Multilayer Networks

Abstract

The chapter “Multilayer Networks” introduces the topic of interconnected multilayer networks, analyzing them from a few fronts: types of multilayer networks, their mathematical description, the dynamics of random walks, and the centrality (versatility) of nodes. Multilayer networks appear naturally in real data as, in many cases, the relationships (links) between the elements (nodes) can be of different kinds. For example, people can be connected through friendship, family relations, or work relations. This structure can be represented as a network formed by three layers, one for each type of relationship, with the same nodes repeated in all layers. Thus, there are different behaviors in each layer, and interactions between them, which result in more realistic but, at the same time, more complex dynamics. This chapter shows how interconnected multilayer topology describes such networks more accurately than edge coloring does and introduces the tensor formalism used to construct them.

Multilayer networks as embodied consciousness interactions. A formal model approach.

“…is the subset of all tuples containing mixed information about the node and the layers. E M ⊆ V M × V M is the set of pairs of possible nodes [12], and L = {L k } d k=1 represents a sequence of sets L k of elementary layers/networks. k indicates the aspect of every one of those sets, being d the total number of aspects.…”

“…To target these mathematical relationships, we present an algebraic interpretation of multilayer networks [12,13], where the abstract tensor structure introduced in [14] appears as perhaps the simplest way to express the semantics of tuples of layers as needed. Multilayer networks are an extension of Graph theory [12], being the latter the mathematical substrate, explicitly or implicitly, of the most influential neuroscientific models of consciousness [15,16]. Therefore, it might be also the simplest and natural choice to generalize mathematical relationships across brain, body and conscious experience.…”

“…red and blue edges, Figure 2A). Moreover, in order to account for specific dynamical overlapping nodes (phenomenon called multiplexity [12]), one may consider specific rotation layers, as depicted in Figure 2B. The contrast between awake and sleep conditions (right).…”

“…Notice that the function m give us the number of times that two specific vertices are connected, while the function col tells us the set of existing colours in those connections. We will make use of m and col to express these features rather than the notation of [12], presented in section 3.1.…”

“…Note that, in the sense of [12] and section 3.1, all the three operations above do not generate any inter-layer edge among layers but rather the intra-layer edges of a (s + q)-aspect layer from previous s-aspect and q-aspect, in the case of . For instance, for m G (uv) = k and m H (uv) = k with col G (uv) = {c 1 , ..., c s } and col H (uv) = c 1 , ..., c q we have m G H (uv) = k + k and col G H (uv) = c 1 , ..., c s , c 1 , ..., c q , where some of the colours may repeat.…”