Author Archives: bolozna

Networks of Nokia investors (1998-2002)

A network of correlations among investors in the Nokia security. Nodes are anonymized investors on the Helsinki stock exchange, divided into three groups: households, financial institutions, and non-financial institutions. Edges are pairwise correlations between the time series of daily net volumes of the individual investors, spanning 1998-2002. Data used in the article:

S Ranganathan, M Kivelä, and J Kanniainen, “Dynamics of investor spanning trees around dot-com bubble”, PLoS ONE 13 (6), e0198807 (2018)

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Licence:  CC0 1.0 Universal (CC0 1.0) Public Domain Dedication

100 years of citation networks

Citation networks between journals with moving time window (10/5 years) starting from 1900 and ending in 2013. Contains metadata on journal classifications. Built from a database of 630 million citations. The are citation networks used in the article:

Darko Hric, Kimmo Kaski, Mikko Kivelä: “Stochastic Block Model Reveals Maps of Citation Patterns and Their Evolution in Time”

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License: Creative Commons Attribution Non Commercial 4.0 International

Finnish social network

Friendship network of around 8k Finnish users in This is the “lastfm” network used in the article:

Toivonen, R., Kovanen, L., Kivelä, M., Onnela, J. P., Saramäki, J., & Kaski, K. (2009). A comparative study of social network models: Network evolution models and nodal attribute models. Social networks, 31(4), 240-254.

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Licence: “Creative Commons Attribution-NonCommercial-ShareAlike 2.0 UK: England & Wales” social network and genders

Friendship network of around 189k users and their genders. This is the “” network used in the article:

A. Asikainen, G. Iñiguez,  J. Ureña-Carrión,  K. Kaski, M. Kivelä. Cumulative effects of triadic closure and homophily in social networks. Science Advances, Vol. 6, no. 19, eaax7310 (2020)

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Licence: “Creative Commons Attribution-NonCommercial-ShareAlike 2.0 UK: England & Wales”

Mapping temporal-network percolation to weighted, static event graphs

M Kivelä, J Cambe, J Saramäki, M Karsai

The dynamics of diffusion-like processes on temporal networks are influenced by correlations in the times of contacts. This influence is particularly strong for processes where the spreading agent has a limited lifetime at nodes: disease spreading (recovery time), diffusion of rumors (lifetime of information), and passenger routing (maximum acceptable time between transfers). We introduce weighted event graphs as a powerful and fast framework for studying connectivity determined by time-respecting paths where the allowed waiting times between contacts have an upper limit. We study percolation on the weighted event graphs and in the underlying temporal networks, with simulated and real-world networks. We show that this type of temporal-network percolation is analogous to directed percolation, and that it can be characterized by multiple order parameters.

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Stochastic block model reveals maps of citation patterns and their evolution in time

D Hric, K Kaski, M Kivelä

In this study we map out the large-scale structure of citation networks of science journals and follow their evolution in time by using stochastic block models (SBMs). The SBM fitting procedures are principled methods that can be used to find hierarchical grouping of journals that show similar incoming and outgoing citations patterns. These methods work directly on the citation network without the need to construct auxiliary networks based on similarity of nodes. We fit the SBMs to the networks of journals we have constructed from the data set of around 630 million citations and find a variety of different types of groups, such as communities, bridges, sources, and sinks. In addition we use a recent generalization of SBMs to determine how much a manually curated classification of journals into subfields of science is related to the group structure of the journal network and how this relationship changes in time. The SBM method tries to find a network of blocks that is the best high-level representation of the network of journals, and we illustrate how these block networks (at various levels of resolution) can be used as maps of science.

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Multilayer Networks Library

Main features

  • Pure Python
  • Can handle general multilayer networks
  • Multilayer networks and multiplex networks (with automatically generated lazy-evaluation coupling edges)
  • Functionality: Analysis, transformations, reading and writing networks, network models etc.
  • Visualization (using Matplotlib or D3)
  • Integration with NetworkX for monoplex network analysis

Get it from Github.



Isomorphisms in Multilayer Networks

M. Kivelä, M. A. Porter

We extend the concept of graph isomorphisms to multilayer networks with any number of “aspects” (i.e., types of layering), and 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 network 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 such that 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 such as motifs, graphlets, structural roles, and network alignment to any multilayer network.

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Estimating inter-event time distributions from finite observation periods in communication networks

M. Kivelä, M. A. Porter

A diverse variety of processes — including recurrent disease episodes, neuron firing, and communication patterns among humans — can be described using inter-event time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected without assuming any particular shape for the IET distribution. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. We analyze several well-known data sets from the literature, and we find that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

Physical Review E 92 052813

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Structure of triadic relations in multiplex networks

E. Cozzo, M. Kivelä, M. De Domenico, A. Solé-Ribalta, A. Arenas, S. Gómez, M. A. Porter, Y. Moreno

Recent advances in the study of networked systems have highlighted that our interconnected world is composed of networks that are coupled to each other through different ‘layers’ that each represent one of many possible subsystems or types of interactions. Nevertheless, it is traditional to aggregate multilayer networks into a single weighted network in order to take advantage of existing tools. This is admittedly convenient, but it is also extremely problematic, as important information can be lost as a result. It is therefore important to develop multilayer generalizations of network concepts. In this paper, we analyze triadic relations and generalize the idea of transitivity to multiplex networks. By focusing on triadic relations, which yield the simplest type of transitivity, we generalize the concept and computation of clustering coefficients to multiplex networks. We show how the layered structure of such networks introduces a new degree of freedom that has a fundamental effect on transitivity. We compute multiplex clustering coefficients for several real multiplex networks and illustrate why one must take great care when generalizing standard network concepts to multiplex networks. We also derive analytical expressions for our clustering coefficients for ensemble averages of networks in a family of random multiplex networks. Our analysis illustrates that social networks have a strong tendency to promote redundancy by closing triads at every layer and that they thereby have a different type of multiplex transitivity from transportation networks, which do not exhibit such a tendency. These insights are invisible if one only studies aggregated networks.

New Journal of Physics 17(7), 073029

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