Grantee: ** Universitat Rovira i Virgili**, Tarragona, Spain

Researcher: Alex Arenas, Ph.D.

Grant Title: Multilevel analysis of complex networked systems

https://doi.org/10.37717/220020325

Program Area: Studying Complex Systems

Grant Type: Scholar Award

Amount: $450,000

Year Awarded: 2012

Duration: 6 years

**Multilevel analysis of complex networked systems**

The discovery of the laws governing the structure and dynamics of complex systems is one of the greatest challenges of modern science. These systems are characterized by the presence of different levels of self-organization, each of which determines the behavior of the next. One of the most important properties of complex systems is the impossibility to predict their overall behavior from the knowledge of their individual properties or as if they were isolated system's components. Understanding and predicting the behavior of these systems is of great importance in contexts as diverse as socio-economic, biological or technological fields. The Science of Complex Systems not only provides a way to answer open questions, but also promotes a new language and the introduction of concepts that will allow the tackling of many current and pressing problems.

One way to address the problem of complexity in such systems is by studying their structure and function. During the last decade, important discoveries show that there is a common pattern of self-organization that emerges over and over again in different complex systems. Somehow, the interactions between system elements (species, individuals, genes, proteins, routers) that occur in real systems (physical, biological, social and communication) give rise to networks that share a large number of common features. These are the so-called ''complex networks". Complex network theory is particularly advantageous to explore several aspects of complexity. The fundamental goals of network science are: to describe the structure of interactions between the components, and to assess the emergent behaviour of many-body systems coupled to the underlying structure. Advances on this theory will improve our understanding and modeling capabilities so that we may control or predict the dynamics and function of complex networked systems. In addition, this approach does not rely on a detailed knowledge of the system's components and therefore allows universal results to be obtained that can be generalized with relative ease (e.g., the study of epidemic spreading processes is equivalent to the spread of computer viruses). For example, biological networks like protein interaction networks share many structural (scale freeness) and dynamical (functional modules) features with other seemingly different systems such as the Internet and interaction patterns in social systems. Thus, systems as diverse as peer-to-peer networks, neural systems, socio-technical phenomena or complex biological networks can be studied within a general unified theoretical and computational framework.

Although outstanding results have already been obtained in this research field, we have not yet progressed enough in basic theoretical aspects and in the application of the generated knowledge to the characterization and exploitation of real multi-level time varying complex systems. Moreover, with the unprecedented amount of data at our disposal nowadays, new challenges arise. Even if the datasets are inherently incomplete and noisy, recent analyses show that most of the current tools are becoming rapidly outdated and are not powerful enough to deal with the complexity of many systems. Current tools fail to keep up with the shifting challenges that complex Information and Communication Technologies (ICT) systems pose. For instance, think of a techno-social system like online social networks in which individuals engage in a multitude of categorical (politics, science, sports, technology, etc) layers, giving to each of them a specific weight. Could we predict or simply understand how likely it is that a given rumor, idea, or belief reaches system-wide proportions? Is there a general mechanism behind such kinds of social contagion or are there different mechanisms unique to each categorical level? Are influential individuals globally defined or instead level dependent? Can we understand what gives rise to social uproar as recently witnessed?

We believe that the only way to answer the aforementioned kind of questions is to explicitly deal with the multi-level nature of complex networked systems. However, almost all of the work to date is based on an 'ordinary" 1-layer or simplex view of the networks in question, where every edge (link) is of the same type and consequently considered at the same temporal and topological scale. Generally speaking, the description of networks so far has been developed using a snapshot of the connectivity (a projection), this connectivity being a reflection of instantaneous interactions or accumulated interactions in a certain time window. This description is limiting when trying to understand the intricate variability of real complex systems, which contain many different time scales and coexisting structural patterns forming the real network of interactions. For example, P2P networks are constantly changing, with some of the connections having a very short lifetime, while others are persistent. Interest groups are constantly being developed and growing, and individual nodes participate through different interests at the same time. An accurate description of such complexity should take into account these differences of interactions and their evolution through time. In the last couple of years, the scientific community on networks theory has focused on this issue and proposed a solution that has been commonly referred to as the multiplex network structure.

A Multiplex network has a variety of different connection types between the nodes of the network. For example, a Facebook user is connected to her Facebook friends through the Facebook network, but she may also be connected to the same people (nodes) through follower relationships in Twitter. In order to understand the structure of the person's true social neighborhood, it is necessary not only to determine the people to whom she is connected, but also to account for the different types of connection (e.g. undirected in Facebook, directed in Twitter) that exist. Networks arising in other applications can also change dynamically in time and contain different types of edges. For example, people have different communication and contact patterns depending on the specific circle (family, work, leisure) they currently reside in, or simply when they are traveling or ill.

All the previous questions and challenges clearly call for an important foundational and methodological transformation of network science. A new framework is required which will accommodate current -- and future -- theoretical and algorithmic needs. To this end, we will adopt a radically new idea: to build a mathematical formalism based on rank-four tensors instead of on adjacency matrices to represent and analyze the most general class of networks, i.e., multiplex networks. This will allow generalizing all existing metrics, algorithms and theoretical tools from ''simple" networks to time-varying, inter-dependent (i.e. ''multiplex") networks. In doing this, we will both generate new mathematical models, and test and validate these models on real data.

In this project, we therefore aim at developing a comprehensive theoretical foundation for studying the structure of multiplex networks, and dynamics that run on these networks. We will generate a general theory, capable of unifying the scattered approaches that currently exist for multiplex network structure, and including these as special cases. We will also develop novel approaches for several types of dynamics on multiplex networks, focusing on analytically tractable cases where the impact of the multiplex nature (in contrast to an ordinary network projection) of real networks has important dynamical effects. Our theoretical approaches will be validated using multiplex social network data.