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graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular
social network A social network is a social structure consisting of a set of social actors (such as individuals or organizations), networks of Dyad (sociology), dyadic ties, and other Social relation, social interactions between actors. The social network per ...
s, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998). Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of the clustering in the network, whereas the local gives an indication of the extent of "clustering" of a single node.


Local clustering coefficient

The local clustering coefficient of a vertex (node) in a graph quantifies how close its
neighbours ''Neighbours'' is an Australian television soap opera that has aired since 18 March 1985. It was created by television executive Reg Watson. The Seven Network commissioned the show following the success of Watson's earlier soap '' Sons and ...
are to being a clique (complete graph). Duncan J. Watts and Steven Strogatz introduced the measure in 1998 to determine whether a graph is a
small-world network A small-world network is a graph characterized by a high clustering coefficient and low distances. In an example of the social network, high clustering implies the high probability that two friends of one person are friends themselves. The l ...
. A graph G=(V,E) formally consists of a set of vertices V and a set of edges E between them. An edge e_ connects vertex v_i with vertex v_j. The
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
N_i for a vertex v_i is defined as its immediately connected neighbours as follows: : N_i = \. We define k_i as the number of vertices, , N_i, , in the neighbourhood, N_i, of vertex v_i. The local clustering coefficient C_i for a vertex v_i is then given by a proportion of the number of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them. For a directed graph, e_ is distinct from e_, and therefore for each neighbourhood N_i there are k_i(k_i-1) links that could exist among the vertices within the neighbourhood (k_i is the number of neighbours of a vertex). Thus, the local clustering coefficient for directed graphs is given as : C_i = \frac. An undirected graph has the property that e_ and e_ are considered identical. Therefore, if a vertex v_i has k_i neighbours, \frac edges could exist among the vertices within the neighbourhood. Thus, the local clustering coefficient for undirected graphs can be defined as : C_i = \frac. Let \lambda_G(v) be the number of triangles on v \in V(G) for undirected graph G. That is, \lambda_G(v) is the number of subgraphs of G with 3 edges and 3 vertices, one of which is v. Let \tau_G(v) be the number of triples on v \in G. That is, \tau_G(v) is the number of subgraphs (not necessarily induced) with 2 edges and 3 vertices, one of which is v and such that v is incident to both edges. Then we can also define the clustering coefficient as : C_i = \frac. It is simple to show that the two preceding definitions are the same, since : \tau_G(v) = C(,2) = \frack_i(k_i-1). These measures are 1 if every neighbour connected to v_i is also connected to every other vertex within the neighbourhood, and 0 if no vertex that is connected to v_i connects to any other vertex that is connected to v_i. Since any graph is fully specified by its
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...
''A'', the local clustering coefficient for a simple undirected graph can be expressed in terms of ''A'' as: : C_i=\frac\sum_ A_A_A_ where: : k_i=\sum_j A_ and ''Ci''=0 when ''ki'' is zero or one. In the above expression, the numerator counts twice the number of complete triangles that vertex ''i'' is involved in. In the denominator, ''ki2'' counts the number of edge pairs that vertex ''i'' is involved in plus the number of single edges traversed twice. ''ki'' is the number of edges connected to vertex i, and subtracting ''ki'' then removes the latter, leaving only a set of edge pairs that could conceivably be connected into triangles. For every such edge pair, there will be another edge pair which could form the same triangle, so the denominator counts twice the number of conceivable triangles that vertex ''i'' could be involved in.


Global clustering coefficient

The global clustering coefficient is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. A triangle graph therefore includes three closed triplets, one centred on each of the nodes ( n.b. this means the three triplets in a triangle come from overlapping selections of nodes). The global clustering coefficient is the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). The first attempt to measure it was made by Luce and Perry (1949). This measure gives an indication of the clustering in the whole network (global), and can be applied to both undirected and directed networks (often called transitivity, see Wasserman and Faust, 1994, page 243). The global clustering coefficient is defined as: : C = \frac. The number of closed triplets has also been referred to as 3 × triangles in the literature, so: : C = \frac. A generalisation to weighted networks was proposed by Opsahl and Panzarasa (2009), and a redefinition to two-mode networks (both binary and weighted) by Opsahl (2009). Since any simple graph is fully specified by its
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...
''A'', the global clustering coefficient for an undirected graph can be expressed in terms of ''A'' as: : C=\frac where: : k_i=\sum_j A_ 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 n : : \bar = \frac\sum_^ C_i. 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 graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...
s (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 The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in Randomness, random systems. Below the threshold a giant connected component (graph theory), connected componen ...
(transmission probability) is given by p_c = \frac, where g_1(z) is the
generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
corresponding to the excess degree distribution. In networks with low clustering, 0 < C \ll 1 , the critical point gets scaled by (1-C)^ such that: p_c = \frac\frac. 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 In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
*
Network theory In mathematics, computer science, and network science, network theory is a part of graph theory. It defines networks as Graph (discrete mathematics), graphs where the vertices or edges possess attributes. Network theory analyses these networks ...
*
Network science Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, Cognitive network, cognitive and semantic networks, and social networks, considering distinct eleme ...
*
Percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
*
Scale free network A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction ''P''(''k'') of nodes in the network having ''k'' connections to other nodes goes for large values of ''k'' as : P( ...
* Small world


References


External links

* {{DEFAULTSORT:Clustering Coefficient Graph invariants Algebraic graph theory Network theory Network analysis