"Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling" (Second Edition)

The unnecessary second edition of the book Temporal Network Theory is now out. It includes a second edition of a chapter that I coauthored. Here are a few details.

Title: Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

Authors: Dane Taylor, Mason A. Porter, and Peter J. Mucha

Abstract: We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change with time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a supracentrality matrix C(ω), where ω controls the extent to which centrality trajectories change with time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of C(ω) represent joint centralities, which simultaneously quantify the importances of every node in every time layer. Inspired by probability theory, we also compute marginal and conditional centralities. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes’ centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large ω.

What Happens in Providence Stays in Providence

I am heading off to Providence to participate in the first roughly 1.5 days of ICERM's workshop on Mathematical Challenges in Neuroscience Network Dynamics.

"Role Detection in Bicycle-Sharing Networks Using Multilayer Stochastic Block Models"

A new paper of mine was published in final form today. The project started in January 2017 as a group project by students in the first course that I ever taught at UCLA. It's taken awhile, but we're finally done!

Title: Role Detection in Bicycle-Sharing Networks Using Multilayer Stochastic Block Models

Authors: Jane Carlen†, Jaume de Dios Pont, CassidyMentus, Shyr-Shea Chang, Stephanie Wang, and Mason A. Porter

Abstract: In urban systems, there is an interdependency between neighborhood roles and transportation patterns between neighborhoods. In this paper, we classify docking stations in bicycle-sharing networks to gain insight into the human mobility patterns of three major cities in the United States. We propose novel time-dependent stochastic block models, with degree-heterogeneous blocks and either mixed or discrete block membership, which classify nodes based on their time-dependent activity patterns. We apply these models to (1) detect the roles of bicycle-sharing stations and (2) describe the traffic within and between blocks of stations over the course of a day. Ourmodels successfully uncover work blocks, home blocks, and other blocks; they also reveal activity patterns that are specific to each city. Our work gives insights for the design and maintenance of bicycle-sharing systems, and it contributes new methodology for community detection in temporal and multilayer networks with heterogeneous degrees.

"A Multilayer Network Model of the Coevolution of the Spread of a Disease and Competing Opinions"

A new paper of mine just came out in final form. Here are some details.

Title: A Multilayer Network Model of the Coevolution of the Spread of a Disease and Competing Opinions

Authors: Kaiyan Peng, Zheng Lu, Vanessa Lin, Michael R. Lindstrom, Christian Parkinson, Chuntian Wang, Andrea L. Bertozzi, Mason A. Porter

Abstract: During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we construct a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. On one layer, we consider the concurrent evolution of two opinions — pro-physical-distancing and anti-physical-distancing — that compete with each other and have mutual immunity to each other. The disease evolves on the other layer, and individuals are less likely (respectively, more likely) to become infected when they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. Through numerical simulations, we illustrate the influence of opinion dynamics on the spread of the disease from complex interactions both between the two conflicting opinions and between the opinions and the disease. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

"Classical and Quantum Random-Walk Centrality Measures in Multilayer Networks"

Another paper of mine just came out in final form. Here are some details about it.

Title: Classical and Quantum Random-Walk Centrality Measures in Multilayer Networks

Authors: Lucas Böttcher and Mason A. Porter

Abstract: Multilayer network analysis is a useful approach for studying networks of entities that interact with each other via multiple relationships. Classifying the importance of nodes and node-layer tuples is an important aspect of the study of multilayer networks. To do this, it is common to calculate various centrality measures, which allow one to rank nodes and node-layers according to a variety of structural features. In this paper, we formulate occupation, PageRank, betweenness, and closeness centralities in terms of node-occupation properties of different types of continuous-time classical and quantum random walks on multilayer networks. We apply our framework to a variety of synthetic and real-world multilayer networks, and we identify notable differences between classical and quantum centrality measures. Our computations give insights into the correlations between certain centralities that are based on random walks and associated centralities that are based on geodesic paths.

"Tunable Eigenvector-Based Centralities for Multiplex and Temporal Networks"

One of my papers just came out in final form. Here are the details.

Title: Tunable Eigenvector-Based Centralities for Multiplex and Temporal Networks

Authors: Dane Taylor, Mason A. Porter, and Peter J. Mucha

"Inference of Edge Correlations in Multilayer Networks"

A new paper of mine came out in final form today. Here are some details.

Title: Inference of Edge Correlations in Multilayer Networks

Authors: A. Roxana Pamfil, Sam D. Howison, and Mason A. Porter

Abstract: Many recent developments in network analysis have focused on multilayer networks, which one can use toencode time-dependent interactions, multiple types of interactions, and other complications that arise in complexsystems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features,such as community structure. A prominent approach for inferring such structures is the employment of multilayerstochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is thesampling of edges in different layers independently, conditioned on the community labels of the nodes. In thispaper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. Wederive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layercorrelation that reflects the similarity between the connectivity patterns in different layers. Finally, we explainhow to use correlated models for edge “prediction” (i.e., inference) in multilayer networks. By incorporating edgecorrelations, we find that prediction accuracy improves both in synthetic networks and in a temporal network ofshoppers who are connected to previously purchased grocery products.

"The Multiplex Nature of Global Financial Contagions"

Our new article came out today. Here are some details.

Title: "The multiplex nature of global financial contagions"

Authors: R. Maria del Rio-Chanona, Yevgeniya Korniyenko, Manasa Patnam, and Mason A. Porter

Abstract: As illustrated by the 2008 global financial crisis, the financial distress of one country can trigger financial distress in other countries. We examine the problem of identifying such “systemically important” countries (i.e., countries whose financial distress can trigger further distress), which is important for assessing global financial stability. Using data on bilateral financial positions that are split by asset type, we build a multiplex global financial network in which nodes represent countries, edges encode cross-country financial assets of various types, and layers represent asset types. We examine the temporal evolution of a measure of node importance known as MultiRank centrality, and we find that several major European countries decrease in rank and that several major Asian countries increase in rank since 2008. We then develop a multiplex threshold model of financial contagions in which a shock can propagate either within a layer or between layers. We find that the number of systemically important countries can be twice as large when we take into account the heterogeneity of financial exposures (i.e., when using a multiplex network) than in a contagion on an associated aggregate global financial network (i.e., on a monolayer network), as is often examined in other studies. We also study the extent to which buffers can reduce the propagation of financial distress. Our analysis suggests that accounting for both intralayer and interlayer propagation of contagions in a multiplex structure of financial assets is important for understanding interconnected financial systems of countries.

"A Framework for the Construction of Generative Models for Mesoscale Structure in Multilayer Networks"

We first posted a version of this article (now known as "The Beast") on arXiv in 2016, and (as of today) we are finally completely DONE! Here are some details.

Title: A Framework for the Construction of Generative Models for Mesoscale Structure in Multilayer Networks

Authors: Marya Bazzi, Lucas G. S. Jeub, Alex Arenas, Sam D. Howison, and Mason A. Porter

Software: You can find code for the model, as well as the outputs of the computational experiments in our paper, at this page.

Abstract: Multilayer networks allow one to represent diverse and coupled connectivity patterns—such as time-dependence, multiple subsystems, or both—that arise in many applications and which are difficult or awkward to incorporate into standard network representations. In the study of multilayer networks, it is important to investigate mesoscale (i.e., intermediate-scale) structures, such as dense sets of nodes known as communities, to discover network features that are not apparent at the microscale or the macroscale. The ill-defined nature of mesoscale structure and its ubiquity in empirical networks make it crucial to develop generative models that can produce the features that one encounters in empirical networks. Key purposes of such models include generating synthetic networks with empirical properties of interest, benchmarking mesoscale-detection methods and algorithms, and inferring structure in empirical multilayer networks. In this paper, we introduce a framework for the construction of generative models for mesoscale structures in multilayer networks. Our framework provides a standardized set of generative models, together with an associated set of principles from which they are derived, for studies of mesoscale structures in multilayer networks. It unifies and generalizes many existing models for mesoscale structures in fully ordered (e.g., temporal) and unordered (e.g., multiplex) multilayer networks. One can also use it to construct generative models for mesoscale structures in partially ordered multilayer networks (e.g., networks that are both temporal and multiplex). Our framework has the ability to produce many features of empirical multilayer networks, and it explicitly incorporates a user-specified dependency structure between layers. We discuss the parameters and properties of our framework, and we illustrate examples of its use with benchmark models for community-detection methods and algorithms in multilayer networks.

"Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling"

A new book chapter in the edited book Temporal Network Theory was just published in final form. Here are some details.

Titles: Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

Authors: Dane Taylor, Mason A. Porter, and Peter J. Mucha

Abstract: We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change in time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a supracentrality matrix ℂ(𝜔), where ω controls the extent to which centrality trajectories change over time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of ℂ(𝜔) represent joint centralities, which simultaneously quantify the importance of every node in every time layer. Inspired by probability theory, we also compute marginal and conditional centralities. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes’ centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large ω.

"The Use of Multilayer Network Analysis in Animal Behaviour"

One of my articles came out in final form today. Here are some details.

Title: The Use of Multilayer Network Analysis in Animal Behaviour

Authors: Kelly R. Finn, Matthew J. Silk, Mason A. Porter, and Noa Pinter-Wollman

Abstract: Network analysis has driven key developments in research on animal behaviour by providing quantitative methods to study the social structures of animal groups and populations. A recent formalism, known as multilayer network analysis, has advanced the study of multifaceted networked systems in many disciplines. It offers novel ways to study and quantify animal behaviour through connected ‘layers’ of interactions. In this article, we review common questions in animal behaviour that can be studied using a multilayer approach, and we link these questions to specific analyses. We outline the types of behavioural data and questions that may be suitable to study using multilayer network analysis. We detail several multilayer methods, which can provide new insights into questions about animal sociality at individual, group, population and evolutionary levels of organization. We give examples for how to implement multilayer methods to demonstrate how taking a multilayer approach can alter inferences about social structure and the positions of individuals within such a structure. Finally, we discuss caveats to undertaking multilayer network analysis in the study of animal social networks, and we call attention to methodological challenges for the application of these approaches. Our aim is to instigate the study of new questions about animal sociality using the new toolbox of multilayer network analysis.


P.S. There is an easter egg in the paper. Let me know when you find it!

"Layer Communities in Multiplex Networks"

The final coordinates of one of my papers finally appeared with its journal coordinates, although a published version was already available in August 2017. It just took a while for the full special issue in which it appeared to be published, so only now to do we have our volume, issue, and page numbers. This is my first publication in Journal of Statistical Physics.

Here are some other details:

Title: Layer Communities in Multiplex Networks

Authors: Ta-Chu Kao and Mason A. Porter

Abstract: Multiplex networks are a type of multilayer network in which entities are connected to each other via multiple types of connections. We propose a method, based on computing pairwise similarities between layers and then doing community detection, for grouping structurally similar layers in multiplex networks. We illustrate our approach using both synthetic and empirical networks, and we are able to find meaningful groups of layers in both cases. For example, we find that airlines that are based in similar geographic locations tend to be grouped together in a multiplex airline network and that related research areas in physics tend to be grouped together in a multiplex collaboration network.

"WHAT IS... a Multilayer Network"

My "WHAT IS..." article on multilayer networks just appeared in published form today in the December 2018 issue of Notices of the American Mathematical Society.

Armin Straub has taken it upon himself to post a comprehensive list of all "WHAT IS..." articles. Mine is the 149th such article.

"Frequency-Based Brain Networks: From a Multiplex Framework to a Full Multilayer Description"

One of my papers just came out in final form. Here are some details.

Title: Frequency-Based Brain Networks: From a Multiplex Framework to a Full Multilayer Description

Authors: Javier M. Buldú and Mason A. Porter

Abstract: We explore how to study dynamical interactions between brain regions by using functional multilayer networks whose layers represent different frequency bands at which a brain operates. Specifically, we investigate the consequences of considering the brain as (i) a multilayer network, in which all brain regions can interact with each other at different frequency bands; and as (ii) a multiplex network, in which interactions between different frequency bands are allowed only within each brain region and not between them. We study the second-smallest eigenvalue λ2 of the combinatorial supra-Laplacian matrix of both the multiplex and multilayer networks, as λ2 has been used previously as an indicator of network synchronizability and as a biomarker for several brain diseases. We show that the heterogeneity of interlayer edge weights and, especially, the fraction of missing edges crucially modify the value of λ2, and we illustrate our results with both synthetic network models and real data obtained from resting-state magnetoencephalography. Our work highlights the differences between using a multiplex approach and a full multilayer approach when studying frequency-based multilayer brain networks.

Bonus: This paper has an easter egg. Can you find it? (Hint: This is Spinal Tap.)

"Isomorphisms in Multilayer Networks"

This paper first appeared on the arXiv in 2015 and was published in advanced access about a year ago. It finally has its final volume and page coordinates, so I am finally writing a blog entry about. At this stage, the paper certainly doesn't feel "new" anymore, but it has some useful ideas in it, and it's also my first paper in an IEEE journal. This paper is also rather unusual for me, in that my coauthor Mikko Kivelä and I decided that the clearest way to present things would be to write the paper in definition–theorem–proof format. I almost never write papers that way. Anyway, here are a few details.

Title: Isomorphisms in Multilayer Networks

Authors: Mikko Kivelä and Mason A. Porter

Abstract: We extend the concept of graph isomorphisms to multilayer networks with any number of “aspects” (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomorphisms for multiplex networks, temporal networks, networks with both of these features, and more. We reduce each of the multilayer network isomorphism problems to a graph isomorphism problem, where the size of the graph isomorphism problem grows linearly with the size of the multilayer network isomorphism problem. One can thus use software that has been developed to solve graph isomorphism problems as a practical means for solving multilayer network isomorphism problems. Our theory lays a foundation for extending many network analysis methods—including motifs, graphlets, structural roles, and network alignment—to any multilayer network.

"Can Multilayer Networks Advance Animal Behavior Research?"

The final version of a new paper of mine now has its final coordinates in a journal. Here are some details.

Title: Can Multilayer Networks Advance Animal Behavior Research?

Authors: Matthew J. Silk, Kelly R. Finn, Mason A. Porter, and Noa Pinter-Wollman

Abstract: Interactions among individual animals — and between these individuals and their environment — yield complex, multifaceted systems. The development of multilayer network analysis offers a promising new approach for studying animal social behavior and its relation to eco-evolutionary dynamics.

'Working Paper' in Collaboration with International Monetary Fund: "Evolution of the Global Financial Network and Contagion: A New Approach"

A 'working paper' from a collaboration with folks from the International Monetary Fund (IMF) came out today. You can download it from this website. We also hope to submit a version of this work to a journal for publication. Here are some details.

Title: Evolution of the Global Financial Network and Contagion: A New Approach

Authors: Yevgeniya Korniyenko, Manasa Patnam, Rita Maria del Rio-Chanon, and Mason A. Porter

Abstract: This paper studies the interconnectedness of the global financial system and its susceptibility to shocks. A novel multilayer network framework is applied to link debt and equity exposures across countries. Use of this approach—that examines simultaneously multiple channels of transmission and their important higher order effects—shows that ignoring the heterogeneity of financial exposures, and simply aggregating all claims, as often done in other studies, can underestimate the extent and effects of financial contagion.The structure of the global financial network has changed since the global financial crisis, impacted by European bank’s deleveraging and higher corporate debt issuance. Still, we find that the structure of the system and contagion remain similar in that network is highly susceptible to shocks from central countries and those with large financial systems (e.g., the USA and the UK). While, individual European countries (excluding the UK) have relatively low impact on shock propagation, the network is highly susceptible to the shocks from the entire euro area. Another important development is the rising role of the Asian countries and the noticeable increase in network susceptibility to shocks from China and Hong Kong SAR economies.

Nice Footnote Notation for a Paper about Bribery Networks

I am amused by the notation for the footnote superscripts in this paper. :)

The Multiplex Social–Slayage Network of Buffy the Vampire Slayer

Here is the multiplex social–slayage network of Buffy the Vampire Slayer.

The Buffy social network: slayage and love links. https://t.co/xwQ0DoKpAG

— Peter Sheridan Dodds (@peterdodds) March 11, 2018

This figure is definite fodder for talks. Also, if somebody sets up the adjacencies, we should compute some centrality and versatility measures.

"Neither Global nor Local: Heterogeneous Connectivity in Spatial Network Structures of World Migration"

One of my papers, which has had a DOI for about half a year, finally has its final publication coordinates. Notably, this is my first paper in a sociology journal. Here are some details.

Title: Neither Global nor Local: Heterogeneous Connectivity in Spatial Network Structures of World Migration

Authors: Valentin Danchev and Mason A. Porterc

Abstract: For a long time, geographic regions were considered the dominant spatial arbiter of international migration of people. However, since the late 1970s, many scholars have argued that movements reach beyond contiguous regions to connect distant, dispersed, and previously disconnected countries across the globe. The precise structure of world migration, however, remains an open question. We apply network analysis that incorporates spatial information to international migration-stock data to examine what multilateral structures of world migration have emerged from the interplay of regional concentration (local cohesion)and global interconnectedness (global cohesion) for the period 1960–2000. In the world migration network (WMN), nodes represent countries located in geographic space, and edges represent migrants froman origin country who live in a destination country during each decade. We characterize the large-scale structure and evolution of the WMN by algorithmically detecting international migration communities (i.e., sets of countries that are densely connected via migration) using a generalized modularity function for spatial, temporal, and directed networks. Our findings for the whole network suggest that movements in the WMN deviate significantly from the regional boundaries of the world and that international migration communities have become globally interconnected over time. However, we observe a strong variability in the distribution of strengths, neighborhood overlaps, and lengths of migration edges in the WMN. This manifests as three types of communities: global, local, and glocal. We find that long-distance movements in global communities bridge multiple non-contiguous countries, whereas local (and, to a lesser extent, glocal) communities remain trapped in contiguous geographic regions (or neighboring regions) for almost the whole period, contributing to a spatially fragmented WMN. Our findings demonstrate that world migration is neither regionally concentrated nor globally interconnected, but instead exhibits a heterogeneous connectivity pattern that channels unequal migration opportunities across the world.