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Dynamic Attack Surfaces: A Queue of Branching Processes
Ismail Cosandal, Abdullah Y. Etcibasi, C. Emre Koksal, Sennur Ulukus
TL;DR
Dynamic attack surfaces combine vulnerabilities’ exposure periods with changing attacker populations, while AI can accelerate both offensive and defensive activity. This paper models those processes with a queue of branching processes and analyzes expected exploits under BDI and RD attacker models. Under BDI, finite expected exploit counts require attacker growth to remain below both discovery and effective patching; proportional AI acceleration can shift the defensive bottleneck and fail to preserve security.
Problem
Attack surfaces evolve as vulnerabilities remain exposed while attacker activity and defensive discovery and patching unfold, dynamics that static vulnerability counts and average patch times do not capture.
Source: §I · p. 1Method
The paper couples resource-constrained vulnerability discovery and patching in a queue with evolving attacker populations, deriving expected exploit counts under BDI and RD models.
Source: §I · p. 2Results
Under BDI, expected exploit counts are finite only when attacker growth remains below both vulnerability discovery and effective patching, while equal proportional AI acceleration need not preserve security.
Source: §I · pp. 1–2Takeaways & Limitations
Security depends on relative attack, discovery, and patching time scales, and equal proportional AI amplification can shift the defensive bottleneck.
Source: §V · p. 6
Abstract
from arXiv · showhide
Modern attack surfaces are inherently dynamic. Vulnerabilities emerge over time and they remain exposed to exploit until defenders discover and patch them. Artificial intelligence (AI) amplifies these dynamics by accelerating both offensive and defensive processes, making modern cybersecurity increasingly a competition among interacting time scales and resources. We capture this competition using a queue of branching processes that couples vulnerability discovery and resource-constrained patching with the evolving attacker population targeting each vulnerability. Our primary risk metric is the cumulative number of exploits a vulnerability experiences before it is patched. We derive the expected cumulative number of exploits under birth-death-immigration (BDI) and rise-decay (RD) models of attacker activity. Under a BDI model of attacker activity, we identify a phase transition: the expected cumulative number of exploits is finite only when attacker growth remains below both the vulnerability discovery rate and the effective patching rate. Across both attacker models, increasing the exploit rate does not necessarily increase the expected number of exploits because faster attacker departure can sufficiently limit the active attacker population. Finally, we show that equal proportional AI amplification of offensive and defensive capabilities need not preserve security: accelerating both sides can shift the defensive bottleneck and move the system into an unbounded-risk regime.
I. INTRODUCTION
The introduction frames attack surfaces as evolving competitions between attacker activity and resource-constrained defense, then presents a queue of branching processes to model them. Its results identify joint discovery-and-patching thresholds for finite expected exploit counts and show that proportional AI acceleration can shift the defensive bottleneck.
I. INTRODUCTION
- AI can accelerate offensive probing and exploitation as well as defensive discovery, prioritization, and patch generation, while static vulnerability counts and average patch times miss the resulting temporal dynamics.The introduction cites a reported 2026 PaperCut campaign in which hundreds of AI agents contributed to exploits reaching the first real-world remote code execution in under four hours and compromising at least 440 exposed instances across 48 countries.Source: §I · p. 1
- Expected cumulative exploits under BDI remain finite only when attacker growth stays below both vulnerability discovery and effective patching rates.Exceeding either defensive rate makes the expectation unbounded, with unusually delayed vulnerabilities driving divergence in the average.Source: §I · p. 1
- Exploit rate and expected exploit count need not increase together, because faster attacker departure can reduce the active population enough to offset added activity.The result identifies discovery and patching as distinct bottlenecks; strengthening only one is insufficient if the other cannot keep pace with attacker growth.Source: §I · pp. 1–2
- Equal proportional acceleration of offensive and defensive AI capabilities need not preserve security when discovery becomes the limiting defense.In the numerical example, patching improves sufficiently, but vulnerability discovery is overtaken by attacker growth.Source: §I · p. 2
- The paper introduces a branching-process queue coupling vulnerability discovery and resource-constrained patching with the evolving attacker population targeting each vulnerability.It derives vulnerability-lifetime distributions and expected cumulative exploit expressions for BDI, establishes a stability condition, and extends the analysis to RD attacker interest that eventually vanishes.Source: §I · p. 2
II. SYSTEM MODEL
The system model places vulnerabilities in a queue and couples each vulnerability’s random lifetime with attacker activity that begins after attacker discovery. It uses BDI and RD attraction processes and measures expected exploits accumulated before patch completion.
II. SYSTEM MODEL
- Vulnerabilities arrive as a Poisson process at rate γ, undergo independent exponential discovery times for defender and attacker, and enter a first-discovered-first-served patching queue with exponential service time.Defender discovery is X ∼Exp(δd), attacker discovery is Y ∼Exp(δa), and patching lasts S ∼Exp(µ).Source: §II · p. 2
- For each vulnerability, N(t) counts active attackers, starts at zero until attacker discovery, and attracts new attackers at intensity λt; each active attacker exploits at rate θ and leaves.The model considers BDI attraction with population-dependent growth plus immigration and RD attraction that decays after attacker discovery.Source: §II · p. 2
- The queue illustration encodes vulnerability creation order, patch-queue membership, and active attacker counts, while empty circles indicate vulnerabilities not yet discovered by attackers.It also illustrates that defender and attacker discovery durations are independent, so a later-created vulnerability can be discovered by the defender first.Source: Figure 1 · p. 2, §II · p. 2
- The risk quantity is the expected number of exploits before patching, expressed through the active attacker process over the vulnerability’s lifetime.In a stable queue, the vulnerability lifetime has a steady-state distribution independent of the branching process N(t).Source: §II · p. 2
III. ANALYSIS
The analysis first derives the distribution of vulnerability lifetimes in the queue, then evaluates expected exploit counts under BDI and RD attacker processes.
III. ANALYSIS
- The analysis obtains vulnerability-lifetime distributions from the queue model and expected exploit counts under BDI and RD branching processes.Source: §III · p. 3
A. Queue Analysis
The queue model requires discovery arrivals to remain below patching capacity for a stable backlog. In steady state, a vulnerability’s lifespan combines discovery delay with its wait and service time, and balanced discovery and patching rates minimize expected exploits under a fixed defense constraint.
A. Queue Analysis
- γ < µ is the stability condition; when γ ≥ µ, the queue length grows without bound.The patching queue is M/M/1, with vulnerability arrival rate γ and service rate µ.Source: A. Queue Analysis · p. 3
- A vulnerability’s lifespan combines its discovery duration with its waiting and patching durations, and in steady state its density is a sum of two exponential distributions.When δd = µ −γ, the lifespan distribution is Erlang-2, obtained as the limit of the unequal-rate case.Source: A. Queue Analysis · p. 3
- Under the constraint µ + δd = b, expected exploits are minimized by choosing δd = (b −γ)/2 when b > γ.The result balances the effective patching rate µ −γ and discovery rate δd.Source: A. Queue Analysis · p. 3
B. Birth-Death-Immigration (BDI) Model
The BDI model represents attacker arrivals through immigration and attacker-to-attacker attraction, with attack growth rate κ = λ −θ. Its expected cumulative exploits are finite only when growth remains below both discovery and effective patching rates, while the effect of exploit rate θ can be non-monotone.
B. Birth-Death-Immigration (BDI) Model
- κ = λ −θ is the BDI attack growth rate, with each active attacker attracting new attackers at rate λ and immigration occurring at rate λ0 = δa.The model assumes the immigration rate equals the attacker discovery rate.Source: B. Birth-Death-Immigration (BDI) Model · p. 3
- The expected cumulative exploits are finite only when δd > κ and µ −γ > κ; otherwise, the expectation is unbounded.The two thresholds correspond to vulnerability discovery and effective patching capacity.Source: B. Birth-Death-Immigration (BDI) Model · p. 4
- Expected exploits can increase, peak, or decrease with θ as λ crosses the thresholds λc and min(δd, µ −γ).For λ ≤ λc the relationship is strictly increasing; above min(δd, µ −γ), it is strictly decreasing except in the stated divergent range.Source: B. Birth-Death-Immigration (BDI) Model · p. 4
C. Rise-Decay (RD) Model
The RD model lets attacker attraction decay exponentially after the first attacker discovers a vulnerability. Its expected exploit count is evaluated over the remaining vulnerability lifetime, and its relationship with exploit rate can increase or peak depending on attraction intensity.
C. Rise-Decay (RD) Model
- In the RD model, attacker attraction decays exponentially over time, with β setting the decay rate.The model shifts its time origin to the first attacker’s discovery.Source: C. Rise-Decay (RD) Model · p. 4
- Σ is the remaining vulnerability lifespan after attacker discovery when discovery precedes patch completion, and zero otherwise.The distribution has a mass at zero, which contributes nothing to expected exploits because D(0) = 0.Source: C. Rise-Decay (RD) Model · p. 4
- Fig. 2 plots average exploits against attraction rate in panel (a) and exploit rate in panels (b) and (c).The BDI and RD exploit-rate panels illustrate the model-specific relationship between exploit rate and average exploits.Source: Figure 2 · p. 5
- In the RD model, expected exploits increase with θ for λ ≤ λc and have a unique maximum for λ > λc.Here λc = δd(µ −γ)/(δd + δa + µ −γ).Source: C. Rise-Decay (RD) Model · p. 5
IV. NUMERICAL RESULTS
The numerical results compare analytical predictions with simulations and examine how attacker activity, defensive resource allocation, and AI scaling affect expected exploits. The AI-scaling example moves from finite to unbounded expected exploits when discovery becomes the limiting defense.
IV. NUMERICAL RESULTS
- Under a fixed defensive resource budget, balancing discovery and patching minimizes expected exploits and expands the range of attacker attraction rates for which that expectation remains finite.The numerical discussion reports this result as verifying Lemma 1.Source: §IV · p. 5
- In the BDI model, increasing the exploit rate raises expected exploits below λc, produces a unique maximum at intermediate λ, and reduces expected exploits when λ exceeds the defensive threshold.The three cases use λ = 0.3, λ = 0.6, and λ = 0.8, respectively.Source: §IV · p. 5
- In the RD model, increasing the exploit rate raises expected exploits below λc, while at λ = 0.5 excessive exploit rates reduce expected exploits by limiting attacker population growth.The analytical and simulation results closely match for this RD experiment.Source: §IV · p. 5
- At baseline, expected cumulative exploits are 34; equal twofold scaling makes them unbounded as attacker growth (0.55) exceeds discovery (0.50), despite patching reaching 1.20.Fig. 3 marks both scenarios and distinguishes finite from unbounded expectation.Source: Figure 3 · p. 6, §IV · p. 6
- The phase boundary is set by the slower defensive mechanism, and expected exploits rise sharply near it, making finite-region systems sensitive to rate uncertainty or temporary attacker surges.The bottleneck switches near (a_o, a_d) = (1.29, 1.07).Source: §IV · p. 6
V. CONCLUSION
The conclusion presents a queue of branching processes that couples vulnerability lifetimes with evolving attacker activity. Its results show that risk depends on relative attack, discovery, and patching rates, and that equal proportional AI scaling can shift the bottleneck and change the operating regime.
V. CONCLUSION
- The queue couples vulnerability lifetimes with attacker populations under BDI and RD models, with exploit counts not necessarily increasing monotonically with the exploit rate.Under BDI, attacker growth must remain below both vulnerability discovery and effective patching for expected exploits to remain finite.Source: §V · p. 6
- Under a fixed defensive resource budget, balancing discovery and patching minimizes expected exploits and expands the range of attacker growth rates the system can sustain.The slower defensive mechanism determines the bottleneck; improving the faster one alone may provide little benefit.Source: §V · p. 6
- Equal proportional AI amplification need not preserve the operating regime: faster patching can leave discovery as the bottleneck and move the system to unbounded expected exploits.Expected exploitation also rises sharply near the phase boundary.Source: §V · p. 6