A unifying framework for measuring weighted rich clubs

Abstract

Network analysis can help uncover meaningful regularities in the organization of complex systems. Among these, rich clubs are a functionally important property of a variety of social, technological and biological networks. Rich clubs emerge when nodes that are somehow prominent or 'rich' (e.g., highly connected) interact preferentially with one another. The identification of rich clubs is non-trivial, especially in weighted networks, and to this end multiple distinct metrics have been proposed. Here we describe a unifying framework for detecting rich clubs which intuitively generalizes various metrics into a single integrated method. This generalization rests upon the explicit incorporation of randomized control networks into the measurement process. We apply this framework to real-life examples, and show that, depending on the selection of randomized controls, different kinds of rich-club structures can be detected, such as topological and weighted rich clubs.

Figure 1. Weighted rich clubs are measured in terms of the weighted connectedness between rich nodes.

The rich club is the set of red (color online) nodes with a richness value above a certain threshold . In this example, the richness parameter is degree and . Size of nodes is proportional to their richness. The rich club is thus the subgraph formed by nodes with degree larger than 3. The weighted connectedness, C, of the rich club is the sum of the weights of the links between the nodes in the subgraph (black lines). The rich-club coefficient ϕ is calculated by dividing the existing weighted connectedness C by the maximal possible weighted connectedness, F. See Eq. 3 and Figure 2.

Figure 2. Nine ways to measure weighted rich clubs, which all simplify to Eq. 5.

The rich-club coefficient ϕ is calculated by comparing the existing weighted connectedness of the rich club to the maximal possible weighted connectedness, F. In each panel, the size of nodes is proportional to their richness, and the width of links to their weight. Red nodes are the members of the rich clubs. Each panel describes an alternate way to define the maximal weighted connectedness, and shows the set of links (black lines) whose collective weight is F. Underlying each metric are different assumptions about how links and weights could, in principle, be alternatively arranged in the network to yield the maximal possible weighted connectedness. Preserving these assumptions in the creation of randomized controls will ensure that the normalized rich-club coefficient ϕ_norm simplifies to Eq. 5. The nine measures in this figure are organized along two dimensions: (i) how many links could contribute to F (rows), and (ii) where the weights associated to these links are drawn from (columns). Left Column (Capped Weight) Assumes links could have any weight up to a specific maximal weight. Middle Column (Globally Selected) Assumes weights are attached to the links, so that the maximal weighted connectivity would be achieved by taking the strongest links from anywhere in the network and placing them inside the rich club. Right Column (Locally Selected) Assumes only links connected to rich nodes can be locally rewired to serve intra-club weighted connectivity. First Row (P Links) Assumes additional links can be added within the club, up to the topological limit P (P = 6 in this example). Second Row (E Links) Assumes the number of links in the club is fixed at the existing number, E (E = 5 in this example). Third Row (All Links) Assumes weights can be redistributed among the links of the network.

Figure 3. Topological and weighted rich-club structures intermingle in networks, but can be distinguished by using different randomized controls.

(a) Visualization of the two weighted networks examined: the white matter network of the human brain (left, data from, link weight: number of fiber tracts) and the global commercial airline network (right, data from, link weight: flights per day). (b–d) The rich-club behavior as measured by applying different metrics to the white matter network of the human brain (left) and the global airline network (right). ϕ_norm is the normalized rich-club coefficient, based on 1, 000 randomized controls. The shaded areas highlight those values for which ϕ_norm is significantly different from 1 (p < .05). Rich-club coefficients were calculated for every unique value of degree in the network. (b) The topological (unweighted) rich-club behavior, based on randomized controls with the same degree sequence as the real network. Both networks show topological rich clubs, in qualitative agreement with what was found in. (c) The weighted rich-club behavior, based on randomized controls characterized by the same topology as the real network, but with uncorrelated weights. (d) The mixed rich-club behavior, based on controls that randomize the topology and also decorrelate the weights. Similar controls were used to measure weighted rich clubs in the networks analyzed in.