Graph distance for complex networks

Quantifying synergy and redundancy between networks

Understanding how different networks relate to each other is key for understanding complex systems. We introduce an intuitive yet powerful framework to disentangle different ways in which networks can be (dis)similar and complementary to each other. We decompose the shortest paths between nodes as uniquely contributed by one source network, or redundantly by either, or synergistically by both together. Our approach considers the networks' full topology, providing insights at multiple levels of resolution: from global statistics to individual paths. Our framework is widely applicable across scientific domains, from public transport to brain networks. In humans and 124 other species, we demonstrate the prevalence of unique contributions by long-range white-matter fibers in structural brain networks. Across species, efficient communication also relies on significantly greater synergy between long-range and short-range fibers than expected by chance. Our framework could find applications for designing network systems or evaluating existing ones.

The spatial distribution of coupling between tau and neurodegeneration in amyloid-β positive mild cognitive impairment

Synergies between amyloid-β (Aβ), tau, and neurodegeneration persist along the Alzheimer's disease (AD) continuum. This study aimed to evaluate the extent of spatial coupling between tau and neurodegeneration (atrophy) and its relation to Aβ positivity in mild cognitive impairment (MCI). Data from 409 participants were included (95 cognitively normal controls, 158 Aβ positive (Aβ+) MCI, and 156 Aβ negative (Aβ-) MCI). Florbetapir PET, Flortaucipir PET, and structural MRI were used as biomarkers for Aβ, tau and atrophy, respectively. Individual correlation matrices for tau load and atrophy were used to layer a multilayer network, with separate layers for tau and atrophy. A measure of coupling between corresponding regions of interest (ROIs) in the tau and atrophy layers was computed, as a function of Aβ positivity. Fewer than 25% of the ROIs across the brain showed heightened coupling between tau and atrophy in Aβ+ , relative to Aβ- MCI. Coupling strengths in the right rostral middle frontal and right paracentral gyri, in particular, mediated the association between Aβ burden and cognition in this sample.

The spatial distribution of coupling between tau and neurodegeneration in amyloid-β positive mild cognitive impairment

Synergies between amyloid-β (Aβ), tau, and neurodegeneration persist along the Alzheimer's disease (AD) continuum. This study aimed to evaluate the extent of spatial coupling between tau and neurodegeneration (atrophy) and its relation to Aβ positivity in mild cognitive impairment (MCI). Data from 409 subjects were included (95 cognitively normal controls, 158 Aβ positive (Aβ+) MCI, and 156 Aβ negative (Aβ-) MCI) Florbetapir PET, Flortaucipir PET, and structural MRI were used as biomarkers for Aβ, tau and atrophy, respectively. Individual correlation matrices for tau load and atrophy were used to layer a multilayer network, with separate layers for tau and atrophy. A measure of coupling between corresponding regions of interest/nodes in the tau and atrophy layers was computed, as a function of Aβ positivity. The extent to which tau-atrophy coupling mediated associations between Aβ burden and cognitive decline was also evaluated. Heightened coupling between tau and atrophy in Aβ+ MCI was found primarily in the entorhinal and hippocampal regions (i.e., in regions corresponding to Braak stages I/II), and to a lesser extent in limbic and neocortical regions (i.e., corresponding to later Braak stages). Coupling strengths in the right middle temporal and inferior temporal gyri mediated the association between Aβ burden and cognition in this sample. Higher coupling between tau and atrophy in Aβ+ MCI is primarily evident in regions corresponding to early Braak stages and relates to overall cognitive decline. Coupling in neocortical regions is more restricted in MCI.

Comparative survey of multigraph integration methods for holistic brain connectivity mapping

One of the greatest scientific challenges in network neuroscience is to create a representative map of a population of heterogeneous brain networks, which acts as a connectional fingerprint. The connectional brain template (CBT), also named network atlas, presents a powerful tool for capturing the most representative and discriminative traits of a given population while preserving its topological patterns. The idea of a CBT is to integrate a population of heterogeneous brain connectivity networks, derived from different neuroimaging modalities or brain views (e.g., structural and functional), into a unified holistic representation. Here we review current state-of-the-art methods designed to estimate well-centered and representative CBT for populations of single-view and multi-view brain networks. We start by reviewing each CBT learning method, then we introduce the evaluation measures to compare CBT representativeness of populations generated by single-view and multigraph integration methods, separately, based on the following criteria: Centeredness, biomarker-reproducibility, node-level similarity, global-level similarity, and distance-based similarity. We demonstrate that the deep graph normalizer (DGN) method significantly outperforms other multi-graph and all single-view integration methods for estimating CBTs using a variety of healthy and disordered datasets in terms of centeredness, reproducibility (i.e., graph-derived biomarkers reproducibility that disentangle the typical from the atypical connectivity variability), and preserving the topological traits at both local and global graph-levels.

Benford Networks

The Benford law applied within complex networks is an interesting area of research. This paper proposes a new algorithm for the generation of a Benford network based on priority rank, and further specifies the formal definition. The condition to be taken into account is the probability density of the node degree. In addition to this first algorithm, an iterative algorithm is proposed based on rewiring. Its development requires the introduction of an ad hoc measure for understanding how far an arbitrary network is from a Benford network. The definition is a semi-distance and does not lead to a distance in mathematical terms, instead serving to identify the Benford network as a class. The semi-distance is a function of the network; it is computationally less expensive than the degree of conformity and serves to set a descent condition for the rewiring. The algorithm stops when it meets the condition that either the network is Benford or the maximum number of iterations is reached. The second condition is needed because only a limited set of densities allow for a Benford network. Another important topic is assortativity and the extremes which can be achieved by constraining the network topology; for this reason, we ran simulations on artificial networks and explored further theoretical settings as preliminary work on models of preferential attachment. Based on our extensive analysis, the first proposed algorithm remains the best one from a computational point of view.

Dyconnmap: Dynamic connectome mapping-A neuroimaging python module

Despite recent progress in the analysis of neuroimaging data sets, our comprehension of the main mechanisms and principles which govern human brain cognition and function remains incomplete. Network neuroscience makes substantial efforts to manipulate these challenges and provide real answers. For the last decade, researchers have been modelling brain structure and function via a graph or network that comprises brain regions that are either anatomically connected via tracts or functionally via a more extensive repertoire of functional associations. Network neuroscience is a relatively new multidisciplinary scientific avenue of the study of complex systems by pursuing novel ways to analyze, map, store and model the essential elements and their interactions in complex neurobiological systems, particularly the human brain, the most complex system in nature. Due to a rapid expansion of neuroimaging data sets' size and complexity, it is essential to propose and adopt new empirical tools to track dynamic patterns between neurons and brain areas and create comprehensive maps. In recent years, there is a rapid growth of scientific interest in moving functional neuroimaging analysis beyond simplified group or time‐averaged approaches and sophisticated algorithms that can capture the time‐varying properties of functional connectivity. We describe algorithms and network metrics that can capture the dynamic evolution of functional connectivity under this perspective. We adopt the word ‘chronnectome’ (integration of the Greek word ‘Chronos’, which means time, and connectome) to describe this specific branch of network neuroscience that explores how mutually informed brain activity correlates across time and brain space in a functional way. We also describe how good temporal mining of temporally evolved dynamic functional networks could give rise to the detection of specific brain states over which our brain evolved. This characteristic supports our complex human mind. The temporal evolution of these brain states and well‐known network metrics could give rise to new analytic trends. Functional brain networks could also increase the multi‐faced nature of the dynamic networks revealing complementary information. Finally, we describe a python module (https://github.com/makism/dyconnmap) which accompanies this article and contains a collection of dynamic complex network analytics and measures and demonstrates its great promise for the study of a healthy subject's repeated fMRI scans.

DReSS: a method to quantitatively describe the influence of structural perturbations on state spaces of genetic regulatory networks

Structures of genetic regulatory networks are not fixed. These structural perturbations can cause changes to the reachability of systems' state spaces. As system structures are related to genotypes and state spaces are related to phenotypes, it is important to study the relationship between structures and state spaces. However, there is still no method can quantitively describe the reachability differences of two state spaces caused by structural perturbations. Therefore, Difference in Reachability between State Spaces (DReSS) is proposed. DReSS index family can quantitively describe differences of reachability, attractor sets between two state spaces and can help find the key structure in a system, which may influence system's state space significantly. First, basic properties of DReSS including non-negativity, symmetry and subadditivity are proved. Then, typical examples are shown to explain the meaning of DReSS and the differences between DReSS and traditional graph distance. Finally, differences of DReSS distribution between real biological regulatory networks and random networks are compared. Results show most structural perturbations in biological networks tend to affect reachability inside and between attractor basins rather than to affect attractor set itself when compared with random networks, which illustrates that most genotype differences tend to influence the proportion of different phenotypes and only a few ones can create new phenotypes. DReSS can provide researchers with a new insight to study the relation between genotypes and phenotypes.

Brain network similarity: methods and applications

Graph theoretical approach has proved an effective tool to understand, characterize, and quantify the complex brain network. However, much less attention has been paid to methods that quantitatively compare two graphs, a crucial issue in the context of brain networks. Comparing brain networks is indeed mandatory in several network neuroscience applications. Here, we discuss the current state of the art, challenges, and a collection of analysis tools that have been developed in recent years to compare brain networks. We first introduce the graph similarity problem in brain network application. We then describe the methodological background of the available metrics and algorithms of comparing graphs, their strengths, and limitations. We also report results obtained in concrete applications from normal brain networks. More precisely, we show the potential use of brain network similarity to build a "network of networks" that may give new insights into the object categorization in the human brain. Additionally, we discuss future directions in terms of network similarity methods and applications.

SimiNet: A Novel Method for Quantifying Brain Network Similarity

Quantifying the similarity between two networks is critical in many applications. A number of algorithms have been proposed to compute graph similarity, mainly based on the properties of nodes and edges. Interestingly, most of these algorithms ignore the physical location of the nodes, which is a key factor in the context of brain networks involving spatially defined functional areas. In this paper, we present a novel algorithm called "SimiNet" for measuring similarity between two graphs whose nodes are defined a priori within a 3D coordinate system. SimiNet provides a quantified index (ranging from 0 to 1) that accounts for node, edge and spatiality features. Complex graphs were simulated to evaluate the performance of SimiNet that is compared with eight state-of-art methods. Results show that SimiNet is able to detect weak spatial variations in compared graphs in addition to computing similarity using both nodes and edges. SimiNet was also applied to real brain networks obtained during a visual recognition task. The algorithm shows high performance to detect spatial variation of brain networks obtained during a naming task of two categories of visual stimuli: animals and tools. A perspective to this work is a better understanding of object categorization in the human brain.

An ANOVA approach for statistical comparisons of brain networks

The study of brain networks has developed extensively over the last couple of decades. By contrast, techniques for the statistical analysis of these networks are less developed. In this paper, we focus on the statistical comparison of brain networks in a nonparametric framework and discuss the associated detection and identification problems. We tested network differences between groups with an analysis of variance (ANOVA) test we developed specifically for networks. We also propose and analyse the behaviour of a new statistical procedure designed to identify different subnetworks. As an example, we show the application of this tool in resting-state fMRI data obtained from the Human Connectome Project. We identify, among other variables, that the amount of sleep the days before the scan is a relevant variable that must be controlled. Finally, we discuss the potential bias in neuroimaging findings that is generated by some behavioural and brain structure variables. Our method can also be applied to other kind of networks such as protein interaction networks, gene networks or social networks.