Random Graphs and Complex Networks T

Transcription

1 Radom Graphs ad Complex Networks T Charalampos E. Tsourakakis Aalto Uiversity Lecture 3 7 September 013

2 Aoucemet Homework 1 is out, due i two weeks from ow. Exercises: Probabilistic iequalities Erdös-Réyi graphs Empirical properties of etworks You eed to do 100 out of 150 poits. You all have to do Problems 1.(b) ad 1.3. If you decide to do everythig i the homework, the extra poits cout as bous. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs / 17

3 Aoucemet You ca fid a list of of suggested papers for your project i the class Web page. Topics of iterest iclude: Stochastic graph models Estimatig models from etwork data Strategic graph models Diffusio Social learig Subgraphs Learig Cuckoo hashig Kidey exchage Fiacial etworks... You ca still propose your ow project. Remider: programmig projects i groups of at most persos. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 3 / 17

4 Overview Figure: I the last lecture we proved that the threshold for coectivity i G(, p) is log. Today, we will see the phase trasitio of the giat compoet of G(, p) where p = 1. We will go over two differet proofs which are based o differet tools. Brachig processes (Lecture otes from Staford available o the Web site) Depth first search (Readigs: Krivelevich-Sudakov paper) C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 4 / 17

5 Phase trasitio Michael Krivelevich Bey Sudakov the phase trasitio i radom graphs a simple proof The Erdős-Réyi paper, which lauched the moder theory of radom graphs, has had eormous ifluece o the developmet of the field ad is geerally cosidered to be a sigle most importat paper i Probabilistic Combiatorics, if ot i all of Combiatorics C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 5 / 17

6 [Krivelevich ad Sudakov, 013] give a simple proof for the trasitio based o ruig depth first search (DFS) o G S : vertices whose exploratio is complete T : uvisited vertices U = V (S T ) : vertices i stack observatio: the set U always spas a path - whe a vertex u is added i U, it happes because u is a eighbor of the last vertex v i U; thus, u augmets the path spaed by U, of which v is the last vertex epoch is the period of time betwee two cosecutive emptyigs of U - each epoch correspods to a coected compoet C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 6 / 17

7 Lemma Let ɛ > 0 be a small eough costat ad let N = ( ) Cosider the sequece X = (X i ) N i=1 of i.i.d. Beroulli radom variables with parameter p 1 let p = 1 ɛ ad k = 7 l ɛ the whp there is o iterval of legth k i [N], i which at least k of the radom variables X i take value 1 let p = 1+ɛ ad N 0 = ɛ the whp N 0 i=1 X i ɛ(1+ɛ) /3 C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 7 / 17

8 Phase trasitio useful tools Lemma (Uio boud) For ay evets A 1,..., A, Pr [A 1... A ] i=1 Pr [A i] Lemma (Chebyshev s iequality) Let X be a radom variable with fiite expectatio E [X ] ad fiite o-zero variace Var [X ]. The for ay t > 0, Pr [ X E [X ] t] Var [X ] t Lemma (Cheroff boud, upper tail) Let 0 < ɛ 1. The, Pr [Bi(, p) (1 + ɛ)p] e ɛ 3 p C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 8 / 17

9 Proof. fix iterval I of legth k i [N], N = ( ) the i I X i Bi(k, p) 1. apply Cheroff boud to the upper tail of B(k, p).. apply uio boud o all (N k + 1) possible itervals of legth k - upper boud the probability of the existece of a violatig iterval (N k + 1)Pr[B(k, p) k] < e ɛ 3 (1 ɛ)k = o(1) sum N 0 i=1 X i distributed biomially (params N 0 ad p) - expectatio: N 0 p = ɛ p = ɛ(1+ɛ) - stadard deviatio of order - applyig Chebyshev s iequality gives the estimate C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 9 / 17

10 Proof. We ru the DFS o a radom iput G G(, p), fixig the order σ o V (G) = [] to be the idetity permutatio. The DFS algorithm is give a sequece of i.i.d. Beroulli(p) radom variables X = (X i ) N i=1. The DFS algorithm gets its i-th query aswered positively if X i = 1, ad aswered egatively otherwise. The obtaied graph is clearly distributed accordig to G(, p). C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 10 / 17

11 Proof. CASE I: p = 1 ɛ assume to the cotrary that G cotais a coected compoet C with more tha k = 7 ɛ l vertices cosider the momet iside this epoch whe the algorithm has foud the (k + 1)-st vertex of C ad is about to move it to U deote S = S C at that momet the S U = k, ad thus the algorithm got exactly k positive aswers to its queries to radom variables X i durig the epoch, with each positive aswer beig resposible for revealig a ew vertex of C, after the first vertex of C was put ito U i the begiig of the epoch. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 11 / 17

12 Proof. at that momet durig the epoch oly pairs of edges touchig S U have bee queried, ad the umber of such pairs is therefore at most ( k ) + k( k) < k - it thus follows that the sequece X cotais a iterval of legth at most k with at least k 1 s iside a cotradictio to Property 1 of our Lemma. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 1 / 17

13 The fact that we have performed N 0 queries implies a upper boud o S. Let s see why. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 13 / 17 Phase trasitio proof sketch Proof. CASE II: p = 1+ɛ Assume that the sequece X satisfies Property of our Lemma. Claim: After the first N 0 = ɛ queries of the DFS algorithm, the set U cotais at least ɛ vertices. This 5 meas: the giat compoet cotais O(f (ɛ)) vertices. The fuctio f (ɛ) = ɛ 5 ca be further improved by tighteig the aalysis of the probabilistic lemma. Check [Krivelevich ad Sudakov, 013], page 6. the logest path is O() sice U forms a path.

14 Proof. CASE II: p = 1+ɛ Assume for the sake of cotradictio S 3. Always U 1 + t i=1 X i. Hece ow U < 3. Combiig the above ad the fact that S, T, U are disjoit sets, we get T > 3. Cotradictio! Why? Hece S < 3. Let s assume ow that U < ɛ 5 for the sake of cotradictio. Clearly, T. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 14 / 17

15 Proof. CASE II: p = 1+ɛ Sice T the algorithm is still revealig the coected compoets of G. Each positive aswer it got resulted i movig a vertex from T to U. By property of the lemma, the umber of positive aswers is at least ɛ(1+ɛ) /3. These positive aswers correspod to S, U, amely S U ɛ(1+ɛ) /3. Sice U ɛ, the S ɛ + 3ɛ 5 10 /3. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 15 / 17

16 Proof. CASE II: p = 1+ɛ All S T pairs betwee S, T have bee queried. However S T > N 0, cotradictio! ( ) ɛ = N 0 S S ɛ 5 ( ) ( ɛ + 3ɛ 10 /3 ɛ ) ɛ + /3 = ɛ + ɛ 0 O(ɛ3 ) > ɛ C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 16 / 17

17 refereces I Krivelevich, M. ad Sudakov, B. (013). The phase trasitio i radom graphs - a simple proof. C.E. Tsourakakis T , Giat compoet Erdös-Réyi graphs 17 / 17