Googling the Brain: Discovering Hierarchical and Asymmetric Network Structures, with Applications in Neuroscience

A hierarchy index for networks in the brain reveals a complex entangled organizational structure

Networks involved in information processing often have their nodes arranged hierarchically, with the majority of connections occurring in adjacent levels. However, despite being an intuitively appealing concept, the hierarchical organization of large networks, such as those in the brain, is difficult to identify, especially in absence of additional information beyond that provided by the connectome. In this paper, we propose a framework to uncover the hierarchical structure of a given network, that identifies the nodes occupying each level as well as the sequential order of the levels. It involves optimizing a metric that we use to quantify the extent of hierarchy present in a network. Applying this measure to various brain networks, ranging from the nervous system of the nematode Caenorhabditis elegans to the human connectome, we unexpectedly find that they exhibit a common network architectural motif intertwining hierarchy and modularity. This suggests that brain networks may have evolved to simultaneously exploit the functional advantages of these two types of organizations, viz., relatively independent modules performing distributed processing in parallel and a hierarchical structure that allows sequential pooling of these multiple processing streams. An intriguing possibility is that this property we report may be common to information processing networks in general.

FairRankVis: A Visual Analytics Framework for Exploring Algorithmic Fairness in Graph Mining Models

Graph mining is an essential component of recommender systems and search engines. Outputs of graph mining models typically provide a ranked list sorted by each item's relevance or utility. However, recent research has identified issues of algorithmic bias in such models, and new graph mining algorithms have been proposed to correct for bias. As such, algorithm developers need tools that can help them uncover potential biases in their models while also exploring the impacts of correcting for biases when employing fairness-aware algorithms. In this paper, we present FairRankVis, a visual analytics framework designed to enable the exploration of multi-class bias in graph mining algorithms. We support both group and individual fairness levels of comparison. Our framework is designed to enable model developers to compare multi-class fairness between algorithms (for example, comparing PageRank with a debiased PageRank algorithm) to assess the impacts of algorithmic debiasing with respect to group and individual fairness. We demonstrate our framework through two usage scenarios inspecting algorithmic fairness.

Graph hierarchy: a novel framework to analyse hierarchical structures in complex networks

Trophic coherence, a measure of a graph's hierarchical organisation, has been shown to be linked to a graph's structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex's ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.

How directed is a directed network?

The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring basal nodes, which limit its applicability. Here we propose simple improved definitions of trophic levels and coherence that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.

Hopf Bifurcations in Directed Acyclic Networks of Linearly Coupled Hindmarsh-Rose Systems

This paper addresses the existence of Hopf bifurcations in a directed acyclic network of neurons, each of them being modeled by a Hindmarsh-Rose (HR) neuronal model. The bifurcation parameter is the small parameter corresponding to the ratio of time scales between the fast and the slow dynamics. We first prove that, under certain hypotheses, the single uncoupled neuron can undergo a Hopf bifurcation. Hopf bifurcation occurrences in a directed acyclic network of HR neurons are then discussed. Numerical simulations are carried out to observe these bifurcations and to illustrate the theoretical results.

Spectral algorithms for heterogeneous biological networks

Spectral methods, which use information relating to eigenvectors, singular vectors and generalized singular vectors, help us to visualize and summarize sets of pairwise interactions. In this work, we motivate and discuss the use of spectral methods by taking a matrix computation view and applying concepts from applied linear algebra. We show that this unified approach is sufficiently flexible to allow multiple sources of network information to be combined. We illustrate the methods on microarray data arising from a large population-based study in human adipose tissue, combined with related information concerning metabolic pathways.