The topology of the directed clique complex as a network invariant

TL;DR

This paper introduces new topological invariants for directed networks using the directed clique complex, enabling mathematical analysis of network shape and dynamics, with applications to neural networks and network invariants.

Contribution

It presents novel algebro-topological invariants based on the directed clique complex, including the Euler characteristic and Betti numbers, for analyzing directed network topology and dynamics.

Findings

Topological features influence neural network evolution.

Euler characteristic can assess functional network features.

Directed clique complex defines a new network invariant.

Abstract

We introduce new algebro-topological invariants of directed networks, based on the topological construction of the directed clique complex. The shape of the underlying directed graph is encoded in a way that can be studied mathematically to obtain network invariants such as the Euler characteristic and the Betti numbers. Two different cases illustrate the application of the Euler characteristic. We investigate how the evolution of a Boolean recurrent artificial neural network is influenced by its topology in a dynamics involving pruning and strengthening of the connections, and to show that the topological features of the directed clique complex influence the dynamical evolution of the network. The second application considers the directed clique complex in a broader framework, to define an invariant of directed networks, the network degree invariant, which is constructed by computing the topological invariant on a sequence of sub-networks filtered by the minimum in- or out-degree of the nodes. The application of the Euler characteristic presented here can be extended to any directed network and provides a new method for the assessment of specific functional features associated with the network topology.