Assessing diversity in multiplex networks

Abstract

Diversity, understood as the variety of different elements or configurations that an extensive system has, is a crucial property that allows maintaining the system’s functionality in a changing environment, where failures, random events or malicious attacks are often unavoidable. Despite the relevance of preserving diversity in the context of ecology, biology, transport, finances, etc., the elements or configurations that more contribute to the diversity are often unknown, and thus, they can not be protected against failures or environmental crises. This is due to the fact that there is no generic framework that allows identifying which elements or configurations have crucial roles in preserving the diversity of the system. Existing methods treat the level of heterogeneity of a system as a measure of its diversity, being unsuitable when systems are composed of a large number of elements with different attributes and types of interactions. Besides, with limited resources, one needs to find the best preservation policy, i.e., one needs to solve an optimization problem. Here we aim to bridge this gap by developing a metric between labeled graphs to compute the diversity of the system, which allows identifying the most relevant components, based on their contribution to a global diversity value. The proposed framework is suitable for large multiplex structures, which are constituted by a set of elements represented as nodes, which have different types of interactions, represented as layers. The proposed method allows us to find, in a genetic network (HIV-1), the elements with the highest diversity values, while in a European airline network, we systematically identify the companies that maximize (and those that less compromise) the variety of options for routes connecting different airports.

Measuring and mitigating behavioural segregation using Call Detail Records

“…While we admit that these unaddressed aspects regarding other socio-cultural factors or different levels of detail of the Communication Network [57] should be considered in subsequent works, the estimations given here can be of practical use in several ways. First, the developed procedure provides estimations as to the level of integration that can be achieved by using social and residential mixing strategies.…”

The Multiplex Efficiency Index: unveiling the Brazilian air transportation multiplex network—BATMN

“…Next we present the -index, which is based on the concept of multiplexed network diversity 6 . Diversity, as a general concept, can be understood as the level of heterogeneity of a system 19 , 20 .…”

“…Formally, a multiplex network with M layers, each one having the same set of nodes N , is represented by adjacency matrices . From these adjacency matrices it is possible to calculate the distance distribution of nodes (NDD) 6 for each node i in the layer ( ). It corresponds to the fraction of nodes that are at distance d (shortest path) of node i in layer ; and the Transition Matrix ( ) provides the probability that the node j in the layer can be reached, in a single step, by a random walker located in node i in .…”

“…In this sense, supported by the definition of diversity of a multiplex network introduced in a recent work 6 , we present the Multiplex Efficiency Index ( -index), which is a modified version of the diversity measure capable of quantifying the connectivity patterns of a multiplex network on a normalized scale from 0 to 1, which is independent of the number of layers.…”

“…There are numerous possibilities of applications of the -index in different areas. For example, to quantify connection differences in social networks 7 , showing user sets that have similar or distinct connections, to analyze networks of criminal relationships 8 , to point out similarities (or differences) between the preferences of a group of people or to verify the protein interactions of the human HIV-1 virus 6 , to check the supply of transportation modalities for a given region 9 , among many others.…”