clustering coefficient for an undirected graph can be expressed in terms of A as: where: and C=0 when the denominator is zero. Network average clustering coefficient As an alternative to the global clustering coefficient, the overall level of clustering in a network is measured by Watts and Strogatz as the average of the local clustering coefficients of all the vertices : It is worth noting that this metric places more weight on the low degree nodes, while the transitivity ratio places more weight on the high degree nodes. A generalisation to weighted networks was proposed by Barrat et al. (2004), and a redefinition to bipartite graphs (also called two-mode networks) by Latapy et al. (2008) and Opsahl (2009). Alternative generalisations to weighted and directed graphs have been provided by Fagiolo (2007) and Clemente and Grassi (2018). This formula is not, by default, defined for graphs with isolated vertices; see Kaiser (2008) and Barmpoutis et al. The networks with the largest possible average clustering coefficient are found to have a modular structure, and at the same time, they have the smallest possible average distance among the different nodes. Percolation of clustered networks For a random tree-like network without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold (transmission probability) is given by , where is the generating function corresponding to the excess degree distribution. In networks with low clustering, , the critical point gets scaled by such that: This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable. For studying the robustness of clustered networks a percolation approach is developed. See also Directed graph Graph theory Network theory Network science Percolation theory Scale free network Small world References External links Graph invariants Algebraic graph theory Network theory Network analysis