Modern social networks frequently encompass multiple distinct types of connectivity information; for instance, explicitly acknowledged friend relationships might complement behavioral measures that link users according to their actions or interests. One way to represent these networks is as multi-layer graphs, where each layer contains a unique set of edges over the same underlying vertices (users). Edges in different layers typically have related but distinct semantics; depending on the application multiple layers might be used to reduce noise through averaging, to perform multifaceted analyses, or a combination of the two. However, it is not obvious how to extend standard graph analysis techniques to the multi-layer setting in a flexible way. In this paper we develop latent variable models and methods for mining multi-layer networks for connectivity patterns based on noisy data.
Recently, there has been a growing interest in the multi-level network problem. Some basic network properties have been extended to the multilevel structure @cite_5 , @cite_24 as well as some results that serve as an extension of single layer concepts, such as multi-level network growth @cite_2 and spreading of epidemics @cite_15 . The metrics that have been proposed attempt to incorporate the dependence of the layers into the statistical framework, which allows for a much richer view of the network. In the same vein, the approach described in this paper performs parameter inference on a multi-level network, incorporating some of the dependence information that the multi-level structure allows.
@cite_24 @cite_5 @cite_15 @cite_2
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There is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously. This is the case for example in social networks where each individual node has different kind of social ties or transportation systems where each location is connected to another location by different types of transport. Many of these multiplex are characterized by a significant overlap of the links in different layers. In this paper we introduce a statistical mechanics framework to describe multiplex ensembles. A multiplex is a system formed by N nodes and M layers of interactions where each node belongs to the M layers at the same time. Each layer @math is formed by a network @math . Here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer. This implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers. Moreover we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplex in these ensembles. Finally we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes. Many real networks are not isolated from each other but form networks of networks, often interrelated in non trivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately, and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations. We propose a modeling framework for growing multiplexes where a node can belong to different networks. We define new measures for multiplexes and we identify a number of relevant ingredients for modeling their evolution such as the coupling between the different layers and the arrival time distribution of nodes. The topology of the multiplex changes significantly in the different cases under consideration, with effects of the arrival time of nodes on the degree distribution, average shortest paths and interdependence.