$\psi_{r}=2\mu[b/\sigma-\sigma^{2}b/(\bm{r}^{\alpha})^{3}]$ (Wagner et al.,
2021), where $\mu=7.5k_{b}T$ is an arbitrary energy scale and $\sigma=b$ is
the separation length at which repulsive forces go to zero. The harmonic
potential between attached beads is given by
$\psi_{s}=K3k_{b}T\bm{(}\bm{r}^{\alpha})^{2}/b^{2}$, where $K=1/2$ is an
arbitrary, dimensionless softening factor. The beads’ positions were
equilibrated using the overdamped steepest descent method of Wagner et al.
(2021). The phenomenological parameters $\mu$, $\sigma$, and $K$ were set so
that stable convergence was always achieved for the initial conditions in
LAMMPS.

Once the initial bead positions were set, the chains were loaded into LAMMPS,
and their beads’ positions were stepped in time according to Eq. (1) per the
methods of Section 2.1. To reach the initial conditions of the study, the
$N^{th}$ beads of each chain were gradually stepped towards the fixed beads at
position $(0,0,0)$ in increments of approximately $(-0.75,0,0)b$ until an end-
to-end separation of $r=b$ was achieved. At each step, the chains were
equilibrated for a duration of $400\tau_{0}$. However, once the initial end-
to-end separation of $r=b$ was reached, the chains with $N=12$, $N=18$, and
$N=36$ Kuhn segments were equilibrated for variable durations of $16\times
10^{3}\tau_{0}$, $32\times 10^{3}\tau_{0}$, and $68\times 10^{3}\tau_{0}$,
respectively. This was done to erase any conformational memory of the
initially stretched state. Once each chain was equilibrated with an end-to-end
length of $b$, it was gradually extended by stepping the position of $N^{th}$
bead by increments of $(0.75,0,0)b$. At each step, the chains were held for
$4\times 10^{3}\tau_{0}$ and the mean end-to-end force was calculated as the
average force of each segment projected onto the chain’s end-to-end axis,
$\bm{e}_{1}$:

$\bar{f}=\langle
N^{-1}\sum_{\alpha}^{N}\bm{f}^{\alpha}\rangle_{t}\cdot\bm{e}_{1},$ (C1)

where $\bm{f}^{\alpha}=-\partial\psi_{b}/\partial\bm{r}^{\alpha}$ is the force
of Kuhn segment $\alpha$ with end-to-end vector $\bm{r}^{\alpha}$, $\psi_{b}$
is the bond potential from Eq. (5), and $\langle\Box\rangle_{t}$ denotes
ensemble averaging over the $4\times 10^{3}\tau_{0}$ duration. This process
was repeated for chains with $N=\\{12,18,36\\}$ Kuhn segments and when the
characteristic energies of Eq. (5) were set to
$E=\\{100,200,400,800\\}k_{b}T$. The distribution of Kuhn lengths remains
within $\pm 10\%$ of $b$ when $E=800k_{b}T$ (Fig. C1.A). Additionally, the
force-extension relations are in good agreement with those predicted by
differentiating Eq. (4) with respect to $r$ when $E=800k_{b}T$ (Fig. C1.B).

Figure C1: Kuhn segment bond potential calibration: (A) End-to-end
distributions of the Kuhn segment lengths of the bead-spring model for
$E^{*}=E/k_{b}T=\\{100,200,400,800\\}$, $L=b$, and $N=36$. (B) Normalized
force versus chain stretch for the Langevin potential of Eq. (4) (dashed
curves) and ensemble average force from the bead-spring model (discrete data)
for various chain lengths ($N=\\{12,24,36\\}$ Kuhn segments) when
$E=800k_{b}T$ and $L=b$.

##  Appendix D Sampling frequency convergence study

Checking for dynamic bonding events incurs a computational cost, however bond
kinetics must be sampled with an adequate frequency, $k_{s}$, to ensure that
bonding opportunities are not missed as stickers oscillate to within distance
$b$ of each other. To check for adequate sampling frequency, a convergence
study was conducted using the single-chain attachment set-up of Section 3.2.
Bond kinetic sampling frequency was swept over
$k_{s}=\\{10,15,20\\}\tau_{0}^{-1}$ and bond kinetic rates and steady state
attached/detached chain fractions were measured for both models. Results are
displayed in Fig. D1. Results do not change significantly as $k_{s}$ is
increased from $15\tau_{0}^{-1}$ to $20\tau_{0}^{-1}$. This is consistent with
the observation that the measured sticker diffusion timescale, $\tau_{s}$ is
consistently $\tau_{s}\approx 0.07\tau_{0}\approx\tau_{0}/14$ per Fig. E1.H.

Figure D1: Sampling frequency convergence of bond dynamics: (A-B) Average (A)
attachment and (B) detachment rates with respect to tether-to-sticker
separation distance, $d/(Nb)$ for bond kinetic sampling frequencies of
$k_{s}=10\tau_{0}^{-1}$ (black), $k_{s}=15\tau_{0}^{-1}$ (brown), and
$k_{s}=20\tau_{0}^{-1}$ (orange). Results are provided for both the bead-
spring (circles) and mesoscale (triangle) models. Solid curves represent the
fitted scaling theory through Eq. (19) with prefactor, $A$, treated as a
fitting parameter. (C-D) Average fractions of (C) attached and (D) detached
chains, plotted with respect to $d/(Nb)$, for the same sampling frequencies as
(A-B). Solid curves represent the predicted steady-state fractions of attached
and detached chains, $f_{a}=k_{a}/(k_{a}+k_{d})$, and $f_{d}=1-f_{a}$,
respectively (Vernerey et al., 2017). Here, $k_{a}$ is the distance-dependent
attachment rate predicted by Eq. (19) while $k_{d}$ is the detachment rate set
a priori.

##  Appendix E Effects of diffusion coefficient

Fig. E1 confirms that modifying the diffusion coefficient, $D_{0}$, has no
effect on the results of either model in normalized time. This is true with
respect to both the monomer diffusion timescale, $\tau_{0}=b^{2}/D_{0}$, and
the resulting Rouse time, $\tau_{r}=\tau_{s}N^{2}$, where $\tau_{s}$ is the
emergent timescale of distal sticker diffusion. Fig. E1H confirms that the
measured sticker diffusion timescale, $\tau_{s}=0.07\tau_{0}$, is relatively
independent of both $D_{0}$ (i.e., $\tau_{0}$) and chain length (i.e., $N$).

Figure E1: Comparing diffusion coefficients: (A-C) MSD, steady state MSD with
respect to $N$, and MSD at the Rouse timescale are presented for the case of
$D_{0}=8\times 10^{-10}$ m2 s-1, respectively. (D-F) The same plots are
depicted for $D_{0}=0.125\times 10^{-10}$ m2 s-1. The time-axes of (A) and (D)
are normalized by the monomer diffusion timescale, $\tau_{0}=b^{2}/D_{0}$,
while the time-axes of (C) and (F) are normalized by the Rouse time,
$\tau_{r}$. For both values of $D_{0}$, the steady-state MSD, $\langle\Delta
r^{2}\rangle_{ss}$, scales linearly with respect to chain length (i.e., $N$).
(G) Rouse time, $\tau_{r}$ and (H) emergent sticker diffusion time,
$\tau_{s}$, both with respect to the prescribed diffusion timescale,
$\tau_{0}$, and chain length via $N$.

Characteristic sticker diffusion times, $\tau_{s}$, and Rouse times,
$\tau_{r}$, were attained by identifying the time at which $\langle\Delta
r^{2}(t)\rangle\geq b^{2}$ (Fig. E2A), and then fitting Eq. (15) to the data
(Fig. E2B).

Figure E2: The Rouse model for sticker diffusion: (A) Raw MSD data from the
bead-spring model for $N\in\\{12,18,24,30,36\\}$ is used to (B) compute the
timescale, $\tau_{s}$, of distal sticker diffusion.

##  Appendix F Validating sticker pairwise kinetic rates

Before examining the effects of chain exploration on bond association, we
confirmed that the discrete model implementation of stochastic bond reactions
reproduced the intrinsic detachment and attachment rates set a priori through
Eqs. (7-9), without the extrinsic influence of diffusion kinetics. To do so,
we simulated pairs of fixed stickers separated by distance $d_{ss}/b=0.5$ that
could bind and unbind to one another (see schematic of Fig. F1A). For adequate
sampling, arrays of $n_{p}=343$ pairs (separated from other pairs by distances
$d_{ss}/b\gg 1$ so that no inter-pair interactions occur) were initialized and
sampled concurrently for a duration of $t=10^{3}\tau_{0}$. Fig. F1B confirms
that the rates of detachment, $k_{d}$, and attachment, $k_{a}$, measured from
the discrete model are in good agreement with those values set by Eqs. (7) and
(8), respectively. Note that here, $\varepsilon_{d}$ was held constant at
$\varepsilon_{d}=0.1k_{b}T$, while $\varepsilon_{a}$ was swept over the order
of $\varepsilon_{a}\in[0.01,10]k_{b}T$. Maintaining a relatively high
dissociation rate (via relatively low $\varepsilon_{d}$) ensured that ample
numbers of dissociation events occurred within reasonable time domains for
adequate statistical sampling of both event types.

Figure F1: Pairwise kinetics of adjacent stickers. (A) Illustration of a pair
of fixed stickers separated by $d/b<1$ at their initial state (top),
immediately after an attachment event (middle), and after the subsequent
detachment event (bottom). (B) Results comparing the discrete model
predictions (discrete data points with error bars) of detachment and
attachment rates to the a priori values of $\varepsilon_{a}$ set through Eq.
(8). Note that $\varepsilon_{d}$ was held constant over all six sets of
simulations, hence the lack of variation in $\bar{k}_{d}$. Error bars
represent SE.

##  Appendix G Extended pairwise kinetics results

To check whether the prefactor, $A$, required for fitting Eq. (19) results
from some effect of repeat attachments, we compare the overall average
attachment rates of Fig. 4D-E (which includes repeat attachment events when
computing the mean) to the average attachment rates computed using only first-
time attachment events, $\bar{k}_{a,1}$, in Fig. G1. While eliminating repeat
attachment events from the computation of $\bar{k}_{a}$ does slightly reduce
the attachment rate in all cases, it does not alter the magnitudes of
$\bar{k}_{a}$ sufficiently to influence the necessity of prefactor $A$ or
explain its trends with respect to $N$ and $\varepsilon_{a}$.

Figure G1: First-time versus overall average attachment rates of a single
bond. Average attachment rates, $\bar{k}_{a}$ (green) and first-time
attachment rates, $\bar{k}_{a,1}$, (yellow) with respect to normalized
separation distance, $\lambda_{c}=d_{ts}/(\sqrt{N}b)$. (A-C), (D-F), and (G-I)
provide data when $\varepsilon_{a}=\\{1,0.1,0.01\\}k_{b}T$, respectively. The
left column (A,D,G), center column (B,E,H), and right column (C,F,I) display
results for the bead-spring model, mesoscale model, and LAMMPS implementation
of Eq. (19), respectively. Data are provided for $N=\\{12,18,36\\}$.

Besides interrogating the associative kinetics of single chains to fixed
stickers, we also evaluate the bead-spring and mesoscale models’ predicted
attachment rates for sets of two chains undergoing pairwise bonding (Fig.
G2A-C). Average attachment rates are plotted with respect to tether-to-tether
separation distance, $d_{tt}$, for $N=\\{12,18,36\\}$ and
$\varepsilon_{a}=0.01k_{b}T$ in Fig. G2D. Prefactor $A$ remains necessary
(Fig. G2E-F), although its value for the two-chain system generally exceeds
that of the single-chain system, for reasons requiring further investigation
in future work.

Figure G2: Pairwise bond kinetics between two chains. (A-C) Illustration of
two adjacent tethered chains with fixed ends (grey) separated by distance,
$d_{tt}$, and free ends (red/blue stickers) that may bind/unbind to one
another. (D) Average normalized attachment rates, $\bar{k}_{a}\tau_{0}$, with
respect to normalized separation distance, $\lambda_{c}=d_{tt}/(\sqrt{2N}b)$,
when $\varepsilon_{a}=0.01k_{b}T$. Error bars represent SE. Best fits of Eq.
(19) for the bead-spring (solid curves) and mesoscale (dashed curves) data are
displayed (where $A$ is a fitting parameter). (E) Prefactor, $A$, for all
chain lengths. Light green, dark green, and grey bars/markers correspond to
$N=12$, $N=18$, and $N=36$, respectively. Error bars represent the 95$\%$
confidence interval. (F) Goodness of fit between the discrete models and
scaling theory, characterized by $R^{2}$.  Table G1: Computational run times
for each model’s pairwise bonding study of Section 3.2 for the parameter
combinations with the lowest ($N=12$, $\varepsilon_{a}=k_{b}T$,
$d_{ts}=0.5Nb$) and highest ($N=36$, $\varepsilon_{a}=0.01k_{b}T$,
$d_{ts}=0.125Nb$) computational costs.

##  Appendix H On prescribing polymer packing fractions

Historically, exact packing fractions – or the more frequently reported free
volume fractions ($1-\phi$) – of polymers have not been consistently defined
within the literature (Consolati et al., 2023). Moreover, they have proved
difficult to estimate experimentally due to the characteristic size of free
volume features (e.g., pores and voids) at the molecular scale. Consequently,
$\phi$ is often characterized by the differential volume fraction,
$\phi_{0}-\phi=\alpha_{f}(T-T_{\infty})$, between a polymer at its current and
Vogel (or reference) temperatures ($T$ and $T_{\infty}$, respectively)
(Rubinstein and Colby, 2003) where $\alpha_{f}$ is the coefficient of thermal
expansion of the free volume, which is on the order of $10^{-4}$ to $10^{-3}$
K-1 for highly crosslinked, rubbery polymers (Marzocca et al., 2013). The
Vogel temperature is commonly taken as 50 K below glass transition
temperature, $T_{g}$, and is typically ascribed as the temperature at which
free volume approximates zero (i.e., $\phi_{0}=1$), which – while an
idealization – provides a pragmatic, relative estimate of empirical free
volume. Experimental efforts that utilize a combination of dynamic mechanical
analysis (DMA), differential scanning calorimetry (DSC), and positron
annihilation lifetime spectroscopy (PALS) have indeed estimated that the
reference, free volume fraction around $T_{g}$ is close to the order of
3-5$\%$ depending on the polymer’s crosslink density (Marzocca et al., 2013).
Based on these values, one would approximate that the polymer packing fraction
remains $\sim 90$-$95\%$ even $10$ to $100$ K above $T_{g}$ in the elastomeric
regime.

However, recent studies conducted using all-atom MD and subsequently
extrapolated machine learning predictions have estimated that the packing
fraction of homopolymers and polyamides at ambient conditions (300 K and 0.1
MPa) are on the order of 55-70$\%$ (Tao et al., 2023). Meanwhile, those of
more broadly defined microporous polymers (with highly variable functional
side group chemistries) at the same ambient conditions range from 55-80$\%$
(Tao et al., 2023). While the work of Tao et al. (2023) does not distinguish
which polymers are in the glassy versus rubbery state when their free volume
fractions are measured, these estimates provide a reasonable upper limit of
polymer packing fraction on the order of 60-70$\%$. Based on these estimates
and the realization that the freely jointed, ideal chain assumption has
diminishing validity at higher packing fractions, we here limit packing
fractions to a maximum value of $\phi\sim 0.5$. Lower limits of the packing
fraction in the models are constrained only by the attainment of percolated,
gel-like networks, which here occurred around $\phi\sim 0.2$.

##  Appendix I Extended ensemble bond dynamics

Average bond detachment rates, $\bar{k}_{d}$, and steady state detached and
attached bond fractions ($\bar{f}_{d}$ and $\bar{f}_{d}$, respectively) from
the study of Section 3.3 are plotted against nominal chain separation,
$\bar{d}$, in Fig. I1.

Figure I1: Dissociative ensemble bond kinetics and steady state bond
fractions. (A) bond detachment rates, $\bar{k}_{d}$, with respect to
normalized chain separation, $d/(Nb)$. Dashed line indicates the value of
$k_{d}$ prescribed a priori. (B-C) Fractions of (B) detached and (C) attached
chains with respect to $d/(Nb)$. Error bars represent standard error of the
mean.

The average attached bond lifetimes, $\tau_{a}$, and renormalized bond
lifetimes, $\bar{\tau}_{rnm}$, from the study of Section 3.3 are plotted with
respect to $\bar{d}$ in Fig. I2. Renormalized bond lifetimes are roughly twice
that of conventionally measured attached bond lifetimes, and the inverse
renormalized bond lifetime is on the order of $\tau_{rnm}^{-1}\sim
10^{-3}\tau_{0}^{-1}$, which his consistent with the order of average partner
exchange rates ($k_{exc}\sim 10^{-3}\tau_{0}^{-1}$ from Fig. 6B), as expected.

Figure I2: Attached ensemble bond lifetimes. Average (A) attached bond
lifetime, $\bar{\tau}_{a}$, and (B) renormalized bond lifetime,
$\bar{\tau}_{rnm}$, with respect to normalized chain separation, $d/(Nb)$.
Error bars represent standard error of the mean.

The data from Fig. 6 are plotted with respect to chain (and therefore sticker)
concentration, $c$, in Fig. I3 to illustrate the collapse of kinetic
attachment rates and pertinent timescales in concentration-space.

Figure I3: Associative kinetic rates versus sticker concentration. (A) Bond
attachment, (B) partner exchange, (C) and repeat attachment rates with respect
to $cb^{3}=(b/d)^{3}$ for both models when $N=\\{12,18,36\\}$. Error bars
represent S.E. Dashed curves in (A-C) represent empirical fits according to
Eq. (21), where ${c}^{-1/3}$ is substituted for $d$.

Empirically fit parameters of Eq. (21) are plotted against chain length in
Fig. I4.

Figure I4: Average, empirically fit parameters. (A) Maximum chain stretches,
$\lambda_{c}^{max}=d_{max}\sqrt{N/2}$, (B) minimum chain concentrations,
$c_{min}=d_{max}^{-1/3}$, (C) and minimum polymer packing fractions,
$\phi_{min}=\pi Nb^{3}/(6d_{max}^{3})$, as well as characteristic (D) bond
attachment rates, $k_{a}^{semi}$, (E) partner exchange rates,
$k_{exc}^{semi}$, and (F) repeat attachment rates, $k_{rpt}^{semi}$ against
$N$ for both models. Error bars represent the $95\%$ confidence interval from
nonlinear, least-squares regression analysis.

##  Appendix J Extended network simulation results

Comparisons of the bead-spring virial stress evolution as computed using Eq.
(22) when the sum is taken over all Kuhn segments versus when it is taken at
the polymer chain end-to-end scale is provided in Fig. J1 for all four
sweeping parameters. Stresses are generally in reasonable agreement, but with
considerably more noise for the Kuhn-scale computed virial stresses than those
computed at the mesh-scale.

Figure J1: Kuhn versus mesh-scale virial stress comparison for bead-spring
model. Virial stress from Eq. (22) versus time for the bead-spring when
computed using all Kuhn segments versus the polymer end-to-end vectors and the
force-extension relation from Fig. C1. Parameters were set as (A)
$\phi=\\{0.2,0.5\\}$, $N=12$, $\dot{\varepsilon}=10^{-2}\tau_{0}^{-1}$, and
$k_{d}=0$; (B) $\phi=0.2$, $N=\\{12,18,36\\}$,
$\dot{\varepsilon}=10^{-2}\tau_{0}^{-1}$, and $k_{d}=0$; (C) $\phi=0.2$,
$N=12$, $\dot{\varepsilon}=\\{1,3,10\\}\times 10^{-2}\tau_{0}^{-1}$, and
$k_{d}=0.1\tau_{0}^{-1}$; and (D) $\phi=0.2$, $N=12$,
$\dot{\varepsilon}=10^{-2}\tau_{0}^{-1}$, and
$k_{d}=\\{10^{-3},10^{-2},10^{-1}\\}\tau_{0}^{-1}$.

CPU run times and required storage per additional step of stored data of the
full network-scale models from Section 4 are provided in Fig. J2.

Figure J2: Computational run times and data storage requirements.  (A) CPU
time and (B) percent reduction in CPU time (from bead-spring to mesoscale
models) with respect to simulated number of polymer chains, $n_{p}$. CPU time
for the bead-spring and mesoscale models scale as
$t_{cpu}\propto[547$s/polymer$]n_{p}$ ($R^{2}=1.00$), and
$t_{cpu}\propto[15$s/polymer$]n_{p}$ ($R^{2}=1.00$), respectively. Percent
reduction in CPU time scales as $\%\Delta t_{cpu}=1-(0.81)n_{p}^{-0.44}$
($R^{2}=1.00$). All simulations were run on 80 cores of a single Ice Lake
(ICX) node using Stampede3. (C) Data storage required per additional step of
data output using LAMMPS’s atom.dump (filled) and bond.dump (open) files.
Storage requirements for the bead-spring and mesoscale models scale
proportionately to $[7.3$ Kb/step/polymer$]n_{p}$ ($R^{2}=1.00)$ and $[0.7$
Kb/step/polymer$]n_{p}$ ($R^{2}=1.00)$, respectively. $N=12$, $\phi=0.5$,
$k_{d}=0.01\tau_{0}^{-1}$, and $\dot{\varepsilon}=0.01\tau_{0}^{-1}$ for all
simulations. Percent reduction in storage requirements is constant and on the
order of $90\%$ for both file types.

Analogous data to that of Fig. 8 in Section 4 are provided in Fig. J3 when
$N=36$.

Figure J3: Mechanical response at the parameter extremes when $\bm{N=36}$.
Normal Cauchy stress in the direction of loading for the bead-spring and
mesoscale models, as well as relative error are plotted with respect to time
at both polymer packing fractions ($\phi=\\{0.2,0.5\\}$), loading rates
($\dot{\varepsilon}=\\{0.01,0.1\\}\tau_{0}^{-1}$), and bond detachment rates
($k_{d}=\\{0,0.1\\}\tau_{0}^{-1}$) when $N=36$. Horizontal dashed lines on
error plots denote zero error.

Fractions of attached and detached chains, as well as the average number of
bonds between two side chains of the same molecule are plotted in Fig. J4 with
respect to time (during initial network equilibration), as measured from the
simulations of Section 4.1. These demonstrate how, despite similar values of
$\bar{f}_{a}$ and $\bar{f}_{d}$, the two modeling approaches predict disparate
degrees of self-attachment for networks of longer chains ($N=36$).

Figure J4: Effects of chain length on network connectivity. (A)
Attached/detached chain fractions and (B) relative error between the models
(bead-spring model as reference) with respect to time during initial network
equilibration. (C) Average degree of intra-molecular dynamic bonds and (D)
relative error between the models (bead-spring model as reference), also with
respect to time during initial network equilibration. Insets of (B,D) display
root mean-square error (RMSE) of (A,C), respectively, with respect to $N$. The
red lines represents the best fit from linear regression analysis with a 95%
confidence interval ($p=0.04$ for (B) and $p=0.19$ for (D)). Although the
degree of intra-molecular attachment appears biphasic with respect to $N$,
RMSE is considerably higher for $N=36$ (RMSE $=15\%$) than the next closest
value (RMSE $=6\%$ for $N=30$).

MSD data and relative error between the bead-spring and mesoscale models from
the simulations of Section 4.1 are plotted with respect to time (during
initial network equilibration) in Fig. J5. These demonstrate how the mesoscale
model captures slower diffusion than that of the bead-spring model for both
sticker and tether nodes.

Figure J5: Effects of chain length on sticker and tether diffusion. (A) MSD
and (B) relative error between the models (bead-spring model as reference)
with respect to time during initial network equilibration for the sticker
nodes. (C-D) The same data as (A-B), except it is provided for the tether
nodes. Insets of (B,D) display RMSE of (A,C), respectively, against $N$. The
red lines represent the best fits from linear regression analysis with a 95%
confidence interval ($p=0.04$ for (B) and $p<0.01$ for (D)), illustrating the
correlation between $N$ and error between the MSD values of both models.

##  Appendix K CRediT authorship contribution statement

RJW: Conceptualization, Methodology, Validation, Investigation, Formal
analysis, Visualization, Software, Data curation, Resources, Writing –
original draft, Writing – review & editing. MNS: Conceptualization, Funding
acquisition, Project administration, Resources, Supervision, Writing –
original draft, Writing – review & editing.

##  Appendix L Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  Appendix M Acknowledgements

This work was supported by the United States National Air Force Office of
Scientific Research (AFOSR) under Contract No. FA9550-22-1-0030 and the United
States National Science Foundation (NSF) under Grant No. CAREER-1653059. This
work used Stampede3 at the Texas Advanced Computing Center (TACC) at The
University of Texas at Austin through allocation MCH230053 from the Advanced
Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS)
program, which is supported by National Science Foundation grants #2138259,
#2138286, #2138307, #2137603, and #2138296 (Boerner et al., 2023). This
content is solely the responsibility of the authors and does not necessarily
represent the official views of AFOSR, NSF, TACC, or ACCESS. The authors would
like to thank Dr. S. Lamont and Professor F. Vernerey (CU Boulder) for
orchestrating the implementation of the Transient Network Theory (TNT) package
into LAMMPS based on the work of Wagner et al. (2021).

## References

  * Chung (2019) Chung D. D. L.,  A review of multifunctional polymer-matrix structural composites,  Composites Part B: Engineering 160 (2019) 644–660. doi:10.1016/j.compositesb.2018.12.117.
  * Rydz et al. (2021) Rydz J., Opálková Šišková A., Zawidlak-Wegrzyńska B., Duale K.,  Chapter 1 - High-performance polymer applications for renewable energy,  in: Devasahayam S., Hussain C. M. (Eds.), Nano Tools and Devices for Enhanced Renewable Energy, Micro and Nano Technologies, Elsevier, 2021, pp. 3–26.
  * Vallejos et al. (2022) Vallejos S., Trigo-López M., Arnaiz A., Miguel Á., Muñoz A., Mendía A., García J. M.,  From classical to advanced use of polymers in food and beverage applications,  Polymers 14 (2022) 4954. doi:10.3390/polym14224954.
  * Ibrahim et al. (2022) Ibrahim I. D., Hamam Y., Sadiku E. R., Ndambuki J. M., Kupolati W. K., Jamiru T., Eze A. A., Snyman J.,  Need for Sustainable Packaging: An Overview,  Polymers 14 (2022) 4430. doi:10.3390/polym14204430.
  * Heinzmann et al. (2016) Heinzmann C., Weder C., Espinosa L. M. d.,  Supramolecular polymer adhesives: advanced materials inspired by nature,  Chemical Society Reviews 45 (2016) 342–358. doi:10.1039/C5CS00477B.
  * Chen et al. (2022) Chen W.-H., Chen Q.-W., Chen Q., Cui C., Duan S., Kang Y., Liu Y., Liu Y., Muhammad W., Shao S., Tang C., Wang J., Wang L., Xiong M.-H., Yin L., Zhang K., Zhang Z., Zhen X., Feng J., Gao C., Gu Z., He C., Ji J., Jiang X., Liu W., Liu Z., Peng H., Shen Y., Shi L., Sun X., Wang H., Wang J., Xiao H., Xu F.-J., Zhong Z., Zhang X.-Z., Chen X.,  Biomedical polymers: synthesis, properties, and applications,  Science China Chemistry 65 (2022) 1010–1075. doi:10.1007/s11426-022-1243-5.
  * Kloxin et al. (2010) Kloxin C. J., Scott T. F., Adzima B. J., Bowman C. N.,  Covalent Adaptable Networks (CANs): A Unique Paradigm in Cross-Linked Polymers,  Macromolecules 43 (2010) 2643–2653. doi:10.1021/ma902596s.
  * Yue et al. (2020) Yue L., Guo H., Kennedy A., Patel A., Gong X., Ju T., Gray T., Manas-Zloczower I.,  Vitrimerization: Converting Thermoset Polymers into Vitrimers,  ACS Macro Letters 9 (2020) 836–842. doi:10.1021/acsmacrolett.0c00299.
  * Shen et al. (2021) Shen T., Song Z., Cai S., Vernerey F. J.,  Nonsteady fracture of transient networks: The case of vitrimer,  Proceedings of the National Academy of Sciences 118 (2021) e2105974118. doi:10.1073/pnas.2105974118.
  * Hubbard et al. (2021) Hubbard A. M., Ren Y., Konkolewicz D., Sarvestani A., Picu C. R., Kedziora G. S., Roy A., Varshney V., Nepal D.,  Vitrimer Transition Temperature Identification: Coupling Various Thermomechanical Methodologies,  ACS Applied Polymer Materials 3 (2021) 1756–1766. doi:10.1021/acsapm.0c01290.
  * Hubbard et al. (2022) Hubbard A. M., Ren Y., Picu C. R., Sarvestani A., Konkolewicz D., Roy A. K., Varshney V., Nepal D.,  Creep Mechanics of Epoxy Vitrimer Materials,  ACS Applied Polymer Materials 4 (2022) 4254–4263. doi:10.1021/acsapm.2c00230.
  * Brunsveld et al. (2001) Brunsveld L., Folmer B. J. B., Meijer E. W., Sijbesma R. P.,  Supramolecular Polymers,  Chemical Reviews 101 (2001) 4071–4098. doi:10.1021/cr990125q.
  * Yount et al. (2005) Yount W. C., Loveless D. M., Craig S. L.,  Strong Means Slow: Dynamic Contributions to the Bulk Mechanical Properties of Supramolecular Networks,  Angewandte Chemie International Edition 44 (2005) 2746–2748. doi:10.1002/anie.200500026.
  * Vidavsky et al. (2020) Vidavsky Y., Buche M. R., Sparrow Z. M., Zhang X., Yang S. J., DiStasio R. A. J., Silberstein M. N.,  Tuning the Mechanical Properties of Metallopolymers via Ligand Interactions: A Combined Experimental and Theoretical Study,  Macromolecules 53 (2020) 2021–2030. doi:10.1021/acs.macromol.9b02756.
  * Mordvinkin et al. (2021) Mordvinkin A., Döhler D., Binder W. H., Colby R. H., Saalwächter K.,  Rheology, Sticky Chain, and Sticker Dynamics of Supramolecular Elastomers Based on Cluster-Forming Telechelic Linear and Star Polymers,  Macromolecules 54 (2021) 5065–5076. doi:10.1021/acs.macromol.1c00655.
  * Xu et al. (2022) Xu L., Fu Y., Wagner R. J., Zou X., He Q., Li T., Pan W., Ding J., Vernerey F. J.,  Thermosensitive P(AAc-co-NIPAm) Hydrogels Display Enhanced Toughness and Self-Healing via Ion–Ligand Interactions,  Macromolecular Rapid Communications 43 (2022) 2200320. doi:10.1002/marc.202200320.
  * Haque et al. (2011) Haque M. A., Kurokawa T., Kamita G., Gong J. P.,  Lamellar Bilayers as Reversible Sacrificial Bonds To Toughen Hydrogel: Hysteresis, Self-Recovery, Fatigue Resistance, and Crack Blunting,  Macromolecules 44 (2011) 8916–8924. doi:10.1021/ma201653t.
  * Gong et al. (2016) Gong Z., Zhang G., Zeng X., Li J., Li G., Huang W., Sun R., Wong C.,  High-Strength, Tough, Fatigue Resistant, and Self-Healing Hydrogel Based on Dual Physically Cross-Linked Network,  ACS Applied Materials & Interfaces 8 (2016) 24030–24037. doi:10.1021/acsami.6b05627.
  * Bai et al. (2018) Bai R., Yang J., Morelle X. P., Yang C., Suo Z.,  Fatigue Fracture of Self-Recovery Hydrogels,  ACS Macro Letters 7 (2018) 312–317. doi:10.1021/acsmacrolett.8b00045.
  * Li and Gong (2022) Li X., Gong J. P.,  Role of dynamic bonds on fatigue threshold of tough hydrogels,  Proceedings of the National Academy of Sciences 119 (2022) e2200678119. doi:10.1073/pnas.2200678119.
  * Tuncaboylu et al. (2011) Tuncaboylu D. C., Sari M., Oppermann W., Okay O.,  Tough and Self-Healing Hydrogels Formed via Hydrophobic Interactions,  Macromolecules 44 (2011) 4997–5005. doi:10.1021/ma200579v.
  * Jeon et al. (2016) Jeon I., Cui J., Illeperuma W. R. K., Aizenberg J., Vlassak J. J.,  Extremely Stretchable and Fast Self-Healing Hydrogels,  Advanced Materials 28 (2016) 4678–4683. doi:10.1002/adma.201600480.
  * Zhang et al. (2019) Zhang H., Wu Y., Yang J., Wang D., Yu P., Lai C. T., Shi A.-C., Wang J., Cui S., Xiang J., Zhao N., Xu J.,  Superstretchable Dynamic Polymer Networks,  Advanced Materials 31 (2019) 1904029. doi:10.1002/adma.201904029.
  * Cai et al. (2022) Cai H., Wang Z., Utomo N. W., Vidavsky Y., Silberstein M. N.,  Highly stretchable ionically crosslinked acrylate elastomers inspired by polyelectrolyte complexes,  Soft Matter 18 (2022) 7679–7688. doi:10.1039/D2SM00755J.
  * Kersey et al. (2006) Kersey F. R., Loveless D. M., Craig S. L.,  A hybrid polymer gel with controlled rates of cross-link rupture and self-repair,  Journal of The Royal Society Interface 4 (2006) 373–380. doi:10.1098/rsif.2006.0187.
  * Brochu et al. (2011) Brochu A. B. W., Craig S. L., Reichert W. M.,  Self-healing biomaterials,  Journal of Biomedical Materials Research Part A 96A (2011) 492–506. doi:10.1002/jbm.a.32987.
  * Li et al. (2016) Li C.-H., Wang C., Keplinger C., Zuo J.-L., Jin L., Sun Y., Zheng P., Cao Y., Lissel F., Linder C., You X.-Z., Bao Z.,  A highly stretchable autonomous self-healing elastomer,  Nature Chemistry 8 (2016) 618–624. doi:10.1038/nchem.2492.
  * Li et al. (2021) Li Z., Yu R., Guo B.,  Shape-Memory and Self-Healing Polymers Based on Dynamic Covalent Bonds and Dynamic Noncovalent Interactions: Synthesis, Mechanism, and Application,  ACS Applied Bio Materials 4 (2021) 5926–5943. doi:10.1021/acsabm.1c00606.
  * Wojtecki et al. (2011) Wojtecki R. J., Meador M. A., Rowan S. J.,  Using the dynamic bond to access macroscopically responsive structurally dynamic polymers,  Nature Materials 10 (2011) 14–27. doi:10.1038/nmat2891.
  * Eom et al. (2021) Eom Y., Kim S.-M., Lee M., Jeon H., Park J., Lee E. S., Hwang S. Y., Park J., Oh D. X.,  Mechano-responsive hydrogen-bonding array of thermoplastic polyurethane elastomer captures both strength and self-healing,  Nature Communications 12 (2021) 621. doi:10.1038/s41467-021-20931-z.
  * Doolan et al. (2023) Doolan J. A., Alesbrook L. S., Baker K., Brown I. R., Williams G. T., Hilton K. L. F., Tabata M., Wozniakiewicz P. J., Hiscock J. R., Goult B. T.,  Next-generation protein-based materials capture and preserve projectiles from supersonic impacts,  Nature Nanotechnology (2023) 1–7. doi:10.1038/s41565-023-01431-1.
  * Ge and Rubinstein (2015) Ge T., Rubinstein M.,  Strong Selective Adsorption of Polymers,  Macromolecules 48 (2015) 3788–3801. doi:10.1021/acs.macromol.5b00586.
  * Yount et al. (2005) Yount W. C., Loveless D. M., Craig S. L.,  Small-Molecule Dynamics and Mechanisms Underlying the Macroscopic Mechanical Properties of Coordinatively Cross-Linked Polymer Networks,  Journal of the American Chemical Society 127 (2005) 14488–14496. doi:10.1021/ja054298a.
  * Zhang et al. (2020) Zhang X., Vidavsky Y., Aharonovich S., Yang S. J., Buche M. R., Diesendruck C. E., Silberstein M. N.,  Bridging experiments and theory: isolating the effects of metal–ligand interactions on viscoelasticity of reversible polymer networks,  Soft Matter 16 (2020) 8591–8601. doi:10.1039/D0SM01115K.
  * Huang et al. (2022) Huang Y., Jayathilaka P. B., Islam M. S., Tanaka C. B., Silberstein M. N., Kilian K. A., Kruzic J. J.,  Structural aspects controlling the mechanical and biological properties of tough, double network hydrogels,  Acta Biomaterialia 138 (2022) 301–312. doi:10.1016/j.actbio.2021.10.044.
  * James and Guth (1943) James H. M., Guth E.,  Theory of the Elastic Properties of Rubber,  The Journal of Chemical Physics 11 (1943) 455–481. doi:10.1063/1.1723785.
  * Flory (1985) Flory P. J.,  Molecular Theory of Rubber Elasticity,  Polymer Journal 17 (1985) 1–12. doi:10.1295/polymj.17.1.
  * Tanaka and Edwards (1992) Tanaka F., Edwards S. F.,  Viscoelastic properties of physically crosslinked networks. 1. Transient network theory,  Macromolecules 25 (1992) 1516–1523. doi:10.1021/ma00031a024.
  * Arruda and Boyce (1993) Arruda E. M., Boyce M. C.,  A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials,  Journal of the Mechanics and Physics of Solids 41 (1993) 389–412. doi:10.1016/0022-5096(93)90013-6.
  * Wu and Van Der Giessen (1993) Wu P. D., Van Der Giessen E.,  On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers,  Journal of the Mechanics and Physics of Solids 41 (1993) 427–456. doi:10.1016/0022-5096(93)90043-F.
  * Miehe et al. (2004) Miehe C., Göktepe S., Lulei F.,  A micro-macro approach to rubber-like materials—Part I: the non-affine micro-sphere model of rubber elasticity,  Journal of the Mechanics and Physics of Solids 52 (2004) 2617–2660. doi:10.1016/j.jmps.2004.03.011.
  * Vernerey et al. (2017) Vernerey F. J., Long R., Brighenti R.,  A statistically-based continuum theory for polymers with transient networks,  Journal of the Mechanics and Physics of Solids 107 (2017) 1–20. doi:10.1016/j.jmps.2017.05.016.
  * Saleh (2015) Saleh O. A.,  Perspective: Single polymer mechanics across the force regimes,  The Journal of Chemical Physics 142 (2015) 194902. doi:10.1063/1.4921348.
  * Flory (1942) Flory P. J.,  Thermodynamics of High Polymer Solutions,  The Journal of Chemical Physics 10 (1942) 51–61. doi:10.1063/1.1723621.
  * Hong et al. (2009) Hong W., Liu Z., Suo Z.,  Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load,  International Journal of Solids and Structures 46 (2009) 3282–3289. doi:10.1016/j.ijsolstr.2009.04.022.
  * Bouklas and Huang (2012) Bouklas N., Huang R.,  Swelling kinetics of polymer gels: comparison of linear and nonlinear theories,  Soft Matter 8 (2012) 8194–8203. doi:10.1039/C2SM25467K.
  * Leibler et al. (1991) Leibler L., Rubinstein M., Colby R. H.,  Dynamics of reversible networks,  Macromolecules 24 (1991) 4701–4707. doi:10.1021/ma00016a034.
  * Stukalin et al. (2013) Stukalin E. B., Cai L.-H., Kumar N. A., Leibler L., Rubinstein M.,  Self-Healing of Unentangled Polymer Networks with Reversible Bonds,  Macromolecules 46 (2013) 7525–7541. doi:10.1021/ma401111n.
  * Bouklas et al. (2015) Bouklas N., Landis C. M., Huang R.,  Effect of Solvent Diffusion on Crack-Tip Fields and Driving Force for Fracture of Hydrogels,  Journal of Applied Mechanics 82 (2015). doi:10.1115/1.4030587.
  * Shen and Vernerey (2020) Shen T., Vernerey F. J.,  Rate-dependent fracture of transient networks,  Journal of the Mechanics and Physics of Solids 143 (2020) 104028. doi:10.1016/j.jmps.2020.104028.
  * Lee et al. (2023) Lee J., Lee S., Chester S. A., Cho H.,  Finite element implementation of a gradient-damage theory for fracture in elastomeric materials,  International Journal of Solids and Structures 279 (2023) 112309. doi:10.1016/j.ijsolstr.2023.112309.
  * Bosnjak et al. (2022) Bosnjak N., Tepermeister M., Silberstein M. N.,  Modeling coupled electrochemical and mechanical behavior of soft ionic materials and ionotronic devices,  Journal of the Mechanics and Physics of Solids 168 (2022) 105014. doi:10.1016/j.jmps.2022.105014.
  * C. Picu (2011) C. Picu R.,  Mechanics of random fiber networks—a review,  Soft Matter 7 (2011) 6768–6785. doi:10.1039/C1SM05022B.
  * Bergström and Boyce (2001) Bergström J. S., Boyce M. C.,  Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity,  Macromolecules 34 (2001) 614–626. doi:10.1021/ma0007942.
  * Somasi et al. (2002) Somasi M., Khomami B., Woo N. J., Hur J. S., Shaqfeh E. S. G.,  Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues,  Journal of Non-Newtonian Fluid Mechanics 108 (2002) 227–255. doi:10.1016/S0377-0257(02)00132-5.
  * Doyle and Underhill (2005) Doyle P. S., Underhill P. T.,  Brownian Dynamics Simulations of Polymers and Soft Matter,  in: Yip S. (Ed.), Handbook of Materials Modeling: Methods, Springer Netherlands, Dordrecht, 2005, pp. 2619–2630.
  * Thompson et al. (2022) Thompson A. P., Aktulga H. M., Berger R., Bolintineanu D. S., Brown W. M., Crozier P. S., Veld in ’t P. J., Kohlmeyer A., Moore S. G., Nguyen T. D., Shan R., Stevens M. J., Tranchida J., Trott C., Plimpton S. J.,  LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales,  Computer Physics Communications 271 (2022) 108171. doi:10.1016/j.cpc.2021.108171.
  * Goodrich et al. (2018) Goodrich C. P., Brenner M. P., Ribbeck K.,  Enhanced diffusion by binding to the crosslinks of a polymer gel,  Nature Communications 9 (2018) 4348. doi:10.1038/s41467-018-06851-5.
  * Evans and Ritchie (1997) Evans E., Ritchie K.,  Dynamic strength of molecular adhesion bonds,  Biophysical Journal 72 (1997) 1541–1555. doi:10.1016/S0006-3495(97)78802-7.
  * Hoy and Fredrickson (2009) Hoy R. S., Fredrickson G. H.,  Thermoreversible associating polymer networks. I. Interplay of thermodynamics, chemical kinetics, and polymer physics,  The Journal of Chemical Physics 131 (2009) 224902. doi:10.1063/1.3268777.
  * Amin et al. (2016) Amin D., Likhtman A. E., Wang Z.,  Dynamics in Supramolecular Polymer Networks Formed by Associating Telechelic Chains,  Macromolecules 49 (2016) 7510–7524. doi:10.1021/acs.macromol.6b00561.
  * Perego and Khabaz (2020) Perego A., Khabaz F.,  Volumetric and Rheological Properties of Vitrimers: A Hybrid Molecular Dynamics and Monte Carlo Simulation Study,  Macromolecules 53 (2020) 8406–8416. doi:10.1021/acs.macromol.0c01423.
  * Zhao et al. (2022) Zhao H., Wei X., Fang Y., Gao K., Yue T., Zhang L., Ganesan V., Meng F., Liu J.,  Molecular Dynamics Simulation of the Structural, Mechanical, and Reprocessing Properties of Vitrimers Based on a Dynamic Covalent Polymer Network,  Macromolecules 55 (2022) 1091–1103. doi:10.1021/acs.macromol.1c02034.
  * Wagner et al. (2024) Wagner R. J., Lamont S. C., White Z. T., Vernerey F. J.,  Catch bond kinetics are instrumental to cohesion of fire ant rafts under load,  Proceedings of the National Academy of Sciences 121 (2024) e2314772121. doi:10.1073/pnas.2314772121.
  * Huang et al. (2023) Huang J., Ramlawi N., Sheridan G. S., Chen C., Ewoldt R. H., Braun P. V., Evans C. M.,  Dynamic Covalent Bond Exchange Enhances Penetrant Diffusion in Dense Vitrimers,  Macromolecules 56 (2023) 1253–1262. doi:10.1021/acs.macromol.2c02547.
  * Taylor et al. (2024) Taylor P. A., Wang J., Ge T., O’Connor T. C., Grest G. S.,  Smoother Surfaces Enhance Diffusion of Nanorods in Entangled Polymer Melts,  Macromolecules 57 (2024) 2482–2489. doi:10.1021/acs.macromol.3c01826.
  * Yang et al. (2015) Yang H., Yu K., Mu X., Shi X., Wei Y., Guo Y., Jerry Qi H.,  A molecular dynamics study of bond exchange reactions in covalent adaptable networks,  Soft Matter 11 (2015) 6305–6317. doi:10.1039/C5SM00942A.
  * Amin and Wang (2020) Amin D., Wang Z.,  Nonlinear rheology and dynamics of supramolecular polymer networks formed by associative telechelic chains under shear and extensional flows,  Journal of Rheology 64 (2020) 581–600. doi:10.1122/1.5120897.
  * Zheng et al. (2021) Zheng X., Yang H., Sun Y., Zhang Y., Guo Y.,  A molecular dynamics simulation on self-healing behavior based on disulfide bond exchange reactions,  Polymer 212 (2021) 123111. doi:10.1016/j.polymer.2020.123111.
  * Agrawal et al. (2016) Agrawal V., Holzworth K., Nantasetphong W., Amirkhizi A. V., Oswald J., Nemat-Nasser S.,  Prediction of viscoelastic properties with coarse-grained molecular dynamics and experimental validation for a benchmark polyurea system,  Journal of Polymer Science Part B: Polymer Physics 54 (2016) 797–810. doi:10.1002/polb.23976.
  * Liu et al. (2020) Liu H., Fu H., Shao X., Cai W., Chipot C.,  Accurate Description of Cation–$\pi$ Interactions in Proteins with a Nonpolarizable Force Field at No Additional Cost,  Journal of Chemical Theory and Computation 16 (2020) 6397–6407. doi:10.1021/acs.jctc.0c00637.
  * Zhang et al. (2023) Zhang X., Dai J., Tepermeister M., Deng Y., Yeo J., Silberstein M. N.,  Understanding How Metal–Ligand Coordination Enables Solvent Free Ionic Conductivity in PDMS,  Macromolecules 56 (2023) 3119–3131. doi:10.1021/acs.macromol.2c02519.
  * Kremer and Grest (1990) Kremer K., Grest G. S.,  Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation,  The Journal of Chemical Physics 92 (1990) 5057–5086. doi:10.1063/1.458541.
  * Hernández Cifre et al. (2003) Hernández Cifre J. G., Barenbrug T. M. A. O. M., Schieber J. D., Brule van den B. H. A. A.,  Brownian dynamics simulation of reversible polymer networks under shear using a non-interacting dumbbell model,  Journal of Non-Newtonian Fluid Mechanics 113 (2003) 73–96. doi:10.1016/S0377-0257(03)00063-6.
  * Sugimura et al. (2013) Sugimura A., Asai M., Matsunaga T., Akagi Y., Sakai T., Noguchi H., Shibayama M.,  Mechanical properties of a polymer network of Tetra-PEG gel,  Polymer Journal 45 (2013) 300–306. doi:10.1038/pj.2012.149.
  * Kothari et al. (2018) Kothari K., Hu Y., Gupta S., Elbanna A.,  Mechanical Response of Two-Dimensional Polymer Networks: Role of Topology, Rate Dependence, and Damage Accumulation,  Journal of Applied Mechanics 85 (2018). doi:10.1115/1.4038883.
  * Wagner et al. (2021) Wagner R. J., Hobbs E., Vernerey F. J.,  A network model of transient polymers: exploring the micromechanics of nonlinear viscoelasticity,  Soft Matter 17 (2021) 8742–8757. doi:10.1039/D1SM00753J.
  * Wyse Jackson et al. (2022) Wyse Jackson T., Michel J., Lwin P., Fortier L. A., Das M., Bonassar L. J., Cohen I.,  Structural origins of cartilage shear mechanics,  Science Advances 8 (2022) eabk2805. doi:10.1126/sciadv.abk2805.
  * Wagner et al. (2022) Wagner R. J., Dai J., Su X., Vernerey F. J.,  A mesoscale model for the micromechanical study of gels,  Journal of the Mechanics and Physics of Solids (2022) 104982. doi:10.1016/j.jmps.2022.104982.
  * Wagner and Vernerey (2023) Wagner R. J., Vernerey F. J.,  Coupled bond dynamics alters relaxation in polymers with multiple intrinsic dissociation rates,  Soft Matter (2023). doi:10.1039/D3SM00014A.
  * Richardson et al. (2019) Richardson B. M., Wilcox D. G., Randolph M. A., Anseth K. S.,  Hydrazone covalent adaptable networks modulate extracellular matrix deposition for cartilage tissue engineering,  Acta Biomaterialia 83 (2019) 71–82. doi:10.1016/j.actbio.2018.11.014.
  * Cruz et al. (2012) Cruz C., Chinesta F., Régnier G.,  Review on the Brownian Dynamics Simulation of Bead-Rod-Spring Models Encountered in Computational Rheology,  Archives of Computational Methods in Engineering 19 (2012) 227–259. doi:10.1007/s11831-012-9072-2.
  * Sliozberg and Chantawansri (2013) Sliozberg Y. R., Chantawansri T. L.,  Computational study of imperfect networks using a coarse-grained model,  The Journal of Chemical Physics 139 (2013) 194904. doi:10.1063/1.4832140.
  * Einstein (1905) Einstein A.,  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,  Annalen der Physik 322 (1905) 549–560. doi:10.1002/andp.19053220806.
  * Rubinstein and Colby (2003) Rubinstein M., Colby R. H., Polymer Physics, - hardcover - michael rubinstein; ralph h. colby - ed., Oxford University Press, 2003.
  * Doi (2013) Doi M., Soft Matter Physics, OUP Oxford, 2013.
  * Cohen (1991) Cohen A.,  A Padé approximant to the inverse Langevin function,  Rheologica Acta 30 (1991) 270–273. doi:10.1007/BF00366640.
  * Rector et al. (1994) Rector D. R., Swol F. V., Henderson J. R.,  Simulation of surfactant solutions,  Molecular Physics (1994). doi:10.1080/00268979400100724.
  * Shimada et al. (2005) Shimada K., Kato H., Saito T., Matsuyama S., Kinugasa S.,  Precise measurement of the self-diffusion coefficient for poly(ethylene glycol) in aqueous solution using uniform oligomers,  The Journal of Chemical Physics 122 (2005) 244914. doi:10.1063/1.1948378.
  * Kravanja et al. (2018) Kravanja G., Škerget M., Knez Ž., Knez Hrnčič M.,  Diffusion coefficients of water and propylene glycol in supercritical CO2 from pendant drop tensiometry,  The Journal of Supercritical Fluids 133 (2018) 1–8. doi:10.1016/j.supflu.2017.09.022.
  * Shi (2021) Shi Q.,  Molecular dynamics simulation of diffusion and separation of CO2/CH4/N2 on MER zeolites,  Journal of Fuel Chemistry and Technology 49 (2021) 1531–1539. doi:10.1016/S1872-5813(21)60095-6.
  * Rouse (1953) Rouse P. E., Jr.,  A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers,  The Journal of Chemical Physics 21 (1953) 1272–1280. doi:10.1063/1.1699180.
  * Eyring (1935) Eyring H.,  The Activated Complex and the Absolute Rate of Chemical Reactions.,  Chemical Reviews 17 (1935) 65–77. doi:10.1021/cr60056a006.
  * Hult et al. (1999) Hult A., Johansson M., Malmström E., Freire J., Burchard W., McLeish T., Milner S., Advances in Polymer Science, Branched Polymers II, Springer, 1999.
  * Bell (1978) Bell G. I.,  Models for the Specific Adhesion of Cells to Cells,  Science 200 (1978) 618–627. doi:10.1126/science.347575.
  * Song et al. (2021) Song Z., Shen T., Vernerey F. J., Cai S.,  Force-dependent bond dissociation explains the rate-dependent fracture of vitrimers,  Soft Matter 17 (2021) 6669–6674. doi:10.1039/D1SM00518A.
  * Buche and Silberstein (2021) Buche M. R., Silberstein M. N.,  Chain breaking in the statistical mechanical constitutive theory of polymer networks,  Journal of the Mechanics and Physics of Solids 156 (2021) 104593. doi:10.1016/j.jmps.2021.104593.
  * Guo et al. (2009) Guo S., Lad N., Ray C., Akhremitchev B. B.,  Association Kinetics from Single Molecule Force Spectroscopy Measurements,  Biophysical Journal 96 (2009) 3412–3422. doi:10.1016/j.bpj.2009.01.031.
  * Bell and Terentjev (2017) Bell S., Terentjev E. M.,  Kinetics of Tethered Ligands Binding to a Surface Receptor,  Macromolecules 50 (2017) 8810–8815. doi:10.1021/acs.macromol.7b01742.
  * Marzocca et al. (2013) Marzocca A. J., Salgueiro W., Somoza A.,  Physical Phenomena Related to Free Volumes in Rubber and Blends,  in: Visakh P. M., Thomas S., Chandra A. K., Mathew A. P. (Eds.), Advances in Elastomers II: Composites and Nanocomposites, Advanced Structured Materials, Springer, Berlin, Heidelberg, 2013, pp. 399–426.
  * Qi et al. (2003) Qi H. J., Joyce K., Boyce M. C.,  Durometer Hardness and the Stress-Strain Behavior of Elastomeric Materials,  Rubber Chemistry and Technology 76 (2003) 419–435. doi:10.5254/1.3547752.
  * Vatankhah-Varnosfaderani et al. (2017) Vatankhah-Varnosfaderani M., Daniel W. F. M., Everhart M. H., Pandya A. A., Liang H., Matyjaszewski K., Dobrynin A. V., Sheiko S. S.,  Mimicking biological stress–strain behaviour with synthetic elastomers,  Nature 549 (2017) 497–501. doi:10.1038/nature23673.
  * Shibayama et al. (2019) Shibayama M., Li X., Sakai T.,  Precision polymer network science with tetra-PEG gels—a decade history and future,  Colloid and Polymer Science 297 (2019) 1–12. doi:10.1007/s00396-018-4423-7.
  * You et al. (2024) You H., Zheng S., Li H., Lam K. Y.,  A model with contact maps at both polymer chain and network scales for tough hydrogels with chain entanglement, hidden length and unconventional network topology,  International Journal of Mechanical Sciences 262 (2024) 108713. doi:10.1016/j.ijmecsci.2023.108713.
  * Herzberg (1955) Herzberg G.,  Molecular Vibrations. The theory of infrared and Raman vibrational spectra. E. Bright Wilson, Jr., J. C. Decius, and Paul C. Cross. McGraw-Hill, New York-London, 1955. xi + 371 pp. Illus. $8.50.,  Science 122 (1955) 422–422. doi:10.1126/science.122.3166.422.a.
  * Yu et al. (2014) Yu K., Ge Q., Qi H. J.,  Effects of stretch induced softening to the free recovery behavior of shape memory polymer composites,  Polymer 55 (2014) 5938–5947. doi:10.1016/j.polymer.2014.06.050.
  * Wanasinghe et al. (2022) Wanasinghe S. V., Dodo O. J., Konkolewicz D.,  Dynamic Bonds: Adaptable Timescales for Responsive Materials,  Angewandte Chemie International Edition 61 (2022) e202206938. doi:10.1002/anie.202206938.
  * Chen et al. (2019) Chen M., Zhou L., Wu Y., Zhao X., Zhang Y.,  Rapid Stress Relaxation and Moderate Temperature of Malleability Enabled by the Synergy of Disulfide Metathesis and Carboxylate Transesterification in Epoxy Vitrimers,  ACS Macro Letters 8 (2019) 255–260. doi:10.1021/acsmacrolett.9b00015.
  * Puthur and Sebastian (2002) Puthur R., Sebastian K. L.,  Theory of polymer breaking under tension,  Physical Review B 66 (2002) 024304. doi:10.1103/PhysRevB.66.024304.
  * Lamont et al. (2021) Lamont S. C., Mulderrig J., Bouklas N., Vernerey F. J.,  Rate-Dependent Damage Mechanics of Polymer Networks with Reversible Bonds,  Macromolecules 54 (2021) 10801–10813. doi:10.1021/acs.macromol.1c01943.
  * Buche et al. (2022) Buche M. R., Silberstein M. N., Grutzik S. J.,  Freely jointed chain models with extensible links,  Physical Review E 106 (2022) 024502. doi:10.1103/PhysRevE.106.024502.
  * Mulderrig et al. (2023) Mulderrig J., Talamini B., Bouklas N.,  A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching,  Journal of the Mechanics and Physics of Solids (2023) 105244.
  * Treloar (1943) Treloar L. R. G.,  The elasticity of a network of long-chain molecules—II,  Transactions of the Faraday Society 39 (1943) 241–246. doi:10.1039/TF9433900241.
  * Okamoto et al. (2011) Okamoto R. J., Clayton E. H., Bayly P. V.,  Viscoelastic properties of soft gels: comparison of magnetic resonance elastography and dynamic shear testing in the shear wave regime,  Physics in Medicine & Biology 56 (2011) 6379. doi:10.1088/0031-9155/56/19/014.
  * Chen et al. (2015) Chen Q., Huang C., Weiss R. A., Colby R. H.,  Viscoelasticity of Reversible Gelation for Ionomers,  Macromolecules 48 (2015) 1221–1230. doi:10.1021/ma502280g.
  * Shabbir et al. (2016) Shabbir A., Javakhishvili I., Cerveny S., Hvilsted S., Skov A. L., Hassager O., Alvarez N. J.,  Linear Viscoelastic and Dielectric Relaxation Response of Unentangled UPy-Based Supramolecular Networks,  Macromolecules 49 (2016) 3899–3910. doi:10.1021/acs.macromol.6b00122.
  * Peter et al. (2021) Peter T., Deal H., Zhao H., He A., Tang C., Lemmon C.,  Rheological characterization of poly-dimethyl siloxane formulations with tunable viscoelastic properties,  RSC Advances (2021). doi:doi.org/10.1039/D1RA03548G.
  * Colby et al. (1998) Colby R. H., Zheng X., Rafailovich M. H., Sokolov J., Peiffer D. G., Schwarz S. A., Strzhemechny Y., Nguyen D.,  Dynamics of Lightly Sulfonated Polystyrene Ionomers,  Physical Review Letters 81 (1998) 3876–3879. doi:10.1103/PhysRevLett.81.3876.
  * Chen et al. (2013) Chen Q., Liang S., Shiau H.-s., Colby R. H.,  Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers,  ACS Macro Letters 2 (2013) 970–974. doi:10.1021/mz400476w.
  * Xie et al. (2024) Xie M.-J., Wang C.-C., Zhang R., Cao J., Tang M.-Z., Xu Y.-X.,  Length effect of crosslinkers on the mechanical properties and dimensional stability of vitrimer elastomers with inhomogeneous networks,  Polymer 290 (2024) 126550. doi:10.1016/j.polymer.2023.126550.
  * Zimm et al. (1953) Zimm B. H., Stockmayer W. H., Fixman M.,  Excluded Volume in Polymer Chains,  The Journal of Chemical Physics 21 (1953) 1716–1723. doi:10.1063/1.1698650.
  * Sun and Faller (2006) Sun Q., Faller R.,  Crossover from Unentangled to Entangled Dynamics in a Systematically Coarse-Grained Polystyrene Melt,  Macromolecules 39 (2006) 812–820. doi:10.1021/ma0514774.
  * Ge et al. (2013) Ge T., Pierce F., Perahia D., Grest G. S., Robbins M. O.,  Molecular Dynamics Simulations of Polymer Welding: Strength from Interfacial Entanglements,  Physical Review Letters 110 (2013) 098301. doi:10.1103/PhysRevLett.110.098301.
  * Schieber and Andreev (2014) Schieber J. D., Andreev M.,  Entangled Polymer Dynamics in Equilibrium and Flow Modeled Through Slip Links,  Annual Review of Chemical and Biomolecular Engineering 5 (2014) 367–381. doi:10.1146/annurev-chembioeng-060713-040252.
  * Masubuchi (2014) Masubuchi Y.,  Simulating the Flow of Entangled Polymers,  Annual Review of Chemical and Biomolecular Engineering 5 (2014) 11–33. doi:10.1146/annurev-chembioeng-060713-040401.
  * Ge et al. (2018) Ge T., Grest G. S., Rubinstein M.,  Nanorheology of Entangled Polymer Melts,  Physical Review Letters 120 (2018) 057801. doi:10.1103/PhysRevLett.120.057801.
  * Kim et al. (2021) Kim J., Zhang G., Shi M., Suo Z.,  Fracture, fatigue, and friction of polymers in which entanglements greatly outnumber cross-links,  Science 374 (2021) 212–216. doi:10.1126/science.abg6320.
  * Steck et al. (2023) Steck J., Kim J., Kutsovsky Y., Suo Z.,  Multiscale stress deconcentration amplifies fatigue resistance of rubber,  Nature 624 (2023) 303–308. doi:10.1038/s41586-023-06782-2.
  * Shi et al. (2023) Shi M., Kim J., Nian G., Suo Z.,  Highly entangled hydrogels with degradable crosslinks,  Extreme Mechanics Letters 59 (2023) 101953. doi:10.1016/j.eml.2022.101953.
  * Ahlawat et al. (2021) Ahlawat V., Rajput S. S., Patil S.,  Elasticity of single flexible polymer chains in good and poor solvents,  Polymer 230 (2021) 124031. doi:10.1016/j.polymer.2021.124031.
  * Liese et al. (2017) Liese S., Gensler M., Krysiak S., Schwarzl R., Achazi A., Paulus B., Hugel T., Rabe J. P., Netz R. R.,  Hydration Effects Turn a Highly Stretched Polymer from an Entropic into an Energetic Spring,  ACS Nano 11 (2017) 702–712. doi:10.1021/acsnano.6b07071.
  * Lee et al. (2008) Lee H., Venable R. M., MacKerell A. D., Pastor R. W.,  Molecular Dynamics Studies of Polyethylene Oxide and Polyethylene Glycol: Hydrodynamic Radius and Shape Anisotropy,  Biophysical Journal 95 (2008) 1590–1599. doi:10.1529/biophysj.108.133025.
  * Consolati et al. (2023) Consolati G., Nichetti D., Quasso F.,  Probing the Free Volume in Polymers by Means of Positron Annihilation Lifetime Spectroscopy,  Polymers 15 (2023) 3128. doi:10.3390/polym15143128.
  * Tao et al. (2023) Tao L., He J., Arbaugh T., McCutcheon J. R., Li Y.,  Machine learning prediction on the fractional free volume of polymer membranes,  Journal of Membrane Science 665 (2023) 121131. doi:10.1016/j.memsci.2022.121131.
  * Boerner et al. (2023) Boerner T. J., Deems S., Furlani T. R., Knuth S. L., Towns J.,  ACCESS: Advancing Innovation: NSF’s Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support,  in: Practice and Experience in Advanced Research Computing, PEARC ’23, Association for Computing Machinery, New York, NY, USA, 2023, pp. 173–176. doi:10.1145/3569951.3597559.


&\, \mathbb{P}\big(R\geq a\omega^2t\big)\nonumber\\\leq &\,\sum^p_{J=1}\mathbb{P}\big(  \sup_{\bar\beta\in \Gamma_J}\langle\epsilon,\bar{X}\bar{\beta}\rangle-\frac{1}{2a}\|\bar{X}\bar{\beta}\|^2_2-\frac{1}{2b}P_{2,0}(\bar\beta;\sqrt{ab}A_1\lambda_0)\geq a\omega^2t\big)\nonumber\\= &\,\sum^p_{J=1}\mathbb{P}\big(  \sup_{\bar\beta\in \Gamma_J}\langle\epsilon,\bar{X}\bar{\beta}\rangle-\frac{1}{2a}\|\bar{X}\bar{\beta}\|^2_2-\frac{1}{4}aA_1^2J\log (ep)\omega^2_{\bar\eta}\geq a\omega^2t\big)\nonumber\\\leq &\,\sum^p_{J=1}\mathbb{P}\big(  \sup_{\bar\beta\in \Gamma_J}\langle\epsilon,\bar{X}\bar{\beta}\rangle-\frac{1}{2a}\|\bar{X}\bar{\beta}\|^2_2-\frac{a}{2}LJ\log (ep)\omega^2\geq a\omega^2t+caA^2_1 J\log (ep)\omega^2\big)\nonumber\\\leq &\,\sum^p_{J=1}C\exp(-ct)\sum^p_{J=1}\exp\{-cA_1^2(J+\log p)\}\nonumber\\\leq &\,C\exp(-ct)p^{-cA^2_1},
\end{align}

where the last inequality is due to the sum of geometric sequence.

Given a  matrix $X\in \mathbb{R}^{n\times pG}$ with   a block form of $X=[X_1,\ldots,X_p]$ with $X_j\in\mathbb{R}^{n\times G},1\leq j\leq p$,
let $X_{\mathcal{J}}$ denote the submatrix formed by the column blocks of $X$ indexed by $\mathcal{J}$. Define $\Gamma_{\mathcal{J}}=\{\alpha\in \mathbb{R}^{pG}: \|\alpha\|_2\leq 1, \alpha\in \mathcal{R}(X_{\mathcal{J}})\}$ and  $\Gamma_{J}=\bigcup_{|\mathcal{J}|=J}\Gamma_{\mathcal{J}}$, where $1\leq J\leq p$.
Let $\epsilon=[\epsilon_i]$ be an $n$-dimensional random vector. (i) Assume $\epsilon$ satisfies $\|\epsilon\|_{\psi_q}\leq \omega$ for some $q\geq 1$. Then for any $t>0$,

\begin{align}
&  \mathbb{P}\big( \sup_{\alpha\in \Gamma_{\mathcal{J}}}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L(JG)^{\frac{2}{q}}\omega^2\geq \omega^2t\big)\leq C\exp\big(-ct^{\frac{q}{2}}\big), \label{lemmaass1medb}\\
&  \mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-L\big[JG+(JG)^{\frac{2}{q}}+(2J)^{\frac{2}{q}}\{\log (ep)\}^{\frac{2}{q}}\big]\omega^2\geq \omega^2t\big)\leq C\exp\big(-ct^{\frac{q}{2}}\big),\label{lemmaass1finalb}
\end{align}

where   $C$ is a universal constant  and  $c,L$ are constants   depending on $q$ only.
(ii) Assume that  $\epsilon_1,\ldots,\epsilon_n$  are independent, centered,  and  $\|\epsilon_i\|_{\psi_q}\leq \omega$ for some $q\in(0,2]$. Then (<ref>) and (<ref>) are replaced by
\begin{align}
&     \mathbb{P}\Big( \sup_{\alpha\in \Gamma_{\mathcal{J}}}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2\geq \omega^2t\Big)\leq C\exp\big(-ct^{\frac{q}{2}}\big), \label{lemmaunionbound1}
\\
&   \mathbb{P}\Big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-L\big[JG+(2J)^{\frac{2}{q}}\{\log(ep)\}^\frac{2}{q}\big]\omega^2\geq \omega^2t\Big)\leq C\exp\big(-ct^{\frac{q}{2}}\big).\label{lemmameanzerobound}
\end{align}

First, we show  (<ref>) under the assumption that $\|\epsilon\|_{\psi_q}\leq \omega$ for some $q\geq 1$.
Define a  centered random vector $\epsilon_c=\epsilon-M$ with  $M=\mathbb{E}[\epsilon]$, then

\begin{align}
\sup_{\alpha\in \Gamma_{\mathcal{J}}} \langle\epsilon,\alpha\rangle\leq\sup_{\alpha\in \Gamma_{\mathcal{J}}}\langle \epsilon_c,\alpha\rangle+\sup_{\alpha\in \Gamma_{\mathcal{J}}}\langle P_{ X_{\mathcal{J}}}M,\alpha\rangle\leq \sup_{\alpha\in \Gamma_\mathcal{J}}\langle\epsilon_c,\alpha\rangle+\|U^TM\|_2,
\end{align}

where  $P_{ X_{\mathcal{J}}}=UU^T$ is the orthogonal projection matrix onto $\mathcal{R}(X_{\mathcal{J}})$ with $\{U_1,\ldots,U_{JG}\}$ as  an orthonormal basis and $\|P_{ X_{\mathcal{J}}}\|_2=1$. We claim that $\|U^TM\|^2_2=\sum^{JG}_{i=1}(\mathbb{E}[U^T_i\epsilon])^2\leq CJG\omega^2$. In fact,
because $\|U^T_i\epsilon\|_{\psi_q}\leq \omega$,

\begin{align}\label{probtomean}
\mathbb{P}\big(|U^T_i\epsilon|>t\omega\big)\leq \int^{+\infty}_{0}\frac{1}{\psi_q(\frac{t\omega}{\omega})}\rd t=\int^{+\infty}_{0}\exp(-t^q)\rd t<\infty,
\end{align}

and so $\mathbb{E}|U^T_i\epsilon|\leq C\omega,\,1\leq i\leq J$.

By definition, $\{\langle\epsilon_c,\alpha\rangle:\alpha\in \Gamma_{\mathcal{J}}\}$ is a (centered)      $\psi_q$-process. The induced metric on $\Gamma_{\mathcal{J}}$ is: $d(\alpha,\alpha')=\omega\|\alpha-\alpha'\|_2$.
To bound the metric entropy $\log (\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)))$, where the $\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d))$ is the smallest cardinality of an $\varepsilon$-net that covers $\Gamma_{\mathcal{J}}$ under the metric $d$, we apply a standard volume argument to get

\begin{align}
\log (\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)))\leq \log  (\frac{C\omega}{\varepsilon})^{JG}=JG\log(C\omega/\varepsilon).
\end{align}

By Theorem 5.36 in [65],  we get

\begin{align}
\mathbb{P}\Big( \sup_{\alpha,\alpha'\in \Gamma_{\mathcal{J}}}|\langle\epsilon_c,\alpha-\alpha'\rangle|-L\int^{D}_{0}\psi^{-1}_q(\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)) \rd \varepsilon \geq Lt\Big)\leq 2\exp\big(-\frac{t^{q}}{D^q}\big),
\end{align}

for any $t>0$, where $D=\sup_{\alpha,\alpha'\in\Gamma_{\mathcal{J}}}d(\alpha,\alpha')=2\omega$. Equivalently,

\begin{align}
\mathbb{P}\Big( \sup_{\alpha,\alpha'\in \Gamma_{\mathcal{J}}}|\langle\epsilon_c,\alpha-\alpha'\rangle|-L\int^{2\omega}_{0}\psi^{-1}_q(\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)) \rd \varepsilon \geq \omega t\Big)\leq C\exp\big(-ct^{q}\big),
\end{align}

which implies

\begin{align}\label{lemma3meanzeromedinq}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_\mathcal{J}}\langle\epsilon_c,\alpha\rangle-L\int^{2\omega}_{0}\psi^{-1}_q(\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)) \rd \varepsilon \geq \omega t\Big)\leq C\exp\big(-ct^{q}\big).
\end{align}

Based on (<ref>) and (<ref>), we obtain

\begin{align}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_\mathcal{J}}\langle\epsilon,\alpha\rangle-L\sqrt{JG}\omega-L\int^{2\omega}_{0}\psi^{-1}_q(\mathcal{N}(\varepsilon,\Gamma_{\mathcal{J}},d)) \rd \varepsilon\geq \omega t\big)\leq C\exp\big(-ct^{q}\big),
\end{align}

\begin{align}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_\mathcal{J}}\langle\epsilon,\alpha\rangle-L\sqrt{JG}\omega-L(JG)^{\frac{1}{q}}\omega\geq \omega t\big)\leq C\exp\big(-ct^{q}\big).
\end{align}

\begin{align}
\mathbb{P}\Big( \sup_{\alpha\in \Gamma_{\mathcal{J}}}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L(JG)^{\frac{2}{q}}\omega^2\geq \omega^2t^2\Big)\leq C\exp\big(-ct^{q}\big),
\end{align}

and letting $s=t^2$ gives   (<ref>). With a union bound, we also obtain

\begin{align}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L(JG)^{\frac{2}{q}}\omega^2\geq \omega^2t\big)\leq &\, {p\choose J}C\exp\big(-ct^{\frac{q}{2}}\big)\nonumber\\\leq &\, C\exp\big\{-ct^{\frac{q}{2}}+J\log(ep)\big\},
\end{align}

from which it follows

\begin{align}
\mathbb{P}\Big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L(JG)^{\frac{2}{q}}\omega^2-L2^{\frac{2}{q}-1}\{J\log (ep)\}^{\frac{2}{q}}\omega^2\geq \omega^2t\Big)\leq C\exp\big(-ct^{\frac{q}{2}}\big).
\end{align}

Next, we prove (<ref>) assuming  $\epsilon_1,\ldots,\epsilon_n$  are independent, centered,  and $\|\epsilon_i\|_{\psi_q}\leq \omega$ for some $0<q\le 2$.
By Hölder's  inequality,

\begin{align}
\sup_{\alpha\in \Gamma_{\mathcal{J}}} \langle\epsilon,\alpha\rangle=\sup_{\alpha\in \Gamma_{\mathcal{J}}}\langle P_{ X_{\mathcal{J}}}\epsilon,\alpha\rangle\leq \|U^T\epsilon\|_2,
\end{align}

where $P_{ X_{\mathcal{J}}}=UU^T$ is the orthogonal projection matrix onto $\mathcal{R}(X_{\mathcal{J}})$ with $\{U_1,\ldots,U_{JG}\}$ as  an orthonormal basis and $\|P_{ X_{\mathcal{J}}}\|_2=1$.
It remains to get a tail bound for $\|U^T\epsilon\|^2_2$.

Noticing that    (i) $\mathbb{E}[\epsilon]=0$, (ii)   $\|\epsilon_i\|_{\psi_q}\leq \omega$ implies    $\mathbb{E}[\epsilon^2_i]\leq C\omega^2$, (iii) $\|UU^T\|_2=1$, and (iv) $Tr(UU^T)=JG$, we apply a generalized Hanson-Wright inequality ([53], Theorem 2.1) to obtain
\begin{align}\label{lemma3mean0medianineq1}
\mathbb{P}\big( \|U^T\epsilon\|^2_2-LJG\omega^2\geq \omega^2t\big)\leq C\exp\big(-ct^{\frac{q}{2}}\big),
\end{align}

where $C$ can be $2$  and  $c,L$ are constants   depending on $q$ only. This results in
\begin{align}\label{lemmamidleboundresult}
\mathbb{P}\Big( \sup_{\alpha\in \Gamma_{\mathcal{J}}}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2\geq \omega^2t\Big)\leq C\exp\big(-ct^{\frac{q}{2}}\big).
\end{align}

Finally, we prove (<ref>). Applying the union bound on  (<ref>) yields

\begin{align}\label{unionbondfirst}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2\geq \omega^2t\big)\leq {p\choose J}C\exp\big(-ct^{\frac{q}{2}}\big)\leq C\exp\big\{-ct^{\frac{q}{2}}+J\log(ep)\big\}.
\end{align}

Let $t'=t^{\frac{q}{2}}-(1/c)J\log(ep)$. Then (<ref>) can be rewritten as

\begin{align}\label{supineq1}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2\geq \omega^2\{t'+\frac{1}{c}J\log (ep)\}^{\frac{2}{q}}\big)\leq C\exp(-ct').
\end{align}

Given $0<q\leq 2$, by the convexity of $x^\frac{2}{q}$ on $\mathbb{R}_+$, we have  $(a+b)^\frac{2}{q}\leq 2^{\frac{2}{q}-1}(a^\frac{2}{q}+b^\frac{2}{q})$ for any $a,b\geq0$, and thus
(<ref>) becomes

\begin{align}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L2^{\frac{2}{q}-1}\{J\log (ep)\}^{\frac{2}{q}}\omega^2\geq 2^{\frac{2}{q}-1}\omega^2(t')^\frac{2}{q}\big)\leq C\exp(-ct'),
\end{align}

or equivalently,

\begin{align}
\mathbb{P}\big( \sup_{\alpha\in \Gamma_J}(\langle\epsilon,\alpha\rangle)^2-LJG\omega^2-L2^{\frac{2}{q}-1}\{J\log (ep)\}^{\frac{2}{q}}\omega^2\geq \omega^2s\big)\leq C\exp\big(-cs^{\frac{q}{2}}\big).
\end{align}

The proof is now complete.

§ FURTHER EXTENSIONS

We showcase two nonparametric statistical applications of skewed pivot-blend, one based on data ranks and the other based on kernels.

Skewed rank-based estimation.
Recently, there has been a lot of interest in rank-based  nonparametric estimation methods that minimize    $\sqrt{12}\sum^n_{i=1}r_i \{\mbox{R}(r_i)/(n+1) -\frac{1}{2} \}$ [33],
where $\mbox{R}(r_i)$ denotes the rank of $r_i$ among $r_1,\ldots,r_n$, or equivalently, the $\ell_1$ loss on the spread of residuals: $%\begin{align}
\frac{1}{ n(n-1) }\sum_{i\neq j}|r_i-r_j|
$ [28, 29, 67].

Applying the technique of U-statistics (with kernel $h(r_i, r_j)= |r_i-r_j|$) can show that the criterion results in estimators with desired asymptotic properties.

The adoption of the symmetric (and relatively robust) $\ell_1$-loss function heavily depends on the assumption that $r_i$ follows an i.i.d. distribution (when assessed against the statistical truth). In turn, it implies that the distribution of  differences  $r_i - r_j$ is symmetrical, as in the case of a double-exponential.

Nonetheless,   the assumption may not  hold in real-world applications beyond the i.i.d. setting. Such deviations can  result in extra skewness of $|r_i - r_j|$ that cannot be captured by an exponential (or half-normal) distribution. We introduce a skewed criterion that operates on the absolute differences $|r_i - r_j|$ and $m, \sigma, \nu>0$:

\begin{align}\label{skewrloss}
&\,\sum_{i\neq j}\Big\{ \big(\frac{|r_i-r_j|-m}{\sigma} + m\big) \,1_{m(1-\sigma)\le |r_i-r_j|\leq m}+ \big(\frac{|r_i-r_j|-m}{\nu} +m\big)\,1_{|r_i-r_j|>m}\Big\}\nonumber\\&\,\quad\quad+ n(n-1)\log \{ \sigma [1-\exp(-m)  ]+\nu \exp(-m) \}.
\end{align}

(<ref>) results from applying  skewed pivot-blend  to the exponential density (instead of   the double-exponential density). A regularization term (such as an $\ell_1$ penalty) can be incorporated to capture structural parsimony.

Kernel-assisted nonparametric skew estimation.
Assume  $y-X\beta^*\sim \mbox{SP}^{(\phi^*)}(\sigma^{*},\nu^{*},m^{*})$, where the functional form of the density  $\phi^{*}$ is also unknown.    We can employ  a backward-forward scheme for    nonparametric estimation, along with  explicit capture of skewness that  aligns with the theme of this paper.

To estimate   $\sigma^{*},\nu^{*},m^{*}$,       kernel may be employed to approximate    $\phi^{*}$, but the data  follow  $\mbox{SP}^{(\phi^*)} $. We can address this by using   backward pivot-blend. Holding $\beta, \sigma, \nu, m$ constant for now, and referring to the definitions of $\tilde{r}_i$ ($1\le i \le n$), $L(m)$, and $R(m)$ in Remark <ref>,  introduce a kernel density estimator with appropriate weights to estimate $\phi^*$ based on the transformed residuals $\tilde r_i$:

\begin{align}\label{rform}
\begin{split}
\phi_K(t;h)=\,\frac{1}{h}\sum^{n}_{i=1}&\Big\{\frac{\nu}{L(m)\nu+R(m)\sigma}K_h\big(\frac{r_i-m}{\sigma}+m-t\big)1_{r_i\leq m}\\&+\frac{\sigma}{L(m)\nu+R(m)\sigma}K_h\big(\frac{r_i-m}{\nu}+m-t\big)1_{r_i>m}\Big\}.
\end{split}
\end{align}

where   $K_h(t)=K\big(t/h\big)$, the kernel function $K(t)$ can be  any continuous symmetric function with $\int^{\infty}_{-\infty} K(t),dt=1$, and $h>0$ represents the bandwidth parameter.
The kernel approach in conjunction with (forward) skewed pivot-blend gives rise to a nonparametric skew-estimation problem, which warrants further investigation.

§ MORE EXPERIMENTS

In this part, we  consider some  densities which may not be unimodal or symmetric (as shown in   Figure <ref>). The experimental setup is the same as in Section  <ref>.

$\phi$: half-normal,  $m^*$: mean

$\phi$: Maxwell,   $m^*$: median

$\phi$: double Rayleigh,   $m^*$:  right mode

Plots of  $\phi$ for Ex 3–Ex 5. The choices  of $m^*$ are indicated in red dashed lines.

Ex 3. (Half-normal with the   mean as the pivotal point): Let $\phi$ be the half-normal density (with scale 1).  We set $n=700 $,   $\kappa=0.1$, $m^*$ as the mean,   $\nu^*=0.1$ and  $\sigma^*=0.2,0.3,0.4$.

Ex 4. (Maxwell-Boltzmann with the  median as the pivotal point): Here, $\phi$ is the Maxwell-Boltzmann density which is widely used in statistical mechanics  [31]. We set   $n=500$,   $\kappa=0.2$, $m^*$ as the median,  $\nu^*=0.1$ and  $\sigma^*=0.4,0.5,0.6$.

Ex 5. (Double Rayleigh with $m^*$ as the  right mode): Let $\phi$ be the double Rayleigh density (with scale 1),    $n=500$, $\kappa=0.1$,   $m^*$ be the  right mode,     $\nu^*=0.1$ and  $\sigma^*=0.2,0.3,0.4$.

13cSkewed half-normal

3c$\sigma^*/\nu^*=2$                              3c$\sigma^*/\nu^*=3$                               3c$\sigma^*/\nu^*=4$

3-5 7-9 11-13

Err($\beta$)      Err($\sigma$)     Err($\nu$)           Err($\beta$)      Err($\sigma$)     Err($\nu$)            Err($\beta$)      Err($\sigma$)     Err($\nu$)

QR$^*$          0.80     —      —          0.80     —       —                   0.80       —         —

BQR$^*$         0.80      0.49      0.62         0.80     0.51       0.64                   0.80       0.53        0.65

AME          0.64      0.99      1.22          0.56     1.0       2.02                   0.48       1.0         2.82

ZQR           0.65     0.97      0.09          0.59     0.95       0.37                   0.52       0.95         0.67

SPEUS          0.14     0.11      0.1          0.08     0.08       0.13                   0.06      0.09         0.17

13cSkewed Maxwell-Boltzmann
3c$\sigma^*/\nu^*=4$                              3c$\sigma^*/\nu^*=5$                               3c$\sigma^*/\nu^*=6$

3-5 7-9 11-13

Err($\beta$)      Err($\sigma$)     Err($\nu$)           Err($\beta$)      Err($\sigma$)     Err($\nu$)            Err($\beta$)      Err($\sigma$)     Err($\nu$)

QR$^*$          1.59     —      —          1.59     —       —                   1.59       —         —

BQR$^*$          1.59     0.41     0.74          1.59     0.43       0.75                   1.59       0.44         0.75

AME           1.47     0.22      0.78          1.46     0.19       0.84                   1.46      0.17        0.91

ZQR          1.49     0.58      0.30         1.48     0.58       0.32                   1.48       0.57         0.34

SPEUS          0.18     0.14      0.20          0.17     0.13      0.21                   0.16       0.12        0.30

13cSkewed double Rayleigh
3c$\sigma^*/\nu^*=2$                              3c$\sigma^*/\nu^*=3$                               3c$\sigma^*/\nu^*=4$

3-5 7-9 11-13

Err($\beta$)      Err($\sigma$)     Err($\nu$)           Err($\beta$)      Err($\sigma$)     Err($\nu$)            Err($\beta$)      Err($\sigma$)     Err($\nu$)

QR$^*$          1.0     —      —          1.0     —       —                   1.0       —         —

BQR$^*$           1.0     0.68      0.27          1.0     0.70       0.26                   1.0      0.71        0.26

AME            0.87     1.78      1.76         0.88     2.02       1.9                   0.89       2.15        1.92

ZQR          0.67     0.50      1.67          0.50     0.48       2.69                   0.32       0.45        3.78

SPEUS          0.22     0.03      0.18          0.15     0.03       0.24                   0.10      0.03      0.26

Performance comparison for skewed half-normal  with pivotal point at the  mean,  skewed Maxwell with pivotal point at the median, and skewed double Rayleigh with pivotal point at the right mode (Ex 3–Ex 5)

Table <ref> illustrates significant issues with conventional methods in capturing skewness, even when asymmetry is centered around the median, mean, or mode. Specifically, QR$^*$ and BQR$^*$ failed to accurately recover the true $\beta^*$ despite using the true quantile as input.  Moreover, in all three cases, AME or AME provided at least one misleading estimate of the scales. In contrast, our proposed method demonstrated remarkable performance, with the associated $\beta$-error being at most $1/3$ of that of the other methods.

We also conducted  experiments  on the air pollution data of
Leeds from 1994 to 1998
[27, 59]. The dataset contain
578 measurements of the daily maximum levels of  ozone  (O3), nitrogen dioxide (NO2), nitrogen oxide (NO), sulfur dioxide (SO2) and particulate matter  (PM10). We forecasted PM10 levels for the summer months (April to July) using  other air pollutants. We considered the Gumbel model and its skewed pivot-blend enhancement SPEUS, in addition to ZQR, AME, and ESN.
The p-values from the Kolmogorov-Smirnov tests for these models were  8e-2, 0.71,  9e-2,  0.20, and 8e-8,  respectively.  The results may appear surprising, considering the popularity of the skewed Gumbel distribution  for modeling such extreme values [12]. Figure <ref> presents   the residual Q-Q plots, demonstrating  that our skew-reinforced method significantly improves data fit compared to the classical Gumbel model.

Q-Q plots of residuals for Gumbel (left) and SPEUS (right) on air pollution  data.

In real-world applications, data scientists frequently confront challenges related to skewness. However,   fitting a   common distribution that inadequately addresses these distortions can lead to subpar model fits and  misleading inferences. The skewed pivot-blend technique refines skewness management in these distributions, thus enhancing the accuracy and reliability of the models.

[1]
Agency for Healthcare Research and Quality (2015).
MEPS HC-181: 2015 Full Year Consolidated Data File.

[2]
Agency for Healthcare Research and Quality (2019).
MEPS HC-216: 2019 Full Year Consolidated Data File.

[3]
Arellano-Valle, R. B., Gómez, H. W., and Quintana, F. A. (2005).
Statistical inference for a general class of asymmetric
Journal of Statistical Planning and Inference, 128(2):427–443.

[4]
Axon, D. R. and Kamel, A. (2021).
Patterns of healthcare expenditures among older United States
adults with pain and different perceived health status.
Healthcare, 9(10):1327.

[5]
Azzalini, A. (1985).
A class of distributions which includes the normal ones.
Scandinavian Journal of Statistics, 171–178.

[6]
Azzalini, A. (2005).
The skew-normal distribution and related multivariate families.
Scandinavian Journal of Statistics, 32(2):159–188.

[7]
Bartlett, P. L., Harvey, N., Liaw, C., and Mehrabian, A. (2019).
Nearly-tight VC-dimension and pseudo-dimension bounds for piecewise
linear neural networks.
The Journal of Machine Learning Research, 20(1):2285–2301.

[8]
Benoit, D. F. and Van den Poel, D. (2017).
bayesQR: A Bayesian approach to quantile regression.
Journal of Statistical Software, 76(1):1–32.

[9]
Bera, A. K., Galvao, A. F., Montes-Rojas, G. V., and Park, S. Y. (2016).
Asymmetric Laplace regression: Maximum likelihood, maximum entropy
and quantile regression.
Journal of Econometric Methods, 5(1):79–101.

[10]
Bernardi, A. and Bernardi, M. (2018).
Two-sided skew and shape dynamic conditional score models.
Mathematical and Statistical Methods for Actuarial Sciences and
Finance, pages 121–124.

[11]
Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009).
Simultaneous analysis of Lasso and Dantzig selector.
The Annals of Statistics, 37(4):1705–1732.

[12]
Boldi, M. O. and Davison, A. C. (2007).
A mixture model for multivariate extremes.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 69(2):217–229.

[13]
Boos, D. D. (1987).
Detecting skewed errors from regression residuals.
Technometrics, 29(1):83–90.

[14]
Bregman, L. M. (1967).
The relaxation method of finding the common point of convex sets and
its application to the solution of problems in convex programming.
USSR Computational Mathematics and Mathematical Physics,

[15]
Chai, H. S. and Bailey, K. R. (2008).
Use of log-skew-normal distribution in analysis of continuous data
with a discrete component at zero.
Statistics in Medicine, 27(18):3643–3655.

[16]
Chang, B. and Joe, H. (2019).
Prediction based on conditional distributions of vine copulas.
Computational Statistics $\&$ Data Analysis, 139:45–63.

[17]
Cooray, K. and Ananda, M. M. (2005).
Modeling actuarial data with a composite lognormal-pareto model.
Scandinavian Actuarial Journal, 2005(5):321–334.

[18]
Cragg, J. G. (1971).
Some statistical models for limited dependent variables with
application to the demand for durable goods.
Econometrica: Journal of the Econometric Society, 829–844.

[19]
Devroye, L., Györfi, L., and Lugosi, G. (1996).
A Probabilistic Theory of Pattern Recognition.
New York: Springer.

[20]
Dominicy, Y. and Sinner, C. (2017).
Distributions and composite models for size-type data.
In Hokimoto, T., editor, Advances in Statistical Methodologies
and Their Application to Real Problems, chapter 8, pages 159–183.
IntechOpen, Rijeka, Croatia.

[21]
Fernández, C. and Steel, M. F. (1998).
On Bayesian modeling of fat tails and skewness.
Journal of the American Statistical Association,

[22]
Giné, E. and Nickl, R. (2015).
Mathematical Foundations of Infinite-Dimensional Statistical
Cambridge: Cambridge University Press.

[23]
Gitman, I., Chen, J., Lei, E., and Dubrawski, A. (2018).
Novel prediction techniques based on clusterwise linear regression.
arXiv preprint:1804.10742.

[24]
Goldberg, P. W. and Jerrum, M. R. (1995).
Bounding the Vapnik-Chervonenkis dimension of concept classes
parameterized by real numbers.
Machine Learning, 18(2-3):131–148.

[25]
Götze, F., Sambale, H., and Sinulis, A. (2021).
Concentration inequalities for polynomials in
$\alpha$-sub-exponential random variables.
Electronic Journal of Probability, 26:1–22.

[26]
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (2011).
Robust Statistics: The Approach Based on Influence Functions.
New Jersey: John Wiley & Sons.

[27]
Heffernan, J. E. and Tawn, J. A. (2004).
A conditional approach for multivariate extreme values.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 66(3):497–546.

[28]
Hettmansperger, T. P. and McKean, J. W. (1978).
Statistical inference based on ranks.
Psychometrika, 43(1):69–79.

[29]
Hettmansperger, T. P. and McKean, J. W. (2010).
Robust Nonparametric Statistical Methods.
Florida: CRC Press.

[30]
Holle, M., Wolff, T., and Herant, M. (2021).
Trends in the concentration and distribution of health care
expenditures in the US, 2001-2018.
JAMA Network Open, 4(9):e2125179–e2125179.

[31]
Huang, K. (2008).
Statistical Mechanics.
New Jersey: John Wiley & Sons.

[32]
Hyndman, R. J. and Grunwald, G. K. (2000).
Applications: Generalized additive modelling of mixed distribution
Markov models with application to Melbourne's rainfall.
Australian $\&$ New Zealand Journal of Statistics,

[33]
Jaeckel, L. A. (1972).
Estimating regression coefficients by minimizing the dispersion of
the residuals.
The Annals of Mathematical Statistics, 43(5):1449–1458.

[34]
Jones, M. (2014).
Generating distributions by transformation of scale.
Statistica Sinica, 749–771.

[35]
Klugman, S. A., Panjer, H. H., and Willmot, G. E. (2012).
Loss models: from data to decisions, volume 715.
John Wiley & Sons.

[36]
Koenker, R. (2009).
Package `quantreg'.

[37]
Koenker, R. and Bassett Jr, G. (1978).
Regression quantiles.
Econometrica: Journal of the Econometric Society, 33–50.

[38]
Kottas, A. and Gelfand, A. E. (2001).
Bayesian semiparametric median regression modeling.
Journal of the American Statistical Association,

[39]
Kuchibhotla, A. K. and Chakrabortty, A. (2022).
Moving beyond sub-Gaussianity in high-dimensional statistics:
Applications in covariance estimation and linear regression.
Information and Inference: A Journal of the IMA, 11:1389–1456.

[40]
Lambert, D. (1992).
Zero-inflated Poisson regression, with an application to defects in
Technometrics, 34(1):1–14.

[41]
Lee, M. J. (1989).
Mode regression.
Journal of Econometrics, 42(3):337–349.

[42]
Linero, A. R., Sinha, D., and Lipsitz, S. R. (2020).
Semiparametric mixed-scale models using shared Bayesian forests.
Biometrics, 76(1):131–144.

[43]
Liu, L., Ma, J. Z., and Johnson, B. A. (2008).
A multi-level two-part random effects model, with application to an
alcohol-dependence study.
Statistics in Medicine, 27(18):3528–3539.

[44]
Ma, Y. and Genton, M. G. (2004).
Flexible class of skew-symmetric distributions.
Scandinavian Journal of Statistics, 31(3):459–468.

[45]
Maronna, R. A., Martin, D. R., and Yohai, V. J. (2006).
Robust Statistics: Theory and Methods.
New Jersey: John Wiley & Sons.

[46]
Mudholkar, G. S. and Hutson, A. D. (2000).
The epsilon-skew-normal distribution for analyzing near-normal data.
Journal of Statistical Planning and Inference, 83(2):291–309.

[47]
Mullahy, J. (1998).
Much ado about two: Reconsidering retransformation and the two-part
model in health econometrics.
Journal of Health Economics, 17(3):247–281.

[48]
Nadarajah, S. and Kotz, S. (2003).
Skewed distributions generated by the normal kernel.
Statistics $\&$ Probability Letters, 65(3):269–277.

[49]
Nash, W. J., Sellers, T. L., Talbot, S. R., Cawthorn, A. J., and Ford, W. B.
The population biology of abalone (Haliotis species) in Tasmania.
i. blacklip abalone (H. rubra) from the North Coast and islands of
Bass Strait.
Sea Fisheries Division, Technical Report, 48:411.

[50]
Olsen, M. K. and Schafer, J. L. (2001).
A two-part random-effects model for semicontinuous longitudinal data.
Journal of the American Statistical Association,

[51]
Rubio, F. J. and Steel, M. F. (2020).
The family of two-piece distributions.
Significance, 17(1):12–13.

[52]
Rubio, F. J. and Steel, M. F. J. (2015).
Bayesian modelling of skewness and kurtosis with two-piece scale and
shape distributions.
Electronic Journal of Statistics, 9(2):1884–1912.

[53]
Sambale, H. (2020).
Some notes on concentration for $\alpha $-subexponential random
arXiv preprint:2002.10761.

[54]
Sarul, L. S. and Sahin, S. (2015).
An application of claim frequency data using zero inflated and hurdle
models in general insurance.
Journal of Business Economics and Finance, 4(4).

[55]
Scollnik, D. (2007).
On composite lognormal-Pareto models.
Scandinavian Actuarial Journal, 2007(1):20–33.

[56]
She, Y. (2016).
On the finite-sample analysis of $\Theta$-estimators.
Electronic Journal of Statistics, 10:1874–1895.

[57]
She, Y. and Tran, H. (2019).
On cross-validation for sparse reduced rank regression.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 81(1):145–161.

[58]
She, Y., Wang, Z., and Jin, J. (2021).
Analysis of generalized bregman surrogate algorithms for nonsmooth
nonconvex statistical learning.
The Annals of Statistics, 49(6):3434–3459.

[59]
Southworth, H., Heffernan, J. E., and Metcalfe, P. (2020).
Package ‘texmex’.

[60]
Stacy, E. W. (1962).
A generalization of the Gamma distribution.
The Annals of Mathematical Statistics, 33(3):1187–1192.

[61]
Tarbath, D. (1999).
Estimates of Growth and Natural Mortality of the Blacklip
Abalone (Haliotis Rubra) in Tasmania.
Technical report series. Tasmanian Aquaculture and Fisheries

[62]
Tobin, J. (1958).
Estimation of relationships for limited dependent variables.
Econometrica: Journal of the Econometric Society, 24–36.

[63]
van de Geer, S. A. and Bühlmann, P. (2009).
On the conditions used to prove oracle results for the Lasso.
Electronic Journal of Statistics, 3:1360–1392.

[64]
van der Vaart, A. W. and Wellner, J. A. (2013).
Weak Convergence and Empirical Processes: With Applications to
Berlin: Springer Science & Business Media.

[65]
Wainwright, M. J. (2019).
High-dimensional Statistics: A Non-Asymptotic Viewpoint.
Cambridge: Cambridge University Press.

[66]
Wang, J., Boyer, J., and Genton, M. G. (2004).
A skew-symmetric representation of multivariate distributions.
Statistica Sinica, 1259–1270.

[67]
Wang, L., Peng, B., Bradic, J., Li, R., and Wu, Y. (2020).
A tuning-free robust and efficient approach to high-dimensional
Journal of the American Statistical Association,

[68]
Yang, T., Gallagher, C. M., and McMahan, C. S. (2019).
A robust regression methodology via M-estimation.
Communications in Statistics-Theory and Methods,

[69]
Yu, K. and Moyeed, R. A. (2001).
Bayesian quantile regression.
Statistics $\&$ Probability Letters, 54(4):437–447.

[70]
Zhu, D. and Zinde-Walsh, V. (2009).
Properties and estimation of asymmetric exponential power
Journal of Econometrics, 148(1):86–99.