# (3) Graph Analysis and Metrics

## Contents

## Intro

*Created by Brighton Ancelin*

This toolbox has four main objectives:

- Help the user import a graph from a file or matrix into MATLAB as a graph object
- Help the user perform common operations and alterations on graph objects
**Provide tools for basic analysis of graph objects and calculation of key metrics**- Provide a variety of different visualization means for graph objects, both time-varying and time-invariant

In this page, we will discuss how to generate a graph metrics struct.

## All-in-One Metrics Struct

A metrics struct contains many different fields of data about a graph. A brief overview of each field can be seen by using the MATLAB command: help getGraphMetrics

There are two main functions associated with graph metrics fields: **getGraphMetrics** and **exportMetricsFile**. The prior can be used to obtain a metrics struct, and the latter can be used to export said metric data into a text file. The latter function will call the prior internally and return the struct, so there's seldom a reason to call the two in succession. An example of proper usage is as follows:

metrics = exportMetricsFile(graph([1,2,3],[2,3,1]),'Title of Graph','filename'); help getGraphMetrics;

$Author Brighton Ancelin Returns a structure of various graph metrics. INPUT: graphObj: Graph object to be analyzed OUTPUT: metrics: Structure with metrics data. Fields: isDirected: true if graph is directed, false if graph is undirected isFullyConn: true if graph is fully, strongly connected. See also: https://en.wikipedia.org/wiki/Connectivity_(graph_theory)#Connected_graph https://en.wikipedia.org/wiki/Strongly_connected_component distances: NxN (N is the number of nodes) matrix where distances(i,j) represents the shortest path distance between node i and node j. distances(i,i) will always equal 0, regardless of self-edges. See also: https://en.wikipedia.org/wiki/Shortest_path_problem nodeCt: Integer number of nodes in the graph. edgeCt: Integer number of edges in the graph. avgPathLength: Average of all shortest paths between distinct nodes in the graph. diameter: Maximum shortest path length in the graph, i.e. maximum value of the aforesaid 'distances' field after matrix linearization. clusterings: Column vector of local clustering coefficients. clusterings(n) will return the local clustering coefficient of the nth node. Nodes with degree 1 or less (i.e. 1 or fewer neighbors) are incapable of forming triangles and are given a default clustering coefficient of 0. See also: https://en.wikipedia.org/wiki/Clustering_coefficient#Local_clustering_coefficient http://www.stevenstrogatz.com/articles/collective-dynamics-of-small-world-networks-pdf avgClustering: Average of all local clustering coefficients as defined above. maxEigenvalue: Maximum eigenvalue of the adjacency matrix. By the Perron-Frobenius Theorem, this eigenvalue has an associated eigenvector whose entries are all nonnegative (for undirected graphs). This is useful for centrality measurements. See also: https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem eigenCentralities: Eigenvector associated with the maxEigenvalue referenced above. eigenCentralities(n) will return the eigencentrality of the nth node. See also: https://en.wikipedia.org/wiki/Eigenvector_centrality degrees: Column vector of degree values. degrees(n) will return the degree of the nth node. See also: https://en.wikipedia.org/wiki/Degree_(graph_theory) degreeCentralities: Column vector of degree centralities, defined as the degree of each node divided by the maximum degree that node could have. degreeCentralities(n) will return the degree centrality of the nth node. See also: https://en.wikipedia.org/wiki/Centrality#Degree_centrality closenessCentralities: Column vector of closeness centralities, defined as the reciprocal of the average of all shortest paths originating from a given node. closenessCentralities(n) will return the closeness centrality of the nth node. See also: https://en.wikipedia.org/wiki/Centrality#Closeness_centrality distanceDistribution: A table of distance distribution data. The 'Distance' vector contains integer values that represent the shortest path lengths. The 'QuantityOfNodePairs' vector contains corresponding quantities of node pairs that have the associated shortest path between them. See also: http://konect.uni-koblenz.de/plots/distance_distribution_plot assortativityByNode: Column vector of assortativity data. assortativityByNode(n) will return the average degree of all neighbors of the nth node. See also: https://en.wikipedia.org/wiki/Assortativity#Neighbor_connectivity assortativityByDegree: Column vector of assortativity data. assortativityByDegree(n) will return the average of the vector assortativityByNode(arr) where arr contains the node indices of all nodes in the graph of degree n. See also: https://en.wikipedia.org/wiki/Assortativity#Neighbor_connectivity GRAPH REQUIREMENTS: - Unweighted