DSR: Metrics for Disruption Success

• Disruption Success Rate (DSR) is a context-dependent metric that quantifies the probability or event rate of successfully managing disruptions across varied systems.
• In MANET routing, DSR is inferred from metrics like packet delivery, throughput, and control overhead, highlighting a protocol's ability to sustain communication despite link failures.
• In astrophysics and tokamak research, DSR gauges outcomes—from stellar tidal disruption rates to plasma survival metrics—emphasizing observable or predictive success.

Disruption Success Rate (DSR) is not a single standardized metric in the cited literature. Instead, the phrase maps onto several formally distinct quantities, depending on the disruption mechanism and the operational notion of success. In MANET routing, it is not defined as a named metric, but the relevant behavior is expressed through packet delivery, throughput, end-to-end delay, and routing overhead under link breaks and mobility (Ramesh et al., 2010). In stellar-dynamical studies of tidal disruption, it corresponds directly to the capture or disruption rate $\dot{C}\equiv dN/dt$ or $\dot N$ of stars entering the loss cone of a black hole (Brockamp et al., 2011, Chen et al., 2012). In optical transient surveys, it denotes the rate at which tidal disruptions produce detectable optical flares, $\dot N_{\rm TDF}$, per galaxy or per comoving volume (Velzen et al., 2014). In tokamak disruption prediction, it can be assembled from the fraction of disruptive discharges that receive a timely and correct alarm, often represented through survival probability, median remaining time $t_{50}$, and expected future lifetime $\tau$ (Tinguely et al., 2019). A plausible implication is that DSR is best treated as a context-dependent success probability or event rate rather than as a universal scalar observable.

1. Terminological scope and formal variants

Across the cited works, the same phrase points to different mathematical objects. The unifying feature is that each object quantifies whether a disruptive process is either successfully handled, successfully produced, or successfully predicted within a specified regime.

Domain	DSR-mapped quantity	Typical units
MANET routing	Packet delivery, throughput, delay, control efficiency under disruptions	ratio, kBps, time
SMBH stellar dynamics	$\dot{C}=dN/dt$ or $\dot N$	yr$^{-1}$, Myr$^{-1}$
Optical TDF surveys	$\dot N_{\rm TDF}$ or $\dot N$0	yr$\dot N$1 galaxy$\dot N$2, yr$\dot N$3 Mpc$\dot N$4
Tokamak prediction	fraction of disruptive discharges with timely alarm; survival-based alarm conditions	probability or fraction

In the stellar-dynamical literature, the mapping is explicit. One paper states that the central quantity

$\dot N$5

is exactly the per-system Disruption Success Rate in that language, and also introduces a per-star hazard-like quantity

$\dot N$6

In the tokamak literature, by contrast, DSR is not introduced explicitly, but the formal ingredients for a DSR-like measure are present: true positives, false negatives, false positives, warning time, survival probability, and threshold-based alarm criteria (Brockamp et al., 2011, Tinguely et al., 2019).

A recurring source of ambiguity is that “DSR” itself is already an established acronym for Dynamic Source Routing in MANET research. In that domain, “Disruption Success Rate” must therefore be inferred from the routing metrics rather than read off from a variable or acronym (Ramesh et al., 2010).

2. MANET routing: disruption handling in DSR, PDSR, and TORA

In the MANET study, standard DSR has two core components: Route Discovery and Route Maintenance. When a source $\dot N$7 wants to send to destination $\dot N$8 and does not already know a route, Route Discovery is invoked. During active forwarding, Route Maintenance detects whether a hop along the source route fails. If a route is broken, the source can try another known route from its cache or invoke Route Discovery again. The paper characterizes this as purely reactive behavior (Ramesh et al., 2010).

Preemptive DSR (PDSR) modifies DSR to anticipate link breaks and prepare backup paths. Its Route Discovery procedure lets the destination collect multiple Route Requests for a quantum time $\dot N$9, select the two best routes, and return both a primary and a backup route. Its Route Monitoring procedure adds signal-strength-based link-failure prediction: if, for a link $\dot N_{\rm TDF}$0,

$\dot N_{\rm TDF}$1

the intermediate node sends the warning “Path likely to be disconnected” to the source. When warned, the source starts using the backup route as well; if it receives an acknowledgement from the destination via the backup route, it switches over from the primary to the backup route, and otherwise initiates a new Route Discovery process. The paper also attributes higher throughput in PDSR to a Data Salvage property: when a link becomes bad, the PDSR node tries to find alternate paths in its local cache, and if found, this path is used to salvage the data packet (Ramesh et al., 2010).

TORA handles disruptions through a different mechanism. It maintains a directed acyclic graph to the destination based on node heights. When a link fails, a node that loses a downstream link raises its own height and broadcasts an UPDATE; neighbors with no downstream links adopt the propagated reference level and reverse their links. If no alternate path exists, a reflected reference level propagates back and may trigger a CLEAR message to delete routes. The paper states that in densely connected networks with many alternate routes, TORA recovers very fast from link failures and new reference levels do not propagate far, whereas in sparsely connected or partitioned networks, reference levels and clear messages propagate widely, losing many packets during the propagation, reflection, and clearing phases (Ramesh et al., 2010).

The paper does not define a metric named Disruption Success Rate, but it identifies the metrics from which such a quantity can be interpreted. These include throughput, percentage of packets delivered, Packet Delivery Fraction,

$\dot N_{\rm TDF}$2

Average End-to-End Delay,

$\dot N_{\rm TDF}$3

Receiving Efficiency, Sending Efficiency, and Sending/Forwarding Efficiency in the Network. The paper explicitly argues that Packet Delivery Ratio or Packet Delivery Fraction, throughput under mobility, end-to-end delay, and control overhead collectively measure how successfully the protocol deals with disruptions (Ramesh et al., 2010).

Quantitatively, the disruption-handling contrast is sharp in some scenarios. In a PDSR Fast, 30-node scenario, throughput is summarized as approximately $\dot N_{\rm TDF}$4–$\dot N_{\rm TDF}$5 kBps, percentage of packets delivered as approximately $\dot N_{\rm TDF}$6–$\dot N_{\rm TDF}$7, and Sending/Forwarding Efficiency as approximately $\dot N_{\rm TDF}$8–$\dot N_{\rm TDF}$9. In a TORA Fast, 30-node scenario, throughput is $t_{50}$0 kBps, percentage of packets delivered is $t_{50}$1, and Sending/Forwarding Efficiency is $t_{50}$2. In the TORA Fast, 10-node case, throughput is $t_{50}$3 and percentage of packets delivered is $t_{50}$4. The paper concludes that PDSR outperforms TORA in terms of the control overhead, provides better data throughput than TORA, and creates new routes faster than TORA, but also that TORA is a better choice than PDSR for densely connected fast moving nodes (Ramesh et al., 2010).

This suggests a MANET-specific DSR interpretation in which success means sustaining delivery through link degradation with minimal control traffic. Under that interpretation, PDSR’s preemptive warnings, backup-route usage, data salvage, and multiple-route discovery increase the probability that communication continues without noticeable interruption, while TORA’s success is conditional on fast-moving, highly connected topologies.

3. Stellar-dynamical DSR as tidal disruption or capture rate

In the direct $t_{50}$5-body study of stars disrupted by supermassive black holes, “capture rate” denotes the number of stars per unit time whose pericentre falls inside a capture radius $t_{50}$6, regardless of whether the star is disrupted outside the horizon or swallowed whole, while “disruption rate” denotes the subset of capture events where the star is tidally disrupted before crossing the event horizon. By choosing $t_{50}$7 to be the tidal radius for a given physical SMBH mass, the numerical captures become tidal disruptions for SMBHs with $t_{50}$8 (Brockamp et al., 2011).

The central quantity is

$t_{50}$9

and the paper explicitly interprets it as a system-level Disruption Success Rate. The corresponding per-star probability per unit time is

$\tau$0

The simulations use a GPU-accelerated modified NBODY6 code with one SMBH particle of mass $\tau$1 in $\tau$2-body units, a Sersic $\tau$3 stellar profile, equal-mass stars, and particle numbers ranging from $\tau$4 to $\tau$5. Three simulation capture radii are explored: $\tau$6 The runs are evolved for $\tau$7 $\tau$8-body time units (Brockamp et al., 2011).

The dynamical underpinning is loss-cone refilling by angular-momentum diffusion. Inside the influence radius, the loss-cone angle obeys

$\tau$9

while the characteristic diffusive deflection per crossing time is

$\dot{C}=dN/dt$0

The critical radius is defined by

$\dot{C}=dN/dt$1

For a number-density profile $\dot{C}=dN/dt$2 in the regime $\dot{C}=dN/dt$3,

$\dot{C}=dN/dt$4

and

$\dot{C}=dN/dt$5

The simulations show that the loss cone is efficiently refilled by two-body relaxation and that the measured rate scales much more steeply with $\dot{C}=dN/dt$6 than the simplest energy-relaxation expectation (Brockamp et al., 2011).

The fitted numerical law is

$\dot{C}=dN/dt$7

with

$\dot{C}=dN/dt$8

For the three capture radii, the measured slopes are $\dot{C}=dN/dt$9, $\dot N$0, and $\dot N$1. The normalization depends on capture radius as

$\dot N$2

A plausible implication is that more populous nuclei are disproportionately more efficient at feeding the black hole through tidal disruptions than simple $\dot N$3 arguments would suggest (Brockamp et al., 2011).

Scaling to real systems uses the $\dot N$4–$\dot N$5 relation

$\dot N$6

and the influence radius

$\dot N$7

The resulting astrophysical rate for solar-type stars is

$\dot N$8

with an alternative calibration

$\dot N$9

The mass dependence is therefore weak, roughly $^{-1}$0 across $^{-1}$1–$^{-1}$2. For an Sgr A*-like SMBH, the deduced tidal disruption rate is

$^{-1}$3

or approximately $^{-1}$4 (Brockamp et al., 2011).

The same paper emphasizes an important distinction between physical capture rate and observable tidal disruption rate. Above the regime in which the tidal radius exceeds the event horizon, numerical captures no longer correspond to luminous tidal disruptions. This distinction reappears in observational-rate papers and is central to any encyclopedia treatment of DSR.

4. Optical DSR as the rate of detectable tidal disruption flares

The observational SDSS Stripe 82 study defines an empirical DSR for inactive galaxies: the rate at which tidal disruptions produce detectable optical flares. Its survey monitors approximately $^{-1}$5 galaxies over $^{-1}$6 yr, yields $^{-1}$7 nuclear flares, and retains two excellent TDF candidates, TDE1 and TDE2, after cuts designed to exclude off-nuclear supernovae and persistent AGN variability (Velzen et al., 2014).

The rate formalism is explicit. The expected number of detected TDFs is

$^{-1}$8

and, under the assumption of a common visible per-galaxy rate,

$^{-1}$9

The efficiency $^{-1}$0 is obtained by simulating the full detection pipeline: catalog-level flux cuts, difference-imaging recovery, real cadence, seasonal gaps, inhomogeneous sampling, and seeing variations. For the preferred empirical models, the mean efficiency is of order $^{-1}$1 when defined over the full $^{-1}$2 yr baseline (Velzen et al., 2014).

Using only the observed portions of the SDSS light curves, without extrapolation, the study derives a model-independent upper limit: $^{-1}$3 Using empirical light-curve models based on TDE1, TDE2, PS1-10jh, and PS1-11af, it obtains a best-estimate rate

$^{-1}$4

Folding this per-galaxy rate with the SDSS $^{-1}$5-band galaxy luminosity function gives an effective galaxy density $^{-1}$6 and a volumetric rate

$^{-1}$7

The authors state that these results apply for galaxies hosting black holes with mass in the range of a few million to $^{-1}$8 solar masses (Velzen et al., 2014).

The paper also treats the visibility cutoff at high black-hole mass. It considers a step-function model,

$^{-1}$9

and an exponential suppression due to direct capture,

$\dot N_{\rm TDF}$0

Using the exponential rather than the step function changes the inferred rate by approximately $\dot N_{\rm TDF}$1–$\dot N_{\rm TDF}$2 (Velzen et al., 2014).

A central interpretive issue is the gap between the observed optical DSR and the true physical tidal disruption rate. For a singular isothermal sphere, the paper quotes

$\dot N_{\rm TDF}$3

which gives approximately $\dot N_{\rm TDF}$4 at $\dot N_{\rm TDF}$5, around $\dot N_{\rm TDF}$6 higher than the empirical optical rate of approximately $\dot N_{\rm TDF}$7. More conservative theoretical estimates based on real surface-brightness profiles yield $\dot N_{\rm TDF}$8 for $\dot N_{\rm TDF}$9, which is compatible with the observed optical DSR. The paper therefore presents two possibilities: either the isothermal-sphere model is not universally applicable, or most physical disruptions fail to produce detectable optical flares because of obscuration, geometry, or non-optical emission (Velzen et al., 2014).

5. Galactic Center DSR and the effect of an intermediate-mass black hole

A second stellar-dynamical use of DSR appears in the Galactic Center IMBH study, where the quantity of interest is the tidal-disruption rate $\dot N$00 or the mass disruption rate $\dot N$01. The paper explicitly identifies this as the probability per unit time that stars in the nuclear cluster are successfully scattered onto orbits with pericenter inside the tidal radius of the central black-hole system (Chen et al., 2012).

For a single SMBH, the loss-cone boundary is set by

$\dot N$02

with tidal radius

$\dot N$03

The energy-resolved loss-cone flux from two-body relaxation is written as

$\dot N$04

where

$\dot N$05

The paper also includes resonant relaxation, with an averaged timescale $\dot N$06 and corresponding flux

$\dot N$07

For the Galactic Center model adopted there, the baseline single-SMBH rate is

$\dot N$08

This is the baseline DSR in the absence of any IMBH (Chen et al., 2012).

An IMBH of mass $\dot N$09 at separation $\dot N$10 from Sgr A* forms a massive black-hole binary and adds coherent torques that refill the loss cone much more efficiently. The characteristic Lidov–Kozai-like timescale is

$\dot N$11

with $\dot N$12. Combining resonant-relaxation and IMBH torques gives a coherent loss-cone filling rate

$\dot N$13

where the authors adopt a crude correction factor $\dot N$14 for the fraction of stars ejected in slingshot encounters (Chen et al., 2012).

The resulting enhancement can be large. The paper states that an IMBH heavier than $\dot N$15 could distinguishably enhance the stellar-disruption rate. For $\dot N$16 at $\dot N$17 pc, the maximum coherent contribution is

$\dot N$18

which is approximately $\dot N$19 times higher than $\dot N$20 and approximately $\dot N$21 times higher than $\dot N$22. Over the parameter range explored, the corresponding number-disruption rates for $\dot N$23 stars span

$\dot N$24

depending on $\dot N$25 and $\dot N$26 (Chen et al., 2012).

The paper then imposes an observational constraint using the fall-back model for stellar debris and the quiescent luminosity of Sgr A*. The most bound debris return time is

$\dot N$27

and the fall-back rate follows

$\dot N$28

Comparing the implied luminosity and infrared flux with Sgr A* observations, the authors argue that no TDE has occurred in the Galactic Center within the last few centuries and adopt the conservative upper bound

$\dot N$29

They conclude that part of the IMBH parameter space, concentrating at the high-mass end, can already be excluded, and that it is crucial to observationally confirm or reject the stellar-disruption rate between $\dot N$30 and $\dot N$31 (Chen et al., 2012).

This is a conceptually different DSR from the optical SDSS rate. Here the quantity is the physical Galactic Center disruption rate itself, not the subset of disruptions that are optically selected in an extragalactic survey.

6. Tokamak disruption prediction: survival-based DSR

In tokamak disruption prediction, DSR is naturally interpreted as the fraction of disruptive discharges that receive a correct alarm with sufficient warning time. The survival-analysis paper does not introduce that acronym explicitly, but it provides the complete formal structure needed to define it (Tinguely et al., 2019).

The starting point is a Random Forest classifier trained on Alcator C-Mod discharges. At each time $\dot N$32, it outputs a disruptivity

$\dot N$33

interpreted as the probability that the instantaneous plasma state belongs to the disruptive class. A class time $\dot N$34 separates disruptive and non-disruptive labels, and a minimum mitigation time $\dot N$35 specifies the lead time required for avoidance or mitigation. For a disruptive discharge, a true positive is an alarm that is triggered before $\dot N$36, a false negative is a missed or late alarm, and a false positive is an alarm on a non-disruptive discharge (Tinguely et al., 2019).

The paper embeds this binary classifier in a survival-analysis framework. The survival function is

$\dot N$37

the hazard function is

$\dot N$38

and the Kaplan–Meier estimator for discrete times gives

$\dot N$39

Interpreting $\dot N$40 as $\dot N$41 and assuming a uniform probability density for time-to-disruption within the disruptive class, the short-interval failure probability becomes

$\dot N$42

which yields

$\dot N$43

Two derived quantities are then introduced: the median remaining time $\dot N$44 defined by

$\dot N$45

and the expected future lifetime

$\dot N$46

These are operational proxies for remaining time to disruption (Tinguely et al., 2019).

The paper compares this framework with a conventional threshold-based Random Forest alarm. In the threshold scheme, the parameters are a high threshold $\dot N$47, a low threshold $\dot N$48, an alarm window $\dot N$49 ms, and a class time $\dot N$50 ms. An alarm is triggered if $\dot N$51 rises above $\dot N$52, remains above $\dot N$53 for at least $\dot N$54, and, for disruptive shots, does so earlier than the class time (Tinguely et al., 2019).

Three illustrative C-Mod discharges show the trade-off. For disruptive shot #1140226013, both the threshold scheme and the survival-analysis approach produce a successful early prediction; $\dot N$55 drops below $\dot N$56 ms and $\dot N$57 for nearly all times with $\dot N$58. For disruptive shot #1150722006, the threshold-based alarm rises above $\dot N$59 only about $\dot N$60 ms before disruption, too late for mitigation, and neither $\dot N$61 nor $\dot N$62 falls below $\dot N$63 at any time; both approaches therefore fail. For non-disruptive shot #1140227018, the threshold rule produces a false positive when $\dot N$64 crosses $\dot N$65, but $\dot N$66 never falls to $\dot N$67, so a survival-based rule using $\dot N$68 would avoid that false alarm (Tinguely et al., 2019).

The paper then states several possible DSR definitions. One is the fraction of disruptive discharges for which there exists a time $\dot N$69 such that the alarm condition holds and $\dot N$70: $\dot N$71 A corresponding false-alarm rate is

$\dot N$72

The paper does not present a global numerical DSR over thousands of shots, but it does argue that survival metrics, especially $\dot N$73, can improve robustness against spurious short-lived spikes in disruptivity while possibly being more conservative for rapid impurity-driven terminations (Tinguely et al., 2019).

A central misconception addressed implicitly by this framework is that a high instantaneous disruptivity is equivalent to a high probability of timely mitigation. The survival-analysis formalism distinguishes those notions: a discharge may briefly cross a disruptivity threshold without implying a short expected future lifetime, and conversely a late rise in disruptivity may be operationally useless even if it correctly identifies the final collapse.