# Impact of Higher-Order Structures in Power Grids’ Graph on Line Outage
Distribution Factor

Nafis Sadik∗, and Mohammad Rasoul Narimani† This work was supported by NSF
under Award Number 2308498.${\ast}$: College of Engineering and Computer
Science, Arkansas State University<EMAIL_ADDRESS>Department
of Electrical and Computer Engineering, California State University Northridge
(CSUN<EMAIL_ADDRESS>
###### Abstract

Power systems often include a specific set of lines that are crucial for the
regular operations of the grid. Identifying the reasons behind the criticality
of these lines is an important challenge in power system studies. When a line
fails, the line outage distribution factor (LODF) quantifies the changes in
power flow on the remaining lines. This paper proposes a network analysis from
a local structural perspective to investigate the impact of local structural
patterns in the underlying graph of power systems on the LODF of individual
lines. In particular, we focus on graphlet analysis to determine the local
structural properties of each line. This research analyzes potential
connections between specific graphlets and the most critical lines based on
their LODF. In this regard, we investigate $N-1$ and $N-2$ contingency
analysis for various test cases and identifies the lines that have the
greatest impact on the LODFs of other lines. We then determine which subgraphs
contain the most significant lines. Our findings reveal that the most critical
lines often belong to subgraphs with a less meshed but more radial structure.
These findings are further validated through various test cases. Particularly,
it is observed that networks with a higher percentage of ring or meshed
subgraphs on their most important line (based on LODF) experience a lower LODF
when that critical line is subject to an outage. Additionally, we investigate
how the LODF of the most critical line varies among different test cases and
examine the subgraph characteristics of those critical lines.

###### Index Terms:

DC load flow, Line outage distribution factor, Graph theory, Graphlets.

##  I Introduction

The power system is comprised of interconnected networks, and the loss of even
one line can potentially lead to a major blackout. As a result, it is crucial
to identify lines that are critical to the network and have the potential to
trigger a cascading failure and even a blackout. For that reason, remedial
actions such as network redesign or transmission line capacity expansion are
required[1]. Power system contingency analysis has lately resurfaced as the
traditional grid is transitioning into a smart grid. Particularly as the grid
is transitioning into a cyber-physical system, grid collapse can be caused by
the collapse of a few essential lines[2, 3].

Numerous efforts have been expended on identifying critical lines in power
systems[4, 5]. Complex network theory has been leveraged by various studies to
identify critical elements in power systems. Betweenness centrality, a measure
based on the shortest path-based between different nodes in a graph, is used
to identify the most vulnerable lines in power system [4]. Other graph
theoretical methods, such as the clustering algorithm, have also been employed
to identify influential buses on the network[6]. Those research approaches
focuses on the topology of the network without considering the physics that
govern the network, i.e. power flow equations, line outage distribution
factor, etc. Line outage distribution factor(LODF) is an indicator that
quantifies the redistribution of power flow amongst surrounding lines when a
line outage happens [7]. It is a linear sensitivity factor that utilizes DC
power flow [7]. LODF plays a vital role in optimal power flow (OPF) studies
[8, 9, 10, 11, 12] by quantifying the impact of line outages on power flow
distribution in the network. It assists in improving system reliability,
congestion management, and optimal transmission planning by providing valuable
insights into the network’s behavior under various operating conditions.

A combination of the line outage distribution factor and complex network
analysis typically yields a better contingency analysis. To extract data from
various levels of contingency, the group betweenness centrality concept can be
used with the line outage distribution factor [13, 14]. Nevertheless, to
better analyze contingency analysis, LODFs among lines can be analyzed with
the local structural properties of those lines. The literature suggests that
there have been very few works that relate subgraph properties with power grid
contingency analysis.

Subgraph analysis is an approach that highlights the local structural
properties of a network. Graphlets are induced connected subgraph. A subgraph
pattern is called a motif when it is found too frequently in a network
compared to random networks[15]. Motif analysis is also leveraged in power
system contingency analysis studies to investigate the impact of the local
structure of a network on power system characteristics. For instance, it has
been discovered that robust and fragile electrical networks exhibit varying
degrees of deterioration of motifs under attack[16]. However, network motifs
represents more global properties rather than local properties of a network.
In particular, it is a measure of how many times a subgraph is found on the
whole network. This paper leveraged the local properties of power system’s
graph, in particular how many times a graphlet can be found in a line, in
analyzing the LODF measure in power system.

The LODF is used in this paper to determine the most critical lines in the
network. In particular, $N-1$ contingency is applied to the test cases to find
the loss of which line yields to higher LODF values. The $N-2$ contingency is
also investigated in this paper. Because of the computational complexity of
$N-2$ contingency analysis for larger test cases, this study only considers
smaller test cases for analysis. Graphlet analysis of those lines is performed
in order to find a relationship between lines with high LODFs and the local
structure of network. It is explored whether any critical lines contain
certain sorts of graphlets. This will expose any vulnerable network components
that have the potential to cause a cascading failure. That will eventually
provide power system operators with crucial recommendations to improve power
system vulnerability in the event of a cascading failure. The following is how
this document is structured. Section II reviews LODF metric. Section III
explains how power networks can be interpreted as complex network. Section IV
presents our methodology. Section V shows the results and Section VI concludes
the paper.

##  II Overview of Line Outage Distribution Factor

This section explains the concept of the line outage distribution factors
(LODFs) that is employed in this study. To compute LODFs, we first need to
consider Injection Shift Factors (ISF). The injection shift factor is a linear
sensitivity factor which measures the sensitivities of active power line flows
due to injection of active power bus injections [17]. Assume a power system
consists of $N$ buses and $L$ lines. The injection shift factor
$\iota^{k}_{lm}$ of line $lm$ represents the sensitivity of the change in
power flow in line $lm$ due to a change in real power injection at bus $k\in
N$. The injection shift factor matrix of a power system can be computed by DC
power flow.

$\displaystyle\underline{\iota}={\underline{B}^{-1}_{n}\underline{A}\underline{B_{l}}}$
(1)

Here, $\underline{B}^{-1}_{n}\in\mathbb{R}^{L\times L}$ represents the
original branch susceptance matrix, $\underline{A}\in\mathbb{R}^{L\times N}$
denotes reduced incidence matrix, and $\underline{B_{l}}\in\mathbb{R}^{L\times
N}$ represents reduced nodal susceptance matrix. Power transmission
distribution factor (PTDF) for line $lm$ for incident $z$ of real power change
$\Delta P$ in bus $c$ to bus $d$ can be defined as,

$\displaystyle{\tau^{z}_{lm}}={\iota^{c}_{lm}-\iota^{d}_{lm}}$ (2)

Consequently, change of power flow in line $lm$ for $\Delta P$ power flow
change in bus $c$ to bus $d$ can be defined in terms of power as,

$\displaystyle{\Delta\omega^{z}_{l_{m}}}={\tau^{z}_{l_{m}}}\Delta P$ (3)

PTDF matrix of the entire system can be obtained using (1).

$\displaystyle{\underline{\tau}_{L}}={\underline{\iota A^{T}}}$ (4)

For outage of line $lo$, LODF of line $lm$ can be considered as
${\phi}^{lo}_{lm}$. If pre-outage real power flow in line $lo$ is
$\omega_{lo}$, change of power flow ${\Delta\omega^{z(lo)}_{lm}}$ in line $lm$
after the outage of line $lo$ can be written in terms of LODF as follows [18]

$\displaystyle{\Delta\omega^{z(lo)}_{lm}={\phi}^{lo}_{lm}{\omega_{lo}}}$ (5)

Using equation (3), equation (5) can be extended as.

$\displaystyle{\phi}^{lo}_{lm}={\frac{{\tau^{z(lo)}_{lm}}}{1-\tau^{z(lo)}_{lo}}}$
(6)

In this paper, we first determine the most critical lines using LODFs. For a
single test case consisting of $N$ buses and $L$ lines, the LODF matrix will
be of size $L\times L$. In particular, for a single line outage, there will be
$L-1$ number of LODFs. The maximum LODF value is recorded among those $L-1$
LODFs. We conduct the $N-1$ contingency analysis for all lines in the network
to determine maximum LODF for all contingencies. After finding LODF matrix, a
limited number of lines with highest and lowest LODF are investigated from
local structure point of view. A similar approach is taken for $N-2$
contingency. Following the outage of a line, it is determined that the outage
of whichever line follows, in conjunction with the first, stresses the network
the most. As a result, some combinations of lines for $N-2$ contingency can be
obtained. Each combination gives a maximum LODF for an $N-2$ contingency. In
this process, LODF helps to determine the most critical lines of a network in
both $N-1$ and $N-2$ contingencies. In following sections, the local network
characteristics of those critical lines are investigated.

##  III Power Grid as a Complex Network

Network Graphlets are building blocks of networks. Analysis of Graphlets is
found to be an indispensable tool for understanding local network structure,
in contrast to measures based on node degree distribution and its functions
that primarily address a global network topology [16]. Examining the local
structure of any grid requires power grid to be modeled as a complex network.
The grid can be represented as an undirected graph $G(V,E)$, where $V$
represents the grid’s buses and $E$ represents the grid’s lines. A graph
$G^{\prime}(V^{\prime},E^{\prime})$ is a subgraph of graph ${G}$,
${G^{\prime}}\subseteq{G}$, only when ${V^{\prime}}\subseteq{V}$ and
${E^{\prime}}\subseteq{E}$. The subgraph $G^{\prime}$ can be called an induced
subgraph of $G$ if $E^{\prime}$ contains all edges $e_{uv}\in E$ such that
$u,v\in V^{\prime}$. Graphs $G^{\prime}$ and $G^{\prime\prime}$ can be called
isomorphic if there exists a bijection $h:V^{\prime}\rightarrow
V^{\prime\prime}$ such that any two adjacent nodes $u,v\in V^{\prime}$ of
$G^{\prime}$ are also adjacent in $G^{\prime\prime}$ after the mapping occurs.
If $G_{k}=(V_{k},E_{k})$ is a k-node subgraph of G and if there exists an
isomorphism between $G_{k}$ and $G^{\prime}$ where $G^{\prime}\in G$, then
there is an occurrence of $G_{k}$ in $G$. In this paper, we only investigate
4-node subgraphs. To this end, we use the FANMOD algorithm, represented in
Algorithm 1, to enumerate subgraphs [19, 20]. In this algorithm, first all
4-node subgraphs are enumerated and then classified. Here, the notation of
$N(\\{v\\})$ means neighbours of nodes $v\in V$. $V_{subgraph}$ means set of
four node connected vertices. Furthermore, if vertex $w\in V/V_{subgraph}$ ,
then $N_{excl}(w,V_{subgraph})$ denotes the exclusive neighbourhood of vertex
$w$, in which no vertices belong to either $V_{subgraph}$ or the neighbourhood
of $V_{subgraph}$. After enumerating subgraphs, we apply subgraph isomorphism
to classify all 4-node subgraphs.

Algorithm 1 $EnumerateSubgraph$

Input: Graph $G(V,E)$

Output: All 4-nodes subgraph in graph $G$

for each vertex $v\in V$ do

$V_{extension}\leftarrow{u\in N({\\{v\\}}):u>v}$

Call $ExtendSubgraph$(${\\{v\\}},V_{extension},v$)

end for

$ExtendSubgraph$(${\\{v\\}},V_{extension},v$)

if $|V_{subgraph}|=4$ then

Output: $G[V_{subgraph}]$

end if

while $V_{extension}\neq\emptyset$ do

Remove an arbitrary chosen vertex w from $V_{extension}$

$V^{\prime}_{\text{extension}}\leftarrow V_{\text{extension}}\cup\\{u\in
N_{\text{excl}}(w,V_{\text{subgraph}}):u>v\\}$

Call ExtendSubgraph$(V_{subgraph}\cup{\\{w\\}},V^{\prime}_{extension},v$)

end while

In general, there are six types of four-node connected subgraphs that can be
found in any power grid network. Fig. 1 shows four-node subgraphs which are
labeled as M1-M6. Subgraphs M1, M2 and M3 represent radial structures, while
M4, M5 and M6 symbolize ring and mesh structures. Different types of subgraph
analysis have been done in the literature. There have been much research on
graphlet analysis in a network. However, the impact of higher-order structures
such as graphlets in power grids’ network is less studied. In this study, our
methodology focuses on how many times a graphlet can be found on a particular
line.

Figure 1: Different type of four node graphlets. In a power grid network it is
very usual to find subgraphs M1 and M2

##  IV Methodology

###  IV-A N-1 Contingency

The LODF calculates the redistribution of a line’s load among the other lines.
However, we observe that for outages of a few particular lines, the LODF of
some other lines are larger. This essentially means that outages on those
critical lines are putting a strain on the network. In other words, for the
outage of those critical lines, LODF impacts on other lines are significantly
higher. In this section, the graphlet characteristics of those critical lines
is observed. In particular, we observe how many times a particular graphlet
can be found on a line. For $N-1$ contingency, the most critical lines of the
IEEE 30-bus network based on LODF impact are illustrated by red lines in Fig
2. Based on the LODF analysis, it can be said that outage on those lines
strain the network most. From a graph theory point of view, graphlet analysis
can be used to determine which 4-node graphlets those critical lines are
belong to. As a consequence, the local structural properties of those critical
lines will be revealed. In this connection, we determine how many times a
particular graphlet is incident on a line. Table I and Table II represent
graphlet characteristics of most and least critical lines based on $N-1$ LODF
analysis for IEEE 30-bus network. When two tables are compared, it is clear
that the local structural characteristics of the two distinct type of lines
differ noticeably. In the table of most critical lines, there are no M3-M6
graphlets in any lines. On the contrary, there is an M4 ring graphlet present
for all lines in the table of least critical lines. In other words, if a line
is part of a ring graphlet M4, its outage is unlikely to have a significant
influence on the network. In addition to that, from figure 2, it is also clear
that there are no critical lines in the graph region where the M4 ring
graphlet exists. As a result, the area is considerably stabilized by the
presence of a ring structure. On the contrary, most of the critical lines are
part of some radial structured graphlet such as M1 and M2. In the following
section, we will observe the local structural properties of lines with the
highest LODF impact among different networks.

Figure 2: IEEE 30-bus system with red lines marked as the most critical lines based on $N-1$ contingency LODF analysis. Lines 15-23 and 21-22 represents $N-2$ contingency combination of lines that produce highest LODF impact. Table I: graphlet characteristics of most critical lines in IEEE 30-bus test system From | To | M1 | M2 | M3 | M4 | M5 | M6
---|---|---|---|---|---|---|---
14 | 15 | 1 | 2 | 5 | 0 | 0 | 0
15 | 23 | 2 | 9 | 1 | 0 | 0 | 0
16 | 17 | 0 | 10 | 0 | 0 | 0 | 0
29 | 30 | 0 | 0 | 2 | 0 | 0 | 0
10 | 21 | 5 | 8 | 6 | 0 | 0 | 0
10 | 17 | 3 | 2 | 1 | 0 | 0 | 0
19 | 20 | 0 | 7 | 0 | 0 | 0 | 0
18 | 19 | 0 | 5 | 0 | 0 | 0 | 0
6 | 28 | 4 | 14 | 7 | 0 | 0 | 0
6 | 8 | 4 | 8 | 7 | 0 | 0 | 0
Table II: graphlet characteristics of least critical lines in IEEE 30-bus test system From | To | M1 | M2 | M3 | M4 | M5 | M6
---|---|---|---|---|---|---|---
5 | 7 | 0 | 7 | 0 | 1 | 0 | 0
2 | 4 | 2 | 8 | 9 | 1 | 0 | 0
6 | 7 | 12 | 14 | 3 | 1 | 0 | 0
2 | 5 | 2 | 9 | 1 | 1 | 0 | 0
2 | 6 | 9 | 16 | 11 | 1 | 0 | 0
3 | 4 | 2 | 12 | 1 | 1 | 0 | 0
1 | 3 | 0 | 4 | 0 | 1 | 0 | 0
1 | 2 | 2 | 9 | 1 | 1 | 0 | 0

###  IV-B N-2 Contingency

In $N-2$ contingency, we observe network performance in the case of an outage
of two lines. In $N-1$ contingency for a network of $L$ lines, we have a LODF
matrix of $L\times\ L$ size. However, for $N-2$ contingency, we have $L$
number of $L-1$ sized matrices. From each of those $L-1$ sized matrices, we
choose the row with highest LODF value. Consequently, we get $L$ combination
of two lines which actually represents lines from $N-2$ contingency. In
particular, that means we first simulate the outage of a single line, then we
observe the outage of which other line strains the network most. In that way,
we get $L$ combination of $N-2$ contingency lines and then we determine which
single combination yields the maximum LODF impact. In Fig 2, line combinations
$15-23$ and $21-22$ has the highest LODF among other combinations. If we look
at the network from a structural point of view, we observe that those two
lines do not belong to any ring or meshed graphlets. In the next section, we
present the results of highest LODF impact line combination from $N-2$
contingency among different networks.

##  V Results

In the previous section, we observed graphlet characteristics of the IEEE
30-bus network. Here, we apply the proposed approach on
“pglib$\\_$793$\\_$goc” and “pglib$\\_$1354$\\_$pegase” as large networks from
the PGLib-OPF benchmark library [21]. For “pglib$\\_$793$\\_$goc” test case,
we observed that it has no M3, M4, M5, M6 graphlets in its most critical
lines, while in it’s least critical lines, M4 graphlets can be found.
Similarly, for the case “pglib$\\_$1354$\\_$pegase”, we noticed that, there is
a very small number of M3, M4, M5, M6 graphlets in it’s most critical lines.
In contrast, most of the least critical lines in this test case are belong to
M3 and M4 graphlets.

Table III: Graphlet characteristics of Most and least critical lines of “pglib$\\_$793$\\_$goc” test system Most Critical Edges |  | Least Critical Edges |
---|---|---|---
M1 | M2 | M3 | M4 | M5 | M6 | M1 | M2 | M3 | M4 | M5 | M6
1 | 9 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 0
4 | 11 | 0 | 0 | 0 | 0 | 3 | 11 | 0 | 1 | 0 | 0
3 | 11 | 0 | 0 | 0 | 0 | 6 | 14 | 0 | 1 | 0 | 0
1 | 3 | 0 | 0 | 0 | 0 | 3 | 6 | 0 | 1 | 0 | 0
1 | 7 | 0 | 0 | 0 | 0 | 3 | 8 | 0 | 1 | 0 | 0
2 | 15 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 1 | 0 | 0
7 | 19 | 0 | 0 | 0 | 0 | 2 | 9 | 0 | 1 | 0 | 0
1 | 6 | 0 | 0 | 0 | 0 | 12 | 29 | 0 | 1 | 0 | 0
6 | 12 | 0 | 0 | 0 | 0 | 2 | 12 | 0 | 2 | 0 | 0
1 | 11 | 0 | 0 | 0 | 0 | 1 | 5 | 0 | 1 | 0 | 0
Table IV: Graphlet characteristics of Most and least critical lines of “pglib$\\_$1354$\\_$pegase” test system Most Critical Edges |  | Least Critical Edges |
---|---|---|---
M1 | M2 | M3 | M4 | M5 | M6 | M1 | M2 | M3 | M4 | M5 | M6
36 | 9 | 1 | 0 | 0 | 0 | 3 | 13 | 0 | 1 | 0 | 0
1 | 3 | 0 | 0 | 0 | 0 | 3 | 11 | 0 | 1 | 0 | 0
3 | 6 | 0 | 0 | 0 | 0 | 9 | 16 | 1 | 2 | 0 | 0
3 | 6 | 0 | 0 | 0 | 0 | 10 | 20 | 1 | 2 | 0 | 0
6 | 22 | 0 | 0 | 0 | 0 | 10 | 20 | 1 | 2 | 0 | 0
13 | 38 | 0 | 0 | 0 | 0 | 26 | 29 | 2 | 1 | 0 | 0
5 | 14 | 0 | 0 | 0 | 0 | 3 | 9 | 1 | 1 | 0 | 0
10 | 18 | 0 | 0 | 0 | 0 | 52 | 28 | 16 | 1 | 0 | 0
11 | 27 | 0 | 0 | 0 | 0 | 4 | 27 | 0 | 2 | 0 | 0
2 | 15 | 0 | 0 | 0 | 0 | 7 | 26 | 0 | 1 | 0 | 0

In order to generalize the finding, we investigate the proposed approach on
different test cases with various sizes. To this end, we first determined the
most critical lines of different test cases using $N-1$ LODF analysis. We then
determined which graphlets they feature and the quantity of each graphlet
present. In particular, total graphlet count of the most important line is
determined. Then it is measured which graphlets account for what percentage of
the entire graphlet count. This gives us the graphlet characteristics of that
line. By taking the percentage of graphlets in a line, we can compare the
results across different networks. The maximum LODF impact of most critical
line of a network varies from network to network. Our objective is to relate
the maximum LODF for a network possible and subsequent graphlet
characteristics of the line for the outage of which the network experiences
the maximum LODF. We present our findings in Fig. 3, where the size of the
bubble represents each graphlet percentage of the most important line of a
network. Also, the color of the bubbles changes with the maximum LODF of that
most important line. From Fig. 3, it is evident that, cases that contain very
high LODF impact line, contain no or very small percentages of ring or meshed
graphlets such as M3, M4, M5 and M6 in their most critical line. Conversely,
there are relatively higher percentages of ring or meshed graphlets in cases
that have relatively low LODF impact lines.

Figure 3: Relation between graphlet characteristics of different networks and
their maximum $N-1$ LODF impacts. Size of the bubble represents graphlet
percentages and color of the bubbles changes with Maximum $N-1$ LODF impact
possible in each cases

From Section IV, we get $L$ combination of two $N-2$ contingency lines for a
network consisted of $L$ lines. The first line corresponds to $N-1$
contingency and the second line corresponds to $N-2$ contingency. For most
combinations, we can observe certain $N-2$ lines recurring. However, we focus
on the $N-2$ contingency line for the combination that produces the maximum
LODF impact. Fig. 4 shows the graphlet analysis of lines for $N-2$
contingency. From Fig. 4 it can be noticed that, cases with very high LODF
impact lines have less percentage of ring or meshed graphlets such as M3, M4,
M5 and M6 in their most critical lines. This results can be easily extended to
higher order contingency analysis and help power system operator to focus on
specific part of the network for finding critical lines. Finding graphlets for
larger test cases can be done offline which helps system operators to quickly
determine potential critical lines in the system.

Figure 4: Relation between graphlet characteristics of different networks and
their maximum $N-2$ LODF impacts. Size of the bubble represents graphlet
percentages and color of the bubbles changes with Maximum $N-2$ LODF impact
possible in each cases

##  VI Conclusion

Graph theoretical analysis has been extensively used to investigate power grid
vulnerabilities. However, graphlet analysis in tandem with grid
vulnerabilities is less studied. To address this gap, this paper first
determines the most important line for $N-1$ contingency based on the LODF
analysis. Then, it focuses on the graphlet characteristics of that critical
line. This strategy is applied to various test cases and graphlet
characteristics of the most important lines of those networks are
investigated. This provides a comparison of the highest LODF possible for a
network and graphlet features of the line for outages in which that LODF
occurs. It is revealed that networks with a higher percentage of ring or
meshed graphlets such as M3, M4, M5, M6 on their most important line
experience less LODF for outage of that line. On the contrary, cases with very
high LODF for outage on its most important line have very little to no
percentages of ring or meshed graphlets in that line. This study also expands
to, $N-2$ contingency where the same type of scenario is observed, which
actually emphasizes the importance of ring or meshed graphlets in power grid
networks for more resilient grids. The proposed graphlet analysis helps the
power system operator to quickly determine possible candidates for critical
lines.

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obtained through a vertex enumeration and are linear. If some $\vec{g}$
depends on a free parameter, a “continuous vertex enumeration” may be possible
and lead to curved bounds. If more free parameters are present, the problem
need to be converted to a programming and solved numerically Li:2021cjv .

  * •

All generators can be interpreted as coming from integrating out a single
heavy particle from the UV spectrum. This connection provides an interface
between positivity and the inverse problem. The only exception is the spin-2
generator(s) in $VV$ scattering. They can only be interpreted as UV completion
at the loop level.

  * •

If a tree-level mapping between $\mathcal{M}^{ijkl}$ and the coefficients are
used, an alternative and easier way to find bounds is to enumerate all
possible heavy particles that couple to scalars and fermions, through the
following types of couplings:

$\displaystyle g_{ijk}MS_{k}\phi_{i}\phi_{j},\quad
g_{ijk}V_{k}^{\mu}\phi_{i}\overleftrightarrow{D}_{\mu}\phi_{j},\quad
g_{ijk}S_{k}\bar{f}_{i}^{c}f_{j},\quad
g_{ijk}V_{k\mu}\bar{f}_{i}\gamma^{\mu}f_{j}$ (516)
$\displaystyle\frac{1}{M}g_{ijk}\left(\bar{f}_{i}^{c}\sigma_{\mu\nu}f_{j}\right)V_{k}^{\mu\nu}$
(517)

Integrating out these $S$ and $V$ particles directly gives all the $\vec{g}$
vectors in the coefficient space. The bounds obtained this way apply not only
to tree-level UV completions, but also to loop-level and strongly coupled
ones. However, this simplified approach does not work for vector operators.

  * •

All extremal bounds for SM parity-conserving self-quartic operators from this
and other works are collected in Section 4.4.

The inverse problem The second half of this paper has been devoted to the
discussion of the inverse problem for tree-level UV completions (Section 5)
and for loop-level UV completions (Section 6). The main findings are presented
in Section 5 and are summarized here:

  * •

Integrating out each heavy particle in the UV spectrum would give rise to
generator vector $\vec{g}$. The total coefficient vector is
$\vec{C}=\sum_{\alpha}w_{\alpha}\vec{g}_{\alpha}$, where $\alpha$ is the
particle type (specified by its spin, charge, irrep, relative couplings with
different SM particles), and $w_{\alpha}=g_{\alpha}^{2}/M_{\alpha}^{4}$. For
measurements at the dim-8 level, we shall discuss a weaker version of the
inverse problem: given the measured $\vec{C}$, how to determine
$\vec{w}\equiv\left(w_{1},w_{2},\cdots\right)$.

  * •

For each value of $\vec{C}$, let $\mathcal{W}$ be the set of feasible solution
for $\vec{w}$, its size $\Delta$ is called the “degeneracy”, which represents
the amount of UV theories consistent with the measured $\vec{C}$. $\Delta$
represents the arbitrariness in finding a UV completion from the measured
coefficients.

  * •

The pattern of the degeneracy in the dim-8 space is nontrivial and is
connected to the positivity bounds. The fact that the positivity cone is
always a salient cone has several interesting physics implications. In
particular, for SMEFTs that live on the boundary, extremality implies:

    * –

If the SMEFT lives on the 0-face, i.e. $\vec{C}=0$, all UV particles can be
ruled out independent of any UV model assumptions. The UV completion is
uniquely determined to be the SM itself. This provides a solid test of the SM.

    * –

If the SMEFT lives on a 1-face, i.e. an extremal ray, the UV completion must
be a “one-particle extension” of the SM, i.e. only one $w_{\alpha}$ can be
nonzero. Heavy particles must all belong to the same type, and their
interaction type (to the SM particles) is uniquely determined.

    * –

If the SMEFT lives on a $k$-face, $k>1$, and there are $l$ generators on that
face, then the dimension of $\mathcal{W}$ is $l-k$. If $l=k$ (e.g. restricted
by symmetries of the theory), the UV particle types (and their interactions
with the SM) are uniquely determined. If $l>k$, the determination is not
unique, but $l-k$ degrees of freedom remain to be fixed.

    * –

To sum up, SMEFTs that live on the boundary of the positivity cone have either
vanishing or very limited degeneracy, meaning that the corresponding UV
particles and their interactions to the SM can be fixed, either uniquely or at
least to a large extent.

  * •

For SMEFTs that live in the interior, $\Delta$ is always finite, and upper
bounds on all $w_{\alpha}$ (or equivalently, exclusion limits for all
particles of type $\alpha$ ) can be set, again independent of any assumptions
on the UV models. This can be done with the geometric trick of Eq. (419),
which works even with an infinite number of $\alpha$.

  * •

All above implications require that a salient positivity cone exists. This is
always true at dim-8, but not at dim-6 . Therefore all these implications are
absent at dim-6.

Finally, Section 6 discusses several cases with loop-induced generators. The
main conclusions are

  * •

Integrating out heavy particle loops or mixed heavy-light particle loops
generates a contribution $\vec{g}_{loop}$ to the amplitude.
$\vec{g}_{loop}\in\mathbf{C}$.

  * •

$\vec{g}_{loop}$ can be computed from two-particle cuts using the dispersion
relation. A partial wave decomposition of the dispersion relation, together
with the direct sum decomposition of irreps, reveals how $\vec{g}_{loop}$ can
be decomposed into a positive sum of the standard generators.

  * •

In general, $\vec{g}_{loop}$ is a sum of several different irreps, and tends
to live in the interior of C. If this is the case, it does not generate
additional degeneracy on the boundary of C. Many conclusions in Section 5 will
still hold (for SMEFTs at the ER, on a $k$-face, or at the original, etc.)

    * –

As a consequence, SMEFTs on the boundary would imply that the loop
contribution can be excluded from the possible UV completion.

    * –

However, the existence of $\vec{g}_{loop}$ does increase the possible
degeneracy inside the C.

    * –

It is still possible that certain loops generate $\vec{g}_{loop}$’s on an ER
or on a face. In this case, new degeneracies will be created on the boundary.

  * •

Without mixed heavy-light particle loops, one can subtract the SM loops and
only focus on heavy particle loops, which are often the leading BSM effects.

  * •

When mixed heavy-light particle loop exists, light particle loops contribute
to the dispersion relation. $\vec{g}_{loop}$ contains information of both BSM
and SM particle states.

    * –

As a result, $\mathcal{M}\left((\epsilon\Lambda)^{2}\right)$ depends on the
scale $\epsilon\Lambda$ and so is $\vec{g}_{loop}.$ For inverse problem, one
should choose a $\epsilon\Lambda$ as large as possible, to increase the purity
of the information from UV.

    * –

As $\epsilon\Lambda$ decreases, $\vec{g}_{loop}$ flows inside $\mathbf{C}$ and
is eventually dominated by IR contributions.

    * –

In any case, the LO BSM effects would be present in other channels at the tree
level, which should be studied first.

Discussions The extremal positivity approach, by itself, is a powerful tool to
determine the exact boundary of UV-completable EFTs, and supersede bounds from
elastic scattering of two factorized and mixed states. The approach has been
introduced in Refs. Zhang:2020jyn ; bellazzini_symmetries_2014 , further
developed by Refs. Yamashita:2020gtt ; Fuks:2020ujk , and systematically
expanded in this work. Many SM examples have been studied and solved in these
works. Several future developments can be foreseen. Cross-quartic operators
can be added to the existing self-quartic bounds cross4paper . Collider tests
of these bounds can be performed at the LHC and future colliders (see, e.g.
Fuks:2020ujk ; Gu:2020ldn  for some initial studies in this direction).
However, obtaining the full set of bounds for all SMEFT operators seems
impractical using this approach. The numerical approach proposed in Ref.
Li:2021cjv  can be a more promising alternative.

The main focus of this paper, however, is the connection between positivity
and the inverse problem. We have shown that the mapping between SMEFTs and
their UV completions show interesting patterns in the dim-8 space (recall that
by dim-8 space we actually mean the $s^{2}$ contributions in 2-by-2
amplitudes). If the SMEFT that describes the nature lives on or near the
boundary of the positivity cone, measuring dim-8 coefficients would provide
crucial information for finding the UV theory, potentially allowing us to
uniquely fixes the UV particle types, their spin, charge, irrep, and couplings
with the SM particles. This feature is absent at dim-6.

We consider this as an important motivation for studying SMEFT at the dim-8
level Li:2020gnx ; Murphy:2020rsh . After all, the goal of precision
measurements is not to fix the SMEFT coefficients, but rather to learn
something about the UV theory. If deviations from the SM are present, we may
be able to measure the dim-6 coefficients accurately, but due to the large
(actually infinite) intrinsic degeneracy, an ignorance about possible UV
models always remains. At dim-8, however, thanks to the existence of the
positivity cone, the degeneracy of UV models is limited, and so even if the
coefficients themselves are not be measured as accurate as dim-6, chances are
that we can actually learn more about the UV, by exploiting the geometry
information encoded in the positivity cones. This, of course, depends on where
exactly the SMEFT is located in the cone, but the possibility to uniquely fix
the UV particle content is already sufficiently attractive.

On the other hand, if signals of deviations from the SM are not observed, we
would need confirm that the new physics cannot exist below certain scales,
depending the precision level of the measurements. We have shown that this
kind of conclusion is not possible at dim-6, because $\vec{C}^{(6)}=0$ is a
point with intrinsic degeneracy. Exclusion limits based on dim-6 coefficients
are possible only when specific UV theories are assumed. In contrast, an
experimental confirmation of $\vec{C}^{(8)}=0$ would unambiguously rule out
all types of UV states, independent of UV assumptions. This is simply because
the origin is an extreme point of the positivity cone. Once we are able to
exclude each heavy state individually, there is no need to further consider
dim-10 operators and beyond.

In practice, investigating the dim-8 parameter space can be challenging for
many reasons. One of them is that there are many dim-8 operators Li:2020gnx ;
Murphy:2020rsh , 993 counting only one flavor and 44807 counting three
flavors. However, we should keep in mind that in our approach we only need to
focus on operators that give rise to 2-to-2 amplitudes with an $E^{4}$
dependence, and the number of these operators is 250 for one generation and
6076 for three. This is still a large number, but is manageable. In addition,
it is always possible to focus on a subset of operators, as we have done in
all the examples in Section 5. More precisely, one first picks a subset of SM
particles, and take all operators that only involve these particles. All the
discussions in Section 5 go through, and from the positivity cone in this
subspace, one learns about how UV states interact with a subset of SM
particles.

Another challenge is that it is unclear whether current and future colliders
have the sensitivity to probe dim-8 effects to a desired precision level. The
fact that we look for $E^{4}/\Lambda^{4}$ rather than $v^{4}/\Lambda^{4}$
effects is a helpful factor, as we expect to reach better sensitivity with
high-mass region measurements or lepton colliders with a large beam energy.
The main difficulty, however, is that dim-6 contributions are supposed to be
the dominant effects, which could naively mask the $E^{4}/\Lambda^{4}$ that we
look for. A possible solution is to perform a global fit including both dim-6
and dim-8 operators. Different energy and angular dependences help to pin down
dim-6 and dim-8 coefficients separately. A simple analysis of this kind has
been applied to $e^{+}e^{-}$ collision for future lepton colliders, and
results do look promising Fuks:2020ujk . For example, model-independent
exclusion limits on all heavy particles can be set, using Eq. (419), to a
scale much larger than the collider energy. Another solution is to look for
specific observables or channels where dim-6 effects are absent. This approach
has been taken in Ref. Gu:2020ldn  for the diphoton channel at future lepton
colliders. While the projected precision is good, a disadvantage is that not
all relevant dim-8 operators can be fixed in this way, and therefore the
geometric information is incomplete. Nevertheless, we believe that more
phenomenological studies in this direction can be interesting and may bring
new opportunities for the bottom-up SMEFT approach.

As a final remark, in this work we have completely ignored the information
from dim-6 coefficients. While the purpose is to emphasize how much we could
learn just by using positivity and extremality at dim-8, in reality one should
always include such information. It is likely that, under certain assumptions,
the UV-completable region in the combined dim-6 and dim-8 space continues to
be a salient cone, and so certain conclusions from this work may be
generalized to the combined space, to develop a better understanding of the
degeneracy in the space of SMEFTs. We will defer this to future works.

## Acknowledgments

We would like to thank Jiayin Gu, Xu Li, Jiang-Hao Yu, Hao Zhang and Shuang-
Yong Zhou for helpful discussions and for editing this paper at the final
stage. CZ is supported by IHEP under Contract No. Y7515540U1.

Note added: The author Cen Zhang unexpectedly passed away while he was putting
some final touches on this paper. He had sent out a mostly finished pdf
version of this paper to most of those in the above list of acknowledgments.
The present paper is recompiled from that the pdf version. To meet the
publication standards, references have been added, minor physical discussions
have been modified, English has been slightly improved and typos have been
corrected by those in the above list.

##  Appendix A  Projectors

Here we give all projectors relevant for examples in Sections 4,5 and 6.
$\textbf{SO(2):}\textbf{2}\otimes\mathbf{2}=\textbf{1}_{S}\oplus\mathbf{1}_{A}\oplus\mathbf{2}$

$\displaystyle P_{\textbf{1}_{S}}^{ijkl}=\frac{1}{2}\delta^{ij}\delta^{kl}$
(518) $\displaystyle
P_{\textbf{1}_{A}}^{ijkl}=\frac{1}{2}\left(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk}\right)$
$\displaystyle
P_{\textbf{2}}^{ijkl}=\frac{1}{2}\left(\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}-\delta^{ij}\delta^{kl}\right)$

$\textbf{SU(2):}\textbf{ 2}\otimes\mathbf{2}=\mathbf{1}\oplus\mathbf{3}$

$\displaystyle P_{\textbf{1}}{}^{ij}{}_{kl}$
$\displaystyle=\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}-\delta_{l}^{i}\delta_{k}^{j}\right)$
(519) $\displaystyle P_{\textbf{3}}{}^{ij}{}_{kl}$
$\displaystyle=\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)$

$\textbf{SU(2):}\textbf{2}\otimes\overline{\mathbf{2}}=\mathbf{1}\oplus\mathbf{3}$

$\displaystyle{P_{\mathbf{1}}}_{jk}^{i}{}^{l}=\frac{1}{2}\delta_{j}^{i}\delta_{k}^{l}$
(520) $\displaystyle
P_{\mathbf{3}}{}_{jk}^{i}{}^{l}=-\frac{1}{2}\delta_{j}^{i}\delta_{k}^{l}+\delta_{k}^{i}\delta_{j}^{l}$

$\textbf{SU(3): 3 }\otimes\textbf{3}=\overline{\textbf{3}}\oplus\mathbf{6}$

$\displaystyle{P_{\overline{\textbf{3}}}}^{ij}_{\ \ kl}$
$\displaystyle=\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}-\delta_{l}^{i}\delta_{k}^{j}\right)$
(521) $\displaystyle{P_{\textbf{6}}}{}^{ij}_{\ \ kl}$
$\displaystyle=\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)$

$\textbf{SU(3): 3 }\otimes\overline{\textbf{3}}=\textbf{1}\oplus\textbf{8}$

$\displaystyle{P_{\mathbf{1}}}_{jk}^{i}{}^{l}=\frac{1}{3}\delta_{j}^{i}\delta_{k}^{l}$
(522)
$\displaystyle{P_{\mathbf{8}}}_{jk}^{i}{}^{l}=-\frac{1}{3}\delta_{j}^{i}\delta_{k}^{l}+\delta_{k}^{i}\delta_{j}^{l}$

##  Appendix B Some expressions for matching fermion loops

Here we list the expressions for the $\mathcal{M}_{x,q}^{S,A}$ defined in Eqs.
(513) and (514):

$\displaystyle\mathcal{M}_{q}^{S}=$
$\displaystyle\frac{y^{4}\left(-2\left(\mathrm{Li}_{2}\left(-\frac{1}{4}\right)+\mathrm{Li}_{2}\left(-\frac{1}{5}\right)+1\right)-4\log^{2}(2)+\log(5)\log\left(\frac{36}{5}\right)\right)}{256\pi^{2}M^{4}}$
(523) $\displaystyle\mathcal{M}_{q}^{A}=$
$\displaystyle\frac{y^{4}\left(6\mathrm{Li}_{2}\left(-\frac{1}{4}\right)+6\mathrm{Li}_{2}\left(-\frac{1}{5}\right)-2+(\log(8)-8)\log(16)+\log(5)(20-6\log(6)+\log(125))\right)}{768\pi^{2}M^{4}}$
(524) $\displaystyle\mathcal{M}_{x}^{S}=$
$\displaystyle\frac{y^{4}}{3072\pi^{2}M^{4}x^{3}}\bigg{[}24x^{3}\left(\mathrm{Li}_{2}(-4x-1)+\mathrm{Li}_{2}(-4x)\right)+4$
$\displaystyle
x\left(x\left(x\left(6\left(\mathrm{Li}_{2}\left(-\frac{1}{4}\right)+\mathrm{Li}_{2}\left(-\frac{1}{5}\right)+1\right)+2\pi^{2}+3\log^{2}(4)-3\log(5)\log\left(\frac{36}{5}\right)\right)-3\right)-3\right)$
$\displaystyle+3\log(4x+1)\left(8x^{3}\log(4x+2)+(3-4x)x+1\right)\bigg{]}$
(525) $\displaystyle\mathcal{M}_{x}^{A}=$
$\displaystyle\frac{y^{4}}{3072\pi^{2}M^{4}x^{3}}\bigg{[}-24x^{3}\left(\mathrm{Li}_{2}(-4x-1)+\mathrm{Li}_{2}(-4x)\right)$
$\displaystyle-4x\left(x^{2}\left(6\mathrm{Li}_{2}\left(-\frac{1}{4}\right)+6\mathrm{Li}_{2}\left(-\frac{1}{5}\right)-2+2\pi^{2}+3\log^{2}(4)+\log(5)(20-6\log(6)+\log(125))\right)\right.$
$\displaystyle+x+1)-64x^{3}\log(x)+\log(4x+1)\left(-24x^{3}\log(4x+2)+(4x(16x+3)+3)x+1\right)\bigg{]}$
(526)

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