### An ensemble perspective on multi-layer networks

, 2015

"... Abstract We study properties of multi-layered, interconnected networks from an ensemble perspective, i.e. we analyze ensembles of multi-layer networks that share similar aggregate characteristics. Using a diffusive process that evolves on a multi-layer network, we analyze how the speed of diffusion ..."

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Abstract We study properties of multi-layered, interconnected networks from an ensemble perspective, i.e. we analyze ensembles of multi-layer networks that share similar aggregate characteristics. Using a diffusive process that evolves on a multi-layer network, we analyze how the speed of diffusion depends on the aggregate characteristics of both intra-and inter-layer connectivity. Through a block-matrix model representing the distinct layers, we construct transition matrices of random walkers on multi-layer networks, and estimate expected properties of multi-layer networks using a mean-field 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 real-world multi-layer complex systems.

### CONTENTS

"... Abstract. This is an incomplete sketch of a theory that produces a Model and a prior on it, from observed data and other explicitly stated prior information. Such a theory shows the potential of explaining the universe around, and inside us. Such a theory is ultimately a theory of ignorance. I cry o ..."

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Abstract. This is an incomplete sketch of a theory that produces a Model and a prior on it, from observed data and other explicitly stated prior information. Such a theory shows the potential of explaining the universe around, and inside us. Such a theory is ultimately a theory of ignorance. I cry out loud: it and bit from not!. Additional information is available online at

### Matching Community Structure Across Online Social Networks

"... The discovery of community structure in networks is a problem of considerable interest in recent years. In online social networks, often times, users are simul-taneously involved in multiple social media sites, some of which share common social relationships. It is of great interest to uncover a sha ..."

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The discovery of community structure in networks is a problem of considerable interest in recent years. In online social networks, often times, users are simul-taneously involved in multiple social media sites, some of which share common social relationships. It is of great interest to uncover a shared community structure across these networks. However, in reality, users typically identify themselves with different usernames across social media sites. This creates a great difficulty in detecting the community structure. In this paper, we explore several approaches for community detection across online social networks with limited knowledge of username alignment across the networks. We refer to the known alignment of usernames as seeds. We investigate strategies for seed selection and its impact on networks with a different fraction of overlapping vertices. The goal is to study the interplay between network topologies and seed selection strategies, and to under-stand how it affects the detected community structure. We also propose several measures to assess the performance of community detection and use them to mea-sure the quality of the detected communities in both Twitter-Twitter networks and Twitter-Instagram networks. 1

### Ranking in interconnected multilayer networks reveals versatile nodes

"... The determination of the most central agents in complex networks is important because they are responsible for a faster propagation of information, epidemics, failures and congestion, among others. A challenging problem is to identify them in networked systems characterized by different types of int ..."

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The determination of the most central agents in complex networks is important because they are responsible for a faster propagation of information, epidemics, failures and congestion, among others. A challenging problem is to identify them in networked systems characterized by different types of interactions, forming interconnected multilayer networks. Here we describe a mathematical framework that allows us to calculate centrality in such networks and rank nodes accordingly, finding the ones that play the most central roles in the cohesion of the whole structure, bridging together different types of relations. These nodes are the most versatile in the multilayer network. We investigate empirical interconnected multilayer networks and show that the approaches based on aggregating—or neglecting—the multilayer structure lead to a wrong identification of the most versatile nodes, overestimating the importance of more marginal agents and demonstrating the power of versatility in predicting their role in diffusive and congestion processes.

### The Hitchhikers Guide to Sharing Graph Data

"... Abstract—A graph is used to represent data in which the relationships between the objects in the data are at least as important as the objects themselves. Over the last two decades nearly a hundred file formats have been proposed or used to provide portable access to such data. This paper seeks to r ..."

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Abstract—A graph is used to represent data in which the relationships between the objects in the data are at least as important as the objects themselves. Over the last two decades nearly a hundred file formats have been proposed or used to provide portable access to such data. This paper seeks to review these formats, and provide some insight to both reduce the ongoing creation of unnecessary formats, and guide the development of new formats where needed. Keywords-Graph, network, XML, GraphML, GML, database, data exchange, data representation.

### COMMUNITY DETECTION IN TEMPORAL MULTILAYER NETWORKS, WITH AN APPLICATION TO CORRELATION NETWORKS∗

"... Networks are a convenient way to represent complex systems of interacting entities. Many networks contain “communities ” of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks re ..."

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Networks are a convenient way to represent complex systems of interacting entities. Many networks contain “communities ” of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the “modularity ” quality function—which is defined by comparing edge weights in an observed network to expected edge weights in a “null network”—is application-dependent. We differentiate between “null networks ” and “null models” in our discussion of modularity maximization, and we highlight that the same null network can correspond to different null models. We then investigate a multilayer modularity-maximization problem to identify communities in temporal networks. Our multilayer analysis only depends on the form of the maximization problem and not on the specific quality function that one chooses. We introduce a diagnostic to measure persistence of community structure in a multilayer network partition. We prove several results that describe how the multilayer maximization problem measures a trade-off between static community structure within layers and larger values of persistence across layers. We also discuss some computational issues that the popular “Louvain ” heuristic faces with

### doi:10.1093/comnet/cnv028 What are essential concepts about networks?

, 2015

"... Networks have become increasingly relevant to everyday life as human society has become increasingly connected. Attaining a basic understanding of networks has thus become a necessary form of literacy for people (and for youths in particular). At the NetSci 2014 conference, we initiated a year-long ..."

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Networks have become increasingly relevant to everyday life as human society has become increasingly connected. Attaining a basic understanding of networks has thus become a necessary form of literacy for people (and for youths in particular). At the NetSci 2014 conference, we initiated a year-long process to develop an educational resource that concisely summarizes essential concepts about networks that can be used by anyone of school age or older. The process involved several brainstorming sessions on one key question: ‘What should every person living in the 21st century know about networks by the time he/she finishes secondary education? ’ Different sessions reached diverse participants, which included professional researchers in network science, educators and high-school students. The generated ideas were connected by the students to construct a concept network. We examined community structure in the concept network to group ideas into a set of important themes, which we refined through discussion into seven essential concepts. The students played a major role in this development process by providing insights and perspectives that were often unrecognized by researchers and educators. The final result, ‘Network Literacy: Essential Concepts and Core Ideas’, is now available as a booklet in several different

### DETECTION OF CORE-PERIPHERY STRUCTURE IN NETWORKS USING SPECTRAL METHODS AND GEODESIC PATHS

, 2014

"... We introduce several novel and computationally ecient methods for detecting “core-periphery structure” in networks. Core-periphery structure is a type of meso-scale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices are well-connected bo ..."

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We introduce several novel and computationally ecient methods for detecting “core-periphery structure” in networks. Core-periphery structure is a type of meso-scale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices are well-connected both among themselves and to peripheral vertices, which are not well-connected to any vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that that vertex is a core vertex. Our second method is based on a low-rank approximation of a network’s adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. Additionally, we design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of real-world data sets.