Formation of stable and efficient social storage cloud

Mane, P amod C.; K ishnamu hy, Naga ajan; Ahuja, Kapil A icle Fo ma ion o s able and e icien social s o age cloud Games P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Mane, P amod C.; K ishnamu hy, Naga ajan; Ahuja, Kapil (2019) : Fo ma ion o s able and e icien social s o age cloud, Games, ISSN 2073-4336, MDPI, Basel, Vol. 10, Iss. 4, pp. 1-17, h ps://doi.o g/10.3390/g10040044 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/219267 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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Mane 1, Naga ajan K ishnamu hy 2and Kapil Ahuja 1,* 1Compu e Science and Enginee ing, Indian Ins i u e o Technology Indo e, Indo e 453552, India; [email p o ec ed] 2Ope a ions Managemen and Quan i a i e Techniques, Indian Ins i u e o Managemen Indo e, Indo e 453556, India; naga ajan@iimid .ac.in *Co espondence: [email p o ec ed] Recei ed: 28 June 2019; Accep ed: 16 Oc obe 2019 ; Published: 1 No embe 2019   Abs ac : In his pape , we s udy he o ma ion o endogenous social s o age cloud in a dynamic se ing, whe e a ional agen s build hei da a backup connec ions s a egically. We p opose a deg ee-dis ance-based u ili y model, which is a combina ion o bene i and cos unc ions. The bene i unc ion o an agen cap u es he expec ed bene i ha he agen ob ains by placing i s da a on o he s’ s o age de ices, gi en he p e ailing da a loss a e in he ne wo k. The cos unc ion o an agen cap u es he cos ha he agen incu s o main ain links in he ne wo k. Wi h his u ili y unc ion, we analyze wha ne wo k is likely o e ol e when agen s hemsel es decide wi h whom hey wan o o m links and wi h whom hey do no . Fu he , we analyze which ne wo ks a e pai wise s able and e icien . We show ha o he p oposed u ili y unc ion, he e always exis s a pai wise s able ne wo k, which is also e icien . We show ha all pai wise s able ne wo ks a e e icien , and hence, he p ice o ana chy is he bes ha is possible. We also s udy he e ec o link addi ion and dele ion be ween a pai o agen s on hei , and o he s’, closeness and s o age a ailabili y. Keywo ds: ne wo k o ma ion; pai wise s abili y; ne wo k ex e nali ies; social s o age cloud; socially-awa e s o age-sha ing MSC: 91A40; 91A80; 91B32; 91B99 JEL Classi ica ion: C72; D62; D85; L86 1. In oduc ion Online da a backup se ices such as BuddyBackup 1 and C ashPlan 2 allow agen s o sha e hei unde -u ilized s o age (disk) space wi h o he s as well as backup hei da a on he s o age space sha ed by o he agen s. In academic discou se, nume ous a chi ec u al p o o ypes o da a backup sys ems ( o example, Social S o age Cloud [ 1 ], F iends o e [ 2 ], F2Box [ 3 ], F iendBox [ 4 ], BlockPa y [ 5 ], and so on) ha e been p oposed. In o de o mi iga e issues like da a secu i y, us , o low quali y o se ices, hese sys ems (se ices) a e le e aging social connec ions. The social connec ions a e ei he exogenous ( ha is, encoded in social g aphs, o ins ance, he Facebook social g aph 3 ) o endogenous (cons uc ed by he agen s [6]). Such social connec ions a e a he co e o hese sys ems. 1h p://www.buddybackup.com (accessed on 21 June 2019). 2h ps://suppo .c ashplan.com (accessed on 21 June 2019). 3h ps://de elope s. acebook.com/docs/g aph-api (accessed on 21 June 2019). Games 2019,10, 44; doi:10.3390/g10040044 www.mdpi.com/jou nal/games Games 2019,10, 44 2 o 17 A ecen ly published su ey [ 7 ] in his ield men ions a ious issues ela ed o social connec ions, such as small iend se s, social closeness quan i ica ion, and so on. These issues a e discussed in he con ex o exogenous social connec ions. The aspec o endogenous social connec ions is no ably lacking. The agen s’ sel -in e es ed beha iou , guided by he cos -bene i ade-o , in building s o age-sha ing connec ions is s ill poo ly unde s ood. Speci ically, wha is no well unde s ood in his con ex is: (1) which ne wo k s uc u e is likely o eme ge when sel -in e es ed agen s cons uc hei s o age sha ing connec ions; (2) whe he he eme ged s o age-sha ing connec ion s uc u e is s able and e icien , o no ; and (3) he impac o link o ma ion be ween wo agen s on hei s o age a ailabili y as well as ha o o he agen s. To ad ance ou unde s anding abou hese aspec s, he e is a need o o mal modeling o endogenous socially-awa e s o age-sha ing ne wo ks, as p e ious s udies ha e ocused exclusi ely on exogenous ne wo ks. This pape s udies he a o emen ioned aspec s by ocusing on social s o age cloud sys ems (a case o socially-awa e esou ce sha ing sys ems). We model social s o age cloud sys ems as an endogenous social s o age cloud by using he ools o ne wo k analysis 4 ,game heo y 5 , and ne wo k o ma ion 6 . Speci ically, we model social s o age cloud sys ems as a s a egic ne wo k o ma ion game, whe e sel -in e es ed agen s decide wi h whom hey wan o o m a connec ion and wi h whom hey do no . Fo his, we de ine he u ili y o agen s in a social s o age cloud by aking in o conside a ion he pa ame e s da a ailu e a e, alue o da a, and cos o main aining social connec ions. In [ 23 ], he au ho s conside a deg ee-based u ili y model, whe e agen s bene i only om di ec neighbo s, and he bene i dec eases wi h an inc ease in he numbe o neighbo s o each neighbou [ 24 ]). The u ili y unc ion we de ine in his s udy is deg ee-dis ance-based, whe e agen s ob ain bene i s om di ec and indi ec neighbo s, bu he bene i dec eases wi h an inc ease in he numbe o di ec and indi ec neighbo s [ 25 ]. Wi h his u ili y unc ion, we s udy he e ec o decisions o addi ion and dele ion o links by pai s o agen s on hei s o age a ailabili y in he ne wo k. We s udy ex e nali ies in he ne wo k, ha is, he e ec o link o ma ion be ween a pai o agen s on he u ili y o he o he agen s. We hen analyze he ne wo k s uc u e ha e ol es due o hese decisions o link addi ion and link dele ion. The ocus o his pape is o s udy ne wo k s abili y, e iciency, and he measu es o p ice o ana chy and p ice o s abili y. Fo he analysis o ne wo k s abili y, we make use o he concep o pai wise s abili y p oposed in [ 26 ]. In ou model, agen s expe ience bo h posi i e and nega i e ex e nali ies, de e mined by s o age a ailabili y. We p o ide necessa y and su icien condi ions o an agen o expe ience posi i e and nega i e ex e nali ies. Fu he , we show ha i da a ailu e a e is less han he a io o cos o main aining he link o da a alue, hen he null ne wo k is he unique pai wise s able as well as an e icien ne wo k. Howe e , i he da a ailu e a e is highe han he a io o he cos o main aining he link o da a alue hen a ne wo k whe e e e y agen has, a mos , a single link is he unique pai wise s able and e icien ne wo k. The s uc u e o he pape is as ollows. Sec ion 2discusses he social s o age cloud model. Sec ion 3s udies he e ec o addi ion and dele ion o a link be ween a pai o agen s on hei closeness and s o age a ailabili y, and ha o o he s. Sec ion 4discusses he cha ac e iza ion o s able ne wo ks, whe e we s udy de ia ion condi ions ha show when agen s ha e incen i es o adding o dele ing a link. Fu he , he sec ion discusses ne wo k s abili y, e iciency, and ine iciency. Sec ion 5concludes he discussion. 4We e e he eade o he body o he pape [8–12] o de ails on cen ali y measu es in ne wo ks. 5 The heo e ical game echniques ha e been qui e success ully used in compu e science [ 13 – 16 ] and o he enginee ing disciplines [17,18]. 6The li e a u e on ne wo k o ma ion is as . The su eys [19–22] explo e many dimension o his opic. Games 2019,10, 44 3 o 17 2. Social S o age Cloud Model In his sec ion, we desc ibe he social s o age cloud model h ough an in e ac ion s uc u e, a s o age-sha ing amewo k and cos -bene i analysis o agen s. 2.1. In e ac ion S uc u e A social s o age cloud g= (A,L) is a s o age-sha ing and da a backup ne wo k ha consis s o a nonemp y se A o N agen s who a e in ol ed in s o age (disk)-space sha ing and da a backup ac i i y; and a se , L , o links ha connec hese agen s. The se L ac s as a communica ion in as uc u e o agen s o sha e hei s o age space wi h o he s and sea ch o s o age space p o ided by o he s. A link, hiji ∈ L , ep esen s a di ec communica ion channel be ween agen s i and j , which is bidi ec ional (and hence, hiji=hjii ). I hiji∈L , we call he agen s i and j as neighbou s in he ne wo k g. The numbe o neighbo s o agen iin gis deno ed by ηi(g). Gi en dis inc agen s a1 , a2 , · · · , an∈ A , i ha1 , a2i , ha2 , a3i , · · · , han−1 , ani ∈ L , hen he e is a pa h Pa1an(g) , om a1 o an , o leng h n− 1. The dis ance dij(g)(= dji(g)) be ween a pai o agen s i and j is he leng h o he sho es pa h connec ing hem in g. A ne wo k g is connec ed i he e exis s a leas one pa h be ween any pai o agen s, o he wise, i is disconnec ed. A pa h o leng h ≥ 2 be ween a pai o agen s is an indi ec communica ion channel be ween hem. The se , G(N), consis s o all possible ne wo ks on Nagen s. Da a s o ed on local s o age space is p one o loss due o mul iple easons such as i us in ec ion, so wa e o ha dwa e ailu e, da a co up ion, and so on. The e o e, each agen wan s o backup i s da a on emo e s o age (disk) space. Fo any agen , da a loss is cos ly. We cap u e his by assuming ha he alue each agen associa es wi h i s da a is quan i iable and gi en. E e y agen (as a da a owne ) s i es o ob aining s o age space p o ided by o he agen s (as s o age p o ide s) in g∈ G(N) . Agen i wan s o backup ¯ bi amoun o da a and sha es ¯ si=∑ j∈A {i} ¯ bj amoun o s o age space. This leads o endogenous social s o age cloud o ma ion, whe e each agen builds i s communica ion channel o seek s o age space om di ec and indi ec communica ion channels. We assume ha each agen has global (comple e) in o ma ion abou he ne wo k s uc u e. A ne wo k g e ol es when agen s pe o m wo ac ions, namely, link addi ion ( g+hiji ) and link dele ion ( g− hiji ). Mu ual consen o a pai o agen s is equi ed o addi ion o a link be ween hem, bu any link can be unila e ally dele ed. Table 1summa izes all no a ions used in his pape . Table 1. No a ion summa y. gsocial s o age cloud. Ase o agen s (o e ices). N he numbe o elemen s in he se A, which is he numbe o agen s in g. Lse o links (o edges). hijilink be ween agen s iand j. ςcos incu ed by each agen o main ain a link. λp obabili y ha an agen loses i s da a. βwo h (o alue) ha each agen has o i s da a. Φi(g)closeness o agen iin g. αij(g)p obabili y ha agen iob ains s o age space om agen jin g. γi(g)p obabili y ha agen iob ains s o age space om a leas one agen in g. ηi(g)neighbo hood size o agen iin g. Also deno es he se o neighbo s o i. Pa1an(g)a pa h om agen a1 o anin gsuch ha ha1,a2i,ha2,a3i,· · · ,han−1,ani ∈ L. dij(g) he leng h o he sho es pa h connec ing agen s iand jin g. g+hijinew link hijiis added o g. g− hijiexis ing link hijiis dele ed om g. G(N) he se o all ne wo ks on Nagen s. ui(g)u ili y o agen iin g. Games 2019,10, 44 4 o 17 2.2. S o age Sha ing Acco ding o [ 1 ], agen s could limi s o age-sha ing wi h hose who a e close o hem in he social cloud. In o de o cap u e his, we make use o he ha monic cen ali y measu e (discussed in [10,27,28]), de ined as ollows: Φi(g) = ∑ j∈g {i} 1 dij(g). (1) We use ha monic cen ali y as i deals wi h disconnec ed ne wo ks as well. In g , an agen j (as a s o age p o ide ) compu es a p obabili y dis ibu ion on all agen s o he pu pose o alloca ing s o age space o agen i∈g(as a da a owne ), as below: αij(g) = 1 dij(g) ∑ j∈g {i} 1 dij(g) =1 dij(g)Φi(g), (2) whe e αij(g)is he p obabili y ha agen iwill ob ain s o age space om agen jin g. Rema k 1. I dij(g) = ∞ , hen αij(g) = 0(and αji(g) = 0). As agen s i and j a e disconnec ed in g , hei chances o ob aining s o age space om each o he is ze o. The p obabili y ha an agen iob ains s o age space om a leas one agen in gis γi(g) = 1−∏ j∈g {i} (1−αij(g)). 2.3. Agen ’s U ili y and Symme y The u ili y o agen i in g is gi en by a unc ion ui:G → R+ . Le u be he he ec o (p o ile) o u ili y unc ions u= (u1 , ..., un) . Thus, we ha e u:G → RN . In o he wo ds, each possible social s o age cloud s uc u e g∈ G leads o a u ili y unc ion p o ile o agen s. We de ine he u ili y o agen s in a social s o age cloud g wi h he ollowing pa ame e s. An agen , i , loses i s da a wi h p obabili y λi∈( 0, 1 ) . The e o e, o minimize his isk o da a loss, agen i aims o backup i s da a on he s o age p o ided by o he s. Fo agen i , βi is he alue o he local da a ha is o be backed up. Agen i ob ains s o age space p o ided by o he s in g , wi h p obabili y γi(g) . Thus, he alue o da a βi , he chance o losing he da a λi , and he chance o ob aining s o age space γi(g)cap u e he expec ed bene i o agen iin g. An agen sea ches o s o age by s aying connec ed in he ne wo k. Di ec as well as indi ec links help agen s o ge s o age space. The di ec link be ween agen s i and j cos s ςi . This cos can be in e p e ed as he cos equi ed o main aining s o age space, in as uc u e, bandwid h, ime, and so on. The cos o main ain an exis ing link and ha o adding (and main aining) a new link a e he same. The e is no addi ional cos o add a new link. Thus, agen i incu s a o al cos o ςiηi(g) in o de o ob ain an expec ed bene i o βiλiγi(g) , in case o da a loss. Bu he ne wo k is o med up on , be o e he da a loss happens. The cos o main ain links is, hence, incu ed e en in he case o no da a loss, whe e he expec ed bene i o iis βi(1−λi). The e o e, gi en he a o emen ioned pa ame e s, he expec ed u ili y is ui(g) = βi(1−λi)+βiλiγi(g)−ςiηi(g). (3) F ee iding (a si ua ion whe e an agen o e s less s o age space, bu consumes mo e) is a widely discussed issue in he li e a u e on pee - o-pee s o age. In o de o deal wi h ee iding, many P2P s o age sys ems ( o example, In e ne Coope a i e Backup Sys em [29], Pee S o e [30], Pas iche [31]) ollow a symme ic s o age-sha ing mechanism, whe e agen s sha e he same amoun o s o age space. Games 2019,10, 44 5 o 17 We de ine a symme ic social s o age cloud gas ollows. De ini ion 1. A symme ic social s o age cloud (SSSC) g is a ne wo k whe e he bene i ( alue) associa ed wi h backed-up da a is he same o all agen s in he ne wo k, ha is, βi=βj (say β ), ςi=ςj (say ς7 ), and λi=λj (say λ8) o all i,j∈ A, and hence, u ili y o each agen i in gis ui(g) = β(1−λ)+βλγi(g)−ςηi(g), (4) whe e λ,β,ς∈(0, 1). Fo u he s udy, we conside he abo e u ili y unc ion (Equa ion (4)). Hence o h, whene e we e e o a ne wo k, o jus g, we mean an SSSC. 2.4. Pai wise S abili y In o de o cha ac e ize endogenously buil social s o age cloud, we adop pai wise s abili y [ 26 ] as a solu ion concep . A ne wo k is pai wise s able i (1) no agen bene i s by dele ing an exis ing link and (2) no wo agen s bene i by adding a new link be ween hem. De ini ion 2. [26] A ne wo k gis pai wise s able i 1. o all i,j∈gsuch ha hiji ∈ g, ui(g)≥ui(g− hiji), and uj(g)≥uj(g− hiji); and 2. o all i,j∈gsuch ha hiji 6∈ g, i ui(g+hiji)>ui(g), hen uj(g+hiji)<uj(g). 3. Ne wo k S uc u e and S o age A ailabili y One o he objec i es o his pape is o unde s and he impac o link addi ion and dele ion on s o age a ailabili y o hose agen s who a e in ol ed in he link addi ion/dele ion as well as hose who a e no . The s o age a ailabili y is de e mined by he dis ances be ween hem and hei closeness ( om Equa ion (2)). The e o e, i s we s udy how addi ion and dele ion o a link impac s he sho es dis ances be ween pai s o agen s and, he e o e, hei closeness. This analysis p o ides a base o unde s anding he e ec o link-addi ion/dele ion on agen s’ s o age a ailabili y in g. 3.1. E ec o Link Al e a ion on Closeness Lemma 1. Suppose hiji 6∈ g. Then, Φi(g+hiji)>Φi(g). P oo . Clea ly, dij(g+hiji)<dij(g) . As hiji 6∈ g , we ha e, dij(g)≥ 2. Also, dij(g+hiji) = 1. Thus, Φi(g)and Φj(g)inc ease by a leas dij(g)−1 dij(g)in g+hiji. Lemma 2. Suppose hiji ∈ g. Then, Φi(g− hiji)<Φi(g). P oo . 1. Le us assume he e is no pa h be ween i and j in g− hiji , hen dij(g− hiji) = ∞ , hus, Φi(g) and Φj(g)dec ease by 1 in g− hiji. 2. Now, le us assume he e exis s a pa h Pij(g− hiji) be ween i and j in g− hiji , he dis ance be ween i and j in g− hiji being a leas 1 mo e han ha in g . Thus, Φi(g) and Φj(g) dec ease by a leas dij(g−hiji)−1 dij(g−hiji)in g− hiji. 7We assume, ς=ςi+ςj 2, ha is, a pai o agen s in ol ed in a link sha e he cos ς. 8Fo simplici y, we assume uni o m da a loss a e λ. Games 2019,10, 44 6 o 17 Lemmas 1and 2show ha , wi h espec o closeness, e e y link bene i s agen s on ei he side o he link. An ac ion o link addi ion o dele ion be ween a pai o agen s no only impac s hei closeness, bu also ha o o he agen s. Now, we s udy he impac o link addi ion o dele ion be ween a pai o agen s (say, iand j) on he closeness o he o he agen s k∈g {i,j}. Lemma 3. Suppose hiji 6∈ g and k∈g {i , j} . Then, Φk(g) = Φk(g+hiji) i and only i dkl(g) = dkl(g+hiji) o all l ∈g. P oo . I dkl(g) = dkl(g+hiji) o all l∈g, hen by Equa ion (1), Φk(g) = Φk(g+hiji). Con e sely, suppose Φk(g) = Φk(g+hiji). I is easy o see ha , i o some l∈g , i dkl(g+hiji)6=dkl(g) , hen dkl(g+hiji)<dkl(g) . (Pa hs in gexis in g+hiji oo). We ha e dkl(g+hiji)≤dkl(g) o all l∈g and, i he e exis s x such ha dkx(g+hiji)<dkx(g) , hen Φk(g)<Φk(g+hiji), a con adic ion. Lemma 4. Suppose hiji ∈ g and k∈g {i , j} . Then, Φk(g) = Φk(g− hiji) i and only i dkl(g) = dkl(g− hiji) o all l ∈g. P oo . As dkl(g− hiji)≥dkl(g) o all l, he p oo ollows in lines simila o ha o Lemma 3. We now show necessa y and su icien condi ions o inc ease in he closeness o agen s who a e no in ol ed in link addi ion o dele ion. Theo em 1. Suppose hiji 6∈ g , and le k be an agen dis inc om i and j . Then, Φk(g)<Φk(g+hiji) i and only i he e exis s a leas one agen l∈g such ha dkl(g)≥ 3and all sho es pa hs Pkl(g+hiji) om k o l in g+hijicon ain hiji. P oo . Le Φk(g)<Φk(g+hiji) . Then, by Lemma 3, he e mus be a leas one agen , say l , such ha dkl(g)>dkl(g+hiji). Suppose i,k, and la e all dis inc . No e ha jmay be he same as l. I possible, le dkl(g)<dki(g) + dij(g) + djl(g) o all l∈g . Then, dkl(g) = dkl(g+hiji) o all l∈g . F om Lemma 1, Φk(g)<Φk(g+hiji) , a con adic ion. The e o e, he e exis s an l∈g such ha dkl(g) = dki(g) + dij(g) + djl(g). As hiji 6∈ g,dij(g)≥2. As k6=i,dik(g)≥1 and j=l. Hence, dkl(g)≥3. Now, dkl(g+hiji) = dki(g+hiji) + dij(g+hiji) + djl(g+hiji) =dki(g) + dij(g+hiji) + djl(g) <dki(g) + dij(g) + djl(g) =dkl(g). I ollows ha e e y sho es pa h be ween k and l in g+hiji con ains hiji . (No e ha i he e exis s a sho es pa h om k o l in g+hiji ha does no con ain hiji , hen his sho es pa h exis s in g oo). Con e sely, le l∈g such ha dkl(g)≥ 3 and all sho es pa hs Pkl(g+hiji) om k o l in g+hiji con ain hiji. Clea ly, Φk(g)≤Φk(g+hiji). I possible, le Φk(g) = Φk(g+hiji) . This means o e e y l in g he e exis s a sho es pa h om k o lin g+hiji ha does no con ain hiji, a con adic ion. The e o e, Φk(g)<Φk(g+hiji). Theo em 2. Suppose hiji ∈ g , and le k be an agen dis inc om i and j . Then, Φk(g− hiji)<Φk(g) i and only i he e exis s a leas one agen l∈g such ha dkl(g)≥ 2and all sho es pa hs Pkl(g) om k o l in g con ain hiji. We skip he p oo as i is simila o he p oo o Theo em 1. Games 2019,10, 44 7 o 17 In subsequen sec ions, we p esen ou esul s due o link addi ion. We p esen ou esul s on link dele ion in Appendix B. 3.2. E ec o Closeness on Dis ances o Agen s No In ol ed in Link Al e a ion In his sec ion, we classi y agen s whose mu ual dis ances om each o he emain he same a e link al e a ion. We use he same o analyze he e ec o closeness on dis ances be ween agen s who a e no in ol ed in he link addi ion o dele ion. Gi en k such ha Φk(g)<Φk(g+hiji) , we use L+ k o deno e he se o all l∈g such ha all sho es pa hs om k o lin g+hijicon ain hiji. We use l+ k o deno e an agen in L+ k. P oposi ion 1. Suppose i , j , and k a e dis inc agen s in g . Suppose l is ano he agen , dis inc om i and k , and suppose Φk(g+hiji)>Φk(g) . I dki(g+hiji)<dkj(g+hiji)≤dkl(g+hiji) , hen dik(g+hiji) = dik(g). P oo . We ha e Φk(g)<Φk(g+hiji) . Then, om Theo em 1, he e exis s l∈g such ha all sho es pa hs Pkl(g+hiji) om k o lin g+hijicon ain hiji. We conside he wo cases j=land j6=l. 1. Suppose j=l . As dki(g+hiji)<dkj(g+hiji) , k obse es i be o e j on all sho es pa hs Pkl(g+hiji). This implies dik(g+hiji) = dik(g). 2. Suppose j6=l . As dki(g+hiji)<dkj(g+hiji)≤dkl(g+hiji) , k obse es i be o e j , and j be o e l , on all sho es pa hs Pkl(g+hiji). This implies dik(g+hiji) = dik(g). De ini ion 3. Suppose hiji 6∈ g and k is an agen such ha Φk(g)<Φk(g+hiji) . A (k , +ij) -sho es -pa h-ne wo k, gk+ ij , is a subne wo k o g+hiji ha consis s o all sho es pa hs om k o l+ kin g+hiji, which con ain hiji, o all l+ k∈L+ k. De ini ion 4. An ( all k , +ij) -sho es -pa h-ne wo k, g+ ij , is S k∈g, Φk(g)<Φk(g+hiji) gk+ ij , he smalles ne wo k consis ing o all (k,+ij)-sho es -pa h-ne wo ks. De ini ion 5. A sub- (i , +) -ne wo k, g+ i o g+ ij , is he induced subne wo k o g+ ij consis ing o all agen s k∈g+ ij such ha dik(g) = dik(g+hiji) . Simila ly, we de ine he sub- (j , +) -ne wo k, g+ j o g+ ij , as he induced subne wo k o g+ ij consis ing o all agen s l ∈g+ ij such ha djl(g) = djl(g+hiji). Re e o Appendix A o an illus a ion o he abo e de ini ions. P oposi ion 2. Fo all k,¯ k∈g+ i, dk¯ k(g) = dk¯ k(g+hiji). P oo . I k , ¯ k∈g+ i hen, om De ini ion 5, dik(g) = dik(g+hiji) and di¯ k(g) = di¯ k(g+hiji) . As k , ¯ k∈g+ ij as well, he e exis s land ¯ lsuch ha dkl(g)>dkl(g+hiji)and d¯ k¯ l(g)>d¯ k¯ l(g+hiji). I is su icien o show ha , gi en ¯ k,¯ lcan ne e be k. I possible, le ¯ l=k . Then, om De ini ion 5, d¯ ki(g) = d¯ ki(g+hiji) implies ¯ k obse es i i s , and subsequen ly j o each k , on all sho es pa hs P¯ kk(g+hiji) om ¯ k o k in g+hiji . Then, dik(g)6= dik(g+hiji) . This is because, i k=j , dik(g)<dik(g+hiji= 1. The e o e, k6∈ g+ i , which is a con adic ion. Now, i k6=j , hen k mus i s isi j , and la e i , o each ¯ k on all sho es pa hs Pk¯ k(g+hiji) om ¯ k o k . This implies dik(g)6=dik(g+hiji) and hence, k6∈ g+ i , again, a con adic ion. Thus, k6=¯ l. We discuss ou esul s on sho es dis ances due o link dele ion in Appendix B.1. Games 2019,10, 44 8 o 17 3.3. E ec o Link Al e a ion on S o age A ailabili y Ou aim he e is o analyze unde wha condi ions agen s’ chance o ob aining s o age space in he ne wo k inc eases o dec eases by adding a new link. We p esen ou esul s in he case o link dele ion in Appendix B.2. Lemma 5. Suppose agen i and j add a di ec link in g and le k6∈ g+ ij . Then, αik(g) = αik(g+hiji) and αjk(g) = αjk(g+hiji). P oo . I agen k6∈ g+ ij hen Φk(g) = Φk(g+hiji) . Thus, dki(g) = dki(g+hiji) . The e o e, om Equa ion (2), αik(g) = αik(g+hiji). A simila p oo holds o j oo. Lemma 6. Suppose agen s i , j , k , and l a e such ha i6=j , j6=k , i6=l , and k6=l . (Agen s i and k may be he same, and agen s j and l may be he same). Suppose hiji 6∈ g, k ∈g+ i, and l ∈g+ j. Then, 1. αkl(g)<αkl(g+hiji), and 2. i 6=k implies ha αik(g)>αik(g+hiji). Simila ly, i j 6=l, hen αjl(g)>αjl(g+hiji). P oo . Re e Appendix C o he p oo . Lemma 7. Le k and ¯ k be agen s in g+ i. Then, αk¯ k(g) = αk¯ k(g+hiji)and α¯ kk(g) = α¯ kk(g+hiji). P oo . The p oo ollows om P oposi ion 2. Theo em 3. Suppose agen s i and j a e such i6=j , and hiji 6∈ g . Then, γi(g)<γi(g+hiji) i and only i ∏ k∈g+ i (1−αik(g+hiji)) ∏ l∈g+ j (1−αil(g)) < ∏ k∈g+ i (1−αik(g)) ∏ l∈g+ j (1−αil(g+hiji)) . Addi ionally, γi(g)<γi(g+hiji)i and only i ∏ k∈g+ i (αik(g+hiji)) ∏ l∈g+ j (αil(g)) > ∏ k∈g+ i (αik(g)) ∏ l∈g+ j (αil(g+hiji)) . P oo . The p oo ollows om Lemmas 5,6, and 7. 3.4. Ex e nali ies In his sec ion, we s udy ex e nali ies, ha is, how a link ha is added be ween a pai o agen s a ec s he u ili y o o he s. (Re e o De ini ion 6). The pa icula o m o ex e nali ies (posi i e, nega i e, o none) is c ucial in de e mining which ne wo k is likely o e ol e and he condi ions unde which i will lead o a s able and e icien ne wo k. De ini ion 6. [ 32 ] Conside a ne wo k, g , wi h agen s i , j∈g such ha i6=j and hiji/∈g . Suppose agen s i and j o m a di ec link hiji. Then, agen k ∈g {i,j}expe iences 1. Posi i e ex e nali ies i uk(g+hiji)>uk(g); 2. Nega i e ex e nali ies i uk(g+hiji)<uk(g); 3. No ex e nali ies i uk(g+hiji) = uk(g). We now show ha he ype o ex e nali ies an agen k∈g expe iences, can be de e mined using condi ions on he s o age a ailabili y, independen o he da a loss a e and he alue ha agen s associa e wi h hei da a. P oposi ion 3. In an SSSC g, an agen k ∈gexpe iences 1. Posi i e ex e nali ies i γk(g+hiji)>γk(g); Games 2019,10, 44 15 o 17 As k∈g+ i and l∈g+ j , dkl(g)>dkl(g+hiji) , hence, Φk(g)<Φk(g+hiji) and Φl(g)<Φl(g+ hiji). I is easy o see ha ne wo k g in Figu e A2a is he one whe e adding link hiji leads o he maximum inc emen in l ’s closeness and he minimum dec emen s in he dis ances be ween l and km , (m= 1, 2 . . . , n− 4, whe e n is he numbe o agen s in g ), he maximum and he minimum being ac oss all ne wo k s uc u es. We conside wo cases, j6=land j=l. Suppose j6=l. Conside he ne wo k g, as shown in Figu e A2a. F om Equa ion (1), Φl(g) = 1 dlj(g)+1 dlx(g)+1 dli(g)+∑ k∈g, dkl(g)=4 1 dkl(g) =1+1 2+1 3+n−4 4=3n+10 12 . Wi hou loss o gene ali y, le k=km o some m∈ { 1, 2, . . . , n− 4 } . Then, om Equa ion (2), αkl(g) = (1 4) Φl(g)=3 3n+10 . I agen s i and j add a di ec link in g , we ha e ne wo k g+hiji , as shown in Figu e A2b. Then, om Equa ions (1) and (2), we ha e Φl(g+hiji) = n+2 3and αkl(g+hiji) = 1 n+2. F om he abo e, clea ly, αkl(g)<αkl(g+hiji). (a)g(b)g+hiji Figu e A2. Ne wo k s uc u e gand g+hijiwi h nagen s. Now, suppose l=j. F om Equa ions (1) and (2) applied o Figu e A2a,b, we ha e Φj(g) = 2n+7 6 , αkj(g) = 2 2n+7 , Φj(g+hiji) = n+2 2, and αkj(g+hiji) = 1 n+2. I is easy see ha αkj(g)<αkj(g+hiji)in his case as well. This comple es he p oo o 1. Appendix D. Expe imen al Resul s We conduc andom expe imen s o answe he ollowing ques ion. Though agen s o m links and backup hei da a wi h adjacen agen s, can any agen s ill lose i s da a? F om Theo em 5, he‘null ne wo k is he unique pai wise s able ne wo k when he cos o add links is “high”, ha is, ς≥β λ , and pai s o agen s wi h links be ween hem is he unique s able ne wo k o he wise. The e o e, as a as o ma ion o ne wo ks is conce ned, we always ob ain one o hese wo ne wo ks, depending on he alues o ς , β , and λ . In ou expe imen , we andomly gene a e ne wo ks o he second ype, namely, pai wise links. We gene a e such ne wo ks on 30 agen s and conside 150 andom scena ios, by gene a ing 10 andom ne wo ks, 5 di e en se s o andomly chosen agen s whose s o age disks ail, o 3 cases, λ= 1%, 2%, and 4%. Ou assump ion on he alues o λ is based on da a om Backblaze 9 on ha d d i e ailu e a es. In e es ingly, in none o he andom cases we gene a ed did agen s on he wo sides o a link ail a he same ime. 9h ps://www.backblaze.com/blog/backblaze-ha d-d i e-s a s-q1-2019/ (accessed on 04 Sep embe 2019). Games 2019,10, 44 16 o 17 Re e ences 1. Cha d, K.; Bubendo e , K.; Ca on, S.; Rana, O.F. Social cloud compu ing: A ision o socially mo i a ed esou ce sha ing. IEEE T ans. Se . Compu . 2012,5, 551–563. [C ossRe ] 2. T an, N.; Chiang, F.; Li, J. E icien coope a i e backup wi h decen alized us managemen . T ans. S o age 2012,8, 8:1–8:25. [C ossRe ] 3. G acia-Tinedo, R.; Sánchez-A igas, M.; Ga cía-López, P. F2Box: Cloudi ying F2F s o age sys ems wi h high a ailabili y co ela ion. 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