Corrected Overlap Weight and Clustering Coefficient

Abstract

We discuss two well known network measures: the overlap weight of an edge and the clustering coefficient of a node. For both of them it turns out that they are not very useful for data analytic task to identify important elements (nodes or links) of a given network. The reason for this is that they attain their largest values on maximal subgraphs of relatively small size that are more probable to appear in a network than that of larger size. We show how the definitions of these measures can be corrected in such a way that they give the expected results. We illustrate the proposed corrected measures by applying them on the US Airports network using the program Pajek.A direct measure of the overlap of an edge e = (u : v) ∈ E in an undirected simple graph G = (V, E) is the number of common neighbors of its end nodes u and v (see Figure 1). It is equal to t(e) -the number of triangles (cycles of length 3) to which the edge e belongs. The edge neighbors subgraph is labeled T (deg(u) − t(e) − 1, t(e), deg(v) − t(e) − 1) -the subgraph in Figure 1 is labeled T (4, 5, 3). There are two problems with this measure:• it is not normalized (bounded to [0, 1]);• it does not consider the 'potentiality' of nodes u and v to form triangles -there are min(deg(u), deg(v)) − 1 − t(e) nodes in the smaller set of neighbors that are not in the other set of neighbors.

Convexity in complex networks

“…Top row of Figure 7 shows different distributions separately for the nodes in the c-core (diamonds) and the periphery (squares). These are the distributions of node degree k and the average geodesic distance to other nodes , i = 1 n−1 ∑ j =i d i j , for C. elegans protein network, and the distribution of corrected node clustering coefficient C µ (Batagelj, 2016), where C µ i = 2t i k i µ and µ is the maximum number of triangles a single edge belongs to, for airline transportation network due to low clustering of the former (see Table 1). Notice that the nodes in the c-core have higher degrees and also clustering coefficient than peripheral nodes, while they also occupy a more central position in the network with lower geodesic distances to other nodes.…”

“…Top row of Figure 7 shows different distributions separately for the nodes in the c-core (diamonds) and the periphery (squares). These are the distributions of node degree k and the average geodesic distance to other nodes , i = 1 n−1 ∑ j =i d i j , for C. elegans protein network, and the distribution of corrected node clustering coefficient C µ (Batagelj, 2016), where…”

Label propagation for clustering

“…Notice that, although the The meta-networks of the Google web graph and the Pennsylvania road network identified by the hierarchical label propagation method. The shades of the nodes are proportional to their corrected clustering coefficient [6], where darker (lighter) means higher (lower). networks are reduced to less than a thousandth of their original size, the group agglomeration process preserves a dense central core of the web graph and a sparse homogeneous topology of the road network [9].…”

“…Figure 1.11 The meta-networks of the Google web graph and the Pennsylvania road network identified by the hierarchical label propagation method. The shades of the nodes are proportional to their corrected clustering coefficient[6], where darker (lighter) means higher (lower).…”