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# Chiral perturbation theory with vector mesons and the pion form factors

Tae-Sun Park<EMAIL_ADDRESS>Center for Exotic Nuclear Studies, Institute
for Basic Science, Daejeon 34126, Korea

###### Abstract

The chiral perturbation theory (ChPT) of pions is extended to include vector
mesons as well as pertinent degrees of freedom. By counting the typical
momentum scale of vector mesons as order of $Q$ and vector meson masses as of
order of $\Lambda_{\chi}$, a consistent theory could be obtained. The explicit
renormalization procedure of the theory is presented for the form factors of
the pion up to one-loop accuracy. The resulting theory prediction for the form
factors is in good agreement with the experimental data for a wide range of
momentum transfers. The vector-meson dominance mechanism is also discussed in
the systematic framework of ChPT.

###### keywords:

chiral perturbation theory , vector mesons , pion form factor

##  1 Introduction

Chiral perturbation theory (ChPT) [1, 2, 3, 4, 5] is a well-proven low-energy
effective field theory (EFT) of QCD, and usually consists of only pions and
nucleons (for the cases where Delta(1232) isobars are also included, see, for
example, Refs. [6, 7, 8]). All other massive degrees of freedom are integrated
out. Integrating out heavy fields is one of the key aspects that make ChPT a
consistent and powerful expansion scheme with respect to powers of
$Q/\Lambda_{\chi}$, where $Q$ stands for the typical momentum scale of the
process and/or the pion mass, which is treated as “light” compared to the
chiral scale $\Lambda_{\chi}\simeq 4\pi f_{\pi}\simeq 1\ \text{GeV}$.

Promoting some of those integrated-out fields as pertinent degrees of freedom
corresponds to a reassessment of a certain class of contributions, and thus
has the potential advantage of widening the validity region and increasing the
accuracy of the theory. The inclusion of the Delta isobars, for example, is in
line with this approach.

In this paper, limiting myself only to the meson sector in SU(2) for the sake
of simplicity, ChPT is extended to include vector mesons that play important
roles in nuclear physics [9, 10]. To this end, I adopt the power counting
scheme where the typical momentum scale and the masses of vector mesons are
counted as of order of $Q$ and $\Lambda_{\chi}$, respectively. The counter-
term Lagrangian that absorbs all the divergences of the loop contributions up
to the next-to-leading order (NLO) is identified, and the resulting theory is
applied to the electric form factors of the pion up to NLO. It will be shown
that the theory is in very good agreement with the experimental data up to
$Q^{2}\sim 1\ \mbox{GeV}^{2}$. The vector-meson dominance (VMD) mechanism [11,
12] will also be discussed in the framework of ChPT.

This manuscript is organized as follows. The formalism for the ChPT with
vector mesons as well as pions is presented in the next section, where the
counting rule and the Lagrangian up-to next-to-leading order Lagrangians are
discussed. with A detailed the renormalization procedure for some relevant
vertex functions is also presented in this section. In Sec. 3, the pion form
factors up to one-loop accuracy are evaluated and compared with experimental
data. Then a final summary is given in Sec. 4.

##  2 Formalism

###  2.1 Power counting rule and the Lagrangian

In this paper, since I am interested in the low-energy processes of pions
where vector mesons appear only in intermediate states, it is natural to adopt
the counting scheme where the typical scale of the momenta carried by the
vector-mesons $p$ is regarded in the order of $Q$, and the vector meson masses
$m_{\rho}$ in the order of $\Lambda_{\chi}$. The corresponding counting rule
has been discussed in Refs. [13, 14], where it is shown that a Feynman diagram
characterized by the number $E_{H}$ ($E_{e}$) of vector mesons (external gauge
fields) and the number of $L$ loops is of the order of $Q^{\nu}$ (modulo
powers of $f_{\pi}$ and $\Lambda_{\chi}$) with

$\displaystyle\nu=2-E_{H}-E_{e}+2L+\sum_{i}\nu_{i},\ \nu_{i}\equiv
d_{i}+h_{i}+e_{i}-2,$ (1)

where $d_{i}$, $h_{i}$ and $e_{i}$ are the number of derivatives and/or pion
masses, vector mesons, and external gauge fields, respectively, attached to
the $i$-th vertex. And the leading-order (LO) Lagrangian consists of terms
with $\nu_{i}=0$,

$\displaystyle{\cal L}_{0}=\accentset{\circ}{f}_{\pi}^{2}\left\langle
i\Delta_{\mu}\,i\Delta^{\mu}\right\rangle+\frac{1}{4}\accentset{\circ}{f}_{\pi}^{2}\left\langle\chi_{+}\right\rangle+\accentset{\circ}{a}\accentset{\circ}{f}_{\pi}^{2}\left\langle\left(g\rho_{\mu}-i\Gamma_{\mu}\right)^{2}\right\rangle,$
(2)

where $\rho_{\mu}\equiv\frac{1}{2}\tau^{a}\rho_{\mu}^{a}$ is the rho-meson
field,

$\displaystyle\Gamma_{\mu}+\Delta_{\mu}$ $\displaystyle=$ $\displaystyle
i\xi^{\dagger}\partial_{\mu}\xi+i\xi^{\dagger}({\cal V}_{\mu}+{\cal
A}_{\mu})\xi,$ $\displaystyle\Gamma_{\mu}-\Delta_{\mu}$ $\displaystyle=$
$\displaystyle i\xi\partial_{\mu}\xi^{\dagger}+i\xi({\cal V}_{\mu}-{\cal
A}_{\mu})\xi^{\dagger},$ $\displaystyle\chi_{+}$ $\displaystyle=$
$\displaystyle\xi^{\dagger}\chi\xi^{\dagger}+\xi\chi^{\dagger}\xi,$ (3)

${\cal V}_{\mu}$, ${\cal A}_{\mu}$ and $\chi$ are external fields: The vacuum
expectation value of $\chi$ is $\accentset{\circ}{m}_{\pi}^{2}$, the bare mass
of the pion squared, while others have vanishing vacuum expectation values.
The bare quantities that should be distinguished from the physical values are
denoted by the “$\circ$” marks. See Ref. [4] for a detailed explanation. Under
the chiral $\text{SU}(N_{f})_{L}\times\text{SU}(N_{f})_{R}$ transformation,
$U$ transforms as $U\to g_{R}Ug_{L}^{\dagger}$, $N_{f}$ is the number of
flavors, and $g_{L}$ and $g_{R}$ are elements of $\text{SU}(N)_{L}$ and
$\text{SU}(N)_{R}$, respectively. And $\xi$ defined as $U=\xi\cdot\xi$, and
transforms non-linearly in terms of $h[g_{R},g_{L},U]$ defined by $\xi\to
g_{R}\xi h^{\dagger}=h\xi g_{L}^{\dagger}$ [1, 2]. And the transformation of
other quantities read $i\Delta_{\mu}\to hi\Delta_{\mu}h^{\dagger}$,
$i\Gamma_{\mu}\to hi\Gamma_{\mu}h^{\dagger}-i\partial_{\mu}h\,h^{\dagger}$,
and $g\rho_{\mu}\to hg\rho_{\mu}h^{\dagger}-i\partial_{\mu}h\,h^{\dagger}$. In
the above equation, $\left\langle X\right\rangle$ stands for the trace of $X$
in the isospin space, $g$ is the rho-meson coupling constant, and the
dimensionless parameter $\accentset{\circ}{a}$ is related to the rho meson
mass, $m_{\rho}^{2}=\accentset{\circ}{a}g^{2}f_{\pi}^{2}+{\cal O}(Q^{2})$.

One might be surprised to find that the LO Lagrangian does not contain the
usual kinetic term for vector mesons, ${\cal
L}_{\rho}^{\text{kin}}=-\frac{1}{2}\left\langle\rho^{\mu\nu}\rho_{\mu\nu}\right\rangle$,
which appears only in the next-to-leading order (NLO) with $\nu_{i}=2$ in this
formalism. Without the kinetic term, the leading-order vector-meson propagator
is momentum-independent, $\frac{1}{m_{\rho}^{2}}g^{\mu\nu}\delta^{ab}$. As a
result, the rho-meson can be trivially integrated out at leading order, and
the remaining theory becomes identical to the usual ChPT. However, at one-loop
or higher-order calculations, the loop corrections cause the vector mesons
introduced as pertinent degrees of freedom to play a non-trivial role.

There are different options for counting $m_{\rho}$ and $p$. To describe the
processes where rho mesons appear as external particles or those that involve
momentum transfer whose size is comparable to $m_{\rho}$, Djukanovic et al.
have developed a power counting scheme [15, 16] where the possible flows of
the external momenta through the internal propagators are considered,
assigning the chiral order of a Feynman diagram as the lowest among all the
possible cases. In this scheme, the rho-meson kinetic term enters in the
leading order and the leading order rho-meson propagator is the familiar
massive vector-meson propagator in the unitary gauge,

$\displaystyle\frac{1}{k^{2}-m_{\rho}^{2}+i0^{+}}\left(-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{m_{\rho}^{2}}\right)\delta^{ab}.$
(4)

However, this propagator behaves poorly in the ultraviolet region. For
example, the diagram given in Fig. 1 with this propagator results in a highly
divergent term of the form
$\frac{1}{d-4}k^{4}(k^{\mu}k^{\nu}-g^{\mu\nu}k^{2})$, where $d$ is the space-
time dimension.

Figure 1: The most divergent one-loop diagram for the rho-meson self-energy.
The double lines are rho-meson propagators.

This problem can be resolved if one adopts the so-called infrared
regularization [17] or the extended-on-mass-shell scheme [18], where the
contribution of the diagram vanishes identically. Another possibility is to
adopt the hidden local gauge symmetry (HLS) where the rho meson is introduced
as a gauge boson of the hidden symmetry [19, 20]. In HLS, one can adopt a
general $R_{\xi}$ gauge propagator, which removes the aforementioned problem.
One-loop calculations in HLS come with however its own complications related
to the gauge-fixing of the hidden symmetry. For a comprehensive discussion of
one-loop calculation in HLS in the chiral limit, see Ref. [20].

The formalism presented in this paper is free from the above-mentioned
divergence, since there is no $\rho\rho\rho$-vertex at LO and thus the
aforementioned problematic one-loop diagram does not appear.

###  2.2 1-loop divergences

In Refs. [3, 4], the 1-loop contributions to the vacuum functional is
evaluated for the chiral perturbation theory with pions. By applying the same
procedure to the cases where vector mesons are also included, one can identify
the counter-term Lagrangian that absorbs all the divergences of the one-loop
graphs with the LO Lagrangian,

$\displaystyle{\cal L}_{ct}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}(d-4)}\Bigg{\\{}\frac{N}{6}\left\langle{\tilde{\Gamma}_{\mu\nu}\tilde{\Gamma}^{\mu\nu}}\right\rangle$
(5) $\displaystyle+$ $\displaystyle
2\left\langle{\Delta^{\mu}\Delta^{\nu}}\right\rangle\left\langle{\Delta_{\mu}\Delta_{\nu}}\right\rangle-\accentset{\circ}{a}^{2}\left\langle{\Delta^{\mu}\phi^{\nu}}\right\rangle\left\langle{\Delta_{\mu}\phi_{\nu}}\right\rangle$
$\displaystyle+\frac{\accentset{\circ}{a}^{4}}{8}\left\langle{\phi^{\mu}\phi^{\nu}}\right\rangle\left\langle{\phi_{\mu}\phi_{\nu}}\right\rangle$
$\displaystyle+$
$\displaystyle\left\langle{W}\right\rangle^{2}+N\left\langle{W^{2}}\right\rangle-\frac{N}{2}\left\langle{\chi_{+}W}\right\rangle-\frac{1}{2}\left\langle{\chi_{+}}\right\rangle\left\langle{W}\right\rangle$
$\displaystyle+$
$\displaystyle\frac{N^{2}-4}{16N}\left\langle{\chi_{+}^{2}}\right\rangle+\frac{N^{2}+2}{16N^{2}}\left\langle{\chi_{+}}\right\rangle^{2}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle\cdots,$

where the ellipses denote non-divergent terms, $N$ is the flavor number ($N=2$
in this work), $\phi_{\mu}\equiv\left(g\rho_{\mu}-i\Gamma_{\mu}\right)$,
$W\equiv\Delta^{\mu}\Delta_{\mu}-\frac{\accentset{\circ}{a}^{2}}{4}\phi^{\mu}\phi_{\mu}$,
and

$\displaystyle\tilde{\Gamma}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\accentset{\circ}{a}}{2}\right)\Gamma_{\mu\nu}-i\frac{\accentset{\circ}{a}}{2}g\rho_{\mu\nu}$
(6)
$\displaystyle+\frac{\accentset{\circ}{a}}{2}\left(1-\frac{\accentset{\circ}{a}}{2}\right)\left[g\rho_{\mu}-i\Gamma_{\mu},\,g\rho_{\nu}-i\Gamma_{\nu}\right]$

with
$\Gamma_{\mu\nu}\equiv\partial_{\mu}\Gamma_{\nu}-\partial_{\nu}\Gamma_{\mu}+\left[\Gamma_{\mu},\,\Gamma_{\nu}\right]$
and
$\rho_{\mu\nu}\equiv\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}-ig\left[\rho_{\mu},\,\rho_{\nu}\right]$.

###  2.3 Renormalization of vertex functions

The renormalization procedure of the vertex functions that are relevant for
the electromagnetic form factors of the pion is described here. The relevant
Lagrangian may be written as

$\displaystyle{\cal L}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}C_{1:1}g^{2}\left\langle{\rho_{\mu\nu}\rho^{\mu\nu}}\right\rangle+\frac{1}{2}C_{1:2}\left\langle{\Gamma_{\mu\nu}\Gamma^{\mu\nu}}\right\rangle$
(7) $\displaystyle-
iC_{1:3}g\left\langle{\rho^{\mu\nu}\Gamma_{\mu\nu}}\right\rangle$
$\displaystyle-
iC_{2:1}g\left\langle{\rho_{\mu\nu}\left[\Delta^{\mu},\Delta^{\nu}\right]}\right\rangle+C_{2:2}\left\langle{\Gamma_{\mu\nu}\left[\Delta^{\mu},\Delta^{\nu}\right]}\right\rangle$
$\displaystyle-
C_{3}\left\langle{\chi_{+}}\right\rangle\left\langle{\Delta^{\mu}\Delta_{\mu}}\right\rangle+C_{4}\left\langle{\chi_{+}}\right\rangle\left\langle{(g\rho_{\mu}-i\Gamma_{\mu})^{2}}\right\rangle$
$\displaystyle+C_{5}\left\langle{\chi_{+}}\right\rangle^{2}+\cdots,$

where the ellipses represent terms that contain four or more fields. I adopt
the convention of Refs. [3, 4] which decompose the low-energy constants (LECs)
into a divergent part and the remaining finite part,

$\displaystyle
C_{i}=\eta_{i}\left[\lambda+\frac{1}{32\pi^{2}}\left(\bar{C}_{i}+\ln\frac{\accentset{\circ}{m}_{\pi}^{2}}{\mu^{2}}\right)\right],$
(8)

where $\mu$ is the scale of the dimensional regularization, and

$\displaystyle\lambda\equiv-\frac{\mu^{d-4}}{32\pi^{2}}\left\\{\frac{2}{4-d}+\ln
4\pi+\Gamma^{\prime}(1)+1\right\\}.$ (9)

The renormalized $\bar{C}_{i}$ are independent of the scale $\mu$, and the
coefficient $\eta_{i}$ can be obtained by comparing Eq.(7) with (5):
$\left\\{\eta_{1:1},\eta_{1:2},\eta_{1:3}\right\\}=\frac{2}{3}\left\\{\frac{\accentset{\circ}{a}^{2}}{4},\left(1-\frac{\accentset{\circ}{a}}{2}\right)^{2},\frac{\accentset{\circ}{a}}{2}\left(1-\frac{\accentset{\circ}{a}}{2}\right)\right\\}$,
$\eta_{2:1}=\eta_{2:2}=0$,
$\left\\{\eta_{3},\eta_{4}\right\\}=\left\\{1,\frac{\accentset{\circ}{a}^{2}}{4}\right\\}$
and $\eta_{5}=\frac{3}{32}$.

####  2.3.1 1PI for ${\pi}_{a}(k)\to{\pi}_{b}(k)$

(a)

(b)

Figure 2: 1PI diagrams for ${\pi}_{a}(k)\to{\pi}_{b}(k)$. The dashed lines
denote pions.

The one-particle irreducible (1PI) vertex for $\pi_{a}(k)\to\pi_{b}(k)$ is the
inverse of the pion propagator $\delta_{ab}\,D_{\pi}(k^{2})$, whose one-loop
diagrams are drawn in Fig. 2. Their contributions are

$\displaystyle D_{\pi}^{-1,(a)}(k^{2})$ $\displaystyle=$ $\displaystyle
2\accentset{\circ}{a}k^{2}\frac{\lambda_{2}}{f_{\pi}^{2}},$ $\displaystyle
D_{\pi}^{-1,(b)}(k^{2})$ $\displaystyle=$
$\displaystyle\left[2(1-\accentset{\circ}{a})k^{2}-\frac{3}{2}\accentset{\circ}{m}_{\pi}^{2}\right]\frac{\lambda_{2}}{f_{\pi}^{2}},$
(10)

where

$\displaystyle\lambda_{2}$ $\displaystyle\equiv$
$\displaystyle-2\accentset{\circ}{m}_{\pi}^{2}\left(\lambda+\frac{1}{32\pi^{2}}\ln\frac{\accentset{\circ}{m}_{\pi}^{2}}{\mu^{2}}\right).$
(11)

There are also tree contributions from ${\cal L}_{0}$ and ${\cal L}_{2}$,

$\displaystyle D_{\pi}^{-1,\text{tree}}(t)$ $\displaystyle=$
$\displaystyle\frac{t}{f_{\pi}^{2}}\left(\accentset{\circ}{f}_{\pi}^{2}+4\accentset{\circ}{m}_{\pi}^{2}C_{3}\right)-\frac{\accentset{\circ}{m}_{\pi}^{2}}{f_{\pi}^{2}}\left(\accentset{\circ}{f}_{\pi}^{2}+32\accentset{\circ}{m}_{\pi}^{2}C_{5}\right)$
(12) $\displaystyle\equiv$ $\displaystyle(1+\delta
Z_{\pi})t-(m_{\pi}^{2}+\delta m_{\pi}^{2}),$

which defines $\delta Z_{\pi}$ and $\delta m_{\pi}^{2}$. The net sum then
reads

$\displaystyle D_{\pi}^{-1}(t)$ $\displaystyle\equiv$ $\displaystyle
D_{\pi}^{-1,(a)}(t)+D_{\pi}^{-1,(b)}(t)+D_{\pi}^{-1,\text{tree}}(t)$ (13)
$\displaystyle=$ $\displaystyle t-m_{\pi}^{2}$

with

$\displaystyle\delta Z_{\pi}$ $\displaystyle=$
$\displaystyle-2\frac{\lambda_{2}}{f_{\pi}^{2}},$ $\displaystyle\delta
m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}\frac{\lambda_{2}}{f_{\pi}^{2}}\accentset{\circ}{m}_{\pi}^{2}\,.$
(14)

Combining Eq.(12) with (14), one can express $f_{\pi}$ and $m_{\pi}$ in terms
of the bare quantities,

$\displaystyle f_{\pi}^{2}$ $\displaystyle=$
$\displaystyle\accentset{\circ}{f}_{\pi}^{2}+\frac{\accentset{\circ}{m}_{\pi}^{2}}{8\pi^{2}}\bar{C}_{3},$
$\displaystyle m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle\accentset{\circ}{m}_{\pi}^{2}-\frac{\accentset{\circ}{m}_{\pi}^{4}}{32\pi^{2}f_{\pi}^{2}}(4\bar{C}_{3}-3\bar{C}_{5}).$
(15)

This is consistent with the results given in Ref.[3], which can be seen as
$C_{3}$ and $C_{5}$ correspond to $\frac{1}{2}l_{4}$ and
$\frac{1}{16}(l_{3}+l_{4})$, respectively.

The 1PI functions for ${\pi}_{b}(k)\to{\cal A}_{a}^{\mu}(k)$ and ${\cal
A}_{a}^{\mu}(k)\to{\cal A}_{b}^{\nu}(k)$ can be evaluated in a similar way:

$\displaystyle\Gamma\left[{\pi}_{a}(k)\to{\cal A}_{b}^{\mu}(k)\right]$
$\displaystyle=$ $\displaystyle ik^{\mu}f_{\pi}\,\delta_{ab},$ (16)
$\displaystyle\Gamma\left[{\cal A}_{a}^{\mu}(k)\to{\cal
A}_{b}^{\nu}(k)\right]$ $\displaystyle=$ $\displaystyle
f_{\pi}^{2}\,g^{\mu\nu}\,\delta_{ab}.$ (17)

####  2.3.2 1PI for ${\cal V}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$

(a)

(b)

(c)

(d)

(e)

Figure 3: 1PI diagrams for ${\cal
V}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$. The wiggled lines denote
external vector fields.

The 1PI diagrams for ${\cal V}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$
are drawn in Fig. 3, and the corresponding vertex function can be written as
$-i\epsilon_{eab}(q_{a}-q_{b})^{\mu}\Gamma_{{\cal V}\pi\pi}(k^{2})$ with

$\displaystyle\Gamma_{{\cal V}\pi\pi}^{(a)}(t)$ $\displaystyle=$
$\displaystyle 3(1-\accentset{\circ}{a})\frac{\lambda_{2}}{f_{\pi}^{2}},$
$\displaystyle\Gamma_{{\cal V}\pi\pi}^{(b)}(t)$ $\displaystyle=$
$\displaystyle-\frac{(2-\accentset{\circ}{a})^{2}}{4f_{\pi}^{2}}\left[\lambda_{2}-tf_{3}(t)\right],$
$\displaystyle\Gamma_{{\cal V}\pi\pi}^{(c)}(t)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\Gamma_{{\cal V}\pi\pi}^{(d+e)}(t)$
$\displaystyle=$ $\displaystyle
2\accentset{\circ}{a}\frac{\lambda_{2}}{f_{\pi}^{2}},$
$\displaystyle\Gamma^{\text{tree}}_{{\cal V}\pi\pi}(t)$ $\displaystyle=$
$\displaystyle 1+\delta
Z_{\pi}-\frac{1}{2f_{\pi}^{2}}\left(\accentset{\circ}{a}\accentset{\circ}{f}_{\pi}^{2}+4\accentset{\circ}{m}_{\pi}^{2}C_{4}\right)$
(18) $\displaystyle+$
$\displaystyle\frac{t}{2f_{\pi}^{2}}\left(C_{1:2}-C_{2:2}\right),$

where the loop function $f_{3}(t)$ is described in the Appendix A.

The net contribution is given as

$\displaystyle\Gamma_{{\cal V}\pi\pi}(t)$ $\displaystyle=$ $\displaystyle
1-\frac{a_{r}}{2}$ (19) $\displaystyle+$
$\displaystyle\frac{t}{2f_{\pi}^{2}}\left[2\left(1-\frac{\accentset{\circ}{a}}{2}\right)^{2}f_{3}(t)+C_{1:2}-C_{2:2}\right]$

with

$\displaystyle
a_{r}=\accentset{\circ}{a}-\frac{m_{\pi}^{2}}{8\pi^{2}}\left(\accentset{\circ}{a}\bar{C}_{3}-\frac{\accentset{\circ}{a}^{2}}{4}\bar{C}_{4}\right).$
(20)

If $\accentset{\circ}{a}=0$, the results become identical to the ChPT without
vector mesons. On the other hand, $a_{r}=2$ (with
$\accentset{\circ}{a}=2+{\cal O}(Q^{2}/\Lambda_{\chi}^{2})$) leads to a
considerable amount of simplification. With this choice, for example,
$\Gamma_{{\cal V}\pi\pi}(0)=1-\frac{a_{r}}{2}$ vanishes, and the photon
coupling to the pion is mediated only by the $1\rho$-exchange diagram drawn in
Fig. 6(b), which is known as the vector-meson dominance (VMD) mechanism [11,
12]. $C_{1:2}$ is in general divergent and to be adjusted to absorb the
divergence coming from $f_{3}(0)$. When $a_{r}=2$, however, the loop
contribution vanishes and $C_{1:2}$ becomes finite. The renormalization
procedure for the linear combination $C_{1:2}-C_{2:2}$ will be discussed in
Section 3.

####  2.3.3 1PI for ${\rho}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$

(a)

(b)

(c)

Figure 4: 1PI diagrams for
${\rho}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$.

The 1PI diagrams for ${\rho}_{e}^{\mu}(k)\to\pi_{a}(q_{a})+\pi_{b}(q_{b})$ are
drawn in Fig. 4, whose vertex function can be written as
$-i\epsilon_{eab}(q_{a}-q_{b})^{\mu}\Gamma_{\rho\pi\pi}(k^{2})$ with

$\displaystyle\Gamma_{\rho\pi\pi}^{(a)}$ $\displaystyle=$
$\displaystyle\frac{\accentset{\circ}{a}g}{2f_{\pi}^{2}}\lambda_{2},$
$\displaystyle\Gamma_{\rho\pi\pi}^{(b)}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\accentset{\circ}{a}}{2}\right)\frac{\accentset{\circ}{a}g}{2f_{\pi}^{2}}\left[-\lambda_{2}+tf_{3}(t)\right],$
$\displaystyle\Gamma_{\rho\pi\pi}^{(c)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Gamma_{\rho\pi\pi}^{\text{tree}}$ $\displaystyle=$
$\displaystyle\frac{g}{2f_{\pi}^{2}}\left\\{\accentset{\circ}{a}\accentset{\circ}{f}_{\pi}^{2}+4\accentset{\circ}{m}_{\pi}^{2}C_{4}+\left(C_{1:3}-C_{2:1}\right)t\right\\},$
(21)

and the net sum reads

$\displaystyle\Gamma_{\rho\pi\pi}(t)=\frac{a_{r}}{2}g+\frac{g}{2f_{\pi}^{2}}t\left[\accentset{\circ}{a}\left(1-\frac{\accentset{\circ}{a}}{2}\right)f_{3}(t)+C_{1:3}-C_{2:1}\right].$
(22)

When $a_{r}=2$, the loop contribution vanishes and one is left with

$\displaystyle\left.\Gamma_{\rho\pi\pi}(t)\right|_{a_{r}=2}$ $\displaystyle=$
$\displaystyle g+\frac{t}{m_{\rho}^{2}}\left(g_{\rho\pi\pi}-g\right),$ (23)

where

$\displaystyle
g_{\rho\pi\pi}\equiv\Gamma_{\rho\pi\pi}(m_{\rho}^{2})=g\left[1+\frac{m_{\rho}^{2}}{2f_{\pi}^{2}}\left(C_{1:3}-C_{2:1}\right)\right]$
(24)

is to be determined. Since
$\Gamma(\rho^{0}\to\pi^{+}\pi^{-})=g_{\rho\pi\pi}^{2}p_{\pi}^{3}/(6\pi
m_{\rho}^{2})$ with $p_{\pi}\equiv\sqrt{m_{\rho}^{2}/4-m_{\pi}^{2}}$, the
experimental data for the rho meson decay width
$\Gamma(\rho^{0}\to\pi^{+}\pi^{-})=(147.8\pm 0.9)$ MeV corresponds to

$\displaystyle g_{\rho\pi\pi}=5.942\pm 0.017.$ (25)

####  2.3.4 1PI for ${\rho}_{a}^{\mu}(k)\to{\rho}_{b}^{\nu}(k)$

(a)

(b)

Figure 5: 1PI diagrams for ${\rho}_{a}^{\mu}(k)\to{\rho}_{b}^{\nu}(k)$.

The vertex function for ${\rho}_{a}^{\mu}(k)\to{\rho}_{b}^{\nu}(k)$ is the
inverse of the rho-meson propagator $\delta_{ab}D_{\rho}^{\mu\nu}(k)$, whose
relevant diagrams are drawn in Fig. 5:

$\displaystyle\left(D_{\rho}^{-1}\right)^{\mu\nu,(a)}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\accentset{\circ}{a}^{2}g^{2}\lambda_{2}\,g^{\mu\nu}$
$\displaystyle+\frac{1}{2}\accentset{\circ}{a}^{2}g^{2}\left(k^{\mu}k^{\nu}-g^{\mu\nu}k^{2}\right)f_{3}(k^{2}),$
$\displaystyle\left(D_{\rho}^{-1}\right)^{\mu\nu,(b)}(k)$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\left(D_{\rho}^{-1}\right)^{\mu\nu,\text{tree}}(k)$
$\displaystyle=$ $\displaystyle
g^{2}\left(\accentset{\circ}{a}\accentset{\circ}{f}_{\pi}^{2}+4\accentset{\circ}{m}_{\pi}^{2}C_{4}\right)g^{\mu\nu}$
(26) $\displaystyle+g^{2}C_{1:1}\left(k^{\mu}k^{\nu}-g^{\mu\nu}k^{2}\right),$

and the resulting propagator is given as

$\displaystyle D^{\mu\nu}_{\rho}(k)$ $\displaystyle=$ $\displaystyle-
D_{\rho}(k^{2})\left(g^{\mu\nu}-A_{\rho}(k^{2})\frac{k^{\mu}k^{\nu}}{a_{r}\,g^{2}f_{\pi}^{2}}\right)$
(27)

with

$\displaystyle D_{\rho}(t)$ $\displaystyle=$
$\displaystyle\left[tA_{\rho}(t)-a_{r}\,g^{2}f_{\pi}^{2}\right]^{-1},$ (28)
$\displaystyle A_{\rho}(t)$ $\displaystyle=$ $\displaystyle
g^{2}C_{1:1}+\frac{1}{2}\accentset{\circ}{a}^{2}g^{2}f_{3}(t).$ (29)

The renormalization condition for $g$ and $C_{1:1}$ may be obtained by
demanding the following structure of the propagator near the pole position,

$\displaystyle\lim_{t\to
m_{\rho}^{2}}D^{-1}_{\rho}(t)=t-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho},$ (30)

where $\Gamma_{\rho}\simeq\Gamma(\rho^{0}\to\pi^{+}\pi^{-})$ is the decay
width of the rho meson. In this work, to reproduce the imaginary part of the
above equation accurately, $\bar{A}_{\rho}(t)\equiv A_{\rho}(t)-A_{\rho}(0)$
is rewritten as

$\displaystyle\bar{A}_{\rho}(t)=2\left[\Gamma_{\rho\pi\pi}(t)\right]^{2}\bar{f}_{3}(t),$
(31)

where the tree order $\rho\pi\pi$ coupling $\frac{a_{r}}{2}g$ is replaced by
the full one-loop order vertex function $\Gamma_{\rho\pi\pi}(t)$. Since
$\Gamma_{\rho\pi\pi}(t)=\frac{a_{r}}{2}g+{\cal O}(Q^{2}/\Lambda_{\chi}^{2})$
when $t={\cal O}(Q^{2})$, this replacement is correct up to one-loop accuracy,
but contains the higher-order contributions needed to reproduce the
experimental $\rho$-meson decay width at the pole position. Then the solution
for $a_{r}\,g^{2}f_{\pi}^{2}=t\Re A_{\rho}(t)$ and $\frac{\partial}{\partial
t}\left(t\Re A_{\rho}(t)\right)=1$ at $t=m_{\rho}^{2}$ reads111If one does not
invoke the replacement given in Eq.(31), the corresponding solution would be
$\displaystyle g$ $\displaystyle=$
$\displaystyle\left(\frac{a_{r}f_{\pi}^{2}}{m_{\rho}^{2}}+\frac{1}{2}a_{r}^{2}m_{\rho}^{2}R_{3}^{\prime}\right)^{-1/2}\simeq
6.25\pm 0.06,$ $\displaystyle A_{\rho}(0)$ $\displaystyle=$
$\displaystyle\frac{a_{r}g^{2}f_{\pi}^{2}}{m_{\rho}^{2}}-\frac{1}{2}a_{r}^{2}g^{2}R_{3}\simeq
1.125\pm 0.035.$ (32)

$\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{m_{\rho}}{\sqrt{a_{r}}f_{\pi}}\Big{[}\frac{\sqrt{a_{r}}g_{\rho\pi\pi}m_{\rho}}{f_{\pi}}R_{3}$
$\displaystyle+\sqrt{1-2g_{\rho\pi\pi}^{2}\left(2R_{3}+m_{\rho}^{2}R_{3}^{\prime}\right)}\Big{]}$
$\displaystyle\simeq$ $\displaystyle 6.21\pm 0.06,$ $\displaystyle
A_{\rho}(0)$ $\displaystyle=$
$\displaystyle\frac{a_{r}g^{2}f_{\pi}^{2}}{m_{\rho}^{2}}-2g_{\rho\pi\pi}^{2}R_{3}\simeq
1.113\pm 0.002,$ (33)

where $R_{3}\equiv\Re\bar{f}_{3}(m_{\rho}^{2})\simeq-0.3395\times 10^{-3}$,
$R_{3}^{\prime}\equiv\Re f_{3}^{\prime}(m_{\rho}^{2})\simeq-1.289\times
10^{-3}/m_{\rho}^{2}$ and $f_{3}^{\prime}(t)\equiv\frac{\partial}{\partial
t}f_{3}(t)$, with $a_{r}=2$, $m_{\rho}=(775.26\pm 0.23)$ MeV and
$f_{\pi}=(92.1\pm 0.8)$ MeV [21].

The low-momentum behavior of the propagator is given as

$\displaystyle\lim_{k^{2}\to
0}D^{\mu\nu}_{\rho}(k)=\frac{1}{A_{\rho}(0)}\frac{1}{\tilde{M}_{\rho}^{2}-k^{2}}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{\tilde{M}_{\rho}^{2}}\right),$
(34)

where

$\displaystyle\tilde{M}_{\rho}\equiv\frac{\sqrt{a_{r}}gf_{\pi}}{\sqrt{A_{\rho}(0)}}\
\simeq(0.989\pm 0.001)\ m_{\rho}.$ (35)

Thus, its overall magnitude is quenched by the factor $1/A_{\rho}(0)\simeq
0.90$, with an “effective mass” which is about 1 percent smaller than physical
mass, $\tilde{M}_{\rho}\simeq 0.99\ m_{\rho}$.

Similarly, the 1PI for ${\rho}_{a}^{\mu}(k)\to{\cal
V}_{b}^{\nu}(k)\equiv\Gamma_{{\rho}{\cal V}}^{\mu\nu}(k)$ reads

$\displaystyle\Gamma_{{\rho}{\cal V}}^{\mu\nu}(k)$ $\displaystyle=$
$\displaystyle-\Gamma_{\rho\cal{V}}(k^{2})g^{\mu\nu}$ (36) $\displaystyle+$
$\displaystyle
g\left[\accentset{\circ}{a}\left(1-\frac{\accentset{\circ}{a}}{2}\right)f_{3}(k^{2})+C_{1:3}\right]k^{\mu}k^{\nu}$

with

$\displaystyle\Gamma_{\rho\cal{V}}(t)=a_{r}gf_{\pi}^{2}+g\left[\accentset{\circ}{a}\left(1-\frac{\accentset{\circ}{a}}{2}\right)f_{3}(t)+C_{1:3}\right]t.$
(37)

$\Gamma_{\rho\cal{V}}(m_{\rho}^{2})$ is related to the $\rho\to e^{+}e^{-}$
decay width as

$\Gamma(\rho\to
e^{+}e^{-})=\frac{8\pi\alpha^{2}}{3m_{\rho}^{4}}\left|\Gamma_{\rho\cal{V}}(m_{\rho}^{2})\right|^{2}\left(1+2\frac{m_{e}^{2}}{m_{\rho}^{2}}\right)p_{e}$

with $p_{e}\equiv\sqrt{m_{\rho}^{2}/4-m_{e}^{2}}$. The experimental value
$\Gamma(\rho\to e^{+}e^{-})=(7.04\pm 0.06)$ keV corresponds to

$\displaystyle\Gamma_{\rho\cal{V}}(m_{\rho}^{2})=(0.1213\pm 0.0005)\
\text{GeV}^{2}$ (38)

and

$\displaystyle\frac{\Gamma_{\rho\cal{V}}(m_{\rho}^{2})}{2f_{\pi}^{2}g_{\rho\pi\pi}}=1.197\pm
0.006.$ (39)

The values of $C_{1:3}$ and $C_{2:1}$ that reproduce the experimental values
for $\Gamma(\rho\to\pi^{+}\pi^{-})$ and $\Gamma(\rho\to e^{+}e^{-})$ can be
obtained by combining Eqs.(24,25,37,38), for the particular choice $a_{r}=2$,

$\displaystyle C_{1:3}$ $\displaystyle=$ $\displaystyle(4.3\pm 1.0)\times
10^{-3},$ $\displaystyle C_{2:1}$ $\displaystyle=$ $\displaystyle(5.5\pm
0.7)\times 10^{-3}.$ (40)

##  3 Electric form factors of the pion

(a)

(b)

Figure 6: Diagrams for the electric form factor of the pion.

The electric form factor of the pion, $F_{\pi}(t)$, is the sum of the 1PI
${\cal V}\pi\pi$ vertex and the one $\rho$-meson-exchange,

$\displaystyle F_{\pi}(t)$ $\displaystyle=$ $\displaystyle\Gamma_{{\cal
V}\pi\pi}(t)$ (41) $\displaystyle+$
$\displaystyle\frac{a_{r}g^{2}f_{\pi}^{2}}{a_{r}g^{2}f_{\pi}^{2}-tA_{\rho}(t)}\frac{\Gamma_{\rho\pi\pi}(t)}{\Gamma_{\rho\pi\pi}(0)}\frac{\Gamma_{\rho{\cal
V}}(t)}{\Gamma_{\rho{\cal V}}(0)}.$

$F_{\pi}(t)$ at small-$t$ is given as $F_{\pi}(t)=1+\frac{1}{6}\langle
r_{\pi}^{2}\rangle t+{\cal O}(t^{2})$ with

$\displaystyle\frac{1}{6}\langle r_{\pi}^{2}\rangle=\Gamma_{{\cal
V}\pi\pi}^{\prime}(0)+\frac{1}{\tilde{M}_{\rho}^{2}}+\frac{1}{2f_{\pi}^{2}}\left(2C_{1:3}-C_{2:1}\right),$
(42)

where $\langle r_{\pi}^{2}\rangle$ is the square of the charge radius of the
pion. The experimental value $\langle r_{\pi}^{2}\rangle=(0.434\pm 0.005)\
\text{fm}^{2}$ [21] with the values of $C_{1:3}$ and $C_{2:1}$ given in
Eq.(40) leads to

$\displaystyle\Gamma_{{\cal V}\pi\pi}(t)$ $\displaystyle=$
$\displaystyle\Gamma_{{\cal V}\pi\pi}^{\prime}(0)\,t,$
$\displaystyle\Gamma_{{\cal V}\pi\pi}^{\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{C_{1:2}-C_{2:2}}{2f_{\pi}^{2}}=(0.000\pm 0.005)\
\text{fm}^{2}.$ (43)

That is, with $a_{r}=2$, one has the complete VMD where not only the leading
tree contribution but also the one-loop correction vanish for $\Gamma_{{\cal
V}\pi\pi}(t)$, and the pion form factor is completely mediated by the vector-
meson exchange.

The resulting $|F_{\pi}(t)|$ with $a_{r}=2$ is drawn in the top panel of Fig.
7, which shows that the theory prediction is in excellent agreement with the
experimental data [22].

Figure 7:  The electric form factors of the pion, $\left|F_{\pi}(t)\right|$,
for the experimental data extracted from the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decay [22] (top) and from the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross-section [23, 24] (bottom). The black solid
lines represent the theory prediction. In the bottom panel, the blue dots are
for the KLOE data [23] while the red dots are for the SND data [24].

This success comes as a surprise because the validity region of the formalism
is limited to the low-energy region $Q<\Lambda_{\chi}$ and fitting to data was
not required, the LECs have been fixed just to reproduce the experimental data
for the charge radius of the pion and the $\rho\to\pi\pi$ and $\rho\to
e^{+}e^{-}$ decay widths while adopting the choice $a_{r}=2$.

###  3.1 $\rho-\omega$ mixing

Contrary to the pion form factor extracted from the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decay that has been discussed so far,
the form factor obtained from $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ [23, 24]
is subject to the $\rho-\omega$ mixing [26, 27, 16]. As discussed in Refs.
[25, 27], the effect of the mixing to the pion form factor is to multiply the
following factor to the rho-meson propagator (or, identically, to the second
term of the pion form factor given in Eq.(41)),

$\displaystyle
1+\frac{F_{\omega}}{F_{\rho}}\frac{\Theta_{\rho\omega}}{t-m_{\omega}^{2}+im_{\omega}\Gamma_{\omega}},$
(44)

where, in the vicinity of $\rho-\omega$ resonance,
$\Theta_{\rho\omega}=2m_{\rho}(m_{u}-m_{d})+e^{2}F_{\rho}F_{\omega}$, and the
vector meson coupling strengths are defined as $\langle
0|j_{\mu}^{\text{em}}|\rho^{0}\rangle=\varepsilon_{\mu}F_{\rho}m_{\rho}$ and
$\langle
0|j_{\mu}^{\text{em}}|\omega\rangle=\varepsilon_{\mu}F_{\omega}m_{\omega}$,
$\varepsilon_{\mu}$ representing the polarization vector. With the values
estimated in [27], ${F_{\omega}}/{F_{\rho}}=\frac{1}{3}$,
$\Theta_{\rho\omega}=(-3.91\pm 0.30)\times 10^{-3}\ \text{GeV}^{2}$, and with
the experimental value [21] $m_{\omega}=782.66\ \text{MeV}$ and
$\Gamma_{\omega}=8.68\ \text{MeV}$, the resulting pion form factor is drawn in
the bottom panel of Fig. 7.

##  4 Summary

In this work, the chiral perturbation theory of pions has been extended to
include vector mesons as well. The employed power counting scheme is to count
the masses and the momentum scale of the vector mesons as ${\cal
O}(\Lambda_{\chi})$ and ${\cal O}(Q)$, respectively. Up to one-loop accuracy,
all the divergences have been identified, and the renormalization procedure
for the vector form factor of the pion has been explicitly demonstrated. For
the parameter $a_{r}$, the special choice $a_{r}=2$ has been adopted as it
simplifies the theory substantially. Interestingly, the theory with $a_{r}=2$
retains the VMD mechanism even at one-loop accuracy. Thus one is left with
$\Gamma_{{\cal V}\pi\pi}(t)=0$, and photons interact with pions only through
vector-meson exchanges.

One-loop order calculations with vector mesons for the pion form factors are
found to be in excellent agreement with the experimental data for a wide range
of momentum transfers. However, this success should be taken with caution, as
the general validity domain of the theory remains $Q\ll\Lambda_{\chi}\sim 1\
\text{GeV}$.

## Acknowledgement

The author thanks Prof. S.-W. Hong and Prof. A.W. Thomas for useful
discussions and their support. He is also grateful to Dr. Joochun (Jason) Park
for his careful reading of the manuscript and helpful comments. This work was
supported by the IBS grant funded by the Korea government (No. IBS-R031-D1).

##  Appendix A Loop function

The loop function $f_{3}(k^{2})$ is defined as [13],

$\displaystyle
f_{3}(k^{2})=\frac{\Gamma(\varepsilon)}{16\pi^{2}}\int_{0}^{1}\\!\\!dz\,z(2z-1)\left(\frac{m_{\pi}^{2}-z(1-z)k^{2}}{4\pi\mu^{2}}\right)^{-\varepsilon}.$

At $k^{2}=0$, it is given as with

$\displaystyle
f_{3}(0)=\frac{\lambda_{0}}{6}=-\frac{1}{6}\left[2\lambda+\frac{1}{16\pi^{2}}\left(\ln\frac{M^{2}}{\mu^{2}}-1\right)\right],$
(45)

and $\bar{f}_{3}(k^{2})\equiv f_{3}(k^{2})-f_{3}(0)$ reads

$\displaystyle\bar{f}_{3}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{2m_{\pi}}^{\infty}\\!\\!dM\,\frac{1}{3M}\left(1-\frac{4m_{\pi}^{2}}{M^{2}}\right)^{3/2}$
(46) $\displaystyle\times\ \frac{k^{2}}{M^{2}-k^{2}(1-y^{2})/4-i0}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\frac{1}{9}+\frac{1}{3\sigma^{3}}\left(\arctan{\sigma}-\sigma\right)\right],$

$\sigma\equiv{k}/{\sqrt{4m_{\pi}^{2}-k^{2}-i0}}$. Its imaginary part reads

$\displaystyle\Im
f_{3}(k^{2})=\frac{1}{96\pi}\left(\frac{k^{2}-4m_{\pi}^{2}}{k^{2}}\right)^{3/2}\theta(k-2m_{\pi}),$
(47)

and $\bar{f}_{3}(m_{\rho}^{2})=(-0.340+2.692i)\times 10^{-3}$ and
$m_{\rho}^{2}f_{3}^{\prime}(m_{\rho}^{2})=(-1.289+0.602i)\times 10^{-3}$.

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[1]Rezvaneh RezapourThese authors contributed equally to this work.

These authors contributed equally to this work.

[1]College of Computing and Informatics, Drexel University, Philadelphia, PA,
United States

2]School of Information, University of South Florida, Tampa, FL, United States

3]School of Information Sciences, University of Illinois at Urbana-Champaign,
Champaign, IL, United States

4]School of Social Sciences and Technology, Technical University of Munich,
Munich, Germany

# Structural Balance in Real-World Social Networks: Incorporating Direction
and Transitivity in Measuring Partial Balance

<EMAIL_ADDRESS>Ly Dinh<EMAIL_ADDRESS>Lan Jiang
<EMAIL_ADDRESS>Jana Diesner<EMAIL_ADDRESS>*  [  [  [

###### Abstract

Structural balance theory predicts that triads in networks gravitate towards
stable configurations. The theory has been verified for undirected graphs.
Since real-world networks are often directed, we introduce a novel method for
considering both transitivity and sign consistency for evaluating partial
balance in signed digraphs. We test our approach on graphs constructed by
using different methods for identifying edge signs: natural language
processing to infer signs from underlying text data, and self-reported survey
data. Our results show that for various social contexts and edge sign
detection methods, partial balance of these digraphs are moderately high,
ranging from 61% to 96%. Our approach not only enhances the theoretical
framework of structural balance but also provides practical insights into the
stability of social networks, enabling a deeper understanding of interpersonal
and group dynamics across different communication platforms.

###### keywords:

structural balance, partial balance, signed directed networks, organizational
communication

##  1 Introduction

Social and communication networks in real world are composed of intricate and
constantly changing interactions between individuals or groups. Network
scholars have examined core principles that explain patterns of social
interactions at various levels of analysis, i.e., node level [1], dyadic level
[2], triadic level [3, 4], subgroup level [5], and the graph level [6]. In
this study, we focus on a fundamental unit of analysis, namely triads, given
that there are patterns of tie formations between three actors that cannot be
explained at any other levels of analysis [7, 8, 9].

Two fundamental theories that explain interactions at the triadic level are
structural balance [10] and transitivity [11], which both model how stable
relationships emerge in groups of three nodes. Theories of structural balance
and transitivity have been applied to examine various social processes of
opinion formation [12], cooperation in groups [13], and maintenance of social
status in organizations [14]. There is also considerable empirical evidence in
support of these theories [15, 16, 17], as well as evidence that falsifies the
premises of the theories [18, 19, 20]. What these studies have in common is
that they highlight the need to consider directionality in the evaluation of
balance for networks where relationships are directed. Forcing a directed
network to an undirected one for balance evaluation would remove additional
configurations of balance between nodes [21, 20], and exclude information
about the direction(s) of influence between nodes [22, 23]. Thus, we take into
account both the sign and direction of edges as properties to evaluate balance
for ten real-world social networks, and empirically assess whether structural
balance holds. Specifically, we assess partial balance [24] for different
types of signed directed relations, namely sentiment, morality, trust,
friendship (preference), and alliance. Examining networks with various types
of signed connections allows us to determine if equilibrium is maintained
within and among different social relationships.

Our empirical assessment considers both transitivity and structural balance as
two operating mechanisms for partial balance in signed and directed networks.
This approach stems from the original formulation of balance [10, 25] stating
that transitivity is a crucial property, which explains how directed and
signed ties are oriented in ways that are consistent with balance. We capture
both transitive and balanced configurations by leveraging the triad census
[26] to extract all transitive triples within a particular triad and calculate
the overall balance with respect to the proportions of balanced triples within
the triad. We also operationalize structural balance as partial balance, given
that complete balance in real-world networks is rare [27, 24]. To compute
partial balance, we consider the relative frequencies of positive and negative
triples within each triad, capturing the extent to which a network tends
towards balance. Altogether, we assess structural balance with transitivity
and partial balance considered across the aforementioned signed relations and
find that partial balance ratios vary, with an average balance ratio of 0.835.

In comparison with the traditional method for determining balance, which
utilizes an undirected approach, our findings indicate a more nuanced
understanding of network dynamics. By incorporating directionality, we reveal
hidden patterns of relationships and influences not captured in undirected
models. Moreover, this directed approach helps in identifying asymmetric
relationships, such as dominance or influence, which are critical in
understanding social hierarchies and power dynamics. We also observe that
different types of relationships exhibit distinct patterns of balance and
transitivity, indicating the importance of context in network analysis. Our
study contributes to a more comprehensive understanding of network theory by
demonstrating the significance of considering both sign and direction in
assessing the structural balance of real-world networks. This approach has
potential applications in various domains, such as organizational behavior,
social media analytics, and political science, where understanding the
dynamics of complex relationships is crucial.

Our research marks a substantial advancement by empirically evaluating
structural balance in a diverse range of real-world social relationships. We
examine five principal types of social relationships, offering a comprehensive
view of partial balance in various social networks. In addition, our findings
reveal a critical insight: the coexistence and concurrent impact of sign
consistency and transitivity in shaping the partial balance of networks. This
finding highlights the intricate interplay between these mechanisms across
different types of relationships, underscoring their pivotal role in network
dynamics.

##  2 Related work

The central tenet of social network analysis is to understand the structures
of relations between sets of objects, such as individuals, groups, or
organizations [28]. Heider’s structural balance theory [10] was the first
formulation of how relationships form between three individuals, or between a
pair of individuals, and their perceptions of or attitudes toward a common
object. Heider initially examined $POX$ triads, where $P$ is a focal
individual, $O$ is a second individual, and $X$ is either an individual or a
common object. He asserted that the relationship between these three entities
is ‘balanced’ if $P$ liked $O$, $O$ liked $X$ and $P$ also liked $X$. On the
other hand, ‘imbalance’ would occur if $P$ liked $O$, $O$ liked $X$, but $P$
did not like $X$. Heider’s primary claim was that balance is a state of
equilibrium, and that individuals in networks strive to move towards and
maintain that equilibrium or balance. Cartwright and Harary (1956) brought
balance theory into the context of signed and directed networks, where
relationships between pairs of nodes can be represented as either a ($+$) or
($-$). They further formalized that a triad within a network is balanced if
the product of the signs of its edges is positive. Subsequently, Holland and
Leinhart [26] introduced the transitivity model which asserted that
transitivity is an important structural property of social relationships in
general, and balance in particular. Specifically, Holland and Leinhart
proposed eight possible types of triads for balance assessment, with
transitivity considered (as shown in Figure 1).

The relationship between transitivity and structural balance is explicitly
stated in Heider’s formulation as: “three positive relations may be considered
psychologically transitive.” This means that “$P$ tends to like $X$ if
$P\mathcal{R}O$ and $O\mathcal{R}X$ are valid at the same time” ($\mathcal{R}$
represents positive relation between two nodes) ([10], p. 109). Thus, Heider
suggested that beyond cognitive consistency as a primary theoretical
mechanism, transitivity may also co-exist in a triad, though his formulation
was at the level of an individual’s perception. Holland and Leinhart [26]
generalized this conjecture further to the context of a social arrangement
with three individuals, represented as a triad of transitive relations (or a
“t-graph”). Feld and Elmore [11], followed by Doreian and Krackhardt [18]
found that differential popularity was another process, beyond cognitive
consistency highlighted in Heider, that explained the presence of transitivity
in balance. In fact, Doreian and Krackhardt [18] found that transitivity and
balance were both present, but rival processes in the Newcomb fraternity
dataset [29]. Hummon and Doreian [30] also found the complexity of the balance
process to involve multiple operating mechanisms, namely transitivity,
balance, and reciprocity. Davis, Holland, and Leinhardt [31] in their studies
of positive relations in triads found that transitivity is a pervasive
property in balanced (all positive) triads. Stix [32] also asserted that
transitivity is an important property of balance, and that “the transitivity
of any structure corresponds to its sensitivity to imbalance” (p. 447).
Altogether, prior literature shows that the relationship between transitivity
and balance is complex; it is still uncertain whether the two are
complementary or rivaling processes. All we know is that they are distinctive
but often co-existing in social relations [32, 30, 26].

Empirical studies on structural balance in real-world social networks offer
mixed evidence regarding the theory. Several studies have found evidence that
structural balance is present in email communication networks [16], sentiment
networks of individuals from the same (and different) nationalities [15], and
ten real-world benchmark network datasets [17] based on several relationship
types such as friendship, trust, alliance relations. In parallel, there are
also a number of studies that did not find evidence for the theory [33, 18].
Doreian and Mrvar [33] in their structural examination of signed and directed
relations between countries from 1946-1999 found that the networks did not
gravitate towards balance, and that balance was not achieved in the network
(in contrary to Aref and Wilson [27]’s finding that this network achieves high
degree of balance ($\gtrapprox$ 0.86). Rawlings and Friedkin [34] found that
some statements of balance hold true (e.g., a friend of a friend is a friend),
while others are violated (e.g., an enemy of an enemy is a friend) in the
context of signed directed networks of sentiment among members in the same
housing community. While evidence for structural balance has been mixed, these
studies show that real-world relationships are complex, where properties such
as signs and direction co-exist.

In this study, we revisit the longstanding literature on transitivity and
balance to calculate structural balance for networks of different signed
relations. Specifically, we calculate balance, with transitivity considered,
using Holland and Leinhart’s triad census [26] and Estrada [35] approach for
analyzing triples. The triad census includes 16 classes of MAN (Mutual,
Asymmetric, Null) triads, representing all possible combinations of directed
ties between three nodes. Specifically, each configuration contains different
combinations of edges, either mutual ($P$ likes $O$ and $O$ likes $P$),
asymmetric ($P$ likes $O$ but $O$ does not like $P$), or null ($P$ and $O$ do
not like each other). The triad census is especially useful for our
calculation, as it contains four triad types that are driven by both
transitivity and balance (as shown in Figure 1). Furthermore, the triad census
includes all the possible triples (containing directed ties) embedded in each
triad type, allowing us to conveniently extract transitive triples and exclude
intransitive (cyclic) triples in our analyses. Another reason to exclude
cyclic triples, besides the fact that they violate the transitivity rule, is
that cycles contained limited information on the process of influence among
relationships [36, 37, 38]. In the next section, we show that these four
configurations are relevant for our operationalization of balanced triads with
transitivity as a co-existing operating mechanism.

Figure 1: Triad census (include only triads which are transitive and
balanced). Example of MAN census label - 030T: Mutual=0, Asymmetric=3, Null=0,
Letter Label: T=Transitive, D=Down, U=Up

Figure 2: Balanced and imbalanced triples

##  3 Balance evaluation for signed directed networks with transitivity
considered

In this section, we present our approach to calculating partial balance in
signed directed networks with respect to transitive triples (typologies of
balanced and imbalanced transitive triples are shown in Figure 2).

###  3.1 Problem definition

Definition 1: Let $G$ be a signed digraph where $G=(D,\sigma)$. $D$ is a
digraph underlying $G$, where ${D}=(V,E)$ and sign function
$\sigma:E\rightarrow\\{+,-\\}$. A triad $T$ in $G$ is a set of three distinct
nodes with (at least) one directed edge between each two of them.

A triple $S$ in signed directed $T$ is a set of three directed edges that
starts from a vertex $V$, follows the direction of edges, and does not return
to the same vertex. In other words, $S$ is transitive and non-cyclic. Every
triple $S$ in a signed digraph $G$ must be transitive in order to be
considered for our balance analysis. Based on triad census [26], we recognize
four types of triads as transitive: (030T, 120D, 120U, and 300). These
specific triad types are deemed transitive due to the presence of only
transitive triples within them. Specifically, the 030T triad contains one
transitive triple, while the 120D and 120U triads each contain two transitive
triples, and the 300 triad contains a total of six transitive triples.

Proposition: We define $T$ a completely balanced triad if and only if every
transitive triple in $T$ is balanced. A transitive triple is positive if it
contains an even number of negative directed edges. Furthermore, we define $T$
a partially balanced triad if it contains at least one negative triple.
Finally, $T$ is completely imbalanced if every transitive triple in $T$ is
imbalanced (or negative thereof).

After calculating balance $B_{T^{(i)}}^{j}$ for each triad $j$ of type $i$, we
compute the weighted balance ratio for the set of all transitive triads of
type $i$ ($B_{T^{(i)}}$). Finally, the overall balance ratio of $G$
($B_{Avg(G)}$) is calculated by averaging the balance ratio of all triads
across a network. A signed digraph $G=(D,\sigma)$ is balanced if all triads
$T$ in $G$ are balanced.

Definition 2: We compare our method for determining structural balance with
the conventional approach, where networks are assumed to be undirected. We
consider $G^{\prime}$ an undirected signed network where
$G^{\prime}=(D^{\prime},\sigma)$, and $D^{\prime}$ is a graph underlying
$G^{\prime}$, where ${D^{\prime}}=(V^{\prime},E^{\prime})$ and sign function
$\sigma:E^{\prime}\rightarrow\\{+,-\\}$. A triad $T^{\prime}$ in $G^{\prime}$
is a set of three distinct nodes with (at least) one edge between each two of
them. The sign of $T^{\prime}$ is the product of the signs of its edges. A
$T^{\prime}$ is balanced if its sign is positive and unbalanced otherwise.
After calculating balance for each $T^{\prime}$, we compute the average
balance ratio in $G^{\prime}$. The overall balance ratio of $G^{\prime}$
($B_{Avg(G^{\prime})}$) is calculated by averaging the balance ratio of all
triads across a network. A signed graph $G^{\prime}=(D^{\prime},\sigma)$ is
balanced if all triads $T^{\prime}$ in $G^{\prime}$ are balanced.

##  4 Empirical Analysis

###  4.1 Data

Several methods such as interviews, surveys, observations, and data mining are
employed to gather social network data, enabling the construction and analysis
of social connections and interactions among entities. Moreover, real-world
communication, whether verbal or nonverbal, written or visual, encompasses a
range of explicit and implicit relationships, including those characterized by
like or dislike, trust or distrust.

For this study, we selected a wide range of datasets with different sizes
(Table 2) to better validate our proposed method for calculating balance in
directed triads. Our selected datasets (ten datasets in total) cover a wide
range of data collection techniques and portray multiple types of
relationships in the networks; namely sentiment, morality, trust, friendship
preference, and alliance. The following section provides more details about
each dataset.

Highland Tribes: The Highland tribes dataset represents an alliance structure
among tribal groups using the Gahuku-Gama system of the Eastern Central
Highlands of New Guinea. The dataset consists of two parts: (1) GAMAPOS
representing alliance (“rova”) relations, and (2) GAMANEG for antagonistic
(“hina”) relations [39, 40].

College House: The college A, B, and C datasets are constructed with respect
to the relationships between three different groups of college girls at an
Eastern college in the United States. Each group consists of approximately 20
members who lived together for at least four months. The girls in each house
were asked to provide evaluations of other girls in the same house based on a
range of behavioral characteristics. These data are then converted into
matrices of choice and rejection for each house [41].

Bitcoin: The Bitcoin OTC and the Bitcoin Alpha datasets both represent trust
between Bitcoin traders on two platforms, called Bitcoin OTC and Bitcoin
Alpha. The transactions on these platforms are anonymous. However, to minimize
fraudulent behavior and to maintain reputation, members of OTC and Alpha rate
others in a scale of -10 (total distrust) to +10 (total trust) in steps of 1
[42].

Enron: The Enron email data is a large-scale, temporal dataset which provides
real-world organizational communication data of a global, U.S. based, former
energy brokerage that went bankrupt in 2001 over a span of 3.5 years. The
communication (email) dataset of 158 employees was released in 2002 by the
FERC [43, 44]. The original dataset went through various edits and
modification over the years. In this study, we use the latest release of the
dataset from 2015 (https://www.cs.cmu.edu/~./enron/).

We disambiguated the email addresses of 558 Enron employees and converted them
to actual names of the people in the Enron dataset [43, 16]. Furthermore, we
used natural language processing techniques to analyze emails and label each
tie (email) based on their valence with respect to moral values [45, 46] and
sentiment [47, 48]. More details on text analysis methods is provided in §A.

Avocado-IT: The Avocado Research Email Collection [49] is provided by the
Linguistic Data Consortium (https://catalog.ldc.upenn.edu/LDC2015T03) and
consists of emails between 279 accounts of a defunct information technology
company referred to as “AvocadoIT”, a pseudonym assigned for anonymity. The
dataset consists of calendars, attachments, contacts, reports, and emails. In
this study, we only used the email communication for studying balance.

For the Avocado dataset, we considered emails that were sent to or from
corporate email addresses (emails ending to @avocadoit.com). Similar to the
Enron dataset, we further used text mining techniques to label each email
based on their valence with respect to moral values [45, 46] and sentiment
[47]. §A provides more details regarding the methods used for analyzing
emails.

###  4.2 Balance analysis based on edge signs and transitivity

To analyze balance in triads, we follow the steps explained in §3.1. Moreover,
we used $NetworkX$, a $Python$ library, to extract instances of four
transitive triads (030T, 120D, 120U, and 300) and analyze balance within each
triad with respect to their triples. For networks with several components, we
first selected the giant component and removed self-loops, pendants, and edges
with neutral (zero) scores from the networks to make the analysis more
efficient.

| Network Measures
---
| Bitcoin-
---
OTC
| Bitcoin-
---
Alpha
| Highland
---
Tribes
| College
---
House A
| College
---
House B
| College
---
House C
Enron | Avocado
Relational type | Trust | Trust | Alliance | Preference | Preference | Preference | Morality | Sentiment | Morality | Sentiment
# of nodes | 5881 | 3783 | 16 | 21 | 17 | 20 | 517 | 518 | 557 | 575
# of edges | 35592 | 24186 | 116 | 94 | 83 | 81 | 7605 | 7510 | 22479 | 23910
# of components | 4 | 5 | 1 | 1 | 1 | 1 | 1 | 19
Giant Component Statistics
# of nodes | 5047 | 3381 | 16 | 21 | 17 | 20 | 494 | 491 | 452 | 402
# of edges | 34759 | 23786 | 116 | 94 | 83 | 80 | 7520 | 7344 | 22953 | 23519
Transitivity | 0.05 | 0.07 | 0.53 | 0.39 | 0.40 | 0.27 | 0.21 | 0.2 | 0.5 | 0.5
Density | 0.001 | 0.002 | 0.48 | 0.22 | 0.31 | 0.21 | 0.031 | 0.03 | 0.11 | 0.14
Average Path Length | 3.44 | 3.49 | 1.54 | 2.21 | 1.95 | 2.34 | 2.53 | 2.56 | 1.7 | 1.6
Clustering Coefficient | 0.21 | 0.20 | 0.54 | 0.35 | 0.37 | 0.25 | 0.46 | 0.46 | 0.62 | 0.68
Table 1: Descriptive network measures for all networks

| Balance ratio (and counts of triads) for each triad type
---|---
Type |  | Bitcoin-
---
OTC
| Bitcoin-
---
Alpha
| Highland
---
Tribes
| College
---
House A
| College
---
House B
| College
---
House C
| Enron-
---
Morality
| Enron-
---
Sentiment
| Avocado-
---
Morality
| Avocado-
---
Sentiment
030T |  | 0.91
---
(3,706)
| 0.82
---
(974)
| 0.00
---
(0)
| 0.67
---
(27)
| 0.43
---
(21)
| 0.82
---
(17)
| 0.91
---
(4,514)
| 0.67
---
(4,238)
| 0.81
---
(8,787)
| 0.76
---
(8,577)
120D |  | 0.85
---
(2,096)
| 0.82
---
(1,142)
| 0.00
---
(0)
| 0.85
---
(13)
| 0.33
---
(6)
| 1.00
---
(8)
| 0.92
---
(2,390)
| 0.68
---
(2,384)
| 0.86
---
(14,111)
| 0.81
---
(14,276)
120U |  | 0.83
---
(2,910)
| 0.87
---
(1,780)
| 0.00
---
(0)
| 1.00
---
(13)
| 0.80
---
(15)
| 1.00
---
(3)
| 0.92
---
(3,615)
| 0.64
---
(3,513)
| 0.87
---
(26,165)
| 0.83
---
(28,615)
300 |  | 0.93
---
(13,752)
| 0.92
---
(9,894)
| 0.87
---
(68)
| 1.00
---
(4)
| 0.88
---
(4)
| 1.00
---
(1)
| 0.94
---
(3,056)
| 0.70
---
(3,056)
| 0.93
---
(124,371)
| 0.90
---
(144,865)
Average Balance (Directed) |  | 0.88
---
(22,464)
| 0.86
---
(13,790)
| 0.87
---
(68)
| 0.88
---
(57)
| 0.61
---
(46)
| 0.96
---
(29)
| 0.92
---
(13,575)
| 0.68
---
(13,191)
| 0.87
---
(173,434)
| 0.82
---
(196,333)
Table 2: Balance counts for different triad types across all networks

##  5 Results

###  5.1 Descriptive network measures

Table 2 shows the structural characteristics of the ten networks with regard
to the network size, transitivity, density, average path length, and
clustering coefficient. We focus on the giant component of each network,
meaning that pendants and isolates are excluded from analysis. The
transitivity score is the highest for the Highland tribes network (0.53) and
lowest for the Bitcoin-OTC network (0.05). High transitivity means the network
has a sizable proportion of triads for balance analysis, whereas low
transitivity suggests the network contains other configurations beyond the
triad (e.g., cycles of length $>$ 3). Density captures the occurrences of
dyads in the network, and we find a close connection between density and
transitivity scores. Highland tribes has the highest density (0.48).This
reveals the connection between the number of dyads and triads present in the
network and how they allude to the interconnectedness between nodes. Measures
of average path length and clustering coefficient reveal the topological
characteristic of a network with respect to the presence of subgroups and
communities. Most networks are found to have low average path length and mid-
range clustering coefficient, meaning that they are exhibiting some
characteristic of a small-world topology (low average path length, high
clustering coefficient), though this needs to be statistically evaluated.
Highland tribes network, Enron-morality, Avocado-morality, and Avocado-
sentiment are three networks that most clearly exhibit the small-world effect.

###  5.2 Partial balance analysis

The average partial balance ratio across all networks (shown in Table 2) is
0.835 (min = 0.61, max = 0.96, s.d = 0.10). The network with the highest
partial balance is College House C (0.96), and the network with the lowest
partial balance is College House B (0.61). We observe that high partial ratios
persist across the different network sizes, with the smallest network having a
balance ratio of 0.87 (Highland tribes), and the largest network having a
balance ratio of 0.88 (Bitcoin OTC). We also examine the proportions of triads
with respect to their balance ratios, as shown in Table 2. For five of the
networks (Bitcoin-OTC, Bitcoin-Alpha, Highland tribes, Avocado-morality,
Avocado-sentiment), triad 300 is the most frequently occurring, and often most
balanced when compared to the other triad types. The prevalence of balanced
triads 300s shows that balance is often present in situations where
individuals initiate and reciprocate the connection, specifically in the cases
of trust (Bitcoin networks) and email communication (Avocado networks). For
the other networks (College Houses A, B, C, Enron-morality, Enron-sentiment),
triad type 030T is the most frequently occurring. The frequent presence of
030T in these networks shows that some relationships are bounded by a “local
hierarchy” - in the context of Enron Sentiment, individual $P$ sends an email
with a positive valance to $O$, $O$ replies to $X$ negatively, thus $P$
replies to $X$ negatively based on the prior interaction between $O$ and $X$.
The prevalence of 030T triads in these particular networks also shows that
reciprocity is lower for certain types of relationships, specifically
friendship preferences, and email communication). On the other hand, trust
relations as observed for the Bitcoin networks have a higher frequency of
reciprocation within triads.

| Directed | Undirected
---|---|---
| \+ + + | \+ - - | \+ + - | \- - - | \+ + + | \+ - - | \+ + - | \- - -
Bitcoin- OTC | 0.85 | 0.07 | 0.07 | 0.004 | 0.71 | 0.16 | 0.12 | 0.01
Bitcoin-Alpha | 0.87 | 0.04 | 0.08 | 0.003 | 0.78 | 0.08 | 0.14 | 0.01
Highland Tribes | 0.28 | 0.59 | 0.03 | 0.10 | 0.28 | 0.59 | 0.03 | 0.10
College House A | 0.42 | 0.45 | 0.05 | 0.08 | 0.23 | 0.48 | 0.14 | 0.15
College House B | 0.20 | 0.48 | 0.15 | 0.17 | 0.08 | 0.44 | 0.23 | 0.26
College House C | 0.33 | 0.60 | 0.02 | 0.05 | 0.19 | 0.69 | 0.04 | 0.08
Enron Morality | 0.92 | 0.01 | 0.07 | 0.0003 | 0.93 | 0.00 | 0.06 | 0.00
Enron Sentiment | 0.58 | 0.11 | 0.30 | 0.01 | 0.39 | 0.39 | 0.21 | 0.01
Avocado Morality | 0.92 | 0.01 | 0.07 | 0.0002 | 0.89 | 0.01 | 0.10 | 0.00
Avocado Sentiment | 0.89 | 0.01 | 0.10 | 0.004 | 0.85 | 0.01 | 0.13 | 0.00

Table 3: Types of signed triples for all networks

###  5.3 Sign analysis of triples

All networks in our dataset contain higher proportions of positive than
negative edges in both directed and undirected networks (results shown in
Table 3). For the majority of directed networks, all-positive triples are most
prevalent, indicating the dominance of strong balance within the network. This
result is consistent with Leskovec, Huttenlocher, and Kleinberg [20]’s finding
that real-world social networks contain 70% to 87% of all-positive triples.
The prevalence of strongly-balanced triples ($+++$) also show evidence in
support of prior work by Doreian and Krackhardt [18] and Davis [31], who all
found transitivity to be a pre-condition for balance when both $P$
$\rightarrow$ $O$ and $O$ $\rightarrow$ $X$ are positive. Our findings also
show two possible outcomes for the pre-transitive condition; there is a higher
likelihood that $P$ $\rightarrow$ $X$ will be positive, but there is also a
possibility for $P$ $\rightarrow$ $O$ to be negative. For the remaining four
networks (Highland Tribes, College House A, B, and C), there is a higher
frequency of ($+--$) triples, showing empirical evidence of signed
transitivity [18] in several real-world networks. These four networks are all
noticeably smaller than other networks, which may suggest that the prominence
of ($+--$) triples may be specific to these social contexts, as opposed to
being generalizable to other networks with the same relational type.

For the imbalanced triples, we find that in six out of ten networks, ($++-$)
triples are more prevalent than ($---$) triples. This finding is in contrast
with [50]’s results, which found that ($++-$) triples are the least prominent
in real-world networks, followed by ($---$) triples. Leskovec et al. [20] also
empirically observed the rarity of ($++-$) triples in real networks. However,
in the case of directed networks where edges are defined in terms of social
status theory, ($++-$) triples are over-represented beyond what is expected by
chance. Easley and Kleinberg [51] also suggested that these two types of
tensions are unique, and suggest that ($++-$) triples are more likely to be
resolved over time. In the context of College House friendship preference, for
example, where $A$ likes $B$, $B$ likes $C$, but $A$ does not like $C$. Person
$A$ may be more likely to reconcile this tension by liking $C$. However, if
there are all negative relations ($---$) between $A$, $B$, and $C$, Davis [52]
argues that it is less likely for two enemies to like each other. Instead, the
three enemies are likely to just stay enemies, and not be involved in the
triadic relationship at all. In the remaining four networks (Highland tribes,
College Houses A, B, and C), we find slightly higher occurrences of one-
negative than all-negative triads. This finding signals the difficulty for the
triads to reconcile the tensions, and thus may be less likely to become more
balanced over time. This is a particularly interesting finding for the case of
Highland Tribes, showing that the alliance structure is rigid and less likely
to change over time. In the context of three College House friendship
networks, friendship preference is not likely to change, despite the fact that
the change could reduce tension for the network as a whole.

Figure 3: Case study: College House C network as a directed network (left) and
undirected network (right)

Szell et al.’s [50] study brings out an important distinction between strong
and weak structural balance. Weak structural balance [52] posits that the
($++-$) triple is the only configuration that is (weakly) imbalanced, while
the ($---$) is considered (weakly) balanced because empirical observations
have found that “an enemy of one’s enemy can also be an enemy.” Additionally,
the all-negative triple is more likely to become balanced, than the one-
negative triple, because it is more likely that two enemies will become
friends because they have a mutual enemy (“an enemy of my enemy is my
friend”). However, as alluded to in Easley and Kleinberg [51], the presence of
the one-negative triple ($++-$) can also signal the possibility of resolving
tension among a friend of a friend. For instance, in the Enron email network,
we observe that $P$ may send a positive email to $X$ in the future, driven by
the positive connection that $P$ has with $O$, and knowing that $O$ sent a
positive email to $X$. Differing findings from the literature suggest that the
sign configurations of triples are complex, especially when directed edges are
considered [20]. In our analysis of these signed and directed triples, we find
that weak balance is not prominent in relationships that are transitive. In
other words, “an enemy of an enemy is an enemy” is not as likely to happen as
a “a friend of a friend is an enemy” connection. It would be worthwhile to
examine the temporal changes of these two triple types over time, and whether
it is more likely that an enemy would become a friend in the ($---$) case, or
that (1) an enemy would become a friend, or (2) a friend becomes an enemy in
the ($++-$) triple. From this perspective, there are two possibilities for the
($++-$) to obtain balance, as opposed to ($---$) triple, which can only be
resolved if an edge becomes positive.

Ultimately, the differences in these results show that (1) the composition of
triples differs across different real-world contexts and relations, and (2)
across directed and undirected realizations of the networks.

Figure 4: Case study: College House B network as a directed network (left) and
undirected network (right). Edge Legends: all indicates the whole network, and
V2 egonet indicates the egonet of node V2.

###  5.4 Utility of directed networks

The overall trend observed across all networks in our study is that the
undirected realizations inflate the count of edges, triples, and consequently,
the triads available for analysis. This inflation occurs because undirected
networks do not require transitivity, meaning that not all connections must
follow a certain pattern of relationships, making more configurations
available for analysis. For instance, in College House A and College House C
networks, the undirected realizations included a notable number of extra
triples which subsequently influenced the balance score. In particular, the
undirected realization of College House A contains 48 extra triples than the
directed version, in which 24 are balanced in sign products, and 24 are
imbalanced. Hence, the resulting balance score is influenced and dropped to
0.72 as opposed to 0.88 in the directed network. The College House C network
(shown in Figure 3 exhibits similar patterns as the College House A network in
terms of the inflated number of triples present in the undirected network.
There are 19 additional triples not presented in the directed network, 16 of
which are balanced, and 3 are imbalanced. For instance, there is a triad that
exists between nodes V4, V9, and V10 in the undirected network but not the
directed network because this particular triad violates the transitivity
assumption. In particular, this triad is a cycle, whereby V4 $\rightarrow$ V9
(+ sign), V9 $\rightarrow$ V10 (-), and V10 $\rightarrow$ V4 (+). Similar
situation occurred with triad V3, V6, V12 where the triad contains one cycle
(V3 $\rightarrow$ V6 (-), V6 $\rightarrow$ V12 (-), V12 $\rightarrow$ V3 (+)).
This particular triad, however, also contains one triad that satisfies both
transitivity and balance conditions (V3 $\rightarrow$ V6 (-), V6 $\rightarrow$
V12 (-), V3 $\rightarrow$ V12 (+)), but the presence of a violating cycle
influences the overall transitivity of the triad.

Undirected realizations also remove edges where there is a discrepancy in the
signs of the incoming and outgoing edges between two nodes, as shown in Figure
5). In the College House B network, we observe the loss of three particular
edges that in turn influence the final balance score of the undirected
realization. As shown in Figure 4, two edges V2 $\rightarrow$ V15 (-), and V2
$\rightarrow$ V17 (-) from the V2 egonet are present in the directed network
but not the undirected network, as they were canceled out due to the mismatch
in edge signs. The directed realizations hence manage to preserve at least an
edge between V2 and V15, for instance, because they are involved in multiple
triadic configurations that satisfy the transitivity and balance assumptions
in the directed network (e.g., V15, V11, V2 as 120D triad; V3, V15, V2 as 120U
triad). There is an exception to V2 $\rightarrow$ V7 edge being present in the
undirected network but not the directed network. This is because, in the
directed network, this edge is not part of any triads that meet the
transitivity requirement. On the other hand, the undirected network keeps this
edge, but this is problematic because there is only an edge between V2
$\rightarrow$ V7, but not V7 $\rightarrow$ V2.

The Highland Tribes network stands out as the only instance where the directed
and undirected networks share identical triad composition and overall balance
scores. This is because the network is completely connected, hence there is a
maximal number of triads that can exist in both network realizations. Such a
network structure is, however, rare in real-world networks. Our case study of
College Houses A, B, and C demonstrates that even the slightest discrepancy in
the edge configurations and their signs when the network is switched from
directed to undirected network can significantly impact the network’s
structure and overall balance score.

| Directed_Partial | Directed_Non-Partial | Undirected
---|---|---|---
| BR | # BT | # IT | BR | # BT | # IT | BR | # BT | # IT
College House A | 0.88 | 46 | 11 | 0.8 | 46 | 11 | 0.74 | 29 | 10
College House B | 0.61 | 29 | 17 | 0.52 | 24 | 22 | 0.44 | 14 | 18
College House C | 0.96 | 26 | 3 | 0.89 | 26 | 3 | 0.92 | 25 | 2
Bitcoin-Alpha | 0.86 | 13069 | 721 | 0.84 | 11649 | 2141 | 0.84 | 2930 | 546
Bitcoin-OTC | 0.88 | 21192 | 1272 | 0.86 | 19447 | 2969 | 0.91 | 3814 | 360
Highland Tribes_sym | 0.87 | 59 | 9 | 0.87 | 59 | 9 | 0.92 | 24 | 2
Enron sentiment | 0.68 | 10444 | 2747 | 0.53 | 6045 | 7165 | 0.69 | 2104 | 949
Avocado_sentiment | 0.82 | 185832 | 10501 | 0.46 | 105222 | 91505 | 0.85 | 8846 | 1495
Avocado_morality | 0.87 | 165639 | 7795 | 0.63 | 117515 | 67781 | 0.84 | 8060 | 1544
Enron morality | 0.92 | 12856 | 719 | 0.79 | 10908 | 2813 | 0.93 | 2831 | 226

Table 4: Comparing Partial and Non-Partial Balance. BR: Balance ratio, BT:
balance triangles, IT: imbalance triangles

Figure 5: Examining differences in network analysis outcomes with and without
considering directionality in real-world data

##  6 Discussion and Conclusion

In this paper, we employ an empirical framework for calculating balance in
signed, transitive digraphs, which was essential to appropriately model and
study balance in real-world communication networks and other networks where
links might be asymmetric. We applied this framework to ten social networks
with five different relational types (trust, alliance, friendship preference,
morality, and sentiment). Our rationale for testing our approach on different
networks was to determine whether mechanisms of structural balance and
transitivity hold true across diverse social contexts. Moreover, prior
research has mainly examined structural balance in signed and undirected
graphs. Our study provided an actionable solution to measure structural
balance in signed digraphs, using principles of transitivity to evaluate the
directionality between edges.

Overall, our findings showed that the amount to which a network was balanced
was strongly impacted by choices of measuring social relations. When the
direction of edges was taken into account, along with sign consistency, we
expected that the overall balance ratio may be different than findings where
only sign consistency was considered [16, 20]. Choices of relational type may
also affect the overall partial balance. Our findings showed that each
relational type captured a different characteristic of a network, as reflected
in the different balance ratios across dimensions of social trust, friendship
preference, alliance relations, morality, and sentiment of email (and online)
communications. While partial balance ratios for all relational types were
about 0.82 and above (balance higher than imbalance), we found the
compositions of triples that make up balanced triads are different depending
on the type of relationship.

The patterns of partial balance that we discovered across the ten real-world
networks offer implications for existing communication and social networks
literature. First, we found that partial balance with transitivity considered
provides nuances into the co-existence of hierarchy, influence, and
reciprocity as mechanisms that influence balance. For instance, if we consider
an undirected and balanced triad ($ABC$) where we know $A$-$B$ is positive,
$B$-$C$ is positive, and $C$-$A$ is positive, the lack of edge direction does
not explain the potential source(s) of balance. Whereas if we are presented
with additional information, such as $A$ $\rightarrow$ $B$ is positive, $B$
$\rightarrow$ $C$ is positive, $A$ $\rightarrow$ $C$ is positive, we observe
that $A$ likely has a positive edge towards $C$ because $B$ has a positive
edge for $C$, hence the process of social influence is captured. Secondly, our
partial balance and sign analyses reveal that triad type 300 is most prominent
in all empirical networks, and specifically this triad 300 is often composed
of all-positive triples. This finding suggests that the processes of
transitivity and balance are salient in all-positive triads, and this is
consistent with Davis, Holland, and Leinhardt [31]’s observations for real-
world small groups. We also found the occurrences of the one-negative triples
to be higher than the all-negative triples, which is in contrast to several
prior studies ([50, 20]). Further investigations are needed on the presence of
the one-negative triad, as the likelihood of resolving the tension is likely
to be unique across different relational types.

Our findings are subject to several limitations. First of all, we only
analyzed network structure and partial balance for the giant component of the
networks in our sample. Our rationale is that only closed triads are
considered for balance and transitivity analysis. Secondly, we consider
partial balance only as a local property, at the triad level, of the network
that has a global impact on the network. In future work, we will extend the
measurement of partial balance and transitivity for longer cycles and examine
if there are areas of tension detected in the network at other levels of
analysis beyond the triad.

## Additional information

The authors declare no competing interests.

##  Appendix A Analysis of Emails using NLP

For the two email datasets, we used NLP methods to extract two types of edge
signs from text data exchanged between the nodes (authors) that form a dyad:
moral values (virtue or vice) and sentiment (positive or negative) [53, 54].
This approach is based on the premise that people’s language use can reflect
their cultural, economic, and ideological backgrounds [55].

To capture moral values in our email data sets, we leveraged the Moral
Foundations Theory (MFT) [45, 56]. MFT can help capture people’s spontaneous
reactions and categorize human behavior into five basic principles
(fairness/cheating, care/harm, authority/subversion, loyalty/betrayal, and
purity/degradation) that are characterized by opposing values (virtues and
vices). To extract moral values from our email data, we used an enhanced
version of MFD111https://doi.org/10.13012/B2IDB-3805242_V1.1 as developed,
introduced, and validated in [46, 57]. The second NLP method we used for
labeling links with signs is sentiment analysis; a technique commonly used for
understanding people’s opinions and affective state [58]. To identify the
sentiment of each email, we leverage a domain-adapted version of Subjectivity
Lexicon, a widely used sentiment lexicon by Wiebe and Riloff [47].

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