Index of /SISTA/afernand

Magnetic Eigenmaps for Visualization of Directed Networks

Micha¨

el Fanuel

∗

, ´

Angela Fern´

andez

†

, Carlos M. Ala´ız

‡

, and Johan A. K. Suykens

§

KU Leuven, ESAT, STADIUS Center. B-3001 Leuven, Belgium.

June 27, 2016

We propose a framework for visualization of directed

net-works relying on the eigenfunctions of the magnetic Laplacian,

called here Magnetic Eigenmaps. The magnetic Laplacian is

a complex deformation of the well-known combinatorial

Lapla-cian. Features such as density of links and directionality

pat-terns are revealed by plotting the phases of the first magnetic

eigenvectors. Directed networks being common in social

sci-ence, biology or computer scisci-ence, our visualization method

may be relevant for the field of complex networks, as well as

applied mathematics and machine learning. Illustrations of our

method are given for both artificial and real-life networks.

1 Introduction

Many objects and problems in neuroscience, biology, social or

computer science are phrased in terms of networks and graphs.

The embedding of data points forming undirected graphs can

be performed using manifold learning methods, among which

we can underline the so-called Laplacian Eigenmaps [1], or

Dif-fusion Maps [2]. In the same spirit, the embedding of a directed

graph originated from the sampling of a vector field on a

man-ifold was studied in [3]. A Laplacian for strongly connected

and aperiodic directed networks was introduced by Chung [4]

in relation with a random walk process, which was used for

visualization for instance in [5]. Actually, Laplacians are very

useful tools for community detection and data visualization. A

common feature of these approaches is the relevance of the

dis-crete or combinatorial Laplacian, and its normalized versions.

In this paper, no assumption on the origin of directed

net-works is needed, so we could deal, for example, with netnet-works of

webpages which are not embedded in any vector space. In

par-ticular, we propose here the use of another Laplacian which

nat-urally exists for a general connected directed network, called the

magnetic Laplacian. Known in the physics literature, it is

ac-tually a vector bundle Laplacian as described by Kenyon [6, 7].

The method that we describe assigns a complex rotation, i.e.,

an element of U(1), to each directed link, and the orientation

of the link determines the direction of the rotation.

2 Magnetic Laplacian and Eigenmaps

Laplacian Eigenmaps have been designed for embedding

undi-rected graphs and for dimensionality reduction of data. Given

a symmetric weight matrix W

(s)

= {w

_ij(s)

}, the maps are

as-sociated to the eigenvectors of the combinatorial Laplacian

L

(0)

= D − W

(s)

where D is the diagonal degree matrix whose

elements are given by the sum of each row of the weight matrix.

In the case of directed networks, the graphs are given by an

unsymmetric weight matrix, W = {w

ij

}. For simplicity, we

∗

_{Email: michael.fanuel@esat.kuleuven.be.}

†

_{Email: angela.fernandez@esat.kuleuven.be.}

‡

_{Email: cmalaiz@esat.kuleuven.be.}

§

_{Email: johan.suykens@esat.kuleuven.be.}

suppose that the weights are binary, i.e. w

ij

= 1 if there is a

link from i to j and w

ij

= 0 otherwise. The weight matrix W

can be decomposed as a symmetric term w

_ij(s)

= (w

ij

+ w

ji

) /2,

that indicates the existence of a connection between i and j,

and a skew-symmetric term, the edge flow a

ij

= −a

ji

, that

encodes the direction of the link, so that a

ij

= 1 if the link

points from i to j.

In this context, the Magnetic Laplacian can be defined as

L

(g)

= D − T

(g)

 W

(s)

_{, where D is the degree matrix}

asso-ciated to the symmetrized weight matrix, 0 ≤ g < 1/2 is an

electric charge parameter that quantifies the importance of the

directional information, and T

(g)

 W

(s)

_{= {exp (i2πga}

ji

) w

(s)ij

}

(notice the Hadamard product ).

The eigenfunctions φ

(g)_k

of the magnetic Laplacian, namely

Magnetic Eigenmaps [8], provide a map of the nodes in C

n

.

They satisfy the optimality criterion of the minimization of

X

i,j

w

(s)_ij







f

i

− e

i2πgaji

_f

j







2

(1)

over the complex functions of the nodes f of unitary norm, so

that the eigenfunction of smallest eigenvalue of L

(g)

, φ

(g)₀

, is the

minimizer of (1).

The magnetic Laplacian, already known in the physics

com-munity [9, 10, 11], is in fact a connection Laplacian [6], as

stud-ied by Singer and Wu [12], except that here we associate to

each link an element of the complex rotation group U(1),

in-stead of an orthogonal matrix of SO(d). Its spectrum is real

and positive semi-definite. Actually, as we have said before, the

magnetic Laplacian depends also on a parameter g, interpreted

as an electric charge, which adds some extra flexibility in the

visualization process. More explicitly, in general, we propose to

choose a quantized charge g = k/m with k /

∈ mZ. The

particu-lar value g = 1/3 is suited in the presence of directed triangles,

while g = 1/4 is relevant in the presence of directed 4-cycles.

We observe empirically that g = 2/5 gives also good results in

the presence of reciprocal links. For more details we refer to [8].

Notice that choosing g > 1/2 would be equivalent to a flip of

all link directions.

Interestingly, the magnetic Laplacian can be interpreted as a

discrete quantum mechanical Hamiltonian of a charged

parti-cle on a network, influenced by a magnetic flux. The objective

function of (1) is the corresponding energy of the wave function

f . Similarly, the combinatorial Laplacian is a discrete

Hamil-tonian of a free particle on a network.

3 Visualization of density and directionality

Going into detail in the relation between the combinatorial

and the magnetic Laplacians, in the case when the edge flow

of the network is given exactly by a certain potential h, i.e.

a

ij

= h

j

− h

i

for any nodes i and j, then the spectrum of

combinatorial Laplacian, and the eigenvectors are related by

φ

(g)_k,i

= e

i2πghi

_φ

(0)

k,i

.

Such a particular case can be characterized by the first

eigen-value, as stated in the following theorem.

Theorem 1. Let us considered a connected graph. The

mag-netic Laplacian L

(g)

has a zero eigenvalue iff there exists a

func-tion h satisfying, for any link {i, j}, a

ij

= h

j

− h

i

.

Proof. Both implications are proved next:

⇒ If L

(g)

_{has a zero eigenvalue, we know that the Rayleigh}

quotient for φ

(g)₀

is

X

i,j

w

_ij(s)







φ

(g) 0,i

− e

i2πgaji

_φ

(g) 0,j







2

= 0,

and therefore, each term in this finite sum has to vanish.

Hence, for any link {i, j}, we have φ

(g)_0,i

= e

i2πgaji

_φ

(g)

0,j

, so

that the eigenfunction has a constant modulus |φ

(g)_0,i

| = c.

Indeed, it can be decomposed as φ

(g)_0,i

= c · e

iθi

_.

Conse-quently, for any link {i, j}, e

i2πgaji

_{= e}

i(θi−θj)

_{, and}

there-fore, 2πga

ji

= θ

i

− θ

j

+ 2πm, with m an integer. Because

a

ji

= −a

ij

by definition, we find that m = 0. Hence, we

choose h

i

= θ

i

/ (2πg).

⇐ If, for any link {i, j}, we have a

ij

= h

j

− h

i

, then we can

write L

(g)

= U

−1

L

(0)

U , with the unitary diagonal matrix

U of matrix elements U

ii

= e

i2πghi

, and where L

(0)

=

D − W

(s)

is the combinatorial Laplacian.

This property may be also understood as a consequence of

the results of [11, 10] in the context of mathematical physics.

Therefore, the function h, interpreted as a potential responsible

for the link directions [13], is effectively encoded in the phase of

the eigenvectors, up to a constant shift. In the ranking context,

h is called a consistent ranking. In practice, obtaining h requires

to solve a linear system or to calculate the pseudo-inverse of

the combinatorial Laplacian as explained in the combinatorial

Hodge theory of [13]. Incidentally, the magnetic Laplacian has

interesting relations to the topology of graphs [10, 11].

In order to introduce now our method, we consider first this

previous particular type of directed networks. Let us further

assume that the symmetrized weight matrix W

(s)

is associated

to a connected undirected graph with a clear cluster structure.

Hence, we have in general that the first eigenvector of the

com-binatorial Laplacian is constant, φ

(0)_0,i

= cst, whereas the sign of

the so-called Fiedler vector, sign(φ

(0)_1,i

), partitions the network

in two dense subgraphs. If a clear cluster structure is present,

then the “phase” of the Fiedler eigenvector is piecewise constant

on the clusters. Therefore, it is interesting to calculate the

com-plex phase of the first eigenvector of the magnetic Laplacian,

i.e. phase(φ

(g)_0,i

) = 2πgh

i

mod 2π, which gives the potential h.

Furthermore, the phase of the second magnetic eigenvector φ

(g)_1,i

is exactly equal to the phase of the first magnetic eigenvector

φ

(g)_0,i

, but shifted by π on the nodes where the Fiedler

eigenvec-tor is negative, sign(φ

(0)_1,i

) = −1. To summarize, in this perfect

case: (i) phase(φ

(g)_0,i

) incorporates the directionality

informa-tion (corresponding to h); and (ii) phase(φ

(g)_1,i

) incorporates the

density information (corresponding to the spectral clustering).

In practice, the edge flow does not satisfy the condition

a

ij

= h

j

− h

i

, and thus the explanation given above is only an

approximation. Actually, due to the inhomogeneous degree

dis-tribution of real-life networks it can be advantageous to

normal-ize the magnetic Laplacian as follows L

(g)_N

= D

−1/2

L

(g)

D

−1/2

,

Figure 1: Identification of the 2-torus with the square [0, 2π] ×

[0, 2π] with identified sides. Notice that the position

of the cuts is arbitrary, and it can be adapted to each

particular dataset for the sake of clarity.

Figure 2: Example of the magnetic eigenmaps plotted over the

3-dimensional 2-torus for a graph with two

commu-nities, corresponding to the dataset ”Political

Blogo-sphere” explained in detail in the applications section.

so that we only have to compute the largest eigenvalues of

D

−1/2

T

(g)

 W

(s)

_D

−1/2

_{. Once the Laplacian is normalized,}

we propose to embed the network by the following mapping

i 7→



phase(φ

(g)_0,i

)

phase(φ

(g)_1,i

)

. . .

phase(φ

(g)_n,i

)



|

.

Notice that the phase operator identifies the angles that

dif-fers by 2π, which means that the geometrical representation,

for the 1-dimensional case, is just a circle, whereas for the

gen-eral n-dimensional case is an n-torus. Hence, for visualization

purposes, the directed network will be embedded on a 2-torus

represented as the square [0, 2π] × [0, 2π] with opposite sides

identified, as shown in Figures 1 and 2. Therefore, the

visual-ization will be symmetric if an eigenvector undergoes a global

rotation in the complex plane (note that given an eigenvector

φ

(g)_k

of the magnetic Laplacian, another eigenvector of the same

eigenvalue is given by e

iα

φ

(g)_k

). We will show empirically that

this low dimensional embedding is able to visualize at the same

time dense regions of links, revealed by the y-axis, and patterns

determined by the link directions, given by the x-axis.

4 Connection with Vector Diffusion Maps

The computation of the eigenvectors of the normalized

mag-netic Laplacian can resemble the Vector Diffusion Maps of

Singer and Wu [12], although in our case we work with a

com-plex and unitary transporter in U(1), instead of with an

orthog-onal transporter. For the sake of completeness, let us outline

the relationship between Magnetic Eigenmaps and Vector

Dif-fusion Maps.

Vector Diffusion Maps describes the transport of a vector

v

j

from the tangent space at point j to the tangent space at

i through an orthogonal transformation O

ji

, followed by an

averaging process over all possible j’s:

(A

VDM

v)

_i

=

1

d

i

X

j

w

(s)_ji

O

ji

v

j

.

(2)

Similarly, for the Magnetic Eigenmaps the transition matrix

A

ME

= D

−1

T

(g)

 W

(s)

describes the transport of a complex

Algorithm 1 Magnetic Eigenmaps Visualization

procedure MEigenmaps(W, g)

Magnetic Laplacian.

W

(s)

← (W + W

|

_{) /2}

I Symmetric weights.

A ← W − W

| I Edge flow.

d

ii

←

P

_j

w

ij(s) I Degree matrix.

t

(g)_ij

← e

i2πgaji _{I Transporter.}

L

(g)

← D − W

(s)

 T

(g) I Magnetic Laplacian.

Normalization.

L

(g)_N

← D

−1/2

_L

(g)

_D

−1/2 I Normalized Laplacian.

Eigendecomposition.

φ

(g)₀

, φ

(g)₁

, . . . ← Eigs(L

(g)N

)

I Eigenvectors.

Mapping.

return

n

phase(φ

(g)m

)

o

n m=0 I Phases.

end procedure

number f

j

along a link between j to i, followed by an averaging

process over all possible j’s connecting i, i.e.

(A

ME

f )

i

=

1

d

i

X

j

w

(s)_ji

t

(g)_ji

f

j

.

(3)

Comparing (2) and (3), the main difference, apart from the

working space (R

n

and C, respectively), resides in the transport

term. In the case of Vector Diffusion Maps, it is an element of

SO(n) determined by Local PCA, and for Magnetic Eigenmaps

it is an element of U(1) which can be tuned by the user through

the parameter g in order to highlight certain properties of the

graph. Moreover, the methodology of both approaches differs in

the way they map the data. On the one hand, Vector Diffusion

Maps follows a natural extension of Diffusion Maps [2], so that

each point is mapped to a matrix defined in terms of the

eigen-values and eigenvectors of the transition matrix. On the other

hand, the proposed magnetic eigenmaps map the points to the

phases of the first eigenvectors of the corresponding Laplacian.

Therefore, although both methods share some similarities,

they are essentially different, and none of them can be seen as

a particular case of the other.

5 Applications

In what follows, we will illustrate how Magnetic Eigenmaps

can be successfully applied to the visualization of directed

graphs over two synthetic examples, and two real-world

net-works. For completeness, we include the procedure to compute

the magnetic eigenmaps for a directed graph, given by the

bi-nary weights W , in Algorithm 1.

In all the examples we will first depict the aspect of the

orig-inal network as a baseline, using for this purpose expert

knowl-edge or a force-directed layout, which is just a way of depicting

graphs using attractive forces between adjacent nodes and

re-pulsive forces between distant nodes.

We will also compare

Magnetic Eigenmaps with Diffusion Maps for the real-world

examples. In this context, the Diffusion Maps embedding is

obtained by computing the algorithm with g = 0, and

plot-ting the first eigenvectors of the corresponding Laplacian

(in-stead of their phases). For the real networks we will also

de-pict the eigenvalues decay. According to Theorem 1, the case

a

ij

= h

j

− h

i

implies a first eigenvalue equal to 0, so in the

cases where the first eigenvalue is near to zero we can interpret

the phases of the first pair of eigenfunctions as a representation

of the directionality and the density of the graph.

5.1 Artificial networks

We propose to visualize first the artificial network with a

run-ning flow of Figure 3a, where the coordinates of the nodes in the

real plane have been chosen according to our knowledge about

the underlying groups. This network, constructed according

to [14], is constituted of three groups of ten nodes (A, B and

C). Two nodes in the same group are linked with a probability

0.5. Any node has also a probability 0.5 to be connected to

a node from another group. Furthermore 90 percent of these

interconnections are directed in the direction of the flow, i.e.

A → B, B → C and C → A. Plotting the real and imaginary

parts of the first eigenvector of the magnetic Laplacian can

indi-cate the presence of a running flow in the network, as illustrated

in Figure 3b. However there could be also dense clusters in the

network which cannot be visualized only thanks to the phase

of the first eigenfunction.

In order to actually visualize the

3 groups and the density information, we use our proposal of

plotting the complex phase of the first eigenfunction versus the

phase of the second eigenfunction of the magnetic Laplacian,

as illustrated in Figure 3c. Notice that the phase of the second

eigenfunction does not distinguish specific dense clusters in the

network, while the phase of the first eigenfunction

(correspond-ing to directionality) is able to separate the three groups. This

is actually true because the network was constructed in that

way.

We are going to consider now an example of network with

a small number of nodes playing a particular role and then a

clear structure with two dense clusters. In the paper of Leicht

and Newman [15], an artificial network of 32 nodes is built as

follows: it consists of two dense groups of 14 nodes with a few

interconnecting links and two pairs of nodes are connected to

the whole network. The first pair has only in-coming links,

while the second pair has only out-going links. An illustration

using the force layout is given in Figure 4a. Plotting the real

and imaginary parts of the first eigenfunction allows to

distin-guish the two pairs from the rest of the network, as showed

in Figure 4b. However, it is more instructive to visualize the

network using the phase of the two first eigenfunctions of the

magnetic Laplacian. Indeed, the two groups and the two pairs

are easily separated in Figure 4c, where the phase of the first

eigenfunction (directionality) is able to separate the two pairs

of disconnected points from the rest of the set, defining three

directionality-groups: green points, the yellow points and the

blue and red points together.

On the other side, the phase

of the second eigenfunction (corresponding to density

informa-tion) shows two groups: the blue and green points versus the

red and yellow points. Combining the information given by

the two phases we are able to easily separate visually the four

groups that we were looking for.

5.2 Real-life networks

In the previous section, we considered directed networks having

known structures either in terms of link directions or link

den-sity. Indeed, Magnetic Eigenmaps is able to provide

simultane-ously information about direction and density, as we illustrate

now also on real-life directed networks where these two aspects

are relevant.

The first data set used represents the network of common

adjective and noun adjacencies for the novel “David

Copper-field” by Charles Dickens, as described by M. Newman [16]. In

this directed graph we have 112 nodes, that represent the most

commonly occurring adjectives and nouns in the book. Edges

connect any pair of words that occur in adjacent position in

the text of the book. From the structure of English, a certain

directional structure can be anticipated, i.e. adjectives are

ex-(a) Graph using the expert knowledge

posi-tions.

(b)

First magnetic eigenvector (Re φ

0

vs.

Im φ

0

).

phase( φ1 ) phase( φ1 ) phase(φ0) phase(φ0)

(c)

Graph over the phase of the magnetic

eigenmaps.

Figure 3: Artificial network with a running flow. The colours indicate the three groups, and the magnetic eigenmaps correspond

to g = 1/4. Figure 3c is shown over the 3-dimensional torus in the SI appendix.

(a) Graph using a force-directed layout.

(b)

First magnetic eigenvector (Re φ

0

vs.

Im φ

0

).

phase( φ1 ) phase( φ1 ) phase(φ0) phase(φ0)

(c)

Graph over the phase of the magnetic

eigenmaps.

Figure 4: Artificial network with two dense clusters and two pairs of nodes with a specific role. The colours indicate the

dense clusters and the hub pairs, and the magnetic eigenmaps correspond to g = 1/4. Figure 4c is shown over the

3-dimensional torus in the SI appendix.

pected to be found before nouns. The graph representation of

this dataset, using a force layout, is shown in Figure 5a, where

the structure can hardly be guessed. However, considering the

phase of the two first magnetic eigenmaps, it is possible to

vi-sualize the presence of two groups in the directed network, as

illustrated in Figure 5b (indeed, the information is provided

mostly by the first coordinate, corresponding to

directional-ity). Finally, for comparison purposes we show in Figure 5c the

first embedding coordinates in the Diffusion Maps case, more

concretely the second and third eigenvectors (the first one is

discarded because it is constant). In this case both classes

ap-peared mixed, so these two diffusion coordinates are not able

to reveal the structure of the data. The reason is that Diffusion

Maps captures the density of the links but not the

directional-ity, while Magnetic Eigenmaps relates both.

In Figure 6 are represented the first eigenvalues of the

com-binatorial Laplacian (g = 0) and the magnetic Laplacian

(g = 2/5). The increase of these eigenvalues is very

contin-uous and homogeneous, presenting just an eigengap between

the first eigenvalue and the second one. This result reinforces

the election of the first and second eigenvectors for this example

as visualization coordinates. Moreover, the differences between

the two spectra, and the nonzero initial eigenvalue, show that

the structure of the graph is not trivial (see Theorem 1).

The second real dataset used in these experiments represents

the political blogosphere in February of 2005, as compiled by

Lada Adamic and Natalie Glance [17].

This directed graph

is composed by 1222 nodes that indicate the political leaning,

meaning left or liberal and right or conservative. We have

re-moved from these data disconnected points. The data on

polit-ical leaning comes from blog directories and some of the blogs

were labelled manually, based on incoming and outgoing links

and posts around the time of the 2004 presidential election in

the USA. The links between blogs were automatically extracted

from a crawl of the front page of the blog. From Figure 7a,

where the network has been depicted using the force layout, it

is already possible to guess the presence of two dense groups of

webpages. The first eigenvalues of the magnetic Laplacian in

Figure 8 instruct us to consider the two first pairs of eigenvalues

in order to visualize two different structures. In Figure 7b, the

magnetic eigenmaps do not distinguish the two classes of nodes,

however we observe that some webpages are less connected to

the rest of the network, whereas in Figure 7c we see the two

classes clearly separated. The latter mapping is also shown in

R

3

over the torus as an illustration in Figure 2. For comparison

purposes, we show also in this case the first embedding

diffu-(a) Graph using a force-directed layout.

phase( φ1 ) phase( φ1 ) phase(φ0) phase(φ0)

(b)

Graph over the phase of the magnetic

eigenmaps.

(c) Second and third diffusion maps (φ

1

vs.

φ

2

).

Figure 5: Word adjacencies example. The colours indicate the class labels: nouns (

•

) and the adjectives (

•

), and the magnetic

eigenmaps correspond to g = 2/5. Figure 5b is shown in more detailed and also over the 3-dimensional torus in the SI

appendix.

2 4 6 8 10 0 0.2 0.4 0.6 g = 0 g = 2/5

Figure 6: First eigenvalues of the combinatorial (g = 0) and

magnetic (g = 2/5) Laplacians, in the case of the

Word Adjacencies network.

sion coordinates in Figures 7d and 7e. In this example, where

the graph structure is clearer than in the previous dataset,

Dif-fusion Maps is able to condense the two classes separately just

using the density of the graph. We would like to highlight that

the distinction between both classes is not very clear when we

select the second and third eigenvectors (the first eigenvector

is again discarded), whereas a cleaner classification structure is

obtained when the fourth and fifth eigenvectors are depicted.

Nevertheless, Magnetic Eigenmaps represents better and in a

neat way the connectivity and density structure.

6 Summary and Discussion

In this paper, we have proposed the use of the eigenvectors of

the magnetic Laplacian, called here Magnetic Eigenmaps, for

the visualization of directed networks. Our work is a natural

extension of the Laplacian Eigenmaps and it is the first

ap-plication of a vector bundle Laplacian in the field of complex

network. Computationally, the method reduces to the

calcula-tion of the eigenvectors of maximal eigenvalues of a Hermitian

matrix, which can be conveniently performed thanks to e.g. the

power method. The advantages of this approach were explained

on artificial and real-life datasets, showing that our method is

able to reveal both the directionality and connectivity patterns

of the networks.

Acknowledgements

• EU: The research leading to these results has received

fund-ing from the European Research Council under the European

Union’s Seventh Framework Programme (FP7/2007-2013) /

ERC AdG A-DATADRIVE-B (290923).

This paper reflects

only the authors’ views, the Union is not liable for any use that

may be made of the contained information. • Research

Coun-cil KUL: GOA/10/09 MaNet, CoE PFV/10/002 (OPTEC),

BIL12/11T; PhD/Postdoc grants.

• Flemish Government:

– FWO: G.0377.12 (Structured systems), G.088114N (Tensor

based data similarity); PhD/Postdoc grants.

– IWT: SBO

POM (100031); PhD/Postdoc grants.

• iMinds Medical

In-formation Technologies SBO 2014. • Belgian Federal Science

Policy Office: IUAP P7/19 (DYSCO, Dynamical systems,

con-trol and optimization, 2012-2017).

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(a) Graph using a force-directed layout.

phase( φ1 ) phase( φ1 ) phase(φ0) phase(φ0)

(b) Graph over the phase of the first pair of

magnetic eigenmaps.

phase( φ3 ) phase( φ3 ) phase(φ2) phase(φ2)

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Figure 7: Political blogosphere example. The colours indicate the class labels: left leaning (

•

) and right leaning (

•

), and the

magnetic eigenmaps correspond to g = 1/4. Figure 7c is shown in more detailed and also over the 3-dimensional torus

in the SI appendix.

2 4 6 8 10 0 0.2 0.4 g = 0 g = 1/4

Figure 8: First eigenvalues of the normalized magnetic

Lapla-cian for g = 1/4 and g = 0 (combinatorial), in the

case of the Political Blogs network.

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eigenmaps for community detection in directed networks.

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[9] M. A. Shubin. Discrete magnetic Laplacian. Comm. Math.

Phys., 164:259–275, 1994.

[10] Y. Colin de Verdi`

ere. Magnetic interpretation of the nodal

defect on graphs. Analysis and PDE, 6(5):1235–1242, 2013.

[11] G. Berkolaiko. Nodal count of graph eigenfunctions via

magnetic perturbations.

Analysis and PDE, 6(5):1213–

1233, 2013.

[12] A. Singer and H.T. Wu.

Vector Diffusion Maps and

the connection Laplacian.

Commun Pure Appl Math.,

65(8):1067–1144, 2012.

[13] X. Jiang, L.-H. Lim, Y. Yao, and Y. Ye. Statistical

rank-ing and combinatorial Hodge theory.

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[14] A. Lancichinetti and S. Fortunato. Benchmarks for testing

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80:016118, 2009.

[15] E. A. Leicht and M. E. J. Newman. Community structure

in directed networks. Phys. Rev. Lett., 100(118703), 2008.

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74:036104, 2006.

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Workshop on the Weblogging Ecosystem, 2005.

SI appendix

Magnetic Eigenmaps over torus

Figure S1: Phase of the magnetic eigenmaps plotted over the 3-dimensional 2-torus for the artificial network with a

running flow (see Figure 3c).

Figure S2: Phase of the magnetic eigenmaps plotted over the 3-dimensional 2-torus for the artificial network with two

dense clusters and two pairs of nodes (see Figure 4c).

Figure S3: Phase of the magnetic eigenmaps plotted over the 3-dimensional 2-torus for the word adjacencies network (see

Figure 5b).

Figure S4: Phase of the magnetic eigenmaps plotted over the 3-dimensional 2-torus for the political blogosphere network

(see Figure 7c).

Detailed graphs for the real-life networks

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