9/14/22, 6:53 AM L5: Statistical Test For The Frequency of a Network Motif: Network Science – CS-7280-O01
L5: Sta s cal Test For The Frequency of a Network Mo f
Suppo se we are given the 16- node networ k G at the left, and we want to examin e if the FFL networ
k motif (shown at the bottom left) appears too frequently. How would we answer this question?
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First, we need to count how many distinct instances of the FFL motif appear in G. One way to do so is to go through each node u that has at least two outgoing edges. For all distinct pairs of nodes v and w that u connects to uni-directionally, we then check whether v and w are also connected with a uni-directional edge. If that is the case (u,v,w) is an FFL instance. Suppose that the count of all FFL instances in the network G is m(G).
We then ask: how many times would the FFL motif take place in a randomly generated network that a) has the same number of nodes with G, and b) each node u in has the same in-degree and out-degree with the corresponding node
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9/14/22, 6:53 AM L5: Statistical Test For The Frequency of a Network Motif: Network Science – CS-7280-O01
u in G? One way to create is to start from G and then randomly rewire it as follows:
Pick a random pair of edges of (u,v) and (w,z)
Rewire them to form two new edges (u,z) and (w,v)
Repeat the previous two steps a large number of times relative to the number of edges in G.
Note that the previous rewiring process generates a network that preserves the in-degree and out-degree of every node in G. We can now count the number of FFL instances in – let us call this count .
The previous process can be repeated for many randomly rewired networks
(say 1000 of them). This will give us an ensemble of networks . We can use the counts to form an empirical distribution of the number of FFL instances that would be expected by chance in networks that have the same number of nodes and the same in-degree and out-degree sequences as G.
We can then compare the count of the given network with the previous empirical distribution to estimate the probability with which the random variable
is larger than in the ensemble of randomized networks. If that probability is very small (say less than 1%) we can have a 99% statistical confidence that the FFL motif is much more common in G than expected by chance.
Similarly, if the probability with which the random variable is smaller than is less than 1%, we can have 99% confidence that the FFL motif is much
less common in G than expected by chance. The magnitude of relative to the average plus (or minus) a standard deviation of the distribution is also useful in quantifying how common a network motif is.
The method we described here is only a special case of the general bootstrapping method in statistics.
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