\section{Results}
\label{sec:results}

\subsection{The transformation of the intraday market, 1992--2026}

Figures~\ref{fig:seasonality_decade} through \ref{fig:epps} document the
environment in which propagation must be measured, on liquidity-screened
universes with strictly lagged membership.

Three secular changes stand out. First, ordinary minute-level variability
collapsed. Mean absolute one-minute returns on the top-1000 universe fell
decade by decade at every minute of the trading day
(Figure~\ref{fig:seasonality_decade}a), and the frequency of one-minute
returns exceeding 1\% fell from about \ExtremeGtOneNineties{} per 10,000 in
the 1990s to \ExtremeGtOneTens{} per 10,000 in the 2010s, a factor of
roughly \ExtremeDropFactor{} (Figure~\ref{fig:extreme_minutes}). A large
part of the 1990s level is bid--ask bounce and price discreteness under
eighth and sixteenth ticks rather than volatility
\citep{roll1984simple,harris1991stock}: a single tick on a \$20 stock is a
0.6\% return, and about half of all one-minute returns in the mid-1990s are
exactly zero (Online Appendix). The decline is nonetheless not only a tick
artifact. Measured from 2002, the first full year after decimalization, the
frequency of $|r|>1\%$ minutes falls from \ExtremeGtOneTwoThousandTwo{} per
10,000 to \ExtremeGtOneTens{} in the 2010s, roughly tenfold, and the
cross-section of daily realized variance shifts down over the same years
(Online Appendix).

Second, the collapse was not monotone in the tails. The frequency of
$|r|>5\%$ minutes rose from \ExtremeGtFivePreCovid{} per 10,000 in 2017--2019
to \ExtremeGtFiveCovid{} in 2020, a factor of \ExtremeCovidFactor{}, and 2008
shows the same signature at a smaller scale (Figure~\ref{fig:extreme_minutes}).
Part of the 2020--2021 level reflects the composition of the top-1000
universe in those years, which dollar-volume ranking populates with
leveraged exchange-traded products and distressed issuers that we cannot
label; the qualitative pattern---extremes became rarer but, when the market
is stressed, arrive in synchronized bursts---survives the exclusion of
heuristically identified non-common tickers and is the subject of
Section~\ref{sec:mechanisms}.

Third, cross-sectional comovement at the one-minute horizon rose from nothing
to most of its half-hour level. In 1996 the mean pairwise correlation of
one-minute returns among the hundred largest stocks was \EppsNinetySixOne{},
against \EppsNinetySixThirty{} at thirty minutes: comovement existed but was
invisible at the minute, the classical Epps effect
\citep{epps1979comovements}. By 2015 the one-minute correlation was
\EppsFifteenOne{} against \EppsFifteenThirty{} at thirty minutes, and 2024
shows the same near-flat profile at a lower level (\EppsTwentyFourOne{}
against \EppsTwentyFourThirty{}; Figure~\ref{fig:epps}). Two mechanisms
produce this flattening and our previous-tick synchronized grids do not
separate them. One is mechanical: nonsynchronous trading and stale prices
attenuate measured high-frequency correlation
\citep{reno2003closer,hayashi2005covariance,zhang2011estimating}, and the
share of stale one-minute prices fell from \ZeroRetNinetySix\% to
\ZeroRetTwentyFour\% between 1996 and 2024. The other is economic: common
information is now impounded across the cross-section within the minute.
\citet{toth2009epps} reach the same conclusion for 1993--2003 and find that
denser trading explains only part of the change. Whatever the mix, the
operational consequence for this paper is the same: one-minute residual
correlations identify economic neighbors far more sharply after 2005 than
before, a point we carry into the interpretation of the network results.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figE3_seasonality.pdf}
\caption{\textbf{Intraday seasonality by decade.} Minute-of-day means during
regular hours, lagged top-1000 dollar-volume universe, split-window and
flagged bars excluded: (a) absolute one-minute log return; (b) share of the
stock-day's volume; (c) high--low log range.}
\label{fig:seasonality_decade}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=.72\textwidth]{figE5_extreme_minutes.pdf}
\caption{\textbf{Extreme one-minute returns.} Annual frequency (log scale) of
one-minute absolute log returns exceeding 1\%, 2\%, and 5\%, lagged top-1000
universe, regular hours. Pre-2001 levels largely reflect eighth/sixteenth
tick bid--ask bounce.}
\label{fig:extreme_minutes}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=.62\textwidth]{figE8_epps.pdf}
\caption{\textbf{The flattening of the Epps curve.} Mean pairwise return
correlation among the top-100 lagged dollar-volume stocks at 1-, 5-, 15-, and
30-minute sampling, previous-tick synchronization within day, for 1996, 2005,
2015, and 2024. Correlations are computed on calendar-clock grids and are
therefore attenuated by nonsynchronous trading at short intervals, most
strongly in 1996.}
\label{fig:epps}
\end{figure}

\subsection{Baseline: isolated shocks and market-wide tail activity}

Table~\ref{tab:baseline_propagation} reports the baseline estimates of
equation~\eqref{eq:dd} for \NEventsReal{} isolated $|z|\ge6$ source events and
\NEventsPlacebo{} matched placebos. The difference-in-differences breadth
response is economically negligible at every horizon: at $h=5$ minutes the
point estimate is \DDBreadthFive$\times10^{-4}$ (bootstrap standard error
\DDBreadthFiveSE$\times10^{-4}$), and at $h=30$ minutes
\DDBreadthThirty$\times10^{-4}$ (\DDBreadthThirtySE$\times10^{-4}$). Against
a placebo base rate of about \BaseBreadthShare{} (Table~\ref{tab:design_checks},
Panel~A), the largest estimate in absolute value implies that an isolated
six-sigma move changes the probability that any given other security shows a
tail exceedance by about a third of one percent of the base rate, with a
negative sign that is statistically distinguishable from zero at $h=30$.
Difference-in-differences intensity estimates are of the same order and
alternate in sign.

\input{tables/table4_baseline_propagation}

The design diagnostics of Table~\ref{tab:design_checks} qualify this null in
three ways. Panel~A shows that real events do not arrive in ordinary minutes:
five minutes before the event, breadth is \PreBreadthReal$\times10^{-4}$ for
real events against \PreBreadthPlacebo$\times10^{-4}$ for placebos, a
difference of \PreBreadthDiff$\times10^{-4}$ (standard error
\PreBreadthDiffSE), and intensity is higher by \PreIntensityDiff{}
(\PreIntensityDiffSE). A six-sigma idiosyncratic move is thus more likely on
days and in minutes when the whole cross-section is already active, even
though the market factor is calm by construction. This is exactly the
ambient level that the inner difference of~\eqref{eq:dd} removes, and it is
the reason the difference-in-differences, not a post-event comparison, is the
baseline; but it also means that the ``no pre-event response'' statement
sometimes made for designs of this kind does not hold here as a statement
about levels, and we do not make it.

Panel~C reports the three estimators side by side. Without pre-differencing,
real events show \PostDiffFive$\times10^{-4}$ more breadth than placebos at
$h=5$, almost exactly the pre-event gap, so that the difference-in-differences
is close to zero. Within deciles of the pre-event level, the gap is
\StratFive$\times10^{-4}$ (\StratFiveSE) at $h=5$ and
\StratThirty$\times10^{-4}$ (\StratThirtySE) at $h=30$. As argued in
Section~\ref{sec:design}, the two estimators bracket the market-wide
response: between zero and roughly \StratShareOfBase\% of the base rate.
Either bound is small relative to the network response of
Section~\ref{sec:mechanisms}, and the conclusion we draw is correspondingly
bounded: isolated idiosyncratic extremes raise market-wide tail activity by at
most a few percent of its base rate within two hours.

Panel~B separates events by the number of source events detected in the same
minute. For the \ShareSingleton\% of events that are alone in their minute,
the difference-in-differences is \DDSingletonFive$\times10^{-4}$
(\DDSingletonFiveSE) at $h=5$ and \DDSingletonThirty$\times10^{-4}$
(\DDSingletonThirtySE) at $h=30$: slightly positive and within two standard
errors of zero. The negative full-sample estimates at longer horizons come
from minutes with three to five co-arriving events
(\DDMultiThirty$\times10^{-4}$, \DDMultiThirtySE, at $h=30$), which are the
minutes with the most elevated ambient activity and the strongest reversion.
The negative sign of the pooled estimate is therefore a mean-reversion
artifact of co-arrival minutes, not evidence that shocks calm the market.

Three further features of Table~\ref{tab:baseline_propagation} deserve
emphasis. First, this is a powered null: the minimum detectable
difference-in-differences at $h=5$ is about \MDEFive$\times10^{-4}$, one
additional responding security per \MDEEventsPer{} events. Second, the null
is uniform in time: Figure~\ref{fig:irf_decade} shows the same flat response
profile in each of the four decades, and the Online Appendix shows it year by
year, so absorption is not a recent acquisition of electronic markets---though
the composition of what is being absorbed changed radically, as the
descriptive results show. Third, Panel~D of Table~\ref{tab:design_checks}
shows that the standard errors are insensitive to the clustering unit for
the market-wide statistics.

\begin{figure}[t]
\centering
\includegraphics[width=.6\textwidth]{figF1_irf_decade.pdf}
\caption{\textbf{Difference-in-differences breadth by decade.} Point estimates of
$\Delta\Delta$ breadth (equation~\eqref{eq:dd}) with 95\% day-block bootstrap
bands, by horizon and decade.}
\label{fig:irf_decade}
\end{figure}

\input{tables/table9_design_checks}

Conditioning does not change the picture (Figure~\ref{fig:gradient_state}):
estimates by shock sign and by calm versus stressed market state are within
two standard errors of zero. Across magnitude bins the point estimates rise
from significantly negative for $6\sigma$--$8\sigma$ events, which are the
most numerous and the most affected by the co-arrival reversion just
described, to a positive but insignificant \DDGradLargest$\times10^{-4}$
(\DDGradLargestSE) for $12\sigma$--$20\sigma$ events. The signed response of
Panel~E is noisy and never significant: the cross-section does not drift in
the direction of the source shock. None of this resurrects market-wide
propagation; it motivates asking where a concentrated response would hide,
which is the subject of the next section.

\begin{figure}[t]
\centering
\includegraphics[width=.66\textwidth]{figF3_gradient_state.pdf}
\caption{\textbf{Shock size, sign, and market state.} $\Delta\Delta$ breadth
at $h=15$ by source-shock magnitude bins, shock sign, and prior-30-minute
market volatility state, with 95\% bootstrap intervals.}
\label{fig:gradient_state}
\end{figure}