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.

1. Centrality in Time-
Dependent Networks
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA

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. 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

4. A Couple of Longstanding Questions in the
Study of Social Networks
◦ I. de Sola Pool & M. Kochen [1978/79], “Contacts and Influence”, Social Networks, 1: 5–51
(though a preprint for 2 decades)
◦ Inspired work of Milgram
theory of human contact nets might help clarify them both.
No such theory exists at present. Sociologists talk of social stratification;
political scientists of influence. These quantitative concepts ought to lend
themselves to a rigorous metric based upon the elementary social events of
man-to-man contact. “Stratification” expresses the probability of two people
in the same stratum meeting and the improbability of two people from dif-
ferent strata meeting. Political access may be expressed as the probability
that there exists an easy chain of contacts leading to the power holder. Yet
such measures of stratification and influence as functions of contacts do not
exist.
What is it that we should like to know about human contact nets?
-~-For any individual we should like to know how many other people he
knows, i.c. his acquaintance volume.
- For a popnfatiorl we want to know the distribution of acquaintance
volumes, the mean and the range between the extremes.
_ We want to know what kinds of people they are who have many con-
tacts and whether those people are also the influentials.
,.- We want to know how the lines of contact are stratified; what is the
structure of the network?
If we know the answers to these questions about individuals and about the
whole population, we can pose questions about the implications for paths
between pairs of individuals.
- How great is the probability that two persons chosen at random from
the population will know each other?
- How great is the chance that they will have a friend in common?
- How great is the chance that the shortest chain between them requires
two intermediaries; i.e., a friend of a friend?
The mere existence of such a minimum chain does not mean, however,
They run into people they know everywhere they go. The experience of
casual contact and the practice of influence are not unrelated. A common
theory of human contact nets might help clarify them both.
No such theory exists at present. Sociologists talk of social stratification;
political scientists of influence. These quantitative concepts ought to lend
themselves to a rigorous metric based upon the elementary social events of
man-to-man contact. “Stratification” expresses the probability of two people
in the same stratum meeting and the improbability of two people from dif-
ferent strata meeting. Political access may be expressed as the probability
that there exists an easy chain of contacts leading to the power holder. Yet
such measures of stratification and influence as functions of contacts do not
exist.
What is it that we should like to know about human contact nets?
-~-For any individual we should like to know how many other people he
knows, i.c. his acquaintance volume.
- For a popnfatiorl we want to know the distribution of acquaintance
volumes, the mean and the range between the extremes.
_ We want to know what kinds of people they are who have many con-
tacts and whether those people are also the influentials.
,.- We want to know how the lines of contact are stratified; what is the
structure of the network?
If we know the answers to these questions about individuals and about the
whole population, we can pose questions about the implications for paths
between pairs of individuals.
- How great is the probability that two persons chosen at random from
the population will know each other?
- How great is the chance that they will have a friend in common?
- How great is the chance that the shortest chain between them requires

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. 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. 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. Example: PageRank ◦ Review article: D. F. Gleich, SIAM Review, 2015
B : adjacency matrix

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. 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. 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. 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. 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. 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. 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. 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. 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. 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

19. Singular Perturbation Expansion
over scores, respectively. We give higher-order expansions in an a
gularity at Infinite Inter-Layer Coupling. To understand
ace (i.e., the eigenspace of the largest eigenvalue) of M(✏) in
rst study the system for ✏ = 0. We write Eq. (2.4) as
M(✏) = B + ✏G ,
A(chain)
⌦ I and G = diag[M(1)
, . . . , M(T )
]. It follows that
To facilitate our discussion of the eigenspace and our subseque
ntroduce the NT ⇥ NT permutation matrix P with entries Pk
T [(k 1) mod N] and Pkl = 0 otherwise, where d·e is the ceiling
permutes node-layer indices so that we can easily go back-and-
ng the node-layer pairs by time and then by physical node index
dering them by physical node index and then by time). In pa
= P I ⌦ A(chain)
PT
. Additionally, because P is a unitary oper
nd the spectral properties of M(0) via the spectral properties o
2
A(chain)
0 0 · · ·
3
first-order-mover scores, respectively. We give high
3.1. Singularity at Infinite Inter-Layer C
inant eigenspace (i.e., the eigenspace of the large
✏ ! 0+
, we first study the system for ✏ = 0. We w
M(✏) = B + ✏G
where B = A(chain)
⌦ I and G = diag[M(1)
, . .
A(chain)
⌦ I. To facilitate our discussion of the eig
lations, we introduce the NT ⇥ NT permutation
l = dk/Ne+T [(k 1) mod N] and Pkl = 0 otherwi
Note that P permutes node-layer indices so that w
tween ordering the node-layer pairs by time and t
versa (i.e., ordering them by physical node index
A(chain)
⌦ I = P I ⌦ A(chain)
PT
. Additionally, b
can understand the spectral properties of M(0) vi
2
6
A(chain)
0
0 A(chain)
node-layer pairs {(i, t)} for the same physical node i across the T network layers.
The identity edges of weight ! attempt to weight the “persistence” of a physical
node through time by enforcing an identification with itself at consecutive times. We
restrict our attention to nonnegative inter-layer coupling ! 0. (One could consider
! < 0 to drive negative coupling between layers, but we do not consider such values
for our applications.) One can construe ! as a parameter to tune interactions between
network layers [6, 9, 81]: in the limit ! ! 0+
, the layers become uncoupled; in the
limit ! ! 1, the layers are so strongly coupled that inter-layer weights dominate the
intra-layer connections.
The case of inter-layer edges only between di↵erent instances of the same physical
node is sometimes called “diagonal coupling,” and the using a constant ! across all
such-interlayer edges is sometimes known as “layer-coupling.” Here we also restrict
ourselves to nearest-neighbor coupling of temporal layers, as we place the identity
inter-layer edges only between node-layer pairs that are adjacent in time, (i, t) and
(i, t ± 1), which results in the block structure in Eq. (2.1). Equivalently, we write
A = diag
h
A(1)
, . . . , A(T )
i
+ A(chain)
⌦ !I , (2.2)
where ⌦ denotes the Kronecker product and A(chain)
is the T ⇥ T adjacency matrix
of an undirected “bucket brigade” (or “chain”) network whose T nodes are each
adjacent to their nearest neighbors along an undirected chain. In this bucket brigade,
A
(chain)
ij = 1 for j = i ± 1 and A
(chain)
ij = 0 otherwise. Although one can choose inter-
layer coupling matrices other than A(chain)
for the inter-layer couplings [70] (and
much of our approach can be generalized to other choices of coupling), we restrict our
attention to nearest-neighbor coupling of layers.
It is tempting to directly apply a standard eigenvector-based centrality formu-
lation to the supra-adjacency matrix A by treating it just like any other adjacency
matrix despite its structure. However, such an approach neglects to respect the funda-
mental distinction between intra-layer edges and inter-layer edges that result from the
block-diagonal structure of A. That is, in such an approach, one treats the inter-layer
couplings (i.e., identity arcs) just like any other edge. In general, however, one needs
to be careful when studying a temporal network using the supra-adjacency matrix
formalism because many basic network properties—some of which carry strong impli-
cations about a static network (e.g., its spectra, connectedness properties, etc.)—do
not naturally carry over without modification to the supra-adjacency matrix. This
issue was discussed for multilayer networks more generally in Refs. [24, 27, 70] and
12 D. TAYLOR et al.
✏ ! 0+
. In Sec. 3.1, we further explore the singularity that arises in the strong-
coupling limit. In Secs. 3.2 and 3.3, we give zeroth- and first-order perturbation
expansions, which lead to principled expressions for time-averaged centralities and
first-order-mover scores, respectively. We give higher-order expansions in an appendix.
3.1. Singularity at Infinite Inter-Layer Coupling. To understand the dom-
inant eigenspace (i.e., the eigenspace of the largest eigenvalue) of M(✏) in the limit
✏ ! 0+
, we first study the system for ✏ = 0. We write Eq. (2.4) as
M(✏) = B + ✏G , (3.1)
where B = A(chain)
⌦ I and G = diag[M(1)
, . . . , M(T )
]. It follows that M(0) =
A(chain)
⌦ I. To facilitate our discussion of the eigenspace and our subsequent calcu-
lations, we introduce the NT ⇥ NT permutation matrix P with entries Pkl = 1 for
l = dk/Ne+T [(k 1) mod N] and Pkl = 0 otherwise, where d·e is the ceiling operator.
Note that P permutes node-layer indices so that we can easily go back-and-forth be-
tween ordering the node-layer pairs by time and then by physical node index, or vice
versa (i.e., ordering them by physical node index and then by time). In particular,
A(chain)
⌦ I = P I ⌦ A(chain)
PT
. Additionally, because P is a unitary operator, one
can understand the spectral properties of M(0) via the spectral properties of
I ⌦ A(chain)
=
2
6
6
6
6
4
A(chain)
0 0 · · ·
0 A(chain)
0
0 0 A(chain) ...
...
...
...
3
7
7
7
7
5
. (3.2)
Eigenvector-Based Centrality Measures for Temporal Networks 13
where we impose u0 = uT +1 = 0 so that Eq. (3.4) holds for all t 2 {1, . . . , T}. We
now let ut / sin n⇡t
T +1 for n 2 {1, . . . , T} to ensure that these boundary conditions are
satisfied. Equation (3.4) then reduces to

2 cos
✓
n⇡
T + 1
◆
(chain)
sin
n⇡t
T + 1
= 0 . (3.5)
For the solution of Eq. (3.5) to be consistent for all t, the term in the square brackets
must be identically 0. Consequently,
(chain)
= 2 cos
✓
n⇡
T + 1
◆
, (3.6)
u(chain)
=
1
p
n

sin
✓
n⇡
T + 1
◆
, sin
✓
2n⇡
T + 1
◆
, . . . , sin
✓
Tn⇡
T + 1
◆ T
, (3.7)
where the normalization constant is n =
PT
t=1 sin2
[n⇡t/(T + 1)]. Setting n = 1
then gives the dominant eigenvalue and its corresponding eigenvector for the matrix
A(chain)
.
Returning to the dominant eigenvalue and eigenspace of Eq. (3.2), we see that the
dominant eigenvalue of both I ⌦ A(chain)
and A(chain)
⌦ I is max = 2 cos [⇡/(T + 1)].
The general solution for the dominant eigenvector of I ⌦ A(chain)
that spans this
eigenspace is
P
j ↵j j. The N dominant eigenvectors of M(0) therefore have the
general form
P
j ↵jP j, where the constants {↵i} must satisfy
P
i ↵2
i = 1 to ensure
A(chain)
.
Returning to the dominant eigenvalue and eigenspace of Eq. (3.2), we see that the
dominant eigenvalue of both I ⌦ A(chain)
and A(chain)
⌦ I is max = 2 cos [⇡/(T + 1)].
The general solution for the dominant eigenvector of I ⌦ A(chain)
that spans this
eigenspace is
P
j ↵j j. The N dominant eigenvectors of M(0) therefore have the
general form
P
j ↵jP j, where the constants {↵i} must satisfy
P
i ↵2
i = 1 to ensure
hat the vector is normalized.
3.2. Zeroth-Order Expansion and Time-Averaged Centrality. In this
section, we study the zeroth-order expansion of the dominant eigenvector (✏) for
he limit ✏ ! 0+
. As we shall now show, the conditional node-layer centralities
{(i, t)} corresponding to a given physical node i become constant across time in this
imit. We refer to these values as the physical nodes’ time-averaged centralities. By
examining the first-order expansion, we show in Eq. (3.14) that one can obtain these
values as the solution to a dominant eigenvalue equation for a matrix of size N ⇥ N.
We consider the dominant eigenvalue equation
max(✏) (✏) = M(✏) (✏) = B (✏) + ✏G (✏) . (3.8)
We expand max(✏) and (✏) for small ✏ by writing max(✏) = 0 + ✏ 1 + · · · and
(✏) = 0 + ✏ 1 + · · · to obtain kth-order approximations: max(✏) ⇡
Pk
j=0 ✏j
j and
(✏) ⇡
Pk
j=0 ✏j
j. Note that we use subscripts to indicate the orders in ✏ of the
erms in the expansion. Our strategy is to develop consistent solutions to Eq. (3.8)
or increasing values of k.
Starting with the first-order approximations, we substitute max(✏) ⇡ 0 + ✏ 1
and (✏) ⇡ 0 + ✏ 1 into Eq. (3.8) and collect the zeroth- and first-order terms in ✏
o obtain
nvalue and eigenspace of Eq. (3.2), we see that the
(chain)
and A(chain)
⌦ I is max = 2 cos [⇡/(T + 1)].
nant eigenvector of I ⌦ A(chain)
that spans this
ominant eigenvectors of M(0) therefore have the
constants {↵i} must satisfy
P
i ↵2
i = 1 to ensure
on and Time-Averaged Centrality. In this
expansion of the dominant eigenvector (✏) for
now show, the conditional node-layer centralities
hysical node i become constant across time in this
the physical nodes’ time-averaged centralities. By
n, we show in Eq. (3.14) that one can obtain these
nt eigenvalue equation for a matrix of size N ⇥ N.
nvalue equation
M(✏) (✏) = B (✏) + ✏G (✏) . (3.8)
mall ✏ by writing max(✏) = 0 + ✏ 1 + · · · and
h-order approximations: max(✏) ⇡
Pk
j=0 ✏j
j and
use subscripts to indicate the orders in ✏ of the
egy is to develop consistent solutions to Eq. (3.8)
pproximations, we substitute max(✏) ⇡ 0 + ✏ 1
and collect the zeroth- and first-order terms in ✏
j
that the vector is normalized.
3.2. Zeroth-Order Expansio
section, we study the zeroth-order e
the limit ✏ ! 0+
. As we shall no
{(i, t)} corresponding to a given phy
limit. We refer to these values as th
examining the first-order expansion,
values as the solution to a dominant
We consider the dominant eigenv
max(✏) (✏) = M
We expand max(✏) and (✏) for sm
(✏) = 0 + ✏ 1 + · · · to obtain kth-
(✏) ⇡
Pk
j=0 ✏j
j. Note that we u
terms in the expansion. Our strateg
for increasing values of k.
Starting with the first-order app
and (✏) ⇡ 0 + ✏ 1 into Eq. (3.8) a
to obtain

20. 0th Order Expansion and Time-Averaged Centrality
of 0 in Eq. (3.11), we obtain
X
j
↵j
T
i PT
GP j = 1
X
j
↵j
T
i PT
P j = 1↵i , (3.13)
T
= I and T
i j = ij, where ij is the Kronecker delta. Letting
Eq. (3.13) corresponds to an N-dimensional eigenvalue equation,
X(1)
↵ = 1↵ , (3.14)
X(1)
has elements
(1)
ij = T
i PT
GP j = 1
1
X
t
M
(t)
ij sin2
✓
⇡t
T + 1
◆
, (3.15)
(⇡t/(T + 1)) is the normalization constant for the dominant eigen-
n by n = 1 in Eq. (3.7). Our assumption that (✏) is nonnegative
any ✏ > 0 ensures that X(1)
is also nonnegative and irreducible.
enius theorem for nonnegative matrices [77], the largest eigenvalue
ltiplicity of one, and its eigenvector ↵ is unique with nonnegative
2 and the footnote 3 therein). We normalize the solution ↵ to
2
i = 1 and substitute the normalized solution into Eq. (3.11) to
rder term 0.
vector 0 is the dominant eigenvector of (✏) and that it gives
+
i i
Using the solution of 0 in Eq. (3.11), we obtain
X
j
↵j
T
i PT
GP j = 1
X
j
↵j
T
i PT
P j = 1↵i , (3.13)
because PT
P = PPT
= I and T
i j = ij, where ij is the Kronecker delta. Letting
↵ = [↵1, . . . , ↵N ]T
, Eq. (3.13) corresponds to an N-dimensional eigenvalue equation,
X(1)
↵ = 1↵ , (3.14)
where the matrix X(1)
has elements
X
(1)
ij = T
i PT
GP j = 1
1
X
t
M
(t)
ij sin2
✓
⇡t
T + 1
◆
, (3.15)
and 1 =
PT
t=1 sin2
(⇡t/(T + 1)) is the normalization constant for the dominant eigen-
vector u(chain)
given by n = 1 in Eq. (3.7). Our assumption that (✏) is nonnegative
and irreducible for any ✏ > 0 ensures that X(1)
is also nonnegative and irreducible.
By the Perron-Frobenius theorem for nonnegative matrices [77], the largest eigenvalue
1 of X(1)
has a multiplicity of one, and its eigenvector ↵ is unique with nonnegative
entries (see Sec. 2.2 and the footnote 3 therein). We normalize the solution ↵ to
Eq. (3.14) by
P
i ↵2
i = 1 and substitute the normalized solution into Eq. (3.11) to
obtain the zeroth-order term 0.
Recall that the vector 0 is the dominant eigenvector of (✏) and that it gives
the joint node-layer centralities in the limit ✏ ! 0+
. By inspection, the elements
of 0 are ↵i sin(⇡t/(T + 1)) for node-layer pair (i, t). It follows that the conditional
centrality of node-layer pair (i, t) is ↵i, independent of the layer t. Importantly, these
{↵i} values arise naturally from our perturbative expansion in the supra-centrality
framework, independently of the value of ✏. By contrast, recall that the marginal
node centralities (MNC) reflect averaging the joint centralities across time layers for
a specific choice of ✏. Accordingly, we hereafter refer to the entry ↵i in the vector ↵
12 D. TAYLOR et al.
✏ ! 0+
. In Sec. 3.1, we further explore the singularity that arises in the strong-
coupling limit. In Secs. 3.2 and 3.3, we give zeroth- and first-order perturbation
expansions, which lead to principled expressions for time-averaged centralities and
first-order-mover scores, respectively. We give higher-order expansions in an appendix.
3.1. Singularity at Infinite Inter-Layer Coupling. To understand the dom-
inant eigenspace (i.e., the eigenspace of the largest eigenvalue) of M(✏) in the limit
✏ ! 0+
, we first study the system for ✏ = 0. We write Eq. (2.4) as
M(✏) = B + ✏G , (3.1)
where B = A(chain)
⌦ I and G = diag[M(1)
, . . . , M(T )
]. It follows that M(0) =
A(chain)
⌦ I. To facilitate our discussion of the eigenspace and our subsequent calcu-
lations, we introduce the NT ⇥ NT permutation matrix P with entries Pkl = 1 for
l = dk/Ne+T [(k 1) mod N] and Pkl = 0 otherwise, where d·e is the ceiling operator.
Note that P permutes node-layer indices so that we can easily go back-and-forth be-
tween ordering the node-layer pairs by time and then by physical node index, or vice
versa (i.e., ordering them by physical node index and then by time). In particular,
A(chain)
⌦ I = P I ⌦ A(chain)
PT
. Additionally, because P is a unitary operator, one
can understand the spectral properties of M(0) via the spectral properties of
I ⌦ A(chain)
=
2
6
6
6
6
4
A(chain)
0 0 · · ·
0 A(chain)
0
0 0 A(chain) ...
...
...
...
3
7
7
7
7
5
. (3.2)
We can see from our above discussion that the base problem at ✏ = 0 de-
couples into N identical chains, where each chain corresponds to a physical node
coupled across the T time layers. Because of the block-diagonal and repeated na-
ture of Eq. (3.2), determining its spectral properties is relatively straightforward:
we obtain them from the eigenvalues and eigenvectors of A(chain)
. The eigenval-
ues of I ⌦ A(chain)
are given by the eigenvalues of A(chain)
, where each eigenvalue
has a multiplicity of N and a corresponding N-dimensional eigenspace spanned by
the vectors based on the eigenvectors of A(chain)
(with appended 0 values in ap-
0 = 2 cos
✓
⇡
T + 1
◆
, 0 =
X
j
↵jP j , (3.11)
where {↵i} are constants that satisfy a constraint that 0 has a magnitude of 1 (i.e.,P
i ↵2
i = 1). We defined i just below Eq. (3.2).
To find the set {↵i} of unique constants that define 0, we need a solvability
condition in the first-order terms. Using the fact that the null space of 0I B is
span(P 1, . . . , P N ) for any physical node i, it follows that (P i)T
( 0I B) 1 = 0,
and left-multiplying Eq. (3.10) by (P i)T
leads to
T
i PT
G 0 = 1
T
i PT
0 . (3.12)
Using the solution of 0 in Eq. (3.11), we obtain
X
j
↵j
T
i PT
GP j = 1
X
j
↵j
T
i PT
P j = 1↵i , (3.13)
because PT
P = PPT
= I and T
i j = ij, where ij is the Kronecker delta. Letting
↵ = [↵1, . . . , ↵N ]T
, Eq. (3.13) corresponds to an N-dimensional eigenvalue equation,
X(1)
↵ = 1↵ , (3.14)
where the matrix X(1)
has elements
X
(1)
ij = T
i PT
GP j = 1
1
X
t
M
(t)
ij sin2
✓
⇡t
T + 1
◆
, (3.15)
and 1 =
PT
t=1 sin2
(⇡t/(T + 1)) is the normalization constant for the dominant eigen-
vector u(chain)
given by n = 1 in Eq. (3.7). Our assumption that (✏) is nonnegative
and irreducible for any ✏ > 0 ensures that X(1)
is also nonnegative and irreducible.
By the Perron-Frobenius theorem for nonnegative matrices [77], the largest eigenvalue
1 of X(1)
has a multiplicity of one, and its eigenvector ↵ is unique with nonnegative
entries (see Sec. 2.2 and the footnote 3 therein). We normalize the solution ↵ to
Eq. (3.14) by
P
i ↵2
i = 1 and substitute the normalized solution into Eq. (3.11) to
obtain the zeroth-order term 0.
Recall that the vector 0 is the dominant eigenvector of (✏) and that it gives
the joint node-layer centralities in the limit ✏ ! 0+
. By inspection, the elements
of 0 are ↵i sin(⇡t/(T + 1)) for node-layer pair (i, t). It follows that the conditional
centrality of node-layer pair (i, t) is ↵i, independent of the layer t. Importantly, these
{↵i} values arise naturally from our perturbative expansion in the supra-centrality
framework, independently of the value of ✏. By contrast, recall that the marginal
node centralities (MNC) reflect averaging the joint centralities across time layers for
One derives expressions for higher-order corrections in a similar way. For example, the coefficient of
the first correction gives a first-mover score.
αi is the time-averaged centrality for entity i

22. Math Departments: Best Authorities
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

23. Tie-Decay Centrality in Continuous Time
◦ W. Ahmad, MAP, & M.
Beguerisse-Díaz [2018], arXiv:
1805.00193
◦ It’s desirable to develop
methods that allow
consideration of continuous
time.
◦ Chopping up data into time
windows is a major issue.
◦ Very important for modeling
◦ Important: we now distinguish
between interactions and ties

24. Ties, Interactions, and Time-Dependent
Networks
◦ Interaction: an interaction between two agents is an event that takes
place at a specific point in time (or during a specific time interval)
◦ A phone call, text message, tweet, etc.
◦ This work: consider only instantaneous interactions
◦ Tie: a tie between two agents is a relationship between them
◦ It can have a weight to represent its strength
◦ Strengthen (or, more generally, change in strength) with repeated interactions,
but they deteriorate in their absence

25. Mathematical Setup
◦ n interacting agents
◦ B(t) = n x n time-dependent, real, non-negative matrix
◦ Entries bij(t) represent the connection strength between agents i and j at time t
◦ To construct a continuous-time temporal network of ties, we make two assumptions about
how ties evolve and how ties strengthen them

26. Yields a Time-Dependent Network
◦ A(t) = instantaneous adjacency matrix (i.e. of interactions) at time t
◦ Technical point: B(t) is a solution of an ODE (not shown)
◦ E.g. we get delta functions from instantaneous interactions, and we know how to solve the ODE. One can
generalize the framework and then analyze a more complicated ODE.

27. Illustration

28. Tie-Decay Generalization of PageRank
◦ = Moore–Penrose pseudo-inverse
◦ c(t) = an n x 1 vector of ‘dangling nodes’ (which have 0 out-degree) at time t
◦ v = vector of teleportation probabilities (time-independent)

29. Updating the Transition Matrix P(t)
◦ No interactions → P + ∆P ≡ P(t + ∆t) = P(t)
◦ If there is a single new interaction during interval ∆t, we get the following expression for ∆P:
◦ Efficient computation of time-dependent PageRank vector using power iteration. We use π(t)
as initial value to obtain π(t+ ∆t)
◦ → Allows efficient computation with data streams (not just data sets)

30. Example: National Health Service (NHS)
Retweet Network
◦ Data set: 5 months of
tweeting activity about NHS
after the controversial Health
and Social Care Act of 2012
◦ Tweets in English with the term
‘NHS’ and look at the 10000
most-active users
◦ Interactions are retweets

32. Conclusions
◦ Clearly, we need to develop more centrality measures, right?
◦ Right?
◦ More seriously: it’s important to generalize ideas that we know (and presumably love) to
time-dependent networks.
◦ This talk: time-dependent generalizations of eigenvector-based centrality measures
◦ Multilayer representation
◦ Continuous time using “tie-decay network” approach
◦ Distinguishes between interactions and ties
◦ Examining continuous time is important for both data and modeling considerations
◦ Ideas for tie-decay continuous-time networks
◦ Include duration of ties, general personalized PageRank and associated community-detection
methods, etc. (There is a ton of stuff to do.)
◦ Key message: Need more studies of networks in continuous time

33. Advertisement: New Journal:
SIAM Journal on Mathematics
of Data Science (SIMODS)
◦ Now accepting submissions!
◦ SIMODS publishes work that advances mathematical,
statistical, and computational methods in the context
of data and information sciences. We invite papers that
present significant advances in this context, including
applications to science, engineering, business, and
medicine.