Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach

Abstract: We study eigenvalue distribution of the adjacency matrix

A^{(N, p, a l p h a)}

of weighted random bipartite graphs

G a m m a = G a m m a_{N, p}

. We assume that the graphs have

N

vertices, the ratio of parts is

f r a c a l p h a 1 - a l p h a

and the average number of edges attached to one vertex is

a l p h a c d o t p

or

(1 - a l p h a) c d o t p

. To each edge of the graph

e_{i j}

we assign a weight given by a random variable

a_{i j}

with the finite second moment. We consider the resolvents

G^{(N, p, a l p h a)} (z)

of

A^{(N, p, a l p h a)}

and study the functions

f_{1, N} (u, z) = f r a c 1 [a l p h a N] s u m_{k = 1}^{[a l p h a N]} e^{- u a_{k}^{2} G_{k k}^{(N, p, a l p h a)} (z)}

and

f_{2, N} (u, z) = f r a c 1 N - [a l p h a N] s u m_{k = [a l p h a N] + 1}^{N} e^{- u a_{k}^{2} G_{k k}^{(N, p, a l p h a)} (z)}

in the limit

N o i n f t y

. We derive closed system of equations that uniquely determine the limiting functions

f_{1} (u, z)

and

f_{2} (u, z)

. This system of equations allow us to prove the existence of the limiting measure

s i g m a_{p, a l p h a}

. The weak convergence in probability of normalized eigenvalue counting measures is proved.

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