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The Configuration Model for Partially Directed Graphs
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0003-0233-0022
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 22015 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 161, no 4, p. 965-985Article in journal (Refereed) Published
Abstract [en]

The configuration model was originally defined for undirected networks and has recently been extended to directed networks. Many empirical networks are however neither undirected nor completely directed, but instead usually partially directed meaning that certain edges are directed and others are undirected. In the paper we define a configuration model for such networks where vertices have in-, out-, and undirected degrees that may be dependent. We prove conditions under which the resulting degree distributions converge to the intended degree distributions. The new model is shown to better approximate several empirical networks compared to undirected and completely directed networks.

Place, publisher, year, edition, pages
2015. Vol. 161, no 4, p. 965-985
Keywords [en]
Configuration model, Partially directed, Semi-directed, Degree distribution, Asymptotic convergence
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-123511DOI: 10.1007/s10955-015-1360-4ISI: 000363257600010OAI: oai:DiVA.org:su-123511DiVA, id: diva2:875122
Funder
Riksbankens Jubileumsfond, P12-0705:1Swedish Research Council, 2009-5759Available from: 2015-11-30 Created: 2015-11-27 Last updated: 2018-04-12Bibliographically approved
In thesis
1. Random networks with weights and directions, and epidemics thereon
Open this publication in new window or tab >>Random networks with weights and directions, and epidemics thereon
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Networks, consisting of nodes and of edges, can be used to model numerous phenomena, e.g, web pages linking to each other or interactions between people in a population. Edges can be directed, such as a one way link from one web page to another, or undirected (bi-directional), such as physical contacts between pairs of people, which potentially could spread an infection either way between them. Edges can also have weights associated with them, in this thesis corresponding to the probability that an infection is transmitted on the edge.

Empirical networks are often only partially known, in the form of ego-centric network data where only a subset of the nodes and the number of adjacent edges of each node have been observed. This situation lends itself well to analysis through the undirected or partially directed configuration model - a random network model where the number of edges of each node (the degree) is given but where the way these edges are connected is random.

The four papers in this thesis are concerned with the properties of the configuration model and with the usefulness of it with respect to its ability to model the spread of epidemics on empirical networks. Paper I proves the asymptotic convergence to a given degree distribution for the partially directed configuration model. In Paper II it is shown that epidemics on some empirical and theoretically constructed networks grow exponentially, similarly to what can be seen on the corresponding configuration models. Finally, in Papers III and IV, large population analytical results for the reproduction number, the probability of a large epidemic outbreak and the final size of such an outbreak are derived assuming a configuration model network with weighted and/or partially directed edges. These results are then evaluated on several large empirical networks upon which epidemics are simulated. We find that on some of these networks the analytical expressions are compatible with the results of the simulations. This makes the model useful as a tool for analyzing such networks.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2018. p. 28
Keywords
Epidemics, Reproduction number, Final size, Large outbreak, Weighted network, Undirected, Partially directed, Configuration model, Copula
National Category
Probability Theory and Statistics Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-154930 (URN)978-91-7797-278-5 (ISBN)978-91-7797-279-2 (ISBN)
Public defence
2018-05-31, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 14:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2009-5759Riksbankens Jubileumsfond, P12-0705:1Swedish Research Council, 2016-04566Swedish Research Council, 2015-05015
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2018-05-07 Created: 2018-04-12 Last updated: 2018-05-04Bibliographically approved

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