Directed Random Graphs with Given Degree Distributions Mariana Olvera-Cravioto Columbia University molvera@ieor.columbia.edu Joint work with Ningyuan Chen July 23th, 2012 ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 1/25
The motivating example: WWW Opte project. Part of the MoMA permanent collection ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 2/25
The World Wide Web ◮ WWW seen as a directed graph (webpages = nodes, links = edges). ◮ Empirical observations: fraction pages > k in-links ∝ k − α , α = 1 . 1 fraction pages > k out-links ∝ k − β , β = 1 . 72 ◮ We want a directed random graph model that matches the degree distributions. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 3/25
Degree distributions ◮ Directed graph on n nodes V = { v 1 , . . . , v n } . ◮ In-degree and out-degree: ◮ m i = in-degree of node v i = number of edges pointing to v i . ◮ d i = out-degree of node v i = number of edges pointing from v i . ◮ ( m , d ) = ( { m i } , { d i } ) is called a bi-degree-sequence. ◮ Target distributions: F = ( f k : k = 0 , 1 , 2 , . . . ) , and G = ( g k : k = 0 , 1 , 2 , . . . ) . ◮ We want the bi-degree-sequence to satisfy: n n 1 1 � � 1( m i = k ) ≈ f k and 1( d i = k ) ≈ g k . n n i =1 i =1 ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 4/25
Simple graphs ◮ Definition: We say that a directed graph is simple if it has no self-loops and at most one edge in each direction between any two nodes. ◮ Definition: We say that ( m , d ) is graphical if there exists a simple directed graph having ( m , d ) as its bi-degree-sequence. ◮ Goal: Choose a graph uniformly at random from all simple graphs having bi-degree-sequence ( m , d ) , where ( m , d ) has approximately the target distributions F and G . ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 5/25
Simple graphs ◮ Definition: We say that a directed graph is simple if it has no self-loops and at most one edge in each direction between any two nodes. ◮ Definition: We say that ( m , d ) is graphical if there exists a simple directed graph having ( m , d ) as its bi-degree-sequence. ◮ Goal: Choose a graph uniformly at random from all simple graphs having bi-degree-sequence ( m , d ) , where ( m , d ) has approximately the target distributions F and G . ◮ Two problems: ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 5/25
Simple graphs ◮ Definition: We say that a directed graph is simple if it has no self-loops and at most one edge in each direction between any two nodes. ◮ Definition: We say that ( m , d ) is graphical if there exists a simple directed graph having ( m , d ) as its bi-degree-sequence. ◮ Goal: Choose a graph uniformly at random from all simple graphs having bi-degree-sequence ( m , d ) , where ( m , d ) has approximately the target distributions F and G . ◮ Two problems: ◮ Construct an appropriate bi-degree-sequence that with high probability will be graphical. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 5/25
Simple graphs ◮ Definition: We say that a directed graph is simple if it has no self-loops and at most one edge in each direction between any two nodes. ◮ Definition: We say that ( m , d ) is graphical if there exists a simple directed graph having ( m , d ) as its bi-degree-sequence. ◮ Goal: Choose a graph uniformly at random from all simple graphs having bi-degree-sequence ( m , d ) , where ( m , d ) has approximately the target distributions F and G . ◮ Two problems: ◮ Construct an appropriate bi-degree-sequence that with high probability will be graphical. ◮ Choose uniformly at random a simple graph from such bi-degree-sequence. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 5/25
The configuration model (Wormald ’78, Bollobas ’80) ◮ For undirected graphs, given a degree sequence d = ( d 1 , . . . , d n ) : ◮ assign to each node v i a number d i of stubs or half-edges; ◮ for the first half-edge of node v 1 choose uniformly at random from all other half-edges, and if the selected half-edge belongs to, say, node v j , draw an edge between node v 1 and v j ; ◮ proceed in the same way for all remaining unpaired half-edges, i.e., choose uniformly from the set of unpaired half-edges and draw an edge between the current node and the node to which the selected half-edge belongs. ◮ The result is a multigraph (e.g., with self-loops and multiple edges) on nodes { v 1 , . . . , v n } . ◮ If we discard any realization that is not simple, we obtain a uniformly chosen simple graph. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 6/25
The directed configuration model ◮ For directed graphs, given a bi-degree-sequence ( m , d ) : ◮ assign to each node v i a number m i of inbound stubs and a number d i of outbound stubs; ◮ pair outbound stubs to inbound stubs to form directed edges by matching to each inbound stub an outbound stub chosen uniformly at random from the set of unpaired outbound stubs. ◮ The result is again a multigraph, but if we discard realizations that have self-loops or multiple edges we obtain a uniformly chosen simple graph. ◮ Questions: ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 7/25
The directed configuration model ◮ For directed graphs, given a bi-degree-sequence ( m , d ) : ◮ assign to each node v i a number m i of inbound stubs and a number d i of outbound stubs; ◮ pair outbound stubs to inbound stubs to form directed edges by matching to each inbound stub an outbound stub chosen uniformly at random from the set of unpaired outbound stubs. ◮ The result is again a multigraph, but if we discard realizations that have self-loops or multiple edges we obtain a uniformly chosen simple graph. ◮ Questions: ◮ What is the probability of the resulting graph being simple? ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 7/25
The directed configuration model ◮ For directed graphs, given a bi-degree-sequence ( m , d ) : ◮ assign to each node v i a number m i of inbound stubs and a number d i of outbound stubs; ◮ pair outbound stubs to inbound stubs to form directed edges by matching to each inbound stub an outbound stub chosen uniformly at random from the set of unpaired outbound stubs. ◮ The result is again a multigraph, but if we discard realizations that have self-loops or multiple edges we obtain a uniformly chosen simple graph. ◮ Questions: ◮ What is the probability of the resulting graph being simple? ◮ Under what conditions is it bounded away from zero as n → ∞ ? ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 7/25
Probability of graph being simple ◮ For the undirected configuration model it is known that if d satisfies certain regularity conditions , the number of self-loops, S n , and the number of multiple edges, M n , satisfy ( S n , M n ) ⇒ ( S, M ) n → ∞ , where S and M are independent Poisson r.v.s. (Janson ’09, Van der Hofstad ’08-’12). ◮ Then, n →∞ P ( graph is simple ) = P ( S = 0 , M = 0) > 0 . lim ◮ The same should be true for the directed version. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 8/25
Regularity conditions Given { ( m n , d n ) } n ∈ N satisfying � n i =1 m ni = � n i =1 d ni for all n , let n P (( N [ n ] , D [ n ] ) = ( i, j )) = 1 � 1( m nk = i, d nk = j ) . n k =1 1. Weak convergence. For some γ, ξ with E [ γ ] = E [ ξ ] > 0 , ( N [ n ] , D [ n ] ) ⇒ ( γ, ξ ) , n → ∞ . 2. Convergence of the first moments. n →∞ E [ N [ n ] ] = E [ γ ] n →∞ E [ D [ n ] ] = E [ ξ ] . lim and lim ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 9/25
Regularity conditions... continued 3. Convergence of the covariance. n →∞ E [ N [ n ] D [ n ] ] = E [ γξ ] . lim 4. Convergence of the second moments. n →∞ E [( N [ n ] ) 2 ] = E [ γ 2 ] n →∞ E [( D [ n ] ) 2 ] = E [ ξ 2 ] . lim and lim ◮ Note: ( N [ n ] , D [ n ] ) denote the in-degree and out-degree of a randomly chosen node. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 10/25
Poisson Limit for Self-Loops and Multiple Edges ◮ Proposition: (Chen, O-C ’12) If { ( m n , d n ) } n ∈ N satisfies the regularity conditions with E [ γ ] = E [ ξ ] = µ > 0 , then ( S n , M n ) ⇒ ( S, M ) as n → ∞ , where S and M are independent Poisson r.v.s with means λ 1 = E [ γξ ] λ 2 = E [ γ ( γ − 1)] E [ ξ ( ξ − 1)] and . 2 µ 2 µ ◮ Proof adapted from the undirected case (Van der Hofstad ’08 -’12). ◮ Theorem: Under the same assumptions, n →∞ P (graph obtained from ( m n , d n ) is simple) = e − λ 1 − λ 2 > 0 . lim ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 11/25
The repeated and erased models ◮ Repeated model: If all four regularity conditions are satisfied, then repeat the random pairing until a simple graph is obtained. ECT, Trento, Italy Directed Random Graphs with Given Degree Distributions 12/25
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