Optimal Intervention in Economic Networks using Influence Maximization Methods
OOptimal Intervention in Economic Networks usingInfluence Maximization Methods *Ariah Klages-Mundt † Andreea Minca ‡ February 4, 2021
Abstract
We consider optimal intervention in the Elliott-Golub-Jackson network model [17] andshow that it can be transformed into an influence maximization problem, interpreted asthe reverse of a default cascade. Our analysis of the optimal intervention problem ex-tends well-established targeting results to the economic network setting, which requiresadditional theoretical steps. We prove several results about optimal intervention: it isNP-hard and additionally hard to approximate to a constant factor in polynomial time.In turn, we show that randomizing failure thresholds leads to a version of the problemwhich is monotone submodular, for which existing powerful approximations in polyno-mial time can be applied. In addition to optimal intervention, we also show practicalconsequences of our analysis to other economic network problems: (1) it is computa-tionally hard to calculate expected values in the economic network, and (2) influencemaximization algorithms can enable efficient importance sampling and stress testing oflarge failure scenarios. We illustrate our results on a network of firms connected throughinput-output linkages inferred from the World Input Output Database.
Following the global crisis due to the Covid-19 medical and economic contagion, govern-ments have unleashed unprecedented macroeconomic stimulus. The variety of proposedstimulus, both in government financing and in monetary policy form, aims to support valuein a shocked global economy. The tools to support value following a systemic shock arethere since the financial crisis, and new ones are being proposed. One difference to the fi-nancial crisis is that the shock originated then from within the financial system and the mainintervention target were systemically important institutions, i.e. those whose failure wouldlead to a large impact on the economy. In this crisis the shock was external and createddisruptions to many economic sectors worldwide. Consequently, intervention is much wider.One lesson learned from the financial crisis is that network effects underpin systemicimportance, which can be measured based on the size of loss cascades, see e.g, [2, 14] or * We thank Sid Banerjee for helpful discussion. This paper is based on work supported by NSF CAREERaward † Cornell University, Center for Applied Mathematics. ‡ Cornell University, Operations Research & Information Engineering. a r X i v : . [ c s . G T ] F e b entrality measures, see [7] and the references therein. Work on systemic risk measures,e.g.,[11, 9, 18, 5], led to different axiomatic frameworks for capital requirements such thataggregate risk is acceptable. Notably, aggregation functions underlying these systemic riskmeasures can account for interconnections. In [3, 4, 10], authors explore optimal capitaland liquidity intervention, and derive insights into the intervention target in stylized core-periphery banking networks subject to the risk of bank runs. Their methods are applied forsmall banking systems. In [1], authors cast the intervention problem in the context of theEisenberg-Noe model [16] as a mixed integer-programming problem, and propose a notionof of (cid:15) -optimality to solve it approximately. They apply their methods to the Korean bankingsystem. In contrast to these past works, our paper focuses on the computational aspect ofoptimal intervention problems, which becomes critical when the number of eligible firms islarge. When entire sectors are hit by shocks rather than a few large institutions, one needsto understand the systemic impact of groups of firms and optimally decide on where tointervene. Such problem quickly becomes computationally hard. The government’s criterionis to maximize the overall value in the system under a budget constraint.Our model abstracts away from the details of interventions, and relies on the notion ofvalue of an organization –firm, sector, country– introduced in [17] in the context of cross-holdings. Without intervention, if the value of the organization drops below a failure thresh-old, there are failure losses and the values of the connected organizations drop as well andso on. This is also in the spirit of the distress notion in [26], which allows for contagionbefore the point of default. The failure threshold is interpreted as the value below which theorganization ceases operations. Intervention can be seen as a way to increase an organi-zation’s value or alternatively lower its failure threshold. Several types of interventions canbe modeled by a decrease in the failure threshold of an organization. Government bailoutscould take the form of equity infusions, as they did in the financial crisis. Central banksare injecting liquidity in the economy via various asset purchase programs, including morerecently corporate debt purchases.It is clear that direct government financing allows firms to survive by directly loweringthe failure threshold. The effect of asset purchase programs(APP) is more subtle. A pointof contention is whether asset purchase programs involve liquidity injection, or whether itinvolves value injection. When central banks can purchase corporate debt they change theoutcome in debt markets. An unavoidable fact of APP is that, whenever the central bankpurchases illiquid assets to intervene in liquidity, it must price those assets in some way.Models are usually used to calculate a ‘fundamental value’. When acting as a lender of lastresort, central banks may essentially accomplish bailout functions.When designing and implementing interventions, it is critical to consider long-term moralhazard effects. Firm default is an important long-term filter that incentivizes strong andcompetent management. The prospect of intervention can disincentivize proper risk man-agement, enabling additional short-term profits to management and equity holders whiletransferring tail risks to government. Note, however, that interventions can be shaped toreduce moral hazard (e.g. by organizing bail-ins by the creditors and thereby diluting equity Arguably, central banks can lower the failure thresholds even without actual liquidity injection: forexample Boeing raised debt in capital markets following the FED’s announcement that they wouldsupport corporate debt markets, see e.g. . This paper.
We construct an economic network intervention model and show how it canbe solved via influence maximization (Section 2). Our analysis extends well-established tar-geting results to the economic network setting, which requires additional theoretical stepsover the classical setting. For instance, the ‘influence matrix’ is more complex ( ∼ the Neu-mann series of the matrix in a linear influence setting that is column-substochastic with zerodiagonals) and the structure of ‘activations’ is more nuanced. We contribute the followingresults, which provide the legwork for adapting powerful targeting algorithms to solve severaleconomic network problems:1. We prove that it is NP-hard to optimize the economic network intervention and addi-tionally hard to approximate to a constant factor in polynomial time (Theorem 1 andCorollary 1).2. We prove that, when modified to consider expected values under random thresholds,the intervention problem is monotone submodular (Theorem 2) and thus admits agreedy polynomial time (1 − /e − (cid:15) ) -approximation (Corollary 2).3. We show that similar results extend to a related problem: identifying large failure cas-cade scenarios. We prove that it is NP-hard to find the worst case failure scenariosgiven a maximum sized aggregate shock to asset values (Theorem 3). Under random-ized thresholds, a similar greedy approximation is applicable.4. We show two practical consequences of Theorem 3 in Section 3.3. (1) It is compu-tationally hard to calculate expected values in the economic network. (2) Influencemaximization algorithms can allow us to find approximate worst case failures, whichwe suggest can enable efficient importance sampling and stress testing.5. We demonstrate a proof-of-concept of optimal intervention approximation applied toeconomic networks constructed from the World Input-Output Database (Section 4).3 Model
In this section, we supplement the Elliot-Golub-Jackson network contagion model [17] toincorporate targeted interventions. We then provide an overview and definition of influencemaximization problems and relate the intervention problem in economic networks to an in-fluence maximization problem.
We define an economic network ( C, D, β, θ , p ) based on the Elliot-Golub-Jackson networkcontagion model as follows:• n firm nodes• m assets owned by firms• p = m × vector of asset prices• D = n × m matrix with D ik ≥ the share of asset k held by firm i (adding to 1)• C = n × n matrix with C ij ≥ the fraction of firm j owned by firm i and 0 along thediagonals• ˆ C = n × n diagonal matrix with ˆ C ii = 1 − (cid:80) j C ji the share of organization i not ownedby another firm in the system• θ = n × vector of failure thresholds for each firm• β = n × n diagonal matrix of extra failure costs for each firm.The matrix C describes the linear cross-holding relationships between firms. If a firm i ’smarket value (defined next) falls below its threshold θ i , it incurs an extra failure cost β ii . Weassume C is column sub-stochastic as otherwise ˆ C − is not well-defined. Notice that thisalso means that I − C is invertible because the spectral radius ρ ( C ) < .The network propagates asset values and defaults across firms in the network. Weillustrate this conceptually in Figure 1. D describes the mapping of underlying assets (bluenodes) to firms (orange nodes). C describes cross-holdings between firms. The breach ofa threshold triggers failure costs, which propagate to other firms through C .Firm book values are given by V = C V + D p − β { v < θ } , where S is the 1-0 valued vector indicating the entries of set S . Notice that book values in-flate the value of underlying assets because asset values are counted multiple times acrossfirms (consequently, (cid:107) V (cid:107) ≥ (cid:107) p (cid:107) and can be arbitrarily large). A more useful measure ofvalue is a scaling of book values by ˆ C , accounting for the ownership share that each firmretains in itself. These are market values, which are given by v = ˆ C V = ˆ C ( I − C ) − ( D p − β { v < θ } ) . In [17], they show that the matrix ˆ C ( I − C ) − is column-stochastic.4 andom-valued assets Nodes own assets, parts of other nodes
Node value = linear function of assets, connected nodes
If node value < threshold, nonlinear default cost incurred
Default costs propagate through holdings
Figure 1: Financial network propagation mechanism.
Lattice of solutions.
As defined, there is always a solution for v . The set of solutionsforms a complete lattice via Tarski’s fixed point theorem. Further, supremum and infimumexist (best and worst cases). The analysis in [17] focuses on the best case solution as othersolutions in the lattice are due to self-fulfilling failures. Intervention lowers thresholds.
Beyond the core model from [17], we add a vector ofintervention payments γ ≥ , which affect the default status of firms. Given an interventionprofile γ , firm i now defaults if V i + γ i < [ ˆ C − θ ] i . This leads to post-intervention market values ˜v = ˆ C ( I − C ) − ( D p − β V + γ < ˆ C − θ ) . An intervention via this mechanism effectively lowers the failure threshold of firms. Thisis consistent with real-world intervention mechanisms as discussed in the introduction.
Our analysis builds on influence propagation research in social networks. This work has his-torically studied processes like diffusion of technological innovation, beliefs, product adop-tion, and viral content. A natural question is how to engineer such a viral cascade giveninformation about the network.A model for this problem is specified as follows:• U is the set of nodes in the network.• f ( S ) a set function that outputs the vector of influence exerted by the activation of nodeset S ⊆ U on each node in U (i.e., f u ( S ) = influence exerted on node u ). We assume f ( ∅ ) = 0 .• w ( S ) outputs an importance weighting of node set S . In the simplest setting, eachnode is weighted by 1. 5 ˜ θ is the vector of thresholds for each node. A node u becomes activated if the influenceexerted on it is ≥ ˜ θ u .• b is the budget for influencing nodes. Integral Influence Maximization, studied in [21], focuses on maximizing the weightednumber of activated nodes by finding an optimal seed set S to activate with payments ofsize ˜ θ u for each u ∈ U subject to budget b . An influence cascade is calculated in stages.Given an initial set of activated nodes S , we construct the set of nodes S i (for i ≥ )activated by the set S i − by adding the nodes u such that f u ( S i − ) ≥ ˜ θ u . The cascade process converges to a final set of activated nodes S . The optimization prob-lem is max S ⊆ U w ( S ) s.t. (cid:88) u ∈ S ˜ θ u ≤ b. Fractional Influence Maximization, studied in [13], is a generalization of the integralcase. In this problem, we choose a payment vector x subject to budget b to exert influ-ence on seed nodes. An influence cascade is again calculated in stages. An initial set ofactivated nodes S is composed of nodes u for which x u ≥ ˜ θ u . We construct the subsequentsets of nodes S i (for i ≥ ) activated by the set S i − by adding the nodes u such that f u ( S i − ) + x u ≥ ˜ θ u . Note that this assumes that direct influence is additive with influence from other vertices inthe network, in the sense that node activated in next stage if and only if this condition satis-fied. The cascade process converges to a final set of activated nodes S . The optimizationproblem is max x ≥ w ( S ) s.t. T x ≤ b where is the all-ones vector. The amounts can be a fraction of the thresholds of the nodes.This allows more efficient use of budget b to influence an effective seed set S . In particular,this takes advantage of the fact that we don’t have to spend as much to influence a nodethat already has partial influence exerted from other influenced nodes.For simple influence models, like the Linear Threshold Model and Triggering Set Model,these problems are NP-hard, as shown in [21] and [13]. Further, they are also hard toapproximate within any general nontrivial factor.However, when we consider a modified problem with randomized thresholds–e.g., ifactivation thresholds for influence are uniform random variables–then the problem changesenough in expectation to lower complexity. In particular, the expected cascade size σ ( S ) := [ w ( S ) | S ] from a given seed set S (with similar definition for σ ( x ) ) is monotone submod-ular and allows a greedy approximation that is provably within (1 − /e ) ≈ of optimal([21],[13]). [23] proved this for more general threshold models and distributions for ˜ θ . Inparticular, letting F u be the distribution function of ˜ θ u , σ ( S ) is monotone submodular giventhat the following functions are monotone submodular: f , w , and F u ◦ f u for all u ∈ U .We define these greedy algorithms formally in Appendix B and discuss influence maxi-mization algorithms further with regard to applications in Section 4. We can interpret the intervention problem in the economic network model as the following:given an impending default cascade, how do we find an optimal intervention to limit defaultsto an acceptable level.To do this, suppose that the set of nodes that would default without intervention is T .Now reduce the system to only look at effects on the nodes in T , while preserving the entirenetwork structure. In particular, define the following• I T = diagonal matrix with I uu = 1 for u ∈ T and otherwise.• Ψ( T ) maps to a system on the non-zero diagonal coordinates of I T . Essentially, Ψ( T ) is the | T | × | U | matrix obtained by dropping zero rows of I T .We can apply the above map to transform the system to look at ¯v := Ψ ˆ C ( I − C ) − ( D p − β v < θ ) . This transformation removes firms that don’t fail without intervention, while preserving thenetworked connections through such nodes. The idea is that among the firms who wouldfail without intervention, some of them will be saved by direct intervention. Their value wouldthen go above the failure threshold and in particular the failure costs are reversed. In areverse causal relation of failure, other firms would be indirectly saved because their valuewould also increase. As we demonstrate in this section, the economic network interventionproblem becomes an influence maximization problem on this transformed system. To sim-plify notation, we will proceed where applicable without the bar notation, but assuming we’reworking in the transformed problem.The direct influence function under the influence maximization version of the problem is f ( S ) = ( I − C ) − β S − (cid:88) u ∈ S I u ( I − C ) − β u . (1)This accounts for the effect on book values across the network of reversing default costsin the S nodes (the first term), which pushes other nodes closer to their influence (or fail-ure reversal) thresholds. In the typical influence maximization problem, a node in S doesnot exert influence on itself. This is complicated in the economic network intervention prob-lem because the reversal of a node’s default has an effect on itself through cross-holdings.However, notice that the intervention γ does not need to cover the cost of β because thiscost disappears when the market value would otherwise be evaluated above threshold. Thesecond term in f ( S ) removes this self-influencing effect from the influence function as it isinstead represented in reduced influence thresholds.7 nfluence thresholds. The influence thresholds under the influence maximization versionof the problem are ˜ θ u = (cid:104) ˆ C − θ − ( I − C ) − ( D p − β T \{ u } ) (cid:105) u (2)for initial defaulting set T and node u ∈ T . This represents how much book values wouldneed to change in order for the failure of u to be reversed . This can also be thought ofas the slack below threshold in the economic network. We can obtain this from taking [ ˆ C − θ − ( I − C ) − ( D p − β T )] u , the divergence of book value from the failure threshold(measured in book value), and subtracting the self-influencing effect described above.An intervention γ in the economic network setting is then analogous to a vector x in thefractional influence setting, and can be defined similarly for integral influence. In the previous section, we set up an economic network intervention model that transformsinto an instance of an influence maximization problem. We now prove theoretical propertiesof the intervention model. We prove that it is NP-hard to optimize the economic networkintervention and additionally hard to approximate to a constant factor in polynomial time.Additionally, we prove that the influence maximization instance of the problem, when mod-ified to consider expected values under random thresholds, is monotone submodular, andthus drawing from results in [23] and [13], admits a (1 − /e − (cid:15) ) -approximation in polynomialtime. In our first result, we show that the optimal economic network intervention problem is NP-hard. Note that this result is not a consequence of influence maximization hardness resultsin, e.g., [21], [13], [20]. While we can transform the economic network intervention modelinto an instance of influence maximization, that does not mean that the general hardness ofinfluence maximization extends to this case.
Theorem 1.
Let ( C, D, β, θ , p ) be a financial system with n firms and deterministic thresh-olds θ , and let ≤ (cid:96) < α ≤ . Suppose αn firms fail in the financial system equilibrium.Then it is NP-hard to determine whether there exists an intervention γ i ≥ with (cid:107) γ (cid:107) ≤ b such that at most (cid:96)n nodes fail after the intervention. [Link to Proof] The proof is a reduction from independent set. A consequence of this is the followingcorollary describing hardness of approximation.
Corollary 1.
Optimal economic network intervention cannot be approximated to within aconstant factor in polynomial time.
We now establish that a modified form of the optimal intervention problem can be well-approximated in polynomial time. The modification incorporates randomized thresholds andreframes the problem to optimize in expectation . For instance, this can be done by treatingthresholds as random variables uniformly distributed over any given uncertainty range. Thiscan be done more generally with different threshold distributions, as we will discuss. Inessence, the combinatorial complexity problems disappear in expectation. We first show that the intervention problem with random thresholds is monotone sub-modular, connecting with results from [23] and [13]. As a result, a greedy hill-climbing algo-rithm provides a (1 − /e − (cid:15) ) -approximation using results from [12, 24].Our next result establishes that the influence function in the intervention problem ismonotone submodular. Prop. 1.
The function f from Eq. 1 is monotone increasing and submodular. [Link to Proof] We need a few assumptions to prove that the objective σ for intervention problem underrandom thresholds is monotone submodular. The first assumption describes the random-ization of thresholds and is necessary for the results of [23] to apply. It allows very generaldistributions of thresholds, an example of which is uniform distributions. Assumption 1.
For u ∈ U , random thresholds θ u are independent with distribution function F u such that F u ◦ f u is monotone increasing submodular. The next assumption is that the influence function f is normalized–no nodes are initiallyactivated/exert influence (i.e., have defaults reversed) if the seed set is empty. This is aneasy assumption to make with fixed thresholds, as we can simply reshape the problem if Since the range of the random variables can be arbitrarily small, this is like saying that the approximationproblem is difficult only on measure 0 sets. θ random in the economicnetwork setting, this is more complicated because the corresponding influence thresholds ˜ θ in (2) could be 0 or negative depending on the realization of thresholds, and, if this occurs,the resulting ˜ θ distributions are not independent. This can be solved in two ways that keepthe initially defaulting nodes technically fixed: (1) the randomization in thresholds can beassociated with ˜ θ ≥ instead of with θ , or (2) the problem can be reformulated: ˜ θ becomesthe positive part in (2), initial defaults are fixed, f ( ∅ ) := 0 , and when ˜ θ u = 0 , u can be addedto the seed set with cost (and so will be added first). Assumption 2.
The influence function f is normalized, i.e. f ( ∅ ) = 0 . The final assumption concerns the function describing node weighting in the objective.The weight function describes how valuable it is to reverse the defaults of a given set ofnodes.In particular, the cardinality function, which weights each node equally, obeys this as-sumption and is consistent with aiming to minimize the number of defaults. A different fixedweighting of nodes would also obey the assumption.
Assumption 3.
The weight function w : 2 U → R + is normalized, monotone, and submodu-lar. Under these assumptions, the intervention objective function–e.g., the expected numberof defaults under a given intervention–is monotone submodular based on results from [23],as formalized in the next result.
Theorem 2.
Given assumptions 1-3 and an instance of the economic network interventionproblem with random thresholds, the σ ( S ) and σ ( x ) functions of the associated integral andfractional influence maximization problems are normalized, monotone, and submodular. [Link to Proof] Then following the application of results in [21], there is a greedy (1 − /e − / poly ( n )) -approximation algorithm for optimizing the expectation, as formalized in the next corollary. Corollary 2.
Given assumptions 1-3, there exists a polynomial-time greedy (1 − /e − (cid:15) ) -approximation for maximizing σ ( S ) and σ ( x ) . [Link to Proof] We now show how these results translate to related economic network problems. We startby showing that it is additionally NP-hard to identify the worst case failure scenarios given amaximum sized aggregate shock to asset values p . Like the intervention problem, there isa (1 − /e − (cid:15) ) -approximation under random thresholds. Theorem 3.
Suppose ( C, D, β, θ , p ) is an instance of an economic network and assetprices evolve to p such that (cid:107) p (cid:107) − (cid:107) p (cid:107) ≤ b for some maximum aggregate shock b > .Let < (cid:96) < . Then it is NP-hard to determine if a failure cascade of size (cid:96) | U | is possible in ( C, D, β, θ , p ) . Link to Proof]
The reduction from independent set again implies a corollary result that the optimum ishard to approximate up to a constant factor in polynomial time. As in the intervention case,when reframed in terms of expectations under random thresholds, a greedy (1 − /e − (cid:15) ) -approximation again applies.We now develop two practical consequences of these results: (1) it is computationallyhard to calculate expected values in the economic network, and (2) approximation methodscan be applied to identify adverse scenarios for the purpose of stress testing. Hardness of calculating expected values.
We next demonstrate a consequence of The-orem 3: it can be computationally hard to calculate expected values of firms in an economicnetwork even if we have perfect information about the underlying setup. Consider a simplesetting in which the prices of underlying assets p are i.i.d. Bernoulli distributed 0-1 withprobability q . The probability that a specific set of b assets fail is (1 − q ) b , which is non-vanishing in the scale of the network and so non-negligible for the calculation of expectedvalue of firms when the problem is large (and potentially computationally complex). Sinceit is NP-hard to determine whether a large failure cascade can occur with that probability,it is in turn NP-hard to determine if the expected value is above some given level. Further,the ability to approximate will depend on the failure costs β in the network, which could bearbitrarily large in the general case, suggesting that approximation is also difficult in generalunder fixed thresholds.This compares to what is typically done in financial models in practice. Firms are typ-ically treated in isolation, i.e., not part of a network model. In this case, firm defaults aretreated as independent or perhaps correlated through a simple copula. Such distributionsof credit risk, such as produced by a Gaussian copula, fail to capture clustering of defaults.The resulting probability that a given fraction of firms default is exponentially unlikely as thenumber of firms grows, and so this computational problem does not arise in those simplemodels. Naturally, the assumption that firm defaults are independent is flawed, and so thecomplexity problems that we describe in calculating expected values may apply in realisticsettings. Identifying extreme default scenarios.
While it is NP-hard to identify the worst casefailure scenarios given a maximum sized aggregate shock in an economic network, it ispossible to identify scenarios that approximate this up to a (1 − /e − (cid:15) ) factor with randomthresholds. As a result, we can apply influence maximization approximation methods toidentify shocks that produce large default cascades of similar size to the worst scenarios. Acommon task in finance is to stress test a financial system subject to aggregate shocks upto a particular size. It is well-known that the straightforward Monte Carlo approach to this willunderestimate risks because random samples are unlikely to contain many of the extremedefault scenarios (e.g., the relevance of importance sampling). Thus these approximationmethods can allow us to sample more large default scenarios for inclusion in stress tests.11 Application to WIOD dataset
To demonstrate the use of our results, we consider an application of influence maximizationalgorithms to an economic network. We construct instances of the economic network in-tervention problem based on the World Input Output Database (WIOD). The data is openlyavailable at . We simulate a number of possible shocks to theresulting network and demonstrate that influence maximization algorithms can derive effec-tive interventions using relatively modest budgets. As we might expect, we see decreasingreturns to scale in the size of the budget.The simulations we perform are intended as a proof of concept of a realistic-lookingsetup based on real underlying data. We stress that many parts of the setup for which datais not available remain stylized: in particular underlying assets, thresholds, failure costs, anddistribution of shocks to underlying asset values. Additionally, there is naturally uncertaintyabout economic network structure as described by the dataset and aggregation effects fromgrouping entire industries of firms into single nodes.
The WIOD dataset (see, e.g., [25]) describes the flow of resources in dollar value betweendifferent economic sectors within different nations (intermediate demand) and national finaldemand (e.g., GDP components, such as consumption, investment, government expen-diture). The dataset includes this information for 2464 distinct economic sectors spreadbetween 28 EU countries and 15 other major countries for the years 2000-2014.We construct an economic network from the 2014 dataset in the following way:• Nodes = n columns in the dataset that refer to economic sectors or final demandcomponents• n × n array of flows between nodes from dataset, with zero rows for final demandcomponents• Transpose components of any negative entries in the array• Scale columns to sum to 1 (inclusive of value added, a row in the dataset that is notincluded in the array) or 0 if a zero column• Zero the diagonals in the array to get C • Fix unnecessarily bad conditioning in C by removing nodes with near zero value added(columns referring to households)• Vector D p = output of each node at basic prices (this is the TOT GO row in the dataset)• Vector θ = ˆ C ( I − C ) − D p − value added, which gives the market value assuming nodefaults − value added.• Diagonal matrix β with diagonal entries . · value added.12he vector D p above represents initial asset values. We sample shocks to these as-set values by sampling a return vector r such that the shocked asset prices are given bythe component-wise multiplication D p · (1 + r ) . The return vector r is sampled from a m -dimensional normal distribution with the following specifications intended to sample a rangeof large deviations:• Common correlation factor ρ = 0 . ,• Marginal distributions have σ = 0 . and drift a = − . ,• Returns bounded by such that r i = max(1 + r i , .Recall that D p are underlying asset prices, and market values will have additional inter-relation and correlation from the network process. As shown in the previous section, there are greedy algorithms with (1 − /e − (cid:15) ) -approximations.The general structure of these greedy algorithms is to start with an empty seed set S and,iteratively, add the node u to S that gives the maximum marginal gain. Since the thresholdsare random, determining the maximum marginal gain in each step involves estimating theexpected size of resulting cascades σ ( S ∪ { u } ) for a number of nodes u . This is typicallydone through Monte Carlo estimation of the expectation. For large networks, these MonteCarlo approximations become prohibitively slow, although still within polynomial time. In practice, heuristic algorithms are used in influence maximization to try to estimate thegreedy algorithm in faster time. For instance
DiscountFrac used in [13] starts with an emptyseed set S and iteratively adds the node u to S that would exert the most total influenceon the remaining unactivated nodes. In particular, given the initial activation set S at thebeginning of a step, DiscountFrac picks the node u that maximizes (cid:107) f ( { u } ) A \{ u } (cid:107) forremaining uninfluenced set A . We define these influence maximization algorithms formallyin Appendix B.In our simulations, we adapt DiscountFrac to choose the node u that maximizes (cid:107) f ( { u } ) A \{ u } (cid:107) ˜ θ u − f u ( S ) , where S is the currently influenced set. This accounts for the cost to influence node u in thecurrent step, given that economic network thresholds can vary significantly in size. We simulate 5000 shocks and apply the adaptation of
DiscountFrac to approximate theresulting optimal intervention problems. In this setting, we explore the effectiveness of arange of targeted intervention sizes. As an area of future research, it would be interesting to examine whether asymptotic results on the size ofthe cascade ´a la [2] could replace part of the Monte Carlo approximations. .0% 0.2% 0.4% 0.6% 0.8% 1.0%Budget % of Total Assets |Dp|0%10%20%30%40%50%60%70%80% % F i r m s D e f a u l t i n g Histogram % Firms Defaulting vs. Intervention Budget
Figure 2: 2-D histogram of simulation results with contour lines.The results of our simulations are represented in a 2-D histogram in Figure 2 depict-ing the percentage of firms defaulting under a range of intervention budget sizes. In thisplot, density of ( x, y ) -pairs is represented in the third dimension (color: light=high density,dark=low density). Additional contour lines are added to better illustrate the densities. Asperhaps expected, the highest density region is near the x -axis, with density increasing withintervention budget size.Figure 3 depicts slices of the above data that show effects more concretely. In partic-ular, we restrict to comparing the effects of a 1% targeted intervention to no intervention.Figure 3a shows histogram densities of firm defaults under the sampled shocks, illustratingthat the 1% intervention effectively reduces the tails of this distribution. Note that 1-D his-tograms in Figure 3a are the leftmost and rightmost verticals of Figure 2 (no intervention and1% intervention respectively).Figure 3b shows histogram densities of defaults averted under the 1% intervention rel-ative to no intervention, which also illustrates the effectiveness. An interesting feature is thebimodal distribution of defaults averted from targeted intervention. One hypothesis to con-sider is that this is a result of the network cluster structure itself: there are several clustersin the network, and firms within the same cluster are more likely to default (or avert defaultfrom a nearby intervention) together.Figure 4 depicts the experimental Tail Value at Risk (TVaR) of default cascade size fordifferent quantiles < q ≤ . TVaR ( q ) is a conditional expectation, conditioned on eventsfalling in the q -th quantile of outcomes:TVaR ( q ; b ) = E (cid:20) | A | ( b ) | U | (cid:12)(cid:12)(cid:12) | A | (0) ≥ VaR (cid:0) | A | (0); q (cid:1)(cid:21) , % 20% 40% 60% 80% % Firms Defaulting C o un t Histogram % Firms Defaulting
No Intervention1% Intervention (a)
0% 5% 10% 15% 20%
Defaults Averted (% Total Firms) C o un t Defaults Averted from 1% Intervention (b)
Figure 3: Histogram densities of defaults under 1% asset value intervention and no inter-vention. E s t . E [ % F i r m s D e f a u lti ng ] Budget % of Total Assets |Dp|
Expected Defaults vs. Intervention Budget
TVaR(q=0.1)TVaR(q=0.2)TVaR(q=0.4)TVaR(q=0.6)TVaR(q=1.0)
Figure 4: Simulation TVaRs with quantiles q for a range of intervention budgets.where | A | ( b ) outputs the number of defaulting firms given budget b , | U | is the number of totalfirms, and VaR ( X ; q ) is the q -quantile of random variable X . Note that in our case q is aquantile of a distribution that is already modeling negative outcomes in these simulations.Also recall that q = 1 gives the unconditional expectation.Figure 4 demonstrates that relatively small budgets effectively reduce systemic risks asmeasured by TVaR. Experimental numbers for the percentage reduction in TVaR using anintervention budget of 1% of initial assets is presented in Table 1. We have shown that the optimal intervention problem is NP-hard under fixed failure thresh-olds. Given a network, one essentially needs to choose a set of firms among those whowould otherwise default and reverse their defaults. The choice of such firms saves the max-15 % Reduction in
TVaR ( q )0 . . . . . Table 1: Percentage change in TVaR with quantile q of default cascade size resulting fromtargeted intervention with budget 1% of total initial assets.imum value. Other related problems are also shown to be computationally hard, even if wehave perfect information about the underlying setup. In particular, given a maximum ag-gregate shock, it is computationally hard to determine if there is a distribution of this shockacross firms leading to a given fraction of the network to fail. In turn, when thresholds arerandom, these problems allow (1 − /e − (cid:15) ) -approximations. Failure thresholds represent thepoints where shareholders of the firm decide to cease the operations and liquidate the asset.In reality thresholds could be based on the expectations of large cascades and large scaleliquidations. Given the complexity issues in assessing which shocks lead to such extremescenarios, it would be interesting to explore further how strategic shareholders would maketheir threshold choices.Using the approximation algorithms, we evaluate the performance of intervention undera large number of shocks. We remark a significant reduction of Tail Value at Risk of thedefault cascade size, even under a small intervention budget relative to total assets. Thiscan be explained by the fact that the solution to the optimal intervention problem unveils ahierarchical or causal structure of defaults, and in practice it selects a relatively small setto directly intervene on. Most of the default cascade is then averted indirectly, by reversingfailure costs and network effects. References [1] A HN , D., AND K IM , K.-K. Optimal intervention under stress scenarios: A case of thekorean financial system. Operations Research Letters 47 , 4 (2019), 257–263.[2] A
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A Proofs
Theorem 1
Proof.
We will reduce from independent set to an instance of the economic network interven-tion problem. Our reduction follows closely that used in [20] for the linear influence model,but we note it requires additional steps to reduce independent set to the economic networkintervention setting, which is a class of instances of more general influence maximizationproblems.In the independent set problem, we are given an undirected graph G = ( U, E ) withnodes U and edges E . Given a number k , we ask if there is an independent set in G of size k . Reduction gadget.
For the reduction, construct a bipartite graph G (cid:48) = ( U ∪ U , E (cid:48) ) asfollows:• Add each node in G to U . Attach thresholds | U | to these nodes.• For each edge { i, j } ∈ E , add a node u to U and add directed edges ( i, u ) , ( j, u ) to E (cid:48) . Attach edge weights | U | and thresholds | U | .18 For each possible pair { i, j } / ∈ E , add two nodes u, w to U and add directed edges ( i, u ) , ( j, w ) to E (cid:48) . Attach influence weights | U | and thresholds | U | .Notice the number of vertices and edges in G (cid:48) : | U ∪ U | = | U | + | E | + 2 (cid:18) | U | − | U | − | E | (cid:19) = | U | − | E | , | E (cid:48) | = | U | − | U | . Set the desired penetration rate in G (cid:48) to ζ = k | U || U | −| E | (this is the fraction of nodes wewant to ‘activate’ or ‘reverse defaults’ of in the economic network). Notice that ζ | U ∪ U | = k | U || U | − | E | | U | − | E | = k | U | , which will be the desired penetration in the reduction graph to correspond to the independentset (which we prove below). Gadget is instance of economic network intervention.
We now show that the problemon G (cid:48) translates to an instance ( C, β, θ , D, p ) of the economic network intervention problem.Let A be the adjacency matrix of G (cid:48) . Since G (cid:48) is a 2-layer DAG, we have A t = 0 for integers t > . Then the Neumann series is ( I − A ) − = I + A. Notice that A is non-negative column-substochastic with zero diagonal. Thus we take C = A , and ˆ C is well-defined. Claim: ( β, θ , D, p ) can be chosen such that, before intervention, all nodes fail with endvalues v = 0 , ˜ θ u = | U | for all u , and β ≥ . Proof of claim:
To find such a ( β, θ , D, p ) , we can setup the following system V = ( I + C )( D p − β U ∪ U ) = 0 θ u > (cid:104) ˆ C ( I + C ) D p (cid:105) u for u ∈ U θ u > (cid:104) ˆ C ( I + C )( D p − β U ) (cid:105) u for u ∈ U ˜ θ u = (cid:104) ˆ C − θ − ( I + C ) D p − Cβ U ∪ U \{ u } (cid:105) u = 1 | U | for all uβ ≥ . The system has the same number of variables as dimensions. Because of the 2-layer DAGstructure, it is simple to see that the system is solvable.Notice that in the equation for ˜ θ is valid. Taking failure set T , we have ˜ θ u = (cid:104) ˆ C − θ − ( I + C )( D p − β T \{ u } ) (cid:105) u = (cid:104) ˆ C − θ − ( I + C ) D p − Cβ T \{ u } (cid:105) u [ Iβ T \{ u } ] u = 0 . Claim:
The effect of reversing defaults S propagates to other nodes through f ( S ) = Cβ S . Proof of claim:
First notice that for all nodes u , (cid:104) ( I + C ) β u (cid:105) u = β u . This is a simple result because C has zero diagonal and the only nonzero entry of u is the u th entry; thus there is contribution from Cβ u for the u th entry.Then we have f ( S ) = ( I + C ) β S − (cid:88) u ∈ S I u ( I + C ) β u = ( I + C ) β S − (cid:88) u ∈ S I u β u = ( I + C ) β S − β S = Cβ S . Claim:
If we reverse the default of a node in U , then its neighbors in U are also saved fromdefault. Proof of claim:
Suppose we reverse the default of u ∈ U . Suppose w ∈ U is a neighborof u . Then w ’s value is affected by [ f ( u )] w = [ Cβ u ] w = β | U | > | U | = ˜ θ w since β ≥ . Thus w ’s default is also reversed.To complete the translation into the economic network intervention problem, define thefollowing: b = k | U | α = 1 (cid:96) = 1 − ζ. In intuitive terms, the corresponding economic network is a 2-layer DAG, in which theonly cross-holdings are the shares in the first layer held by the second layer. In this case,the interactions are quite simple, described solely by C . In this network, every node startsin default. We can pay ˜ θ = | U | to reverse a node’s default (“activate” a node). Our budget is b and we can choose at most k nodes to influence. Reduction to integral case.
We first consider the integral case and then extend to thefractional case. We want to select a subset S of k nodes from G (cid:48) such that, if we providepayments equal to their ˜ θ , a cascade of reverse-defaults occurs of size at least ζ | U ∪ U | (i.e., at most (cid:96) | U ∪ U | nodes fail after intervention). This occurs if and only if G has anindependent set of size k , as we prove next. 20irst, note that sets S ⊆ U always dominate sets S ⊆ U ∪ U with S (cid:40) U . Thisis because, by construction, activating any node in U in turn activates its neighbors in U whereas activating a node in U does not activate its neighbors in U . Since each node in U has a neighbor in U , it always makes sense to activate such a neighbor instead of theconsidered node in U . Thus it is sufficient to consider only solutions in U . Notice that thisextends to the fractional case since threshold-crossing payments are of the same size fornodes in U and U .Each node in U has | U | − neighbors, and two nodes in V share a neighbor if and onlyif they are neighbors in G . So if we pick the subset S ⊆ U , the size of the default reversecascade is default reverses = | U || S | − |{{ i, j } ∈ E | i, j ∈ S }| . E.g., if no nodes in S are connected in G , then the second term is and each activated nodeactivates itself and | U | − unique nodes in U for a total of | S | + ( | U | − | S | = | U || S | nodes.The number of activations is ≥ (cid:96) | U ∪ U | = k | U | if and only if ∀ u, v ∈ S , { u, v } / ∈ E ,which is that case if and only if there is an independent set of size k in G . Reduction to fractional case.
Notice that this easily extends to the fractional case. In thiscase, we want to find payments such that (cid:80) i γ i ≤ b = k | U | and we save ζ | U ∪ U | nodes fromfailure (i.e., at most (cid:96) | U ∪ U | nodes fail after intervention). In G (cid:48) , all edges and thresholdshave value | U | . Given the structure of G (cid:48) , optimal node payments will obey γ i ∈ { , | U | } . Thisis because a payment to a node in U is again always better than a payment to a node in U (same argument as before), and any payment smaller than | U | will result in no activationin U , and hence no subsequent effect on U . Thus there is one-to-one correspondencebetween optimal integral solutions and optimal fractional solutions. Thus the fractional caseis NP-hard in general. Proposition 1
Proof.
To simplify notation, define A := ( I − C ) − β .(Monotonicity) Let T ⊂ U and u ∈ U \ T . Then we have f ( T ∪ { u } ) = A T ∪{ u } − (cid:88) j ∈ T ∪{ u } I j A j = A T − (cid:88) j ∈ T I j A j + A u − I u A u = f ( T ) + A u − I u A u . Since A is non-negative, the second term is ≥ . The third term only affects the u thcomponent, and then only cancels the contribution of the second term. Thus we have f ( T ∪ { u } ) ≥ f ( T ) .(Submodularity) Let S ⊆ T ⊆ U and u ∈ U \ T . From the above equations, we have f ( T ∪ { u } ) − f ( T ) = A u − I u A u . and similarly with S . Thus the submodularity condition f ( S ∪ { u } ) − f ( S ) ≥ f ( T ∪ { u } ) − f ( T ) holds with equivalence.21 heorem 2 Proof.
Recall that the intervention problem is equivalent to an instance of an influence max-imization problem. By assumption, w is normalized, monotone, and submodular, and f isnormalized. And by Prop. 1, f is monotone and submodular. Notice that the interventionproblem is easily normalized (in a different sense) so as to restrict each f i and ˜ θ i to therange [0 , . Then by Theorem 1 in [23], the integral influence problem has σ ( S ) normal-ized, monotone, and submodular. And by Theorems 2-3 in [13], the fractional influenceproblem has σ ( x ) normalized, monotone, and submodular (note that these definitions aremodified to describe non-set functions in the fractional case). Corollary 2
Proof.
This follows using the same application of results as in [21]. In particular, the re-sults of [12],[24] show that a greedy hill-climbing algorithm approximates the optimum ofmonotone submodular problems to within a factor of (1 − /e ) . Given that σ has to be ap-proximated, the result can be extended to show that for any (cid:15) > , there is δ > such that byusing (1 + δ ) -approximate values for the σ function, we obtain a (1 − /e − (cid:15) ) -approximation.For the fractional case, this uses Theorem 4 in [13]. Theorem 3
Proof.
First consider a specific subclass of economic network instances. We will reduceindependent set to an instance of this subclass. The subclass has the following properties:• Asset prices p take values in { , } .• D is row-sub-stochastic, such that a firm’s underlying assets can be valued at most 1.• C = 0 , in which case ˆ C = I and ( I − C ) − = I .• β = 0 , in which case a firm’s value is in [0 , .• b is an integer.As a result, the shock to be chosen in our problem, if applied to asset i , can change it’s pricefrom to . The problem at hand is now to find a set of b assets that, if set to 0, cause (cid:96) | U | firms to default.Next consider a reformulation of the network process into a bipartite graph G (cid:48) as follows:• Add nodes for each underlying asset. Denote these nodes U .22 Add nodes for each firm. Denote these nodes U .• For each u ∈ U , add a weighted directed edge from u to nodes in U according tothe matrix D . The weights here represent the effect of the asset on the book values offirms that own those assets in the simple setting with C = 0 .Assume the assets in U are initially set to 1. If an asset is changed to 0, (negative) influenceis exerted on its connections in U via D , lowering those firms’ values. If enough (negative)influence is exerted on a firm in U , its value decreases below threshold, triggering default.The equivalent problem is to find a set of b nodes in U such that, if set to 0, cause (cid:96) | U | firmsto default.To reduce from independent set, we can follow essentially the same reduction as in The-orem 1 to a process on a bipartite graph like above. With appropriate definition of parame-ters, this is an instance of the subclass of economic networks above. And thus independentset reduces to economic network maximum shock problem. B Algorithms
The algorithms below use the following problem setting:• f ( S ) outputs the vector of influence exerted by the activation of node set S on eachnode. In the analysis, we define f to give the linear threshold model.• w ( S ) outputs a weight of node set S . In the analysis, we define w to weight each nodeby 1.• Θ = uniform [0 , n is the distribution for node thresholds ( n =number of nodes).• b = budget. Algorithm 1
CalcIntCascade ( S ; f, θ ) Require: set S , set function f , thresholds θ Initialize S ← ∅ , S ← S , i ← while S i (cid:54) = S i − do S i +1 = { node v | f ( S i )[ v ] ≥ θ [ v ] } ∪ S i i ← i + 1 end whilereturn S i lgorithm 2 ˆ σ ( S ) estimate of σ ( S ) for integral influence Require: set S , set function f , weight function w , thresholds distr. Θ , sample size k =10 , Initialize σ ← for i ≤ k do Sample θ ∼ Θ T, = CalcIntCascade (cid:16) S ; f, θ (cid:17) σ ← σ + w ( T ) end forreturn σ/k Algorithm 3
GreedyIntInfMax = Greedy algorithm for integral influence maximization
Require: set function f , weight function w , thresholds distr. Θ , budget b Initialize S ← ∅ , i ← while | S i | < b dofor node v / ∈ S i do q [ v ] = ˆ σ (cid:16) S i ∪ { v } ; f, Θ , w (cid:17) end for S i +1 ← S i ∪ { arg max q } , i ← i + 1 end whileif | S i | ≤ b thenreturn S i elsereturn S i − end ifAlgorithm 4 CalcFracCascade ( x ; f, θ ) Require: vector x , set function f , thresholds θ Initialize S ← ∅ , i ← S ← { node v | x [ v ] ≥ θ [ v ] } while S i (cid:54) = S i − do S i +1 = { node v | f ( S i )[ v ] + x [ v ] ≥ θ [ v ] } i ← i + 1 end whilereturn S i lgorithm 5 ˆ σ ( x ) estimate of σ ( x ) for fractional influence Require: vector x , set function f , weight function w , thresholds distr. Θ , sample size k =10 , Initialize σ ← for i ≤ k do Sample θ ∼ Θ T = CalcFracCascade (cid:16) x ; f, θ (cid:17) σ ← σ + w ( T ) end forreturn σ/k Algorithm 6
GreedyFracInfMax = Greedy algorithm for fractional influence maximization
Require: set function f , weight function w , thresholds distr. Θ , budget b Initialize x ← , i ← while T x i < b do S i = { node v | x i [ v ] > } for node v / ∈ S i do x v = x i + (cid:16) θ max [ v ] − Γ + ( v, S i ) (cid:17) v q [ v ] = ˆ σ (cid:16) x v ; f, Θ , w (cid:17) end for u = arg max qx i + ← x i + (cid:16) θ max [ u ] − Γ + ( u, S i ) (cid:17) u , i ← i + 1 end whileif T x i ≤ b thenreturn x i elsereturn x i − end ifAlgorithm 7 Γ + ( v, A ) = total sum of weight of edges from set A to node v Require: set A , set function f , node v return f ( A )[ v ] Algorithm 8 Γ − ( v, A ) = total sum of weight of edges from node v to set A Require: set A , set function f , node v return TA f ( { v } ) lgorithm 9 DiscountFrac heuristic algorithm
Require: set function f , weight function w , thresholds distr. Θ , budget b Initialize x ← , i ← while T x i < b do S i = { node v | x i [ v ] > } for node v / ∈ S i do q [ v ] = Γ − ( v, V \ S i ) end for u = arg max qx i + ← x i + (cid:16) θ max [ u ] − Γ + ( u, S i ) (cid:17) u , i ← i + 1 end whileif T x i ≤ b thenreturn x i elsereturn x i − end ifend if