network reliability teams

Abstract: We present an algorithm that computes the structural signature and the mean time-to-failure (MTTF) of a monotone binary system at every one of its states, rather than only at the state in which all components work. It makes no assumption on the structure function and evaluates it exactly once per state, as many times as needed for the signature of the complete system alone, so computing the MTTF everywhere costs only a factor polynomial in the number of components over computing it once. As a by-product, it identifies all minimal path sets. We assume exchangeable component failure times whose survivor count is a continuous-time Markov chain. This setting admits simultaneous failures, as in the Lévy-frailty Marshall-Olkin, binomial failure rate and lethal-shock models, where the simplifications available for single failures do not apply. We use the per-state MTTF to define threshold repair policies. We show that the per-state signature is a strictly finer decision criterion than the MTTF: signature dominance at a common failure count implies MTTF dominance, but not conversely. Under the Lévy-frailty Marshall-Olkin model the MTTF is monotone in the state, which yields the long-run cost rate of every threshold policy in a single pass. Experiments on 22 benchmark networks with up to 30 components and 31 failure models show that the best MTTF-threshold policy has a median excess cost of 0.36% over the optimal state-based policy, and that it is never worse than the best policy that repairs after a fixed number of failures.

Abstract: We investigate Strong Data-Processing Inequality (SDPI) constants for Rényi Divergences on finite spaces. We study their dependence on the Rényi order $α$, proving that they are non-decreasing and that their scaling by $(α-1)$ is convex for $α\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the Rényi-SDPI constant. For $α\in[0,1]$, we further prove equality with the $χ^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $\varphi$-Divergences. At finite orders, the Rényi-SDPI constants are bounded above and below through comparisons with the $χ^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects Rényi-LDP to Rényi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which Rényi-SDPIs improve on classical $χ^2$-based bounds by arbitrarily large factors.

Abstract: Adversarial consensus simulators require not only a model of message delivery but also a model of time. For timeout-sensitive protocols, coupling protocol-time progression to scheduler activity can allow communication scheduling to influence when failure detectors expire. We study this interaction in VeriProtocol using controlled Multi-Paxos and Raft experiments. In the primary Multi-Paxos study, we compare EventCoupled timing, where every scheduler opportunity advances protocol time, with RoundTick timing, which decouples individual scheduler events from timeout progression. At N=5, increasing the bounded message-delay budget from K=6 to K=14 increased the observed EventCoupled leadership-instability rate from 5\% to 100\% across the tested grid, while no additional valid leader elections occurred in any of the corresponding 140 RoundTick executions. At K=0, neither model produced additional elections. Matched-trace analysis identifies the mechanism: under EventCoupled timing, scheduler activity used to perturb communication also directly advances failure-detector state. Under RoundTick, comparable scheduler activity does not imply equivalent protocol-time progression. A targeted heartbeat-delay control confirms that RoundTick does not suppress timeout-driven elections; an additional election remains reachable when leader contact is unavailable across sufficient logical time.

Abstract: This paper investigates a fundamental question in stripeless erasure coding based on symmetric balanced incomplete block designs (SBIBDs): how many replicas per object are precisely required to guarantee recovery from any set of at most $p$ node failures? We refine the known sufficient recovery guarantee for the generalized SBIBD $(v,k,λ)$ construction. One replica is necessary and sufficient for $p=1$, and $q=λ(p-1)+2$ replicas ($q\le k$) guarantee recovery for $p\ge2$. Recovery takes one round for $p=2$ and at most two rounds in general. To study the tightness of this count, we define the universal replica threshold $q^\ast(A,p)$ for a fixed zero-diagonal SBIBD representative~$A$. We determine $q^\ast(A,p)$ for $p=1$ and $p=2$. For $p\ge3$, we identify pairwise separated sets and private target permutations as the structures determining whether the sufficient count is tight or can be further reduced. These conditions give exact thresholds for all but the case where $λ>1$ and $A$ contains no pairwise separated set of size~$p$. We improve the bounds for this remaining case and leave its exact threshold open. Finally, we give sufficient conditions for pairwise separated sets and show how affinity relabeling can realize private target permutations.

Abstract: We consider the question of designing a universal family of sets $F \subset 2^{[n]}$ such that for any function $f:2^{[n]} \to R_{\geq 0}$ in a certain class, we have $$\max_{S \in F} f(S) \geq c(n) \cdot \max_{S \subset [n]} f(S).$$ We prove that there is a family of subpolynomial size such that for any nonnegative submodular function, $c(n) = Ω(\frac{\log \log n}{\log n})$, and there is a family of logarithmic size such that $c(n) = Ω(\frac{1}{\log n})$. We also prove that pairwise independence (which achieves a constant factor for graph cut functions), or even $k$-wise independence, does not imply a bound better than $O(\frac{1}{\sqrt{\log n}})$ for submodular functions. On the other hand, we prove that for any polynomially representable subclass of nonnegative submodular functions (such as the matroid connectivity functions for matroid representable over $F_q$), a constant-factor universal family of polynomial size always exists. For absolute XOS functions (a class that we introduce, in the form $f(S) = \max_i |\sum_{j \in S} w_{ij} + c_i|$ where $w_{ij}, c_i \in R$), we design a family of polynomial size such that $c(n) \geq \sqrt{\frac{\log n}{n}}$, and prove that there is no polynomial-size family achieving a factor better than $O(\sqrt{\frac{\log n}{n}})$.

Abstract: We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a $\left(\frac12-\varepsilon\right)$-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with $O(\log n)$ layers and weak diameter $O(\log n)$. We give two applications in the $\mathsf{LOCAL}$ model. For every constant $\varepsilon>0$, we obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for list $\left(\frac32+\varepsilon\right)Δ$-edge coloring on graphs of maximum degree $Δ\geqΔ_0(\varepsilon)$; for bipartite graphs, the result holds for all $Δ$. For every constant $0<\varepsilon<1/4$, we also obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for a $\left(\frac14-\varepsilon\right)$-locally balanced cut, where every vertex has at least a $\left(\frac14-\varepsilon\right)$-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in $O(\log^2 n)$ randomized rounds using Miller--Peng--Xu low-diameter clustering and a simple trimming procedure, and deterministically in $\widetilde O(\log^2 n)$ rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].