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...of instability of a market-based 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, w...

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... the discussion in Section 5. 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 =-=[13, 16, 17, 36]-=-, disease spreading in urban networks [18, 26, 27], percolation models in physics and mathematics [48] and other types of contagion spreads [11, 12]. However, the model for shock propagation in financ...

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...threats. The concept of instability of a market-based 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 econom...

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...dge of H and the size of the solution, namely |V1|+ |V2|, is minimum. For notational simplicity, let n= |V left|+ |V right|. The following result is a consequence of Raz’s parallel repetition theorem =-=[40, 47]-=-. Theorem 14.2. [40] Let L be any language in NP and 0 < δ < 1 be any constant. Then, there exists a reduction running in npoly(log n) time that, given an input instance x of L, produces an instance o...

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... bounded node degrees is large. 6.1.2 Homogeneous Networks, DUAL-STABT,Φ,κ Any T , 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 in-arborescence and every node can be individually shocked to ...

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...ith maximum connectivity since better-connected 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 m...

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... the discussion in Section 5. 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 =-=[13, 16, 17, 36]-=-, disease spreading in urban networks [18, 26, 27], percolation models in physics and mathematics [48] and other types of contagion spreads [11, 12]. However, the model for shock propagation in financ...

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... 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 DUAL-STABT,Φ,κ . The dual stability index of a network G can then be defined as DSI∗(G,T,κ) = max V ′⊆V : |V ′|=κ ∣∣ infl(V ′)/κ ∣∣ The dual stability...

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...y 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 =-=[5, 19, 31, 45, 49]-=-, to name a few). As commented by researchers such as [5, 46]: the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types...

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...y 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 =-=[5, 19, 31, 45, 49]-=-, to name a few). As commented by researchers such as [5, 46]: the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types...

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...g networks, i.e., – Are such instabilities systemic, e.g., caused by a repeal of Glass-Steagall 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...

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...a subset of nodes V ′ ⊆ V such that every edge is incident to at least one node in V ′. The goal is then to find a node cover V ′ ⊆V such that |V ′| is minimized. This problem is known to be APX-hard =-=[10]-=-. v1 v2 v3 v4 v5v6 e2,3 G = (V,F) u1 u′1 u2 u2 ′ u3 u′3 u4 u′4 u5 u′5 u6 u′6 e1,2 e1,4 e1,6 e2,3 {v2,v3} e2,5 e3,4 e3,5 e4,5 e5,6 sink nodes super-source nodes −→G = (−→V ,−→F ) Figure 4: A 3-regular ...

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...tions 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 c...

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...th 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 [25, 46], define two appropriate stability measures and investigate the computational c...

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... the past in several other contexts such as influence maximization in social networks [13, 16, 17, 36], disease spreading in urban networks [18, 26, 27], 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...

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...dge of H and the size of the solution, namely |V1|+ |V2|, is minimum. For notational simplicity, let n= |V left|+ |V right|. The following result is a consequence of Raz’s parallel repetition theorem =-=[40, 47]-=-. Theorem 14.2. [40] Let L be any language in NP and 0 < δ < 1 be any constant. Then, there exists a reduction running in npoly(log n) time that, given an input instance x of L, produces an instance o...

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...vering sets in polynomial time. Instead, we reformulate the problem to that of computing an optimal solution of a polynomial-size 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 T > 1, approximation hardness and exact algorithm The APX...

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...y 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 =-=[5, 19, 31, 45, 49]-=-, to name a few). As commented by researchers such as [5, 46]: the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types...

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...m 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 c...

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...ating 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 ...

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... the discussion in Section 5. 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 =-=[13, 16, 17, 36]-=-, disease spreading in urban networks [18, 26, 27], percolation models in physics and mathematics [48] and other types of contagion spreads [11, 12]. However, the model for shock propagation in financ...

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.... 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 str...

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...es V ′ ⊂ V such that every node in V \V ′ is incident on at least one edge whose other end-point is in V ′. It is known that DOMIN-SAT is equivalent to the minimum set-cover problem under L-reduction =-=[9]-=-, and thus cannot be approximated within a factor of (1− ε) lnn unless NP⊆ DTIME(nlog logn) [28]. Consider an instance G = (V,F) of DOMIN-SET with n nodes and m edges, and let OPT denote the size of a...

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... DSI∗(G,2,κ) cannot be approximated within a large approximation factor provided a complexitytheoretic assumption is satisfied. To understand this assumption, we recall the following definitions from =-=[6]-=-. A random (m,n,d) hyper-graph H is a random hyper-graph of n nodes, m hyper-edges each having having exactly d nodes obtained by choosing each hyper-edge independently and uniformly at random. For ou...

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...s influence maximization in social networks [13, 16, 17, 36], disease spreading in urban networks [18, 26, 27], 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 ...

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...ions 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 ma...

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...f beneficial or harmful attributes have been investigated in the past in several other contexts such as influence maximization in social networks [13, 16, 17, 36], disease spreading in urban networks =-=[18, 26, 27]-=-, 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 fundamental...

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...tments 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 c...

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...ed before, definition of a precise stability measure and analysis of its computational complexity issues for stability calculation were not provided for these models before. 7 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 netwo...

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...y 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 =-=[5, 19, 31, 45, 49]-=-, to name a few). As commented by researchers such as [5, 46]: the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types...

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...f beneficial or harmful attributes have been investigated in the past in several other contexts such as influence maximization in social networks [13, 16, 17, 36], disease spreading in urban networks =-=[18, 26, 27]-=-, 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 fundamental...

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...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 =-=[25, 46]-=-, 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 proof...

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....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, 25, 39, 46]-=- have already commented, graph-theoretic frameworks may provide a powerful tool for analyzing stability of banking and other financial networks. We provide and use a mathematically precise abstraction...

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...dels before. 7 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 ex...

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...ugh 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 cross-holding of deposits, can be a source of contagious breakdowns by investigating how depositors, who receive a private signal about fundamental...

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...esilience 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 trade-off between the be...

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... the discussion in Section 5. 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 =-=[13, 16, 17, 36]-=-, disease spreading in urban networks [18, 26, 27], percolation models in physics and mathematics [48] and other types of contagion spreads [11, 12]. However, the model for shock propagation in financ...

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...re 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 ...

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...y 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 =-=[5, 19, 31, 45, 49]-=-, to name a few). As commented by researchers such as [5, 46]: the modelling challenge in studying banking networks lies not so much in analyzing a model that is flexible enough to represent all types...

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....g., Theorem 14.1 shows a polylogarithmic fluctuation even if the ratio E/I is large. 13 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 700000 combinations of networks types and parameters, and uncovered many interesting insights into the relationships of...