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Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Spectra of Random Graphs Ü ¡ À (Xiao-Dong Zhang) Shanghai Jiao Tong University Shanghai 200240, P. R. China [email protected] May 26th 2012 in East China Normal University 2012c‘Åã†E, äï?¬ Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background The Erdos-Renyi model Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background The Erdos-Renyi model Chung’s Model (General random graph with expected degree sequence) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background The Erdos-Renyi model Chung’s Model (General random graph with expected degree sequence) The eigenvalues of directed random graphs Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background The Erdos-Renyi model Chung’s Model (General random graph with expected degree sequence) The eigenvalues of directed random graphs Generally directed random graph model Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Outline Definition and background The Erdos-Renyi model Chung’s Model (General random graph with expected degree sequence) The eigenvalues of directed random graphs Generally directed random graph model Main results Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Definition and background G = (V (G), E(G)) a simple graph, vertex set V (G) = {v1 , · · · , vn } edge set E(G) = {e1 , · · · , em }. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Definition and background G = (V (G), E(G)) a simple graph, vertex set V (G) = {v1 , · · · , vn } edge set E(G) = {e1 , · · · , em }. D(G) = diag(d1 , · · · , dn ) : degree diagonal matrix di : degree of vertex vi (the number of edges incident to vi ). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Definition and background G = (V (G), E(G)) a simple graph, vertex set V (G) = {v1 , · · · , vn } edge set E(G) = {e1 , · · · , em }. D(G) = diag(d1 , · · · , dn ) : degree diagonal matrix di : degree of vertex vi (the number of edges incident to vi ). There are several matrices associated with a graph Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Definition and background G = (V (G), E(G)) a simple graph, vertex set V (G) = {v1 , · · · , vn } edge set E(G) = {e1 , · · · , em }. D(G) = diag(d1 , · · · , dn ) : degree diagonal matrix di : degree of vertex vi (the number of edges incident to vi ). There are several matrices associated with a graph A(G) = (aij ) : Adjacency matrix of G, aij = 1 if vi ∼ vj and aij = 0 otherwise. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Definition and background G = (V (G), E(G)) a simple graph, vertex set V (G) = {v1 , · · · , vn } edge set E(G) = {e1 , · · · , em }. D(G) = diag(d1 , · · · , dn ) : degree diagonal matrix di : degree of vertex vi (the number of edges incident to vi ). There are several matrices associated with a graph A(G) = (aij ) : Adjacency matrix of G, aij = 1 if vi ∼ vj and aij = 0 otherwise. A(G) is a nonnegative symmetric (0, 1) matrix with the zeros on the main diagonal. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of A(G) are denoted by λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of A(G) are denoted by λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G). The eigenvalues are sensitive to the maximum degree, which is a local property. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Laplacian and Normal Laplacian Laplacian (Combinatorial) L(G) = D(G) − A(G) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Laplacian and Normal Laplacian Laplacian (Combinatorial) L(G) = D(G) − A(G) The Laplacian eigenvalues of L(G) are µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G) = 0. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Laplacian and Normal Laplacian Laplacian (Combinatorial) L(G) = D(G) − A(G) The Laplacian eigenvalues of L(G) are µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G) = 0. Normal Laplacian L(G) = D−1/2 L(G)D−1/2 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Laplacian and Normal Laplacian Laplacian (Combinatorial) L(G) = D(G) − A(G) The Laplacian eigenvalues of L(G) are µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G) = 0. Normal Laplacian L(G) = D−1/2 L(G)D−1/2 The Normal Laplacian eigenvalues of L(G) : ν1 (G) ≥ ν2 (G) ≥ · · · ≥ νn (G) = 0. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Laplacian and Normal Laplacian Laplacian (Combinatorial) L(G) = D(G) − A(G) The Laplacian eigenvalues of L(G) are µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G) = 0. Normal Laplacian L(G) = D−1/2 L(G)D−1/2 The Normal Laplacian eigenvalues of L(G) : ν1 (G) ≥ ν2 (G) ≥ · · · ≥ νn (G) = 0. The adjacency eigenvalues are sensitive to the maximum degree, which is a local property, which the normal Laplacian are global property. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref For example v3 c v2 e2 c e3 e1 c v4 c e4 c v1 A(G) = 0 1 0 1 1 1 0 1 0 0 e5 0 1 0 1 0 1 0 1 0 0 v5 1 0 0 0 0 . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of A(G) are 2.1358 > 0.6622 > 0 > −0.6622 > −2.1358 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of A(G) are 2.1358 > 0.6622 > 0 > −0.6622 > −2.1358 the Laplacian eigenvalues are Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of A(G) are 2.1358 > 0.6622 > 0 > −0.6622 > −2.1358 the Laplacian eigenvalues are The normal Laplacian eigenvalues are Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The Erdos-Renyi model The Erdos-Renyi model: Gn,p For a given p, 0 < p < 1, each potential edge of G is chosen with probability p, independently of other edges. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The Erdos-Renyi model The Erdos-Renyi model: Gn,p For a given p, 0 < p < 1, each potential edge of G is chosen with probability p, independently of other edges. The adjacency matrix of the random graph G(n, p) can be viewed as a random symmetric matrix whose diagonal entries are zeros and whose entries above the diagonal are i.i.d random variables, each taking value with probability p and value 0 with probability q = 1 − p. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref For p(n) >> log n, λ1 = (1 + o(1))np. In fact denote by ∆(G) = max{d(v), v ∈ V (G)} P v∈V (G) d(v) ≤ λ1 ≤ ∆(G) n Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref For p(n) >> log n, λ1 = (1 + o(1))np. In fact denote by ∆(G) = max{d(v), v ∈ V (G)} P v∈V (G) d(v) ≤ λ1 ≤ ∆(G) n P d(v) If p(n) >> log n, then v∈V n(G) asymptotically to (1 + o(1))np. and ∆(G) are both Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Further, Furedi and Komlos(1981) showed that the following result Theorem 1 Let G be a random graph in G(n, p). If p is constant (which is independent with n, then λ1 has a normal distribution asymptotically, with expectation (n − 1)p + q and variance 2pq. With probability tending to 1, √ max |λi (G) ≤ 2 pqn + O(n1/3 logn). i≥2 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Generally, Krivelevich and Sudakov(2003) proved that the following result: Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Generally, Krivelevich and Sudakov(2003) proved that the following result: Theorem 2 Let G be a random graph in ER model and ∆ be the maximum degree of G. Then almost surly the largest eigenvalue of the adjacency matrix of G satisfies √ λ1 = (1 + o(1)) max{ ∆, np}, √ where the o(1) term tends to zero as max{ ∆, np} tends to infinity. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Random Subgraph of the Hypercube Further, Soshnikov and Sudakov studied that random subgraph of the hypercube. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Random Subgraph of the Hypercube Further, Soshnikov and Sudakov studied that random subgraph of the hypercube. Hypercube: Qn = (V, E), where the vertex set are all vectors V = {(a1 , · · · , an ) : ai = 0, or1} and two vectors are adjacent if they differ in exactly one coordinate. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Random Subgraph of the Hypercube Further, Soshnikov and Sudakov studied that random subgraph of the hypercube. Hypercube: Qn = (V, E), where the vertex set are all vectors V = {(a1 , · · · , an ) : ai = 0, or1} and two vectors are adjacent if they differ in exactly one coordinate. random subgraph of Qn : each edge of Qn appears randomly and independently with probability p(n). In other words, percolation in Qn . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Random Subgraph of the Hypercube Theorem 3 Let G be a random subgraph of Qn . Then almost surly the largest eigenvalue of the adjacency matrix of G satisfies √ λ1 = (1 + o(1)) max{ ∆, np}, √ where the o(1) term tends to zero as max{ ∆, np} tends to infinity. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Question Question: Let H be a regular graph and G be a random subgraph of H. Is the above result still hold ? Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Chung’s Model Chung’s Model ( General random graph model): For a given sequence ω = (ω1 , · · · , ωn ), define a general random graph model G(ω) as follows: Each potential edge between vertex vi and vj is chosen with probability ωi ωj ρ and is independent of other edges. Where 2 ωmax = max ωi2 < i n X X ωk , k Pn 1 ωi2 e ρ= P , ω= ωi = , d = Pi=1 . n ρn j ωj i=1 ωi i=1 1 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The above conditions makes the 0 ≤ ωi ωj ρ ≤ 1 and therefore can be as a probability pij = ωi ωj ρ Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The above conditions makes the 0 ≤ ωi ωj ρ ≤ 1 and therefore can be as a probability pij = ωi ωj ρ The expected degree at a vertex vi is exactly ωi Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The above conditions makes the 0 ≤ ωi ωj ρ ≤ 1 and therefore can be as a probability pij = ωi ωj ρ The expected degree at a vertex vi is exactly ωi The classical random graph Gn,p is a random graph with expected degree sequence ω = (np, · · · , np). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Basic properties of G(ω) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Basic properties of G(ω) Theorem 4 For a graph G in G(ω), √ 2 P rob(di > ωi − c ω) ≥ 1 − e−c /2 , √ P rob(di < ωi + c ωi ) > 1 − exp{− c2 }. √ 2(1 + c/(3 ωi )) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Basic properties of G(ω) Theorem 4 For a graph G in G(ω), √ 2 P rob(di > ωi − c ω) ≥ 1 − e−c /2 , √ P rob(di < ωi + c ωi ) > 1 − exp{− c2 }. √ 2(1 + c/(3 ωi )) Theorem 5 For G in G(ω), almost surely all vertices vi satisfy √ |di − ωi | ≤ 2( ωi + log n). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The largest eigenvalue of Chung’s Model Chung, Lu and Vu (2003) proved the following result Theorem 6 For aPgraph G in G(ω), denoted by ∆ the maximum degree and n ωi2 de = Pi=1 . n i i=1 ω√ e (1) If d > ∆ log n, then the largest eigenvalue of G is almost e surely (1 + o(1))d. √ (2) If ∆ > delog2 n, then the largest eigenvalue of G is almost √ surely (1 + o(1)) ∆. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Further, they showed Theorem 7 The largest eigenvalue of a random graph in G(ω) is at most √ p e 7 log nmax{ ∆, d}. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Remarks and Problems From Theorem 7, we may see that the largest eigenvalues of √ G in G(ω) is roughly equal to de or ∆ if one of them is much larger than the other. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Remarks and Problems From Theorem 7, we may see that the largest eigenvalues of √ G in G(ω) is roughly equal to de or ∆ if one of them is much larger than the other. On the other hand, for Erdos-Renyi model, is has been proved that the largest eigenvalues of G in G(ω) is equal to √ e ∆}. It is natural to expect that the result (1 + o(1)) max{d, holds for Chung’s model. However, the conjecture does not holds: Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Example: For given ∆ > log2 n and d is constant. Choose the expected degree sequence as follows: There are n1 = ∆nd 3/2 = 0(n) vertices with weight ∆. The remaining vertices have weight d. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Example: For given ∆ > log2 n and d is constant. Choose the expected degree sequence as follows: There are n1 = ∆nd 3/2 = 0(n) vertices with weight ∆. The remaining vertices have weight d. Then the sum of the degrees of vertices are equal to n1 ∆ + (n − n1 )d ≈ nd de = n1 ∆2 ρ + (n − n1 )d2 ρ ≈ √ ∆. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The random graph is defined with this special degree sequence. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The random graph is defined with this special degree sequence. Theorem The largest eigenvalue of the adjacency matrix of G(ω) is almost surely at least √ √ 1 + 5√ (1 − o(1)) ∆ > 1.618 ∆. 2 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Conjecture 8 For a random graph G in G(ω), the largest eigenvalue of G is almost surely upper bound by √ (1 + o(1))(de + if de + √ ∆ is sufficiently large. ∆) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Faloutsos, Faloutsos and Faloutsos (1999) discovered the graph adjacency eigenvalues do not follow the semi-circle law. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Faloutsos, Faloutsos and Faloutsos (1999) discovered the graph adjacency eigenvalues do not follow the semi-circle law. They conjectured that the eigenvalues of the adjacency matrices have a power law distribution with own exponent different from the exponent of the graph. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Faloutsos, Faloutsos and Faloutsos (1999) discovered the graph adjacency eigenvalues do not follow the semi-circle law. They conjectured that the eigenvalues of the adjacency matrices have a power law distribution with own exponent different from the exponent of the graph. Mihail and Papapdimitriou (2002) showed that the eigenvalues of the power law graphs with exponent β are distributed according to a power law for β > 3. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Faloutsos, Faloutsos and Faloutsos (1999) discovered the graph adjacency eigenvalues do not follow the semi-circle law. They conjectured that the eigenvalues of the adjacency matrices have a power law distribution with own exponent different from the exponent of the graph. Mihail and Papapdimitriou (2002) showed that the eigenvalues of the power law graphs with exponent β are distributed according to a power law for β > 3. Chung and Lu further proved the following result Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Theorem 9 For a random graph G in G(ω), if the k−th largest expected √ e 2 n and ∆2 md, e then almost degree ∆k satisfies ∆k > dlog k surely the i−th largest eigenvalue of a random graph in G(ω) is √ (1 + o(1)) ∆i for all 1 ≤ i ≤ k. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Let the degree sequence G(ω) = (w1 , · · · , wn ), wi = ci−1/(β−1) f or i0 ≤ i ≤ n + i0 . β−1 d(β − 2) β − 2 β−1 1 dn , i0 = n . c= β−1 ∆(β − 1) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Let the degree sequence G(ω) = (w1 , · · · , wn ), wi = ci−1/(β−1) f or i0 ≤ i ≤ n + i0 . β−1 d(β − 2) β − 2 β−1 1 dn , i0 = n . c= β−1 ∆(β − 1) G is power law : The number of vertices degree k is proportional to k −β Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs e The second order average degree d, (β−2)2 d (β−1)(β−3) (1 + o(1)), if β > 3 1 2∆ d ln d (1 + o(1)) if β = 3 de = 2 3−β (β−1)∆ d (β−2)2 (1 + o(1)), if 2 < β < 3, (β−1)(3−β) d(β−2) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of Power Law graphs Theorem 10 d For k < n( ∆ log )β−1 and β > 2.5, almost surely the k−th largest n eigenvalues of the random power law graph G with exponent β have power law distribution with exponent 2β − 1, and λk ≈ p ∆i ∝ (i + i0 − 1)−1/((2β−1)−1) , provided that ∆ is large enough. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The Semicircle Law for graphs Wigner proved the most celebrated result in the field of random matrix theory which described the limiting behavior of the bulk of the spectrum of random symmetric matrices and is called semicircle law. This result can be applied to random graph. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The Semicircle Law for graphs Theorem 11 Let G be a random graph of order n in E − R model and λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of the adjacency matrix G. Let WA,n (x) ≡ Then the number of eigenvalues less than or equal to x . n √ lim WA,n (x2σ n) = W (x), n→∞ in probability, where W is an absolutely continuous distribution function with density ( 2 (1 − x2 )1/2 f or |x| ≤ 1 π ω(x) = 0 f or |x| > 1. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref No Semicircle Law for Power graphs The semicircle law for eigenvalues of the power law does not hold. In fact, the distribution of eigenvalues of power law graphs follows power law distribution with different exponent. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref No Semicircle Law for Power graphs The semicircle law for eigenvalues of the power law does not hold. In fact, the distribution of eigenvalues of power law graphs follows power law distribution with different exponent. How about another eigenvalues (for Normal Laplacian eigenvalues) of the power law graphs? Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref No Semicircle Law for Power graphs The semicircle law for eigenvalues of the power law does not hold. In fact, the distribution of eigenvalues of power law graphs follows power law distribution with different exponent. How about another eigenvalues (for Normal Laplacian eigenvalues) of the power law graphs? Chung, Lu, Vu proved that The normal Laplacian eigenvalues satisfy the semicircle law under the condition that the minimum expected degree is relatively large. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The Semicircle Law for Normal Laplacian eigenvalu Let D denote by the diagonal matrix with the (i, i) netry having wi , the expected degree of the i− vertex. Let 2 √ √ √ √ C = ( √ )−1 (D−1/2 AD−1/2 −( w1 ρ, · · · , wn ρ)T ( w1 ρ, · · · , wn ρ)). w Let N (x) be the number of eigenvalues of C less than x and Wn (x) = n−1 N (x). Theorem 12 (Chung, Lu, Vu (2003)) For random graphs in G(W ) with √ satisfying wmin w. Then Wn (x) tends to W (x) in probability as n tends to infinity Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of directed random graphs Directed random graph model: Γn,p : For a given p,0 < p < 1, and vertices {v1 , · · · , vn }, each potential directed edge (vi , vj ) of Γn,p is chosen with probability p for i 6= j, independently of other directed edges. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The eigenvalues of directed random graphs Directed random graph model: Γn,p : For a given p,0 < p < 1, and vertices {v1 , · · · , vn }, each potential directed edge (vi , vj ) of Γn,p is chosen with probability p for i 6= j, independently of other directed edges. Juasz in 1982 consider the largest eigenvalue of directed random graphs and showed the following result: Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Theorem Let D be a directed graph in Γn,p . If p is independent of n, then the largest eigenvalue of D is almost surely np. In fact, for any ε > 0 and δ > 0, we have P rob(| λ1 − p| > δn−1/2+ε ) −→ 0, as n → ∞ n Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref A general directed random model Γ(R, S): For given two sequences R = (r1 , · · · , rn ) and S = (s1 , · · · , sn ) Pn Pn satisfying i=1 ri = i=1 si ≡ τ1 , we define a general directed random graph model Γ(R, S) as follows: Each potential direct edge from vi to vj is chosen with probability riτsj and is independent of other directed edges for i 6= j. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Here, we must assume that rmax smax < n X i=1 ri . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Here, we must assume that rmax smax < n X ri . i=1 the above condition guarantees viewed as a probability, ri sj τ ≤ 1 and therefore can be P rob(aij = 1) = ri sj τ Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The expected out-degree at vertex vi is exactly X P rob(vi → vj ) = ri j and the expected in-degree at vertex vi is exactly X j P rob(vj → vi ) = si Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref The expected out-degree at vertex vi is exactly X P rob(vi → vj ) = ri j and the expected in-degree at vertex vi is exactly X P rob(vj → vi ) = si j The classical directed random graph Γn,p is a directed random graph Γ(R, S) with R = S = (np, · · · , np). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Main Results Theorem Let D be a random digraph in −(R, S) with R = (r1 , · · · , rn ), S = (s1 , · · · , sn ) and Pn Pn i=1 ri = i=1 si = m. Let ∆ = max{ri , 1 ≤ i ≤ n} and Pn 1 d = m j=1 rj sj . Then the spectral radius of D satisfies √ q √ 1 P rob(ρ(D) ≤ d + 2 ∆ ln n + 3 ∆ ln n(d + ln n)) ≤ √ . n In other words, the spectral radius of a random graph D is almost surely at most q √ √ d + 2 ∆ ln n + 3 ∆ ln n(d + ln n). Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Lemma Let Xi be independent random variables satisfying Pn |Xi | ≤ M . If X = i=1 Xi , then P rob(|X − E(X)| > a) ≤ exp{ −a2 }. 2(V ar(X) + M a/3) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Lemma Let Xi be independent random variables with P rob(Xi = 1) = pi , P rob(Xi = 0) = 1 − pi . Let Pn X = i=1 ai Xi . Then Pn (1) E(X) = i=1 ai pi Pn (2) V ar(X) = i=1 a2i pi . (3) t2 } P rob(X < E(X) − t) exp{− V ar(X) Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Lemma Let A be an n × n nonnegative matrix. For any positive constants c1 , · · · , cn , the largest eigenvalue λ(A) satisfies Pn j=1 aij cj λ(A) ≤ max { } 1≤i≤n ci Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Let A(D) = (aij ) be the adjacency matrix of D. Then aij is independent random variable. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Let A(D) = (aij ) be the adjacency matrix of D. Then aij is independent random variable. √ Let x = ∆ ln n, and ci = ri for ri > x and ci = x for otherwise. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Let A(D) = (aij ) be the adjacency matrix of D. Then aij is independent random variable. √ Let x = ∆ ln n, and ci = ri for ri > x and ci = x for otherwise. Pn let Xi = j=1 ccji aij . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Case 1 ri > x. Then E(Xi ) = n X cj j=1 ci E(aij ) X cj ri sj cj ri sj + ci m c m rj ≤x i X rj X xsj m+ ≤ d + x. = s m rj >x j r ≤x = X rj > x j Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref ν = n X cj ri sj j=1 = m X c2j ri sj rj >x = ci c2i m + X c2j ri sj rj ≤x c2i m x2 X sj 1 X rj2 sj + ri rj >x m ri r ≤x m j ≤ ≤ n ∆ X rj sj ri j=1 m ∆d + x. x 2 + x ri Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Case 2: ri > x. We have E(Xi ) ≤ d + x and ν≤ ∆d + x. x By Lemma, we have λ2 P rob(Xi ≥ E(Xi ) + λ) ≤ e− 2(ν+aλ/3) . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Since ∆ ln n = x2 , λ2 3 ≥ ln n 2(ν + aλ/3) 2 is equivalent to λ ≥ = ! r ∆ ln n ∆ ln n 2 + ( ) + 12ν ln n x x √ 1 x + x2 + 12ν ln n 2 1 2 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Choose λ=x+ q 3x(d + ln n). Then λ = ≥ ≥ q 1 x + (x + 2 3x(d + ln n) ) 2 √ 1 x + (x + 12ν ln n ) 2 √ 1 x + x2 + 12ν ln n . 2 Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Therefore P rob(Xi ≥ E(Xi ) + x + q 3x(d + ln n)) ≤ n−3/2 . Hence we have q √ √ P rob(Xi ≥ d + 2 ∆ ln n + 3 ∆ ln n(d + ln n)) q √ √ ≤ P rob(Xi ≥ E(Xi ) + ∆ ln n + 3 ∆ ln n(d + ln n)) ≤ n−3/2 . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref By Lemma √ q √ P rob(ρ(D) ≥ d + 2 ∆ ln n + 3 ∆ ln n(d + ln n)) q √ n X √ ≤ P rob(Xi ≥ d + 2 ∆ ln n + 3 ∆ ln n(d + ln n)) i=1 −1/2 ≤ n . Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref References B. Bollobás Random graphs, second ed., Cambridge University Press, Cambridge, 2001. F. Chung , L. Lu and V. Vu, Eigenvalues of random power law graphs,, Annals of Combinatorics, 7(2003) 21-33. F. Chung and L. Y. Lu, Complex graphs and networks AMS Publications, 2006. Outline Definition and background E-R Model The eigenvalues of directed random graphs A general directed random model Ref Thank you for attention!