Mason A. PorterUniversity of Oxford @masonporter http://people.maths.ox.ac.uk/porterm/ (go to Twitter for link to slides on slideshare)

�  MAP & J. P. Gleeson, “Dynamical Systems on Networks: A Tutorial”, arXiv:1403.7663 (2015) !�  Also see the section where we point to several other survey, review, and

tutorial articles. !

�  P. Holme & J. Saramäki, “Temporal Networks”, Phys. Rep., Vol. 519, No. 3: 97–125 (2012). !

�  M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and MAP, “Multilayer Networks”, J. Cplx. Net., Vol. 2, No. 3: 203–271 (2015). !�  Accompanying tutorial slides that go through this article:

http://www.slideshare.net/masonporter/multilayer-tutorialnetsci2014slightlyupdated!

�  Introduction !

�  Dynamical Systems on Networks !�  Example: Watts threshold model and the helpfulness of random-graph

ensembles !

�  Temporal Networks !�  Dynamics on temporal networks !

�  Random walks on temporal networks !

�  Adaptive voter model!

�  Multilayer representation of temporal networks !

�  Eigenvector-based centralities on temporal networks !

�  Community structure in temporal networks !

�  Multilayer Networks!�  Multiplex networks, networks of networks, and all that jazz !

�  Dynamical systems on multilayer networks !

�  Conclusions!

�  From structure to dynamics, but also “from structure to function”!

�  Networks are ubiquitous, and numerous different types of dynamics occur on networks. !

�  The structure of networks can have a major influence on dynamical processes that occur on networks. !

�  What should we be measuring to study network structure and dynamical systems on networks?!

�  Random-graph ensembles are very useful helping to achieve a better understanding of dynamical process. !�  Mean-field theories and their generalizations yield lower-dimensional

systems. When are such lower-dimensional systems good approximations for dynamical processes on networks?!

�  Has been prominent in all recent Snowbird dynamics conferences!

�  Many sessions in the NS15 workshop (including Bassett’s IP) !

�  Invited talks at DS15 with some relation to network dynamics: !�  Motter (IP1, this morning): control of dynamics on networks !

�  Bertozzi (IP2): mathematics of crime !

�  Ermentrout (IP4): large-scale activity in the brain !

�  Gore (IP5): Cooperation, cheating, and collapse in biological populations !

�  Moehlis (IP6): Brain control!

�  Many minisymposia. A subset: MS5, MS25, MS27, MS30 (all at the same time), MS51, MS66, MS68, MS74, MS76, MS84, MS88, MS89, MS102, MS115, MS117, MS127!

�  States of the nodes (or edges) change much faster than network structure !�  è Dynamics on static networks !

�  Structure of network changes much faster than states of nodes (or edges) !�  è Dynamics of network (“Temporal networks”) !

�  Comparable timescales !�  è Dynamics of network coupled to dynamics on networks !

�  Also: maybe network structure changes so fast that only some properties (but not the “microscopic” connections between individual entities) can be measured reliably, so consider dynamical system on a random-graph ensemble that preserves those properties. !�  è also dynamics on networks (but mean properties over an ensemble of

graphs) !

How does network structure affect dynamics (and vice versa)?

�  Toy (percolation-like) model for social influence !�  D. S. Watts, PNAS, 2002 !�  Each node j has a (frozen) threshold Rj drawn from some

distribution and can be in one of two states (0 or 1) !�  Choose a seed fraction ρ(0) of nodes (e.g. uniformly at random)

to initially be in state 1 (“infected”, “active”, etc.) !�  Updating can be either: !

�  Synchronous: discrete time; update all nodes at once !

�  Asynchronous: “continuous” time; update some fraction of nodes in time step dt!

�  Update rule: Compare fraction of infected neighbors (m/kj) to Rj. Node j becomes infected if m/kj ≥ Rj. Otherwise no change. !�  Variant (Centola-Macy): Compare number of active neighbors (m) rather than

fraction of active neighbors!

�  Monotonicity: Nodes in state 1 stay there forever. !

Response functions for many types of dynamical processes: J. P. Gleeson, PRX, Vol. 3, 021004 (2013)

(see our tutorial article and references therein)

Response functions for many types of dynamical processes: J. P. Gleeson, PRX, Vol. 3, 021004 (2013)

S. Melnik, J. A. Ward, J. P. Gleeson, & MAP, “Multi-Stage Complex Contagions”, Chaos, Vol. 23, No. 1: 013124 (2013)

Passive (S0) !

Active (S1) Hyper-active (S2)

Influences neighbors Influences neighbors, but with bonus influence compared to Active nodes

No influence

Note: S2 ⊆ S1 but Si ⊈ S0 for i > 0

� Peer pressure = total influence experienced by a degree-k node !� P = (m1 + βm2)/k !

�  m1 = number of neighbors in S1 !

�  m2 = number of neighbors in S2 !

� β = bonus influence (β = 0 è only S0 and S1 ; no S2 state) !

� Update step: Node j becomes Si-active if Pj ≥ Rj,i !� 2 different thresholds for each node; chosen

from some distributions !

�  Recall: one can use a convenient random-graph ensemble to gain a better understanding of a dynamical process on a network !

�  (z1,z2)-regular random graphs !�  Have precise knowledge of when nodes have state changes !

�  Fix degree distribution P(k) and possibly also fix joint degree-degree distribution P(k,k’) !�  Otherwise connect uniformly at random!

�  Example: !�  Half of nodes have degree z1 = 4 and the other half have degree z2 = 24 !

�  Ensemble in which each node has on average all but one neighbor from its own degree class !

�  Assume all nodes have identical thresholds R1 = 0.2 and R2 = 0.8 !

�  Consider the case in which S2 activations drive S1 activations. !

D. R. Taylor, et al. arXiv: 1408.1168 (gave a talk this morning)

Holme & Saramäki, Phys. Rep. 2012

Hol

me

& S

aram

äki,

Phys

. Rep

. 201

2

�  Activity times of nodes !

�  Interevent times of edges !�  Naïve aggregation to obtain a static network assumes Poisson statics, but

the times series are bursty (see Bertozzi’s talk) !

�  You are making major assumptions when aggregating to construct a static network, so we need to keep the temporal event statistics in mind!

�  How do the temporal dynamics of nodes and edges affect dynamical processes (e.g., disease spread) on temporal networks?!

�  Ordering of contacts, concurrency of contacts, etc. !

�  Continuous versus discrete time !

�  How do we generalize ideas like measures of node and edge importance (“centrality” measures), community structure, and so on?!

�  T. Hoffmann, MAP, & R. Lambiotte, Phys. Rev. E, Vol 86, No. 4: 046102 (2012) !

�  Let’s consider a 3-node example with different waiting-time statistics for the three edges !GENERALIZED MASTER EQUATIONS FOR NON-POISSON . . . PHYSICAL REVIEW E 86, 046102 (2012)

2 3

1

a

c

bta = 1 tb = 12

tc = 13

FIG. 2. An undirected network with N = 3 nodes and no self-loops. The waiting-time distributions for the edges a, b, and c areexponential, uniform, and Rayleigh, respectively. We denote thecorresponding means of these distributions by ⟨ta⟩, ⟨tb⟩, and ⟨tc⟩,respectively. We show the mean of each distribution using a verticaldashed line.

We plot the temporal evolution of the walker densities in Fig. 3to illustrate the differences between the two processes. Weobtain the walker densities from numerical simulations of therandom walks with all walkers located initially at node 1. (Seethe appendix for computational details.) The system relaxestoward a stationary solution in both cases, but the stationarysolution clearly depends on the nature of the WTDs, as walkerstend to be underrepresented on node 1 for the non-Poissondynamics.

In general, the value of the walker density is not sufficientto define the state of the system. The distribution of restingtimes—that is, the times that a walker spends on a node beforemaking a step—also needs to be specified. Consequently, thewalker density temporarily departs from its steady-state value

FIG. 3. Random-walker density on each node as a function oftime obtained from numerical computations for Poisson (upper panel)and non-Poisson (lower panel) processes. In the former case, wealso plot the analytical solution of the rate equation. The error barsare smaller than the widths of the curves used for plotting. For thenon-Poisson example, we obtained the steady-state walker densityfrom equation (27). The relaxation towards stationarity exhibits kinksin the dynamics that originate from the noncontinuity of the WTDs.

FIG. 4. Random-walker density on each node of the graphillustrated in Fig. 2 as a function of time when the initial conditionis the steady-state solution. Due to the time-dependent nature of thedynamics, the system exhibits transient dynamics before returning toits steady-state solution. The error bars are smaller than the widths ofthe curves used for plotting.

if a process is restarted with its initial walker density equal tothe steady-state value (see Fig. 4). However, a Poisson processhas no memory—as indicated in Eq. (22), its memory kernelis proportional to a delta function—so its temporal evolutionis independent of the distribution of resting times. This canalso be understood as a consequence of a unique property ofexponential distributions: The probability for an event to occurin a time interval dt is independent of the time since the processstarted [38]. Setting the walker density to its steady-state valuefor a Poisson process is thus sufficient for the walker densityto remain steady, which is the result expected from the rateequations (24).

V. EXAMPLES: THEORY AND COMPUTATIONS

We consider the steady-state solutions for two differentexample network topologies (using networks with N = 103

nodes) to demonstrate the consequences of considering non-Poisson processes. We again use undirected networks asexamples.

In general, the steady-state solution of a random walk isgiven by p = D⟨t⟩x, where x is the dominant eigenvector ofthe stochastic matrix T . However, for a Poisson process, thesteady-state solution is pi = 1

Nfor all i because the rate of

change of all walker densities vanishes according to the rateequations (24):

dni

dt= 1

N

⎛

⎝∑

j

λij − "i

⎞

⎠ = 0,

where we have used λij = λji . Consequently, the steady-statesolution of a Poisson random walk on an undirected networkis uniform regardless of the topology of the network and thetransition rates associated with the edges.

For non-Poisson random walks, let us consider situationsfor which the WTDs of all edges are identical. In otherwords, ψij (t) = ψ(t)Aij , where Aij is the binary adjacencymatrix. Because of this assumption, we can obtain analyticalexpressions for steady-state solution and thereby demonstratethat the steady-state solutions are nontrivial despite therestrictions placed on the network and the WTDs.

046102-7

GENERALIZED MASTER EQUATIONS FOR NON-POISSON . . . PHYSICAL REVIEW E 86, 046102 (2012)

2 3

1

a

c

bta = 1 tb = 12

tc = 13

FIG. 2. An undirected network with N = 3 nodes and no self-loops. The waiting-time distributions for the edges a, b, and c areexponential, uniform, and Rayleigh, respectively. We denote thecorresponding means of these distributions by ⟨ta⟩, ⟨tb⟩, and ⟨tc⟩,respectively. We show the mean of each distribution using a verticaldashed line.

We plot the temporal evolution of the walker densities in Fig. 3to illustrate the differences between the two processes. Weobtain the walker densities from numerical simulations of therandom walks with all walkers located initially at node 1. (Seethe appendix for computational details.) The system relaxestoward a stationary solution in both cases, but the stationarysolution clearly depends on the nature of the WTDs, as walkerstend to be underrepresented on node 1 for the non-Poissondynamics.

In general, the value of the walker density is not sufficientto define the state of the system. The distribution of restingtimes—that is, the times that a walker spends on a node beforemaking a step—also needs to be specified. Consequently, thewalker density temporarily departs from its steady-state value

FIG. 3. Random-walker density on each node as a function oftime obtained from numerical computations for Poisson (upper panel)and non-Poisson (lower panel) processes. In the former case, wealso plot the analytical solution of the rate equation. The error barsare smaller than the widths of the curves used for plotting. For thenon-Poisson example, we obtained the steady-state walker densityfrom equation (27). The relaxation towards stationarity exhibits kinksin the dynamics that originate from the noncontinuity of the WTDs.

FIG. 4. Random-walker density on each node of the graphillustrated in Fig. 2 as a function of time when the initial conditionis the steady-state solution. Due to the time-dependent nature of thedynamics, the system exhibits transient dynamics before returning toits steady-state solution. The error bars are smaller than the widths ofthe curves used for plotting.

if a process is restarted with its initial walker density equal tothe steady-state value (see Fig. 4). However, a Poisson processhas no memory—as indicated in Eq. (22), its memory kernelis proportional to a delta function—so its temporal evolutionis independent of the distribution of resting times. This canalso be understood as a consequence of a unique property ofexponential distributions: The probability for an event to occurin a time interval dt is independent of the time since the processstarted [38]. Setting the walker density to its steady-state valuefor a Poisson process is thus sufficient for the walker densityto remain steady, which is the result expected from the rateequations (24).

V. EXAMPLES: THEORY AND COMPUTATIONS

We consider the steady-state solutions for two differentexample network topologies (using networks with N = 103

nodes) to demonstrate the consequences of considering non-Poisson processes. We again use undirected networks asexamples.

In general, the steady-state solution of a random walk isgiven by p = D⟨t⟩x, where x is the dominant eigenvector ofthe stochastic matrix T . However, for a Poisson process, thesteady-state solution is pi = 1

Nfor all i because the rate of

change of all walker densities vanishes according to the rateequations (24):

dni

dt= 1

N

⎛

⎝∑

j

λij − "i

⎞

⎠ = 0,

where we have used λij = λji . Consequently, the steady-statesolution of a Poisson random walk on an undirected networkis uniform regardless of the topology of the network and thetransition rates associated with the edges.

For non-Poisson random walks, let us consider situationsfor which the WTDs of all edges are identical. In otherwords, ψij (t) = ψ(t)Aij , where Aij is the binary adjacencymatrix. Because of this assumption, we can obtain analyticalexpressions for steady-state solution and thereby demonstratethat the steady-state solutions are nontrivial despite therestrictions placed on the network and the WTDs.

046102-7

Poisson

Non-Poisson

�  P. Holme & M. E. J. Newman,Phys. Rev. E, Vol. 74, No. 5: 056108 (2006); R. Durrett et al., PNAS, Vol. 109: 3682–3687 (2012). !

�  Note: see survey and review articles by Thilo Gross and his various collaborators on adaptive networks (part of compiled list of surveys in my tutorial article) !

�  The mechanism in Durrett et al. (modified version of Holme-Newman model): !�  Nodes with opinion 0 and 1!

�  Edges are picked randomly!

�  If the opinions of the incident nodes are different (“discordant” edge), then one (picked randomly) imitates the other with probability 1 – α. Otherwise, the edge is broken and one of the nodes connects to some other randomly-chosen individual!

�  Case (i): connects to random individual with the same opinion !

�  Case (ii): connects to any random individual!

�  Evolution stops when there are no more discordant edges. !

�  Of interest: ρ = fraction of minority individuals at steady state !�  How does ρ depend on α and on initial fraction (u) of nodes with opinion 1?!

�  The phase-transition structure of these two models differs dramatically! !�  See paper for analytics (some

mathematically rigorous) and numerics to illustrate the contrasting situations!

�  Case (i): rewire to random individual with same opinion !�  There is a critical value α = αc,

which is independent of u (initial number of 1s), such that ρ≈u for α>αc and ρ≈0 for α<αc!

�  Case (ii): rewire to random individual!�  Now αc = αc(u) in the phase

transition !

�  Useful: Derive approximate equations for quantities such as the fraction of 0-1 edges versus time !

Simulations of the voter model are done on a finite set, typi-cally the torus ðZmodLÞd. In this setting, the behavior of thevoter model is “trivial” because it is a finite Markov chain withtwo absorbing states, all ones and all zeros. As the next resultshows (see Cox and Greven, ref. 45), the voter model has inter-esting behavior along the road to absorption.

Theorem. If the voter model on the torus in d ≥ 3 starts from productmeasure with density p, then at timeNt it looks locally like νθðtÞ wherethe density θt changes according to the Wright–Fisher diffusionprocess

dθt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβd · 2θtð1 − θtÞ

pdBt

and βd is the probability that two random walks starting from neigh-boring sites never hit.

In the next section we will describe conjectures for the evolvingvoter model that are analogues of the last theorem. To preparefor stating our conjectures, note that (i) although the voter modelon the torus does not have a nontrivial stationary distribution,it does have a one parameter family of “quasi-stationary distribu-tions” that look locally like νp, and (ii) the quantity under thesquare root in the Wright–Fisher diffusion is, by results of Holleyand Liggett (43), the expected value of N10∕M under νθðtÞ.

ConjecturesOur next goal is to use simulation results to formulate the ana-logues of the Cox and Greven result (45) for our two evolvingvoter models, beginning with the more interesting rewire-to-random case. Fig. 4 shows results from simulations of the systemwith α ¼ 0.5. The initial graph is Erdös–Rényi with N ¼ 10;000vertices and average degree λ ¼ 4. Observations of the pairðN1∕N;N10∕MÞ are plotted every 1,000 steps starting from den-sities u ¼ 0.2, 0.35, 0.5, 0.65, and 0.8. The plotted points convergequickly to a curve that is approximately (fitting to a parabola)1.707xð1 − xÞ − 0.1867 and then diffuse along the curve until theyhit the axis near 0.125 or 0.875. Thus the final fraction with theminority opinion ρ ≈ 0.125, a value that agrees with the universalcurve in Fig. 2 at α ¼ 0.5.

The fact that, after the initial transient, N10∕M is a function ofN1∕N supports the conjecture that the evolving voter model has aone parameter family of quasi-stationary distributions, for if thisis true, then the values of all of the graph statistics can be com-puted from N1∕N. To further test this conjecture, we examined

the joint distribution of the opinions at three sites. Let Nijk be thenumber of oriented triples x-y-z of adjacent sites having states i, j,k, respectively. Note for example, in the 010 case, this will countall such triples twice, but this is the approach taken in the theoryof limits of dense graphs (46), where the general statistic is thenumber of homomorphisms of some small graph (labeled by onesand zeros in our case) into the random graph being studied.

Fig. 5 shows a plot of N010∕N versus N1∕N. After an initialtransient, the observed values stay close to a curve that is wellapproximated by a cubic. Simulations of the other Nijk show si-milar behavior. Because the numbers of 010 triples must vanishwhen the number of 1-0 edges do, the fitted cubic shares tworoots with the quadratic approximating the graph of N10∕M ver-sus N1∕N. This quadratic curve (see again Fig. 4 for α ¼ 0.5) isfundamental to our understanding of the observed system beha-vior, and we hereafter refer to it as the “arch.”

The phenomena just described for α ¼ 0.5 also hold for othervalues of α. Fig. 6 shows the arches that correspond toα ¼ 0.1;0.2;…;0.7. Numerical results show that the curves arewell approximated by cαuð1 − uÞ − bα. Let ðvðαÞ;1 − vðαÞÞ bethe “support interval” where the arch has positive values. Simula-tions show that if u < vðαÞ, then the simulated curve rapidly goesalmost straight down and hits the axis where N10 ¼ 0.

Conjecture 1. In the rewire-to-random model, if α < αcð1∕2Þ andvðαÞ < u ≤ 1∕2, then starting from product measure with densityu of ones, the evolving voter model converges rapidly to a quasi-stationary distribution να;u. At time tN, the evolving voter modellooks locally like να;θðtÞ where the density changes according to a gen-eralized Wright–Fisher diffusion process

dθt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 − αÞ½cαθtð1 − θtÞ − bα%

pdBt

until θt reaches vðαÞ or 1 − vðαÞ.Here the quantity under the square root is ð1 − αÞN10∕M with

ð1 − αÞ equal to the fraction of steps that are voter steps becauserewiring steps do not change the number of ones.

If Conjecture 1 is true, then the universal curve in Fig. 2 hasρðα;0.5Þ ¼ vðαÞ for α < αcð0.5Þ. When α is close to αcð0.5Þ,vðαÞ ≈ 1∕2, so when the evolving voter model hits N10 ¼ 0 bothopinions are held by large groups, and the graph splits into twogiant connected components (that is, their size is proportional to

Fig. 4. Plot of N10∕M versus N1∕N when α ¼ 0.5 in the rewire-to-randomcase. Five simulations starting from u ¼ 0.2, 0.35, 0.5, 0.65, and 0.8 areplotted in different colors. These results are from graphs withN ¼ 10;000 ver-tices and plotted every 1,000 steps.

Fig. 5. Plot of N010∕N versus N1∕N when α ¼ 0.5 in the rewire-to-randomcase. All simulations start at u ¼ 0.5 because multiple runs from one startingpoint are enough to explore all of the arch. These results are from graphswith N ¼ 100;000 vertices and plotted every 10,000 steps.

3684 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1200709109 Durrett et al.

Any source of time series that you want: neuroscience, finance, epidemics, output of a dynamical system, etc.

•  Simple idea: Glue common nodes across “slices” (i.e. “layers”)

•  E.g., consecutive layers only, as in upper right

•  P. J. Mucha, T. Richardson, K. Macon, MAP, & J.-P. Onnela, “Community Structure in Time-Dependent, Multiscale, and Multiplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010)

�  Interlayer edge strength ω represents the strength of persistence in a node’s “trajectory” through time !

•  Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, & S. D. Howison, “Community Detection in Temporal Multilayer Networks, and its Application to Correlation Networks”, arXiv:1501.00040 !

�  One can measure the relative importances of a network’s nodes, edges, or other substructures, by calculating “centrality” measures. !

�  Numerous centrality measures: degree, betweenness (on many geodesic paths), closeness (short distance to many nodes), PageRank, etc.!

�  Fun fact: developing new centrality measures is among the top 10 most popular activities of network scientists (or at least it seems like it) !

�  J. M. Kleinberg, “Authoritative sources in a hyperlinked environment”, Journal of the ACM, Vol. 46: 604–632 (1999) !

�  Intuition: A node (e.g., Web page) 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) !�  A good hub points to good authorities, and a good authority is pointed to by

good hubs !

�  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 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 score xi and authority score yi !

�  It works not just for Web pages but also mathematics departments !

�  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!

•  We consider MPG data in the US from 1973–2010 (data from 10/09)

•  Example: Danny Abrams earned a PhD from Cornell and leter supervised a student at Northwestern. •  è Directed edge of unit weight

from Cornell University to Northwestern University

•  A school 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 Northwestern edge after Abrams supervises a PhD student)

•  Hubs and authorities should change in time

Mathematical genealogy and departmentprestige

Sean A. Myers,1 Peter J. Mucha,1 and Mason A. Porter21Department of Mathematics, University of North Carolina,Chapel Hill, North Carolina 27599, USA2Mathematical Institute, University of Oxford, OX1 3LB, UK(Received 1 July 2011; published online 20 December 2011)[doi:10.1063/1.3668043]

The Mathematics Genealogy Project (http://www.genealogy.ams.org/) is a database of over 150 000 scholarswith advanced degrees in mathematics and related fields.Entries include dissertation titles, adviser(s), graduationyears, degree-granting institutions, and advisees. The MGPis popular among mathematicians, and it can be used to traceacademic lineages through luminaries like Courant, Hilbert,and Wiener to historical predecessors such as Gauss, Euler,and even Kant. For example, MGP data was used recently tostudy the role of mentorship in protege performance.1

We consider recent branches of this mathematical fam-ily tree by projecting the MGP data for degrees granted since1973 onto a network whose nodes represent academic insti-tutions in the United States. An individual who earns a doc-torate from institution A (during the selected period) andlater advises students at institution B is represented by adirected edge of unit weight pointing from B to A. The totaledge weight from B to A counts the number of such advisers.

This network representation can be used to estimate themathematical prestige of each university using various“centrality” scores2 of the corresponding node (see Fig. 1).We represent “hub” and “authority” scores3 using node sizeand color (red to blue), respectively. Institutions with highauthority scores have high-valued hubs pointing to them, andhigh-valued hub nodes point to high-valued authorities. Auniversity with a high authority score is a strong source ofprestigious Ph.D. students and a university with a high hubscore is a strong destination. In the legend of Fig. 1, we listthe top 20 institutions in order of their authority scores.

We use a “geographically inspired” layout to balance nodelocations and node overlap. A Kamada-Kawai visualization4

places the high-authority universities in the network’s center.In Fig. 2, we compare authority scores with three rankings

of mathematics departments5–7 for the 58 universities thatappear in the top 40 of at least one of the rankings or have oneof the top-40 authority scores. As expected, higher authorityscores correlate with higher prestige (i.e., smaller rank numbers).However, scatter is obviously present, particularly with the 2010National Research Council (NRC) rankings.

We thank Mitch Keller at the MGP for providing data.This work was supported by the NSF (PJM: DMS-0645369)and the James S. McDonnell Foundation (MAP: #220020177).

1R. D. Malmgren, J. M. Ottino, and L. A. N. Amaral, Nature 465, 622(2010).

2M. E. J. Newman, Networks: An Introduction (Oxford University Press,Oxford, UK, 2010).

3J. M. Kleinberg, J. ACM 46, 604 (1999).4T. Kamada and S. Kawai, Inf. Process. Lett. 31, 7 (1988).5National Research Council 1995, http://www.ams.org/notices/199512/nrctables.pdf.

6National Research Council 2010 (rank order of medians of the S-rankingranges; original release), http://graduate-school.phds.org/rankings/mathematics.

7US News & World Report 2010, http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools/top-mathematics-programs/rankings.

FIG. 1. (Color) Visualizations of a mathematics genealogy network.

FIG. 2. (Color) Rankings versus authority scores.

1054-1500/2011/21(4)/041104/1/$30.00 VC 2011 American Institute of Physics21, 041104-1

CHAOS 21, 041104 (2011)

Downloaded 20 Dec 2011 to 129.67.66.135. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

Hubs: node size Authorities: node color

Mathematical genealogy and departmentprestige

Sean A. Myers,1 Peter J. Mucha,1 and Mason A. Porter21Department of Mathematics, University of North Carolina,Chapel Hill, North Carolina 27599, USA2Mathematical Institute, University of Oxford, OX1 3LB, UK(Received 1 July 2011; published online 20 December 2011)[doi:10.1063/1.3668043]

The Mathematics Genealogy Project (http://www.genealogy.ams.org/) is a database of over 150 000 scholarswith advanced degrees in mathematics and related fields.Entries include dissertation titles, adviser(s), graduationyears, degree-granting institutions, and advisees. The MGPis popular among mathematicians, and it can be used to traceacademic lineages through luminaries like Courant, Hilbert,and Wiener to historical predecessors such as Gauss, Euler,and even Kant. For example, MGP data was used recently tostudy the role of mentorship in protege performance.1

We consider recent branches of this mathematical fam-ily tree by projecting the MGP data for degrees granted since1973 onto a network whose nodes represent academic insti-tutions in the United States. An individual who earns a doc-torate from institution A (during the selected period) andlater advises students at institution B is represented by adirected edge of unit weight pointing from B to A. The totaledge weight from B to A counts the number of such advisers.

This network representation can be used to estimate themathematical prestige of each university using various“centrality” scores2 of the corresponding node (see Fig. 1).We represent “hub” and “authority” scores3 using node sizeand color (red to blue), respectively. Institutions with highauthority scores have high-valued hubs pointing to them, andhigh-valued hub nodes point to high-valued authorities. Auniversity with a high authority score is a strong source ofprestigious Ph.D. students and a university with a high hubscore is a strong destination. In the legend of Fig. 1, we listthe top 20 institutions in order of their authority scores.

We use a “geographically inspired” layout to balance nodelocations and node overlap. A Kamada-Kawai visualization4

places the high-authority universities in the network’s center.In Fig. 2, we compare authority scores with three rankings

of mathematics departments5–7 for the 58 universities thatappear in the top 40 of at least one of the rankings or have oneof the top-40 authority scores. As expected, higher authorityscores correlate with higher prestige (i.e., smaller rank numbers).However, scatter is obviously present, particularly with the 2010National Research Council (NRC) rankings.

We thank Mitch Keller at the MGP for providing data.This work was supported by the NSF (PJM: DMS-0645369)and the James S. McDonnell Foundation (MAP: #220020177).

1R. D. Malmgren, J. M. Ottino, and L. A. N. Amaral, Nature 465, 622(2010).

2M. E. J. Newman, Networks: An Introduction (Oxford University Press,Oxford, UK, 2010).

3J. M. Kleinberg, J. ACM 46, 604 (1999).4T. Kamada and S. Kawai, Inf. Process. Lett. 31, 7 (1988).5National Research Council 1995, http://www.ams.org/notices/199512/nrctables.pdf.

6National Research Council 2010 (rank order of medians of the S-rankingranges; original release), http://graduate-school.phds.org/rankings/mathematics.

7US News & World Report 2010, http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools/top-mathematics-programs/rankings.

FIG. 1. (Color) Visualizations of a mathematics genealogy network.

FIG. 2. (Color) Rankings versus authority scores.

1054-1500/2011/21(4)/041104/1/$30.00 VC 2011 American Institute of Physics21, 041104-1

CHAOS 21, 041104 (2011)

Downloaded 20 Dec 2011 to 129.67.66.135. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

�  Let’s now try to do this with a temporal network. !�  S. A. Myers, D. S. Taylor, E. A. Leicht, A. Clauset, MAP, & P. J. Mucha,

“Eigenvector-based Centrality Measures for Temporal Networks”, in preparation (coming soon) !

�  Use a multilayer representation (c = 1/ω); we show only 4 layers for illustration !

�  More generally: One can use any eigenvector-based centrality in the diagonal blocks. !

MGP:%1946A2010%(“Take%2”)%A,beLer,definiJon,of,temporal,authority,

•  Eigenvector%Centrality%of%

�  Multilayer network with adjacency-tensor elements Aijt!�  Directed edge from university i to university j for a specific person’s PhD

granted at time t (multi-edges give weights). !

�  231 US universities, T = 65 time layers (1946–2010) !

�  Use perturbation theory (in c) to derive time-averaged centralities (coefficient at leading order) and first movers (next order) !�  We can talk privately if you want to see precise definitions.!

18 S. A. MYERS et al.

Table 4.1Top 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 0.66852 Berkeley 0.27223 Stanford 0.22954 Princeton 0.18035 Illinois 0.16456 Cornell 0.16427 Harvard 0.16288 UW 0.15909 Michigan 0.152110 UCLA 0.1456

Rank University mi

1 MIT 688.622 Berkeley 299.073 Princeton 248.724 Stanford 241.715 Georgia Tech 189.346 Maryland 186.657 Harvard 185.348 CUNY 182.599 Cornell 180.5010 Yale 159.11

map: do we want to indicate what any of those other papers do withthe 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 thateach faculty member graduated with his/her Ph.D. degree. We thus construct amultilayer network of the MGP Ph.D. exchange using elements Aijt that indicate adirected edge from university i to university j at time t to represent a doctoral degreein the MGP data from university j in year t who later advised at least one student(who also must appear in the MGP data) at university i. Edges can be weighted toaccount for multiple doctorates from university j in year t who later advise studentsat university i. Note that this construction aligns edge directions in a way that isopposite to the flow of people, so a node with large in-degree (i.e., with many graduateswho advise students elsewhere) is considered both an academic authority as well asan authority with respect to HITS centrality. We restrict our attention to movementbetween N = 231 U.S. universities, and we construct the weighted adjacency tensorA(MGP) over T = 65 time layers that represent Ph.D. graduation years 1945–2010.Note that one can define the multilayer network in more intricate ways (e.g., bynormalizing edge weights using the number of graduates and in many other ways)and examine how the results vary for di↵erent choices, though this lies far outside ofthe scope of the present manuscript.

pjm: Fix this sentence map: what do you think of my sentence; also,do we want to bring up the year gap? e.g. my first PhD student fromOxford is multiple years after I arrived at Oxford? my gut feeling is ’no’because we don’t discuss individuals’ trajectories

Given A(MGP), we begin by identifying the universities that have the largesttime-averaged authority centralities {↵i} from the eigenvectors of the matrix M1

⇤defined in Eq. (3.15). We summarize these authority values in Table 4.1, and we notethat the most central universities according this measure are all widely-accepted top-tier programs in mathematics. The time-averaged authorities identify the four mostcentral mathematics universities for this time period as MIT, Berkeley, Stanford, andPrinceton.

Although the results in Table 4.1 are interesting, time-averaged centrality (by def-inition) does not provide information about temporal trajectories of the universities’authorities, and this is the type of idea that we seek to explore. We thus calculate the

20 S. A. MYERS et al.

1945 1965 1985 200510

−3

10−2

10−1

Georgia Tech condit ional centrality

t ime (t )

conditionalcentrality

α i

(a)

1945 1965 1985 200510

−6

10−4

10−2

Georgia Tech joint centrality

t ime (t )

jointcentrality

ε=0.0001

ε=0.001

ε=0.01

ε=0.1

ε=1

α i

(b)

Fig. 4.2. Centrality trajectories for Georgia Tech illustrate that one can construe ✏ as a tuningparameter that controls how much centrality can vary between neighboring time layers. (a) To studythe trajectory of university centralities across time, we examine the conditional centrality, whichrescales the centrality of a node in time layer t according to the marginal centrality of that layer.For su�ciently small ✏, we observe a steady increase in ranking with time for Georgia Tech. Notethat one can tune the size and smoothness of this change by varying ✏. If ✏ is too large (e.g.,✏ � 0.01), then the coupling between layers is so weak that the conditional centrality of a physicalnode at time t and t+1 are no longer similar in value. As ✏ ! 0+, the conditional centrality limitsto the stationary, time-average ranking given by ↵i (horizontal dashed line), but we still observesignificant variation even at ✏ = 0.0001. (b) We plot the joint centrality of the T node-layer pairsthat correspond to Georgia Tech across the time layers for several ✏. Note for small ✏ that thenode’s joint centralities are determined by the chain of identity edges (which leads to the sinusoidaldependence given by Eq. (3.7) with n = 1). For large ✏, Georgia Tech has its highest joint centralityin time layer t = 1982, which occurs because this time layer dominates the joint centrality rankingsfor this and all other universities when ✏ is too large.

(iii) an intermediate regime in which the centralities behave di↵erently than expectedfor either the strong or weak coupling regime. In Fig. 4.1(b), we show results for✏ = 0.5, and we observe that the universities tend to have slowly-varying centralitytrajectories. However, the choice of ✏ should depend both on application and on thequestion of interest. The reason is that one can interpret the value of inter-layercoupling ✏ > 0 as a tuning parameter for controlling the temporal variation of nodes’centrality trajectories. As shown in the following experiment, an appropriate scalemust be selected.

In Fig. 4.2 we study centrality trajectories for Georgia Tech for several choicesof ✏ 2 {10�4, 10�3, 10�2, 10�1, 100}. Fig. 4.2(a) shows the conditional centrality forGeorgia Tech versus time t. Recall that the conditional centrality of the node-layerpair (i, t) indicates the centrality of physical node i with respect to the other physicalnodes at that particular time t. In addition, we also plot by the dashed line ↵i, whichis the conditional centrality of Georgia Tech in the limit that ✏ ! 0+. Note that forsmall but nonzero ✏ (e.g., ✏ = 10�3), similar to the trajectory for ✏ = 0+ the trajectoryis relatively smooth across time—that is, the conditional centrality of Georgia Techat times t and t+1 are approximately equal. Alternatively, as ✏ increases, we observethat this continuity is lost. For example, when ✏ � 10�1 the conditional centralityof Georgia Tech at times t and t + 1 are typically very dissimilar, which we find tobe a property that is consistent with conditional centralities in the limit ✏ ! 1.While the regimes of very small and very large ✏ are can be studied by comparingto the limiting cases ✏ ! 0+ and ✏ ! 1, respectively, we find that the intermediatethat corresponds to the transition between these is not straightforward and demandsfurther work. Hence the boundaries between these qualitative regimes are not clear

Eigenvector-Based Centralities for Temporal Networks 19

1 501

50

t ime-averaged centrality rank

first-m

overrank

F irst-movers and centrality

1 2271

227

CUNY

(a)

MIT

Georgia Tech

1945 1965 1985 2005

10−2

10−1

100

t ime (t )

conditionalcentrality

C entrality tra jectorie s for ϵ = 10−4

MITBerkeleyPrincetonStanfordGeorgia TechCUNY

(b)

Fig. 4.1. Time-averaged (i.e., conditional) node authority scores for the Mathematics GenealogyProject data [4]. (a) We plot the first-order mover ranking of nodes (i.e., ranked according to {mi})versus the time-averaged ranking of nodes (i.e., ranked according to {↵i}). As shown by the inset,nodes with large time-averaged rank tend to also have large first-order mover rank (e.g., MIT ranksfirst in both). However, there are nodes that have a much higher first-order mover rank than time-averaged rank (e.g., Georgia Tech and CUNY). (b) We plot the conditional rankings of nodes versustime (i.e., the ranking of nodes versus time normalized by the centrality of each time layer) to trackuniversity centrality trajectories across time. We show results for the seven top ranked first-ordermovers. Note that most of these top first-order movers are also top time-averaged authorities (e.g.,MIT). In contrast, Georgia Tech and CUNY rank in the top six of the first-order movers ranking,but they are in the lower reaches of the top 40 for the time-averaged ranking. As expected, thisranking di↵erence reflects that their centrality trajectories exhibit a significant change over time.Georgia Tech rises in rank during t 2 [1965, 1985], whereas CUNY drops during this time period.

first-order mover scores {mi} from Eq. (3.26). We show these values in Fig. 4.1(a) byplotting university ranking according to {mi} versus its ranking according to {↵i}.Note in the bottom left corner that MIT is ranked first for both quantitites, and thatin general there is a strong linear correlation between rank according to ↵i and rankaccording to mi. Intuitively, this suggests that shifts in centrality include a naturale↵ect that is directly related to the centrality score itself. (In other words, large cen-trality values tend to also include large fluctuations, whereas small centrality valuestypically experience only small fluctuations.) Deviations from the observed nearly-linear relation indicate universities whose centrality trajectory is more variable acrosstime, and it is interesting to look at these universities in more detail for potentially in-teresting insights. For example, the universities with large mi rank but small ↵i rankinclude Georgia Tech and CUNY, and it is known that Georgia Tech’s mathematicsdepartment transitioned from a primarily serve-oriented department to a much moreresearch-oriented department starting in the 1980s. map: need a reference forthe above sentence!; I avoided mentioning Jack Hale by name, but thisis the ”Jack Hale e↵ect” as it were In Fig. 4.1(b), we plot the conditional au-thority centralities at ✏ = 10�4 of universities versus time for the seven universitieswith the largest first-order mover scores mi. This includes the universities with toptime-averaged centrality, as well as Georgia Tech and CUNY (which have lower time-averaged centralities, but have very high first-order mover scores). As we expect, theconditional centralities for Georgia Tech and CUNY change drastically across time,whereas the trajectories for the others remain relatively constant across time.

As we showed in Sec. 2.3 for a synthetic network, the choice of inter-layer couplingstrength ✏ strongly e↵ects the temporal behavior of a node’s centrality trajectory,and in general we expect three qualitative regimes: (i) the strong coupling ✏ ! 0+

regime that which we studied in Sec. 3; (ii) a weak coupling regime ✏ ! 1; and

�  Generalization of modularity maximization to “multislice” networks (multilayer networks with “diagonal coupling”) !�  P. J. Mucha, T. Richardson, K. Macon, MAP, & J.-P. Onnela, “Community

Structure in Time-Dependent, Multiscale, and Multiplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010) !

�  “Diagonal coupling”: interlayer edges only between corresponding entities in different layers!

Puck Rombach

•  Find communities algorithmically by optimizing “multislice modularity”!–  We derived this function in Mucha et al, 2010 !

•  Laplacian dynamics: find communities based on how long random walkers are trapped there. Exponentiate and then linearize to derive modularity. !

•  Generalizes derivation of monoplex modularity from R. Lambiotte, J.-C. Delvenne, &. M Barahona, arXiv:0812.1770 (now published, with updates, in TSNE, 2015) !•  Different spreading weights on different types of edges !

–  Node x in layer r is a different node-layer from node x in layer s !

�  Aijs = number of times i and j voted the same in Congress s divided by the total number bills on which they both voted in layer s (one layer = one 2-year Congress) !

�  Can get insights into party realignments (gray areas) !

P. J. Mucha & M. A. Porter, Chaos, Vol. 20, No. 4, 041108 (2010)

�  Dani Bassett’s IP in the NS15 meeting!

�  fMRI data: network from correlated time series !

�  Examine role of modularity in human learning by identifying dynamic changes in modular organization over multiple time scales !

�  Main result: flexibility, as measured by allegiance of nodes to communities, in one session predicts amount of learning in subsequent session !

4

functions W (l)AB, where l 2 {1, 2}). Because we use all of the

global interactions, we obtain much more spiral-like commu-nities (see Figs. S6–S9 of the SM [5]), compared to blob-likecommunities shown in Fig. 2.

Results for Snapshots of Drifter Data.—GH:Where is thisfrom? Irina Rypina, maybe. MAP:we also need to brieflydescribe what is in the data, so that readers know whatwe’re talking about; the first sentence below comes acrossas a non sequitur SHL:I think we need more informationfrom Irina Rypina (starting this section with such infor-mation would make the transition much smoother), as wellas the permission to use it in our paper. In Fig. S10 of theSM [5], we show the drifter trajectories. The drifters are re-leased at di↵erent times, and their trajectories are recordedfor di↵erent time intervals. (See Fig. S11 of the SM [5].) Forconsistency, we need a common time interval [tinit, tfinal] overwhich the trajectories are known for all drifters. The earliestinitial time is tinit = 0.2 (days). For the final time tfinal, weconsider several values: tfinal = 0.3, tfinal = 1.5, and tfinal = 3.0(days). For each final time, we only include drifters whosetime interval of recorded trajectory contains the entire interval[tinit, tfinal] are as nodes in a network. Consequently, the num-ber of network nodes is smaller for later tfinal. (See Fig. S12of the SM [5].)

For all of the drifters, we take the common initial time tinitand the final time tfinal as we described in the previous para-graph. The weight between two drifters A and B is based onrelative dispersion W (1)

AB defined in Eq. (1), where |ri(A, B)| and|r f (A, B)|, respectively, give the initial and final Euclidean dis-tances in the 2D (longitude, latitude) plane [25]. Note that thedeformation-gradient tensor F(A) in Eq. (3) is unknown for thedrifter data, so we cannot use W (2)

AB in Eq. (3) in those cases.The modularity for the NG null model is given by Eq. (4). InFig. S12 of the SM [5], we show the detected communities forfour di↵erent values of the resolution parameter �.

MAP:the discussion of what a community is above seemsrather redundant with earlier discussions? I suspect spacemay be an issue, in which case it should be shortened toonly what we need, but it also seems like we may wish toshorten it to minimize redundancy anyway; I am trying tofigure out what it’s contributing

SHL:I commented out that part, and I moved the Gen-Louvain part to the methodology section because GenLou-vain is used throughout the work.

MAP:should time-dependent community structure start anew section in order to emphasize it?

SHL:I separated the sections.Results for Time-Dependent Drifter Data.—To gain in-

sights into the dynamics of LCSs, it is also helpful to ex-amine time-dependent community structure. To do this,we will split up the full temporal dynamics into a se-ries of intervals. The simplest approach is to divide thetime interval [tinit, tfinal] uniformly into S “layers” usinga temporal resolution of tres. This yields the time in-tervals {[t0 = tinit, t1 = t0 + tres], [t1, t2], · · · , [tS�1, tS = tfinal]},and similar to W (1)

AB in Eq. (2), we define the weight between

nodes A and B in layer s 2 {1, . . . , S } as

W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)

where rt(A, B) is the Euclidean distance between thenodes A and B at time t [see Eq. (1)]. First, we treateach layer separately and detect communities as be-fore. We set tinit = 0.1 days and tfinal = 3.1 days. InFig. S13 of the SM [5], we show results for � = 1.We divide the time interval [tinit = 0.1, tfinal = 3.1]is uniformly into S = 30 non-overlapping intervals{[t0 = tinit = 0.1, 0.2), [0.2, 0.3), · · · , [3.0, tS = tfinal = 3.1]}with a temporal resolution of tres = 0.1.

We then apply multilayer community detection [26] to ourtime-dependent networks by using a generalized version ofthe modularity function in Eq. (4) for multilayer networks:

Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,

(7)where A and B index nodes (i.e., fluid elements) as in the orig-inal modularity function in Eq. (4), and s and r index the timelayers in the multilayer network. (See Refs. [27, 28] for a re-view of multilayer networks.) That is, for each layer s, thereexists a separate network described by the adjacency-matrixelements W (1)

ABs. The quantity W (1)ABs, which describes the

strength of the interaction between fluid elements A and B attime s, is thus an element of an adjacency tensor [26, 27, 29].The intralayer interactions W (1)

ABs , 0 if nodes A and B areconnected in layer s, and W (1)

ABs = 0 otherwise. Additionally,kAs =

PB W (1)

ABs, we normalize in each layer s separately usingms =

PAB W (1)

ABs, and �s is the resolution parameter in layer s.To connect fluid elements to themselves when they are presentin multiple layers, we use the interlayer interactions TBsr , 0

0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index

FIG. 3. (Color online) Multilayer (time-dependent) communitystructure of drifters for tinit = 0.1 day, tfinal = 3.1 day, and tres = 0.1.The resolution-parameter value is � = 1, and the interlayer couplingstrength is ! = 25. The vertical axis gives the initial point of timelayers ts in [ts, ts+1) where s = 0, 1, . . . , 29. MAP:I was unable todiscern what was meant by the description, and I am not entirelysure of the axis label either, so we need to adjust phrasing SHL:Ihope it’s clear now, and the horizontal axis indicates the node index.We indicate di↵erent communities by using di↵erent colors and sym-bols (also used in Fig. 4). The solid horizontal lines in the interiorcorrespond to the snapshots in Fig. 4.

S. H. Lee, M. Farazmand, G. Haller, & MAP, “Finding Lagrangian Coherent Structures Using Community Detection”, in preparation Adjacencies constructed using, for example, nearest-neighbor interactions in relative dispersion of different fluid elements 3

(a) (b)

FIG. 2. (Color online) Ten communities (each of a di↵erent color), which we detect algorithmically from a network constructed from nearest-neighbor interactions, from the simulated data that we show in Fig. 1. Panels (a) and (b), respectively, show the fluid elements at the initial andfinal times. We detect the communities using the relative dispersion W (1)

AB in Eq. (1) and the modularity QNG in Eq. (4). The resolution-parametervalue is � = 0.005. See Figs. S2–S5 in SM [5] for similar results using the modularity QLN and various resolution-parameter values.

detect the set {gA | A 2 V} of communities, where node Ais assigned to community gA, such that modularity is maxi-mized. We use di↵erent null models for the relative disper-sion W (1)

AB (which is symmetric) and the deformation-gradienttensor W (2)

AB (which is not).For the relative dispersion between nodes A and B, we use

the modularity QNG for the Newman–Girvan (NG) null model[8, 20]:

QNG =1

2m

X

AB

W (1)

AB � �kAkB

2m

!� (gA, gB) , (4)

where kA =P

B W (1)AB =

PB W (1)

BA is the sum of weights corre-sponding to the interactions of A, the quantity 2m =

PA kA is

the total sum of weights in all of the interactions, � is a reso-lution parameter, and �(gA, gB) = 1 if A and B are in the samecommunity and 0 if they are not. The normalization constant1/(2m) enforces Q 2 [�1, 1]. Larger resolution-parameter val-ues � tend to result in smaller communities (with respect to thetypical number of nodes in a community).

For the deformation-gradient tensor between nodes A andB, we use the modularity QLN with the Leicht–Newman (LN)null model [21]:

QLN =1m

X

AB

0BBBB@W (2)

AB � �kin

A koutB

m

1CCCCA � (gA, gB) , (5)

where kinA =

PB W (2)

BA (respectively, koutA =

PB W (2)

AB) is the sumof incoming (respectively, outgoing) weights correspondingto the interactions of A, and m =

PA kin

A =P

A koutA is the total

sum of weights for all of the interactions. (This sum is nec-essarily the same for both incoming and outgoing weights.)To detect the communities for both Eqs. (4) and (5), we usethe GenLouvain code [22], which is an implementation of amethod that is similar to the locally-greedy Louvain methodof Ref. [23].

Results for Simulation Data.—The turbulence-simulationdata consists of 512 ⇥ 512 grid points = 262144 nodes andtheir interactions given by Eqs. (1) and (3). For computationaltractability, we only consider the four nearest-neighbor inter-actions in the 2D grid system. This corresponds to setting allof the interactions between non-nearest-neighboring nodes inEqs. (1) and (3) to zero. The initial time is tinit = 0, and thefinal time is tfinal = 50 in the simulation time units. In Fig. 2,we show the ten detected communities using W (1)

AB [see Eq. (1)]with � = 0.005 [24]. For small values of �, one detects a smallnumber of large communities. They have blob-like structuresthat do not undergo substantial filamentation. Larger valuesof �, however, yield a larger number of communities, whichexhibit spiral patterns that correspond to vortex filaments. InFigs. S2–S5 of the SM [5], we show the dependence of thedetected communities for a set of values of � and using theweight function W (2)

AB. We find that the weight functions W (1)AB

and W (2)AB yield similar communities for the same value of �.

In Figs. S6–S9 of the SM [5], we present results for sam-pled grid points in which we use every fourth element (whichyields a 128⇥128 grid points and thus 16384 nodes) along thex and y axes. In this case, we also include all of their globalinteractions (i.e., all of the nonzero elements in the weight

�  M. Sarzynska, E. A. Leivht, G. Chowell, and MAP, “Null Models for Community Detection in Spatially-Embedded, Temporal Networks”, arXiv:1407.6297 (2015) !

�  Networks constructed from correlations in time series !

�  Generalizing null models to incorporate spatial information and exploring the results for different null models !

�  Below: aggregated community structure versus consensus communities for provinces from multilayer community structure !

29 of 46

(a) (b) (c) (d)

FIG. 15. Province-level algorithmic community structure, which we obtain by maximizing modularity, for the static andmultilayer dengue fever correlation networks. We color the provinces according to their community assignments. Whiteprovinces are ones in which our data does not include any reported cases of dengue fever in the indicated time window.(a) NG null model that is fully aggregated (i.e., t = 1 and D = 779) with a resolution-parameter value of g = 1. (b)NG null model that is fully aggregated with g = 1.1. (c) NG null model in a multilayer network with province-levelcommunities that we obtain from the multilayer network with a time window of width D = 60. (d) Correlation nullmodel in a multilayer network with province-level communities that we obtain with a time window of width for D = 60.

terns. The NG null model finds one more pattern type (nodes with late onset of disease, as weillustrate in Fig. 17) than the correlation null model.

(a) (b)

FIG. 16. Membership of the consensus province-level communities, which we computing by maximizing modularity, inmultilayer dengue fever networks for g = 1. In panels (a) and (b), we compare the climate composition of the communi-ties using (a) the NG null model and (b) a correlation null model. We order communities according to their size, and thehorizontal axis gives the community number.

5. Conclusions

In conclusion, we examined time-dependent community structure and the effect of different nullmodels — including ones that incorporate spatial information — on the results of modularitymaximization. We conducted our computational experiments using novel synthetic benchmarkspatial networks and correlation networks constructed from spatiotemporal dengue fever inci-dence data in provinces of Peru (a system that is strongly influenced by spatial effects). Wecompared our results for the standard Newman-Girvan null model versus two null models thatincorporate spatial information: a gravity null model [26] and a novel radiation null model. Wealso compared the NG null model on disease-correlation networks with a recently-developed

Flora Meng

�  M Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, & MAP, “Multilayer Networks”, Journal of Complex Networks, 2(3): 203–271, 2014. !

�  S. Boccaletti, et al., “Structure and Function of Multilayer Networks”, Physics Reports, Vol. 544: 1–122 (2014). !

•  Definition of a multilayer network M –  M = (VM,EM,V,L)

•  V: set of nodes –  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 (“monoplex”) networks

•  Adjacency tensor for unweighted case:

•  Elements of adjacency tensor: –  Auvαβ = Auvα1β1 … αdβd = 1 iff ((u,α), (v,β)) is an element of

EM (else Auvαβ = 0)

•  Important note: ‘padding’ layers with empty nodes –  One needs to distinguish between a node not present in a layer

and nodes existing but edges not present (use a supplementary tensor with labels for edges that could exist), as this is important for normalization in many quantities.

� Example: multilayer clustering coefficient!� Our approach: Cozzo et al.,

!�  Use the idea of multilayer walks. Keep track of returning to

entity i (possibly in a different layer from where we started) separately for 1 total layer, 2 total layers, 3 total layers (and in principle more). !

� Insight: Need different types of transitivity for different types of multiplex networks. !�  Example: transportation versus social networks !

�  There are several different clustering coefficients for monoplex weighted networks, and this situation is even more extreme for multilayer networks. !

�  Need to separately keep track of different types of “elementary cycles”, which all traverse three intra-layer but have different numbers of inter-layer edges !

�  Again: Keep track of particular types of multilayer walks !

�  Basic question: How do multilayer structures affect dynamical systems on networks?!�  Effects of multiplexity? (colored edges) !

�  Effects of interconnectedness? (colored nodes) !

�  Important goal: Find new phenomena that cannot occur without multilayer structures. !�  Example: Speeding up versus slowing down spreading?!

�  Example: Multiplexity-induced correlations in dynamics?!

�  Example: Effect of different costs for changing layers?!

�  We need to keep track of different types of network ties simultaneously!�  Example: spread of ebola coupled to spread of fear of ebola (only one of

these can occur online) !

�  Networks structure is really interesting, but it’s important to emphasize dynamics !�  Dynamical systems on networks !

�  Dynamics of the networks themselves !

�  Even when studying network structure, it is useful think about what structural considerations can potentially say about dynamics (e.g., random graphs, random walks, etc.) !

�  Multilayer networks and multilayer representations of temporal networks offer new avenues of explorations !�  E.g., time-dependent measures of node importance (centralities), time-

dependent community structure, effects of multilayer structures (e.g., correlations between properties of different layers) on dynamical processes !

�  In summary: You say you want some evolution? You know it’s gonna’ be alright! !

•  Mathematical Biosciences Institute, The Ohio State University, USA !

•  Semester program on “Dynamics of Biologically Inspired Networks” !–  http://mbi.osu.edu/programs/emphasis-programs/spring-2016-dynamics-biologically-inspired-networks/!

–  Focuses on theoretical questions on networks that arise from biology !

•  Four awesome workshops: (1) dynamics of networks with special properties, (2) interplay of stochastics and deterministic dynamics, (3) generalized network structures and dynamics,(4) control and observability of network dynamics !

•  March 21–25, 2016 !•  http://mbi.osu.edu/event/?id=898 !