Title: Stability and resilience of alliances: A network approach
Subject: Journal of Public Economics, 256 (2026) 105610. doi:10.1016/j.jpubeco.2026.105610
Author: Arnold Polanski
Keywords: Public goods,Networks,Alliances,NATO

A. Polanski and S. Cortes-Corrales	Journal of Public Economics 256 (2026) 105610

Journal of Public Economics 256 (2026) 105610

ARTICLE INFO

ABSTRACT

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Journal of Public Economics
journal homepage: www.elsevier.com/locate/pubec
Stability and resilience of alliances: A network approach
Arnold Polanski^a,∗ iD, Sebastian Cortes-Corrales^b
^a University of East Anglia, School of Economics, Norwich, UK
^b University of Birmingham, Department of Economics, Birmingham, UK
HIGHLIGHTS
•		Allies and their adversaries are embedded in a node- and edge- weighted network.
•		We relate the resilience and stability of alliances to graph-theoretical concepts.
•		We define two alliance stability concepts and link them to the network’s topology.
•		We evaluate NATO’s stability, resilience, and measures designed to strengthen it.

JEL classification:
D74
H42
H56
L14
Keywords:
Public goods
Networks
Alliances
NATO

Public-goods models of alliances examine the trade-off between the benefits of a collectively provided good and the private costs of contribution. This line of inquiry prompts a further question: what makes alliances endure? We contribute by characterizing alliance stability and resilience using graph-theoretic concepts, offering a structural explanation for why some alliances persist while others do not. Our framework embeds alliances in node- and edge-weighted networks: links among allies generate positive externalities, whereas links between allies and adversaries capture negative externalities. We relate the resilience of alliances – and of their members – to the members’ core numbers within the alliance-induced subnetwork. We then analyze strategic stability when members choose the magnitude of the externalities through costly contributions and the redistribution of spillovers. Finally, we apply the framework to assess the resilience and stability of the North Atlantic Alliance.

1		. Introduction
Alliances are mechanisms through which individuals, organizations, and states can pursue common goals that would be difficult or impossible to achieve independently. They occur at all levels of society – from personal relationships to nation-states – and serve to advance mutual interests. Alliance formation is often considered essential to social existence, providing security and cooperation in the face of external threats (Walt, 1985, Bowles and Gintis, 2011, Arkush et al., 2025). Alliances generate collective benefits similar to public goods. An example is collective security in a military alliance; when two countries form a defensive pact, the resulting deterrence or protection benefits both parties simultaneously without the option of excluding either (Plümper and Neumayer, 2015). Following the seminal work of Olson and Zeckhauser (1966), a substantial body of literature has explored the incentives that lead alliance members to free-ride on others. Yet this line of inquiry has also prompted another fundamental question: what makes alliances endure?
The present work advances the literature by characterizing alliance stability and resilience through graph-theoretic concepts, offering a structural perspective on why some alliances persist while others unravel. We study a setting in which agents can impose both positive and negative externalities on one another, represented by a directed nodeand edge-weighted (NEW) network: nodes are agents and edges encode the magnitude and sign of the externality transmitted. These externalities accumulate into each agent’s stock of a network good (NG) - a transferable benefit that circulates through the network and mitigates adverse shocks. Each agent also contributes to their own NG via a self-loop. The sign of a link indicates whether its externality is beneficial, neutral, or detrimental.
Within this framework, an alliance is a subset of agents connected exclusively by positive links, so that each member benefits every other member and themselves. The links among alliance members form the alliance network, the internal structure of mutual positive influence. In this network, externalities originating outside the alliance are absorbed

^∗ Corresponding author.
Email address: A.Polanski@uea.ac.uk (A. Polanski).
https://doi.org/10.1016/j.jpubeco.2026.105610
Received 13 March 2025; Received in revised form 16 February 2026; Accepted 19 February 2026
Available online 24 February 2026
0047-2727/© 2026 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/).

Fig. 1. Alliances within an exogenous network. Note: Network 𝐺 is formed by two disjoint alliances 𝐴_1 = {𝑁_1 , 𝑁_2 } (blue) and 𝐴_2 = {𝑁_3, 𝑁_4 } (green). The number next to each connection indicates the intensity and sign of the externality between agents 𝑖 and 𝑗 . The externalities in 𝐺 are symmetric but, typically, the weight of the link (𝑖𝑗) will be different from the weight of the link (𝑗𝑖). An absent link indicates no externality. Note that the negative externality between 𝑁_1 and 𝑁_4 - indicated by the dotted link - is absorbed into the self-loops in the two alliance networks. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

into members’ self-loops, thereby internalizing influences from the broader environment. In an alliance network, an agent’s NG equals the sum of all externalities directed to that agent from all agents (allies and non-allies), including the self-loop. Higher NG indicates greater resilience to adverse shocks; conversely, when NG falls below a critical threshold, the agent’s survival in the network is at risk.
In Fig. 1, we present an environment with four agents, 𝑁_1 ,…, 𝑁_4, that form two alliances, 𝐴_1 and 𝐴_2, and are embedded in a network 𝐺. Note that the negative externalities between agents 𝑁_1 and 𝑁_4 , indicated by the dotted line, are incorporated into the respective alliance networks through adjustments to self-links.
We assume that a member fails and is removed from the alliance network along with all its connections if its NG is below an exogenous failure threshold. The removal of a member leads to the loss of all NG transfers to and from that member. Crucially, it reduces the positive externalities received by other members, which may in turn cause their NG to fall below the threshold. This dynamic can trigger a cascading sequence of failures. Consequently, we define a member as vulnerable if they fail either in the initial alliance network or in any subsequent subnetwork that emerges after the failure of other members. Otherwise, this member is resilient. Similarly, an alliance is vulnerable (resilient) when at least one (none) of its members is vulnerable.
In the theoretical section, we first relate the resilience of an alliance and its individual members to the concept of core numbers in the induced alliance network. Originally introduced by Seidman (1983), core numbers measure the connectivity of nodes relative to their neighbors. In an undirected and unweighted graph, the core number of a node is defined as the largest integer 𝑘 such that the node belongs to the 𝑘-core of the network - that is, to its maximal subgraph obtained by iteratively removing nodes with degree less than 𝑘. In our setting, we generalize this concept to NEW networks. As a result, core numbers in our framework are not necessarily integers and may take negative values, reflecting the presence of negative externalities.
Given a fixed failure threshold and an alliance network, our first result establishes that an ally is resilient whenever its core number in the alliance network is at least as large as the failure threshold; otherwise, it is vulnerable. An alliance is vulnerable if at least one of its members has a core number below the threshold; otherwise, it is resilient. This result also implies that each ally is vulnerable or resilient for specific failure thresholds, which allows for a meaningful ranking of agents. In particular, an ally with a lower core number is more vulnerable than one with a higher core number, regardless of the failure threshold.
We then extend our model by assuming that the network externality (NG transfer) is the product of an individual contribution and a spillover parameter. Each alliance member determines their outgoing externalities by selecting a costly contribution level and, potentially, the distribution of their spillover parameters. To better reflect real-world conditions, we also introduce uncertainty by assuming that the failure threshold is drawn from a publicly known distribution. Economically, members trade off marginal resilience benefits from higher contributions against marginal costs of contributing. The random failure threshold in this context can be viewed as an unpredictable shock intensity. Members are resilient, which results in a higher utility, only if their NG stocks exceed the realized shock. Contributions increase the likelihood of being resilient by raising both the contributor’s own NG stock and that of other allies through positive spillovers.
We examine two stability concepts in settings where each alliance member seeks to maximize a state-dependent utility, with an agent’s binary state determined by whether it is resilient or vulnerable. An alliance is contribution stable if no member has an incentive to unilaterally change their contribution, given fixed spillover parameters. Alternatively, an alliance is contribution-transfer stable if no member wishes to alter their contribution and/or their spillover parameters, provided the sum of the latter remains constant. When there is no need to distinguish between these two stability concepts, we will simply call an alliance stable.
We show that stability imposes specific conditions on the core numbers in the alliance network. When the common failure threshold follows a uniform distribution, with the length of the support interval representing uncertainty, the core numbers of all members in a stable alliance must lie within this interval. Moreover, when the utility differential between resilience and vulnerability is sufficiently large and/or uncertainty is sufficiently low, then all core numbers in a stable alliance converge to a common value equal to the highest feasible failure threshold. In a contribution-transfer stable alliance, this value equals the common degree, implying that the alliance network must be regular. Interestingly, stable alliances are fragile, as all members face a positive probability of failure if any member marginally reduces their contribution.
In the empirical part, we apply our framework to assess the resilience and stability of the North Atlantic Treaty Organization (NATO) under the simplifying assumption that it faces a single adversary. The underlying NEW network is constructed using countries’ military expenditures (NG contributions) and normalized distances between capitals (spillover parameters). Based on this network, we identify NATO’s most and least resilient members – those with the highest and lowest core numbers, respectively – and demonstrate that NATO is neither contribution stable nor contribution-transfer stable. Finally, we discuss the potential impact of various measures aimed at enhancing the Alliance’s resilience.
(see, e.g., Bramoullé and Kranton, 2007; Allouch, 2015; Kinateder and Merlino, 2017). Within this literature strand Elliott and Golub (2019) study weighted, directed networks describing heterogeneous marginal externalities of a costly effort. We also model non-rival heterogeneous benefits of a costly contribution with weighted networks but, instead of focusing on Pareto efficiency of outcomes as they do, we consider their stability.
This work is also related to the literature on conflict networks (for an overview, see Dziubiński et al., 2016), which examines how strategic decisions are shaped by the structure of interdependent conflicts. Franke and Öztürk (2015) introduce a seminal framework with identical agents embedded in a conflict network, where each connection represents a bilateral conflict between two agents. Following this, Xu et al. (2019, 2022) and Cortes-Corrales and Gorny (2024) generalize their model by considering multilateral contests on a given hypergraph or heterogeneous agents. These studies emphasize how different network structures influence equilibrium behavior amid multiple threats. In a similar environment, Hiller (2017) proposes a model of signed network formation, in which agents form positive links (friendships) and negative links (enmities). Using this framework, Dong et al. (2024) experimentally investigate the dynamics of alliance formation and conflict. They find evidence supporting the theoretical comparative statics in the laboratory.
There is also extensive literature, discussed in Konishi and Pan (2021), on contests among endogenously formed groups that study how group size and group sharing rules affect incentives to contribute. König et al. (2017) consider a population of groups, where each pair of groups can be in one of three states: alliance, enmity, or neutrality. They study how such a network of alliances and enmities affects the intensity of a conflict and how the latter can be reduced through different measures. Taking a dynamic perspective, Bloch (2012) provides an excellent survey of the literature on alliance formation and stability in political science and economics. Moreover, he proposes some new results by considering a unified model of conflict that encompasses most of the literature on public and private good contests. Focusing specifically on military alliances, Jackson and Nei (2015) investigate the role of networks of alliances in preventing wars. They show that international trade induces peaceful and (myopically) stable networks, which do not exist without trade. Building on their work, Cai and Kimya (2023) study farsighted stability as opposed to myopic stability and show that peaceful alliance networks can be constructed with farsighted agents. Considering conflicts as dynamic processes, Konrad and Morath (2025) study the expansion of military alliances led by two great powers at opposite ends of a geographical spectrum. In a similar vein, Konrad and Kovenock (2005) consider a tug-of-war game in which two hegemonic players repeatedly interact in conquest battles. Dziubiński et al. (2021) address a similar conquest game played on a contiguity network through which adjacent countries are connected.

See Sánchez-Pagés (2007) or Konrad and Kovenock (2009) for some examples of the incentives in alliance formation in contests.

Our empirical study on the stability of NATO is closely related to George and Sandler (2022). Like ours, their analysis accounts for the relative geographical locations of NATO allies. The formation and breakdown of alliances have also been studied experimentally (for example Ke et al., 2015; Herbst et al., 2015; Send, 2020).
Last but not least, this work draws heavily on graph-theoretical literature and, in particular, on seminal works on degenerate graphs (Lick and White, 1970) and 𝑘-cores (Seidman, 1983). A more recent work by Kong et al. (2019) presents a comprehensive review of theoretical advances in 𝑘-cores and includes some interdisciplinary applications. Similar to our work, Polanski (2024) relates 𝑘-cores to the provision of public goods in networks, and Gagnon and Goyal (2017) to markets and inequality in the presence of network externalities.
3		. Definitions and notation
3.1		. Node- and edge-weighted networks
A node- and edge-weighted (NEW) graph (network) 𝐺 with the set of nodes 𝑁 (𝐺) = {1,…, 𝑛} is represented by an 𝑛 × 𝑛 adjacency matrix 𝐺 = {𝑔_𝑖𝑘 }_𝑖,𝑘∈𝑁(𝐺), where 𝑔_𝑖𝑖 ∈ R denotes the weight of node 𝑖 and, for each pair of nodes 𝑖 ≠ 𝑘, the weight of their link is denoted by 𝑔_𝑖𝑘 ∈ R. In our setting, 𝑔_𝑖𝑘 stands for the externality that 𝑖 exerts on 𝑘, or for the amount of a network good (NG) transferred from 𝑖 to 𝑘. This externality can be either positive, negative or nonexistent whenever 𝑔_𝑖𝑘 > 0, 𝑔_𝑖𝑘 < 0 or 𝑔_𝑖𝑘 = 0, respectively. When the matrix 𝐺 is symmetric (asymmetric), the corresponding network is said to be undirected (directed). The (in-) degree of node 𝑖 ∈ 𝑁 (𝐺) is
𝛿𝑖(𝐺) ≡ ^∑ 𝑔𝑘𝑖. (1)
𝑘∈𝑁 (𝐺)
Note that this definition also includes the weight 𝑔_𝑖𝑖 of the self-link. Let the minimum degree in 𝐺 be 𝛿(𝐺) ≡ min_𝑖∈𝑁(𝐺) 𝛿_𝑖(𝐺). A network is regular when all its nodes have the same degree.
3.2		. Alliance and alliance networks
An alliance in a NEW network 𝐺 is a subset of agents from 𝑁 (𝐺) that exert a strictly positive externality on (or contribute a strictly positive amount to the NG) each other. These agents must then be interconnected by links with strictly positive weights.
Definition 1 (Alliance). A subset 𝐴 ⊆ 𝑁 (𝐺) of agents is called an alliance in network 𝐺 when 𝑔_𝑖𝑘 > 0 for all (not necessarily distinct) 𝑖, 𝑘 ∈ 𝐴.
The external impact on the member 𝑖 of the alliance 𝐴 is the accumulated externality of all agents in 𝑁 (𝐺) that are not members of this alliance,
𝜃_𝑖^𝐴(𝐺) ≡ ^∑ 𝑔𝑘𝑖. (2)
𝑘∈𝑁(𝐺)∖𝐴
When this number is negative, we refer to its absolute value as the (external) threat to the member 𝑖 ∈ 𝐴. We write 𝜃_𝑖^𝐴 omitting the reference to 𝐺 when it is clear from the context. The members of an alliance induce an alliance network. This network inherits all the links that connect different allies in 𝐺, while augmenting the self-links of allies by external impacts.
Definition 2 (Alliance network). An alliance network 𝐺^𝐴 in network 𝐺 has the set of nodes 𝐴 = 𝑁(𝐺^𝐴) ⊆ 𝑁 (𝐺) and link weights computed from the link weights in 𝐺:
𝑔_𝑖^𝐴_𝑘 = 𝑔_𝑖𝑘, 𝑔_𝑖^𝐴_𝑖 = 𝑔_𝑖𝑖 + 𝜃_𝑖^𝐴, ∀𝑖, 𝑘 ∈ 𝐴, 𝑖 ≠ 𝑘. (3)
In particular, the alliance network 𝐺^𝑁(𝐺) induced by the grand coalition 𝑁 (𝐺) is identical to 𝐺. From expressions (2) and (3), it is also clear that the degrees of all members of alliance 𝐴 are the same in network 𝐺 and in the induced alliance network 𝐺^𝐴, i.e., 𝛿_𝑖 (𝐺) = 𝛿_𝑖(𝐺^𝐴) for all 𝑖 ∈ 𝐴 and any 𝐴 ⊆ 𝑁 (𝐺). Fig. 2 extends our initial illustration by presenting the network’s structure in matrix form. Specifically, it displays the adjacency matrices corresponding to the node- and edge-weighted (NEW) network and to the induced alliance networks depicted in Fig. 1.
4		. Failure, vulnerability and resilience
In our framework, each member of an alliance 𝐴 ⊆ 𝑁 (𝐺) needs a minimum amount of the NG to survive. If this amount (i.e., this member 𝑖’s degree in 𝐺^𝐴) falls below a fixed threshold, this member “fails”, which implies stopping contributions to all other agents. The failed ally is eliminated from 𝐺^𝐴, which results in the deletion of all its links with other agents. We assume that the minimum amount of the NG needed for survival is exogenous and the same for all members, who may or may not know it.
Assumption 1. For a given failure threshold 𝜅 ∈ R and alliance 𝐴 ⊆ 𝑁 (𝐺) in network 𝐺, a member 𝑖 ∈ 𝐴 fails in the alliance network 𝐺^𝐴 ⇔ 𝛿_𝑖(𝐺^𝐴) < 𝜅. A failed member stops all her NG transfers to other members.
Moreover, the transfers to a failed ally are sunk and cannot be redirected to other agents. Therefore, not only are the externalities exerted by a failed ally lost, but also all externalities that she was receiving. Lost positive externalities of a failed ally may cause another ally to fall below the failure threshold possibly triggering a domino effect. This leads to the following definitions of vulnerability and resilience.
Definition 3 (Vulnerability/Resilience). For a given threshold 𝜅 ∈ R and an alliance 𝐴 ⊆ 𝑁 (𝐺) in network 𝐺, a member of this alliance is vulnerable when they fail in the alliance network 𝐺^𝐴 or in a subnetwork of it that results after failure of other members. Otherwise, this member is resilient. The alliance 𝐴 is vulnerable (resilient) when at least one (none) of its members is vulnerable.
An alliance (its members) is vulnerable for some values of 𝜅 but not for others. However, if the alliance 𝐴 (its member 𝑖 ∈ 𝐴) is vulnerable for a particular 𝜅, it is also vulnerable for any 𝜅^′ > 𝜅. Similarly, if 𝐴 (𝑖 ∈ 𝐴) is resilient for a particular 𝜅, it is also resilient for any 𝜅^′ < 𝜅. This property allows for a direct comparison of any two members of an alliance or members of different alliances without knowing the value of the failure threshold. We say that a member 𝑖 ∈ 𝐴 is more vulnerable (more resilient) than another member 𝑗 ∈ 𝐴 when the set of threshold values 𝜅 for which 𝑖 is vulnerable (resilient) strictly contains the set of threshold values 𝜅 for which 𝑗 is vulnerable (resilient), that is, when failure of 𝑗 implies failure of 𝑖.
It turns out that vulnerability (resilience) is closely related to the 𝑘‑core concept in graph theory introduced by Seidman (1983). He defines the 𝑘-core, 𝑐_𝑘 (𝐺), of an undirected graph 𝐺 as its largest subgraph with a minimum degree of at least 𝑘 that can be found by repeatedly deleting (in any order) all vertices of a degree lower than 𝑘. Seidman (1983) proposed a recursive pruning algorithm to find the 𝑘-core in an undirected graph 𝐺:

. Related literature
Following the seminal paper by Olson and Zeckhauser (1966), an important strand of the literature, reviewed by Sandler and Hartley (2001), focuses on the incentives for members of an alliance to contribute to a (pure) public good. Similar to other contexts, also in alliances members tend to free-ride on the resources of others. This phenomenon leads to the classical public good problem and results in a subopti- mal total contribution (e.g., Bergstrom et al., 1986; Pecorino, 2015; Goldstein, 1995; Alley, 2021). A related exploitation hypothesis states that stronger/richer members tend to bear a disproportionate share of the costs with weaker/poorer members relying on their contributions (see, e.g. Bruce, 1990; Andreoni and McGuire, 1993; Sandler, 2015). Other contributions to the discourse on alliances in the context of public goods models include Sandler and Hartley (2001), Garfinkel (2004), Walt (2009) and Scott (2010). A substantial part of this literature discusses the application of public goods models to NATO (e.g., Hilton and Anh, 1991; Plümper and Neumayer, 2015; Kim and Sandler, 2020). Although alliances are mostly seen as providers of a global public good (such as nuclear deterrence), our work is closer to studies on local public goods, that is, public goods that benefit individual agents to different degrees

𝐺 if 𝑁 (𝐺) = ∅ 𝑜𝑟 𝛿(𝐺) ≥ 𝑘, 𝑐_𝑘 (𝐺) =
𝑐_𝑘 𝐺∖{𝑖 ∈ 𝑁 (𝐺) ∶ 𝛿_𝑖 (𝐺) < 𝑘} otherwise.
(4)
This algorithm is widely used across a broad range of scientific fields including biology, network science, computer science, ecology, economics, social sciences, among others (see Kong et al. (2019) for a review). It is easy to verify that the 𝑘-core is unique and, hence, well defined, and that the 𝑘-cores are nested: 𝑐_𝑘′ (𝐺) ⊆ 𝑐_𝑘(𝐺) when 𝑘^′ ≥ 𝑘. Graph 𝐺 is called a k-core when 𝑐_𝑘 (𝐺) = 𝐺. It is worth noting that the 𝑘-core is the end step of a dynamic pruning process that takes a finite number of steps. A crucial role in our analysis is played by the concept of core numbers. The core number (coreness), 𝑟_𝑖 (𝐺), of node 𝑖 ∈ 𝑁 (𝐺) is the largest value for 𝑘 such that 𝑖 ∈ 𝑐_𝑘 (𝐺). Although Seidman (1983) defined his concepts for unweighted and undirected networks, we applied them here to NEW alliance networks. Then, we simplify the notation by writing 𝑐_𝜅^𝐴 instead of 𝑐_𝜅 (𝐺^𝐴) for the 𝜅-core in the alliance network 𝐺^𝐴, and 𝑟_𝑖^𝐴 instead of 𝑟_𝑖(𝐺^𝐴) for agent 𝑖’s core number in this network. Moreover, we define the minimum core number in the alliance network 𝐺^𝐴 as 𝑟(𝐺^𝐴) ≡ min_𝑖∈𝑁(𝐺) 𝑟_𝑖(𝐺) and often write 𝑟^𝐴 instead of 𝑟(𝐺^𝐴). It can be easily verified that 𝑟^𝐴 = 𝛿^𝐴 and 𝑟_𝑖(𝐺) ≤ 𝛿_𝑖(𝐺) for any node 𝑖 ∈ 𝑁 (𝐺).
By way of illustration, consider the network in Fig. 3. Suppose we wish to identify the 3-core. The procedure is as follows: first, identify all nodes with degree less than 3 and remove them along with their

Fig. 2. Alliances within a NEW network. Note: 𝐴_1 = {𝑁_1 , 𝑁_2 } (blue) and 𝐴_2 = {𝑁_3, 𝑁_4 } (green) are two disjoint alliances. The number next to each connection indicate intensity and sign of the externality between agents 𝑖 and 𝑗 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. 3-core and degrees.

associated links. Then, in the residual network, again remove all nodes whose degree is less than 3. This process is repeated iteratively until no such nodes remain. The set of nodes that persists through all iterations constitutes the 3-core of the network. In Fig. 3, only a single iteration is required to identify the 3-core of the network. However, in general, finding a 𝑘-core may require several iterations. It is worth noting that the initial network supports at most a 3-core; any 𝑘-core with 𝑘 > 3 is empty, even though nodes 𝑁_3 , 𝑁_4 and 𝑁_5 have degrees greater than 3.
First, we show that the resilience and vulnerability of an alliance – and of its members – are closely related to the concepts of 𝑘‑cores and core numbers in the induced alliance network.
Proposition 1. Given threshold 𝜅 ∈ R and alliance 𝐴 ⊆ 𝑁 (𝐺) in network 𝐺:
•		Member 𝑖 ∈ 𝐴 is resilient (vulnerable) ⇔ 𝑖 ∈ 𝑐_𝜅^𝐴 ⇔ 𝑟_𝑖^𝐴 ≥ 𝜅 (𝑖 ∉ 𝑐_𝜅^𝐴 ⇔ 𝑟_𝑖^𝐴 < 𝜅).
•		Alliance 𝐴 is resilient (vulnerable) ⇔ 𝐴 ⊆ 𝑐_𝜅^𝐴 ⇔ 𝑟^𝐴 ≥ 𝜅 (𝐴 ⊈ 𝑐_𝜅^𝐴 ⇔ 𝑟^𝐴 < 𝜅).
All proofs are relegated to Appendix A.
Member 𝑖 ∈ 𝐴 is resilient when it belongs to the 𝜅-core of the alliance network 𝐺^𝐴 or, equivalently, when its core number 𝑟_𝑖^𝐴 in 𝐺^𝐴 weakly exceeds 𝜅. In this case, there is no sequence of members of 𝐴, including 𝑖, that can be sequentially removed from 𝐺^𝐴 so that, when removed, each has a degree less than 𝜅. While a node’s degree in an alliance network is the stock of effective protection it receives against external threats, its core number is the resilience floor in this network after accounting for knock-on failures. In fact, given an exogenous failure threshold 𝜅, the relationship between resilience and the 𝜅-core allows us to focus on the final outcome of the pruning process described in Expression (4). This is convenient because the concept of stability – the main focus of this paper as defined in Section 5 – is only meaningful once the pruning process ends.
From the resilience of all members follows the resilience of the entire alliance. In other words, alliance 𝐴 in network 𝐺 is resilient when its minimum core number (degree) in 𝐺^𝐴 is greater or equal to 𝜅. Notice, that even though the external threat (negative externalities) to agent 𝑖 determines only the degree of this agent in 𝐺 and in 𝐺^𝐴, it can affect the core numbers of all members in 𝐺^𝐴 . Moreover, we can order allies with respect to their resilience without knowing the failure threshold as shown in the next corollary.
Corollary 1. Given alliance 𝐴 ⊆ 𝑁 (𝐺), its member 𝑖 ∈ 𝐴 is more vulnerable (or more resilient) than another member 𝑘 ∈ 𝐴 ⇔ 𝑟_𝑖^𝐴 < 𝑟_𝑘^𝐴 (𝑟_𝑖^𝐴 > 𝑟_𝑘^𝐴).
The core number 𝑟_𝑖^𝐴 of member 𝑖 ∈ 𝐴 indicates a (relative) ability of this agent to withstand failure. Therefore, we often equate the resilience of a member with its core number. Similarly, we refer to the minimum core number 𝑟^𝐴 in the alliance 𝐴 as resilience of this alliance. Fig. D.1 in the Appendix shows an alliance that faces two adversaries and the core numbers of its members.
A key question in this context is which network structure best enhances an alliance’s resilience. To address this meaningfully, we must constrain the sum of the members’ degrees within the network. Without this limitation, the alliance’s resilience would be unbounded if members’ degrees could be arbitrarily increased.
Proposition 2. Assume that the alliance network 𝐺^𝐴 is regular – that is, 𝛿_𝑖 (𝐺^𝐴) = 𝛿_𝑗 (𝐺^𝐴) for all 𝑖, 𝑗 ∈ 𝐴 ⊆ 𝑁 (𝐺). Given constant external impacts on the members of alliance 𝐴, there is no other alliance network 𝐺^̃ ^𝐴 with ^∑_𝑘∈𝐴 𝛿_𝑘 (𝐺^̃ ^𝐴) ≤ ^∑_𝑘∈𝐴 𝛿_𝑘(𝐺^𝐴) that increases the resilience of this alliance.
Recall that the resilience of an alliance is determined by the minimum core number (or degree) in its alliance network. If there is a member with a degree higher than this minimum, the resilience of the alliance can be enhanced by taking a small amount of the NG from this member and redistributing it among the allies with the smallest degrees. Consequently, an alliance achieves maximum resilience only when all its members have equal degrees. Economically, equalizing degrees implements a risk-sharing rule that maximizes the alliance’s minimum resilience – leveling protection where it is most scarce.
5		. Stability of alliances
In this section, we introduce a strategic role for the agents in 𝐺 by assuming that the externality (NG transfer) 𝑔_𝑖𝑘 in the NEW network 𝐺 is the product of 𝑖’s individual contribution 𝑐_𝑖 ∈ R_+ and a spillover parameter 𝑡_𝑖𝑘 ∈ R, that is, 𝑔_𝑖𝑘 = 𝑐_𝑖𝑡_𝑖𝑘. Because we do not impose any restrictions on the spillover parameters, 𝑔_𝑖𝑘 may be negative, in which case the NG is taken away from agent 𝑘 (i.e., agent 𝑘’s NG decreases). In this scenario, the matrix 𝐺 = 𝐷(𝐜) ⋅ 𝑇, where 𝑇 = {𝑡_𝑖𝑘}_𝑖,𝑘∈𝑁(𝐺) is the matrix of spillover parameters, 𝐜 = {𝑐_𝑖}_𝑖∈𝑁(𝐺) is the vector of individual contributions, and 𝐷(𝐜) is the diagonal matrix with the elements of 𝐜 on its main diagonal. The (in-)degree of node 𝑖 ∈ 𝑁 (𝐺) defined in (1) boils down to,
𝛿_𝑖 (𝐺) = ^∑ 𝑔_𝑘𝑖 = ^∑ 𝑐_𝑘𝑡_𝑘𝑖.	(5)
𝑘∈𝑁 (𝐺)	𝑘∈𝑁(𝐺)
Note that 𝑐_𝑖 and 𝑡_𝑖𝑖 must be strictly positive for each alliance member because 𝑐_𝑖 ∈ R_+ and 𝑔_𝑖𝑖 = 𝑐_𝑖𝑡_𝑖𝑖 > 0 by Definition 1 of an alliance. Then, it follows for the spillover parameters 𝑡_𝑖𝑘 = ^𝑔𝑖𝑘 > 0 as 𝑔_𝑖𝑘 > 0 for all 𝑖𝑘	𝑐_𝑖	𝑖𝑘
𝑖, 𝑘 ∈ 𝐴 by the same definition.
To enhance realism, we assume that the failure threshold is unknown, while all agents share the belief that it follows a continuous cumulative distribution function (CDF) Φ(⋅). By Proposition 1, member 𝑖 ∈ 𝐴 ⊆ 𝑁 (𝐺) is resilient if and only if its core number in the alliance network 𝐺^𝐴 satisfies 𝑟_𝑖^𝐴 ≥ 𝜅. Hence,
𝑃 𝑟(𝑖 is resilient) = 𝑃 𝑟(𝜅 ≤ 𝑟_𝑖^𝐴) = Φ(𝑟_𝑖^𝐴). (6)
We also note that 𝑟_𝑖^𝐴 strictly increases with 𝑖’s contribution 𝑐_𝑖 . This is because a marginal change 𝑑𝑐_𝑖 > 0 increases the degree of each member of 𝐴 by at least min_𝑘∈𝐴 𝑡_𝑖𝑘⋅𝑑𝑐_𝑖 > 0 and, then, 𝑖’s core number 𝑟_𝑖^𝐴 by at least the same amount. Therefore, we often express the core number of agent 𝑖 and the probability that 𝑖 is resilient as a function of 𝑖’s contribution, that is, we write 𝑟_𝑖^𝐴(𝑐_𝑖) and Φ(𝑟_𝑖^𝐴(𝑐_𝑖)), respectively.
In this setting, each alliance member maximizes its state-dependent expected utility (see, e.g., Debreu, 1959; Karni, 1985) by choosing a costly contribution that increases the probability of being resilient, which, ceteris paribus, yields higher utility than being vulnerable. On the other hand, a higher contribution lowers the utility in either state through its direct cost. Specifically, each member 𝑖 ∈ 𝐴 in network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 maximizes,
𝜋𝑖(𝐺^𝐴, 𝑐𝑖) ≡ Φ(𝑟_𝑖^𝐴(𝑐𝑖)) ⋅ ~~𝑢~~𝑖(𝐺^𝐴, 𝑐𝑖) + (1 - Φ(𝑟_𝑖^𝐴(𝑐𝑖))) ⋅ 𝑢_𝑖(𝐺^𝐴, 𝑐𝑖), (7) where ~~𝑢~~_𝑖 (𝑢_𝑖) is 𝑖^′𝑠 utility when resilient (vulnerable) weighted with the corresponding probability. Eq. (7) highlights a central trade-off: costly contributions increase the probability of resilience on the one hand, while reducing utility on the other. Specifically, we impose the following structure on the resilience probability and on the utility functions.
Assumption 2.
𝜅 - 𝜅
1.		Φ(𝜅) = max 0, min	, 1 with -∞ < 𝜅 < ~~𝜅~~ < ∞.
𝜅 - 𝜅
2.		~~𝑢~~_𝑖(𝐺^𝐴,𝑐_𝑖) = 𝑢_𝑖(𝐺^𝐴, 𝑐_𝑖) + 𝛽, 𝛽 > 0, and -∞ < ^𝑑~~𝑢~~𝑖 ^(_𝑑^𝐺_𝑐^𝐴,𝑐𝑖) = ^𝑑𝑢𝑖(_𝑑^𝐺_𝑐^𝐴,𝑐𝑖) < 0, ∀𝑐_𝑖 ≥ 0, 𝑖 ∈ 𝐴.	^𝑖	^𝑖
Firstly, we assume that the threshold 𝜅 is uniformly distributed between some finite 𝜅 and ~~𝜅~~, we use the length of the support interval (~~𝜅~~-𝜅) > 0 as a measure of uncertainty. Secondly, we require that the utility of a resilient agent is higher than the utility of a vulnerable agent by a constant amount of 𝛽 > 0. We will refer to the parameter 𝛽 as resilience premium. Naturally, we also assume that an agent’s utility diminishes in her contribution (i.e., contributions are costly).
For a given network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 , we consider the incentives of the members of alliance 𝐴 ⊆ 𝑁 (𝐺) to contribute or redistribute the NG. We assume that each member 𝑖 ∈ 𝐴 observes 𝐺, as well as her contribution 𝑐_𝑖 and her spillover parameters (𝑡_𝑖𝑘)_𝑘∈𝐴, but not necessarily 𝑐_𝑗 and (𝑡_𝑗𝑘)_𝑘∈𝐴 for 𝑗 ≠ 𝑖. Given the strict monotonicity of 𝑟_𝑖^𝐴(𝑐_𝑖), we define ~~𝑐~~_𝑖^𝐴 ≥ 0 as the minimum contribution of the member 𝑖 ∈ 𝐴 such that 𝑟_𝑖^𝐴 (~~𝑐~~_𝑖^𝐴) ≥ ~~𝜅~~ for a given profile of contributions of the other members. As ~~𝜅~~ is finite by Assumption 2, so is ~~𝑐~~ _𝑖^𝐴 for each 𝑖 ∈ 𝐴.
First, we focus on contributions and assume that each ally 𝑖 ∈ 𝐴 can change her contribution 𝑐_𝑖 but cannot modify her spillover parameters (𝑡_𝑖𝑘)_𝑘∈𝐴. The fixed spillover matrix among allies may result from a formal treaty that spells out their mutual obligations, or reflects their spatial locations. For example, in our empirical exercise in Section 6, we use geographic distances among allies to construct a fixed spillover matrix.
Definition 4 (Contribution Stability). An alliance 𝐴 in the network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 is contribution stable (CS) when no member 𝑖 ∈ 𝐴 can increase her payoff 𝜋_𝑖 (𝐺^𝐴, 𝑐_𝑖) by unilaterally changing her contribution 𝑐_𝑖.
This definition links the stability of an alliance to the payoff maximization of its members. In many cases, alliances also allow for flexibility in managing transfers between members. In the following definition, we allow each member 𝑖 ∈ 𝐴 to choose her contribution
𝑐_𝑖 and to modify her spillover parameters (𝑡_𝑖𝑘)_𝑘∈𝐴 . However, we will require that each member 𝑖 ∈ 𝐴 can only redistribute her spillover parameters among her allies without changing their total value. Otherwise, 𝑖 could always increase her payoff by increasing all 𝑡_𝑖𝑘. This leads to the following definition.
Definition 5 (Contribution-Transfer Stability). An alliance 𝐴 in the network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 is contribution-transfer stable (CTS) when no member 𝑖 ∈ 𝐴 can increase her payoff 𝜋_𝑖(𝐺^𝐴, 𝑐_𝑖) by unilaterally changing her contribution 𝑐_𝑖 and/or by redistributing her spillover parameters (𝑡_𝑖𝑘)_𝑘∈𝐴 while keeping ^∑_𝑘∈𝐴 𝑡_𝑖𝑘 constant.
Notice that our two notions of stability are closely related. On the one hand, CTS allows alliance members to change unilaterally their contributions 𝑐_𝑖 and/or the spillover parameters. Accordingly, a member may (i) adjust 𝑐_𝑖 and reallocate spillovers, (ii) keep 𝑐_𝑖 fixed and reallocate spillovers, or (iii) adjust 𝑐_𝑖 while keeping the original spillover parameters. Case (iii) coincides with our definition of CS (Definition 4). It therefore follows that any CTS alliance is also CS. When there is no need to distinguish between the different stability concepts, we will simply call an alliance stable.
In the following results, we characterize stable alliances in terms of the core numbers of their members and other topological properties of the corresponding alliance networks. These results hold under Assumption 2 regarding the probability of resilience and the utility functions.
Proposition 3. For each member 𝑖 ∈ 𝐴 of a CS alliance 𝐴 in network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 holds 𝑟_𝑖^𝐴 ∈ [𝜅, ~~𝜅~~] and, in particular, 𝑟_𝑖^𝐴 = ~~𝜅~~ if
𝛽	𝑑~~𝑢~~_𝑖(𝐺^𝐴, 𝑐_𝑖)∕𝑑𝑐_𝑖 >-
𝜅 - 𝜅	𝑑𝑟_𝑖^𝐴(𝑐_𝑖)∕𝑑𝑐_𝑖

∀𝑐𝑖 ∈ [0, ~~𝑐~~_𝑖^𝐴].

(8)

The range of core numbers in a CS alliance is bounded from below by 𝜅 and from above by ~~𝜅~~ . This follows from the simple observation that any member 𝑖 ∈ 𝐴 with 𝑟_𝑖^𝐴 > ~~𝜅~~ could decrease her costly contribution 𝑐_𝑖 and, therefore, her core number 𝑟_𝑖^𝐴 until the latter reaches 𝑟_𝑖^𝐴 = ~~𝜅~~ (or 𝑐_𝑖 = 0). As this would not affect 𝑖’s resilience (but possibly that of other members of 𝐴), 𝑖’s utility defined in Eq. (7) unambiguously increases. Similarly, a member 𝑖 with 𝑟_𝑖^𝐴 ≤ 𝜅 could reduce her contribution to zero without affecting her resilience. Note that the contribution 𝑐_𝑖 must be strictly positive for each member 𝑖 ∈ 𝐴 as 𝑐_𝑖 = 0 implies 𝑔_𝑖𝑘 = 𝑐_𝑖𝑡_𝑖𝑘 = 0 for all 𝑘 ∈ 𝐴, which is incompatible with Definition 1 of an alliance.
Moreover, when the resilience premium is large relative to uncertainty, each member optimally pushes their core number to the upper bound ~~𝜅~~, yielding knife-edge fragility consistent with public-goods freeriding around a common threshold. Therefore, members ensure that they do not fail for any feasible failure threshold. However, if any of these members marginally decreases her contribution, and consequently all the core numbers, all members will fail with a positive probability. In this sense, a CS alliance is fragile. This fragility results because the common core number ~~𝜅~~ implies that all members of 𝐴 can be successively removed from 𝐺^𝐴 so that when removed, each has a degree at most ~~𝜅~~.
We also note that there is typically a multiplicity of contribution profiles that can sustain a CS alliance. Consequently, CS alliances can suffer from an exploitation problem with some members bearing a disproportionate share of the costs and other members relying on their contributions.
Proposition 3 also holds for CTS alliances as the latter are clearly contribution stable. However, when the ratio of resilience premium to uncertainty is sufficiently high, we can strengthen this result for members of a contribution-transfer stable alliance. We show next that these members not only have the same core number, but also that this number is equal to their common degree.
Proposition 4. For each member 𝑖 ∈ 𝐴 of a CTS alliance 𝐴 in network 𝐺 = 𝐷(𝐜) ⋅ 𝑇 holds 𝑟_𝑖^𝐴 ∈ [𝜅, ~~𝜅~~] and, in particular, 𝑟_𝑖^𝐴 = 𝛿_𝑖^𝐴 = ~~𝜅~~ if
_𝜅 _-𝛽 _𝜅 _>-𝑑~~𝑢~~_𝑑𝑖(_𝑟𝐺_𝑖𝐴(𝐴_𝑐,_𝑖)𝑐_∕𝑖)_𝑑∕_𝑐𝑑_𝑖𝑐𝑖_, _∀𝑐𝑖∈[0,~~𝑐~~𝑖𝐴_]

(9)

Hence, when the ratio of the resilience premium to uncertainty is sufficiently high, the alliance network of a CTS alliance is regular. Allowing spillover reallocation then aligns incentives and yields regular networks – a result with a direct welfare interpretation: by Proposition 2, an alliance attains its maximum resilience precisely when its alliance network is regular. This is a notable case in which stability coincides with optimality. As with contribution-stable alliances, there is typically a multiplicity of contribution profiles and spillover matrices that can sustain a CTS alliance.
6		. Empirical study: NATO vs Russian Federation
In the empirical part of the paper, we apply the network approach to study the resilience and stability of the North Atlantic Treaty Organization (NATO). We consider the network good in this case as security or deterrence, calculated from the military resources available to allies and their adversaries, where the resources of the latter detract from the security of the former. In this context, our key Assumption 1 means that all allies need a common (but typically unknown) level of deterrence to counterbalance the threat of aggression by an adversary. The importance of sustaining a robust (and costly) level of deterrence for preserving peace has been recognized from ancient Roman generals to modern military strategists.

For example, the Roman general Publius Flavius Vegetius Renatus is credited with the maxim (adapted from his “De Re Militarī”): Si vis pacem para bellum (if you want peace, prepare for war). Similarly, Carl von Clausewitz, the 19th century Prussian general and military theorist, wrote in his political-military analysis “Vom Kriege”: “Weil die Politik den Krieg als zynisches Mittel einsetzt, kann es nurmehr Abschreckung geben oder den Kampf auf Leben und Tod.” (Because politics employs war as a cynical means, there can only be deterrence or the struggle to the death.)

Specifically, we consider a NEW network with nodes comprising the NATO members in 2022 and, for simplicity, just one NATO adver- sary—the Russian Federation (RF). Although NATO allies face threats and challenges from different state and non-state actors, the RF is “the most significant and direct threat to their security” according to their Madrid Summit Declaration from June 2022. For the construction of the relevant network 𝐺 = 𝐷(𝐜) ⋅ 𝑇, we use military expenditures (ME) and geographical proximity of the countries. Specifically, the contribution 𝑐_𝑖 is set to the military expenditure of country 𝑖 in 2022 (in millions of constant 2015 US dollars), while the element 𝑡_𝑖𝑘 of the matrix 𝑇 is calculated from the circle distance 𝑑_𝑖𝑘 (in 1000 km) between the capitals of countries 𝑖 and 𝑘. We use the Gaussian kernel exp (-𝑑_𝑖^2_𝑘) to measure their proximity, which can be thought of as a normalization of the Euclidean distance to values lying in the unit interval. Specifically, for any pair of NATO countries 𝑖 and 𝑘, we set the spillover parameters 𝑡_𝑖𝑘 = 𝑡_𝑘𝑖 = exp (-𝑑_𝑖^2_𝑘) and interpret it as the share of the ME of one country committed to the defense of the other country in the event of aggression. In particular, 𝑡_𝑖𝑖 = 1 as a country would use 100% of its resources to defend itself. Similarly, the geographic proximity between the RF and a NATO country proxies for the threat posed by these countries to each other. In this case, the Gaussian kernel enters the matrix 𝑇 with a negative sign as it detracts from the security of both these countries. We note that the construction of network 𝐺 may be plausible when a potential aggression is directed against a single country but less so when multiple countries can be attacked. In the latter case, a “double counting” of military resources of an attacked country 𝑖 occurs, as they would be entirely used for self-defense (𝑡_𝑖𝑖 = 1) but also to help an attacked ally 𝑘 (𝑡_𝑖𝑘 = exp (-𝑑_𝑖^2_𝑘) > 0). To account for a scenario with multiple attacks, one could normalize each row 𝑖 of the matrix 𝑇 so that ^∑_𝑘∈𝐴 𝑡_𝑖𝑘 = 1.

Data sources: ME of NATO members: https://www.nato.int/cps/en/natohq/ news_197050.htm. ME of the RF: https://milex.sipri.org/sipri. Capital cities distances: http://ksgleditsch.com/data-5.html.

Fig. 4. Capital city distances to Moscow (km) vs core numbers in the alliance network (left panel) and the percentages of GDP spent on defense in 2022 (right panel) for the European NATO members.

Fig. 4 relates the resilience (core numbers) and the percentage of the GDP spent on ME for the European NATO members against the distance (km) of their capital cities to Moscow (see Table C.1 in the Appendix). Unsurprisingly, the left panel of Fig. 4 shows that the most vulnerable members of the Alliance are the Baltic states: Estonia, Latvia, and Lithuania, which not only have the lowest core numbers, but are also geographically closest to the RF. Estonia, in particular, is NATO’s “weakest link” with the lowest degree and core number, 𝛿_𝐸𝐸 (𝐺) = 𝑟_𝐸𝐸 (𝐺^𝐴) = 7626.29, which is then equal to the resilience of the whole Alliance. At the other extreme, France, the UK, the Netherlands, Luxembourg, and Belgium are the most resilient European NATO members. Interestingly, these countries would be a good candidate for a contribution stable alliance as their core numbers are (almost) identical. However, as a whole, NATO is neither contribution stable nor contribution-transfer stable when we assume vanishing uncertainty (see Propositions 3 and 4). Contribution stability would require that all members align along a horizontal line in the left panel of Fig. 4. Contribution-transfer stability would imply that the degrees and core numbers of all members collapse to a single point in the degree-core number plane.

We omit Fig. 4 the American NATO members as their core numbers are far higher than those of their European counterparts, e.g., 𝛿_𝑈𝑆 = 715, 779, 𝑟_𝑈𝑆 = 703, 363.

Unlike the resilience of countries that tends to increase with their distance from the capital of the RF, the share of their GDP spent on defense decreases with this distance (right panel in Fig. 4). Especially, former Soviet Baltic republics and Warsaw Pact members have been investing heavily in their defense, even before the Russian invasion of Ukraine in February 2022.

Greece appears to be an outlier in the latter diagram, as it is mostly concerned about a conflict with its direct neighbor rather than a potential threat from Russia. See also: “Falling stars: Europe could be torn apart by new divisions”, The Economist, 9th of January 2025.

6.1		. How to strengthen an alliance?
Here, we focus on measures that increase resilience in the context of our model and empirical study. Firstly, an obvious step is to increase the ME of (some) members. In fact, this was the objective of the NATO Heads of State and Government agreement in 2014 to commit 2% of their national GDP to defense spending. It is clear in our framework that an increase in ME by any member increases the degrees of all members and, hence, the resilience of the Alliance. However, the impact varies greatly between allies, given the construction of the spillover matrix from geographical distances. For example, any increase in ME by the United States has only negligible impact on the core number of the “weakest link” Estonia, which determines the resilience of the Alliance. In our model, this is due to the distance of 6980 km that separates Washington DC from Tallinn, leading to a vanishing spillover parameter exp (-6.98^2) ≈ 0. On the other hand, an increase of 1% in Germany’s ME increases Estonia’s degree and its core number by 0.34% due to the relative proximity of Berlin to Tallinn (exp (-1.045^2) ≈ 0.34). Therefore, increases in ME targeted at some pivotal countries are more likely to improve the resilience of the Alliance than a common goal on defense spending.
Secondly, resilience can be improved by changing the spillover parameters. The use of geographical distances in our empirical exercise to capture military spillovers is obviously a heroic simplification. Arguably, the effective distance between two locations, individuals, or countries depends on the relevant context. For example, economic distance can be measured by the transport cost (e.g., Isard, 1956), while social distance can be measured by the amount and type of interaction between individuals or groups (e.g., the social distance scale of Bogardus, 1926). In the context of our empirical study, the effective distances indicate the ease of moving military hardware and personnel between allies or the difficulty of moving them between an ally and an adversary. While keeping members’ contributions constant, the resilience of an alliance can be improved by shortening the effective distances among allies, while lengthening this distance between a member and an adversary. Practically, this manifests itself in some recent actions taken by NATO members. For example, Finland, Sweden, and Norway agreed in June 2024 to establish a military transport corridor in the Nordic region by upgrading bridges, railroads, roads, and communication infrastructure, while Poland unveiled details of fortification of its eastern borders in May 2024.
Finally, we consider how an accession of new members affects resilience. In our model, the resilience of all old members and of a new member (weakly) increases after the accession of the latter (considered originally as a single-country alliance). Interestingly, this does not imply that the resilience of the enlarged alliance also increases. As an example, consider the possible accession of the Republic of Georgia. Due to its proximity to the RF, low ME, and its remoteness from the other NATO members (except Turkey), this country would become the “weakest link” in the Alliance with a core number that is below that of the current weakest link Estonia. As the core number of the weakest member determines the resilience of an alliance, NATO would be less resilient after this enlargement.
7		. Conclusions
This work complements the literature on public good models of alliances. An alliance in our model is embedded in a node- and edge- weighted network, and we relate the resilience of its members to the core numbers of nodes in the subnetwork induced by them. Moreover, we analyze the trade-off between resilience and costly contributions and/or redistribution of the spillovers, and propose two concepts of strategic stability that address this tension. We show how these concepts relate to the topological properties of the induced alliance network. In an empirical application, we analyze the resilience and stability of the North Atlantic Alliance and discuss the impact of different measures aimed at strengthening it.
Future research could build on this framework in several directions. A promising avenue is the generalization of the concepts of 𝑘-core and core numbers to weighted 𝑘-cores and weighted core numbers (see
Appendix B), allowing heterogeneous failure thresholds. Such a generalization would better capture asymmetries in members’ abilities to withstand external threats, thereby offering a more nuanced measure of resilience in NEW networks. Extending the analysis to dynamic or evolving alliance networks – where members can adjust their contributions, spillovers, or alliances over time – would provide insights into the long-term stability of cooperative structures under uncertain shocks. Another fruitful line of inquiry would be to incorporate multilayer or multiplex network architectures, reflecting the coexistence of military, economic, and political ties among allies. Moreover, exploring empirical applications beyond military alliances, such as corporate consortia, climate coalitions, or cybersecurity partnerships, could test the robustness and versatility of the theoretical results. Finally, developing computational and experimental approaches to validate the model’s predictions would enhance its empirical relevance and help identify the mechanisms that promote or hinder resilience in complex alliance systems.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix A . Proofs
Proof 1. Proposition 1: By the Definition 3, a member 𝑖 ∈ 𝐴 ⊆ 𝑁 (𝐺) in network 𝐺 is vulnerable for a given failure threshold 𝜅, when 𝑖 fails in 𝐺, or in a network that ensues from 𝐺 after failure of other agents who are all members of 𝐴. This can only happen when there is a sequence of members of 𝐴, including 𝑖, that can be deleted iteratively from 𝐺 so that when deleted each has a degree less than 𝜅 . Such a sequence exists in 𝐺 if and only if it also exists in the alliance network 𝐺^𝐴 because the degree of a member 𝑖 ∈ 𝐴 is the same in 𝐺 and in 𝐺^𝐴, and also in the respective subnetworks of 𝐺 and 𝐺^𝐴 that result after removing other members of 𝐴. The existence of such a sequence is equivalent to 𝑖 ∉ 𝑐_𝜅 (𝐺) and 𝑖 ∉ 𝑐_𝜅 (𝐺^𝐴) by the formula (4).
On the other hand, a member 𝑖 ∈ 𝐴 is resilient for a given failure threshold 𝜅 when such a sequence does not exist, which is equivalent to 𝑖 ∈ 𝑐_𝜅 (𝐺^𝐴) but not necessarily to 𝑖 ∈ 𝑐_𝜅 (𝐺). The situation, where a resilient agent 𝑖 ∉ 𝑐_𝜅 (𝐺) arises when there is a sequence of agents that includes 𝑖 and non-members of 𝐴 that can be deleted iteratively from 𝐺 so that when deleted each agent has a degree less than 𝜅.
The core number 𝑟_𝑖(𝐺^𝐴) is the largest value of 𝜅 such that 𝑖 ∈ 𝑐_𝑘(𝐺^𝐴). Hence, 𝑖 ∈ 𝑐_𝜅^𝐴 ⇔ 𝑟_𝑖^𝐴 ≥ 𝜅 because 𝑘-cores are nested, i.e., 𝑐_𝑘^𝐴_′ ⊆ 𝑐_𝑘^𝐴 when 𝑘^′ ≥ 𝑘. Conversely, it holds that 𝑖 ∉ 𝑐_𝜅^𝐴 ⇔ 𝑟_𝑖^𝐴 < 𝜅.
The proof of the claim that an alliance 𝐴 is resilient (vulnerable) in network 𝐺 ⇔ 𝐴 ⊆ 𝑐_𝜅^𝐴 ⇔ 𝑟^𝐴 ≥ 𝜅 (𝐴 ⊈ 𝑐_𝜅^𝐴 ⇔ 𝑟^𝐴 < 𝜅) follows directly from the Definition 3 which states that an alliance is vulnerable (resilient) when at least one (none) of its members is vulnerable.
Proof 2. Corollary 1: From (the proof of) Proposition 1 and the fact that k-cores are nested it follows that a member 𝑘 ∈ 𝐴 is resilient ⇔ 𝑘 ∈ 𝑐_𝜅^𝐴 for all 𝜅 ≤ 𝑟_𝑘^𝐴 . Therefore, 𝑟_𝑖^𝐴 < 𝑟_𝑗^𝐴 holds if and only if the set of threshold values 𝜅 for which 𝑗 is resilient strictly contains the set of threshold values 𝜅 for which 𝑖 is resilient and, hence, to 𝑗 being more resilient than 𝑖. In particular, 𝑗 but not 𝑖 is resilient when 𝜅 ∈ (𝑟_𝑖^𝐴, 𝑟_𝑗^𝐴]. By a symmetric argument, 𝑖 is more vulnerable than 𝑗 when 𝑟_𝑖^𝐴 < 𝑟_𝑗^𝐴 .
Proof 3. Proposition 2: For the sake of contradiction, assume that alliance network 𝐺^𝐴 maximizes the resilience 𝑟^𝐴 = 𝛿(𝐺^𝐴) of alliance 𝐴 ⊆ 𝑁 (𝐺) and 𝛿_𝑖(𝐺^𝐴) = 𝛿_𝑖 (𝐺) > 𝑟^𝐴 for some member 𝑖 ∈ 𝐴. We obtain a new network 𝐺^̂ from 𝐺 by changing this member’s links only,
𝑔̂ = 𝑔 - 𝜖 > 0, 𝑔̂ = 𝑔 + ^𝜖 > 𝑔 , (A.1) 𝑠
for some 𝜖 > 0, and for any member 𝑘 ≠ 𝑖 among the 𝑠 members with the degree 𝛿_𝑘(𝐺) = 𝛿_𝑘(𝐺^𝐴) = 𝑟^𝐴 (recall that 𝑔_𝑖𝑖 > 0 for each member 𝑖 of an alliance, so there exists an 𝜖 > 0 ∶ 𝑔_𝑖𝑖 - 𝜖 > 0). If 𝜖 > 0 is sufficiently small, the minimum degree in 𝐺^̂ is greater than the minimum degree in 𝐺 while ^∑_𝑘∈𝐴 𝛿_𝑘(𝐺) = ^∑_𝑘∈𝐴 𝛿_𝑘 (𝐺^̂). As the degrees of alliance members are the same in 𝐺^̂ and in the induced alliance network 𝐺^̂ ^𝐴 and the external impacts do not change, we conclude that the resilience of alliance 𝐴 in 𝐺^̂ ^𝐴, which is equal to the minimum degree in this network, is greater than in 𝐺^𝐴 . At the same time, ^∑_𝑘∈𝐴 𝛿_𝑘 (𝐺^̂ ^𝐴) = ^∑_𝑘∈𝐴 𝛿_𝑘(𝐺^𝐴), which contradicts our initial assumption that network 𝐺^𝐴 with 𝛿_𝑖 (𝐺^𝐴) > 𝑟^𝐴 maximizes 𝐴’s resilience.
Proof 4. Proposition 3:
For any member 𝑖 of a CS alliance 𝐴 in the underlying network 𝐺 = 𝐷(𝐜) ⋅ 𝑇, the following holds:
1.		The core number 𝑟_𝑖^𝐴 in the alliance network 𝐺^𝐴 is a continuous, strictly increasing function of the contribution 𝑐_𝑖 . This is because a marginal change 𝑑𝑐_𝑖 > 0 increases the degree of any member of 𝐴 by at least min_𝑘∈𝐴 𝑡_𝑖𝑘 ⋅ 𝑑𝑐_𝑖 > 0 and then 𝑖’s core number 𝑟_𝑖^𝐴 by at 𝑑𝑟^𝐴(𝑐_𝑖)
least the same amount. Hence, ^𝑖	≥ min_𝑘∈𝐴 𝑡_𝑖𝑘 > 0.
𝑑 𝑐_𝑖
2.		The contribution 𝑐_𝑖 > 0 as 𝑐_𝑖 = 0 ⇒ 𝑔_𝑖𝑘 = 𝑐_𝑖𝑡_𝑖𝑘 = 0 for all 𝑘 ∈ 𝐴 ⇒ 𝐴 is not an alliance.
3.		The core number 𝑟_𝑖^𝐴 (𝑐_𝑖) ∈ [𝜅, ~~𝜅~~]. Otherwise, 𝑖 could decrease 𝑟_𝑖^𝐴(𝑐_𝑖) by reducing 𝑐_𝑖 until it reaches 𝑟_𝑖^𝐴(0) or 𝑟_𝑖^𝐴(𝑐_𝑖) = ~~𝜅~~. This would not change 𝑖’s resilience, but would lower her contribution cost. Note that 𝑟_𝑖^𝐴(0) < ~~𝜅~~ as otherwise 𝑐_𝑖 = 0, which contradicts the fact that 𝑖 belongs to an alliance.
4.		From 1-3, it follows for 𝑖’s contribution 𝑐_𝑖 that 0 ≤ 𝑐 _𝑖^𝐴 ≤ 𝑐_𝑖 ≤ ~~𝑐~~ _𝑖^𝐴 < ∞, where 𝑐_𝑖^𝐴 (~~𝑐~~_𝑖^𝐴) is the minimum contribution of 𝑖 ∈ 𝐴 such that 𝑟_𝑖^𝐴 (𝑐_𝑖^𝐴) ≥ 𝜅 (𝑟_𝑖^𝐴 (~~𝑐~~_𝑖^𝐴) ≥ ~~𝜅~~) for the given contribution profile of the other members. Note that ~~𝑐~~_𝑖^𝐴 < ∞ because ~~𝜅~~ < ∞.
Now, using the Assumption 2, we can write the expected utility function (7) in the form,
𝜋𝑖(𝐺^𝐴,𝑐𝑖)=^𝑟𝑖𝐴(𝑐𝑖)-𝜅⋅~~𝑢~~𝑖(𝐺^𝐴,𝑐𝑖)+~~^𝜅~~-𝑟𝑖𝐴(𝑐𝑖) 𝜅 - 𝜅	𝜅 - 𝜅
with its derivative w.r.t. 𝑐_𝑖 boiling down to,
𝑑𝜋𝑖(𝐺^𝐴, 𝑐𝑖)	𝑑𝑟𝑖^𝐴(𝑐𝑖) 𝛽	𝑑~~𝑢~~𝑖(𝐺^𝐴, 𝑐𝑖) =+ 𝑑𝑐_𝑖	𝑑𝑐_𝑖 𝜅 - 𝜅	𝑑𝑐_𝑖
This derivative is strictly positive when
𝛽	𝑑~~𝑢~~_𝑖(𝐺^𝐴, 𝑐_𝑖)∕𝑑𝑐_𝑖 >-
𝜅 - 𝜅	𝑑𝑟_𝑖^𝐴(𝑐_𝑖)∕𝑑𝑐_𝑖

⋅ {~~𝑢~~𝑖(𝐺^𝐴,𝑐𝑖)-𝛽},

(A.2)

(A.4)

Note that the expression on the r.h.s. is finite and strictly positive because 0 < 𝑑𝑟_𝑖^𝐴(𝑐_𝑖)∕𝑑𝑐_𝑖 < ∞ and -∞ < 𝑑~~𝑢~~_𝑖(𝐺^𝐴, 𝑐_𝑖)∕𝑑𝑐_𝑖 < 0. Therefore, when the ratio of the resilience premium 𝛽 to uncertainty (~~𝜅~~ - 𝜅) is sufficiently high, each member 𝑖 ∈ 𝐴 chooses her contribution at 𝑐_𝑖 = ~~𝑐~~ _𝑖^𝐴 ensuring 𝑟_𝑖^𝐴 ≥ 𝑘. Moreover, by Definition 1 of an alliance, ~~𝑐~~ _𝑖^𝐴 must be strictly positive for each member 𝑖 ∈ 𝐴, which implies 𝑟_𝑖^𝐴 = 𝑘.
Proof 5. Proposition 4: As a CTS alliance 𝐴 in the network 𝐺 is also CS it must satisfy Proposition 3. In particular, 𝑟_𝑖^𝐴 = 𝑟^𝐴 = ~~𝜅~~ for each member 𝑖 ∈ 𝐴 when the ratio of resilience premium to uncertainty is sufficiently high. For the sake of contradiction, assume in this case that the alliance network 𝐺^𝐴 is not regular, that is, 𝛿_𝑖^𝐴 > 𝛿^𝐴 = 𝑟^𝐴 = ~~𝜅~~ for some 𝑖 ∈ 𝐴 (recall that the minimum degree is equal to the minimum core number in 𝐴). Then 𝑖 could decrease 𝑡_𝑖𝑖 marginally to 𝑡^′_𝑖𝑖 (recall that
(𝑡_𝑖𝑖 - 𝑡_𝑖^′_𝑖)
𝑡_𝑖𝑖 > 0 in an alliance) and set 𝑡^′ = 𝑡_𝑖𝑘 +	^𝑖𝑖 for each member ^𝑖𝑖	𝑖𝑘	^𝑖	𝑠
𝑘 among the 𝑠 members with the degree equal to ~~𝜅~~, while keeping her own degree 𝛿_𝑖^𝐴 strictly greater than 𝛿^𝐴 = 𝑟^𝐴 . Then, the minimum degree in 𝐴 and, hence, the minimum core number rises above ~~𝜅~~. Then, 𝑖 could
decrease 𝑐_𝑖 until 𝑟_𝑖^𝐴 = ~~𝜅~~, which would restore her original resilience but at a lower contribution cost.
Appendix B . Weighted k-core
Here, we present a possible generalization of the definitions of 𝑘-core and core numbers to better capture asymmetries in members’ abilities to withstand external threats. Given a vector of node weights 𝝎 = (𝑤_1 ,…, 𝑤_𝑛), we define the weighted k-core in the network 𝐺 with 𝑛 nodes by the recursive formula,

𝑐_𝑘^𝑤(𝐺) =

𝐺	if𝑁(𝐺) = ∅ or 𝛿_𝑖(𝐺) ≥ 𝑘 ⋅ 𝑤_𝑖, ∀𝑖 ∈ 𝑁(𝐺),
𝑐_𝑘^𝑤 ^(𝐺∖{𝑖 ∈ 𝑁 (𝐺) ∶ 𝛿_𝑖(𝐺) < 𝑘 ⋅ 𝑤_𝑖 }^)	otherwise.
(B.1)
Note that the k-core is a special case with 𝝎 = (1,…, 1).
Moreover, we can define the weighted core number 𝑟_𝑖^𝑤(𝐺) of node 𝑖 ∈ 𝑁 (𝐺) as the largest value of 𝑘 such that 𝑖 ∈ 𝑐_𝑘^𝑤(𝐺). In analogy to the k-core, it holds that 𝑟_𝑖^𝑤(𝐺) ≤ 𝑤_𝑖𝛿_𝑖(𝐺).

Appendix C . Tables
Table C.1
Country information for NATO members and the Russian Federation. Degrees and core numbers are computed from circle distances between capital cities and military expenditures in 2022 (expressed in millions of 2015 US dollars). Column 𝑑_𝑅𝑈. contains the circle distances (in km) from Moscow to the capital cities of NATO countries.

Appendix D . Figures

Country | Code | Degree | Core Nr | ME | ME/GDP | 𝑑𝑅𝑈.
Albania | AL | 61,531 | 46,401 | 173 | 1.62 | 2031
Belgium | BE | 188,788 | 122,929 | 6106 | 1.18 | 2294
Bulgaria | BG | 52,803 | 34,541 | 982 | 1.67 | 1782
Canada | CA | 433,388 | 433,388 | 21,187 | 1.27 | 7215
Croatia | HR | 108,220 | 73,707 | 1136 | 2.03 | 1890
Czechia | CZ | 146,278 | 109,091 | 2892 | 1.33 | 1654
Denmark | DK | 130,806 | 109,091 | 4872 | 1.39 | 1592
Estonia | EE | 7626 | 7626 | 616 | 2.34 | 898
France | FR | 176,350 | 122,928 | 49,554 | 1.9 | 2530
Germany | DE | 150,503 | 114,982 | 54,613 | 1.44 | 1621
Greece | GR | 38,025 | 29,497 | 8221 | 3.76 | 2170
Hungary | HU | 97,079 | 73,707 | 2892 | 1.55 | 1563
Italy | IT | 83,894 | 73,708 | 30,139 | 1.54 | 2399
Latvia | LV | 24,744 | 22,980 | 672 | 2.1 | 854
Lithuania | LT | 21,524 | 21,071 | 1284 | 2.36 | 791
Luxembourg | LU | 188,494 | 121,961 | 404 | 0.58 | 2195
North Macedonia | MK | 59,884 | 42,980 | 184 | 1.78 | 1869
Montenegro | ME | 71,652 | 57,530 | 68 | 1.75 | 1936
Norway | NO | 80,556 | 73,708 | 6521 | 1.55 | 1677
Netherlands | NL | 183,370 | 122,929 | 12,867 | 1.65 | 2173
Poland | PL | 77,587 | 71,955 | 14,066 | 2.42 | 1138
Portugal | PT | 28,138 | 28,138 | 3223 | 1.44 | 3948
Romania | RO | 43,010 | 29,497 | 4005 | 1.99 | 1477
Slovak Republic | SK | 117,940 | 80,423 | 1855 | 2 | 1611
Slovenia | SL | 125,427 | 83,257 | 692 | 1.22 | 1909
Spain | ES | 60,035 | 57,529 | 15,347 | 1.01 | 3452
Türkiye | TR | 24,420 | 24,344 | 16,203 | 1.22 | 1812
United Kingdom | GB | 171,960 | 122,929 | 73,357 | 2.12 | 2512
United States | US | 715,779 | 703,363 | 703,362 | 3.47 | 7873
Russia | RU | – | – | 77,551 | 5.96 | 0
Fig. D.1. An alliance with two adversaries. Note: Network 𝐺 contains the alliance 𝐴 = {1, 2, 3} and its adversaries {4, 5}. Members of the alliance are connected by solid blue lines, while negative impacts of their adversaries are indicated by red dotted lines. Assume 𝑔_𝑖𝑘 = 𝑔_𝑖𝑖 = ^1 for all 𝑖, 𝑘 ∈ 𝐴, and 𝑔_41 = 𝑔_42 = 𝑔_51 = - ^1 . Then, the resilience of the members is indicated by their core numbers: agent 3 is more resilient than agent 2, who is more resilient than agent 1. Only for the most vulnerable agent 1, the core number and the degree are both equal to zero. Hence, the alliance 𝐴 is resilient for all 𝜅 ≤ 0. When the uncertainty vanishes and the highest possible failure threshold ~~𝜅~~ = 0, then the alliance 𝐴 is not CS because, for example, agent 3 could profitably reduce her contribution to 𝑐_3 = 0. This would ensure 𝑟_3^𝐴 ≥ ~~𝜅~~ = 0 and increase her payoff (7). In fact, 𝐴 would not be even an alliance in this case as membership requires 𝑐_3 > 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Data availability
Data will be made available on request.
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^2 For a subgraph 𝐺^′ of graph 𝐺 it holds that 𝑁(𝐺^′) ⊆ 𝑁 (𝐺) and 𝐸(𝐺^′) ⊆ 𝐸(𝐺) where 𝑖𝑗 ∈ 𝐸(𝐺^′) implies that 𝑖, 𝑗 ∈ 𝑁 (𝐺^′).