My presentation at NetSci 2018 on Portrait Divergence, a new approach to comparing networks that is simple, general-purpose, and easy to interpret. The preprint: https://arxiv.org/abs/1804.03665 The code: https://github.com/bagrow/portrait-divergence

1. An information-theoretic, all-scales
approach to comparing networks
Jim Bagrow
bagrow.com
NetSci 2018
2018-06-15
james.bagrow@uvm.edu
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The University of Vermont
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The University of Vermont
The University of Vermont
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VERMONT
COMPLEX SYSTEMS CENTER

2. Problem: Comparing networks
distance(G1, G2) similarity(G1, G2)
n network, which encodes structural information; provides
networks; and allows for rigorous statistical comparison
uch as percolation can be visualized using animation.
ic networks
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3. Many applications
Anomaly detection Modeling
Temporal networks
Biological networks
Aggregation
Experiments
and many more

4. Many methods
PHYSICAL REVIEW E 86, 036104 (2012)
Taxonomies of networks from community structure
Jukka-Pekka Onnela,1,2,3,4,*
Daniel J. Fenn,4,5,*
Stephen Reid,3
Mason A. Porter,4,6
Peter J. Mucha,7
Mark D. Fricker,4,8
and Nick S. Jones3,4,9,10
1
Department of Biostatistics, Harvard School of Public Health, Boston, Massachusetts 02115, USA
2
Department of Health Care Policy, Harvard Medical School, Boston, Massachusetts 02115, USA
3
Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
4
CABDyN Complexity Centre, University of Oxford, Oxford OX1 1HP, United Kingdom
5
Mathematical and Computational Finance Group, University of Oxford, Oxford OX1 3LB, United Kingdom
6
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, OX1 3LB, United Kingdom
7
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics and Institute for Advanced Materials,
Nanoscience & Technology, University of North Carolina, Chapel Hill, North Carolina 27599, USA
8
Department of Plant Sciences, University of Oxford, South Parks Road, Oxford OX1 3RB, United Kingdom
9
Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, Oxford OX1 3QU, United Kingdom
10
Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
(Received 30 November 2011; published 10 September 2012)
The study of networks has become a substantial interdisciplinary endeavor that encompasses myriad disciplines
in the natural, social, and information sciences. Here we introduce a framework for constructing taxonomies of
networks based on their structural similarities. These networks can arise from any of numerous sources: They
can be empirical or synthetic, they can arise from multiple realizations of a single process (either empirical
or synthetic), they can represent entirely different systems in different disciplines, etc. Because mesoscopic
properties of networks are hypothesized to be important for network function, we base our comparisons on
summaries of network community structures. Although we use a specific method for uncovering network
communities, much of the introduced framework is independent of that choice. After introducing the framework,
we apply it to construct a taxonomy for 746 networks and demonstrate that our approach usefully identifies
similar networks. We also construct taxonomies within individual categories of networks, and we thereby expose
nontrivial structure. For example, we create taxonomies for similarity networks constructed from both political
voting data and financial data. We also construct network taxonomies to compare the social structures of 100
Facebook networks and the growth structures produced by different types of fungi.
DOI: 10.1103/PhysRevE.86.036104 PACS number(s): 89.75.Hc
I. INTRODUCTION
Although there is a long tradition of scholarship on
etworks, the last two decades have witnessed substantial
dvances in network science due to developments in physics,
athematics, computer science, sociology, and numerous
her disciplines [1,2]. Given that the questions asked by
searchers in different fields can be surprisingly similar, it
ould be useful to be able to highlight similarities in network
ructures across disciplines in a systematic way. One way to
by the periodic table of elements in chemistry and phylogenetic
trees of organisms in biology [3]. It is plausible that an
organization of networks has the potential to shed light
on mechanisms for generating networks, reveal how an
unknown network should be treated once one has discerned
its position in a taxonomy, or help identify a network
family’s anomalous members. Further potential applications
of network taxonomies include unsupervised study of multiple
realizations of a given model process (e.g., characterizing
the similarities and differences of many different networks
ARTICLE
Received 3 Jul 2014 | Accepted 7 Mar 2015 | Published 23 Apr 2015
Structural reducibility of multilayer networks
Manlio De Domenico1,*, Vincenzo Nicosia2,*, Alexandre Arenas1 & Vito Latora2,3
Many complex systems can be represented as networks consisting of distinct types of
interactions, which can be categorized as links belonging to different layers. For example, a
good description of the full protein–protein interactome requires, for some organisms, up to
seven distinct network layers, accounting for different genetic and physical interactions, each
containing thousands of protein–protein relationships. A fundamental open question is then
how many layers are indeed necessary to accurately represent the structure of a multilayered
complex system. Here we introduce a method based on quantum theory to reduce the
number of layers to a minimum while maximizing the distinguishability between the multi-
layer network and the corresponding aggregated graph. We validate our approach on
synthetic benchmarks and we show that the number of informative layers in some real
multilayer networks of protein–genetic interactions, social, economical and transportation
systems can be reduced by up to 75%.
DOI: 10.1038/ncomms7864
Comparing Brain Networks of Different Size and
Connectivity Density Using Graph Theory
Bernadette C. M. van Wijk1
*, Cornelis J. Stam2
, Andreas Daffertshofer1
1 Research Institute MOVE, VU University Amsterdam, Amsterdam, The Netherlands, 2 Department of Clinical Neurophysiology, VU University Medical Center, Amsterdam,
The Netherlands
Abstract
Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its
use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the
number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of
network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between
empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various
approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed
thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N
and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-
significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates
but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and
small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to
correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local
network structures including exponential random graph models and motif counting. We show that none of the here-
investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.
Citation: van Wijk BCM, Stam CJ, Daffertshofer A (2010) Comparing Brain Networks of Different Size and Connectivity Density Using Graph Theory. PLoS
ONE 5(10): e13701. doi:10.1371/journal.pone.0013701
Editor: Olaf Sporns, Indiana University, United States of America
Received December 22, 2009; Accepted October 7, 2010; Published October 28, 2010
Copyright: ß 2010 van Wijk et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was financially supported by the Netherlands Organisation for Scientific Research (NWO, http://www.nwo.nl) grant numbers 021-002-047
and 452-04-332. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: b.vanwijk@fbw.vu.nl
Vol. 23 ECCB 2006, pages e177–e183
doi:10.1093/bioinformatics/btl301BIOINFORMATICS
Biological network comparison using graphlet degree
distribution
Natasˇa Przˇulj
Computer Science Department, University of California, Irvine, CA 92697-3425, USA
ABSTRACT
Motivation: Analogous to biological sequence comparison, compar-
ing cellular networks is an important problem that could provide insight
into biological understanding and therapeutics. For technical reasons,
comparing large networks is computationally infeasible, and thus
heuristics,suchasthedegreedistribution,clusteringcoefficient,diame-
ter, and relative graphlet frequency distribution have been sought. It
is easy to demonstrate that two networks are different by simply show-
ing a short list of properties in which they differ. It is much harder to
show that two networks are similar, as it requires demonstrating their
Therefore, analogous to the BLAST heuristic (Altschul et al., 1990)
for biological sequence comparison, we need to design a heuristic
tool for the full-scale comparison of large cellular networks
(Berg and Lassig, 2004). The current network comparisons consist
of heuristics falling into two major classes: (1) global heuristics,
such as counting the number of connections between various parts
of the network (the ‘degree distribution’), computing the average
density of node neighborhoods (the ‘clustering coefficients’), or the
average length of shortest paths between all pairs of nodes (the
‘diameter’); and (2) local heuristics that measure relative distance
DELTACON: Principled Massive-Graph Similarity Function
with Attribution
DANAI KOUTRA, Computer Science and Engineering, University of Michigan, Ann Arbor∗
NEIL SHAH, Computer Science Department, Carnegie Mellon University
JOSHUA T. VOGELSTEIN, Department of Biomedical Engineering & Institute of Computational
Medicine, Johns Hopkins University Child Mind Institute
BRIAN GALLAGHER, Lawrence Livermore National Laboratory
CHRISTOS FALOUTSOS, Computer Science Department, Carnegie Mellon University
How much has a network changed since yesterday? How different is the wiring of Bob’s brain (a left-handed
male) and Alice’s brain (a right-handed female), and how is it different? Graph similarity with given node
correspondence, i.e., the detection of changes in the connectivity of graphs, arises in numerous settings.
In this work, we formally state the axioms and desired properties of the graph similarity functions, and
evaluate when state-of-the-art methods fail to detect crucial connectivity changes in graphs. We propose
DELTACON, a principled, intuitive, and scalable algorithm that assesses the similarity between two graphs
on the same nodes (e.g., employees of a company, customers of a mobile carrier). In conjunction, we propose
DELTACON-ATTR, a related approach that enables attribution of change or dissimilarity to responsible nodes
and edges. Experiments on various synthetic and real graphs showcase the advantages of our method over
existing similarity measures. Finally, we employ DELTACON and DELTACON-ATTR on real applications: (a) we
+ many more!

5. Assume node correspondence?
G1 = (V, E1)
G2 = (V, E2)
G3 = (V, E3)

6. Assume node correspondence?
G1 = (V, E1)
G2 = (V, E2)
G3 = (V, E3)

7. Assume node correspondence?
G1 = (V, E1)
G2 = (V, E2)
G3 = (V, E3)
I will not assume this

8. Network portraits (2008)
March 2008
EPL, 81 (2008) 68004 www.epljournal.org
doi: 10.1209/0295-5075/81/68004
Portraits of complex networks
J. P. Bagrow1
, E. M. Bollt2,1
, J. D. Skufca2
and D. ben-Avraham1
1
Department of Physics, Clarkson University - Potsdam, NY 13699-5820, USA
2
Department of Math and Computer Science, Clarkson University - Potsdam, NY 13699-5815, USA
received 21 November 2007; accepted in final form 28 January 2008
published online 27 February 2008
PACS 89.75.Hc – Networks and genealogical trees
PACS 02.10.Ox – Combinatorics; graph theory
PACS 89.75.-k – Complex systems
Abstract – We propose a method for characterizing large complex networks by introducing a
new matrix structure, unique for a given network, which encodes structural information; provides
useful visualization, even for very large networks; and allows for rigorous statistical comparison
between networks. Dynamic processes such as percolation can be visualized using animation.
Copyright c⃝ EPLA, 2008
Introduction. – Large, complex stochastic networks
14
Portraits are a matrix aggregating the
shortest paths distributions
Computed with breadth-first search (for
example) starting at every node

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Network Portrait

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<latexit sha1_base64="63MHvBGniSuUarlZWLEd+sXH1JY=">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</latexit>
Network Portrait
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Degree distribution
Number of nodes
Number of edges
Number of
shortest paths
Diameter

11. Kent
OhioState
Michigan
WesternMichigan
PennState
Wisconsin
BowlingGreenState
Ohio
Purdue
MichiganState
NorthernIllinois
Temple
Navy
BallState
Syracuse
VirginiaTech
EasternMichigan
BostonCollege
WestVirginia
NorthCarolina
MiamiOhio
CentralMichigan
MiamiFlorida
Marshall
Akron
Rutgers
Pittsburgh
Buffalo
Northwestern
Toledo
Indiana
IowaState
BrighamYoung
SouthernMethodist
FresnoState
Hawaii
MiddleTennesseeState
Connecticut
NorthCarolinaState
Duke
CentralFlorida
Army
Cincinnati
Mississippi
Memphis
LouisianaState
MississippiState
AlabamaBirmingham
SouthernMississippi
Arkansas
SouthCarolina
Louisville
Tennessee
Kentucky
Florida
Georgia
GeorgiaTech
FloridaState
Virginia
Clemson
WakeForest
Vanderbilt
EastCarolina
Maryland
Auburn
LouisianaMonroe
Alabama
Houston
Tulane
NotreDame
LouisianaTech
Tulsa
Missouri
Iowa
Illinois
Minnesota
TexasChristian
Colorado
Nevada
NewMexico
SanDiegoState
ColoradoState
Utah
Nebraska
TexasA&M
TexasElPaso
TexasTech
NorthTexas
Oklahoma
Texas
UtahState
NevadaLasVegas
Baylor
Kansas
OklahomaState
BoiseState
Wyoming
Rice
UCLA
AirForce
ArkansasState
Idaho
KansasState
SanJoseState
NewMexicoState
LouisianaLafayette
SouthernCalifornia
WashingtonState
ArizonaState
Stanford
Arizona
OregonState
Oregon
Washington
California
<latexit sha1_base64="mpvC3+1FOTxb2ByJGRlRKgamKNI=">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</latexit>
the number of nodes with
k nodes at distance l<latexit sha1_base64="q7idF2pYCcL0HWmntev8P5/y4To=">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</latexit> <latexit sha1_base64="MmdChoW2smf/xxB83DvHrCYgySs=">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</latexit>
<latexit sha1_base64="63MHvBGniSuUarlZWLEd+sXH1JY=">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</latexit>
Network Portrait

12. k=0
The graph diameter d is
d = max{` | B`,k > 0 for k > 0}.
The shortest path distribution is also captured: the number
of shortest paths of length ` is 1
2
PN
k=0 kB`,k. And the
portraits of random graphs present very differently from
highly ordered structures such as lattices (Fig. 1).
One of the most important properties of portraits is that
they are a graph invariant:
Definition 2.1. A graph invariant is a property of a graph that
is invariant under graph isomorphism, i.e., it is a function f such
that f(G) = f(H) whenever G and H are isomorphic graphs.
Theorem 2.1. The network portrait (Eq. (1)) is a graph invariant.
Proof. Let f : VG ! VH be a vertex bijection between
two graphs G = (VG, EG) and H = (VH, EH) such
that the number of edges between every pair of vertices
(i, j) in G equals the number of edges between their im-
ages (f(i), f(j)) in H. Then G and H are isomorphic.
Let `G(i, j) be the length of the shortest path between
nodes i and j in G. For two isomorphic graphs G and
1. Note that a distance ` = 0 is admissible, with two nodes i and j at
distance 0 when i = j. This means that the matrix B so defined has a
zeroth row. It also has a zeroth column, as there may be nodes that have
zero nodes at some distance `. This occurs for nodes with eccentricity
less than the graph diameter.
The row
tween c
allows a
a two-sa
spondin
underly
differen
graph c
eter gra
Lastly,
using a
between
where
is a wei
heavily
Whi
paring g
we did
2
r a network comparison measure. In Sec. 4
measure to both synthetic networks (random
bles) and real-world datasets (multilayer bi-
temporal social networks), demonstrating its
on practical problems of interest. Lastly, we
ec. 5 with a discussion of our results and future
RK PORTRAITS
traits were introduced in [25] as a way to
H, `G(i, j) = `H(f(i), f(j)) for all i and j in G, since the
shortest path tuples (i, . . . , j) in G and (f(i), . . . , f(j)) in
H are the same length. All elements in the matrix B(G) are
computed by aggregating the values of `G(i, j). Therefore,
B(G) = B(H).
Note that the converse is not necessarily true: that
f(G) = f(H) does not imply that G and H are isomor-
phic. As a counter-example, the non-isomorphic distance-
regular dodecahedral and Desargues graphs have equal
portraits [25].
Portraits are a graph invariant

13. An information-theoretic, all-scales approach to comparing
networks
James P. Bagrow1,2,* and Erik M. Bollt3
1Department of Mathematics & Statistics, University of Vermont, Burlington, VT, United States
2Vermont Complex Systems Center, University of Vermont, Burlington, VT, United States
3Department of Mathematics, Clarkson University, Potsdam, NY, United States
*Corresponding author. Email: james.bagrow@uvm.edu, Homepage: bagrow.com
April 10, 2018
Abstract As network research becomes more sophisticated, it is more common than ever for researchers to
find themselves not studying a single network but needing to analyze sets of networks. An important task when
Bagrow and Bollt, under review, 2018

14. <latexit sha1_base64="fmOBjJFSIvlewgnNT1wKNab7Myw=">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</latexit>
Comparing networks by
comparing portraits
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Graph 1 Graph 2
Rows of the portrait B (B') encode probability distributions:

15. <latexit sha1_base64="fmOBjJFSIvlewgnNT1wKNab7Myw=">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</latexit>
Comparing networks by
comparing portraits
<latexit sha1_base64="krT5lXMMZifEs2kbfuNGw0JRN+8=">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</latexit>
Graph 1 Graph 2
Rows of the portrait B (B') encode probability distributions:
Invites information-theoretic comparison per row:
(KL-divergence)
<latexit sha1_base64="qpMEm1gdu5b8ofMvux673MjOYbo=">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</latexit>

16. Concerns with just KL
1. Undefined if ∃ k such that P(k) > 0 and Q(k) = 0
2. KL is not symmetric and does not define a distance
3. Want a single comparison measure? Need to aggregate
max(d, d') + 1 divergences (diameters d, d')
<latexit sha1_base64="jxtwg6Fk2w/qcaXDlRToWe0ee3M=">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</latexit>

17. We propose
<latexit sha1_base64="utWHLcQN3K+oJt9iFjJbWRyjXaE=">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</latexit>
1. Choose two nodes in G uniformly at random (w/
replacement)
Prob they are at distance ℓ:
2. Combine with P(k | ℓ) → aggregation
<latexit sha1_base64="6iBda0LcXHIFWU7NLjir71OcviI=">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</latexit>

18. We propose
<latexit sha1_base64="utWHLcQN3K+oJt9iFjJbWRyjXaE=">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</latexit>
1. Choose two nodes in G uniformly at random (w/
replacement)
Prob they are at distance ℓ:
2. Combine with P(k | ℓ) → aggregation
<latexit sha1_base64="6iBda0LcXHIFWU7NLjir71OcviI=">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</latexit>

19. <latexit sha1_base64="6iBda0LcXHIFWU7NLjir71OcviI=">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</latexit>
(and likewise for Graph 2 → Q(k, ℓ) )

20. <latexit sha1_base64="6iBda0LcXHIFWU7NLjir71OcviI=">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</latexit>
(and likewise for Graph 2 → Q(k, ℓ) )
<latexit sha1_base64="LJS9bVoP8Kp+nnC0dm2iYL2J+Do=">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</latexit>
Portrait divergence
where

21. <latexit sha1_base64="6sHZflHBxHVNoRnI43dLcyyxLSY=">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</latexit>
<latexit sha1_base64="6iBda0LcXHIFWU7NLjir71OcviI=">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</latexit>
(and likewise for Graph 2 → Q(k, ℓ) )
<latexit sha1_base64="LJS9bVoP8Kp+nnC0dm2iYL2J+Do=">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</latexit>
Portrait divergence
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where

22. 1. Undefined if ∃ k such that P(k) > 0
and Q(k) = 0
2. KL is not symmetric and does not
define a distance
3. Want a single comparison measure?
Need to aggregate max(d, d') + 1
divergences (diameters d, d')
<latexit sha1_base64="LJS9bVoP8Kp+nnC0dm2iYL2J+Do=">AAADXHicbVJRaxNBEN4mVWtqNVXwxZfVUEgxCbmg2JdCwWrVRmxJ0xay17DZm0vW7u2du3vVsL2f5K/xSVD/ieBekoekceBg+Oab+XbmvkEiuDbN5s+VQnH11u07a3dL6/c27j8obz481XGqGHRZLGJ1PqAaBJfQNdwIOE8U0Ggg4Gxw+Tqvn12B0jyWJ2acgB/RoeQhZ9Q4qF8+IPufdfWghg8uLEkUjyDbxgS+pPwKk1BRZr3MtjJy2LZHmf2Y4ec34eMcLvXLlWajOQm8nHizpIJmcdTfXPlLgpilEUjDBNW65zUT41uqDGcCshJJNSSUXdIh9FwqaQTat5ONM7zlkACHsXKfNHiClrbmWiyNtB5HA0eNqBnppWKO/rfYS02441suk9SAZFOtMBXYxDi/IA64AmbE2CWUKe6ei9mIupsYd+fSolAqOYsDqE/UFmVyqK7NWMDu+84n/wZ1kXvi+TZfNH/P/GGshK/mmw4XR0/R6Zz5KToMIKSpMDXNqIBgt7Hz0rdvuaIdKrXTJK6e+8Hu90neriL7oZPlBTeRxVFEZeB+ea/lWwJSpwpylrWH5CluZ0RAaKq44mFSu74mNVxpYaL4cGS2M3dFfNiuB9x5cei2gNww3k17LCenrYbXbHjHLyp7b2bWWUNP0DNURR56hfbQO3SEuoih7+gH+oV+F/4UV4vrxY0ptbAy63mEFqL4+B/BmRLc</latexit>
Portrait divergence
Concerns
Solved by Jensen-Shannon
Divergence
Solved by P(k | ℓ) aggregation

23. Results

24. Model ensembles
Portrait divergence〈DJS〉
Erdős-Rényi
Barabási-Albert

25. Effects of rewiring
Erdős-Rényi Barabási-Albert
Number of rewirings
〈DJS〉
N = 300
⟨k⟩ = 3
N = 300
m = 3
Goriginal vs Grewired

26. Multilayer networks

27. Multilayer networks
Stark et al., Nucleic Acids Res., 2006 Chen et al., PNAS, 2006

28. Collaboration
See also: Klug & Bagrow, 2016

29. github.com/ibm

30. 2013 2014 2015 2016 2017
A
B C
Temporal network

31. 2013 2014 2015 2016 2017
A
B C
Temporal network

32. 2013 2014 2015 2016 2017
A
B C
Temporal network

33. A
Weighted
Networks
Weighted Portrait:

34. Summary
• Network comparison is a common problem—many
methods exist
• We introduce Portrait Divergence, a simple, general-
purpose, easy-to-interpret comparison measure
• Portrait Divergence reveals differences in network models
and provides useful information about multilayer and
temporal networks
Bagrow and Bollt,
An information-theoretic, all-scales
approach to comparing networks,
arXiv:1804.03665 (2018)
Code available:
github.com/bagrow/portrait-
divergence
Links to both at:
bagrow.com

35. Summary
• Network comparison is a common problem—many
methods exist
• We introduce Portrait Divergence, a simple, general-
purpose, easy-to-interpret comparison measure
• Portrait Divergence reveals differences in network models
and provides useful information about multilayer and
temporal networks
Bagrow and Bollt,
An information-theoretic, all-scales
approach to comparing networks,
arXiv:1804.03665 (2018)
Code available:
github.com/bagrow/portrait-
divergence
Links to both at:
bagrow.com

36. Thanks:
IIS-1447634
An information-theoretic, all-scales
approach to comparing networks
The niversity
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The University of Vermont
The University of Vermont
The University of Vermont
Solid logo can be made to take on tint
of background color.
The University of Vermont
The University of Vermont
Logo used on the web
VERMONT
COMPLEX SYSTEMS CENTER
Erik Bollt
Clarkson University

37. • Network comparison is a common problem—many
methods exist
• We introduce Portrait Divergence, a simple, general-
purpose, easy-to-interpret comparison measure
• Portrait Divergence reveals differences in network models
and provides useful information about multilayer and
temporal networks
Bagrow and Bollt,
An information-theoretic, all-scales
approach to comparing networks,
arXiv:1804.03665 (2018)
Code available:
github.com/bagrow/portrait-
divergence
Links to both at:
bagrow.com
Thanks!

38. • Network comparison is a common problem—many
methods exist
• We introduce Portrait Divergence, a simple, general-
purpose, easy-to-interpret comparison measure
• Portrait Divergence reveals differences in network models
and provides useful information about multilayer and
temporal networks
Bagrow and Bollt,
An information-theoretic, all-scales
approach to comparing networks,
arXiv:1804.03665 (2018)
Code available:
github.com/bagrow/portrait-
divergence
Links to both at:
bagrow.com
An information-theoretic, all-scales
approach to comparing networks