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# This dierential equation has a solution di t m m log t

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Unformatted text preview: d di ( t ) m =, dt t since each new node at each time spreads its m new links randomly over the t existing nodes at time t . This diﬀerential equation has a solution di (t ) = m + m log �t � i . From this solution, we derive an approximation to the degree distribution. 12 Networks: Lecture 6 “Expected” Degree Distribution We ﬁrst note that the expected degrees of nodes are increasing over time. If we ask how many nodes have degree ≤ 100 and we know that a node born at time τ has degree = 100 at time t , then we are equivalently asking how many nodes were born on or after time τ . This implies that at time t , the fraction of nodes having degree less than or equal to 100 would be t −τ . t For any d and any time t , let i (d ) be a node such that di (d ) (t ) = d . The i (d ) resulting cumulative distribution function then is Ft (d ) = 1 − t . Applying this technique to the uniform attachment model, we solve for i (d ) such that �t� d −m i (d ) d = m + m log , which yields = e− m , i (d ) t and therefore the distribution function Ft (d ) = 1 − e − d −m m . This is an exponential distribution with support from m to inﬁnity and a mean degree of 2m. 13 Networks: Lecture 6 Preferential Attachment Model Nodes are born over time and indexed by their date of birth. Assume that the system starts with a group of m nodes all connected to one another. Each node upon birth forms m (undirected) edges with pre-existing nodes. Instead of selecting m nodes uniformly at random, it attaches to nodes with probabilities proportional to their degrees. For example, if an existing node has 3 times as many links as some other existing node, then it is 3 times as likely to be linked to by the newborn node. Thus, the probability that an existing node i receives a new link to the newborn node at time t is m times i ’s degree relative to the overall degree of all existing nodes at time t , or d (t ) m t i . ∑j =1 dj (t ) 14 Networks: Lecture 6 Preferential Attachment Model Since there are tm total links at time t in the system, it follows that ∑t=1 dj (t ) = 2tm. Therefore, the probability that node i gets a new link in j d (t ) time t is i2t . Hence, we can write down the evolution of expected degrees in continuous time as d di ( t ) d (t ) =i , dt 2t with initial condition di (i ) = m (assuming degree is a continuous variable). This equation has a solution: di (t ) = m � t �1/2 . i As before, expected degrees of nodes are increasing over time. Hence to ﬁnd the fraction of nodes with...
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