Download Spectra of Random Graphs

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!