Papachristou and Rahimian

Production Networks Resilience

Production Networks Resilience: Cascading Failures, Power Laws and Optimal Interventions

Marios Papachristou
Cornell University<EMAIL_ADDRESS>M. Amin Rahimian
University of Pittsburgh<EMAIL_ADDRESS>
Problem definition:  The pandemic crisis made apparent the need to take a holistic view of production networks to understand risk factors that arise due to their many interdependent units. We investigate the structural factors contributing to the severity of cascading failures in production networks, how to quantify the effect of these risk factors with resilience metrics, and how to design and evaluate interventions to improve resilience.

We propose a node percolation process on production networks that model product suppliers failing independently due to exogenous, systemic shocks and causing other products to fail as their production requirements are unmet. We first show that the size of the cascading failures follows a power law in random directed acyclic graphs, whose topology encodes the natural ordering of products from simple raw materials to complex products. This motivates the need for a resilience metric, which we define as the maximum magnitude shock the production network can withstand with only a small fraction of products failing. Next, we study the resilience of several architectures, such as trees, parallel products, and random trellis, and classify them as resilient or fragile depending on their topological attributes. In the next step, we give general lower bounds and optimal interventions for improving resilience as a function of Katz centralities. Finally, we empirically calculate the resilience metric and study interventions in various real-world networks to validate our theoretical results.

Managerial implications: We offer new thoughts and theories on reasoning about supply chain resilience based on topological attributes such as source dependency and product complexity and propose novel, theory-informed metrics that we evaluate empirically on real-world networks. Our theories also inform intervention designs for improving resilience, not just from a firm’s perspective but also on a national scale.

Production networks, Resilience, Cascading failures, Interventions
This version: August 27, 2024.

MP was partially supported by a LinkedIn Ph.D. Fellowship, a grant from the A.G. Leventis Foundation, and a grant from the Gerondelis Foundation. MAR acknowledges support from a Pitt Momentum Funds Grant on Socially Responsible Data Collection and Network Intervention Designs. Data and code can be found at:

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University of Chicago Press Chicago, IL

journaltitleNBER Macroeconomics Annual
titleNetworks and the macroeconomy: An empirical exploration

Wiley Online Library

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journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

journaltitleJournal of operations Management
titleThe organizational antecedents of a firm’s supply chain agility for risk mitigation and response

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

Cambridge University Press

journaltitleProbability in the Engineering and Informational Sciences
titleA loading-dependent model of probabilistic cascading failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

Taylor & Francis

journaltitleIISE Transactions
titleA structured overview of insights and opportunities for enhancing supply chain resilience

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleOperations Research
titleDisruption risk mitigation in supply chains: The risk exposure index revisited

journaltitleJournal of Cleaner Production
titleThe Circular Economy–A new sustainability paradigm?

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

titleSupply chain disruptions: Theory and practice of managing risk

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journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

titleA new status index derived from sociometric analysis

Wiley Online Library

journaltitleProduction and Operations Management
titleManaging disruption risks in supply chains

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

journaltitleApplied network science
titleNetwork science approach to modelling the topology and robustness of supply chain networks: a review and perspective

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

journaltitleHarvard Business Review
titleFrom superstorms to factory fires: Managing unpredictable supply chain disruptions

Wiley Online Library

journaltitleProduction and Operations Management
titleIncreasing supply chain robustness through process flexibility and inventory

titleIdentifying risks and mitigating disruptions in the automotive supply chain

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journaltitleReview of International Economics
titleAn illustrated user guide to the world input–output database: the case of global automotive production

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleManufacturing & Service Operations Management
titleData set—real-world multiechelon supply chains used for inventory optimization

Wiley Online Library

titleThe network origins of aggregate fluctuations

journaltitleAmerican Economic Review
titleSystemic risk and stability in financial networks

journaltitleOperations Research Letters
titleOptimal intervention under stress scenarios: A case of the Korean financial system

journaltitlepreprint arXiv:1907.09943
titleYield uncertainty and strategic formation of supply chain networks

World Scientific

journaltitleInternational Journal of Theoretical and applied finance
titleStress testing the resilience of financial networks

journaltitleJournal of Banking & Finance
titleContagious synchronization and endogenous network formation in financial networks

journaltitleTinbergen Institute Discussion Paper
titleThe formation of financial networks

journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

Nature Publishing Group

journaltitleNature Scientific Reports
titleDebtrank: Too central to fail? financial networks, the fed and systemic risk

journaltitleJournal of economic dynamics and control
titleLiaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

journaltitleManagement Science
titleContagion in financial networks: a threat index

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

journaltitleAvailable at SSRN 3034406
titleNetwork hazard and bailouts

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

booktitleInternational Conference on Current Trends in Theory and Practice of Computer Science
titlePayment Scheduling in the Interval Debt Model

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Wiley Online Library

titleOn reliability of graphs with node failures

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

Wiley Online Library

journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

Public Library of Science San Francisco, CA USA

journaltitlePloS one
titleThe propagation of economic impacts through supply chains: The case of a mega-city lockdown to prevent the spread of COVID-19

journaltitleDistorted Investment Incentives, Regulation, and Systemic Risk Measurement (March 1, 2019)
titleWhat makes financial networks special? distorted investment incentives, regulation, and systemic risk measurement

journaltitlepreprint arXiv:2212.06808
titleIncentive-Aware Models of Dynamic Financial Networks

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

booktitleProceedings of the Web Conference 2022
titleAllocating Stimulus Checks in Times of Crisis

booktitleInternational Conference on Web and Internet Economics
titleDefault ambiguity: finding the best solution to the clearing problem

journaltitlepreprint arXiv:2005.10585
titleProduction networks and epidemic spreading: How to restart the UK economy?

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

UT Austin

journaltitleM375T/M396C: Topics in Complex Networks Course
titleGalton-Watson Branching Process

journaltitleGames and Economic Behavior
titleFree and perfectly safe but only partially effective vaccines can harm everyone

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleDiscrete mathematics
titleUniformly optimal graphs in some classes of graphs with node failures

Online Appendix

§ ANALYTICAL BOUND ON X TO ENSURE PRFGEQK FOR RDAG(K, P)

We can convert the statement of <ref> to a high-probability statement, if we require $\Pr [F \ge \varepsilon K]$ to be $O(1/K)$. Since $g(x, K, p, \varepsilon)$ is an increasing function of $x$, we are asking to find the largest possible $x$ such that $g(x, K, p, \varepsilon) \le \frac {C} {K}$.

In order to get an analytically tractable expression for $x$, we use the fact that $\log t \le t$ for all $t > 0$ and get that <ref> becomes

\begin{align*}
g(x, K, p, \varepsilon) & \le x^n \left [ 1 - \varepsilon + \frac {1} {K \log \left ( \frac 1 {1 - p} \right )} \left ( \frac {1 - (1 - x^n)(1 - p)^{K}} {1 - (1 - x^n)(1 - p)^{\varepsilon K}} \right ) \right ] \le x^n \left [ 1 - \varepsilon + \frac {1} {K \log \left ( \frac 1 {1 - p} \right )} \left ( \frac {1} {1 - (1 - x^n)(1 - p)^0} \right ) \right ] \\ & = x^n \left ( 1 - \varepsilon \right ) + \frac {1} {K \log \left ( \frac 1 {1 - p} \right )} = \overline g (x, K, p, \varepsilon)
\end{align*}

If $p$ is constant, then, choosing $x = \left ( \frac {1} {K \log \left ( \frac {1} {1 - p} \right ) (1- \varepsilon)} \right )^{1/n},$ makes $\overline g (x, K, p, \varepsilon) \le \frac {2} {\log \left ( \frac {1} {1 - p} \right ) K} = O \left ( \frac 1 K \right )$.

§ OMITTED PROOFS

§.§ Proof of theorem:powerlaws

Let $G \sim \mathsf{rdag}(K, p)$, with nodes $i_1, \dots, i_K$ (in this order). Let $P_{k, f}$ be the probability of having $f$ distinct failures in the random DAG with $k$ nodes conditioned on a failure on node 1. We have that $P_{1, 1} = 1$ and $P_{k, f} = 0$ for $f > i$ and $f < 1$. To devise a recurrence formula for $P_{i, f}$ note that for the $i$-th node we have the following:

* $i_k$ is affected by the cascade. That happens if at least one connection to $f - 1$ infected nodes upto node $i_{k - 1}$, or if $i_k$ fails due to percolation. This happens with probability $\big \{ [ 1 - (1 - p)^{f - 1} ] + x^n - [ 1 - (1 - p)^{f - 1} ] x^n \big \} P_{k - 1, f - 1} = \big [ 1 - (1 - p)^{f - 1}(1 - x^n) \big ]P_{k - 1, f - 1}$.

* $i_{k}$ is not affected by the cascade. That means that $i_k$ has $\ge 1$ functional supplier, and no connection exists from the $f$ infected nodes. That happens with probability $(1 - p)^f (1 - x^n) P_{k - 1, f}$.

This yields the following recurrence,

\begin{align} \label{eq:recurrence}
P_{k, f} = \left [ 1 - (1 - p)^{f - 1}(1 - x^n) \right ]P_{k - 1, f - 1} +   (1 - p)^f (1 - x^n) P_{k - 1, f}.
\end{align}

To determine the distribution of $F$ in $\mathsf{rdag}(K, p)$, we assume that the cascade can start at any node with equal probability $1 / K$ and that the probability of failure is conditioned on the choice of a node is $x^n$. Also, since a cascade in $\mathsf{rdag}(K, p)$ starting from node 1 is the same as starting from node $i$ in $\mathsf{rdag}(K + i - 1, p)$, the distribution obeys the following,

\begin{align}
\Pr [F = f] = \frac {x^n} {K} \sum_{k \in [K]} P_{k, f}
\end{align}

We let $Q_{K, f} =  \sum_{k \in [K]} P_{k, f}$, so that $\Pr [F = f] = \frac {x^n} {K} Q_{K, f}$. Summing <ref> for $k \in [K]$ and using the definition of $Q_{K, f}$ yields a recurrence relation for $Q_{K, f}$, i.e. $Q_{K, f} = (1 - p)^f (1 - x^n) Q_{K - 1, f} + [ 1 - (1 - p)^{f - 1} ] (1 - x^n) Q_{K - 1, f - 1}$. We take the limit for $K$ large, we let $q_f = \lim_{K \to \infty} Q_{K, f}$, and solve the recurrence $q_{f} = (1 - p)^f (1 - x^n) q_{f} + [ 1 - (1 - p)^{f - 1} ] (1 - x^n) q_{f - 1}$ to get $q_f = \frac {1} {1 - (1 - x^n) (1 - p)^f}$. Since $e^x \ge x$, we have that $(1 - p)^f \le \log (1 - p) f$ and subsequently $1 - (1 - x^n) (1 - p)^f \le f \left (1 + (1 - x^n) \log \left ( \frac 1 {1 - p} \right ) \right )$. Therefore, for sufficiently large $K$,

\begin{align*}
\small
\Pr [F = f] & \asymp \frac {x^n q_f} {K} \asymp \frac {x^n} {K(1 - (1 - x^n) (1 - p)^f)} \ge \underbrace {\frac {x^n} {K \left (1 + (1 - x^n) \log \left ( \frac 1 {1 - p} \right ) \right )}}_{C(K, p, x, n) > 0} \frac 1 f.
\end{align*}

§.§ Proof of lemma:upperboundresilience

If $S(x)$ is the number of surviving products for percolation probability $x$, and $x_1 \le x_2$ are two percolation probabilities, a straightforward coupling argument shows that $S(x_1) \ge S(x_2)$, and subsequently for every $s \in [0, K]$ we have that $\Pr_{x = x_1} [S \ge s] \ge \Pr_{x = x_2} [S \ge s]$. Now, in order to arrive at a contradiction, let $\overline R_{\cG}(\varepsilon) \le R_\cG$, and $s = (1 - \varepsilon)K$. Then $1 - 1 / K \le \Pr_{x = R_{\cG}(\varepsilon)} [S \ge (1 - \varepsilon) K] \le \Pr_{x = \overline R_{\cG}(\varepsilon)} [S \ge (1 - \varepsilon) K] \le 1 / 2$ which yields a contradiction.

§.§ Proof of theorem:parallelproducts

Lower Bound. For $\cC$, let $\varepsilon \in (0, 1)$. If $F_{\cR}$ (resp. $F_{\cC}$) is the number of failed raw materials (resp. complex products), we have that $\{ F_{\cC} \ge \varepsilon K \} \implies \{ F_{\cR} \ge {\varepsilon K} / d \} $. Let $\delta = \frac {1} {x^n} \sqrt {\frac {\log K} {2R}}$ and let $\frac {\varepsilon K} {d} = (1 + \delta) \ev {} {F_R} = (1 + \delta) \rho x^n$. We apply the one-sided Chernoff bound and get $\Pr [F_{\cC} \ge \varepsilon K] \le \Pr \left [F_{\cR} \ge \frac {\varepsilon K} {d} \right ] = \Pr \left [F_{\cR} \ge (1 + \delta) \ev {} {F_{\cR}} \right ] \le e^{-2 \delta^2 \ev {} {F_{\cR}}^2 / R} = \frac 1 K$. Finally, by resolving the last equation $(1 + \delta)\rho x^n = \frac {\varepsilon K} {d}$, we get that $x = \left ( \frac {\varepsilon K} {\rho d} + \sqrt {\frac {\log K} {2\rho}} \right )^{1/n}$. Also, we have that $\rho \le mK$ and therefore $R_{\cC}(\varepsilon) \ge \left ( \frac {\varepsilon} {dm} + \sqrt {\frac {\log K} {2mK}} \right )^{1/n}$. If $\varepsilon, m$ and $d$ are independent of $K$ then for $K \to \infty$ we have that $R_{\cC}(\varepsilon) \ge \left ( \frac {\varepsilon} {md} \right )^{1/n} > 0$.

For $\cG$, the analysis is similar to the above. For brevity, we give the analysis in expectation (it is easy to extend it to an analysis in high probability): If on expectation $\ev {} {F_{\cR}} = \rho x^n$ raw materials fail, that implies that at most $\ev {} {F} = \ev {} {F_{\cR}} + \ev {} {F_\cC} \le \rho x^n + d \rho x^n = (d + 1) \rho x^n \le m K x^n (d + 1)$ total products fail in expectation. We want the fraction of failed products to be at least $\varepsilon (K + \rho) \ge \varepsilon K / 2$. Therefore by solving the inequality, we get that the lower bound in the resilience is $\left ( \frac {K} {2m (d + 1)} \right )^{1/n}$. The high-probability analysis would be similar to the above case, yield and extra additive $\sqrt {\frac {\log(K / 2)} {2mK}}$ factor.

Upper Bound. For $\cC$, to derive the upper bound, we first bound $\ev {} {S_{\cC}}$. It is easy to see that due to the linearity of expectation $\ev {} {S_{\cC}} = K (1 - x^n)^m$. Thus by Markov's inequality we have that $\Pr [S_{\cC} \ge (1 - \varepsilon) K] \le \frac {\ev {} {S}} {(1 - \varepsilon) K} \le \frac {(1 - x^n)^m} {1 - \varepsilon}$. To make this probability $1 / 2$ it suffices to set $x = \left ( 1 - \left ( \frac {1 - \varepsilon} {2} \right )^{1/m} \right )^{1/n}$, thus from <ref> this establishes an upper bound on $R_{\cC}(\varepsilon)$.

For $\cG$, we proceed similarly by showing that the number of expected products is $\ev {} {S} = \rho (1 - x^n) + K (1 - x^n)^m \le mK (1 - x^n) + K(1 - x^n)^m \le K (m + 1) (1 - x^n)$. Similarly to the above, from <ref> we get that the upper bound on $R_{\cG}$ is $\left ( 1 - \frac {1 - \varepsilon} {2 (m + 1)} \right )^{1/n}$.

§.§ Proof of theorem:treeresilience

Lower Bound. Depending on the range of $m$ we have two choices

* Case where $m = 1$. For every $\tau \in [D]$ we have that $\Pr[S \ge D - \tau] = \Pr \left [ \bigcap_{d > \tau} \{ Z_i = 1 \} \right ] = \Pr [Z_1 = 1] \Pr [Z_2 = 1 | Z_1 = 1] \dots = \prod_{d > \tau} (1 - x^n) = (1 - x^n)^{D - \tau}$. We let $\tau = \varepsilon D$ for some $\varepsilon \in (0, 1)$ and thus $\Pr [S \ge (1 - \varepsilon) D] = (1 - x^n)^{(1 - \varepsilon)D}$. We want to make this probability at least $1 - 1 / D$, and therefore, the resilience of the path graph is $R_{\cG}(\varepsilon) \ge \left ( 1 - \left ( 1 - \frac 1 D \right )^{\frac 1 {(1 - \varepsilon) D)}} \right)^{1/n}$. Since $K = D$ we get the desired result.

* Case where $m \ge 2$. Let $\cK_d$ be the products of tier $d$. We let $\tau = \sup \{ d \in [D] : \exists i \in \cK_d : Z_i = 0 \}$ be the bottom-most tier for which a product failure happens. If at level $\tau$ a failure happens, all levels above $\tau$ are deactivated. The probability that all products up to tier $\tau$ operate is given by

\begin{align*}
\Pr [\text{all products up to tier $\tau$ operate}] & = \Pr \left [ \bigcap_{d > \tau} \bigcap_{i \in \cK_d} \{ Z_i = 1 \} \right ] = \prod_{d = D}^{\tau + 1} \Pr \left [ \bigcap_{i \in \cK_d} \{ Z_i = 1 \} \bigg | \bigcap_{d' > d} \bigcap_{i \in \cK_{d'}} \{ Z_i = 1 \} \right ] \\
& = \prod_{d = D}^{\tau + 1} (1 - x^n)^{m^d} = \left ( 1 - x^n \right )^{\sum_{d = D}^{\tau + 1} m^d} = \left ( 1 - x^n \right )^{\frac {m^D - m^\tau} {m - 1}}
\end{align*}

Also, $\{ \text{all products up to tier $\tau$ operate} \} \implies \{ S \ge \frac {m^D - m^\tau} {m - 1} \}$. Therefore, the tail probability of $S$ for $\tau \in [D]$ is given by $\Pr \left [S \ge \frac {m^D - m^\tau} {m - 1} \right ] = \Pr \left [ S \ge \underbrace {\left ( 1 - \frac {m^\tau} {m^D} \right )}_{:= 1 - \varepsilon}  \frac {m^D} {m - 1}  \right ] \ge \left ( 1 - x^n \right )^{(1 - \varepsilon) \frac {m^D} {m - 1}}$. For large enough $D$ we approach the continuous distribution and thus $\Pr [S \ge (1 - \varepsilon) K] \ge (1 - x^n)^{(1 - \varepsilon)K}$. Letting the above be at least $1 - 1 / K$ we get that $R_{\cG}(\varepsilon) \ge \left [ 1 - \left ( 1 - \frac 1 {K} \right )^{\frac {1} {(1 - \varepsilon) K}} \right ]^{1/n}$.

Upper Bound. To derive an upper bound, we have the following cases, depending on the value of $m$

* Case $m = 1$. We follow the same logic as the $m \ge 2$ case, and upper bound $\ev {} {S} \le \sum_{d \ge 0} (1 - x^n)^d = \frac {1} {x^n}$ which yields an upper bound $R_{\cG}(\varepsilon) < \left ( \frac {2} {D (1 - \varepsilon)}  \right )^{1/n} \to 0$ as $D \to \infty$.

* Case $m \ge 2$. By Markov's Inequality we get that $\Pr [S \ge (1 - \varepsilon) K] \le \frac {\ev {} {S}} {(1 - \varepsilon) K}$. <ref> (proved in Appendix <ref>), implies that $\ev {} {S} \le \frac {K D x^n} {2}$, thus $\Pr [S \ge (1 - \varepsilon) K] \le \frac {K D x^n} {2 (1 - \varepsilon) K}$. To make the RHS equal to $1 / 2$, it suffices to pick $x = \left ( \frac {1 - \varepsilon} {D} \right )^{1/n}$. By <ref> we get that $R_{\cG}(\varepsilon) < \left ( \frac {1 - \varepsilon} {D} \right )^{1/n} \to 0$ as $D \to \infty$.

§.§ Proof of theorem:gwresilience

In order to prove <ref>, we first prove this auxiliary lemma:

For $\tau$ finite, $\frac {\one \{ \mu > 1 \}} {\log \mu} < \alpha < \frac 1 2$ , and $0 < \beta < \one \{ \mu < 1 \} + \one \{ \mu > 1 \} \frac {\log \mu - 1} {\mu} $, let

\begin{align*}
\phi(z) & = z \frac {\mu^\tau z^\tau - 1} {\mu z - 1} - \alpha \frac {\mu^\tau - 1} {\mu - 1}, \text{ for } z \neq \frac 1 \mu, \quad
\psi(z) & = \frac {\mu^\tau - 1} {\mu - 1} - z \frac {\mu^\tau z^\tau - 1} {\mu z - 1} - \beta, \text{ for } z \neq \frac 1 \mu.
\end{align*}

* If $\mu < 1$, then there exist $z_1, z_2 \in (0, 1)$ such that $\phi(z_1) = \psi(z_2) = 0$.

* If $\mu > e^2$, then there exists $z_1 \in (1/\mu, 1)$ such that $\phi(z_1) = 0$.

* If $\mu > e$, then there exists $z_2 \in (1/\mu, 1)$ such that $\phi(z_2) = 0$.

Analysis for $\phi(z)$. We do case analysis:

* If $\mu < 1$ then $\phi$ is defined everywhere in $[0, 1]$ and is also continuous. It is also easy to prove that $\phi$ is increasing in $[0, 1]$ since its the product of two non-negative increasing functions, $z$ and  $ \frac {(\mu z)^\tau - 1} {\mu z - 1} =\sum_{i = 0}^{\tau - 1} (\mu z)^i$. Moreover, note that $\phi(0) < 0$ and $\phi(1) > 0$. Therefore, there exists a unique solution $z_1 \in (0, 1)$ such that $\phi(z_1) = 0$.

* If $\mu > e^2$, we study $\phi$ in $(1/\mu, 1]$. Again, $\phi$ is increasing (for the same reason as above), continuous in $(1/\mu, 1]$, and has $\phi(1) > 0$. We also have that, by using L'Hôspital's rule,

\begin{align*}
\lim_{z \to 1 / \mu} \frac {\mu^\tau z^\tau - 1 } {\mu z - 1} & = \lim_{z \to 1 / \mu} \frac {(\mu^\tau z^\tau - 1)'} {(\mu z - 1)'} = \lim_{z \to 1 / \mu} \frac {\mu^\tau \tau z^{\tau - 1}} {\mu} = \tau \implies \\
\lim_{z \to 1 / \mu} \phi(z) & = \frac {\tau} {\mu} - \alpha \frac {\mu^\tau - 1} {\mu - 1} < \frac {\tau (1 - \alpha \log \mu)} {\mu} < 0 \text{ for } \alpha > \frac {1} {\log \mu}.
\end{align*}

Therefore, for $\alpha \in (1/\log \mu, 1/2)$, there exists a unique solution $z_1 \in (1/\mu, 1]$ such that $\phi(z_1) = 0$.

Analysis for $\psi(z)$. Note that $\psi$ is a decreasing function of $z$. We do case analysis:

* If $\mu < 1$, then $\psi$ is defined everywhere in $[0, 1]$ and is continuous in $[0, 1]$. We have that $\psi(0) > 0$ and $\psi(1) < 0$ therefore there exists a unique solution $z_2$ such that $\psi(z_2) = 0$.

* If $\mu > e$, then $\psi$ is decreasing and contunuous in $(1/\mu, 1]$, with $\psi(1) < 0$. We also have that $\lim_{z \to 1 / \mu} \psi(z)  = \frac {\mu^\tau - 1} {\mu - 1} - \frac {\tau} {\mu} - \beta > \frac {\tau (\log \mu - 1 - \mu \beta)} {\mu} > 0 \text { for } \beta < \frac {\log \mu - 1} {\mu}$.

Subsequently we prove <ref>:

§.§.§ Proof of <ref>.

Upper Bound. Let $\tau = \inf \{ d \ge 1 : |\cK_d| = 0 \}$ be the extinction time of the GW process. In order to establish an upper bound on the resilience it suffices to set the expected number of surviving products to be at most $\frac {1 - \varepsilon} {2} \ev {\cG} {K}$, since by Markov's inequality the probability of a fraction of at least $(1 - \varepsilon)$-fraction of products surviving would be at most $1 / 2$ and by <ref> we would get an upper bound on the resilience $R_{\cG}(\varepsilon)$; namely, $\Pr_{\cG, x} [S \ge (1 - \varepsilon) \ev {\cG} {K}] \le \frac {\ev {\cG, x} {S}} {(1 - \varepsilon) \ev {\cG} {K}} \le \frac 1 2$. Conditioned on $Z_1 = 1$, which happens with probability $1 - x^n$, the surviving products grow as a GW process with mean $\mu_x = (1 - x^n) \mu$. Therefore, the condition $\ev {\cG, x} {\cS} = \frac {1 - \varepsilon} {2} \ev {\cG} {K}$, conditioned on the extinction time being $\tau$, is equivalent to

\begin{align}
(1 - x^n) \frac {\mu_x^\tau - 1} {\mu_x - 1} & \le \frac {1 - \varepsilon} {2} \frac {\mu^\tau - 1} {\mu - 1} \iff (1 - x^n) \frac {\mu^\tau (1 - x^n)^\tau - 1} {\mu (1 - x^n) - 1}  \le \frac {1 - \varepsilon} {2} \frac {\mu^\tau - 1} {\mu - 1}  \label{eq:upper_bound_gw}
\end{align}

We have the following cases

* If $\mu < 1$ then $\Pr [\tau < \infty] = 1$ (i.e., the process goes extinct after a finite number of steps), then the upper bound on the resilience is always finite due to <ref> which can be found by numerically solving <ref>.

* If $\mu (1 - x^n) > 1$, then $\Pr [\tau = \infty] > 0$ and in this case <ref> is only feasible if and only iff $x = 0$, at which case the upper bound on the resilience is 0, and the GW process is not resilient. If $\tau < \infty$, which happens with non-zero probability, the  upper bound on the resilience is finite when $\mu > e^2$ due to <ref>.

For a specific triplet $(\mu, \tau, \varepsilon)$, let $\overline x(\mu, \tau, \varepsilon)$ be the smallest possible solution to <ref>, which exists for $\mu \in (0, 1) \cup (e^2, \infty)$ due to <ref>. Then the expected upper bound on the resilience $\ev {\cG} {\overline \cR_{\cG} (\varepsilon)}$, can be expressed as $\ev {\cG} {\overline \cR_{\cG} (\varepsilon)} = \ev {\tau} {\overline \cR_{\cG} (\varepsilon)} = \sum_{1 \le k < \infty} \Pr [\tau = k] \overline x(\mu, \tau, \varepsilon) > 0$.

Lower Bound. Similarly to the upper bound, in order to devise a lower bound, it suffices to set $\ev {\cG} {K} - \ev {\cG} {S}$ to be at most $\varepsilon$, since, again, by Markov's inequality, we are going to get that the probability that at least a $\varepsilon$-fraction of products fails is at most $\frac {1} {\ev {\cG} {K}}$; namely, $\Pr_{\cG, x} [F \ge \varepsilon \ev {\cG} {K}] \le \frac {\ev {\cG, x} {F}} {\varepsilon \ev {\cG} {K}} \le \frac 1 {\ev {\cG} {K}}$. This yields

\begin{align}
\frac {\mu^\tau - 1} {\mu - 1} - (1 - x^n) \frac {\mu_x^\tau - 1} {\mu_x - 1}  \le \varepsilon  \iff         \frac {\mu^\tau - 1} {\mu - 1} - (1 - x^n) \frac {\mu^\tau (1 - x^n)^\tau - 1} {\mu (1 - x^n) - 1}  \le \varepsilon \label{eq:lower_bound_gw}
\end{align}

Similarly to the upper bound, we have the following cases,

* In the subcritical regime $\mu < 1$, we can again prove that the lower bound is always finite due to <ref>.

* In the supercritical regime $\mu (1 - x^n) > 1$, we have that when $\tau < \infty$, which happens with positive probability then for $\mu > e$ from <ref> we get the existence of the resilience. When $\tau = \infty$, we have again that the only way <ref> can hold is iff $x = 0$.

For a specific triplet $(\mu, \tau, \varepsilon)$, let $\underline x(\mu, \tau, \varepsilon)$ be the largest possible solution to <ref>, which exists for $\mu \in (0, 1) \cup (e, \infty)$ due to <ref>. Then the expected lower bound on the resilience $\ev {\cG} {\underline \cR_{\cG} (\varepsilon)}$, can be expressed as
$\ev {\cG} {\underline \cR_{\cG} (\varepsilon)} = \ev {\tau} {\underline \cR_{\cG} (\varepsilon)} = \sum_{1 \le k < \infty} \Pr [\tau = k] \underline x(\mu, \tau, \varepsilon) > 0$.

Determining $\Pr [\tau < \infty]$ when $\mu (1 - x^n) > 1$. It is known from the analysis of GW processes (see, e.g., <cit.>) that the extinction probability $\Pr [\tau < \infty]$ can be found as the smallest solution $\eta \in [0, 1]$ to the fixed-point equation $\eta = G_{\cD}(\eta)$ where $G_{\cD}(s) = \ev {\xi \sim \cD} {e^{s \xi}}$ is the moment generating function of the branching distribution $\cD$.

§.§ Proof of theorem:generalub

For $i \in \cK$ let $\beta_i = \Pr [Z_i = 0] \in [0, 1]$. By the union bound, we have that

\begin{align*}
\beta_i & = \Pr [(\exists j \in \cN(i) : (i, j) \text{ is operational } \wedge Z_j = 0) \vee (\forall s \in \cS(i) Y_{is} = 0)] \\
& \le \Pr [\exists j \in \cN(i) : (i, j) \text{ is operational } \wedge Z_j = 0] + \Pr [\forall s \in \cS(i) Y_{is} = 0] \\
& \le y \sum_{j \in \cN(i)} \beta_j + x^n.
\end{align*}

The number of failed products equals $\ev {} {F} = \sum_{i \in \cK} \beta_i$. Thus, finding the upper bound on $\ev {} {F}$ corresponds to solving the following LP,

\begin{align*}
\max _{\beta \in [\zero, \one]} \quad & \sum_{i \in \cK} \beta_i \qquad \text{s.t.} \quad & \beta \le y A^T \beta + x^n \one.
\end{align*}

When $y \| A^T \|_1 \le 1$, which is equivalent to $y \le \frac {1} {\Delta}$, this problem resembles the financial clearing problem of <cit.>, and from Lemma 4 of <cit.>, we know that we can also compute $\beta$ by solving the fixed point equation $\beta = \one \wedge (y A^T \beta + x^n \one) = \Phi(\beta).$ The set of fixed points $\mathsf{FIX} (\Phi) = \{ \beta \in [\zero, \one] : \beta = \Phi(\beta) \}$ has a greatest fixed point $\overline \beta^*$ and a least fixed point $\underline \beta^*$, due to the Knaster-Tarski Theorem.

§.§ Proof of prop:katz

First of all, it is easy to see that $\Phi(\beta)$ is a contraction under the assumption $y < \frac 1 \Delta$, and, thus, by Banach's fixed point theorem, $\Phi(\beta)$ has a unique fixed point $\beta^*$. Finally, note that, for any $i \in \cK$ we have that $\left ( y A^T \beta^* + x^n \one \right )_i = y \sum_{j \in \cN(i)} \beta_j^* + x^n < y \Delta + (1 - y \Delta) < 1$. Therefore, the fixed point equation simplifies to $\beta^* = y A^T \beta^* + x^n \one$, which has a solution $\beta^* = x^n (I - y A^T )^{-1} \one$, which corresponds to the Katz centrality, since $yA$ is substochastic, and subsequently, $I - yA^T$ is invertible. Finally, from Markov's inequality $\Pr [F \ge \varepsilon K] \le \frac {x^n \one^T (I - yA^T)^{-1} \one} {\varepsilon K} = \frac {x^n \| \mathsf{vec} ((I - yA^T)^{-1}) \|_1} {\varepsilon K}$, where the last equality holds since $(I - yA^T)^{-1}$ has non-negative elements (since $A^T$ is an adjacency matrix and $(I - yA^T)^{-1}$ can be expressed as Neumann series). To make the above $1 / K$ is suffices to pick, $x = \left ( \frac {\varepsilon} {\| \mathsf{vec} ((I - yA^T)^{-1} \|_1} \right )^{1/n}$, hence the lower bound in $R_{\cG}(\varepsilon)$.

§.§ Proof of prop:intervention

Since $0 < y < \frac 1 {\Delta}$ and $0 < x < (1 - y\Delta)^{1/n}$, the optimal solution of the internal maximization equals $\hat \beta(t) = x (I - yA^T)^{-1} (\one - t)$. Substituting that in the objective function of <ref>, we get that

\begin{align*}
\hat t = \arg \min_{t \in \{ 0, 1 \}^n } \one^T \hat \beta(t) & = \arg \min_{t \in \{ 0, 1 \}^n} \one^T (I - yA^T)^{-1} (\one - t) = \arg \min_{t \in \{0, 1\}^n} (\one - t)^T ((I - yA^T)^{-1})^{T} \one \\
& = \arg \min_{t \in \{ 0, 1 \}^n} (\one - t)^T (I - yA)^{-1} \one = \arg \min_{t \in \{ 0, 1 \}^n } \gamma_{\mathsf{Katz}}^T(\cG^R,y) (\one - t)
\end{align*}

where, we remind that $\gamma_{\mathsf{Katz}}(\cG^R,y) = (I - yA)^{-1} \one$ is the vector of Katz centralities for the reverse graph $\cG^R$. The third equality is true since $(I - yA^T)^{-T} = \left ( \sum_{k \ge 0} (yA^T)^k \right )^T = \sum_{k \ge 0} (yA)^k \overset {y < \frac 1 {\Delta_R}} {=} (I - yA)^{-1}$. It is easy to observe that, by the rearrangement inequality, the optimal solution would be to intervene to the top-$T$ nodes in terms of Katz centrality in $\cG^R$.

§ UPPER AND LOWER BOUNDS ON EVS FOR THE M-ARY TREE

Let $q_d$ be the probability that a product in tier $d$ can be produced. Then

\begin{equation}
q_{d} = \begin{cases} \left ( 1 - x^n \right )^{\frac{m^{D - d + 1} - 1} {m - 1}}, & m \ge 2 \\
\left ( 1 - x^n \right )^{D - d + 1}, & m = 1
\end{cases}
\end{equation}

Let$ q_d = \Pr [\text{a product in tier $d$ can be produced}] = \Pr [\exists \text{a functional supplier at tier $d$}]$. To calculate $q_d$, note that all inputs for a product node at tier $d$ succeed with probability $q_{d + 1}^m$ and then the probability that at least one supplier is functionally conditioned on the all the inputs working is $1 - x^n$. That yields the following recurrence relation $q_d = q_{d + 1}^m (1 - x^n)$ with $q_{D + 1} = 1$. Solving this recurrence relation, we get that for $d \in [D]$, $q_{D - d} = \left ( 1 - x^n \right )^{\sum_{l = 0}^d m^l}$. This yields to  $q_{d} = \begin{cases} \left ( 1 - x^n \right )^{\frac {m^{D - d + 1} - 1} {m - 1}}, & m \ge 2 \\
\left ( 1 - x^n \right )^{D - d + 1}, & m = 1

\end{cases}

§.§ Proof of prop:daglbsparse

Let $\cG$ be a DAG, and let $\beta^*$ be an optimal solution to <ref>, and let the link operation probability $y$ be a value to be determined later. Assume that $v_1, \dots, v_K$ is a topological ordering of the DAG from raw materials to more complex materials. We have that for the $i+1$-th vertex: $\beta_i^* = y \sum_{j \in \cN(v_{i})} \beta_j^* + x^n \le y \sum_{j \le i} \beta_j^*$.

It is easy to see (by induction) that the solution to the above recurrence is: $\beta_i^*  \le (1 + y)^{i - 1} x^n$

which is tight when all the possible $\binom K 2$ edges in the DAG exist. Therefore, $\ev {} {F} \le \sum_{i \in \cK} \beta_i^* \le x^n \sum_{i = 1}^K (1 + y)^{i - 1} \le \frac {x^n (1 + y)^K} {y} \le \frac {K x^n e^{K y}} {Ky}$. This yields the first part of the proof. For the second part we want to minimize the RHS so that we maximize the lower bound on the resilience. Hence, we choose $z = K y > 0$ so that $\frac {e^z} {z}$ is minimized, which happens at $z = 1$, and equivalently $y = \frac 1 K$, at which case we have that $\ev {} {F} \le K x^n \min_{z > 0} \frac {e^z} {z} = K x^n e$. By Markov's inequality we can get, similarly to <ref>, that $R_{\cG}(\varepsilon) \ge \left ( \frac {\varepsilon} {e K} \right )^{1/n}$.

Under the tree structure and $m \ge 2$ the expected size obeys $\ev {} {S} \le \frac {K x^n (D - 1)} {2} = U(x)$.

From <ref> we have that $q_d =\left ( 1 - x^n \right )^{\frac {m^{D - d + 1} - 1} {m - 1}}$. Using the inequality $(1 - t)^a \le \frac 1 {1 + ta}$ for $a > 0$ and $t \in (0, 1)$ we get that

\begin{align*}
\ev {} {S} & = \sum_{d = 1}^D m^{d - 1} q_d \le \sum_{d = 1}^D m^{d - 1} \frac {1} {1 + x^n \frac {m^{D - d + 1}} {2(m - 1)}} \asymp \int_{t = 1}^D \frac {m^{t - 1}} {1 + x^n \frac {m^{D - t + 1}} {2(m - 1)}}.
\end{align*}

By letting $u = \frac {x^n m^{D - t + 1}} {2 (m - 1)}$ we get that the above integral equals

\begin{align*}
& \frac {m^D x^n} {2 \log m (m - 1)} \log \left ( \frac {(1 - x^n)(1 - p)^{\varepsilon K} (1 + (1 - x^n)(1 - p)^{K})} {(1 - x^n)(1 - p)^{K} (1 + (1 - x^n)(1 - p)^{\varepsilon K})} \right ) \bigg |_{u_2 = x^n m^D / 2(m-1)}^{u_1 = x^n m / 2(m-1)} \\ & \le \frac {m^D x^n} {2 \log m (m - 1)} \log \left ( m^{D - 1} \right ) \le \frac {K x^n (D - 1)} {2} = U(x).
\end{align*}

Under the tree structure and $m \ge 2$ the expected cascade size obeys $\ev {} {S} \ge K (1 - x^n(D - 1)) = L(x)$ where $K = m^D - 1$ is the number of products.

From <ref> we have that $q_d =\left ( 1 - x^n \right )^{\frac{m^{D - d + 1} - 1} {m - 1}}$. By Bernoulli's inequality $q_d \ge 1 - x^n \left (\frac {m^{D - d + 1} - 1} {m - 1} \right)$.

Since every level has $m^{d - 1}$ nodes we have that

\begin{align*}
\ev {} {S} & = \sum_{d = 1}^D m^{d - 1} q_d \ge \sum_{d = 1}^D m^{d - 1} \left [ 1 - x^n \left (\frac {m^{D - d + 1} - 1} {m - 1} \right) \right ] = K (1 + x^n) - x^n \frac {1} {m - 1} \sum_{d = 1}^D m^{d - 1} m^{D - d + 1} \\ &  = K (1 + x^n) - x^n \frac {1} {m - 1} \sum_{d = 1}^D m^D = K (1 + x^n) - x^n D (K - 1) \ge K (1 - x^n (D - 1)).
\end{align*}

§ RANDOM WIDTH-W TRELLIS ARCHITECTURES

[transform shape, scale=0.7]
[x=0, y=0]u1
[x=0, y=1,color=pink]u2

[x=1.5, y=0]v1
[x=1.5, y=1, color=pink]v2

[x=3, y=0, color=pink]w1
[x=3, y=1]w2

Instance of random width-$w$ trellis with $w = 2, D =3, K = 6$. Failures are drawn in pink.

We analyze the resilience of the random width-$w$ trellis depicted in <ref>. The width-$w$ trellis, $\mathsf{rt}(K, p, w)$ is a graph with $K = w D$ nodes, that consists of $D$ tiers of products $\cK_1, \dots, \cK_D$ and each tier has size/width $|\cK_d|= w$ for all $d \in [D]$. The edges are generated randomly and independently with probability $p$ if $i \in \cK_d, j \in \cK_{d + 1}, d \in [D - 1]$ and do not exist otherwise. We prove the following,

Let $\cG \sim \mathsf{rt}(K, p, w)$ be a random width-$w$ trellis with $K = wD$ nodes. Then the resiliency of $\cG$ satisfies,

\begin{align}
R_{\cG}(\varepsilon) \ge \begin{cases} \Omega \left ( \left ( \frac {\varepsilon w^2} {K^2} \right )^{1/n} \right ) = \Omega \left ( \left ( \frac {\varepsilon } {D^2} \right )^{1/n} \right ), & \text{for } pw \le 1 \\ \left ( \frac {\varepsilon w (pw - 1)} {K (pw)^{K / w}} \right )^{1/n} = \left ( \frac {\varepsilon  (pw - 1)} {D (pw)^{K / w}} \right )^{1/n}, & \text{for } pw > 1 \end{cases}, \quad
R_{\cG} (\varepsilon) \le \begin{cases}
\left ( \frac {(1 + \varepsilon)w (1  -pw)} {2} \right )^{1/n}, & \text{for } pw < 1 \\
O \left ( \left ( \frac {w^2} {K}  \right )^{1/n} \right ) = O \left ( \left ( \frac {w} {D}  \right )^{1/n} \right ), & \text{for } pw = 1 \\
O \left ( \left ( \frac {w} {K} \right )^{1/n} \right ) = O \left ( \left ( \frac {1} {D} \right )^{1/n} \right ), & \text{for } pw > 1
\end{cases}
\end{align}

* If $pw \le 1$, $K, w \to \infty$, and $K / w \to D < \infty$, then $\cG$ is resilient.

* If $pw \ge 1$, $K, D \to \infty$, and $K / D \to w < \infty$, then $\cG$ is fragile.

Lower Bound. To determine a lower bound in $\cR_{\cG}(\varepsilon)$ we let $\gamma_d$ be the probability that a node at tier $d \in [D]$ fails (due to symmetry the probability is the same for each product of a given tier). The expected number of failures is $\ev {} {F} = D \sum_{d \in [D]} \gamma_d$. By applying the union bound – similarly to <ref> – we get that $\gamma_{d + 1} \le p w \gamma_d + x^n$. Solving the recurrence, yields  $\gamma_d \le x^n \left ( (pw)^{d - 1} + \frac {1} {1 - pw} \right )$ for $pw \neq 1$, and $\gamma_d \le d x^n$ for $pw = 1$. Summing up everything,

\begin{align*}
\ev {} {F} = \frac {K} {w} \sum_{d \in [D]} \gamma_d \le \begin{cases} \frac {K x^n} {w} \frac {(pw)^{K / w}} {pw - 1}, & \text{for } pw > 1  \\ \frac {K^2 x^n} {w^2} \frac {1} {1 - pw}, & \text{for } pw < 1 \\ \frac {K^2 x^2} {w^2}, & \text{for } pw = 1 \end{cases} \le \begin{cases} \frac {K x^n} {w} \frac {(pw)^{K / w}} {pw - 1}, & \text{for } pw > 1  \\ C \frac {K^2 x^n} {w^2} , & \text{for } pw \le 1 \end{cases}
\end{align*}

Therefore, letting $\ev {} {F} = \varepsilon$ yields the following lower bounds for the resilience: $\underline R_{\cG} (\varepsilon) = \begin{cases} \Omega \left ( \left ( \frac {\varepsilon w^2} {K^2} \right )^{1/n} \right ), & \text{for } pw \le 1 \\ \left ( \frac {\varepsilon w (pw - 1)} {K (pw)^{K / w}} \right )^{1/n}, & \text{for } pw > 1 \end{cases}$. If $w$ is constant, as $K \to \infty$ we have that $\lim_{K \to \infty} \underline R_{\cG}(\varepsilon) = 0$ for all $p \in (0, 1]$, and $w \in \mathbb N$. Now, if $w, K \to \infty$ but $K / w \to D$ where $D$ is finite, then $\cG$ is resilient  with resilience $\Omega \left ( \left ( \frac {\varepsilon} {D^2} \right )^{1/n} \right )$ for $pw \le 1$. If $pw > 1$, then it is easy to prove that the lower bound goes to 0.

Upper Bound. To construct an upper bound, we need a lower bound on $\ev {} {F}$. Let $f_{d}$ be the expected number of failures conditioned on any node in tier $d$ failing solely (and spontaneously). As base case we have that $f_D = 1$. We have that a cascade starting from any node at tier $d$ causes at least $a_{d, k}$ failures on expectation at an offset $k$ from tier $d$. We have that for all $d$, $a_{d, 0} = 1$, and that $a_{d, k + 1} \ge \min \{ w, pw a_{d, k}  \}$ since the failures are amplified by a factor of $pw$ at each level, up to being $w$, since the width of the trellis is $w$. Subsequently $f_d \ge \sum_{k = 0}^{D - d} a_{d, k}$.

For $pw = 1$ we have that $a_{d, k + 1} \ge \min \{ w, a_{d, k} \}$, and since $a_{d, 0} = 1$ we have that $a_{d, k} = 1$ for all $k$.

For $pw < 1$  we have that at every new offset the number of failures is decreasing by a factor of $pw$. Therefore, we deduce that $a_{d, k + 1} \ge (pw) a_{d, k}$, and since $a_{d, 0} = 1$ we have that $a_{d, k} \ge (pw)^{k}$.

For $pw > 1$, for $k \le \frac {w} {\log (pw)}$ we have that at least $(pw)^k \ge k \log (pw)$ nodes are influenced and are at most $w$. For $k > \frac {w} {\log (pw)}$, $w$ nodes are infected at every step on expectation.

Therefore, we get that $f_d \ge  \sum_{k = 0}^{D - d} a_{d, k} \ge \begin{cases}
\frac {1 - (pw)^{D - d + 1}} {1 - pw}, & \text{for } pw < 1 \\
D - d + 1, & \text{for } pw = 1 \\
\sum_{k = 0}^{w / \log (pw)}
\log (pw) k + w \left ( D - d - \frac {w} {\log (pw)} \right ). & \text{for } pw > 1
\end{cases}.$

Therefore the lower bound on $\ev {} {F}$ is $\ev {} {F} \ge x^n \sum_{d = 1}^D f_d$.  We have the following three cases

* If $pw = 1$ then by a change of summation indices $\ev {} {F} \ge x^n \sum_{k = 1}^D k = x^n \frac {D (D + 1)} {2} \ge \frac {x^n D^2} {2} = \frac {x^n K^2} {2w^2}$.

* If $pw < 1$ then, by change of summation indices we have that $\ev {} {F} \ge \frac {x^n} {1 - pw} \sum_{k = 1}^D(D - k + 1) (pw)^{k - 1} \ge \frac {x^n D} {1 - pw}$.

* If $pw > 1$, we have that for $w \ge 2$,

\begin{align*}
\ev {} {F} & \ge x^n \left \{ \log (pw) \sum_{k = 0}^{w / \log(pw)} k + w \left (D - d - \frac {w} {\log (pw)} \right ) \right \} \ge x^n \sum_{d = 1}^D \left \{ \frac {w^2 / 2 - w} {\log (pw)} + w(D - d) \right \} \\
& \ge x^n w \sum_{d = 1}^D (D - d) \ge x^n w \frac {D^2} {2}  = \frac {x^n K^2} {2 w}
\end{align*}

We want to make each of the above lower bounds equal to $\frac {1 + \varepsilon} {2} K$, since that would give an $\frac {1 - \varepsilon}{2} K$ upper bound on $\ev {} {S}$, and due to <ref> we get the upper bound
$\overline R_{\cG}(\varepsilon)$: $\overline R_G(\varepsilon) = \begin{cases}
\left ( \frac {(1 + \varepsilon)w (1  -pw)} {2} \right )^{1/n}, & \text{for } pw < 1 \\
O \left ( \left ( \frac {w^2} {K}  \right )^{1/n} \right ), & \text{for } pw = 1 \\
O \left ( \left ( \frac {w} {K} \right )^{1/n} \right ), & \text{for } pw > 1
\end{cases}$. Therefore, if $pw \ge 1$, $K, D \to \infty$ and $K / D \to w$ with $w$ finite, then $\cG$ is fragile.

§ SUPPLIER HETEROGENEITY & OPTIMAL INTERVENTIONS

Here we give a proof of the derivation of the optimal intervention mechanism through supplier heterogeneity presented in <ref>. Namely, we increase the number of suppliers of product $i$, from $n$ to $n + \nu_i$ for some $0 \le \nu_i \le \overline \nu_i$, subject to a budget $\sum_{i \in \cK} \nu_i \le N$, to decrease its self-percolation probability from $x^n$ to $x^{n + \nu_i}$. Under the Assumptions of <ref>, the optimization problem is:

\begin{align*}
\hat \nu = \arg \min_{\zero \le \nu \le \overline \nu, \one^T \nu \le N} x^n \gamma_{\mathsf{Katz}}^T (\cG^R,y) x^{\odot \nu} = \arg \min_{\zero \le \nu \le \overline \nu, \one^T \nu \le N} \gamma_{\mathsf{Katz}}^T (\cG^R,y) x^{\odot \nu},
\end{align*}
where $x^{\odot \nu}$ is the Hadamard power (elementwise power) of $x$ raised to the vector $\nu$, i.e.  $x^{\odot \nu} = (x^{\nu_1}, \dots, x^{\nu_K})^T$. Since $x^\nu$ is a decreasing function of $\nu$, the optimal policy is to order the nodes in decreasing order $\pi$ in terms of their Katz centrality in $\cG^R$ and then try to put as many suppliers as possible in $\pi(1), \pi(2), \dots, $ while respecting the budget constraint $N$ at the same time. This yields the following optimal policy  $\hat \nu_{\pi(i)} = \left ( \overline \nu_{\pi(i)} \wedge \left ( N - \sum_{j < i} \hat \nu_{\pi(j)} \right ) \right )^+.$ where $(a)^+ = \max \{a, 0 \}$.

§ EXTENDED RELATED WORK

Supply Chain Contagion. There have been multiple works on production networks in macroeconomics and how shocks in production networks propagate through the production network's input-output relations, see a comprehensive survey by <cit.>. One of the earliest works dates back to <cit.> introduces a multi-sector model that extends the earlier Long-Plosser model <cit.> and argues that sector-specific shocks are, in fact, affected by the graph topology between producing sectors. Such arguments contrast the prior arguments of <cit.>, which argue that small microeconomic shocks would significantly affect the economy.
In the 2008 financial crisis, the ex-CEO of Ford, Alan Mulally, argued that the collapse of GM or Chrysler would significantly impact Ford's production capabilities for non-trivial amounts of time.
The works of <cit.> build on the above observation and shows that even small shocks lead to cascades that can have devastating effects on the economy. Specifically, <cit.> argues that firm-level idiosyncratic shocks can translate into large fluctuations in the production network when the firm sizes are heavy-tailed, and, in the sequel, <cit.> replaced <cit.>'s analysis on the firm size with the intersectoral network, building on the Long-Plosser model. <cit.> introduces a supply-chain model in which when a firm cannot satisfy its orders, it rations its production to the firms that depend on it via the Input-Output matrix. The model of <cit.> has been used to study supply chain effects of the COVID-19 pandemic <cit.>. Finally, the recent work of <cit.> studies the propagation of shocks in a network (see <ref>).
Our work is related to the above works and attempts to study the propagation of individual shocks at the supplier level to the (aggregate) production network through the definition of a resilience metric.

Seeding Problems in Supply Chains. In a different flavor from the literature that we discussed above, the work of <cit.> study how to find the least costly set of firms to target as early adopters of a tractability technology in a supply chain network, where hyperedges model individual supply chains.

The connections to our paper involve the spread of information in a networked  environment through interventions. However, in this paper, we do not focus on building algorithms for interventions; rather, we provide metrics to assess the vulnerability of nodes in a supply chain network, which can be informative for interventions.

Node Percolation.  <cit.> study node percolation processes, wherein graph nodes fail independently with probability $p$. Their goal is to find the graphs – among graphs with a fixed number of nodes and edges – such that the probability that the induced subgraph (after percolation) is connected is maximized.

Resilience and Risk Contagion in Financial Networks. In a different context, similar models have been used to study financial networks, and optimal allocations in the presence of shocks <cit.> and financial network formation and risk, see, e.g., <cit.>. Our work is related to the above works and attempts to study the propagation of individual shocks at the supplier level to the (aggregate) production network through the definition of a resilience metric.  <cit.> study financial networks and state that contagion in financial networks has a phase transition: for small enough shocks, a densely connected financial network is more stable, and, on the contrary, for large enough shocks, a densely connected system is a more fragile financial system. In a similar spirit, <cit.> and <cit.> characterize the size of defaults under financial contagion in random graphs. They show that firms that contribute the most to network instability are highly connected and have highly contagious links. Moreover, it has also been shown – see, e.g., <cit.>, and the references therein – that risky nodes in financial networks are connected to centrality measures.

Cascading Failures and Emergence of Power Laws. There has been a large body of literature on cascading failures in networks and how the cascade distributions behave as power laws in social networks <cit.>, and power grids, see e.g., <cit.>). We bring the perspective of supply chains and production networks to this literature and offer new insights on how complexity of products and their interdependence affect production network resilience.

§ EXPERIMENTS ADDENDUM: WORLD I-O TABLES

Country      Size ($K$)     Avg. Degree     Density     Min/Max In-degree     Min/Max Out-degree      AUC

USA     55     54.00     1.000     54     54     0.052

Japan     56     45.33     0.824     0 – 50     0 – 50      0.058

G. Britain     56     52.05     0.946     0 – 54     0 – 54      0.052

China     56     37.89     0.688     0 – 46     0 – 46      0.078

Indonesia     56     35.75     0.65     0 – 46     0 – 46      0.078

India     56     29.57     0.537     0 – 41     0 – 43     0.095

Network Statistics and AUC for the world economies. The edge density is computed as $\frac {|\cE(\cG)|} {K^2 - K}$.

[Estimating $R_{\cG}(\varepsilon)$]
[Optimal Interventions]
World Economy Input-Output Networks. We set the number of suppliers for each product to $n = 1$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

University of Chicago Press Chicago, IL

journaltitleNBER Macroeconomics Annual
titleNetworks and the macroeconomy: An empirical exploration

Wiley Online Library

titleThe network origins of aggregate fluctuations

journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

journaltitleJournal of operations Management
titleThe organizational antecedents of a firm’s supply chain agility for risk mitigation and response

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

Cambridge University Press

journaltitleProbability in the Engineering and Informational Sciences
titleA loading-dependent model of probabilistic cascading failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

Taylor & Francis

journaltitleIISE Transactions
titleA structured overview of insights and opportunities for enhancing supply chain resilience

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleOperations Research
titleDisruption risk mitigation in supply chains: The risk exposure index revisited

journaltitleJournal of Cleaner Production
titleThe Circular Economy–A new sustainability paradigm?

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

titleSupply chain disruptions: Theory and practice of managing risk

Wiley Online Library

journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

titleA new status index derived from sociometric analysis

Wiley Online Library

journaltitleProduction and Operations Management
titleManaging disruption risks in supply chains

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

journaltitleApplied network science
titleNetwork science approach to modelling the topology and robustness of supply chain networks: a review and perspective

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

journaltitleHarvard Business Review
titleFrom superstorms to factory fires: Managing unpredictable supply chain disruptions

Wiley Online Library

journaltitleProduction and Operations Management
titleIncreasing supply chain robustness through process flexibility and inventory

titleIdentifying risks and mitigating disruptions in the automotive supply chain

Wiley Online Library

journaltitleReview of International Economics
titleAn illustrated user guide to the world input–output database: the case of global automotive production

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleManufacturing & Service Operations Management
titleData set—real-world multiechelon supply chains used for inventory optimization

Wiley Online Library

titleThe network origins of aggregate fluctuations

journaltitleAmerican Economic Review
titleSystemic risk and stability in financial networks

journaltitleOperations Research Letters
titleOptimal intervention under stress scenarios: A case of the Korean financial system

journaltitlepreprint arXiv:1907.09943
titleYield uncertainty and strategic formation of supply chain networks

World Scientific

journaltitleInternational Journal of Theoretical and applied finance
titleStress testing the resilience of financial networks

journaltitleJournal of Banking & Finance
titleContagious synchronization and endogenous network formation in financial networks

journaltitleTinbergen Institute Discussion Paper
titleThe formation of financial networks

journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

Nature Publishing Group

journaltitleNature Scientific Reports
titleDebtrank: Too central to fail? financial networks, the fed and systemic risk

journaltitleJournal of economic dynamics and control
titleLiaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

journaltitleManagement Science
titleContagion in financial networks: a threat index

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

journaltitleAvailable at SSRN 3034406
titleNetwork hazard and bailouts

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

booktitleInternational Conference on Current Trends in Theory and Practice of Computer Science
titlePayment Scheduling in the Interval Debt Model

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Wiley Online Library

titleOn reliability of graphs with node failures

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

Wiley Online Library

journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

Public Library of Science San Francisco, CA USA

journaltitlePloS one
titleThe propagation of economic impacts through supply chains: The case of a mega-city lockdown to prevent the spread of COVID-19

journaltitleDistorted Investment Incentives, Regulation, and Systemic Risk Measurement (March 1, 2019)
titleWhat makes financial networks special? distorted investment incentives, regulation, and systemic risk measurement

journaltitlepreprint arXiv:2212.06808
titleIncentive-Aware Models of Dynamic Financial Networks

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

booktitleProceedings of the Web Conference 2022
titleAllocating Stimulus Checks in Times of Crisis

booktitleInternational Conference on Web and Internet Economics
titleDefault ambiguity: finding the best solution to the clearing problem

journaltitlepreprint arXiv:2005.10585
titleProduction networks and epidemic spreading: How to restart the UK economy?

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

UT Austin

journaltitleM375T/M396C: Topics in Complex Networks Course
titleGalton-Watson Branching Process

journaltitleGames and Economic Behavior
titleFree and perfectly safe but only partially effective vaccines can harm everyone

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleDiscrete mathematics
titleUniformly optimal graphs in some classes of graphs with node failures

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

University of Chicago Press Chicago, IL

journaltitleNBER Macroeconomics Annual
titleNetworks and the macroeconomy: An empirical exploration

Wiley Online Library

titleThe network origins of aggregate fluctuations

journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

journaltitleJournal of operations Management
titleThe organizational antecedents of a firm’s supply chain agility for risk mitigation and response

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

Cambridge University Press

journaltitleProbability in the Engineering and Informational Sciences
titleA loading-dependent model of probabilistic cascading failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

Taylor & Francis

journaltitleIISE Transactions
titleA structured overview of insights and opportunities for enhancing supply chain resilience

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleOperations Research
titleDisruption risk mitigation in supply chains: The risk exposure index revisited

journaltitleJournal of Cleaner Production
titleThe Circular Economy–A new sustainability paradigm?

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

titleSupply chain disruptions: Theory and practice of managing risk

Wiley Online Library

journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

titleA new status index derived from sociometric analysis

Wiley Online Library

journaltitleProduction and Operations Management
titleManaging disruption risks in supply chains

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

journaltitleApplied network science
titleNetwork science approach to modelling the topology and robustness of supply chain networks: a review and perspective

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

journaltitleHarvard Business Review
titleFrom superstorms to factory fires: Managing unpredictable supply chain disruptions

Wiley Online Library

journaltitleProduction and Operations Management
titleIncreasing supply chain robustness through process flexibility and inventory

titleIdentifying risks and mitigating disruptions in the automotive supply chain

Wiley Online Library

journaltitleReview of International Economics
titleAn illustrated user guide to the world input–output database: the case of global automotive production

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleManufacturing & Service Operations Management
titleData set—real-world multiechelon supply chains used for inventory optimization

Wiley Online Library

titleThe network origins of aggregate fluctuations

journaltitleAmerican Economic Review
titleSystemic risk and stability in financial networks

journaltitleOperations Research Letters
titleOptimal intervention under stress scenarios: A case of the Korean financial system

journaltitlepreprint arXiv:1907.09943
titleYield uncertainty and strategic formation of supply chain networks

World Scientific

journaltitleInternational Journal of Theoretical and applied finance
titleStress testing the resilience of financial networks

journaltitleJournal of Banking & Finance
titleContagious synchronization and endogenous network formation in financial networks

journaltitleTinbergen Institute Discussion Paper
titleThe formation of financial networks

journaltitleSIAM Journal on Financial Mathematics
titleRisk-dependent centrality in economic and financial networks

Nature Publishing Group

journaltitleNature Scientific Reports
titleDebtrank: Too central to fail? financial networks, the fed and systemic risk

journaltitleJournal of economic dynamics and control
titleLiaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk

journaltitleManagement Science
titleSupply disruptions and optimal network structures

journaltitlepreprint arXiv:2104.14818
titleTraceability Technology Adoption in Supply Chain Networks

ACM New York, NY, USA

journaltitleACM Transactions on Economics and Computation (TEAC)
titleNetwork formation in the presence of contagious risk

Annual Reviews

journaltitleAnnual Review of Economics
titleProduction networks: A primer

journaltitleManagement Science
titleContagion in financial networks: a threat index

booktitle37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the
titleA branching process approximation to cascading load-dependent system failure

journaltitleManagement Science
titleSystemic risk in financial systems

journaltitleAmerican Economic Review
titleSupply network formation and fragility

journaltitleAvailable at SSRN 3034406
titleNetwork hazard and bailouts

journaltitleEuropean Economic Review
titleNetwork formation and systemic risk

booktitleInternational Conference on Current Trends in Theory and Practice of Computer Science
titlePayment Scheduling in the Interval Debt Model

Wiley Online Library

titleThe granular origins of aggregate fluctuations

journaltitleJournal of Banking & Finance
titleHow likely is contagion in financial networks?

Wiley Online Library

titleOn reliability of graphs with node failures

Nature Publishing Group

journaltitleNature Human Behaviour
titleGlobal supply-chain effects of COVID-19 control measures

Wiley Online Library

journaltitleRisk Analysis: An International Journal
titleAn adaptive regional input-output model and its application to the assessment of the economic cost of Katrina

journaltitleReview of Economic Dynamics
titleCyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

Public Library of Science San Francisco, CA USA

journaltitlePloS one
titleThe propagation of economic impacts through supply chains: The case of a mega-city lockdown to prevent the spread of COVID-19

journaltitleDistorted Investment Incentives, Regulation, and Systemic Risk Measurement (March 1, 2019)
titleWhat makes financial networks special? distorted investment incentives, regulation, and systemic risk measurement

journaltitlepreprint arXiv:2212.06808
titleIncentive-Aware Models of Dynamic Financial Networks

booktitleProceedings of the 2007 SIAM international conference on data mining
titlePatterns of cascading behavior in large blog graphs

The University of Chicago Press

journaltitleJournal of political Economy
titleReal business cycles

Macmillan Basingstoke

journaltitleEssential readings in economics
titleUnderstanding business cycles

journaltitlePhysical Review Letters
titleEmergence of scale-free blackout sizes in power grids

journaltitlepreprint arXiv:2205.13394
titleDynamic Interventions for Networked Contagions

booktitleProceedings of the Web Conference 2022
titleAllocating Stimulus Checks in Times of Crisis

booktitleInternational Conference on Web and Internet Economics
titleDefault ambiguity: finding the best solution to the clearing problem

journaltitlepreprint arXiv:2005.10585
titleProduction networks and epidemic spreading: How to restart the UK economy?

booktitleInternational Conference on Complex Networks and their Applications
titleClearing algorithms and network centrality

UT Austin

journaltitleM375T/M396C: Topics in Complex Networks Course
titleGalton-Watson Branching Process

journaltitleGames and Economic Behavior
titleFree and perfectly safe but only partially effective vaccines can harm everyone

journaltitleEconomics of Disasters and Climate Change
titleThe Impacts of the Coronavirus on the Economy of the United States

booktitleProceedings of the 26th International Conference on World Wide Web
titleWhy do cascade sizes follow a power-law?

journaltitleDiscrete mathematics
titleUniformly optimal graphs in some classes of graphs with node failures