Network Measures Essay
The average degree is a measure of how many edges are in a set. In network measure, it helps in the average degree of all nodes given that it shows how many adjacent nodes the network has in average. It is calculated only when the network has not less than one edge connecting the nodes. In calculation of the average degree, two identified nodes which are adjacent i. e. I, j. Where I and j are not equal, then the set of all neighbor nodes will be denoted as a degree.
Thus in order to find the average degree, all degrees are summed up and later divided by the total number of nodes in the network. The average degree explains the type of network topology whether structured or random . In a direct network, the average degrees of a node must be equal both in and out, though the average degree is not very informative concerning a network thus the degree distribution is always the alternative. In addition, the average degree of a graph say Y is a measure of many edges are present in set E compared to the number of nodes in set N. Given that each edge is incident to two nodes and counts in the degree of the two nodes, then the average degree of a graph will be 2*,E,/N, .
Average Path Length The average path length is the average number of steps along the shortest paths for all the possible pairs of network nodes in respect to the concept of network topology. Thus, average path length is simply a measure of the effectiveness of information on a network. The possible examples of average path length are the typical number of clicks which will lead a person from one website to another website.
In addition, it helps in distinguishing an easily negotiable network from one which is complicated and inefficient which ought to be short in order for it to be effective and desirable. In fundamental nature, the average path length is the possible path length even though a network may have several remotely connected nodes which are adjacent to each other . The average length path is highly applied in life in many aspects; the shortest average length path facilitates quick transfer of information thus reducing cost in the World Wide Web.
The shortest average length path in most real networks have led to the concept of a small world given that everyone is connected to everyone else through the shortest path. In a metabolic network, the efficiency of mass transfer can be judged by studying its average path length and in the case of a power grid; less losses will be realized if only the shortest length path could be chosen. In order to find the average path length, the length between all node pairs are calculated and the short paths are selected and if a node is not reachable from node row, the length of the path between them is then considered to be zero.
Therefore, when constructing the average path length, the average values of all the non-zero elements of the matrix path are taken and the average obtained from all the paths of all the nodes is thus referred to as the average path length. Given that there may be adjacent nodes at each level, all the visited neighbor node is marked and all its adjacent are placed in a stack a process that repeats itself until the stacks becomes empty. These great and practical applications have led to development of models which all tend to focus on the shortest average path length.
Furthermore, according to small world network, it claims that the average path length changes proportionately to log n, where n is the total number of nodes in the network. Thus the average path length depends on the size of the system but does not drastically change with the size of the system. Through link efficiency, the trade off between the number of links and the number of nodes in the average path length can be captured. . In summary, average path length is the middling number of connections which are needed to link pairs of vertices in a given network thus bridging long distances in a network.
Average clustering coefficient When the average clustering coefficient is significantly higher than the random graph constructed on the same set of nodes, then a graph is considered to be a small world. The average clustering coefficient is the average number of edges connecting the adjacent nodes of the vertex divided by the average of the maximum number of edges . In times when the clustering coefficient is small, then the possibility of the adjacent nodes to form connections is less.
The average cluster coefficient elaborates on whether the structured network is either scale free or small world. Thus the average clustering coefficient of the graph is the average of all clustering coefficient over all the nodes in the graph which measures the degree of transitivity of a graph implying that the adjacent nodes themselves are likely to be neighbors. The average clustering coefficient of a network plays a big role in the globe and that’s why it has been addressed by network growing models.
The average clustering coefficient for math co-authorship network is comparatively higher with C = 0. 46, than the corresponding same random graph size and average degree which has C = 5. 9*10^-5 . In essence, the average clustering coefficient can be denoted as C = pCb, where Cb is the clustering coefficient of the simple one mode projection of the bipartite graph and p is the proportionate probability. In a nutshell, the average clustering coefficient is independent of the size of the network but they are affected by probability.
In addition, the value is affected by the presence of isolated vertices in the network given that the clustering coefficient value of isolated vertices is zero thus the presence of which decreases the average clustering coefficient of network. References  Lewis, G. T, Network Science: Theory and Practice. Hoboken, NJ: John Wiley and Sons, 2009.  Snasel, V. , Abraham, A. & Hassanien, A. , Computational Social Network Analysis: Trends, Tools and Research Advances. New York: Springer, 2009.