# Sensitivity Enhancement of Third-order Nonlinearity
Measurement in THz Frequency Range

A. Nabilkova, A. Ismagilov, M. Melnik, M. Zhukova, M. Guselnikov, S. Kozlov,
A. Tcypkin This paper was produced by the IEEE Publication Technology Group.
They are in Piscataway, NJ.Manuscript received April 19, 2021; revised August
16, 2021. Corresponding author is A. Tcypkin.
A. Nabilkova, A. Ismagilov, M. Melnik, M. Zhukova, M. Guselnikov, S. Kozlov,
A. Tcypkin are with Research and Education Center of Photonics and Optical IT,
ITMO University (e-mail<EMAIL_ADDRESS>
###### Abstract

In this article for the first time we applied the eclipse Z-scan technique
which is by one order of magnitude more sensitive than the conventional Z-scan
technique to measure LiNbO3 crystal’s nonlinear refractive index in the THz
range. The obtained value of LiNbO3 nonlinear refractive index is estimated to
be 5$\pm$2$\times$10-11cm2/W which is commensurate with known results. This
value correlates with the theoretically calculated nonlinear refractive index
coefficient of vibrational nature. The influence of thermal nonlinearity on
the experimental results can be neglected, since the estimated temperature
induced refractive index change $\Delta$nT equals to 2.6$\times$10-5, while
addition $\Delta$nnl from optical nonlinearity is 2.9$\times$10-3. The
demonstrated heightened sensitivity of eclipse Z-scan allows to hold promise
for the properties evaluation of materials exhibiting lower nonlinear
refractive indices, thus expanding its applicability in characterizing diverse
nonlinear optical materials.

###### Index Terms:

Article submission, IEEE, IEEEtran, journal, LaTeX, paper, template,
typesetting.

††publicationid: pubid: 0000–0000/00$00.00 © 2021 IEEE

##  I Introduction

Nonlinear characteristics of materials such as nonlinear refractive index
($n_{2}$) and absorption coefficients in the terahertz (THz) frequency range
have already been experimentally and theoretically investigated for a number
of liquids [1, 2, 3, 4, 5], vapors [6, 4, 7] and crystals [8, 9, 10, 11, 12,
13, 14, 15]. Within the existing amount of work, the experimental method for
determining the value of $n_{2}$ is used along with theoretical approach [2,
1, 16] to prove the validity of the theory [17] and stimulate the development
of fundamental research for the evaluation of the coefficient $n_{2}$ [18, 8].

Besides, various experimental techniques are used to investigate nonlinear
properties of materials in the THz range, each with its unique setup and
specific objectives. Examples of such techniques include: Z-scan [19, 20, 21,
22, 23], widely employed for nonlinearities extraction across all spectral
ranges, 4f-system Z-scan [24, 25, 26], pump-probe with time delay [10, 27, 28,
29, 30], four wave mixing [31], nonlinear ellipse rotation [32, 33], and full-
phase analysis [4]. These techniques primarily leverage self-induced lensing
phenomena to explore and quantify the nonlinear properties of materials.

It is important to note the orders of magnitude of previously measured and
estimated nonlinear refractive indices. For liquids, these values are of the
order of 10${}^{-10}-$10-9cm2/W [1, 34, 23], for solids these values vary from
10-13cm2/W to 10-11cm2/W [35, 36, 37]. In particular, for lithium niobate
(LiNbO3) crystal the value of $n_{2}$ in the terahertz region is deduced to be
10-11cm2/W [38, 39].

The described nonlinearities for solids are couple orders of magnitude lower
than for liquids. The sensitivity of the conventional laboratory tabletop
Z-scan setup [2] becomes insufficient for the comprehensive investigation of
the nonlinearities that possessed by solids. In particular, the robust
interferometric sensitivity characteristic of the Z-scan method, which
facilitates the determination of alterations in transmittance due to phase
change, emanates from the interference (diffraction) effects engendered by
distinct segments of the spatial profile within the far field [19].

Only a few unique techniques, such as one consisting of single-mode Fabry-
Pérot microcavity with detection by phase contrast imaging technique using a
4f system [15] or high-powered sources like FEL [40], are suitable for
evaluating nonlinear refractive index $n_{2}$ of LiNbO3 in the THz range.
Another technique is based on the phenomenon of third harmonic generation
disappearing during single-cycle THz pulse propagation in cubic nonlinear
medium. It is replaced by the generation of radiation at quadruple frequencies
and by the position of the dip in the region of triple frequencies and the
peak of quadruple frequency it is possible to evaluate the nonlinear
refractive index of LiNbO3 [39].

One of the alternative method to increase sensitivity of the conventional
Z-scan measurements is the eclipse Z-scan [41, 42]. The system’s sensitivity
is enhanced by selectively registering signal counts within the beam
peripheries. This is achieved through the obstruction of the central region
due to implying disk aperture that mask the central segment of the laser beam.
The higher sensitivity to transmission variations derives from the manner
light bypasses the obstructing disk, thereby enabling the detection of an
”eclipse” and facilitating assessments of the third-order optical nonlinearity
inherent in materials. Notably, the application of the eclipse Z-scan method
in the context of the THz frequencies represents a novel endeavor,
necessitating adaptations to accommodate this spectral domain.

In this work we introduce an advancement in measuring the third-order
nonlinearity in the THz spectral range with eclipse Z-scan method. With a one-
order sensitivity enhancement over conventional Z-scan techniques, this
approach has yielded a remarkable LiNbO3 nonlinear refractive index value of
5$\pm$2$\times$10-11cm2/W. The observed value of the refractive index change
induced by temperature variations ($\Delta$nT), estimated as 2.6$\times$10-5,
and the dominant refractive index change ($\Delta$nnl) attributed to optical
nonlinearity, measured as 2.9$\times$10-3, assume that the observed nonlinear
behavior cannot arise from thermal effects. It could be stated that the
nonlinearity obtained has a vibrational nature, since the value obtained
coincides with the value theoretically calculated earlier [1, 36] according to
the analytical model [8]. In addition, a comparative analysis of the
sensitivities of the conventional and eclipse Z-scan techniques is presented,
offering insights into their capabilities.

##  II Experiment

With a view to reach an order-of-magnitude enhancement of the signal-to-noise
ratio of the conventional Z-scan technique, the disk [43, 42] instead of the
aperture is implemented into setup according to pioneering works in mentioned
direction [41, 19]. The application of the disk allows using both open and
closed aperture geometry in the same way as aperture does. Here, the nonlinear
refractive index $n_{2}$ of lithium niobate is experimentally determined
utilizing the eclipse Z-scan technique in conjunction with a pulsed THz
radiation source.

The experimental setup is illustrated in Fig. 1. The THz source operates at a
repetition rate of 1 kHz. The vertically polarized THz pulse displays a
duration of 1 ps, a pulse energy of 400 nJ (with a peak electric field
strength of 0.27 MV/cm), and a spectral range extending from 0.1 to 2.5 THz
[1]. The full width at half maximum (FWHM) of the collimated THz beam is
determined using the knife-edge method, yielding values of 8.5 mm for the
x-axis and 5.6 mm for the y-axis, as reported in [1]. The beam is tightly
focused onto the sample via the PM1 mirror (the focal length of the parabolic
PM1 mirror is 12.7 mm), resulting in a beam diameter of approximately 1.5 mm
at the focal point. The beam diameter values are verified by the knife-edge
method with a precision of 100 $\mu$m. The peak intensity at focus reaches
0.5$\times 10^{8}$ W/cm2. Then the THz radiation is collimated by the
parabolic mirror PM2 and focused by the lens with focal length of 50 mm. The
slight ellipticity of the beam observed in the experiment should not impact
the closed aperture Z-scan traces [44]. The metal disk is installed in front
of the lens. The diameter of disk is selected relative to the required
sensitivity of the eclipse Z-scan technique. Detection is performed using a
bolometer (Gentec-EO THZ5B-BL-DZ-D0).

Figure 1:  Experimental setup for the eclipse Z-scan measurements. The THz
pulse is produced through optical rectification within a MgO:LiNbO3 crystal
(Tera-AX Avesta project [45]), sourced from a 35 femtosecond duration pulse
with a central wavelength of 790 nm and a pulse energy of 1 mJ originating
from a Ti:Sapphire femtosecond laser with a regenerative amplifier (Regulus
Avesta project). Parabolic mirrors (PM1, PM2) are used to focus and collimate
the THz pulse. The modulator is used to synchronize the bolometer and the THz
radiation source. The disk can be moved from the beam to the open position. A
bolometer (Gentec-EO THZ5B-BL-DZ-D0) is used to measure the power of THz
radiation.

##  III Results

###  III-A Experimental data of eclipse Z-scan

In order to conduct a comparative analysis of the experimental curves obtained
with the eclipse Z-scan and conventional Z-scan methods, the 10% transmittance
disk was replaced with a 2% transmittance aperture. Fig. 2 (a) shows
unnormalized closed aperture Z-scan and eclipse Z-scan (EZ-scan) curves
obtained experimentally. Fig 2 (b) presents results of the open aperture
measurements, which were carried out without disk to exclude the influence of
the material’s absorption properties. It can be seen that the amplitude of the
Z-scan curve fluctuations, where the peak and valley should be, is comparable
to the noise of the curves for the EZ-scan. Notably, the difference between
the peak to valley in the field E registered here for the EZ-scan curves is
more than 10 times greater than for the conventional Z-scan curves.

Figure 2:  (a) Comparison of the experimental results from closed aperture
measurement of Z-scan and eclipse Z-scan (EZ-scan). All obtained curves are
normalized to the corresponding mean value to show the difference between
$\Delta T$ and order of sensitivity boosted.(b) Experimental results for the
case of open aperture measurements without disk.

Fig 3 shows the experimental results of the closed aperture eclipse Z-scan
(EZ-scan) measurements normalized to open aperture measurements, compared to
the analytical modeling of the EZ-scan. As can be seen in Fig. 2 (a) and 3,
the ends of the curve point upward may be an analytical artifact arising from
dividing the closed aperture curve by the open aperture curve due to the
presence of noise in both curves.

These experimental results demonstrate that the difference between the
normalized peak and valley transmittance $\Delta T$ is found to be 0.021.
Thus, using the empirical linear relationship between $\Delta T$ and nonlinear
phase shift $\Delta\Phi_{0}$ [41, 42] we can extract the nonlinear refractive
index $n_{2}$ of LiNbO3 crystal by following equation:

$n_{2}={\frac{\Delta T}{0.68I_{in}}\times\frac{(1-S)^{0.44}}{kL_{\alpha}}}$
(1)

Substituting the experimentally obtained value of $\Delta T$ we found $n_{2}$
= 5$\pm$2$\times$10-11 cm2/W.

###  III-B Theoretical model of eclipse Z-scan

The convenience and precision of applying mathematical expressions [19] for
the THz range and pulsed few-cycle radiation have already been shown [2, 1,
21]. In our simulations, the field distribution $E$ assuming a TEM00 Gaussian
beam of waist radius $w_{o}$ traveling in the $+z$ direction is given by:

$E(r,t,z)=E_{0}(t)\frac{w_{0}}{w(z)}\exp\left({\frac{-r^{2}}{w(z)^{2}}}-{\frac{ikr^{2}}{2R(z)}}\right)e^{-i\phi(r,t,z)}$
(2)

where $E_{0}(t)$ is the initial temporal envelope of the input field
$E_{0}\sin(2\pi\nu_{0}t)$, $w(z)^{2}=w^{2}_{0}(1+z^{2}/z^{2}_{0})$ is the beam
radius, $R(z)=z(1+z^{2}/z^{2}_{0})$ is the radius of curvature of the wave
front at $Z$, $z_{0}=kw^{2}_{0}/2$ is the diffraction length of the beam, and
$k=2\pi/\lambda$ is the wave vector. The $e^{-i\phi(z,t)}$ term contains all
the radically uniform phase variations. In the thin sample slowly varying
envelope approximation, the electric field amplitude only changes due to the
absorption, while the phase of the electric field due to the nonlinear
refraction resulting in nonlinear phase term $e^{-i\Delta\phi(r,t,z)}$.
Gaussian decomposition (GD) method is implemented according to [19] to
calculate the complex field distribution $E_{a}$ in the aperture plane. So
$E_{a}$ is derived as:

$\begin{gathered}E_{a}(r,t,z)=E(z,t)\exp(-\alpha L/2)\times\\\
\times\sum_{m=0}^{+\infty}{\frac{[-i\Delta\phi_{0}(z,t)]^{m}}{m!}}{\frac{w_{m0}}{w_{m}}}\exp({\frac{-r^{2}}{w_{m}^{2}}}-{\frac{ikr^{2}}{2R_{m}}}+iQ_{m})\end{gathered}$
(3)

where
$\Delta\phi_{0}(z,t)=\Delta\Phi_{0}(t)/(1+z^{2}/z_{0}^{2}),\Delta\Phi_{0}(t)=k\Delta
n_{0}(t)L$ is the nonlinear phase shift. The GD method involves decomposing of
the electric field at the sample exit plane into Gaussian beams using a Taylor
series expansion of the nonlinear phase term. This approach is based on the
assumption of small phase changes, allowing to take into consideration only
the initial terms of the expansion. Therefore, defining $d$ as the propagation
distance in free space from the sample to the disk, and $g=1+d/R(z)$, other
parameters in Eq. 3 according to [19] are expressed as:

Figure 3:  Comparison of the experimental results of the Z-scan method for the
pulsed broadband THz radiation for the LiNbO3 crystal with an analytical
Z-scan curve. The analytical curve is calculated using Eq. 10.

$\displaystyle w_{m0}^{2}=w_{z}^{2}/(2m+1)$ (4) $\displaystyle
d_{m}=kw_{m0}^{2}/2$ (5) $\displaystyle
w_{m}^{2}=w_{m0}^{2}\left(g^{2}+d^{2}/d_{m}^{2}\right)$ (6) $\displaystyle
R_{m}=d\left(1-\frac{g}{g^{2}+d^{2}/d_{m}^{2}}\right)^{-1}$ (7)
$\displaystyle\theta_{m}=\tan^{-1}\left(\frac{d/d_{m}}{g}\right)$ (8)

The transmitted through the disk power is obtained by spatially integrating
$E_{a}(r,t,z)$ from the disk radius $r_{d}$ to the beam radius on the aperture
$w_{a}$ giving:

$P_{T}(\Delta\Phi_{0}(t))=c\epsilon_{0}N_{0}\pi\int_{r_{d}}^{w_{a}}|E_{a}(r,t)|^{2}rdr$
(9)

Integrating up to beam radius on the aperture $w_{a}$ allows neglecting the
impact of ”noise”, where the beam profile fluctuates around zero. Thus, the
normalized transmittance can be calculated as

$T(z)={\frac{\int_{-\infty}^{+\infty}P_{T}(\Delta\Phi_{0}(t))dt}{S\int_{-\infty}^{+\infty}P_{i}(t)dt}}$
(10)

where $P_{i}$(t)=${\pi w_{0}^{2}I_{0}(t)/2}$ is the instantaneous input power
(within the sample), $S$=1-$\exp(-2{r_{d}^{2}/w_{a}^{2}})$ is the disk linear
transmittance.

Figure 4:  Comparison of the analytical results of the Z-scan and eclipse
Z-scan (EZ-scan) modeling.

By gathering the setup parameters and the beam characteristics from the
experiment, the nonlinear refractive index $n_{2}$ value calculated from 1,
and substituting them into equation 10 (and nested equations), we can obtain
the normalized transmittance T from the coordinates of crystal displacement z.
Fig. 4 shows a comparison of analytical eclipse and conventional Z-scan
curves, demonstrating the order of sensitivity enhancement for the former
method. The Z-scan curves are obtained with the same theoretical model,
however Eq. 9 is adjusted to perform integration over the aperture: from 0 to
$r_{a}$.

##  IV Discussion

The experimental as well as the analytical findings underscore the efficiency
of employing eclipse Z-scan instead of conventional Z-scan method, culminating
in an enhancement in the sensitivity of THz systems by an order of magnitude.
The achieved order of nonlinearity, approximately 10-11 cm2/W, stands as a
noteworthy accomplishment. As was mentioned, the diameter of the metal disk,
which determines the parameter S, was chosen on the basis of two
considerations: the sensitivity of the eclipse Z-scan method and the
sensitivity of the detector. As a result, a disk with a diameter that covers
90% of the incident radiation was chosen, which allowed us to perform
measurements with a signal-to-noise ratio of 10 and to obtain a significant
drop in the Z-scan curve.

Certain types of light sources can induce cumulative interactions, such as
pulsed THz systems with large pulse durations or high repetition rates. In
such systems, thermal effects may occur. In the context of determining the
nonlinear refractive index of liquids [2, 1, 3], temperature-induced nonlinear
effects were deemed negligible, as each pulse interact with a refreshed
surface of liquid jet. However, for crystals, consideration of temperature
fluctuations during the experiment becomes mandatory for a comprehensive
evaluation of their influence on the refractive index. Strong thermal effects
can contribute to the refractive index by inducing temperature fluctuations
through non-radiative interactions within the system. The upper bound
evaluation was made for temperature nonlinear refractive index contribution
$n_{2_{T}}$ in comparison with the contribution of vibrational nonlinear
refractive index $n_{2_{nl}}$ to the change in the refractive index $\Delta
n$.

To estimate the thermal effects per laser repetition rate, we use the
following expression $n_{2_{th}}=\frac{dn}{dT}\frac{\alpha R^{2}}{\kappa}$
[46]. Here $\frac{dn}{dT}$ displays the speed of refractive index change with
temperature variation, it is found to be 7.03$\times$10-4 1/K [9]. $\alpha$ =
9 cm-1 denotes the linear absorption coefficient of the material, $\kappa$ =
0.046 W/(cm$\times$K) stands for the thermal conductivity, and $R$ = 0.7 mm is
a radius of a circular laser beam in focus.

The response time $\tau_{r}$ for the thermal nonlinearity is given by
$\tau_{r}\approx\frac{\rho CR^{2}}{\kappa}$ [47], so thermally induced $\Delta
n$ per one repetition of pulses would be
$\frac{n_{2~{}th}I\tau_{p}}{\tau_{r}}$, because interaction of the crystal
with the beam occurs for about pulse duration $\tau_{p}$, which is equal to 1
ps. Here, material’s density $\rho$ is 4700 kg/m3, heat capacity $C$ is 628
J/(kg K).

The energy per one repetition can be estimated as $Q=\epsilon Pm$, where pulse
energy $P$ is equal to 400 nJ, emissivity of the crystal $\epsilon$ is 0.6,
and number of pulses per one repetition $m$, controlled by chopper’s speed, is
40. If we assume that all obtained energy is used for heating, then the
crystal heats up according to the equation $\Delta T_{h}=\frac{Q}{Cm}$, where
$m=\rho\pi R^{2}L$ is mass of illuminated area of the crystal. Then, during
the time when the chopper closes the laser, the crystal cools by an amount
$\Delta T_{c}=\frac{\tau hS\Delta T_{h}}{mC}$, where $S$ is an illuminated
area and $h$ is a heat transfer coefficient equal to 10 W/(m2 K). Accordingly,
the temperature $\Delta T$ to which the crystal is heated during one passage
will be equal to $\Delta T_{h}$ \- $\Delta T_{c}$. The overall heating
temperature of the sample for the entire duration of the experiment is
0.1$\times$ 10-3 K.

As an upper estimate, we assume that the temperature change $\Delta T$ is
constant while the spot size of the laser beam on the crystal should change as
the sample is displaced through the beam focus in the $z$ direction. The
sample passes this path in 10 seconds, so the crystal has time to interact
with radiation and heat up by $\Delta T$ in 250 repetitions, according to
repetition rate of laser and frequency of chopper. Therefore, accumulative
thermal contribution to refractive index change $\Delta n_{th}$ is 2.6$\times$
10-5, while contribution of vibrational nonlinearity $\Delta n_{nl}$ is
2.9$\times$ 10-3.

##  V Conclusion

This study shed a light on an adjustment to tried and tested in THz range
Z-scan technique, that offer to gain sensitivity of experiment. Eclipse Z-scan
methodology was used for assessing third-order nonlinearity within the THz
spectral domain. Demonstrating a sensitivity enhancement of one order compared
to conventional Z-scan methodologies [2, 1], our refinement has unveiled a
noteworthy nonlinear refractive index value $n_{2}$ for LiNbO3 crystal of
5$\pm$2$\times$10-11cm2/W. Additionally, the theoretical approach, that
describes the behavior of light in a crystal when conducting eclipse Z-scan
measurement, was modified to justify the obtained value of $n_{2}$. Our
results show that experimentally obtained Z-scan measurement curves and
simulations of self-focusing processes in LiNbO3 crystal are in good
agreement. Notably, our investigation into temperature-induced variations in
refractive index ($\Delta$nT), estimated as 2.6$\times$10-5, alongside the
predominant refractive index change ($\Delta$nnl) attributed to optical
nonlinearity, measured at 2.9$\times$10-3, suggests that the observed
nonlinear phenomena mostly ascribed to optical nature. This assertion is
supported by the congruence between our measured nonlinearity and previously
theorized values [1, 36], as predicted by the analytical framework proposed by
[8], indicating a vibrational underpinning to the observed nonlinear behavior.
Furthermore, a comparative evaluation of the sensitivities inherent to
conventional and eclipse Z-scan methods is provided experimentally and
analytically, affording valuable insights into their respective efficiency and
potential applications.

The results presented emphasize the importance of exploring the nonlinear
refractive index of diverse media to identify materials exhibiting high
nonlinearity, thereby facilitating the advancement of nonlinear terahertz
optics. The nonlinear refractive index plays a pivotal role in determining the
optical response of materials, enabling the manipulation of light for various
applications in photonics and optoelectronics. Devices such as semiconductor
optical amplifiers, flip-flops, switchers, optical transistors, and modulators
have been identified as examples of systems with high nonlinear refractive
index, showcasing the practical relevance of investigating this property. By
exploring and harnessing the nonlinear refractive index of materials,
researchers can unlock new avenues for enhancing the performance and
functionality of optical devices. This research not only contributes to
expanding our fundamental understanding of light-matter interactions but also
holds immense promise for driving innovation in the field of nonlinear optics,
for example, developing devices for creating all-optical computer.

## Acknowledgments

All authors acknowledge support from the Russian Science Foundation (Grant No.
24-22-00084).

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# Globalness Detection in Online Social Network

Yu-Cheng Lin, Chun-Ming Lai, S. Felix Wu and George A. Barnett University of
California, Davis
Email<EMAIL_ADDRESS>
###### Abstract

Classification problems have made significant progress due to the maturity of
artificial intelligence (AI). However, differentiating items from categories
without noticeable boundaries is still a huge challenge for machines – which
is also crucial for machines to be intelligent. In order to study the fuzzy
concept on classification, we define and propose a globalness detection with
the four-stage operational flow. We then demonstrate our framework on Facebook
public pages inter-like graph with their geo-location. Our prediction
algorithm achieves high precision (89%) and recall (88%) of local pages. We
evaluate the results on both states and countries level, finding that the
global node ratios are relatively high in those states (NY, CA) having large
and international cities. Several global nodes examples have also been shown
and studied in this paper. It is our hope that our results unveil the perfect
value from every classification problem and provide a better understanding of
global and local nodes in Online Social Networks (OSNs).

##  I Introduction

Thanks to the development of graphics processing unit (GPU), artificial
intelligence (AI) and machine learning (ML) have achieved great advances in
this decade. Many research fields have made great strides, especially in
quantitative and cognitive aspects with the efficiency and effectiveness
provided by AI and ML. However, most research problems in social informatics
addressing human cognition and behaviors are not applicable with category
identification, which has been a critical topic in machine learning. In most
cases, it is relatively challenging and arbitrary to classify an item into
just a single category since there exist vague items which could be classified
as neither category A nor B. For example, discussion messages in social media
can be conservative, liberal, or impartial – which means these messages are
not polarized enough that they can be labeled as either liberal or
conservative [1].

Conversely, another special case is that an item could be classified into one
of the multiple categories – all seem to be reasonable. For instance, a
restaurant page (e.g. Applebee at Davis, displayed in Figure 1) could be
classified as an American-style restaurant, bar, grill restaurant, and even a
steak restaurant at the same time. In general, most labelled tasks require
manual feature selection and interpretation, which facilitates the recent
popularity of deep and reinforcement learning. In the learning process, each
item is given a combination of probability distribution of all possible
categories [2]. Nonetheless, we still need to determine the classification
results by the category with the highest probability ultimately.

We seek to complement this fundamental problem of classification by proposing
the framework of globalness detection – finding the items which does not
belong to a specific category. This framework is especially useful in social
science field since human perception is not just a simple binary
classification problem . Sometimes we are confident to perform near-perfect
algorithms and get results with large-scale data training and validation. For
example, in pattern recognition, dogs should not be identified as birds
because dogs have tails and four legs while birds have wings and two legs –
the distinction is clear among different categories. Instead, in most cases
when considering social informatics research problems, there are usually no
clear definition and support vectors to build a convincing classifier.

In this paper, we try to approach this classification challenge by uncovering
the global nodes in OSNs. We define global nodes as those nodes, instead of
being local, having an unbiased relationship with the nodes in multiple
categories. Our objective is to build an accurate classifier which can
identify both local and global nodes with high precision. In our methodology,
we employ anchor nodes as the center of localization to the corresponding
categories, which is similar to the idea applied in sensor networks [3]. The
detailed operational flow is described in Section II.

Globalness detection in OSNs (e.g. Facebook) has two directions of potential
applications. First, understanding the location of your target audience is a
primary tool in the marketing strategy toolbox. In this way, locality
differentiation between various Facebook pages is crucial to ensuring that
marketers can understand their audiences as discrete geo-located elements. For
instance, BMW has distinct advertisements based on elements such as geography,
culture, and language [4]. Another example involves media broadcasters; CBS
News has divided its viewing audiences into separate groups based on
geographic data. They exploit this classification as a virtual demarcation
line in order to broadcast different messages to different communities [5]. In
short, locality differentiation facilitates the subtle division of information
for multiple audiences.

Second, there are always positive and negative supporters for a specific
standing in OSNs. It’s relatively accessible to find these polarized group
members from the hybrid approach combining content and networks features [6,
7]. However, from the information entropy’s point of view, the neutral group
members or weakly supporting group members provide much more valuable
arguments. Our novel methodology offers a new avenue into this open research
problem.

The rest of this paper is organized as the following. In Section II, the
concept of globalness is characterized and defined. Then we presents the
proposed operational flow to discover global nodes. Section III provides a
detailed description for our dataset. Two experimental cases are studied in
Section IV to illustrate the power of our platform. In Section V, we summarize
past research on node analysis in OSN and data collection of location
information. Finally, Section VI offers a summary of our work.

Figure 1: The restaurant chain Applebee’s fan page at Davis, CA

##  II Operationalizing Globalness

For the computational social science, it’s important to operationalize an
intuitively fuzzy notion – making it quantitatively distinguishable in terms
of empirical observations. Since globalness is not a measurable variable,
determining an appropriate approach is an important part of the study.
Specifically, we try to answer these two research questions:

RQ1: What does it mean for an item (node) to be global in online social
network graph?

RQ2: How can we detect these global nodes in online social network graph?

In the following, we start from an informal characterization and then propose
an innovative method to operationalize this phenomenon. With the methodology
of the operational flow, we are able to detect these global nodes in large-
scale.

###  II-A Characterizing Concept of Globalness

  * •

Globalness is a property of the relationship of this node with not only the
category it belongs to, but also with other categories. This follows the
intuition that comparing to the local nodes, the global nodes are not inclined
towards a few categories.

  * •

Globalness is not an absolute concept and is defined with respect to a given
context: the nodes of a graph and a given set of (two or more) parties. For
instance, a Facebook page could be global from Midwest’s point of view, but
could be local from the perspective of the United States (US).

###  II-B Operational Definition of Globalness

From our observation, the global nodes in OSNs are those nodes which have
connections with multiple categories (classes or regions) and do not lean
toward a specific category. For instance, given all the Facebook pages located
within the US, some of them have characteristics which are similar to the
pages in the other countries. We propose an operational definition of this
concept by the following formula:

$\begin{split}Global\\_node:=\\{p|p\in C_{i}{}\land(\sum_{j\in
C}\delta_{pj})>=N_{global\\_threshold}\\}\\\ \end{split}$ (1)

$\begin{split}\delta_{pj}=\begin{cases}1,&\Delta_{pj}\leq\min_{k\in
C}(\Delta_{pk})+\epsilon\\\ 0,&\text{otherwise}\end{cases}\end{split}$ (2)

$\begin{split}\Delta_{pk}=\sum_{c\in C\land c\neq
k}\omega_{c}||D_{pc}||\end{split}$ (3)

where $p$ is the global node which originates from the class $C_{i}{}$;
$\Delta_{pk}$ is the weighted error when node $p$ is classified as the class
$C_{k}{}$ and $D_{pc}{}$ is the error distance to a specific class. In short,
abstractly global nodes are those nodes which could be classified as anyone of
the multiple classes within a small error distance. Figure 2 demonstrates an
example flow to detect global pages in the US.

###  II-C Operational Flow of Globalness Detection

In order to find the global nodes in OSNs, we devise a four-stage operational
flow of globalness detection.

####  II-C1 Making hypothesis

Define globalness relationship between two affiliations. Specifically, there
are mainly two groups: A and non-A (others). Note that while this definition
can be intuitively extended to relationship among more than two affiliations,
for the sake of simplicity, we only consider relationship between two
affiliations in this work.

####  II-C2 Deciding anchor nodes

For each class, pick one anchor node which can exemplify its own class –
namely, choose the representative ”local” nodes at this stage.

####  II-C3 Selecting biased data

In the training process, we reinforce the distinction between local nodes and
global nodes by polarizing the selection of training samples. For the local
nodes, we choose the nodes with local tendency in class A; for the global
nodes, we pick the nodes with global propensity in class non-A.

####  II-C4 Detecting global nodes by classification

At this stage, we train the model by the biased data. We identify these global
nodes by the trained classifier. The global nodes are the nodes which are
identified as belonging to the class ”others (OT)” by the classifier – which
means that these nodes do not belong to their original class, and thus are so-
called global nodes by our operational definition.

Figure 2: Global page detection flow for the US pages

##  III Dataset

In this paper, our experiments from the operational flow of global detection
are materialized on the Facebook public pages.

###  III-A Facebook Public Page Graph

Formally, we model the Facebook public pages as a page-like graph $G=(V,E)$
wherein V is a set of pages and E is a set of directed edges presenting ”Like”
relationship which would be seen by the community participants.

The page host has the privilege to disclose the geographic location of the
page. We crawled 38,831,367 pages in total. Among these pages, 15,115,963
(38.93%) pages have specific location information. There are 2,430,873 pages
proclaimed as in the United States (US) and 12,685,090 non-US pages. We took
the pages with declared location information of country and city as ground
truth data. Then we map the city to the corresponding state where it resides.
Few pages are excluded because their city names exist in multiple states,
which can result in ambiguous city-to-state mapping. There are 29,849 cities
in total in the US.

###  III-B Anchor Node Selection

Seed selection is an important technique in graph analysis and also exploited
in combating fake contents (e.g. web spam and review manipulation) [8, 9].

In Section II-C, the second step of the operational flow is to decide anchor
nodes representative of their own categories. These anchor nodes have to be as
local as possible such that the distances to these nodes can afford the
authentic tendency towards locality.

In our dataset, we select all subsidiary pages of ”OnlyInYourState.com” as the
anchor nodes. For example, ”Only_In_Delaware” is part of ”OnlyInYourState.com”
and mostly connects the local communities in Delaware. In addition, since
California is much larger than the other states in the perspectives of
population and economy, ”OnlyInYourState.com” splits California into Northern
and Southern regions. As a result, both ”Only_In_Northern_California” and
”Only_In_Southern_California” are used as anchor nodes. Furthermore, since
”Only_In_Idaho” had been registered, OnlyInYourState.com named its Idaho
counterpart as ”Idaho_Only” instead. In sum, we have 51 anchor nodes in total
for this dataset.

##  IV Case Study: Globalness in Facebook Pages

To illustrate the methodology we proposed to detect global nodes in OSN, in
this section we investigate two cases utilizing Facebook public pages as the
graph nodes.

###  IV-A Implementation on Facebook Public Pages

With the anchor node as the center of each state, we adopt the breadth-first
search (BFS) to calculate the distance from each anchor node. First, we define
the state distance vector (SDV) to represent the hop distances, for each
specific page, to the anchor pages. The following equation shows SDV for each
page:

$\begin{split}SDV(Page)=[\qquad\qquad\qquad\qquad\\\
[IHOP(Page,S_{i}),\qquad\qquad\\\ OHOP(Page,S_{i})]:\qquad\quad\\\
i\in{1,...,N_{anchored\\_pages}}]\end{split}$ (4)

where $IHOP(Page,S_{i})$ denotes the hop distance between the page and the
seed $S_{i}$, using the inward edges as connection for BFS; $OHOP(Page,S_{i})$
denotes the hop distance between the page and the seed $S_{i}$, using the
outward edges as connection for BFS. In addition, we define the minimum hop
distance $MHOP$ as the minimum of all hop distances of a page:

$\begin{split}MHOP=Min(Min(IHOP),Min(OHOP))\end{split}$ (5)

where $IHOP$ is the set of $IHOP(Page,S_{i})$ and $OHOP$ is the set of
$OHOP(Page,S_{i})$.

From our observation, in some cases, an anchor page can connect with those
pages far beyond where it resides within a small hop distance. This
probability is not high; however, this kind of connection noise would highly
reduce the prediction accuracy. To address this problem, we take the
neighborhood location probability into consideration as the auxiliary
features. The state neighborhood probability (SNP) presents the location
information of the adjacent pages of a specific page. The definition of SNP is
shown below:

$\begin{split}SNP(Page)=[\qquad\qquad\qquad\qquad\\\
[INP(Page,R_{i}),\qquad\qquad\\\ ONP(Page,R_{i})]:\qquad\quad\\\
i\in{1,...,N_{region}}]\qquad\quad\end{split}$ (6)

where $INP(Page,R_{i})$ denotes the inward neighborhood location probability
between this page and the adjacent pages belonging to the region $R_{i}$;
$ONP(Page,R_{i})$ denotes the outward neighborhood location probability
between the page and the adjacent pages belonging to the region $R_{i}$. The
equations are as the following:

$\begin{split}INP(Page,R_{i})=\dfrac{IE(Page,R_{i})}{\sum_{i}IE(Page,R_{i})}\\\
ONP(Page,R_{i})=\dfrac{OE(Page,R_{i})}{\sum_{i}OE(Page,R_{i})}\\\ \end{split}$
(7)

where $IE(Page,R_{i})$ is the number of the inward edges between this page and
the adjacent pages belonging to the region $R_{i}$; $OE(Page,R_{i})$ is the
number of the outward edges between this page and the adjacent pages belonging
to the region $R_{i}$.

Figure 3: Comparison of state-wise global page percentage

###  IV-B State-wise Globalness Detection

To investigate the global nodes in the state-wise perspective, we select
Facebook pages in the threes states of the Midwest region: Illinois (IL),
Indiana (IN), Iowa (IA).

####  IV-B1 Making hypothesis

For IL globalness detection, we define global pages (OT) in IL as those pages
labeled as IL, but behave as those pages which reside in non-IL states (e.g.
IN and IA in this experiment setting) and have state-wise global tendency –
which means a balanced relationship with all three states (IL, IN, and IA).

####  IV-B2 Deciding anchor nodes

As described in Section III-B, we pick the three anchor nodes in IL, IN, and
IA.

####  IV-B3 Selecting biased data

We choose the pages labeled as IL and $\\#MHOP<=N_{local\\_threshold}=1$ as
the local node training samples; we pick the pages labeled as others (i.e. IN
and IA) and $\\#MHOP>=N_{global\\_threshold}=3$ as the global node training
samples.

####  IV-B4 Detecting global nodes by classification

The trained classifier is deployed to recognize the global pages in IL. There
are two categories in this classifier: IL and ”others (OT)” representing the
global pages.

The same globalness detection flow is applied accordingly to detect global
nodes in both IN and IA. Figure 3 compares the global page percentage in the
three states (IL, IN, and IA). 31.56% of the pages in IL are global pages,
which are at least 20% more than the counterparts in IN and IA. This result
suggests that IL has much more pages which actively interact with other pages
among the three core states in the Midwest [10] – which supports the fact that
IL, home of Chicago, is the center of the Midwest region.

Figure 4: Global page distribution by state Figure 5: Global page ratio w.r.t
population density by state

###  IV-C Country-wise Globalness Detection

In this section, we explore the possibility of applying our methodology onto
globalness detection in the country-wise aspects. As an example, the US global
pages are distinguished in the following steps, as shown in Figure 2.

####  IV-C1 Making hypothesis

We define global pages (OT) in the US as those pages labeled as states in the
US, but behave as those pages which are outside the US and have global
propensity.

####  IV-C2 Deciding anchor nodes

As described in Section III-B, we choose all 51 anchor nodes for the 50 states
in the US.

####  IV-C3 Selecting biased data

We choose the pages labeled as US and $\\#MHOP<=N_{local\\_threshold}=2$ as
the local node training samples; we pick the pages labeled as others (non-US)
and $\\#MHOP>=N_{global\\_threshold}=5$ as the global node training samples.
Both local threshold $N_{local\\_threshold}$ and global threshold
$N_{global\\_threshold}$ could be adjusted to make the biased selection more
polarized.

####  IV-C4 Detecting global nodes by classification

The trained classifier is deployed to identify the global pages in US. There
are 51 categories in this classifier: fifty US states and ”others (OT)”
representing the global pages.

Figure 4 presents the global page distribution by state. The proportion of
global pages in all states is 1.16% on average. Not surprisingly, NY has the
highest global page ratio (4.72%). This result is consistent with NY’s image
as a global ’melting pot.’ CA is the other state whose global page ratio is
higher than 2%, which shows its global diversity. From demographics’ point of
view, we rank states by the ratio of $(\\#Global\enspace
pages)\,/\,(Population\enspace density)$ [11]. As Figure 5 shows, AS ranks the
1st (59.2), which manifests its endeavor in travel promotion, and the
followings are the most populous states: CA, NY, and TX.

###  IV-D Evaluation

In both cases in the above experiments, we’ve applied three different ML
models: Naive Bayes [12], Adaboost [13], and Random Forests [14]. The Random
Forests model gets the best performance. Hence we only presents the results
based on this model in this paper.

In Section IV-B, it’s a simplified classification problem, and our classifiers
get 100% precision and recall rates when recognizing local pages in IL, IN,
and IA. There are 50 state categories to identify in Section IV-C, which is a
much more realistic and complex experiment. Our classifier gets 89% precision
and 88% recall rate. The detailed prediction results by state are provided in
[15].

Next, we evaluate the accuracy of globalness detection for our approach. To
that end, we performed a simple experiment: We randomly sampled 15 pages from
each state. Then one of the authors manually verified whether these 15 pages
can be really classified as global page by human judgment. We found that 100%
of these pages can be classified as global pages, which have connections with
pages in multiple other states and behave like pages outside the US.

It is important to highlight that our methodology was not designed to capture
all of the global pages. In addition, the parameter settings of the thresholds
give flexibility to trade-off between precision and recall for globalness
detection. Our approach aimed at building a high precision prediction platform
for both local and global nodes.

In sum, there are primarily two kinds of global pages: the commercial pages
which intend to connect multiple regions for promotion purposes, and the
immigrant communities which are closely linked with exotic contents. For
instance, the page shown in Figure 6 is Tabasco’s fan page; Figure 7 (page ID
= 1627668764132931) is identified as a global page – located in NY, but highly
related to the Middle East culture.

Figure 6: Demonstration of a global page located in LA Figure 7: Demonstration
of a global page located in NY

###  IV-E Stability

In Section IV-C, our experimental results are based on the anchor nodes of the
subsidiary pages from ”OnlyInYourState.com.” However, from our assumption, the
anchor nodes are not unique as long as they are local enough to provide
accurate distance measurement in BFS process. Then what’s the stability of our
globalness detection results?

From our observation, state university pages are also a possible source set of
anchor nodes since local students tend to attend the state universities in
their residence states. Therefore, to make a fair comparison, we pick one
state university page for Northern California, one state university page for
Southern California, and one state university page for the other states. Our
classifier gets 88% precision and 88% recall rate. These comparable results
suggest that the anchor set of state university pages is also a valid anchor
set.

From the perspective of platform stability, we compare the global pages
distinguished from the two sets of anchor nodes. In sum, there are around 60%
global pages overlapped between these two sets. This high repetition rate
proves the stability of our globalness detection platform with distinct anchor
sets.

##  V Related Work

###  V-A Node Analysis in Online Social Network

In OSN, users, pages, and posts/comments are nodes comprising a multi-layer
composite social network graph. Most research papers analyze the nodes in a
single layer, with another layer as a supplement. Firstly, Ugander _et al._
analyzed the Facebook user graph from network structure properties [16].
Second, Hong _et al._ focused on the analysis of the Facebook public page
graph with the data-driven methodology [17]. Last but not least, the political
polarity of posts/comments in OSN (e.g. Twitter) is also a popular topic many
researchers have been working on [1, 6, 7, 18, 19].

###  V-B Location Information

The ground truth location information plays a key role in model building and
evaluation of geo-location prediction. There are two methodologies to acquire
this information: direct collection and indirect extraction.

First, direct collection relies on the OSN platform to fetch data. With a non-
required text field, guided in the case of Facebook and freeform in the case
of Twitter, both services provide users with the option to proclaim their
locations. This field can facilitate the generation of location labels from
country, state, to city level. According to Hecht’s report, 34% of users
provide invalid geographic information in this location field on Twitter [20].
Another more accurate way is to adopt the coordinate location instead of the
named location [21].

Second, indirect extraction uses crowdsourcing to tag the data. Huang _et al._
adopted a composite approach with the language detection tools, an R library,
as well as the Yahoo! and Twitter API to make the crowdsourcing job easier for
people to label data [22].

Since the majority of public pages are commercial in nature, and as such are
managed by page hosts who have a desire to make these pages thriving, they
tend to use more accurate location data in order to ensure more viable
commercial traffic [17]. As users have no such intrinsic motivation to provide
accurate location data, this means the quality of location attribute data is
much higher for pages than for users. For this reason we used direct
collection to obtain location data on Facebook.

##  VI Conclusion

In this paper, we first characterized and defined the concept of globalness to
answer RQ1: What does it mean for an item (node) to be global in online social
network graph? Secondly, an operational flow of globalness detection was
introduced to answer RQ2: How can we detect these global nodes in online
social network graph? Based on this flow, we examined two cases to find global
nodes in Facebook pages. Our results are encouraging and reasonable,
indicating that this framework can indeed uncover global nodes in the OSN
graph. For future work, we are also eager to further explore other possible
applications including social media’s political standing (e.g. liberal,
conservative, or neutral), from the perspective of relationship. In addition,
since one primary source of global pages is related to commercial promotion,
the judgement of globalness could also be taken as a feature of malicious link
detection in Facebook pages [23].

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