Introduction to the Modeling and Analysis of Complex Systems

18.6. MEAN-FIELD APPROXIMATION ON SCALE-FREE NETWORKS 421latter is still p i , but the former is no longer represented by a single mean field. We needto consider all possible degrees that the neighbor might have, and then aggregate themall to obtain an overall average probability for the neighbor to be in the infected state. So,here is a more elaborated probability:(1 − q(k))(1 − ∑ k ′ P n (k ′ |k)q(k ′ )p i) k(18.30)Here, k ′ is the neighbor’s degree, and the summation is to be conducted for all possiblevalues of k ′ . P n (k ′ |k) is the conditional probability for a neighbor of a node with degreek to have degree k ′ . This could be any probability distribution, which could represent assortativeor disassortative network topology. But if we assume that the network is neitherassortative nor disassortative, then P n (k ′ |k) doesn’t depend on k at all, so it becomes justP n (k ′ ): the neighbor degree distribution.Wait a minute. Do we really need such a special distribution for neighbors’ degrees?The neighbors are just ordinary nodes, after all, so can’t we use the original degree distributionP (k ′ ) instead of P n (k ′ )? As strange as it may sound, the answer is an astonishingNO. This is one of the most puzzling phenomena on networks, but your neighbors arenot quite ordinary people. The average degree of neighbors is actually higher than theaverage degree of all nodes, which is often phrased as “your friends have more friendsthan you do.” As briefly discussed in Section 16.2, this is called the friendship paradox,first reported by sociologist Scott Feld in the early 1990s [68].We can analytically obtain the neighbor degree distribution for non-assortative networks.Imagine that you randomly pick one edge from a network, trace it to one of its endnodes, and measure its degree. If you repeat this many times, then the distribution youget is the neighbor degree distribution. This operation is essentially to randomly chooseone “hand,” i.e, half of an edge, from the entire network. The total number of hands attachedto the nodes with degree k ′ is given by k ′ nP (k ′ ), and if you sum this over all k ′ , youwill obtain the total number of hands in the network. Therefore, if the sampling is purelyrandom, the probability for a neighbor (i.e., a node attached to a randomly chosen hand)to have degree k ′ is given byP n (k ′ ) = k′ nP (k ′ )∑kk ′ nP (k ′ ) = k′ P (k ′ )∑′ kk ′ P (k ′ ) = k′〈k〉 P (k′ ), (18.31)′where 〈k〉 is the average degree. As you can clearly see in this result, the neighbor degreedistribution is a modified degree distribution so that it is proportional to degree k ′ . Thisshows that higher-degree nodes have a greater chance to appear as neighbors. If we