An application of neighbourhoods in digraphs to the classification of binary dynamics

Abstract

A binary state on a graph means an assignment of binary values to its vertices. A time dependent sequence of binary states is referred to as binary dynamics. We describe a method for the classification of binary dynamics of digraphs, using particular choices of closed neighbourhoods. Our motivation and application comes from neuroscience, where a directed graph is an abstraction of neurons and their connections, and where the simplification of large amounts of data is key to any computation. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. We consider existing and introduce new real-valued functions on closed neighbourhoods, comparing them by their ability to accurately classify different binary dynamics. We describe a classification algorithm that uses two parameters and sets up a machine learning pipeline. We demonstrate the effectiveness of the method on simulated activity on a digital reconstruction of cortical tissue of a rat, and on a non-biological random graph with similar density.

Heterogeneous and non-random cortical connectivity undergirds efficient, robust and reliable neural codes

“…A value of 1 indicates the original connectome has the same percentage of reciprocal connections than its corresponding control; values above/below 1 represent over/under expression; vertical bars show the standard deviation. All values are significantly different than the mean with p-values under 3 × 10 − 1 55 for a two sided one sample t-test. C: Overexpression of simplex motifs with respect to 10 random controls, which by design match the counts of the original network for dimensions 0 and 1.…”

“…We have split the parameters into three categories: Activity, which considers only the activity of the neighborhoods and not their network structure, Topology, which consider a topological metric of the active subgraphs and Spectral, which consider a spectral property of the active subgraphs. Detailed definitions of all network the metrics can be found in Conceição et al 55 .…”

Simplicial $q$-connectivity of directed graphs with applications to network analysis

“…The microcircuit also facilitates simulation of neuronal activity. Recently, various simulated activities were classified with high accuracy using feature vectors constructed from the network struc-ture [13,33]. The microcircuit is obtainable from [31].…”

“…In topological data analysis (TDA) particularly the advent of applying topological tools to questions in neuroscience has spawned interest in constructing topological spaces out of digraphs, developing computational tools for obtaining topological information, and using these to understand networks and phenomena they support. For a progression of works on these ideas, see [13,29,32,33]. Our main example of a topological space on a digraph G is the directed flag complex, which is constructed from the directed cliques of G. For example, a 2-simplex is given by an ordered sequence of vertices (v 0 , v 1 , v 2 ) whenever any ordered pair (v i , v j ), for i < j, is a directed edge in G. By construction the simplices are endowed with an inherent directionality.…”

Hochschild homology, and a persistent approach via connectivity digraphs

“…This type of application in neuroscience has particularly ignited the interest in applied topology and topological data analysis (TDA), together with the subsequent development of computational tools [54,61]. For an application in classifying network (brain) dynamics, see the recent work [23]. One of the main techniques adopted in TDA is persistent homology (PH), which has been employed not just in neuroscience and neuroimaging [16,20,48,50,51,67], but also in fields like endoscopy analysis [28], angiography [6], pulmonary diseases [10], finance [31], fingerprint classification [30], image classification [25], to name a few.…”