Resource Availability in the Social Cloud: An Economics Perspective
RResource Availability in the Social Cloud: An Economics Perspective
Pramod C. Mane a , Nagarajan Krishnamurthy b and Kapil Ahuja c a Department of Computer Science and Engineering, National Institute of Technology Raipur, Raipur, India. b Operations Management and Quantitative Techniques, Indian Institute of Management Indore, Indore, India. c Department of Computer Science and Engineering, Indian Institute of Technology Indore, Indore, India.
A B S T R A C T
This paper focuses on social cloud formation, where agents are involved in a closeness-based conditional resource sharing and build their resource sharing network themselves.The objectives of this paper are: (1) to investigate the impact of agents’ decisions oflink addition and deletion on their local and global resource availability, (2) to analyzespillover effects in terms of the impact of link addition between a pair of agents on others’utility, (3) to study the role of agents’ closeness in determining what type of spillovereffects these agents experience in the network, and (4) to model the choices of agents thatsuggest with whom they want to add links in the social cloud. The findings include thefollowing. Firstly, agents’ decision of link addition (deletion) increases (decreases) theirlocal resource availability. However, these observations do not hold in the case of globalresource availability. Secondly, in a connected network, agents experience either positiveor negative spillover effect and there is no case with no spillover effects. Agents observeno spillover effects if and only if the network is disconnected and consists of more thantwo components (sub-networks). Furthermore, if there is no change in the closeness of anagent (not involved in link addition) due to a newly added link, then the agent experiencesnegative spillover effect. Although an increase in the closeness of agents is necessary inorder to experience positive spillover effects, the condition is not sufficient. By focusingon parameters such as closeness and shortest distances, we provide conditions under whichagents choose to add links so as to maximise their resource availability.
1. Introduction
The idea of the Social Cloud (Chard et al., 2012, 2010,2015) has received much attention in the last few years.These systems take advantage of social connections to of-fer a secure and reliable way of resource sharing betweenagents. Researchers believe that exploiting social connec-tions (in the form of social networks) can aid dealing withvarious issues like resource sharing policies and mecha-nisms (Caton et al., 2014; Zhang and van der Schaar,2013), trust (Caton et al., 2012; Mohaisen et al., 2014),and incentivising resource sharing (Punceva et al., 2015;Haas et al., 2013). In this context, social connections areeither exogenous or endogenous. Exogenous social con-nections are those social connections which are extractedfrom an online social network (for example, Facebook) inthe form of a social graph. Whereas endogenous socialconnections are those social connections which are con-structed in the context of social cloud application (for ex-ample, BuddyBackup ), where agents are decision makerswho build their resource sharing connections.Recent research trends in social cloud have led to twodirections. One trend focuses on the role of exogenous so-cial connections in defining quality of services (for exam-ple, data availability, reliability), and trust in social cloud.For example, Zuo and Iamnitchi (2016) show that a smallset of friends play a crucial role in deterring the quality [email protected] (P.C. Mane); [email protected] (N.Krishnamurthy); [email protected] (K. Ahuja) of services and workload balance. Another research trend(Mane et al., 2020b, 2019, 2014, 2020a; Moscibroda et al.,2011) focuses on endogenous social connections in termsof resource sharing network formation, the stability andefficiency analysis of these networks, and analysis of exter-nalities in these resource sharing networks. For example,Mane et al. (2020b) study social storage cloud formationin a strategic setting, where self-interested agents builda storage resource sharing network for maximizing theirrespective utilities. They show that for the given degree-based utility of agents in this social cloud setting, agentsalways form the 𝜂 -regular network, where each agent has 𝜂 neighbors.However, the present literature on social cloud hasleft behind several aspects, for example, 1) the impact oflink addition and deletion on resource availability of thosewho are involved in the link addition and deletion, 2) theimpact of link addition between a pair of agents on otheragents’ utilities, 3) choice modelling that captures withwhom self-interested agents add new social connectionsand delete existing social connections. This paper aimsto fill these gaps.Following are the objectives of this study. The firstobjective is to study the impact of agents’ decision of linkaddition and deletion on their local as well as global re-source availability. We show that for the utility modelproposed in (Mane et al., 2014), the local resource avail-ability of a pair of agents increases by adding a link anddecreases by deleting the link between them. However,in the case of global resource availability, the same is nottrue. The second objective is to analyze agents’ local re- P. C. Mane et al.:
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Page 1 of 11 a r X i v : . [ c s . G T ] J a n esource Availability in the Social Cloud: An Economics Perspective source availability in the context of their local connections.We find that, in the case of link addition, agents never ob-serve an increase in their local resource availability fromtheir neighbors. The opposite is true in the case of linkdeletion. The third objective is to analyze the impact oflink addition between a pair of agents on the utility of oth-ers. This aspect can be outlined in terms of the spillovereffect. Mane et al. (2019) provide a necessary and suffi-cient condition under which an agent experiences positiveor negative spillover effect. In Mane et al. (2020a), theauthors follow an empirical approach to study the role ofnetwork structure and size in determining spillover. In thispaper, we throw attention on the role of agents’ closenessin determining the kind of spillover effects they experiencedue to a newly added link in the network. We show thatan increase in closeness is necessary, but not sufficient,for an agent to experience positive spillover effect. Weshow that in the two diameter network, agents always ex-perience negative spillover effect. The fourth objective isto understand the preferences of agents in link addition.That is, with whom agents prefer to add links in a net-work. For this, we provide a set of conditions by takingthe distances and closeness of agents into consideration.
2. The Social Cloud Model
For the sake of completeness, we first describe thesocial cloud model presented in Mane et al. (2014). Asocial cloud can be seen as a socially-aware resource shar-ing network 𝔤 = { 𝐀 , 𝐋 } that consists of a non-empty set 𝐀 of 𝑛 agents and a set 𝐋 of 𝓁 undirected links connect-ing these agents. One can view the set 𝐋 as a platformthat facilitates agents to share their computing resources,such as disk space and computing power, with others, andsearch for resources shared by others. An undirected link ⟨ 𝑖𝑗 ⟩ ∈ 𝐋 represents a direct communication channel be-tween agents 𝑖 and 𝑗 in 𝔤 . In other words, agents 𝑖 and 𝑗 are neighbours in 𝔤 . The set 𝜂 𝑖 ( 𝔤 ) represents the numberof neighbors of agent 𝑖 in 𝔤 .A path in 𝔤 connecting agents 𝑖 and 𝑖 𝑛 is a sequence ofdistinct agents ( 𝑖 , 𝑖 , ..., 𝑖 𝑛 ) such that ⟨ 𝑖 𝑖 ⟩ , ⟨ 𝑖 𝑖 ⟩ , … , ⟨ 𝑖 𝑛 −1 𝑖 𝑛 ⟩ ∈ 𝐋 . The length of a path is the number of links thatthe path contains. A shortest path between agent 𝑖 andagent 𝑗 is a path between 𝑖 and 𝑗 that has the least length,among all paths between 𝑖 and 𝑗 . The distance between 𝑖 and 𝑗 , 𝑑 𝑖𝑗 ( 𝔤 ) , is the length of the shortest path betweenthem. We say, 𝑖 and 𝑗 are 𝑑 𝑖𝑗 ( 𝔤 ) hops away from eachother. The diameter, 𝔤 , of 𝔤 is the maximum distancebetween any pair of agents. The radius of 𝔤 is the min-imum distance between any pair of agents. A network 𝔤 is connected if there exists at least one path betweenany pair of agents; otherwise it is disconnected. A dis-connected network 𝔤 is a collection of two or moredis-joint components (sub-networks) 𝔤 ( 𝔠 ) , 𝔤 ( 𝔠 ) ⋯ 𝔤 ( 𝔠 𝑧 ) suchthat 𝔠 ∪ 𝔠 ∪ ⋯ ∪ 𝔠 𝑧 = 𝐀 , and 𝔠 𝑥 ∩ 𝔠 𝑦 = ∅ for all 𝑥, 𝑦 ∈ {1 , , ⋯ , 𝑧 } , 𝑥 ≠ 𝑦 , such that any pair of agents 𝑖 and 𝑗 are connected if and only if they are elements of the same set 𝔠 𝑥 .A network 𝔤 evolves when agents perform two oper-ations, namely, (1) link addition, where agents 𝑖, 𝑗 in 𝔤 , ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 , mutually add the link ⟨ 𝑖𝑗 ⟩ , resulting in the net-work 𝔤 + ⟨ 𝑖𝑗 ⟩ , and (2) link deletion, where agents 𝑘, 𝑙 in 𝔤 , ⟨ 𝑘𝑙 ⟩ ∈ 𝔤 , unilaterally or mutually delete ⟨ 𝑘𝑙 ⟩ to give 𝔤 − ⟨ 𝑘𝑙 ⟩ .Henceforth, we refer to a socially-aware resource shar-ing network as a resource sharing network, and wheneverwe refer to 𝔤 , we mean a resource sharing network. In 𝔤 , agents who have underutilised resources share thesame with other agents who want to use the resourcesto accomplish their computational tasks. For example,an agent may need to backup its data and may use thestorage space shared by another agent in 𝔤 . Now, we statethe basic assumptions on which the social cloud modelstands. Assumption 1.
Agents in 𝔤 share the same kind of resource.We denote the resource by 𝔯 . Assumption 2.
In the prevailing 𝔤 , an agent has under-utilised resource 𝔯 with probability 𝑝 and needs to performa computational task using the resource with probability 𝑞 . Assumption 3.
An agent plays either the role of a resourceprovider or that of a resource consumer, with probabilities 𝑝 (1 − 𝑞 ) and 𝑞 (1 − 𝑝 ) , respectively. Assumption 4.
Each agent has global information, that is,each agent is aware of the network structure 𝔤 and the pre-vailing resource sharing situation in 𝔤 . In 𝔤 , agents perform closeness based resource sharing(Chard et al., 2012). For example, agents could limitresource sharing with those agents who are close to them.The notion of how close an agent is to all other agents canbe captured by the harmonic centrality measure, discussedin (Boldi and Vigna, 2014; Opsahl et al., 2010; Marchioriand Latora, 2000), defined as follows: Φ 𝑖 ( 𝔤 ) = ∑ 𝑗 ∈ 𝔤⧵ { 𝑖 } 𝑑 𝑖𝑗 ( 𝔤 ) (1) Φ 𝑖 ( 𝔤 ) is called the closeness of 𝑖 in 𝔤 . Harmonic centralityhandles ∞ smoothly, and hence, this centrality measuredeals with disconnected networks too.In 𝔤 , an agent 𝑗 ∈ 𝔤 (who acts as a resource provider)computes a probability distribution on all agents for thepurpose of allocating the resource to agent 𝑖 ∈ 𝔤 (whoacts as a resource consumer), as given below: 𝛼 𝑖𝑗 ( 𝔤 ) = 𝑝 (1 − 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑗 ∈ 𝔤⧵ { 𝑖 } 1 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑝 (1 − 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑖 ( 𝔤 ) . (2) In other words, 𝛼 𝑖𝑗 ( 𝔤 ) is the probability that agent 𝑖 willobtain the resource from agent 𝑗 in 𝔤 . P. C. Mane et al.:
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Remark 1. If 𝑑 𝑖𝑗 ( 𝔤 ) = ∞ then 𝛼 𝑖𝑗 ( 𝔤 ) = 0 ( = 𝛼 𝑗𝑖 ( 𝔤 ) ). Asagents 𝑖 and 𝑗 are disconnected in 𝔤 their chances of obtain-ing the resource from each other is nil. The probability that agent 𝑖 obtains the resource fromat least one agent in 𝔤 is as follows: 𝛾 𝑖 ( 𝔤 ) = 1 − ∏ 𝑗 ∈ 𝔤⧵ { 𝑖 } (1 − 𝛼 𝑖𝑗 ( 𝔤 )) . (3) Definition 1.
We call 𝛼 𝑖𝑗 ( 𝔤 ) the local resource availabilityof 𝑖 from 𝑗 in 𝔤 , and 𝛾 𝑖 ( 𝔤 ) , the global resource availability of 𝑖 in 𝔤 . An agent’s chance of obtaining the resource from an-other agent is determined by, first, the distance betweenthe agent (who wants the resource) and the other agent(who may provide the resource), and second, the otheragent’s closeness. Hence, it is important to get to knowhow a newly added link affects the distance between pairsof agents and their closeness.
Remark 2.
Suppose 𝑘, 𝑙 are distinct agents in 𝔤 such that ⟨ 𝑘𝑙 ⟩ ∉ 𝔤 . Then,either 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , for all 𝑖 ∈ 𝔤 ⧵ { 𝑘, 𝑙 } , 𝑗 ∈ 𝔤 ,or 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) for some 𝑖 ∈ 𝔤 ⧵ { 𝑘, 𝑙 } , 𝑗 ∈ 𝔤 ,and 𝑑 𝑖𝑗 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) for others.Similarly,either Φ 𝑖 ( 𝔤 ) = Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , for all 𝑖 ∈ 𝔤 ⧵ { 𝑘, 𝑙 } ,or Φ 𝑖 ( 𝔤 ) = Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , for some 𝑖 ∈ 𝔤 ⧵ { 𝑘, 𝑙 } and Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , for others. Lemma 1.
Suppose 𝑖, 𝑗, 𝑘, 𝑙 are distinct agents in 𝔤 such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 and ⟨ 𝑘𝑙 ⟩ ∉ 𝔤 . If 𝑑 𝑖𝑗 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) then Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) and Φ 𝑗 ( 𝔤 ) < Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) .Proof. 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) ≤ 𝑑 𝑖𝑗 ( 𝔤 )−1 . On adding ⟨ 𝑘𝑙 ⟩ , even if thedistances of all other agents remain unchanged, the closenessof both 𝑖 and 𝑗 will increase by at least 𝑑 𝑖𝑗 ( 𝔤 )−1 − 𝑑 𝑖𝑗 ( 𝔤 ) , or 𝑑 𝑖𝑗 ( 𝔤 )( 𝑑 𝑖𝑗 ( 𝔤 )−1) . Due to Remark 2 and Lemma 1, we study an agent’sprobability of obtaining the resource in 𝔤 by taking thefollowing cases into the consideration. 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) and Φ 𝑖 ( 𝔤 ) = Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) and Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . 𝑑 𝑖𝑗 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) and Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . The utility of agent 𝑖 in 𝔤 is given by a function 𝑢 𝑖 ∶ → ℝ + , where is the set of all possible networks on 𝑛 agents. 𝑢 ∶ → ℝ 𝑛 gives the the vector (profile) of utilityfunctions 𝑢 = ( 𝑢 , ..., 𝑢 𝑛 ) . In other words, each possibleresource sharing network structure ( 𝔤 ⊆ ) generates apayoff for each agent.In 𝔤 , an agent 𝑖 gains benefit 𝜃 𝑖 and 𝜉 𝑖 by accomplish-ing a computational task and by providing their resourceto others, respectively. An agent 𝑖 ∈ 𝔤 gains 𝜉 𝑖 with prob-ability 𝑝 (1 − 𝑞 ) . An agent 𝑖 ’s expected benefit 𝜃 𝑖 depends on whether the agent has its own resource or depends onthe resource availability in 𝔤 . Note that, with probability 𝑝𝑞 , the agent is self-reliant and does not depend on theother agents in 𝔤 . However, with probability 𝑞 (1 − 𝑝 ) , theagent wants the resource but does not have it and, hence,seeks the resource from other agents in 𝔤 .An agent seeks resources in 𝔤 by maintaining directlinks. Each agent 𝑖 pays cost 𝜍 𝑖 for each of their directlinks in 𝔤 . Thus, 𝑖 incurs a total cost of 𝜂 𝑖 ( 𝔤 ) × 𝜍 𝑖 in 𝔤 .The cost 𝜍 𝑖 can be interpreted as the effort or time thatagent 𝑖 spends to maintain an active connection (or link).We consider that agents 𝑖 and 𝑗 share the link additioncost equally, that is, 𝜍 = 𝜍 𝑖 + 𝜍 𝑗 .Then, for a given resource sharing network 𝔤 , the ex-pected payoff of agent 𝑖 is 𝑢 𝑖 ( 𝔤 ) = 𝑝 (1 − 𝑞 ) 𝜉 𝑖 + 𝑞 [ 𝑝 + (1 − 𝑝 ) 𝛾 𝑖 ( 𝔤 )] 𝜃 𝑖 − 𝜍𝜂 𝑖 ( 𝔤 ) . (4)
3. Resource Availability
In this section, we discuss our results on the local re-source availability of agent 𝑖 from another agent 𝑗 in thenetwork 𝔤 , that is, the probability that agent 𝑖 will obtainthe resource from agent 𝑗 in 𝔤 . In this section, we discuss the effect of link additionand deletion on the local resource availability.
Lemma 2.
Suppose 𝔤 and 𝔤 are resource sharing net-works, and suppose 𝑖, 𝑗 ∈ 𝔤 ∩ 𝔤 . Then 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑗 ( 𝔤 ) ifand only if 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) .Proof. 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑗 ( 𝔤 ) , if and only if 𝑝 (1− 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 )( 𝑑𝑖𝑗 ( 𝔤 + ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑𝑗𝑘 ( 𝔤 ) > 𝑝 (1− 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 )( 𝑑𝑖𝑗 ( 𝔤 + ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑𝑗𝑘 ( 𝔤 ) ,if and only if 𝑑 𝑖𝑗 ( 𝔤 )( 𝑑 𝑖𝑗 ( 𝔤 ) + ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) ) > 𝑑 𝑖𝑗 ( 𝔤 )( 𝑑 𝑖𝑗 ( 𝔤 ) + ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) ) ,if and only if 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤 ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) . Proposition 1.
Suppose 𝑖 and 𝑗 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . Then, 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > 𝛼 𝑖𝑗 ( 𝔤 ) .Proof. Owing to Lemma 2, it suffices to show that 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 𝑑 𝑗𝑘 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ∑ 𝑘 ∈ 𝔤 + ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 𝑑 𝑗𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) (5) P. C. Mane et al.:
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Page 3 of 11esource Availability in the Social Cloud: An Economics Perspective (a) Network 𝔤 (b) Network 𝔤 + ⟨ 𝑖𝑗 ⟩ Figure 1:
Link addition and local resource availability
Note that 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 1 and 𝑑 𝑖𝑗 ( 𝔤 ) ∈ {2 , , … , } .It suffices to check that Inequality (5) holds in the fol-lowing three cases.1. 𝑑 𝑖𝑗 ( 𝔤 ) = ∞ . That is, 𝑖 and 𝑗 are not connected in 𝔤 .Inequality (5) clearly holds.2. ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) = ∑ 𝑘 ∈ 𝔤 + ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . That is, addi-tion of link ⟨ 𝑖𝑗 ⟩ does not change the distance between 𝑗 and any other agent 𝑘 , except 𝑖 . It is easy to see thatInequality (5) holds in this case too.3. ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) < ∑ 𝑘 ∈ 𝔤 + ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . This happenswhen the addition of link ⟨ 𝑖𝑗 ⟩ changes the distance be-tween 𝑗 and at least one 𝑘 (besides 𝑖 ). Shortest pathsbetween 𝑗 and every such 𝑘 in 𝔤 + ⟨ 𝑖𝑗 ⟩ are, clearly,shorter than those in 𝔤 . To show that the Left HandSide of Inequality (5) is greater than the Right HandSide, we show that it holds for the worst possible caseof the above inequality (given 𝑛 ). This happens when 𝑑 𝑖𝑗 ( 𝔤 ) is the minimum possible, that is, , and 𝔤 , with 𝑛 agents, is as shown in Figure 1(a). Agent 𝑖 is con-nected with 𝑛 − 2 agents, agent 𝑗 has a single neigh-bour 𝑘 , and 𝑘 is an intermediary between 𝑖 and 𝑗 . In 𝔤 , Φ 𝑗 ( 𝔤 ) = 𝑛 +36 and 𝑑 𝑖𝑗 ( 𝔤 ) = 2 . Suppose agents 𝑖 and 𝑗 add a direct link, resulting in the network 𝔤 + ⟨ 𝑖𝑗 ⟩ asshown in Figure 1(b). Here, Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝑛 +12 . Wehave, ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) = 𝑛 and ∑ 𝑘 ∈ 𝔤 + ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝑛 −12 . The Left Hand Side of Inequality (5) = 𝑛 ,which is clearly greater than its Right Hand Side, 𝑛 −12 . The above result shows that agents always improvetheir local resource availabilities by adding new resourcesharing connections. We, now, show that an agent’s deci-sion to delete an existing resource sharing connection de-creases the local resource availability of the pair of agentsfrom each other.
Proposition 2.
Suppose 𝑖 and 𝑗 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . Then, 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑗 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . Proof. Owing to Lemma 2, it suffices to show that 𝑑 𝑖𝑗 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) ∑ 𝑘 ∈ 𝔤 − ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 𝑑 𝑗𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) > 𝑑 𝑖𝑗 ( 𝔤 ) ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 𝑑 𝑗𝑘 ( 𝔤 ) (6) We know that 𝑑 𝑖𝑗 ( 𝔤 ) = 1 , 𝑑 𝑖𝑗 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) ∈ {2 , , …} , and ≤ ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) ≤ 𝑛 − 2 , 0 when 𝑗 is isolated and 𝑛 − 2 when 𝑗 is connected to all 𝑘 .It suffices to check that Inequality (6) holds in the fol-lowing three cases.1. 𝑑 𝑖𝑗 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) = ∞ . That is, ⟨ 𝑖𝑗 ⟩ is the only path be-tween 𝑖 and 𝑗 in 𝔤 . Inequality (6), clearly, holds in thiscase.2. ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) = ∑ 𝑘 ∈ 𝔤 − ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) .That is, dele-tion of link ⟨ 𝑖𝑗 ⟩ does not change the distance between 𝑗 and any other agent 𝑘 except 𝑖 . It is easy to see thatInequality (6) holds in this case too.3. ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) > ∑ 𝑘 ∈ 𝔤 − ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) .To show that the Left Hand Side of Inequality (6) isgreater than the Right Hand Side, we show that it holdsfor the worst possible case of the above inequality (given 𝑛 ). Refer Figure 2(a). Here, ∑ 𝑘 ∈ 𝔤⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 ) = 𝑛 −12 . In 𝔤 − ⟨ 𝑖𝑗 ⟩ , 𝑖 has links with all agents except 𝑗 . Agent 𝑗 has a link to at least one other agent, for otherwisewe have Case 1. 𝑑 𝑖𝑗 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) = 2 , the greatest lowerbound on the distance, as ⟨ 𝑖𝑗 ⟩ has been deleted. ReferFigure 2(b). Here, ∑ 𝑘 ∈ 𝔤 − ⟨ 𝑖𝑗 ⟩ ⧵ { 𝑖,𝑗 } 1 𝑑 𝑗𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) = 𝑛 .Inequality (6) holds in this case too. Theorem 1.
Suppose 𝑖 and 𝑗 are distinct agents in 𝔤 . Then,the link ⟨ 𝑖𝑗 ⟩ is always strictly beneficial to both 𝑖 and 𝑗 , withrespect to local resource availabilities from each other.Proof. Follows from Propositions 1 and 2.
In the previous section, we saw that an agent 𝑖 im-proves its local resource availability from another agent P. C. Mane et al.:
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Page 4 of 11esource Availability in the Social Cloud: An Economics Perspective (a) Network 𝔤 (b) Network 𝔤 − ⟨ 𝑖𝑗 ⟩ Figure 2:
Link deletion and local resource availability 𝑗 by adding link ⟨ 𝑖𝑗 ⟩ . However, this newly added linkdecreases agent 𝑖 ’s local resource availability from its ex-isting neighbors 𝑘 who are at least three hops away from 𝑗 . For neighbors 𝑘 who are less than three hops awayfrom 𝑗 , agent 𝑖 ’s local resource availability from them re-mains the same. Similarly, while an agent’s local resourceavailability from another agent decreases if their existinglink is deleted, the agent’s local resource availability fromits existing neighbors who are at least three hops awayincreases, and remains the same for the other neighbors.We prove these results below. Proposition 3.
Suppose 𝑖, 𝑗 and 𝑘 are distinct agents in 𝔤 such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 and 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) . Then, the following hold: If 𝑘 ∈ 𝜂 𝑗 ( 𝔤 ) , then 𝛼 𝑖𝑘 ( 𝔤 ) = 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) = 2 , then 𝛼 𝑖𝑘 ( 𝔤 ) = 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) > , then 𝛼 𝑖𝑘 ( 𝔤 ) > 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) .Proof. Suppose 𝑑 𝑘𝑗 ( 𝔤 ) > . Then, Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > Φ 𝑘 ( 𝔤 ) , as 𝑑 𝑘𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 2 < 𝑑 𝑘𝑗 ( 𝔤 ) , the new shortest path between 𝑘 and 𝑗 , in 𝔤 + ⟨ 𝑖𝑗 ⟩ , being the path with the two links ⟨ 𝑘𝑖 ⟩ and ⟨ 𝑖𝑗 ⟩ . Therefore, from Eq. (2), 𝛼 𝑖𝑘 ( 𝔤 ) = 𝑝 (1− 𝑞 )Φ 𝑘 ( 𝔤 ) > 𝑝 (1− 𝑞 )Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , proving .If 𝑑 𝑘𝑗 ( 𝔤 ) ≤ , then 𝑑 𝑘𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝑑 𝑘𝑗 ( 𝔤 ) and, hence, Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = Φ 𝑘 ( 𝔤 ) , implying 𝛼 𝑖𝑘 ( 𝔤 ) = 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ,proving and . Similar results hold for agent 𝑘 ’s resource availabilityfrom agent 𝑖 too, as stated in the following corollary. Corollary 1.
Suppose 𝑖, 𝑗 and 𝑘 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 and 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) . Then, the following hold: If 𝑘 ∈ 𝜂 𝑗 ( 𝔤 ) , then 𝛼 𝑘𝑖 ( 𝔤 ) = 𝛼 𝑘𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) = 2 , then 𝛼 𝑘𝑖 ( 𝔤 ) = 𝛼 𝑘𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) > , then 𝛼 𝑘𝑖 ( 𝔤 ) > 𝛼 𝑘𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . We have the following results on the aggregate localresource availability, aggregated over all neighbors of 𝑖 in 𝔤 . Corollary 2.
Suppose 𝔤 is a two-diameter network where 𝑖 and 𝑗 are distinct agents such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . Then, ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) 𝛼 𝑖𝑘 ( 𝔤 ) = ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ⧵ { 𝑗 } 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . Corollary 3.
Suppose 𝑖 and 𝑗 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . If there exists at least one agent 𝑘 in 𝔤 , differentfrom 𝑖 , which is more than two hops away from 𝑗 , then ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) 𝛼 𝑖𝑘 ( 𝔤 ) > ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ⧵ { 𝑗 } 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . We, now, see that an agent’s local connections in-crease its aggregate local resource availability, when theagent deletes a link with one of the neighbors who is atleast three hops away from the other neighbors.
Proposition 4.
Suppose 𝑖, 𝑗 and 𝑘 are distinct agents in 𝔤 such that ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 and 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) . Then, the following hold: If 𝑘 ∈ 𝜂 𝑗 ( 𝔤 ) , then 𝛼 𝑖𝑘 ( 𝔤 ) = 𝛼 𝑖𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) = 2 , then 𝛼 𝑖𝑘 ( 𝔤 ) = 𝛼 𝑖𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) > , then 𝛼 𝑖𝑘 ( 𝔤 ) < 𝛼 𝑖𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) .Proof. The result can be proved in lines similar to the proofof Proposition 3.
Corollary 4.
Suppose 𝑖, 𝑗 and 𝑘 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 and 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) . Then, the following hold: If 𝑘 ∈ 𝜂 𝑗 ( 𝔤 ) , then 𝛼 𝑘𝑖 ( 𝔤 ) = 𝛼 𝑘𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) = 2 , then 𝛼 𝑘𝑖 ( 𝔤 ) = 𝛼 𝑘𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . If 𝑑 𝑘𝑗 ( 𝔤 ) > , then 𝛼 𝑘𝑖 ( 𝔤 ) > 𝛼 𝑘𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . Corollary 5.
Suppose 𝔤 is a two-diameter network where 𝑖 and 𝑗 are distinct agents. Suppose ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . Then, ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) ⧵ { 𝑗 } 𝛼 𝑖𝑘 ( 𝔤 ) = ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) 𝛼 𝑖𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . Corollary 6.
Suppose 𝑖 and 𝑗 are distinct agents in 𝔤 suchthat ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . If there exists at least one agent 𝑘 in 𝔤 , differentfrom 𝑖 , which is more than two hops away from 𝑗 , then ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) ⧵ { 𝑗 } 𝛼 𝑖𝑘 ( 𝔤 ) < ∏ 𝑘 ∈ 𝜂 𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) 𝛼 𝑖𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) . 𝔤 is a two diameter network if ≤ 𝑑 𝑖𝑗 ( 𝔤 ) ≤ , for all 𝑖, 𝑗 ∈ 𝔤 . P. C. Mane et al.:
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Page 5 of 11esource Availability in the Social Cloud: An Economics Perspective (a) Network 𝔤 = 𝔤 ′ + ⟨ 𝑖𝑗 ⟩ (b) Network 𝔤 ′ = 𝔤 − ⟨ 𝑖𝑗 ⟩ Figure 3:
Link addition and deletion
Theorem 2.
Suppose 𝑖 , 𝑗 and 𝑘 are distinct agents in 𝔤 , suchthat 𝑘 ∈ 𝜂 𝑖 ( 𝔤 ) . Then, the link ⟨ 𝑖𝑗 ⟩ is always strictly beneficialto 𝑖 as well as 𝑘 , with respect to local resource availabilitiesfrom each other, if and only if 𝑑 𝑘𝑗 > .Proof. Follows from Propositions 3 and 4.
In this section, we discuss our results on the globalresource availability of agent 𝑖 in 𝔤 which is the probabilitythat 𝑖 obtains the resource from at least one agent in 𝔤 . In this section, we study the effect of link addition anddeletion on the global resource availability of agents whoare involved in these actions. Global resource availabilityof agents not involved in the link addition or deletion, thatis, spillover effect, is discussed in the next section.In Section 3.1.1, we saw that the local resource avail-abilities of both agents involved in link addition increase,and decrease for both in the case of link deletion. How-ever, this is not true when we look at the global resourceavailability of the agents involved. To understand this, weconsider the following example.
Example 1.
Consider the network 𝔤 as shown in Figure 3(a).If agent 𝑖 decides to delete the existing link with agent 𝑗 , wehave the network 𝔤 − ⟨ 𝑖𝑗 ⟩ as shown in Figure 3(b). The re-source availabilities of agents in these networks are tabu-lated in Table 1 under "Link Deletion". This table showsthat agent 𝑖 benefits by deleting an existing link with 𝑗 as itsresource availability increases by . . However, agent 𝑗 's resource availability decreases by . .To understand that link addition is not beneficial for anagent, we reverse the above situation. That is, we have net-work 𝔤 ′ as shown in Figure 3(b). Now, if agent 𝑖 decides toadd a direct link with agent 𝑗 , we have network 𝔤 ′ + ⟨ 𝑖𝑗 ⟩ ,as shown in Figure 3(a), as a result. The resource avail-abilities of agents in these networks are tabulated in Table1 under "Link Addition". Here, 𝑖 's resource availability de-creases by . on adding a direct link with 𝑗 , whereas 𝑗 's resource availability increases by . . In this section, our focus is to understand the effectthat link addition or deletion between a pair of agents has,on the global resource availability of the other agents, thatis, the spillover effect.
Definition 2. (Jackson, 2008) Suppose 𝑖, 𝑗 are distinct agentsin 𝔤 , such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . If agents 𝑖 and 𝑗 add the link ⟨ 𝑖𝑗 ⟩ ,then, agent 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } experiences positive spillover effect due to link addition if 𝑢 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > 𝑢 𝑘 ( 𝔤 ) , negative spillover effect due to link addition if 𝑢 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) < 𝑢 𝑘 ( 𝔤 ) , and no spillover effect due to link addition if 𝑢 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝑢 𝑘 ( 𝔤 ) . Mane et al. (2019) show that the externalities (spillovereffect) that agents experience is determined by their globalresource availability. For the sake of completeness, westate the result in the context of resource sharing net-works.
Proposition 5.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . If 𝑖 and 𝑗 add the link ⟨ 𝑖𝑗 ⟩ then, agent 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } experiences positive spillover effect due to link addition if 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > 𝛾 𝑘 ( 𝔤 ) , negative spillover effect due to link addition if 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) < 𝛾 𝑘 ( 𝔤 ) , and no spillover effect due to link addition if 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝛾 𝑘 ( 𝔤 ) .Proof. Adding ⟨ 𝑖𝑗 ⟩ does not change the neighbourhood sizeof 𝑘 . Therefore, from Eq. 4, 𝛾 𝑘 ( 𝔤 ) increases on adding ⟨ 𝑖𝑗 ⟩ if and only if 𝑢 𝑘 ( 𝔤 ) increases. Here, we also define the spillover effect when a link isdeleted and show a similar result.
Definition 3.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , such that ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . If agents 𝑖 and 𝑗 delete the link ⟨ 𝑖𝑗 ⟩ then, agent 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } experiences positive spillover effect due to link deletion if 𝑢 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) > 𝑢 𝑘 ( 𝔤 ) , P. C. Mane et al.:
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Page 6 of 11esource Availability in the Social Cloud: An Economics Perspective
Table 1
Link deletion and addition, and global resource availabilityLink Deletion Link AdditionAgent 𝛾 𝑖 ( 𝔤 ) 𝛾 𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) 𝛾 𝑖 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) − 𝛾 𝑖 ( 𝔤 ) 𝛾 𝑖 ( 𝔤 ′ ) 𝛾 𝑖 ( 𝔤 ′ + ⟨ 𝑖𝑗 ⟩ ) 𝛾 𝑖 ( 𝔤 ′ + ⟨ 𝑖𝑗 ⟩ ) − 𝛾 𝑖 ( 𝔤 ′ ) 𝑎 . . . . . . . . . . . . . . . . . . . . . 𝑡 𝑖 𝑘 𝑗 negative spillover effect due to link deletion if 𝑢 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) < 𝑢 𝑘 ( 𝔤 ) , and no spillover effect due to link deletion if 𝑢 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) = 𝑢 𝑘 ( 𝔤 ) . Proposition 6.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , suchthat ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . If agents 𝑖 and 𝑗 delete the link ⟨ 𝑖𝑗 ⟩ then,agent 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } experiences positive spillover effect due to link deletion if 𝛾 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) > 𝛾 𝑘 ( 𝔤 ) , negative spillover effect due to link deletion if 𝛾 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) < 𝛾 𝑘 ( 𝔤 ) , and no spillover effect due to link deletion if 𝛾 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) = 𝛾 𝑘 ( 𝔤 ) .Proof. Similar to the proof of Proposition 5.
Now, with the above background, we study how anewly added link affects the chance of an agent obtainingthe resource from the other agents.
Lemma 3.
Suppose 𝑖, 𝑗, 𝑘, 𝑙 are distinct agents in 𝔤 , suchthat ⟨ 𝑘𝑙 ⟩ ∉ 𝔤 . Then, If 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , the following hold: (a) If Φ 𝑗 ( 𝔤 ) = Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) then 𝛼 𝑖𝑗 ( 𝔤 ) = 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . (b) If Φ 𝑗 ( 𝔤 ) < Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) then 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . If 𝑑 𝑖𝑗 ( 𝔤 ) > 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) , the following hold: (a) If 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) 𝑑 𝑖𝑗 ( 𝔤 ) > Φ 𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) then 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . (b) If 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) 𝑑 𝑖𝑗 ( 𝔤 ) < Φ 𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) then 𝛼 𝑖𝑗 ( 𝔤 ) < 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) .Proof. The proof follows as 𝛼 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 ) and 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) = 𝑑 𝑖𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ )Φ 𝑗 ( 𝔤 + ⟨ 𝑘𝑙 ⟩ ) . Further, in this paper, we study the role of close-ness in determining spillover effects of agents, and showthat agents always experience either positive or negativespillover effect, and the case of "no spillover effect" neverhappens.
Proposition 7.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , suchthat ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . An agent 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } experiences negativespillover effect on addition of the link ⟨ 𝑖𝑗 ⟩ , if Φ 𝑘 ( 𝔤 ) = Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) .Proof. Let agent 𝑖 and 𝑗 add a direct link in 𝔤 . This newlink ⟨ 𝑖𝑗 ⟩ reduces their distance by at least , and by at most 𝑑 𝑖𝑗 ( 𝔤 ) − 1 in 𝔤 + ⟨ 𝑖𝑗 ⟩ . Thus, their closeness increases by atleast 𝑑 𝑖𝑗 ( 𝔤 )−1 𝑑 𝑖𝑗 ( 𝔤 ) in 𝔤 + ⟨ 𝑖𝑗 ⟩ .Therefore, Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) and Φ 𝑗 ( 𝔤 ) < Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) .As Φ 𝑘 ( 𝔤 ) = Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , we have 𝑑 𝑘𝑙 ( 𝔤 ) = 𝑑 𝑘𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ,for all 𝑙 ∈ 𝔤 . In particular, 𝑑 𝑖𝑘 ( 𝔤 ) = 𝑑 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) .Therefore, 𝛼 𝑘𝑖 ( 𝔤 ) > 𝛼 𝑘𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . Similarly, 𝛼 𝑘𝑗 ( 𝔤 ) > 𝛼 𝑘𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) . Hence, (1− 𝛼 𝑘𝑖 ( 𝔤 ))(1− 𝛼 𝑘𝑗 ( 𝔤 )) < (1− 𝛼 𝑘𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ))(1− 𝛼 𝑘𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ )) .This implies 𝛾 𝑘 ( 𝔤 ) > 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , as 𝛼 𝑘𝑙 ( 𝔤 ) ≥ 𝛼 𝑘𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for all 𝑙 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } too (since Φ 𝑙 ( 𝔤 ) ≤ Φ 𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ). Corollary 7.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . Φ 𝑘 ( 𝔤 ) = Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for all 𝑘 ∈ 𝔤 ⧵ { 𝑖, 𝑗 } andhence, all agents experience only negative spillover effect if ⟨ 𝑖𝑗 ⟩ is added. Remark 3.
Proposition 7 shows that an increase in agents’closeness is necessary for them to experience positive spillovereffect. However, the increase in agents’ closeness is not asufficient condition for positive spillover effect, as demon-strated by the following example.
Example 2.
Consider the Pedgetts’s Florentine_Families net-work (Breiger and Pattison, 1986) (showing business andmarital ties of 16 agents) generated by Social Network Visu-alizer tool from its known data set, as shown in Figure 4(a).Call this network 𝔤 . Suppose Medici and Strozzi add a linkin 𝔤 resulting in the network 𝔤 ′ = 𝔤 + ⟨ 𝑀𝑒𝑑𝑖𝑐𝑖, 𝑆𝑡𝑟𝑜𝑧𝑧𝑖 ⟩ , asshown in Figure 4(b). From Eq. (1), Φ 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ) = 7 . and Φ 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ′ ) = 8 . . However, from Eq. (3), 𝛾 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ) =0 . and 𝛾 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ′ ) = 0 . . This indicates that, although https://socnetv.org/ P. C. Mane et al.:
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Page 7 of 11esource Availability in the Social Cloud: An Economics Perspective (a) Network 𝔤 = 𝔤 ′ − ⟨ 𝑀𝑒𝑑𝑖𝑐𝑖, 𝑆𝑡𝑟𝑜𝑧𝑧𝑖 ⟩ (b) Network 𝔤 ′ = 𝔤 + ⟨ 𝑀𝑒𝑑𝑖𝑐𝑖, 𝑆𝑡𝑟𝑜𝑧𝑧𝑖 ⟩ Figure 4:
Spillover in resource sharing network 𝔤 the closeness of Albizzi increases in 𝔤 ′ , its global resourceavailability decreases. Similar is true in the case of Ginoriand Pazzi. On the contrary, the newly added link betweenMedici and Strozzi increases not only Bischeri’s closeness(from Φ 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ) = 7 . to Φ 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ′ ) = 7 . ), but alsoits global resource availability (from 𝛾 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ) = 0 . to 𝛾 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ′ ) = 0 . ). Similar holds for Acciaiuodi, Peruzziand Salviati too. Note that 𝛾 𝑃 𝑢𝑐𝑐𝑖 ( 𝔤 ) = 𝛾 𝑃 𝑢𝑐𝑐𝑖 ( 𝔤 ′ ) = 0 . We provide similar results for spillover effects due tolink deletion too.
Proposition 8.
Suppose 𝑖, 𝑗, 𝑘 are distinct agents in 𝔤 , suchthat ⟨ 𝑖𝑗 ⟩ ∈ 𝔤 . 𝑘 experiences positive spillover effect due tolink deletion, on deletion of ⟨ 𝑖𝑗 ⟩ , if Φ 𝑘 ( 𝔤 ) = Φ 𝑘 ( 𝔤 − ⟨ 𝑖𝑗 ⟩ ) .Proof. The result can be proved in lines similar to the proofof Proposition 7.
Example 3.
Consider the resource sharing network, 𝔤 ′ , shownin Figure 4(b). Let Medici and Strozzi delete the link be-tween them in 𝔤 ′ resulting in the resource sharing network 𝔤 ′ − ⟨ 𝑀𝑒𝑑𝑖𝑐𝑖, 𝑆𝑡𝑟𝑜𝑧𝑧𝑖 ⟩ ( 𝔤 , as shown in Figure 4(a). FromEq. (1), Φ 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ′ ) = 8 . and Φ 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ) = 7 . . How-ever, from Eq. (3), 𝛾 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ′ ) = 0 . and 𝛾 𝐴𝑙𝑏𝑖𝑧𝑧𝑖 ( 𝔤 ) =0 . . This indicates that, although the closeness of Albizzidecreases in 𝔤 ′ − ⟨ 𝑖𝑗 ⟩ , its global resource availability in-creases. Similar is true in the case of agents Ginori andPazzi. On the contrary, the link deletion between Medici andStrozzi in 𝔤 ′ decreases not only agent Bischeri’s closeness(from Φ 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ′ ) = 7 . to Φ 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ) = 7 . ), but alsoits global resource availability (from 𝛾 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ′ ) = 0 . to 𝛾 𝐵𝑖𝑠𝑐ℎ𝑒𝑟𝑖 ( 𝔤 ). We have a similar observation for Acciaiuodi,Peruzzi and Salviati too. 𝛾 𝑃 𝑢𝑐𝑐𝑖 ( 𝔤 ′ ) = 𝛾 𝑃 𝑢𝑐𝑐𝑖 ( 𝔤 ) = 0 . Now, we show that in any connected resource sharingnetwork with three or more agents, all agents experienceeither positive or negative spillover effects, and there is nocase where any agent experiences no spillover effect.
Theorem 3.
Suppose 𝔤 is connected and has three or moreagents. Suppose 𝑘 is any agent in 𝔤 . Agent 𝑘 always expe-riences either positive or negative spillover effect, and it isnot possible that 𝑘 experiences no spillover effect. Proof. We prove the result for spillover effect due to link ad-dition, and a similar proof works for spillover effect due tolink deletion too.Suppose 𝑖 , 𝑗 and 𝑘 are distinct agents in 𝔤 , such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . If possible, let 𝑘 have no spillover effect when ⟨ 𝑖𝑗 ⟩ is added. This means 𝛾 𝑘 ( 𝔤 ) = 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , by Proposi-tion 5.Therefore, by Eq. (3), 𝛼 𝑘𝑙 ( 𝔤 ) = 𝛼 𝑘𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , for all 𝑙 ∈ 𝔤 . (Note that 𝛼 𝑘𝑙 ( 𝔤 ) ≥ 𝛼 𝑘𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) , for all 𝑙 ∈ 𝔤 ).This implies that 𝑑 𝑘𝑙 ( 𝔤 ) = 𝑑 𝑘𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) and Φ 𝑙 ( 𝔤 ) =Φ 𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for all 𝑙 ∈ 𝔤 , from Equations (1) and (2).The link addition between agents 𝑖 and 𝑗 in 𝔤 decreasestheir distance by 𝑑 𝑖𝑗 ( 𝔤 )−1 in 𝔤 + ⟨ 𝑖𝑗 ⟩ , and thus, the closenessof both 𝑖 and 𝑗 in 𝔤 + ⟨ 𝑖𝑗 ⟩ increases by 𝑑 𝑖𝑗 ( 𝔤 )−1 .Hence, Φ 𝑖 ( 𝔤 ) < Φ 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) and Φ 𝑗 ( 𝔤 ) < Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) ,a contradiction to our deduction that Φ 𝑙 ( 𝔤 ) = Φ 𝑙 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for all 𝑙 ∈ 𝔤 .Therefore, our assumption that 𝑘 has no spillover effectis incorrect. In other words, 𝑘 has either positive or negativespillover effect. Corollary 8.
Suppose 𝔤 is disconnected with at least threedisjoint components. Suppose 𝑖, 𝑗 , and 𝑘 are three distinctagents in 𝔤 such that 𝑖 ∈ 𝔤 ( 𝔠 𝑥 ) , 𝑗 ∈ 𝔤 ( 𝔠 𝑦 ) , and 𝑘 ∈ 𝔤 ( 𝔠 𝑧 ) ,where 𝔠 𝑥 , 𝔠 𝑦 and 𝔠 𝑧 are disjoint. Suppose agents 𝑖 and 𝑗 add a direct link in 𝔤 , then 𝛾 𝑘 ( 𝔤 ) = 𝛾 𝑘 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for all 𝑘 ∉ 𝔤 ( 𝔠 𝑥 ) , 𝔤 ( 𝔠 𝑦 ) .
4. Choice Modelling
To understand the relation between agents’ distancefrom each other, and local as well as global resource avail-abilities, we, first, discuss the following example.
P. C. Mane et al.:
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Page 8 of 11esource Availability in the Social Cloud: An Economics Perspective (a) 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) (b) 𝛾 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) Figure 5:
Local and global resource availabilities of agent 𝑖 in the ring network Example 4.
Suppose 𝔤 is a ring network . Then, for all 𝑖 ∈ 𝔤 , Φ 𝑖 ( 𝔤 ) = { + + … + 𝑁 ) , if N is odd + + … + 𝑁 −1 ) + 𝑁 , if N is even.Suppose 𝑖 ∈ 𝔤 . For 𝑗 ∈ 𝔤 , suppose ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . We com-pute the local resource availability, 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) of 𝑖 from 𝑗 ,and the global resource availability 𝛾 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) for different 𝑗 in increasing order of the distance between 𝑖 and 𝑗 in 𝔤 .Figures 5(a) and 5(b) show that as the distance between 𝑖 and 𝑗 increases, 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) and 𝛾 𝑖 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) also increase. We, now, discuss the relation between local resourceavailability and distance as well as closeness.
Lemma 4.
Suppose 𝑖, 𝑗 are distinct agents in 𝔤 , such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 . The local resource availability of agent 𝑖 fromagent 𝑗 increases with decrease in the distance, 𝑑 𝑖𝑗 ( 𝔤 ) , be-tween them.Proof. For any 𝑖, 𝑗 ∈ 𝔤 , 𝑖 ≠ 𝑗 , < 𝑑 𝑖𝑗 ( 𝔤 ) < 𝑑 𝑖𝑗 ( 𝔤 )+1 . Thenfrom Eq. (2), 𝑝 (1− 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 ) > 𝑝 (1− 𝑞 )( 𝑑 𝑖𝑗 ( 𝔤 )+1)Φ 𝑗 ( 𝔤 ) . Lemma 5.
Suppose 𝑖, 𝑗, 𝑘 are distinct agents in 𝔤 , such that ⟨ 𝑖𝑗 ⟩ , ⟨ 𝑖𝑘 ⟩ ∉ 𝔤 . If 𝑑 𝑖𝑗 ( 𝔤 ) > 𝑑 𝑖𝑘 ( 𝔤 ) then, 𝛼 𝑖𝑗 ( 𝔤 ) < 𝛼 𝑖𝑘 ( 𝔤 ) .Proof. Follows from Lemma 4.
Lemma 6.
Agent 𝑖 in 𝔤 obtains maximum local resourceavailability from that 𝑘 ∈ 𝔤 who is least close to others (thatis, with the least harmonic centrality).Proof. Let 𝑗, 𝑘 ∈ 𝔤 and < Φ 𝑘 ( 𝔤 ) < Φ 𝑗 ( 𝔤 ) , and 𝑑 𝑖𝑗 ( 𝔤 ) = 𝑑 𝑖𝑘 ( 𝔤 ) . Then from Eq. (2), 𝑝 (1− 𝑞 ) 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 ) > 𝑝 (1− 𝑞 ) 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 ) . We showed, in Lemmas 5 and 6, that the local resourceavailability of an agent from another agent increases with A ring network 𝔤 is a connected network where 𝜂 𝑖 ( 𝔤 ) = 2 for all 𝑖 ∈ 𝔤 . decrease in the distance between them and that maxi-mum local resource availability is obtained from the agentwith the least closeness (that is, least harmonic central-ity). We, now, look at the relation between the globalresource availability of an agent and its closeness. Lemma 7.
Agent 𝑖 in 𝔤 maximizes its global resource avail-ability by maximizing its own closeness or equivalently, byminimizing its distance with others.Proof. The proof is in lines similar to that of Lemma 6.
We now discuss results that show which agent to adda link to, so as to maximize the local resource availability.Such results are difficult to establish for global resourceavailability because, as seen in Example 1, if agents 𝑖 and 𝑗 add or delete the link between them, the global resourceavailability of one of them may increase while that of theother may decrease. Proposition 9.
Suppose 𝑖 is an agent in 𝔤 . Across all agents 𝑗 in 𝔤 such that ⟨ 𝑖𝑗 ⟩ ∉ 𝔤 , suppose 𝑖 chooses 𝑗 = 𝑗 to which toadd a link, then 𝑗 maximizes the local resource availabilityof 𝑖 from 𝑗 in 𝔤 + ⟨ 𝑖𝑗 ⟩ if and only if 𝑗 is the agent (or one ofthe agents) whose closeness is the least.Proof. Suppose agents 𝑘 and 𝑙 ∈ 𝔤 are such that ⟨ 𝑖𝑘 ⟩ ∉ 𝔤 and ⟨ 𝑖𝑙 ⟩ ∉ 𝔤 . Agent 𝑖 prefers 𝑙 over 𝑘 to add a link, if andonly if 𝛼 𝑖𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) > 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) , if and only if 𝑑 𝑖𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ )Φ 𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) > 𝑑 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ )Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) .We have 𝑑 𝑖𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) = 𝑑 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) = 1 . Hence, 𝛼 𝑖𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) > 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) , if and only if 𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) > 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) , if and only if Φ 𝑙 ( 𝔤 + ⟨ 𝑖𝑙 ⟩ ) < Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) . P. C. Mane et al.:
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Lemma 8.
Suppose 𝑖, 𝑗, 𝑘 are distinct agents in 𝔤 . With re-spect to the local resource availability, 𝑖 prefers 𝑗 over 𝑘 towhich to add a link, if and only if Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ )−Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ )Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ )Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 )− 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 )[ 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 )][ 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 )] .Proof. 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) − 𝛼 𝑖𝑗 ( 𝔤 ) > 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) − 𝛼 𝑖𝑘 ( 𝔤 ) ,if and only if 𝛼 𝑖𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) − 𝛼 𝑖𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) > 𝛼 𝑖𝑗 ( 𝔤 ) − 𝛼 𝑖𝑘 ( 𝔤 ) ,if and only if 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) − 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ ) > 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 ) − 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 ) ,if and only if Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ )−Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ )Φ 𝑘 ( 𝔤 + ⟨ 𝑖𝑘 ⟩ )Φ 𝑗 ( 𝔤 + ⟨ 𝑖𝑗 ⟩ ) > 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 )− 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 )[ 𝑑 𝑖𝑘 ( 𝔤 )Φ 𝑘 ( 𝔤 )][ 𝑑 𝑖𝑗 ( 𝔤 )Φ 𝑗 ( 𝔤 )] . Theorem 4.
Suppose 𝑖, 𝑗, 𝑘 are distinct agents in 𝔤 . Withrespect to the local resource availability, 𝑖 prefers to add a link with 𝑗 over 𝑘 if 𝑗 ’s closeness in 𝔤 + ⟨ 𝑖𝑗 ⟩ is less than that of 𝑘 in 𝔤 + ⟨ 𝑖𝑘 ⟩ , given that,in 𝔤 , both 𝑗 and 𝑘 have the same closeness and are atthe same distance from 𝑖 . 𝑖 prefers to add a link with 𝑗 over 𝑘 if, in 𝔤 , 𝑗 ’s distancefrom 𝑖 is more than that of 𝑘 from 𝑖 , given that, 𝑗 and 𝑘 have the same closeness in 𝔤 + ⟨ 𝑖𝑗 ⟩ and 𝔤 + ⟨ 𝑖𝑘 ⟩ ,respectively, as well as in 𝔤 . 𝑖 prefers to add a link with 𝑗 over 𝑘 if 𝑗 ’s closeness isless than that of 𝑘 in 𝔤 , given that, 𝑗 and 𝑘 have thesame closeness in 𝔤 + ⟨ 𝑖𝑗 ⟩ and 𝔤 + ⟨ 𝑖𝑘 ⟩ , respectively,and they are at the same distance from 𝑖 in 𝔤 .Proof. Follows from Lemma 8.
5. Conclusion
The focus of this study is on endogenous social cloudformation. In this social cloud setting, agents build theirresource sharing network to maximize their utility and per-form closeness-based conditional resource sharing. Theaim of this study is to fill the research gap in the socialcloud literature by analyzing the impact of link additionbetween a pair of agents on their resource availability andthat of others. This study provides a theoretical investi-gation of the impact of link addition on agents’ resourceavailability. It further studies choice modelling, which cap-tures the preferences over agents in link addition. In otherwords, this study focuses on the agent(s) in the networkthat is (are) preferred by other agents for link addition,so as to maximize their utility (in terms of their resourceavailability).Our approach of understanding spillover effect (thatis, the impact of link addition between a pair of agents onothers’ resource availability) is different from the existingapproaches discussed in the social cloud literature. Weanalyze the relation between closeness and spillover. Weshow that, for an agent to observe positive spillover effect,it is necessary, but not sufficient, to increase its closeness. Our study does not find a reason for this. This is one ofthe limitations of the study. Despite this limitation, ourstudy has several implications and applications.Our study provides considerable insight into resourceavailability in endogenous social cloud setting. This anal-ysis will help in defining resource sharing and allocationpolicy framework in endogenous social cloud systems likeBuddyBackup. For example, in these settings, the pol-icy making may include the recommendation of friends(as backup partners) in link addition to avoid negativespillover effect. If an agent wants to select another agentfor data backup, then the policy makers can recommend abackup partner who is far from the agent so that negativespillover can be reduced.
Declaration of Competing Interest
The authors declare that they have no known compet-ing financial interests or personal relationships that couldhave appeared to influence the work reported in this pa-per.
Conflicts of Interest
The authors declare that they have no conflict of in-terest.
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