Upgrade Report - Department of Informatics - King's College London

7 GRAPH GENERATION 20(a) Constant m = 3 Varying s(b) Constant s = 50 Varying mFigure 7: Comparison of different parameter effects on undirected random walk graphs N = 500007.2 Preferential attachment and message propagationThis model is essentially a combination of the well known preferential attachment model with an intuitiveflavor of the Twitter link creation procedure. The idea is that there are two ways of creating links. The firstway is the traditional undirected preferential attachment on a directed graph where a new vertex joins andconnects to m other vertices with probability proportional to each vertex’s total degree. The alternativeway is that an existing vertex activates and begins transmitting a message down to its out-edges. Eachrecipient of that message has a probability p to retransmit that message. After the process has died out (orreached a maximum number of hops) then from all the vertices which retransmitted, m of them choose tocreate a link to the originator of the message.One variation of this model consists of two phases. The first phase is the growth phase, which is thesame process as in the preferential attachment model. The second variation of the model alternates betweena preferential attachment step and a message propagation step until the graph reaches a desired size.The the message propagation (or densify) phase, which consists of N steps (N being the number ofvertices of the graph), performs the following per step:1. Given the vertex u i where i is the current step (i ≤ N) add < u i , 0 > to a queue Q where h = 0represents the current number of hops h2. Let q =< u, h > be an entry in our queue. Do the following while the queue Q is not empty• Remove q from Q.• ∀v j : ∃e j = (u, v j ) : With probability p add v j to a list L and < t i , h + 1 > to Q ∀e i = (u, t i )• The above step will be performed until Q is empty.3. Select m vertices {w 0 , · · · , w m } UAR from L and create edges e j = (w j , u i ) 0 < j ≤ mHere we would like to point out that during first step of the message propagation (or densify) procedurewe expect pm vertices to retransmit the message. If pm ≥ 1 then with high probability the process willnever terminate. For that reason we have two different ways of ensuring termination. The first is to neveradd the same vertex to Q or L more than once, which would mean that it will terminate after at most NMsteps (where M is the number of edges). The second is to have an upper bound on the maximum hopsh max for which we will add items in the queue, which will constrain the number of steps to p ∑ h maxi=1m i . Inpractice both these methods are used to optimize both for time and space performance where h max = 16 inthe example we will be presenting.The first variation of the above process generates power-law degree distributions on both the out-degreesand in-degrees as seen below. Additionally some sort of community structure is created with the modularityQ ranging between 0.25 and 0.5. It is empirically observed that p affects Q but we have not yet determinedhow and why this occurs. However intuitively we assume that the aspect of the model which generatesthe community structure is the densification and we would expect that values of p such that pm = 1 willresult in a higher success rate of message propagations while limiting the number of hops that the message