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# Centrality in Time- Dependent Networks

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My slides for my keynote talk at the NetSci 2018 (#NetSci2018) conference in Paris, France (June 2018). This talk will take place on Thursday 13 June in the morning.

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### Centrality in Time- Dependent Networks

1. 1. Centrality in Time- Dependent Networks Mason A. Porter (@masonporter) Department of Mathematics, UCLA
2. 2. Outline ◦ Introduction ◦ Motivation ◦ Centrality in Time-Independent Networks ◦ Centrality in Temporal Networks ◦ Eigenvector-Based Centralities in Multilayer Representations of Time-Dependent Networks ◦ “Tie-Decay” Networks in Continuous Time ◦ Example: Generalization of PageRank Centrality ◦ Conclusions
3. 3. The Top-5 Hobbies of Network Scientists ◦ 5. Citing papers based on preferential attachment (and/or possibly about preferential attachment) ◦ 4. Arguing about power laws ◦ 3. Community detection ◦ 2. Finding ways to get the Zachary Karate Club network into their talks ◦ 1. Developing new centrality measures
5. 5. Some classical notions of centrality ◦ Degree ◦ Closeness centrality ◦ Betweenness centrality and its many variants ◦ Eigenvector-based ◦ Solutions of eigenvalue problems ◦ Eigenvector centrality ◦ PageRank ◦ Hubs and authorities ◦ … ◦ Katz centrality, communicability, etc. ◦ … ◦ …
6. 6. Example: Betweenness Centrality ◦ Which nodes (or edges, or perhaps other substructures) are on a lot of short paths between nodes? ◦ Example: shortest (“geodesic”) paths ◦ Geodesic node betweenness centrality is the number of shortest (“geodesic”) paths through node i divided by the total number of geodesic paths (common convention: i, j, m distinct): ◦ Similar formula for geodesic edge betweenness ◦ One can also deﬁne notions of betweenness based on ideas like random walks (or by restricting to particular paths in useful ways).
7. 7. Example: Eigenvector Centrality ◦ Leading eigenvector v of the adjacency matrix A. The entries of v, which has strictly positive entries by the Perron–Frobenius theorem, give the eigenvector centralities of the nodes.
8. 8. Example: PageRank ◦ Review article: D. F. Gleich, SIAM Review, 2015 B : adjacency matrix
9. 9. Example: Hubs and Authorities ◦ J. M. Kleinberg, Journal of the ACM, Vol. 46: 604–632 (1999) ◦ Intuition: A Web page (node) is a good hub if it has many hyperlinks (out-edges) to important nodes, and a node is a good authority if many important nodes have hyperlinks to it (in-edges) ◦ Imagine a random walker surﬁng the Web. It should spend a lot of time on important Web pages. Steady-state populations of an ensemble of walkers satisfy the eigenvalue problem: ◦ x = aAy ; y = bATx è ATAy = λy & AATx = λx, where λ = 1/(ab) ◦ Leading eigenvalue λ1 (strictly positive) gives strictly positive authority vector x and hub vector y (leading eigenvectors) ◦ Node i has hub centrality xi and authority centrality yi
10. 10. Example application of Hubs and Authorities: Measuring the Quality of Programs in the Mathematical Sciences ◦ Apply the same idea to mathematics departments based on the flow of Ph.D. students ◦ S. A. Meyer, P. J. Mucha, & MAP, “Mathematical genealogy and department prestige”, Chaos, Vol. 21: 041104 (2011) ◦ One-page paper in Gallery of Nonlinear Images ◦ Data from Mathematics Genealogy Project (MGP)
11. 11. MGP with hubs and authorities • We consider MPG data in the US from 1973–2010 (data from 10/09) • Example: Peter Mucha earned a PhD from Princeton and later supervised students at Georgia Tech and UNC Chapel Hill. • è Directed edge of unit weight from Princeton to UNC (and also from Princeton to Georgia Tech) • (Note: several additional students not listed) • A university is a good authority if it hires students from good hubs, and a university is good hub if its students are hired by good authorities. • Caveats • Our measurement has a time delay (only have the PrincetonèUNC edge after Peter supervises a PhD student there) • Hubs and authorities should change in time
12. 12. Geographically-Inspired Visualization Mathematical genealogy and department prestige Sean A. Myers,1 Peter J. Mucha,1 and Mason A. Porter2 1 Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA 2 Mathematical Institute, University of Oxford, OX1 3LB, UK FIG. 1. (Color) Visualizations of a mathematics genealogy network. CHAOS 21, 041104 (2011) Hubs: node size Authorities: node color
13. 13. How do our rankings do? artment son A. Porter2 arolina, 3LB, UK ber 2011) (http://www. 50 000 scholars related ﬁelds. s), graduation ees. The MGP be used to trace ourant, Hilbert, s Gauss, Euler, We use a “geographically inspired” layout to balance node locations and node overlap. A Kamada-Kawai visualization4 or) Visualizations of a mathematics genealogy network. FIG. 2. (Color) Rankings versus authority scores.
14. 14. Generalizing Centralities to Time- Dependent Networks ◦ There have been numerous efforts to generalize centrality measures to time-dependent networks using various approaches. ◦ For some discussions, see the review articles on temporal networks by P. Holme & J. Saramaki (2012) and P. Holme (2015) ◦ A very small selection of examples (using references cited in Taylor et al. 2017) ◦ Calculate centrality from networks constructed from independent time windows: D. Braha & Y. Bar-Yam, 2006 (and many others) ◦ Calculate centrality in a time-independent network constructed from time-respecting paths in a temporal network: G. Kossinets, J. M. Kleinberg, and D. J. Watts (2008) ◦ Calculate (for PageRank) in a way that counteracts age bias: D. Walker, X. Xie, K.-K. Yan, and S. Maslov (2007) ◦ Generalizations, from numerous perspective, of many common centrality measures: betweenness, eigenvector, PageRank, Katz, communicability, etc. [See Taylor et al. for many references.] ◦ Continuous-time approach for Katz centrality (where the generalization was devised speciﬁcally for Katz centrality): P. Grindrod and D. J. Higham, A Dynamical Systems View of Network Centrality, Proc. Royal Soc. A (2014), Vol. 470, 20130835 ◦ Helped inspire our work in Ahmad et al., where our goal was to come up with a ﬂexible formulation for studying temporal networks in continuous time, using PageRank centrality as an example.
15. 15. Using a Multilayer Framework for Eigenvector- Based Centralities for Time-Dependent Networks ◦ D. Taylor, S. A. Myers, A. Clauset, MAP, & P. J. Mucha, “Eigenvector-based Centrality Measures for Temporal Networks”, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2017 ◦ Uses a multilayer representation of time- dependent networks to study important “central” entities and their dynamics over time M. Kivelä et al., 2014ZKCC Network
16. 16. Supra-adjacency Matrix (‘flattened’ linear-algebraic representation) • Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, & S. D. Howison [2016] Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 14(1): 1–41 13 Layer 1 11 21 31 Layer 2 12 22 32 Layer 3 13 23 33 ! 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0 1 1 ! 0 0 0 0 0 1 0 0 0 ! 0 0 0 0 1 0 0 0 0 ! 0 0 0 ! 0 0 0 1 1 ! 0 0 0 ! 0 1 0 1 0 ! 0 0 0 ! 1 1 0 0 0 ! 0 0 0 ! 0 0 0 1 0 0 0 0 0 ! 0 1 0 1 0 0 0 0 0 ! 0 1 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Fig. 3.1. Example of (left) a multilayer network with unweighted intra-layer connections (solid lines) and uniformly weighted inter-layer connections (dashed curves) and (right) its corresponding adjacency matrix. (The adjacency matrix that corresponds to a multilayer network is sometimes called a “supra-adjacency matrix” in the network-science literature [39].) or an adjacency matrix to represent a multilayer network.) The generalization in [49] consists of applying the function in (2.16) to the N|T |-node multilayer network:
17. 17. Multilayer Construction to Examine Temporal Eigenvector-Based Centralities ◦ Math department rankings change in time, so we want to consider centrality measures that change in time ◦ E.g. via a multilayer representation of a temporal network ◦ “Multislice” network with adjacency tensor elements Aijt ◦ Directed intralayer edge from university i to university j at time t for a speciﬁc person’s PhD granted at time t for a person who later advised a student at i (multi-edges give weights) ◦ E.g. Peter Mucha yields a Georgia TechèPrinceton edge and a UNC Chapel HillèPrinceton edge for t = 1998 ◦ Use a multilayer network with “diagonal” and ordinal interlayer coupling ◦ 231 US universities, T = 65 time layers (1946–2010)
18. 18. Construct a Supra-centrality Matrix ◦ E.g. M(t) = A(t)[A(t)]T to examine hubs and authorities ◦ A different choice of M(t) gives a temporal generalization of a different eigenvector-based centrality measure ◦ ε = 1/ω ; t indexes the layers ◦ A singular perturbation from the ε è 0 (strong coupling) limit yields an averaged (time-independent) centrality, then a “first mover” perturbation term, and then higher- order corrections 2.2. Inter-Layer Coupling of Centrality Matrices. To avoid that arise from ignoring the distinction between inter-layer edges and in we deﬁne a somewhat more nuanced generalization of eigenvector-bas To preserve the special role of inter-layer edges, we directly couple the defne the eigenvector-based centrality measure within each temporal dinary adjacency matrices for eigenvector centrality). That is, one eigenvector-based centrality in terms of some matrix M that is a f adjacency matrix A. For example, hub and authority scores are the lea tors of the matrices AAT and AT A, respectively (using the convention Aij indicate i ! j edges). Letting M(t) denote the centrality matrix couple these centrality matrices with inter-layer couplings of strength ! supra-centrality matrix M(✏) = 2 6 6 6 6 6 4 ✏M(1) I 0 · · · I ✏M(2) I ... 0 I ✏M(3) ... ... ... ... ... 3 7 7 7 7 7 5 . For notational convenience, we deﬁne the supra-centrality matrix us factor ✏ = 1/! rather than the coupling weight !. [That is, we re by a factor ✏ to obtain Eq. (2.4).] We study the dominant eigenvect corresponds to the largest eigenvalue max(✏) [i.e, M(✏) (✏) = max entries of the dominant eigenvector give the centralities of the node-la this represents the centrality of physical node i at time t. As we Sec. 2.3, we refer to this type of centrality as a “joint node-layer cent it reﬂects the centrality of both the physical node i and the time layer One can interpret the parameter ✏ > 0 as a tuning parameter tha strongly a given physical node’s centrality is coupled to itself between ne