On the Computational Complexity of Measuring Global Stability of Banking Networks^{†}^{†}thanks: Talks based on these results were given or will be given at the annual New York Computer Science and Economics Day, New York University, September 16, 2011, at the IndustrialAcademic Workshop on Optimization in Finance and Risk Management, October 34, 2011, Fields Institute, Toronto, Canada, and at the Mathematical Finance theme, 2012 Annual Meeting of the Canadian Applied and Industrial Mathematics Society, July 2428, 2012.
Abstract
Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global markets is the socalled financial contagion, namely the spreadings of instabilities or failures of individual components of the network to other, perhaps healthier, components. This leads to a natural question of whether the regulatory authorities could have predicted and perhaps mitigated the current economic crisis by effective computations of some stability measure of the banking networks. Motivated by such observations, we consider the problem of defining and evaluating stabilities of both homogeneous and heterogeneous banking networks against propagation of synchronous idiosyncratic shocks given to a subset of banks. We formalize the homogeneous banking network model of Nier et al. [46] and its corresponding heterogeneous version, formalize the synchronous shock propagation procedures outlined in [46, 25], define two appropriate stability measures and investigate the computational complexities of evaluating these measures for various network topologies and parameters of interest. Our results and proofs also shed some light on the properties of topologies and parameters of the network that may lead to higher or lower stabilities.
1 Introduction and Motivation
In marketbased economies, financial systems perform important financial intermediation functions of borrowing from surplus units and lending to deficit units. Financial stability is the ability of the financial systems to absorb shocks and perform its key functions, even in stressful situations. Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The concept of instability of a marketbased financial system due to factors such as debt financing of investments can be traced back to earlier works of the economists such as Irving Fisher [29] and John Keynes [37] during the 1930’s Great Depression era. Subsequently, some economists such as Hyman Minsky [44] have argued that:
Such instabilities are inherent in many modern capitalist economies.
In this paper, we investigate systemic instabilities of the banking networks, an important component of modern capitalist economies of many countries. The financial crisis in the current and past decade has shown that an important component of instability in global financial markets is the socalled financial contagion, namely the spreadings of instabilities or failures of individual components of the network to other, perhaps healthier, components. The general topic of interest in this paper, motivated by the global economic crisis in the current and the past decade, is the phenomenon of financial contagion in the context of banking networks, and is related to the following natural extension of the question posed by Minsky and others:

What is the true characterization of such instabilities of banking networks, i.e.,

Are such instabilities systemic, e.g., caused by a repeal of GlassSteagall act with subsequent development of specific properties of banking networks that allowed a ripple effect [14]?

Or, are such instabilities caused just by a few banks that were “too big to fail” and/or “a few individually greedy executives” ?

To investigate these types of questions, one must first settle the following issues:

What is the precise model of the banking network that is studied?

How exactly failures of individual banks propagated through the network to other banks?

What is an appropriate stability measure and what are the computational properties of such a measure?
As prior researchers such as Allen and Babus [1] pointed out,graphtheoretic concepts provide a conceptual framework within which various patterns of connections between banks can be described and analyzed in a meaningful way by modeling banking networks as a directed network in which nodes represent the banks and the links represent the direct exposures between banks. Such a networkbased approach to studying financial systems is particularly important for assessing financial stability, and in capturing the externalities that the risk associated with a single or small group of institutions may create for the entire system. Conceptually, links between banks have two opposing effects on contagion:

More interbank links increase the opportunity for spreading failures to other banks [32]: when one region of the network suffers from a crisis, another region also incurs a loss because their claims on the troubled region fall in value and, if this spillover effect is strong enough, it can cause a crisis in adjacent regions.

More interbank links provide banks with a form of coinsurance against uncertain liquidity flows [2], i.e., banks can insure against the liquidity shocks by exchanging deposits through links in the network.
2 The Banking Network Model
2.1 Rationale Behind the Model
As several prior researchers such as [1, 46, 25, 39] have already commented, graphtheoretic frameworks may provide a powerful tool for analyzing stability of banking and other financial networks. We provide and use a mathematically precise abstraction of a banking network model as outlined in [46] and elsewhere. The same or very similar version of the graphtheoretic loss propagation model used in this paper has also been extensively used by prior researchers in finance, economics and banking industry to study various properties and research questions involving banking systems similar to what is studied in this paper (e.g., see [31, 49, 45, 5, 19], to name a few). As commented by researchers such as [46, 5]:
the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types of insolvency cascades, but in studying a model that can mimic the empirical properties of these different types of networks.
A loss propagation model such as the one discussed here and elsewhere such as in [31, 49, 45, 5, 19] conceptualises the main characteristics of a financial system using network theory by relating the cascading behavior of financial networks both to the local properties of the nodes and to the underlying topology of the network, allowing us to vary continuously the key parameters of the network.
2.2 Homogeneous Networks: Balance Sheets and Parameters for Banks
We provide a precise abstraction of the model as outlined in [46] which builds up on the works of Eboli [25]. The network is modeled by a weighted directed graph of nodes and directed edges, where each node corresponds to a bank () and each directed edge indicates that has an agreement to lend money to . Let and denote the indegree and the outdegree of node . The model has the following parameters:
total external asset, total interbank exposure, total asset 
percentage of equity to asset, weight of edge , severity of shock () 
Now, we describe the balance sheet for a node (i.e., for )^{1}^{1}1This model assumes that all the depositors are insured for their deposits, e.g., in United States the Federal Deposit Insurance Corporation provides such an insurance up to a maximum level. Thus, we will omit the parameters for all in the rest of the paper when using the model. Similarly, quantities (which depend on the ’s) are also only necessary in writing the balance sheet equation and will not be used subsequently.:
Assets  Liabilities 

(balance sheet equation) 
Note that the homogeneous model is completely described by the tuple of parameters .
2.3 Balance Sheets and Parameters for Heterogeneous Networks
The heterogeneous version of the model is the same as its’ homogeneous counterpart as described above, except that the shares of interbank exposures and external assets for different banks may be different. Formally, the following modifications are done in the homogeneous model:

denotes the weight of the edge along with the constraint that .

, and .

for some along with the constraint . Since , this gives . Consequently, now equals .
Denoting the dimensional vector of ’s by and the dimensional vector of ’s by , the heterogeneous model is completely described by the tuple of parameters .
Illustration of calculations of balance sheet parameters
We illustrate the calculation of relevant parameters of the balance sheet of banks for the simple banking network shown in Fig. 1.
(a) Homogeneous version of the network

.

.

.

.

.

, , , , .
(b) Heterogeneous version of the network
Suppose that 95% of is distributed equally on the two banks and , and the rest 5% of is distributed equally on the remaining three banks. Thus:
Suppose that 95% of is distributed equally on the three edges , and the remaining 5% of is distributed equally on the remaining four edges . Then,
2.4 Idiosyncratic Shock [46, 25]
As in [46], our initial failures are caused by idiosyncratic shocks which can occur due to operations risks (frauds) or credit risks, and has the effect of reducing the external assets of a selected subset of banks perhaps causing them to default. While aggregated or correlated shocks affecting all banks simultaneously is relevant in practice, idiosyncratic shocks are a cleaner way to study the stability of the topology of the banking network. Formally, we select a nonempty subset of nodes (banks) . For all nodes , we simultaneously decrease their external assets from by , where the parameter determines the “severity” of the shock. As a result, the new net worth of becomes . The effect of this shock is as follows:

If , continues to operate but with a lower net worth of .

If , defaults (i.e., stops functioning).
; 
(* start the shock at on nodes in *) 
if then else 
(* shock propagation at times *) 
while do 
(* transmit loss to next time step *) 
(* remove from network if it is to fail at this step *) 
endwhile 
2.5 Propagation of an Idiosyncratic Shock [46, 25]
We use the notation to denote at time , and to denote any . Let be the set of nodes that have not failed at time and let be the corresponding nodeinduced subgraph of at time with and denote the indegree and outdegree of a node in the graph . In a continuoustime model, the shock propagates as follows:

, if , and otherwise.

If a banks equity ever becomes negative, it fails subsequently, i.e., .

A failed bank at time affects the net worth (equity) of all banks that gave loan to in the following manner. For each edge in the network at time , the equity is decreased by an amount^{2}^{2}2If then the depositors incur a loss of , but as already mentioned before this model assumes that all the depositors are insured for their deposits. of . Thus, the shock propagation is defined by the following differential equation:
An intuitive explanation of the two quantities inside the summation in the above equation is as follows. The term distributes the loss of equity of a bank equitably among its creditors that have not failed yet. The term ensures that the total loss propagated is no more than the total interbank exposure of the failed bank.
A discretetime version of the above can be obtained by the obvious method of quantizing time and replacing the partial differential equations by “difference equations”. With appropriate normalizations, the discretetime model for shock propagation is described by a synchronous iterative procedure shown in Table 1 where denotes the discrete time step at which the synchronous update is done ().
A simplified illustration of the effect of idiosyncratic shocks
Consider the case when the model is homogeneous and the topology of the graph is inarborescence, i.e., a directed rooted tree where all edges are oriented towards the root. Consider two nodes such that and (see Fig. 2). Suppose that at time the node is shocked and consequently it defaults. The amount of shock transmitted from to is
Since , we have
Assuming and , the above expression simplifies to
Suppose that . Then, . Consequently, one can observe the following:

If , then and node will surely fail at time .

If and then and node will surely not fail at time .
2.6 Parameter Simplification
We can assume without loss of generality that in the homogeneous shock propagation model . To observe this, if , then we can divide each of the quantities , , and by ; it is easy to see that the outcome of the shock propagation procedure in Table 1 remains the same. Moreover, we will ignore the balance sheet equation since has no effect in shock propagation.
3 Related Prior Works on Financial Networks
Although there is a large amount of literature on stability of financial systems in general and banking systems in particular, much of the prior research is on the empirical side or applicable to smallsize networks. Two main categories of prior researches can be summarized as follows. The particular model used in this paper is the model of Nier et al. [46]. As stated before, definition of a precise stability measure and analysis of its computational complexity issues for stability calculation were not provided for these models before.
Network formation
Babus [7] proposed a model in which banks form links with each other as an insurance mechanism to reduce the risk of contagion. In contrast, Castiglionesi and Navarro [15] studied decentralization of the network of banks that is optimal from the perspective of a social planner. In a setting in which banks invest on behalf of depositors and there are positive network externalities on the investment returns, fragility arises when “not sufficiently capitalized” banks gamble with depositors’ money. When the probability of bankruptcy is low, the decentralized solution wellapproximates the first objective of Babus.
Contagion spread in networks
Although ordinarily one would expect the risk of contagion to be larger in a highly interconnected banking system, some empirical simulations indicate that shocks have an extremely complex effect on the network stability in the sense that higher connectivity among banks may sometimes lead to lower risk of contagion.
Allen and Gale [2] studied how a banking system may respond to contagion when banks are connected under different network structures, and found that, in a setting where consumers have the liquidity preferences as introduced by Diamond and Dybvig [23] and have random liquidity needs, banks perfectly insure against liquidity fluctuations by exchanging interbank deposits, but the connections created by swapping deposits expose the entire system to contagion. Allen and Gale concluded that incomplete networks are more prone to contagion than networks with maximum connectivity since betterconnected networks are more resilient via transfer of proportion of the losses in one bank’s portfolio to more banks through interbank agreements. Freixas et al. [30] explored the case of banks that face liquidity fluctuations due to the uncertainty about consumers withdrawing funds. Gai and Kapadia [32] argued that the higher is the connectivity among banks the more will be the contagion effect during crisis. Haldane [34] suggested that contagion should be measured based on the interconnectedness of each institution within the financial system. Liedorp et al. [42] investigated if interconnectedness in the interbank market is a channel through which banks affect each others riskiness, and argued that both large lending and borrowing shares in interbank markets increase the riskiness of banks active in the dutch banking market.
Dasgupta [21] explored how linkages between banks, represented by crossholding of deposits, can be a source of contagious breakdowns by investigating how depositors, who receive a private signal about fundamentals of banks, may want to withdraw their deposits if they believe that enough other depositors will do the same. Lagunoff and Schreft [41] considered a model in which agents are linked in the sense that the return on an agents’ portfolio depends on the portfolio allocations of other agents. Iazzetta and Manna [35] used network topology analysis on monthly data on deposits exchange to gain more insight into the way a liquidity crisis spreads. Nier et al. [46] explored the dependency of systemic risks on the structure of the banking system via network theoretic approach and the resilience of such a system to contagious defaults. Kleindorfer et al. [39] argued that network analyses can play a crucial role in understanding many important phenomena in finance. Corbo and Demange [20] explored, given the exogenous default of set of banks, the relationship of the structure of interbank connections to the contagion risk of defaults. Babus [8] studied how the tradeoff between the benefits and the costs of being linked changes depending on the network structure, and observed that, when the network is maximal, liquidity can be redistributed in the system to make the risk of contagion minimal.
4 The Stability and Dual Stability Indices
A banking network is called dead if all the banks in the network have failed. Consider a given homogeneous or heterogeneous banking network or . For , let
The Stability Index
The optimal stability index of a network is defined as
For estimation of this measure, we assume that it is possible for the network to fail, i.e., . Thus, , and the higher the stability index is, the better is the stability of the network against an idiosyncratic shock. We thus arrive at the natural computational problem Stab. We denote an optimal subset of nodes that is a solution of Problem Stab by , i.e., . Note that if then the Stab finds a minimum subset of nodes which, when shocked, will eventually cause the death of the network in an arbitrary number of time steps.
Input: a banking network with shocking parameter ,  Input: a banking network with shocking parameter , 

and an integer  and two integers 
Valid solution: A subset such that  Valid solution: A subset such that 
Objective: minimize  Objective: maximize 
Stability of banking network ()  Dual Stability of banking network () 
The Dual Stability Index
Many coveringtype minimization problems in combinatorics have a natural maximization dual in which one fixes apriori the number of covering sets and then finds a maximum number of elements that can be covered with these many sets. For example, the usual dual of the minimum set covering problem is the maximum coverage problem [38]. Analogously, we define a dual stability problem DualStab. The dual stability index of a network can then be defined as
The dual stability measure is of particular interest when , i.e., the entire network cannot be made to fail. In this case, a natural goal is to find out if a significant portion of the nodes in the network can be failed by shocking a limited number of nodes of ; this is captured by the definition of .
Violent Death vs. Slow Poisoning
In our results, we distinguish two cases of death of a network:
 violent death ()

The network is dead by the very next step after the shock.
 slow poisoning (any )

The network may not be dead immediately but dies eventually.
4.1 Rationale Behind the Stability Measures
Although it is possible to think of many other alternate measures of stability for networks than the ones defined in this paper, the measures introduced here are in tune with the ideas that references [46, 25] directly (and, some other references such as [31, 49] implicitly) used to empirically study their networks by shocking only a few (sometimes one) node. Thus, a rationale in defining the stability measures in the above manner is to follow the cue provided by other researchers in the banking industry in studying models such as in this paper instead of creating a completely new measure that may be out of sync with ideas used by prior researchers and therefore could be subject to criticisms.
5 Comparison with Other Models for Attribute Propagation in Networks
Models for propagation of beneficial or harmful attributes have been investigated in the past in several other contexts such as influence maximization in social networks [36, 17, 16, 13], disease spreading in urban networks [27, 18, 26], percolation models in physics and mathematics [48] and other types of contagion spreads [11, 12]. However, the model for shock propagation in financial network discussed in this paper is fundamentally very different from all these models. For example, the cascade models of failure considered in [11, 12] are probabilistic models of failure propagation of a more generic nature, and thus not very useful to study failure propagation via interlocked balance sheets of financial institutions (as is the case in OTC derivatives markets). Some distinguishing features of our model include:
Almost all of these models include a trivial solution in which the attribute spreads to the entire network if we inject each node individually with the attribute. This is not the case with our model: a node may not fail when shocked, and the network may not be dead if all nodes are shocked. For example, consider the network in Fig. 3 (i).

Suppose that all the nodes are shocked. Then, the following events happen.

Node (and similarly node ) fails at since .

Node also fails at since .

Node (and similarly node ) do not fail at since and its equity stays at .

At , node (and similarly node ) receives a shock from node of the amount . Thus, nodes and do not fail. Since no new nodes fail during , the network does not become dead.


However, suppose that only nodes and are shocked. Then, the following events happen.

Node (and similarly node ) fails at since .

At , node receives a shock of the amount . Thus, node fails at .

At , node (and similarly node ) receives a shock of the amount . Thus, both these nodes fail at and the entire network is dead.

As the above example shows, if shocking a subset of nodes makes a network dead, adding more nodes to this subset may not necessarily lead to the death of the network, and the stability measure is neither monotone nor submodular. Similarly, it is also possible to exhibit banking networks such that to make the entire network fail:

it may be necessary to shock a node even if it does not fail since shocking such a node “weakens” it by decreasing its equity, and

it may be necessary to shock a node even if it fails due to shocks given to other nodes.
The complexity of the computational aspects of many previous attribute propagation models arise due to the presence of cycles in the graph; for example, see [16] for polynomialtime solutions of some of these problems when the underlying graph does not have a cycle. In contrast, our computational problems are may be hard even when the given graph is acyclic; instead, a key component of computational complexity arises due to two or more directed paths sharing a node.
6 Overview of Our Results and Their Implications on Banking Networks
Table 2 summarizes our results, where the notation denotes a constantdegree polynomial in variables . Our results for heterogeneous networks show that the problem of computing stability indices for them is harder than that for homogeneous networks, as one would naturally expect.
6.1 Brief Overview of Proof Techniques
6.1.1 Homogeneous Networks, Stab
, approximation hardness and approximation algorithm
The reduction for approximation hardness is from a corresponding inapproximability result for the dominating set problem for graphs. The logarithmic approximation almost matches the lower bound. Even though this algorithmic problem can be cast as a covering problem, one cannot explicitly enumerate exponentially many covering sets in polynomial time. Instead, we reformulate the problem to that of computing an optimal solution of a polynomialsize integer linear programming (ILP), and then use the greedy approach of [24] for approximation. A careful calculation of the size of the coefficients of the ILP ensures that we have the desired approximation bound.
Any , approximation hardness and exact algorithm
The hardness result, which holds even if the degrees of all nodes are small constants, is via a reduction from the node cover problem for regular graphs. Technical complications in the reduction arise from making sure that the generated graph instance of Stab is acyclic, no new nodes fail for any , but the network can be dead without each node being individually shocked. If the network is a rooted inarborescence and every node can be individually shocked to fail, then we design an time exact algorithm via dynamic programming; as a by product it also follows that the value of the stability index of this kind of network with bounded node degrees is large.
6.1.2 Homogeneous Networks, DualStab
Any , approximation hardness and exact algorithm
For hardness, we translate a lower bound for the maximum coverage problem [28]. The reduction relies on the fact that in dual stability measure every node of the network need not fail. If the given graph is a rooted inarborescence and every node can be individually shocked to fail, we provide an time exact algorithm via dynamic programming.
6.1.3 Heterogeneous Networks, Stab
Any , approximation hardness
The reduction is from a corresponding inapproximability result for the minimum set covering problem. Unlike homogeneous networks, unequal shares of the total external assets by various banks allows us to encode an instance of set cover by “equalizing” effects of nodes.
The approximation algorithm uses linear program in Theorem 9.1 with more careful calculations.
Any , approximation hardness
This stronger polylogarithmic inapproximability result than that in Theorem 12.1 is obtained by a reduction from Minrep, a graphtheoretic abstraction of two prover multiround protocol for any problem in . Many technical complications in the reduction, culminating to a set of symbolic linear equations between the parameters that we must satisfy. Intuitively, the two provers in Minrep correspond to two nodes in the network that cooperate to fail to another specified set of nodes.
6.1.4 Heterogeneous Networks, DualStab,
approximation hardness
The reduction for this stronger inapproximability result is from the densest hypergraph problem.
6.2 Implications of Our Results on Banking Networks
6.2.1 Effects of Topological Connectivity
Though researchers agree that the connectivity of banking networks affects its stability [2, 32], the conclusions drawn are mixed, namely some researchers conclude that lesser connectivity implies more susceptibility to contagion whereas other researchers conclude in the opposite. Based on our results and their proofs, we found that topological connectivity does play a significant role in stability of the network in the following complex manner.
Even acyclic networks display complex stability behavior: Sometimes a cause of the instability of a banking network is attributed to cyclical dependencies of borrowing and lending mechanisms among major banks, e.g., banks , and borrowing from banks , and , respectively. Our results show that computing the stability measures may be difficult even without the presence of such cycles. Indeed, larger inapproximability results, especially for heterogeneous networks, are possible because slight change in network parameters can cause a large change in the stability measure. On the other hand, acyclic smalldegree rooted inarborescence networks exhibit higher values of the stability measure, e.g., if the maximum indegree of any node in a rooted inarborescence is and the shock parameter is no more than twice the value of the percentage of equity to assets , then by Theorem 11.1 .
Intersection of borrowing chains may cause lower stability: By a borrowing chain we mean a directed path from a node to another node , indicating that bank effectively borrowed from bank through a sequence of successive intermediaries. Now, assume that there is another directed path from to another node . Then, failure of and propagates the resulting shocks to and, if the shocks arrive at the same step, then the total shock received by bank is the addition of these two shocks, which in turn passes this “amplified” shock to other nodes in the network.
Based on these kinds of observations, it can be reasonably inferred that homogeneous networks with topologies more like a smalldegree inarborescence have higher stabilities, whereas networks of other types of topologies may have lower stabilities even if the topologies are acyclic. For example, as we observe later, when , and , we get and the network cannot be put to death without shocking more than of the nodes.
6.2.2 Effects of Ratio of External to Internal Assets () and percentage of equity to assets () for Homogeneous Networks
As our relevant results and their proofs show, lower values of and may cause the network stability to be extremely sensitive with respect to variations of other parameters of a homogeneous network. For example, in the proof of Theorem 8.1 we have , leading to variation of the stability index by a logarithmic factor; however, in the proof of Theorem 10.1 we have and leading to much smaller variation of the stability index.
6.2.3 Homogeneous vs. Heterogeneous Networks
Our results and proofs show that heterogeneous networks of banks with diverse equities tend to exhibit wider fluctuations of the stability index with respect to parameters, e.g., Theorem 14.1 shows a polylogarithmic fluctuation even if the ratio is large.
6.2.4 Further Empirical Study
Subsequent to writing this paper, DasGupta and Kaligounder in a separate article [22] performed a thorough empirical analysis of the stability measure over more than combinations of networks types and parameters, and uncovered many interesting insights into the relationships of the stability with other relevant parameters of the network, such as:
 Effect of uneven distribution of assets:

Networks where all banks have roughly the same external assets are more stable over similar networks in which fewer banks have a disproportionately higher external assets compared to the remaining banks, and failures of those banks with higher assets contribute more damage to the stability of the network. Furthermore, networks in which fewer banks have a disproportionately higher external assets compared to the remaining banks has a minimal instability even if their equity to asset ratio is large and comparable to loss of external assets. This is not the case for networks where all banks have roughly the same external assets. Thus, in summary, they concluded that banks with disproportionately large external assets (“banks that are too big”) affect the stability of the entire banking network in an adverse manner.
 Effect of connectivity:

For banking networks where all banks have roughly the same amount of external assets, higher connectivity leads to lower stability. In contrast, for banking networks in which few banks have disproportionately higher external assets compared to the remaining banks, higher connectivity leads to higher global stability.
 Correlated versus random failures:

Correlated initial failures of banks causes more damage to the entire banking network as opposed to just random initial failures of banks.
 Phase transition properties of global stability:

The global stability exhibits several sharp phase transitions for various banking networks within certain parameter ranges.
7 Preliminary Observations on Shock Propagation
Proposition 7.1.
Let be the given (homogeneous or heterogeneous) banking network. Then, the following are true:
 (a)

If for some , then node must be given a shock (and, must fail due to this shock) for the entire network to fail.
 (b)

Let be the number of edges in the longest directed simple path in . Then, no new node fails at any time .
 (c)

We can assume without loss of generality that is weakly connected, i.e., the unoriented version of is connected.
Proof.
(a) Since , no part of any shock given to any other nodes in the network can reach . Thus, the network of , namely stays strictly positive (since ) and node never fails.
(b) Let be the latest time a node of failed, and let be the set of nodes that failed at time . Then, is a partition of . For every , add directed edges from a node to a node if was last node that transmitted any part of the shock to before failed. Note that is also an edge of and for every node there must be an edge for some node . Thus, has a path of length at least .
(c) This holds since otherwise the stability measures can be computed separately on each weakly connected component. ∎
8 Homogeneous Networks, Stab, Logarithmic Inapproximability
Theorem 8.1.
cannot be approximated in polynomial time within a factor of , for any constant , unless .
Proof.
The dominating set problem for an undirected graph (DOMINSET) is defined as follows: given an undirected graph with nodes, find a minimum cardinality subset of nodes such that every node in is incident on at least one edge whose other endpoint is in . It is known that DOMINSAT is equivalent to the minimum setcover problem under Lreduction [9], and thus cannot be approximated within a factor of unless [28].
Consider an instance of DOMINSET with nodes and edges, and let denote the size of an optimal solution for this instance. Our (directed) banking network is obtained from by replacing each undirected edge by two directed edges and . Thus we have for every node . We set the global parameters as follows: , and .
For a node , let be the set of neighbors of in . We claim that if a node is shocked at time , then all nodes in in fail at time . Indeed, suppose that is shocked at . Then, surely fails because
Now, consider and consider a node such that has not failed but a node failed at time . Then, node surely fails because
Thus, we have a – correspondence between the solutions of DOMINSET and death of , namely is a solution of DOMINSET if and only if shocking the nodes in makes fail at time . ∎
9 Homogeneous Networks, Stab, Logarithmic Approximation
Theorem 9.1.
Stab admits a polynomialtime algorithm with approximation ratio .
Proof.
Suppose that for some node . Then, there exists an optimal solution in which we do not shock the node . Indeed, if was shocked, the equity of increases from to and does not propagate any shock to other nodes. Thus, if still fails at , then it also fails at if it was not shocked.
Let denote the set of nodes that we will select for shocking, and, for every node , let be defined as: Then, our problem reduces to a covering problem of the following type:
find a minimum cardinality subset such that, for every node , .
Note that we cannot even explicitly enumerate, for a node , all subsets such that , since there are exponentially many such subsets. Let the binary variable be the indicator variable for a node for inclusion in . However, we can reformulate our problem as the following integer linear programming problem:
(1)  
Let . We can rewrite each constraint as to ensure that every nonzero entry is at least . Since the coefficients of the constraints and the objective function are all positive real numbers, (1) can be approximated by the greedy algorithm described in [24, Theorem 4.1] with an approximation ratio of . Now, observe that: