The Effects of Edge Weights on Correlating Dynamical Networks

Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients

Christian Moewes and Rudolf Kruse

Faculty of Computer Science, Otto-von-Guericke University, Universit

atsplatz 2, 39106 Magdeburg, Germany

Keywords:

Dynamical Networks, Regression, Vector Autoregression Weighted Graph.

Abstract:

We are interested in the regression analysis of dynamical networks. Our goal is to predict real-valued function

values from a given observation which is manifested as series of graphs. Every observation is described

by a set of dependent variables that we want to predict using the dynamical graphs. These graphs change

their edges over time, while the set of nodes is assumed to be constant. Such settings can be found in many

real-world applications, e.g., communication networks, brain connectivity, microblogging. We apply several

measures to every graph in the series to globally describe its evolution. The resulting multivariate time series

is used to learn vector autoregressive (VAR) models. The parameters of these models can be used to correlate

them with the dependent variables. The graph measures typically depend on the type of edges, i.e., weighted

or unweighted. So do the VAR models and thus the regression results. In this paper we argue that it is

beneﬁcial to keep edge weights in this setting. To support this claim, we analyze electroencephalographic

(EEG) networks from patients suffering from visual ﬁeld defects. The edge weights are in the unit interval and

might be thresholded. We show that dynamical network models for weighted edges lead to similar regression

performances compared to those of unweighted graphs.

1 INTRODUCTION

Complex networks be constructed from any set of

objects that “interact” with each other. So, based

on the large number different objects with this prop-

erty, complex networks can be found in nearly ev-

ery part of our world, e.g., in social science (Wasser-

mann and Faust, 1994), computer science (Faloutsos

et al., 1999), medicine (Pereira-Leal et al., 2004), bi-

ology (Fischhoff et al., 2007), neuroscience (Sporns,

2010), and the World Wide Web (Kleinberg et al.,

1999). Many researchers in these and other ﬁelds

think that the analysis of complex networks may be

useful to extract their intrinsic regularities, causal re-

lations, and other useful pieces of information.

Probably the most complex network in the uni-

verse is the human brain. Since (Varela et al., 2001),

if not earlier, it became very common in neuroscience

to analyze functional networks. One property of these

complex networks is their nature to dynamically—

sometimes even chaotically—change their edges over

time. They are typically obtained from neuroimaging

methods, e.g., electroencephalography (EEG), elec-

trocorticography (ECoG), magnetoencephalography

(MEG), or functional magnetic resonance imaging

(fMRI). These methods record the activity of different

brain regions—on the skull, on the brain meninges, or

even inside the brain.

Whenever two brain regions, i.e., the nodes of the

brain graph, are co-active, they are functionally con-

nected to each other. These connections lead to a

complex brain network which represents a high-level

abstraction of the biological nervous cells. The analy-

sis of these brain networks has already led to a better

understanding of the functionality of different brain

centers and the brain as a whole (Sporns, 2010).

Most research on complex graphs—no matter if

neuroscientiﬁc or not—has mainly focused on static

graphs. Such a static network is simply obtained by

averaging the connections over time. Of course, it is

far easier to analyze static networks than a series of

networks. Nevertheless averaging diminishes an im-

portant property, i.e., the dynamics of such complex

networks.

We argue that the dynamical nature of networks

can also be exploited to analyze complex networks

and some of their functions. Some ﬁrst ﬁndings sup-

port this claim, at least for “damaged” human brain

webs: In (Moewes et al., 2013) patients’ EEG has

been correlated to the size of visual ﬁeld deﬁcits that

279

Moewes C. and Kruse R..

The Effects of Edge Weights on Correlating Dynamical Networks - Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients.

DOI: 10.5220/0004641402790284

In Proceedings of the 5th International Joint Conference on Computational Intelligence (FCTA-2013), pages 279-284

ISBN: 978-989-8565-77-8

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

resulted from optic nerve damages (W

ust et al., 2002).

The authors identiﬁed a relation between the extend

of the vision loss and the dynamics of the functional

brain connectivity. More precisely, the authors of this

paper propose the ﬁrst model-based approach to cap-

ture the relevant features of a dynamical complex net-

work (Moewes et al., 2013).

In the above-mentioned work of the authors, un-

weighted graphs have been constructed to facilitate

the computation of the network models. There-

fore some patient-independent threshold for the edge

weights had to be identiﬁed. The question we ad-

dress in this paper is whether the tedious and some-

what arbitrary choice can be omitted. We thus claim

that it is beneﬁcial to use weighted graphs instead of

unweighted ones.

To support this hypothesis, we analyze the same

kind of EEG networks used in (Moewes et al., 2013).

We apply different thresholds to the weighted edges to

obtain an unweighted graph. Eventually we show that

dynamical network models for weighted edges lead

to similar if not better regression results than those of

unweighted graphs. Thus the tedious and somewhat

arbitrary choice of an edge weight threshold is super-

ﬂuous.

The rest of the paper is organized as follows. Sec-

tion 2 introduces the concept of dynamical networks

to describe functional brain connectivity. Section 3

describes the experimental setup we have used to

compare the regression performance of weighted and

unweighted graphs. The results of this experiment are

listed in Section 4. Finally, Section 5 concludes the

paper.

2 DYNAMICAL NETWORK

ANALYSIS

Having deﬁned the functional connectivity of two

brain regions, the corresponding dynamical brain net-

work can be obtained. We remark that functional

connectivity is just a statistical relationship between

brain regions without implying any causal coher-

ence (Pearl, 2009). There exist many ways to com-

pute functional connectivity. We refer the interested

reader to (Wendling et al., 2009) for a review on EEG

connectivity measures.

2.1 Synchronization Likelihood

Out of these methods, we have used the synchroniza-

tion likelihood (SL) (Stam and van Dijk, 2002) since

it has been used in the literature to study the relation-

ship between structural network damage and func-

tional connectivity (Stam et al., 2007).

Consider a multivariate time series (e.g. a multi-

channel EEG recording) of length N with n variables.

Let measurement x

i,k

be observed at timestamp i in

channel k. For the SL, a time-delay embedding is

computed by

i,k



i,k

i+L,k

i+2·L,k

,...,x

i+(m−1)·L,k



where L is the lag and m the dimension of the embed-

ding. The state vectors X

i,k

shall capture the relevant

patterns of the signal.

Now consider two channels A,B. The probability

that X

i,k

are closer to each other than ε is

i,k

2(W

−W

)

∑

<|i− j|<W

θ(ε − d(X

i,k

j,k

))

where d is the Euclidean distance (or any other dis-

tance measure). For each k and i, a critical distance

i,k

is computed such that P

i,k

= p

ref

whereas p

ref

 1

is some user-deﬁned threshold. For each pair of points

in time (i, j) within W

< |i − j| < W

, the number of

channels H

i, j

for which d(X

i,k

j,k

) < ε

i,k

is computed

i, j

= θ(ε

i,A

− d(X

i,A

j,A

)) + θ(ε

i,B

− d(X

i,B

j,B

))

where θ(x) = 0 if x ≤ 0 and θ(x) = 1 for x > 0. Then

the synchronization likelihood is then given by

ref

−W

)

∑

<|i− j|<W

i, j

− 1). (1)

Now the ﬁnal brain graph can be computed. Note that

only two of these the parameters are needed to com-

pute SL if prior information about the frequency range

and temporal resolution of the signal are given (Mon-

tez et al., 2006).

2.2 Graph Measures

We remark that any functional connectivity measure

based on a sliding window will change its value over

time. So do the edge weights of the corresponding

brain graph. In (Moewes et al., 2013) the authors

described these changes by learning how graph mea-

sures evolve.

Therefore we consider a set of different graph

measures, i.e., the number of cliques, the density, and

the squared distance between the current graph and

the previous one (Bunke, 1997).

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

280

2.3 Vector Autoregressive Models

Vector autoregressive (VAR) models (L

utkepohl,

2005) have been shown to capture a high proportion

of variance in these data.

A VAR model with p lags is given by

= c +

∑

i=1

t−i

+ ε

where c is a constant, A

is a matrix storing the rela-

tionships between every pair of variable at point t − i,

and ε

is white noise. Flattening the coefﬁcients A

to a feature vector is beneﬁcial for statistical learning

methods.

3 EXPERIMENTS

In our experiments we used EEG data from 33 vi-

sually impaired subjects suffering from optic nerve

damages (W

ust et al., 2002). For every patient, sev-

eral optometric tests had been performed to describe

the location and the size of the optic nerve damage

for both eyes (Sabel et al., 2011). Among them are

high resolution perimetry (HRP) (Kasten et al., 1998),

static perimetry, kinetic perimetry, and the assessment

of visual acuity (Bailey and Lovie, 1976). Based on

these tests an expert deﬁned the following clinical

variables relevant to quantify the vision loss of both

eyes:

• detection accuracy in HRP visual ﬁeld (%),

• foveal threshold in static perimetry (dB),

• mean threshold in static perimetry (whole 30

◦

vi-

sual ﬁeld, dB),

• mean eccentricity in kinetic perimetry (

◦

• visual acuity of near vision (LogMAR scale),

• visual acuity of far vision (LogMAR scale).

Nearly every optometric test is very tiring, time-

consuming, and error-prone. Each one of them served

as variable assumed to be depended from the dynam-

ics in the EEG. It is our motivation to ﬁnd good corre-

lates of the EEG signal and these tests. If successful,

a good prediction model may determine the size of

the optic nerve damage by solely looking at EEG data

that can be recorded much faster.

To preprocess the EEG data we did the following

steps in EEGLAB (Delorme and Makeig, 2004):

• manually removal of noisy time frames at begin-

ning/end of each recording,

• removal of uncommon EEG channels across all

subjects (28 were used),

• high-pass ﬁltering with cutoff frequency at 1 Hz to

remove slow movements,

• notch ﬁltering 50 Hz to remove the European

power line frequency,

• low-pass ﬁltering with cutoff frequency at 95Hz,

• re-referencing by the average electrode,

• down-sampling to 250Hz to reduce the costs of

SL computation,

• manual removal of biological artifacts using in-

dependent component analysis (Makeig et al.,

1996).

Biological artifacts that stem from electromyographic

(EMG) or electrocardiograph (EKG) signal appear as

noise in the recorded EEG signal in all variations. To

remove EMG and ECG signals ICA was applied to

very carefully remove noisy components.

We applied FIR ﬁlters to obtain the conventional

frequency bands. They are associated with differ-

ent brain states (Edwards, 2007). These bands are

δ: f ∈ (1, 4] Hz, θ: f ∈ (4,7] Hz, α: f ∈ [8,12] Hz,

β: f ∈ [13,30] Hz, and γ: f ∈ [30, 50] Hz. We ex-

pect the optometric variables to explainable by the dy-

namics of functional connectivity in these frequency

bands. Functional connectivity was computed by

the synchronization likelihood (Stam and van Dijk,

2002): We used an outer window length of W

= 3s

and a reference probability of p

ref

= 0.02. To cap-

ture most of the dynamics, the sliding window shifted

every 0.5 s, i.e. an overlay of

/6. Note that the analy-

sis of the averaged graphs did not result in any useful

model (Held et al., 2012).

We applied the measures mentioned in Section 2

to every brain graph. We used the Python package

igraph (Cs

ardi and Nepusz, 2006) to accomplish this

task. This resulted in a multivariate time series for

each subject and each frequency band. Every time

series was then ﬁtted by a VAR model with p = 1,2

for simplicity. Eventually we obtained p · 3 · 3 = 9 and

18 parameters, respectively, describing the dynamics

of the corresponding multivariate time series.

4 RESULTS

We used ordinary least-squares regression and com-

puted its leave-one-out (LOO) estimation of the true

error. All models have been ﬁt using the Python pack-

ages sklearn (Pedregosa et al., 2011) and statsmod-

els (Seabold and Perktold, 2010). Both the VAR pa-

rameters and the optometric variables have been z-

score normalized, i.e., we subtracted the mean and di-

vided by the standard deviation. The regression per-

TheEffectsofEdgeWeightsonCorrelatingDynamicalNetworks-ComparingUnweightedandWeightedBrainGraphsof

nervusopticusPatients

281

Table 1: Mean-squared errors (MSE) of the VAR models to describe the vision loss of the right eye. The results from the

weighted networks are shown on the left-hand side, the ones from the unweighted network are shown on the right-hand side.

weighted δ θ α β γ

HRP DA 0.963 1.297 1.055 1.403 2.015

SP FT 1.335 1.447 1.330 1.142 1.869

SP MT 1.134 1.542 0.897 1.436 2.087

KP ME 1.259 1.559 1.369 1.354 1.844

VA N log 1.235 1.229 1.385 1.313 1.532

VA F log 1.209 1.283 1.348 1.254 1.431

SL ≥ 0.5 δ θ α β γ

HRP DA 1.337 1.291 1.448 1.204 2.440

SP FT 1.202 0.941 1.571 2.159 1.627

SP MT 1.197 1.140 1.396 2.261 1.949

KP ME 1.318 1.601 1.435 2.329 1.245

VA N log 1.215 1.365 1.483 2.115 1.779

VA F log 1.181 1.411 1.580 2.656 1.411

Table 2: Mean-squared errors (MSE) of the VAR models to describe the vision loss of the right eye. The results from two

unweighted networks that used different thresholds are shown on both sides of the table.

SL ≥ 0.7 δ θ α β γ

HRP DA 1.371 1.580 1.144 1.148 0.979

SP FT 1.314 1.275 1.385 1.070 0.807

SP MT 1.336 1.546 1.282 0.857 1.055

KP ME 1.493 1.518 1.626 0.971 1.081

VA N log 1.270 1.321 1.292 1.183 0.752

VA F log 1.256 1.242 1.325 1.295 0.838

SL ≥ 0.2 δ θ α β γ

HRP DA 0.963 1.485 1.290 1.422 1.187

SP FT 1.335 1.375 1.751 0.992 1.175

SP MT 1.134 1.280 1.130 1.358 1.244

KP ME 1.259 1.459 1.476 1.575 1.428

VA N log 1.235 1.143 1.959 0.733 1.136

VA F log 1.209 1.169 2.199 0.758 1.014

Table 3: Mean-squared errors (MSE) of the VAR models to describe the vision loss of the left eye. The results from the

weighted networks are shown on the left-hand side, the ones from the unweighted network are shown on the right-hand side.

weighted δ θ α β γ

HRP DA 1.580 1.572 1.450 1.629 1.397

SP FT 1.511 1.239 1.089 1.873 1.647

SP MT 1.259 1.559 1.369 1.354 1.844

KP ME 1.534 1.103 1.305 1.456 1.471

VA N log 1.541 1.140 1.298 2.193 1.578

VA F log 1.425 1.117 1.433 1.913 1.682

SL ≥ 0.5 δ θ α β γ

HRP DA 1.164 0.760 1.398 1.217 1.054

SP FT 1.490 1.178 1.253 1.368 1.434

SP MT 1.080 1.021 1.065 1.109 1.189

KP ME 1.286 1.011 1.221 1.047 1.222

VA N log 1.568 1.064 1.299 1.328 1.347

VA F log 1.551 1.465 1.412 1.527 1.359

Table 4: Mean-squared errors (MSE) of the VAR models to describe the vision loss of the left eye. The results from two

unweighted networks that used different thresholds are shown on both sides of the table.

SL ≥ 0.7 δ θ α β γ

HRP DA 1.357 1.259 1.401 0.665 3.353

SP FT 1.423 1.524 1.283 1.698 1.412

SP MT 1.230 1.267 1.122 1.502 1.056

KP ME 1.283 0.987 1.324 1.037 0.992

VA N log 1.484 1.173 1.413 1.520 1.391

VA F log 1.454 1.325 1.387 1.414 1.378

SL ≥ 0.2 δ θ α β γ

HRP DA 1.580 0.967 1.174 1.755 1.783

SP FT 1.511 1.128 1.451 1.166 1.234

SP MT 1.291 0.931 0.745 1.344 1.367

KP ME 1.534 1.106 1.471 1.651 1.299

VA N log 1.541 1.091 1.726 1.367 1.124

VA F log 1.425 1.257 1.717 1.340 1.240

formance of the normalized data was measured by the

mean-squared error (MSE).

Tables 1 and 2 on the next page summarize our

analyzes for the right eye. Weighted networks lead to

MSE that are competitive to the ones of the thresh-

olded networks. For low-frequency bands (i.e., δ,

θ, α), the weighted networks produce even smaller

errors. This is in line with the previous results in

(Moewes et al., 2013) where 8 subjects less have been

used. Equivalently, the results for the left eye are

shown in Tables 3 and 4. They do conﬁrm the ﬁnd-

ings.

5 CONCLUSIONS

We have advocated not to threshold edge weights

when analyzing dynamical networks. The choice of

a threshold to obtain an unweighted graphs can be

omitted if the graphs are used for regression models.

To show this we have analyzed EEG networks from

patients suffering from visual ﬁeld defects. We have

thresholded the edge weights using several different

values. Still, the models describing the weighted dy-

namical network lead to similar errors compared to

those of unweighted graphs.

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282

We plan to enlarge the data set to further increase

the quality of our models. We also want to establish a

way to combine the optometric tests of both eyes. A

comparison of brain networks to classical EEG rep-

resentations is in preparation. Furthermore we are

working on the creation of different data sets, e.g.,

communication networks and microblogs, to general-

ize our ﬁndings.

ACKNOWLEDGEMENTS

The ﬁrst author thanks Carolin Gall and her students

from the Medical Faculty for collecting the EEG data.

We give thanks to Hermann Hinrichs from the Medi-

cal Faculty for pointing out several hints to preprocess

the EEG data. Last not least we thank Bernhard A.

Sabel and Michał Bola for fruitful discussions while

preparing this paper.

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