--- title: 'DSR: Metrics for Disruption Success' url: https://www.emergentmind.com/topics/disruption-success-rate-dsr type: topic --- # DSR: Metrics for Disruption 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 [1101.0209]. 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 [1108.2270, 1211.4609]. 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 [1407.6425]. 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 $\tau$ [1907.04291]. 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 $\mathcal R_{\rm TDF}$ | yr$^{-1}$ galaxy$^{-1}$, yr$^{-1}$ Mpc$^{-3}$ | | 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 \[ \dot{C}\equiv \frac{dN}{dt} \] is exactly the per-system Disruption Success Rate in that language, and also introduces a per-star hazard-like quantity \[ \lambda_{\rm DSR}\equiv \frac{\dot C}{N_*}. \] 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 [1108.2270, 1907.04291]. 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 [1101.0209]. ## 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 $S$ wants to send to destination $D$ 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 [1101.0209]. 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 $q$, 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 $(i,j)$, \[ P_r(i,j) < T, \] 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 [1101.0209]. 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 [1101.0209]. 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, \[ P = \frac{1}{C}\sum_f \frac{R_f}{N_f}, \] Average End-to-End Delay, \[ D = \frac{1}{N}\sum_i (r_i-s_i), \] 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 [1101.0209]. Quantitatively, the disruption-handling contrast is sharp in some scenarios. In a PDSR Fast, 30-node scenario, throughput is summarized as approximately $392$–$417$ kBps, percentage of packets delivered as approximately $93.9$–$99.7\%$, and Sending/Forwarding Efficiency as approximately $99.97$–$99.99\%$. In a TORA Fast, 30-node scenario, throughput is $245.7$ kBps, percentage of packets delivered is $60.2\%$, and Sending/Forwarding Efficiency is $95.9\%$. In the TORA Fast, 10-node case, throughput is $0$ and percentage of packets delivered is $0$. 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 [1101.0209]. 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 $N$-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 $r_{\rm cap}$, 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 $r_{\rm cap}$ to be the tidal radius for a given physical SMBH mass, the numerical captures become tidal disruptions for SMBHs with $M_{\rm BH}\lesssim 10^7M_\odot$ [1108.2270]. The central quantity is \[ \dot C \equiv \frac{dN}{dt}, \] and the paper explicitly interprets it as a system-level Disruption Success Rate. The corresponding per-star probability per unit time is \[ \lambda_{\rm DSR}\equiv \frac{\dot C}{N_*}. \] The simulations use a GPU-accelerated modified NBODY6 code with one SMBH particle of mass $M_{\rm BH}=0.01$ in $N$-body units, a Sersic $n=4$ stellar profile, equal-mass stars, and particle numbers ranging from $10^3$ to $5\times10^5$. Three simulation capture radii are explored: \[ r_{\rm cap}^{\rm sim}=2\times10^{-7},\;4\times10^{-7},\;8\times10^{-7}. \] The runs are evolved for $100$ $N$-body time units [1108.2270]. The dynamical underpinning is loss-cone refilling by angular-momentum diffusion. Inside the influence radius, the loss-cone angle obeys \[ \theta_{\rm lc}\propto \left(\frac{2r_{\rm cap}}{3r}\right)^{1/2}, \] while the characteristic diffusive deflection per crossing time is \[ \theta_{\rm Diff}\propto \left(\frac{t_{\rm cross}}{t_{\rm rel}}\right)^{1/2}. \] The critical radius is defined by \[ \frac{\theta_{\rm lc}}{\theta_{\rm Diff}}\Big|_{r=r_{\rm crit}} = 1. \] For a number-density profile $n(r)=n_0r^\alpha$ in the regime $r_{\rm crit}10^8M_\odot, \end{cases} \] and an exponential suppression due to direct capture, \[ \dot N_i=\dot N\,\exp\!\left[-\left(\frac{M_{\rm BH}}{3\times10^7M_\odot}\right)^{0.9}\right]. \] Using the exponential rather than the step function changes the inferred rate by approximately $40\%$–$50\%$ [1407.6425]. 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 \[ \dot N_{\rm theory}=9.9\times10^{-4} \left(\frac{M_{\rm BH}}{10^6M_\odot}\right)^{-0.32}\,{\rm yr}^{-1}, \] which gives approximately $4\times10^{-4}\,{\rm yr}^{-1}\,{\rm galaxy}^{-1}$ at $M_{\rm BH}\sim10^7M_\odot$, around $20\times$ higher than the empirical optical rate of approximately $2\times10^{-5}\,{\rm yr}^{-1}\,{\rm galaxy}^{-1}$. More conservative theoretical estimates based on real surface-brightness profiles yield $(1-20)\times10^{-5}\,{\rm yr}^{-1}$ for $M_{\rm BH}\sim10^7M_\odot$, 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 [1407.6425]. ## 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 $\dot N$ or the mass disruption rate $\dot M$. 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 [1211.4609]. For a single SMBH, the loss-cone boundary is set by \[ J_{\rm lc}\simeq \sqrt{2GM_\bullet r_t}, \] with tidal radius \[ r_t \simeq r_*\left(\frac{M_\bullet}{m_*}\right)^{1/3}. \] The energy-resolved loss-cone flux from two-body relaxation is written as \[ \mathcal F_{\rm 2b}(\mathcal E)\,d\mathcal E= \frac{j_D^2(\mathcal E)\,n(\mathcal E)\,d\mathcal E}{P(\mathcal E)}, \] where \[ j_D^2\equiv \min\left[ \frac{J_{\rm lc}^2}{J_c^2}, \frac{(J_D/J_c)^2}{\ln(J_c/J_{\rm lc})} \right]. \] The paper also includes resonant relaxation, with an averaged timescale $\bar T_{\rm RR}(\mathcal E)$ and corresponding flux \[ \mathcal F_{\rm RR}(\mathcal E)\,d\mathcal E= \frac{n(\mathcal E)\,d\mathcal E}{\bar T_{\rm RR}(\mathcal E)}. \] For the Galactic Center model adopted there, the baseline single-SMBH rate is \[ \dot N_{\rm single}\simeq \dot N_{\rm 2b}+\dot N_{\rm RR} \simeq 3.5\times10^{-5}\,{\rm yr}^{-1} +3.5\times10^{-6}\,{\rm yr}^{-1} \sim 4\times10^{-5}\,{\rm yr}^{-1}. \] This is the baseline DSR in the absence of any IMBH [1211.4609]. An IMBH of mass $M_{\rm IMBH}=m$ at separation $d$ 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 \[ T_K= \begin{cases} \displaystyle \frac{2}{3\pi q}\left(\frac{a}{d}\right)^{-3}P(a), & a\le d/2,\\[1.2ex] \displaystyle \frac{16\sqrt2}{3\pi q}\left(\frac{a}{d}\right)^{1/2}P(a), & a>d/2, \end{cases} \] with $q=m/M_\bullet$. Combining resonant-relaxation and IMBH torques gives a coherent loss-cone filling rate \[ \mathcal F_{\rm co}(\mathcal E)\,d\mathcal E = \frac{(1-f_{\rm ej})\,n(\mathcal E)\,d\mathcal E}{\bar T_{\rm co}(\mathcal E)}, \] where the authors adopt a crude correction factor $f_{\rm ej}\simeq0.5$ for the fraction of stars ejected in slingshot encounters [1211.4609]. The resulting enhancement can be large. The paper states that an IMBH heavier than $2000\,M_\odot$ could distinguishably enhance the stellar-disruption rate. For $M_{\rm IMBH}=10^4M_\odot$ at $d\simeq0.07$ pc, the maximum coherent contribution is \[ \dot N_{\rm co}\simeq1.1\times10^{-3}\,{\rm yr}^{-1}, \] which is approximately $30$ times higher than $\dot N_{\rm 2b}$ and approximately $300$ times higher than $\dot N_{\rm RR}$. Over the parameter range explored, the corresponding number-disruption rates for $1M_\odot$ stars span \[ \dot N \simeq 8\times10^{-5} - 10^{-2}\ {\rm yr}^{-1}, \] depending on $M_{\rm IMBH}$ and $d$ [1211.4609]. 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 \[ t_{\rm min}\simeq 0.22\,k^{-3/2}\beta^{-3} \left(\frac{m_*}{M_\odot}\right)^{-1} \left(\frac{r_*}{R_\odot}\right)^{3/2}\,{\rm yr}, \] and the fall-back rate follows \[ \dot M_{\rm fb}(t)\propto (t-t_D)^{-5/3}. \] 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 \[ \dot N_{\rm TDE}\lesssim 0.002\,{\rm yr}^{-1}. \] 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 $10^{-4}$ and $10^{-2}\,{\rm yr}^{-1}$ [1211.4609]. 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 [1907.04291]. The starting point is a Random Forest classifier trained on Alcator C-Mod discharges. At each time $t$, it outputs a disruptivity \[ \mathcal D(t)\in[0,1], \] interpreted as the probability that the instantaneous plasma state belongs to the disruptive class. A class time $\Delta t_c$ separates disruptive and non-disruptive labels, and a minimum mitigation time $\Delta t_{\rm mit}$ specifies the lead time required for avoidance or mitigation. For a disruptive discharge, a true positive is an alarm that is triggered before $t_{\rm disr}-\Delta t_{\rm mit}$, a false negative is a missed or late alarm, and a false positive is an alarm on a non-disruptive discharge [1907.04291]. The paper embeds this binary classifier in a survival-analysis framework. The survival function is \[ S(t)=P(T>t), \] the hazard function is \[ h(t)=-\frac{1}{S(t)}\frac{dS(t)}{dt}, \] and the Kaplan–Meier estimator for discrete times gives \[ S(t_n)=\prod_{i=0}^{n}P(T>t_i\mid T\ge t_i) =\prod_{i=0}^{n}(1-P_{i\rightarrow i+1}). \] Interpreting $\mathcal D(t)$ as $P(D)$ and assuming a uniform probability density for time-to-disruption within the disruptive class, the short-interval failure probability becomes \[ P_{t\rightarrow t+\delta t}\approx \mathcal D(t)\frac{\delta t}{\Delta}, \] which yields \[ S(t+\Delta t\mid t) = \prod_{j=0}^{n} \left[ 1-\mathcal D(t+j\delta t)\frac{\delta t}{\Delta} \right], \qquad \Delta t=n\delta t\le\Delta. \] Two derived quantities are then introduced: the median remaining time $t_{50}$ defined by \[ S(t+t_{50}\mid t)=0.5, \] and the expected future lifetime \[ \tau=\int_0^\infty S(t+t'\mid t)\,dt'. \] These are operational proxies for remaining time to disruption [1907.04291]. The paper compares this framework with a conventional threshold-based Random Forest alarm. In the threshold scheme, the parameters are a high threshold $\mathcal D_H=0.35$, a low threshold $\mathcal D_L=0.05$, an alarm window $\Delta t_{\rm alarm}=5$ ms, and a class time $\Delta=325$ ms. An alarm is triggered if $\mathcal D(t)$ rises above $\mathcal D_H$, remains above $\mathcal D_L$ for at least $\Delta t_{\rm alarm}$, and, for disruptive shots, does so earlier than the class time [1907.04291]. 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; $t_{50}$ drops below $\Delta=325$ ms and $\tau\approx\Delta$ for nearly all times with $t_{\rm disr}-t<\Delta$. For disruptive shot \#1150722006, the threshold-based alarm rises above $\mathcal D_H$ only about $20$ ms before disruption, too late for mitigation, and neither $t_{50}$ nor $\tau$ falls below $\Delta$ at any time; both approaches therefore fail. For non-disruptive shot \#1140227018, the threshold rule produces a false positive when $\mathcal D(t)$ crosses $0.35$, but $\tau(t)$ never falls to $\Delta$, so a survival-based rule using $\tau\le\Delta$ would avoid that false alarm [1907.04291]. The paper then states several possible DSR definitions. One is the fraction of disruptive discharges for which there exists a time $t$ such that the alarm condition holds and $t_{\rm disr}-t\ge\Delta t_{\rm mit}$: \[ {\rm DSR} = \frac{\#\{\text{disruptive discharges with success}\}}{N_{\rm disr}}. \] A corresponding false-alarm rate is \[ {\rm FAR} = \frac{\text{number of non-disruptive discharges with alarm}}{N_{\rm nodisr}}. \] The paper does not present a global numerical DSR over thousands of shots, but it does argue that survival metrics, especially $\tau$, can improve robustness against spurious short-lived spikes in disruptivity while possibly being more conservative for rapid impurity-driven terminations [1907.04291]. 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. Source: https://www.emergentmind.com/topics/disruption-success-rate-dsr