Multilayered networks
Multilayered networks consist of layers of several networks, where nodes appear in at least one of these layers. The networks are both connected by intralayer links (links in one layer) as well as interlayer links (links between layers). This can be seen in social networks, where multiple types of social ties exist at the same time (private or professional). In real communication networks, such as a peertopeer network, one can draw a logical network (the connectedness of peers with each other) as well as a physical network (the way peers are connected through cables, hubs and data centres).
The research on single layer networks is mostly a simplification of the realworld: a social network is multilayered since we have different networks based on the type of relation with other individuals. Although different layers of a network are mostly partly separated, it is interesting to estimate diffusion and failures propagation between nodes, based on the properties of intralayer and interlayer links.
Furthermore, we have developed Multinet.js, a visualization framework for large, dynamic and multilayered graphs, that works directly in the browser. The source code is also available on Github.
Besides actively doing research in this field, we have prepared a volume with a collection of works, edited by Antonios Garas, which highlights and summarizing recent developments on network theory with respect to multilayered networks. The book is titled "Interconnected Networks", and appeared in Springer's "Complexity" series. http://www.springer.com/physics/complexity?SGWID=04061961277470
Here follows the Table of Contents, summarizing the list of contributions.
 A tipping point in the structural formation of interconnected networks
Alex Arenas and Filippo Radicchi
 Multilayer networks: metrics and spectral properties
Emanuele Cozzo, Guilherme Ferraz de Arruda, Francisco A.Rodrigues and Yamir Moreno
 An ensemble perspective on multilayer networks
Nicolas Wider, Antonios Garas, Ingo Scholtes, Frank Schweitzer
 Interconnecting networks: the role of connector links
J. M. Buldu, R. SevillaEscoboza, J. Aguirre, D. Papo and R. Gutierrez
 Vulnerability of interdependent networks and networks of networks
Michael M. Danziger, Louis M. Shekhtman, Amir Bashan, Yehiel Berezin and Shlomo Havlin
 A unified approach to percolation processes on multiplex networks
G. J. Baxter, D. Cellai, S. N. Dorogovtsev, A. V. Goltsev and J.F. F. Mendes
 How much interconnected should networks be for cooperation to thrive?
Zhen Wang, Attila Szolnoki, Matjaz Perc
 The Cacophony of Interconnected Networks
V. H. P. Louzada, N. A. M. Araujo, J. S. Andrade Jr, and H. J. Herrmann
 Several multiplexes in the same city: The role of socioeconomic differences in urban mobility
Laura Lotero, Alessio Cadillo, Rafael Hurtado and Jesus GomezGardenes
 The weak core and the structure of elites in social multiplex networks
Bernat CorominasMurtra and Stefan Thurner
 Interbank markets and multiplex networks: centrality measures and statistical null models
Leonardo Bargigli, Giovanni di Iasio, Luigi Infante, Fabrizio Lillo and Federico Pierobon
 The financial system as a nexus of interconnected networks
Stefano Battiston, Guido Caldarelli, Marco D’Errico
Selected Publications
Multiplex Network Regression: How do relations drive interactions?

[2017]

Casiraghi, Giona

arXiv eprint
pages: 117

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Abstract We introduce a statistical method to investigate the impact of dyadic relations on complex networks generated from repeated interactions. It is based on generalised hypergeometric ensembles, a class of statistical network ensembles developed recently. We represent different types of known relations between system elements by weighted graphs, separated in the different layers of a multiplex network. With our method we can regress the influence of each relational layer, the independent variables, on the interaction counts, the dependent variables. Moreover, we can test the statistical significance of the relations as explanatory variables for the observed interactions. To demonstrate the power of our approach and its broad applicability, we will present examples based on synthetic and empirical data.
Systemic risk in multiplex networks with asymmetric coupling and threshold feedback

[2016]

Burkholz, Rebekka;
Leduc, Matt;
Garas, Antonios;
Schweitzer, Frank

Physica D,
pages: 6472,
volume: 323324

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Abstract We study cascades on a twolayer multiplex network, with asymmetric feedback that depends on the coupling strength between the layers. Based on an analytical branching process approximation, we calculate the systemic risk measured by the final fraction of failed nodes on a reference layer. The results are compared with the case of a single layer network that is an aggregated representation of the two layers. We find that systemic risk in the twolayer network is smaller than in the aggregated one only if the coupling strength between the two layers is small. Above a critical coupling strength, systemic risk is increased because of the mutual amplification of cascades in the two layers. We even observe sharp phase transitions in the cascade size that are less pronounced on the aggregated layer. Our insights can be applied to a scenario where firms decide whether they want to split their business into a less risky core business and a more risky subsidiary business. In most cases, this may lead to a drastic increase of systemic risk, which is underestimated in an aggregated approach.
Value of peripheral nodes in controlling multilayer scalefree networks

[2016]

Zhang, Yan;
Garas, Antonios;
Schweitzer, Frank

Physical Review E,
pages: 012309,
volume: 93

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Abstract We analyze the controllability of a twolayer network, where driver nodes can be chosen randomly only from one layer. Each layer contains a scalefree network with directed links and the node dynamics depends on the incoming links from other nodes. We combine the indegree and outdegree values to assign an importance value w to each node, and distinguish between peripheral nodes with low w and central nodes with high w. Based on numerical simulations, we find that the controllable part of the network is larger when choosing low w nodes to connect the two layers. The control is as efficient when peripheral nodes are driver nodes as it is for the case of more central nodes. However, if we assume a cost to utilize nodes that is proportional to their overall degree, utilizing peripheral nodes to connect the two layers or to act as driver nodes is not only the most costefficient solution, it is also the one that performs best in controlling the twolayer network among the different interconnecting strategies we have tested.
Generalized Hypergeometric Ensembles: Statistical Hypothesis Testing in Complex Networks

[2016]

Casiraghi, Giona;
Nanumyan, Vahan;
Scholtes, Ingo;
Schweitzer, Frank

ArXiv eprints

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Abstract Statistical ensembles define probability spaces of all networks consistent with given aggregate statistics and have become instrumental in the analysis of relational data on networked systems. Their numerical and analytical study provides the foundation for the inference of topological patterns, the definition of networkanalytic measures, as well as for model selection and statistical hypothesis testing. Contributing to the foundation of these important data science techniques, in this article we introduce generalized hypergeometric ensembles, a framework of analytically tractable statistical ensembles of finite, directed and weighted networks. This framework can be interpreted as a generalization of the classical configuration model, which is commonly used to randomly generate networks with a given degree sequence or distribution. Our generalization rests on the introduction of dyadic link propensities, which capture the degreecorrected tendencies of pairs of nodes to form edges between each other. Studying empirical and synthetic data, we show that our approach provides broad perspectives for community detection, model selection and statistical hypothesis testing.
An ensemble perspective on multilayer networks

[2016]

Wider, Nicolas;
Garas, Antonios;
Scholtes, Ingo;
Schweitzer, Frank

Interconnected Networks

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Abstract We study properties of multilayered, interconnected networks from an ensemble perspective, i.e. we analyze ensembles of multilayer networks that share similar aggregate characteristics. Using a diffusive process that evolves on a multilayer network, we analyze how the speed of diffusion depends on the aggregate characteristics of both intra and interlayer connectivity. Through a blockmatrix model representing the distinct layers, we construct transition matrices of random walkers on multilayer networks, and estimate expected properties of multilayer networks using a meanfield approach. In addition, we quantify and explore conditions on the link topology that allow to estimate the ensemble average by only considering aggregate statistics of the layers. Our approach can be used when only partial information is available, like it is usually the case for realworld multilayer complex systems.
ReactionDiffusion Processes on Interconnected ScaleFree Networks

[2015]

Garas, Antonios

Physical Review E,
pages: 020801(R),
volume: 92

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Abstract We study the twoparticle annihilation reaction A+B→∅ on interconnected scalefree networks, using different interconnecting strategies. We explore how the mixing of particles and the process evolution are influenced by the number of interconnecting links, by their functional properties, and by the interconnectivity strategies in use. We show that the reaction rates on this system are faster than what was observed in other topologies, due to the better particle mixing that suppresses the segregation effect, in line with previous studies performed on single scalefree networks.
Predicting Scientific Success Based on Coauthorship Networks

[2014]

Sarigol, Emre;
Pfitzner, Rene;
Scholtes, Ingo;
Garas, Antonios;
Schweitzer, Frank

EPJ Data Science,
pages: 9,
volume: 3

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Abstract We address the question to what extent the success of scientific articles is due to social influence. Analyzing a data set of over 100000 publications from the field of Computer Science, we study how centrality in the coauthorship network differs between authors who have highly cited papers and those who do not. We further show that a machine learning classifier, based only on coauthorship network centrality measures at time of publication, is able to predict with high precision whether an article will be highly cited five years after publication. By this we provide quantitative insight into the social dimension of scientific publishing  challenging the perception of citations as an objective, socially unbiased measure of scientific success.
Engineering and Mastering Interwoven Systems

[2014]

Tomforde, Sven;
Hähner, Jörg;
Seebach, Hella;
Reif, W.  E.;
Sick, Bernhard;
Wacker, Arno;
Scholtes, Ingo

Proceedings of ARCS 2014  27th International Conference on Architecture of Computing

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Abstract Networked systems are becoming increasingly complex in development and operation. Due to this complexity, it is mostly impossible to follow a simple sequential designdeployuse cycle. Instead, development and operation will become more evolutionary in nature. Additionally, one can observe that individual complex systems are coupled with each other, even though this has never been intended in the early development of these systems. As a result, we are facing interwoven systems – multiple open timevariant systems are coupled and interact having, e.g., different goals and objectives as well as changing system and communication structure. Based on and extending the idea of composing Systems of Systems, this article identiﬁes challenges that are becoming increasingly apparent as the inevitable integration of systems progresses.
The Social Dimension of Information Ranking: A Discussion of Research Challenges and Approaches

[2014]

Schweitzer, Frank;
Scholtes, Ingo;
Pfitzner, Rene

Socioinformatics  The Social Impact of Interactions between Humans and IT

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Abstract The extraction of relevant knowledge from the increasingly large amount
of information available in information repositories is one of the big challenges of our
time. Although it is clear that the social and the information layer of collaborative
knowledge spaces like the World Wide Web (WWW), scholarly publication databases
or Online Social Networks (OSNs) are inherently coupled and thus inseparable, the
question how the ranking and retrieval of information is inﬂuenced by the structure
and dynamics of the social systems that create it has been addressed at most partially.
In this talk, we will highlight associated research questions and challenges from an
ethical, social and computer science perspective and introduce a multiplex network
perspective that integrates both the social and the semantic layer of social information
systems.
