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# Mathematics and Social Networks

These slides are for my talk for the Somerville College Mathematics Reunion ("Somerville Maths Reunion", 6/24/17): http://www.some.ox.ac.uk/event/somerville-maths-reunion/

These slides are for my talk for the Somerville College Mathematics Reunion ("Somerville Maths Reunion", 6/24/17): http://www.some.ox.ac.uk/event/somerville-maths-reunion/

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## Weitere Verwandte Inhalte

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### Mathematics and Social Networks

1. 1. Mathematics and Social Networks Mason A. Porter (@masonporter) Department of Mathematics, UCLA (2007–2016: Tutor in Applied Mathematics, Somerville College)
2. 2. Outline • Introduction and a Few Ideas • Pictures of social networks • Types of networks and mathematical representations • Some old questions about social networks • What is a complex system? • Small worlds • Which nodes are important? • Multilayer Networks • Introduction • Which nodes are important? • Conclusions
3. 3. Introduction • Some motivation, general ideas, and examples
4. 4. • A network has nodes (representing entities), which are connected by edges (representing ties between the entities). The simplest type of network is a graph. • Example: • Members of a karate club connected to others in the club with whom they hung out (left figure) • “The harmonic oscillator of network science” Zachary Karate Club
5. 5. Gangs in Los Angeles
6. 6. Rabbit Warren
7. 7. Types of Networks • Binary networks: 1 if there is a connection and 0 if there isn’t • Weighted networks: Some value if there is a connection (representing strength of connection) and otherwise 0 • Directed networks: awkward? • Bipartite networks: only nodes of different types are connected to each other (e.g. an actor connected to a movie in which he/she appeared) • Time-dependent networks: nodes and/or edges (existence and/or weights) are time-dependent • Multiplex networks: more than one type of edge • Spatial networks: embedded in space • More…
8. 8.  Representing a Network  Adjacency matrix A  This example: binary (“unweighted”)  Aij = 1 if there is a connection between nodes i and j  Aij = 0 if no connection  How do we generalize this representation to weighted, directed, and bipartite examples? Representing a Network
10. 10. Spread of “Fake News” on Social Networks
11. 11. What is a Complex System? • Complexity*: • “The behaviour shown by Complex Systems” • Complex System*: • “A System Whose Behaviour Exhibits Complexity” • I wrote a brief introduction for a general audience on Quora: https://www.quora.com/How-do-I-explain-to-non-mathematical- people-what-non-linear-and-complex-systems-mean * Neil Johnson, Two’s Company,Three is Complexity, Oneworld Publications, 2007.
12. 12. What is a Complex System? My definition follows the way that the US Supreme Court once defined pornography: “I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description [‘complex systems’]; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it.” (adapted from Justice Potter Stewart)
13. 13. What are complex systems? No rigorous definition; some potential features/components*: • The system has a collection of many interacting objects or “agents” • These objects may be affected by memory or feedback • The system may be “open” (influenced by environment) • The system exhibits “emergent phenomena” • Emergent phenomena arise without a central controller * Neil Johnson, Two’s Company,Three is Complexity, Oneworld Publications, 2007.
14. 14. Two Fundamental Ideas in Complex Systems • Emergence: Finding (possibly unexpected) order in high- dimensional systems (large systems of interacting components) • We seek examples of “emergence” in the study of networks. • Chaos: Finding (possibly unexpected) disorder in low- dimensional systems • E.g. Lorenz attractor
15. 15. Networks are Complex Systems • We want to somehow summarize the information in networks to learn something about them. • Important entities? • Important interactions? • Dense sets (“communities”) of entities? • Perhaps connected sparsely to other dense sets? • Sets of behaviorally similar entities? • Structural bottlenecks to dynamical processes (e.g. disease or rumor spreading) • Where should you start a rumor to maximize how far it spreads? And how does this depend on the network structure? • Voting dynamics on networks (“echo chambers” and “majority illusion”) • Etc.
16. 16. 6 Degrees of Karin Erdmann
17. 17. 6 Degrees of Karin Erdmann
18. 18. 6 Degrees of Karin Erdmann
19. 19. 6 Degrees of Karin Erdmann
20. 20. Watts–Strogatz “Small World” Model • Watts–Strogatz ‘small world’ model (reviewed in MAP, Scholarpedia, 2012) • Start with a 1D ring of nodes but connect each node to its k nearest neighbors on each side. Then either rewire nodes with probability p (original model) or add ‘shortcut’ with probability p (Newman–Watts variant). • What happens as p increases from 0?
21. 21. Regime with both short mean geodesic path length (“small world”) and high mean clustering coefficient A small number of shortcuts has a small effect on local clustering, but very quickly shortens the mean geodesic distance from scaling linearly to scaling logarithmically with the number of nodes. Note: Navigation of networks efficiently is even more amazing than the fact that the world is often small.
22. 22. Determining Important (“Central”) Nodes 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. Equilibrium populations of an ensemble of walkers satisfy an 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
23. 23. Application: Ranking Mathematics Programs • 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
24. 24. Hubs and Authorities Among US Mathematics Programs • We consider MPG data in the US from 1973– 2010 (data from 10/09) • Example: I earned a PhD from Cornell and subsequently supervised students at Oxford and UCLA. • è Directed edge of unit weight from Cornell to Oxford (and also from Cornell to UCLA) • 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 Cornell è UCLA edge after I supervise a PhD student there; normally there’s a delay) • Eventually, there will be an edge from Oxford to University of Vermont when Puck Rombach graduates her first Ph.D. student. (I want grandstudents!) • Hubs and authorities should change in time
25. 25. 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, FIG. 1. (Color) Visualizations of a mathematics genealogy network. CHAOS 21, 041104 (2011) Hubs: node size Authorities: node color
26. 26. How do our rankings do? rtment n A. Porter2 olina, LB, UK 2011) (http://www. 000 scholars elated ﬁelds. graduation s. The MGP used to trace rant, Hilbert, We use a “geographically inspired” layout to balance node locations and node overlap. A Kamada-Kawai visualization4 Visualizations of a mathematics genealogy network. FIG. 2. (Color) Rankings versus authority scores.
27. 27. Multilayer Networks Review Article: M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson,Y. Moreno, & MAP, “Multilayer Networks”, Journal of Complex Networks, 2(3): 203–271, 2014.
28. 28. What is a Multilayer Network?
29. 29. General Form of a Multilayer Network • Definition of a multilayer network M – M = (VM,EM,V,L) • V: set of nodes (“entities”) – As in ordinary graphs • L: sequence of sets of possible layers – One set for each additional “aspect” d ≥ 0 beyond an ordinary network (examples: d = 1 in schematic on this page; d = 2 on last page) • VM: set of tuples that represent node-layers • EM: multilayer edge set that connects these tuples • Note 1: allow weighted multilayer networks by mapping edges to real numbers with w: EM èR • Note 2: d = 0 yields the usual single-layer (“monolayer”) networks
30. 30. Supra-Adjacency Matrices (from “flattening” tensors) • 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: N|T | ✓ ◆
31. 31. Example: Multiplex Network (e.g. multirelational social network) • Monster movement in the game “Munchkin Quest”
32. 32. My Ego-Centric Multiplex Network (edge-colored multigraph)
33. 33. Example: Interconnected Network (e.g. UK infrastucture) (CourtesyofScottThacker,ITRC,UniversityofOxford)
34. 34. Time-dependent centrality from multilayer representation of time-dependent network • Math department rankings change in time, so we need centrality measures that change in time • E.g. via a multilayer representation of a temporal network • Multilayer 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. I would yield an OxfordèCornell edge and a UCLAèCornell edge for t = 2002 • Though neither of these is actually in the analyzed data set • Use a multilayer network with “diagonal” and ordinal interlayer coupling • 231 US universities, T = 65 time layers (1946–2010)
35. 35. Ranking Math Programs: Best Authorities • Note: We construct a “supra-centrality matrix” and do a singular perturbation expansion (and examine the coefficients of the expansion). 18 S. A. MYERS et al. Table 4.1 Top centralities and ﬁrst-order movers for universities in the MGP [4]. Top Time-Averaged Centralities Top First-Order Mover Scores Rank University ↵i 1 MIT 0.6685 2 Berkeley 0.2722 3 Stanford 0.2295 4 Princeton 0.1803 5 Illinois 0.1645 6 Cornell 0.1642 7 Harvard 0.1628 8 UW 0.1590 9 Michigan 0.1521 10 UCLA 0.1456 Rank University mi 1 MIT 688.62 2 Berkeley 299.07 3 Princeton 248.72 4 Stanford 241.71 5 Georgia Tech 189.34 6 Maryland 186.65 7 Harvard 185.34 8 CUNY 182.59 9 Cornell 180.50 10 Yale 159.11 map: do we want to indicate what any of those other papers do with the MGP data?drt: my vote is no. keep things as brief as possible. We extend our previous consideration of this data [71] by keeping the year that each faculty member graduated with his/her Ph.D. degree. We thus construct a multilayer network of the MGP Ph.D. exchange using elements Aijt that indicate a directed edge from university i to university j at time t to represent a doctoral degree
36. 36. Conclusions
37. 37. Conclusions • Mathematical ideas have long been used in the study of social networks. • ”Network science” is a vibrant and ever-growing area of mathematics. • Note: major connections to data science, graph theory, probability, statistics, dynamical systems, statistical mechanics, optimization, computer science, and more • The study of “multilayer networks”, currently the most prominent area of network science, allows one to examine (and integrate) heterogeneous types of information.