Networks/Theory IV

There is no doubt that our lives are dependant on a diverse range of critical systems and infrastructure that are becoming increasingly complex. This raises difficult questions around how do we ensure the safety of these complex engineered systems (where by engineered we mean that they are at least partly deliberately designed), when they exhibit many if not all of the characteristics of complex systems [1, 2]? These properties include emergence, selforganisation, non-linearity and so on.

Nature offers a wealth of systems where basic unities multiply and non-linearly interact with each other to eventually exhibit a synchronous behavior. In particular, biology displays a vast assortment of examples, from myocyte cells coordinating the heartbeats to the synchronous firing pattern of coupled neurons. The importance of self-organized collective behaviors in many biological systems has triggered the interest of scientists since long time, leading to the study of stability and, consequently, robustness of the synchronous state.

Cluster synchronization (a type of synchronization where different groups of nodes in the system of coupled oscillators follow distinct synchronized trajectories) on networks is a broadly analyzed phenomenon characterizing behaviors with wide areas of applicability from neuroscience to consensus dynamics. Analyzing cluster synchronization can be used to understand phenomena such as remote synchronization and the emergence of chimera states.

Random walks on networks, which are usually formulated as first-order Markov processes, are a prevailing tool for constructing various stochastic algorithms to obtain information from network data. The node2vec algorithm [1] provides a flexible biased random walk framework for graph machine learning, which can be applied to tasks such as node classification and link prediction. The behavior of node2vec in these applications is contingent on two parameters that control how a random walker explores the entire network.

In this article we combine the novel mathematical concept of a hierachical hypergraph with the idea of ruled-based stochastic processes. The hierachical hypergraph structural concept is immensely important when dealing with the mathematical abstraction of the idea of 'multi-scale composition' in general. Classical graphs as systems descriptions, and therefore also network theory, lack the ability to describe the structure of multi-scale systems. However, hierachical hypergraphs can describe such a system completely in a consistent way.