a disconnection syndrome, motivating the use of brain network measures to detect changes in whole-brain resting state functional connectivity (FC). We investigated changes in FC within and among resting state networks (RSN) across four different stages in the Alzheimer's disease continuum. FC changes were examined in two independent cohorts of individuals (84 and 58 individuals, respectively) each comprising control, subjective cognitive decline, mild cognitive impairment and Alzheimer's dementia groups. For each participant, FC was computed as a matrix of Pearson correlations between pairs of time series from 278 gray matter brain regions. We determined significant differences in FC modular organization with two distinct approaches, network contingency analysis and multiresolution consensus clustering. Network contingency analysis identified RSN sub-blocks that differed significantly across clinical groups. Multiresolution consensus clustering identified differences in the stability of modules across multiple spatial scales. Significant modules were further tested for statistical association with memory and executive function cognitive domain scores. Across both analytic approaches and in both participant cohorts, the findings converged on a pattern of FC that varied systematically with diagnosis within the frontoparietal network (FP) and between the FP network and default mode network (DMN). Disturbances of modular organization were manifest as greater internal coherence of the FP network and stronger coupling between FP and DMN, resulting in less segregation of these two networks. Our findings suggest that the pattern of interactions within and between specific RSNs offers new insight into the functional disruption that occurs across the Alzheimer's disease spectrum

Network morphospace

The structure of complex networks has attracted much attention in recent years. It has been noted that many real-world examples of networked systems share a set of common architectural features. This raises important questions about their origin, for example whether such network attributes reflect common design principles or constraints imposed by selectional forces that have shaped the evolution of network topology. Is it possible to place the many patterns and forms of complex networks into a common space that reveals their relations, and what are the main rules and driving forces that determine which positions in such a space are occupied by systems that have actually evolved? We suggest that these questions can be addressed by combining concepts from two currently relatively unconnected fields. One is theoretical morphology, which has conceptualized the relations between morphological traits defined by mathematical models of biological form. The second is network science, which provides numerous quantitative tools to measure and classify different patterns of local and global network architecture across disparate types of systems. Here, we explore a new theoretical concept that lies at the intersection between both fields, the ‘network morphospace’. Defined by axes that represent specific network traits, each point within such a space represents a location occupied by networks that share a set of common ‘morphological’ characteristics related to aspects of their connectivity. Mapping a network morphospace reveals the extent to which the space is filled by existing networks, thus allowing a distinction between actual and impossible designs and highlighting the generative potential of rules and constraints that pervade the evolution of complex systems