Cutting Resilient Networks Shi Cecilia Sherman Holmgren Xing - PowerPoint PPT Presentation

Cutting Resilient Networks Shi Cecilia Sherman Holmgren Xing Cai Fiona Uppsala ) University C Sweden Luc Devroye Mcgill University

Cutting Resilient Networks Paths k and Some Trees cut on - Shi Cecilia Sherman Holmgren Xing Cai Fiona Uppsala ) University C Sweden Luc Devroye Mcgill University

The Model -

Rooted Graph A graph rooted has one . the node Labelled root as . boss ) ( root . models Can be viewed as • networks criminal for , cells botnets terrorist on , ) networks ( manic P2P ions . T muscle .

Destroying Networks a We where do know is not So the boss we . boss I Choose unit node a ) ( root . . . it Remove at random . . It the graph becomes 2 . keep only disconnected , the the containing component boss . I until the 3 root Repeat muscle . . removed is .

Destroying Resilient Network a Assume each IN node has E K We backups . The unit I Choose root node at a . . t of random Remove one . back ( its ) up cut it once if Remove he node all 2 a . backups gone are . only the component 3 Keep . the containing root . the until 4 Repeat root is . gone .

Example

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Example ill the is root une gone .

Example mostly Kian ) We - care of the needed number cuts to the for end process to . This how ' ' hand ' ' to measures the network destroy .

Cutting Rooted Trees -

Trees Cutting Rooted take of We rave ← the tree as rock of the graph .

The k=I case In . with Let rooted be vertices tree a n . In KC IID of the needed Let be destroy cuts to norm • . . C In ) studied has k been in K for I - . - ) Cayley 4970 Mein Moon and trees - C Complete ) Janson 2004 Binary trees - . ) Galton Conditional Watson C Janson 2006 trees - - . - Berry Addario Holmgren C 2014 ) Bro utin , , . ) Split C 2011 Binary Search Trees and Trees Holmgren 2012 - , . ) ) ( Dr their and Moon al C 2009 RRT 1974 - mota ee . .

k An model equivalent I - - I I # . time \ node give each We time Stamp u a - \ .. in 'd Exp Cl ) To - . 9 - 0 1.8 we node at out u . a if Tv still the is in tree u . y 0.7 1.3 1.6 Each still time - are we uniform cutting random a node . from ) Svante Idea 2004 C comes .

Records Tv root time in U is at c- tree I I . ←→ ETA before of died Tv No ancestor v 0.9 1.8 - - min In L I . . up 0.7 0.6 1.3 U : I simply Tv u ) Janson call or record a . X. X record C- a # of # of records cuts =

Generalize Lk k I to > qz Gi ,vT . 05 1. 1.82 Each node timestamps get U • → , , Tiu , Tzu Exp a ) iid . - . . , CIN . 0.31 0.80 Gru Tiu r Game D Let . = s 1.40 3.22 ~ . it Cut Gru still time in at u v • is ↳ 0.83 tree 0.10 1.9 Time dies a y . 1. 29 3.6 2.35 Gru min C Gru u LV w : X. X ← record I - ( simply v ) Gru Call such a or X x. C- record 2- record r an - . = ÷ , of Number records r - . ← - C kn K ) ) In cuts →

k path cut on - a -

An cut path graph The 3 simplest . 8 - . KCA path Bu all graphs Let nodes be n a a . 4 0.9 Gln of nodes For n , K ( k ) 0.7 2 C → path i. the is easiest e. a to , . Which ? the graph hardest cut is Quiz to : 1. I 5 KUPD-H.cn# K ( En ) of # records For =L ~ k , 0.6 I unit . in permutation round . . . t 's ( Rt ) ) T dy It ECK ( normal ) , ) Nco t = . . . , ✓ TgcnJ login Hn = -

Simulation for k=2 . ✓ 227=108 Looks like normal a . Tried prove to . failed But !

More simulations k=z . 9 to = * distribution normal This be cannot a . The expectation Fr is • order . } the ? find Can constant we The order • variance is n .

The Expectation moment node - . f O node I • - . . i → • that Let the Ir indicator be ill is • it a . • i record r . . Every node above ill . Then after :} dies x : • = . " . I ?" i . EC to . . . ) Ir £ I - it eicaamck - ⇒ - ) i , - > = it node , O # - of Gr constant : Density , it , : Summing this • up • ← Only records I - . h " ' - C Rn ) ) FE Tt C , i ) matter E Ir Mar Kr C n = = . . . , For k=2 simulation soon by • 2. , f Fan ( K , ( Hn ) ) 2.5066in E - =

' ( The Variance moment - A node O f • ii. - . . about only We records • I care - • . :•o • : I Similar i expectation • to • *"¥ ' ' t n # • \ . it node Rather complicated constant . [ simulation 0.66 by when k node 2 n jtl • = • } . a C En ) ) ( Var CK ) 0.651 t 2K n n ~ 2 . - , . . be harder Higher to ! moments • seem

The distribution limit Rn Rn , Dies Dies at at RnB at 2 , , I ' f I f Diggs . ! . . ntl = ? Pmi Pn the of the Let be records path breaks • where position k a ( a , - , . , a. the - Png they die Let be Rna time Rn . " . • , , , , Conditioning Puri , Rn Bnj Pu Rn on . ' • , , . , I , , , , 2 -7T Pay Rnj Pt ( Expel ) ) ) ( breaks ' Bin C segment me = . , , . . , off - - . T It of Prob of being nodes record records t a - one - . between Pn ; Png - i , .

The distribution limit Up Uz Us be Unit ( I ] Let iid • . . Expel ? o . - , , , . Pn . Pn U -~UzPn Uz = ,Pn = n Pnp - , , . , , , , too • Koo • • ' . he Ei be also • Let Ez E3 iid approximate We . . . can , . , - th - k k 'T ( Pn Ck ! Ck ! E.) , ) Ea ) Rn Rn U , th Ez n = = ! - - , - , , a , , , - - ' t " Pn , when when dies Pn dies - , . , # Brij rescaled KCP a d→ I Bn Bp : = the pl 7 - D= ' [ J ALL function of rated E Ui Ez moments Us - . . . . , . . , , , . also converge . .

simulation The simulation suggests the of function Bh We do have density not • . that distribution close it's normal But • to a . that everything looks normal normal ! Not is •

Complete Binary Trees , Conditional Galton Watson - Trees and many more , -

The for Landscape k 32 current . EJP Sherman Cai Dev roye 2019 Holmgren , , , . complete 2018 resilient networks Holmgren arxiv Cutting binary Cai trees - , . . . The model k and deterministic random Ber in cut trees - za z un . , if for the would have arxiv boat ? Holmgren been trip Cai C on not , . the A note asymptotic on the Lerch expansion 's . Lopez Transcendent Cai J . L . , . Transforms and 2019 Integral . Special functions .

The challenge Can graph ? be studied in I random even cut any • - Cannot record • use more any . 413 with Possible Gn - ' - • candidate th p n = : ,p AW almost • The Op Cl ) giant is plus tree a , the edges choose unit giant We root in rand . . . C kei ) takes Simulation it • average suggest on , . I giants.IT 1. 42 the cut giant to should be for 1.25 this ? Can • aw we prove trees .

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