topostats
topostats is a package of programs that calculate various statistics on network topologies (that is, on graphs). The computed statistics are defined in the paper "Lessons from Three Views of the Internet Topology: Technical Report", which also gives a sample of the computed statistics. The topostats package computes most but not all of these statistics, and this package does not itself make any plots.
Download
Also available on GitHub
Description of results
The statistics currently output by these tools are:

Number of nodes
The count of unique nodes found in the graph 
Number of edges
The count of undirected edges between nodes found in the graph 
Average node degree
The sum of the degrees for each node, divided by the number of nodes 
Maximum node degree
The largest degree found across all nodes 
Degree distribution exponent
The calculated exponent of a power law distribution that best fits the distribution of degrees in the graph 
Powerlaw maximum degree
The natural cutoff value of the power law distribution that best fits the distribution of degrees in the graph, calculated from the degree distribution exponent 
Normalized average average neighbor degree
The value of each node's average neighbor degree (the sum of its neighbors' degrees, divided by the number of neighbors), summed over all nodes and divided by the number of nodes, then further divided by the number of nodes minus 1 
Normalized maximum average neighbor degree
The maximum value of average neighbor degree (the sum of a node's neighbors' degrees, divided by the number of neighbors) across all nodes, divided by the number of nodes minus 1 
Assortative coefficient
The Pearson correlation coefficient, which indicates the amount of relationship between nodes of similar or dissimilar degrees 
Mean clustering
The local clustering of each node (defined as the ratio of the number of links between a node's neighbors to the maximum possible number of links), summed over all nodes and divided by the total number of nodes. 
Clustering coefficient
The percentage of 3cycles (caused by three nodes that are all connected to each other) among all the connected node triplets in the graph 
Top clique size
The largest number of nodes, when sorted by descending degree, that are completely connected to each other 
Minimum node coreness
The smallest coreness (defined as the maximum k for which, after nodes with degree k are iteratively removed from the graph, the node is still in the graph) across all nodes 
Average node coreness
The coreness (defined as the maximum k for which, after nodes with degree k are iteratively removed from the graph, the node is still in the graph) of all nodes, divided by the number of nodes 
Maximum node coreness
The largest coreness (defined as the maximum k for which, after nodes with degree k are iteratively removed from the graph, the node is still in the graph) across all nodes 
Core size
The number of nodes that have maximum coreness (defined above) 
Minimum degree in core
The smallest degree (prior to node removal) found amongst those nodes with maximum coreness (defined above) 
Fringe size
The number of nodes that have minimum coreness (defined above) 
Maximum degree in fringe
The largest degree (prior to node removal) found amongst those nodes with minimum coreness (defined above) 
Average distance
The number of hops between all pairs of nodes, summed and divided by the number of node pairs. 
Standard deviation of distance
The standard deviation of the distribution of the number of hops between all pairs of nodes 
Average eccentricity
The eccentricity (defined as the maximum distance from one node to any other node) of each node, summed and divided by the number of nodes 
Graph radius
The minimum eccentricity (defined above) across all nodes 
Minimum degree in graph center
The smallest degree amongst all nodes that have minimum eccentricity (defined above) 
Maximum degree in graph periphery
The largest degree amongst all nodes that have maximum eccentricity (defined above) 
Minimum node betweeness
The smallest betweenness (which measures how often a node or edge is on the shortest path between other nodes) across all nodes 
Average node betweeness
The betweenness (which measures how often a node or edge is on the shortest path between other nodes) of all nodes, summed and divided by the number of nodes 
Maximum node betweeness
The largest betweenness (which measures how often a node or edge is on the shortest path between other nodes) across all nodes 
Minimum edge betweeness
The smallest betweenness (which measures how often a node or edge is on the shortest path between other nodes) across all edges 
Average edge betweeness
The betweenness (which measures how often a node or edge is on the shortest path between other nodes) of all edges, summed and divided by the number of edges 
Maximum edge betweeness
The largest betweenness (which measures how often a node or edge is on the shortest path between other nodes) across all edges
Tools and example output
The included tools are
 topology_stats: calculates many simple statistics,
 distance: calculates statistics on the distance between all nodes in the graph,
 betweenness: calculates statistics on the betweenness of all nodes/links in the graph, and
 components: a utility for viewing all connected components in a graph and for optionally extracting the largest connected component into a separate graph file.
Shown below is the output of these tools on the skitter dataset from the data supplement to "Lessons from Three Views of the Internet Topology: Technical Report".
topology_stats output:
Number of nodes: 9204 Number of edges: 28959 Avg node degree: 6.29269882659713 Max node degree: 2070 Degree dist exponent (via CCDF) [warning: can be inaccurate]: 2.13036587884151 Powerlaw maximum degree [warning: can be inaccurate]: 3212 Normalized avg avg neighbor degree: 0.0469423442421394 Normalized max avg neighbor degree: 0.0530352021237445 Assortative coefficient: 0.235624710744895 Mean clustering: 0.456656295004884 Clustering coefficient: 0.0257810954106928 Top clique size: 16 Min node coreness: 0 Avg node coreness: 2.22761842677097 Max node coreness: 27 Core size: 47 Min degree in core: 68 Fringe size: 2460 Max degree in fringe: 5
distance output:
loaded 9204 nodes, 28959 undirected links, 42352206 pairs ... decreasing radius from 0 to 5 with node 1 ... raising diameter from 0 to 5 with node 1 ... raising diameter from 5 to 6 with node 9 ... decreasing radius from 5 to 4 with node 22 ... raising diameter from 6 to 7 with node 3287 ... decreasing radius from 4 to 1 with node 21372 average distance = 3.115 std deviation of distance = 0.635 average eccentricity = 5.108 graph radius = 1 graph diameter = 7 min degree in graph center = 1 max degree in graph periphery = 1
betweenness output:
loaded 9204 nodes, 28959 undirected links, 42352206 pairs min node betweenness = 0.0000e+00 average node betweenness = 2.2997e04 max node betweenness = 2.4114e01 min edge betweenness = 2.3617e08 average edge betweenness = 1.0760e04 max edge betweenness = 8.6025e03
components output:
loaded 9204 nodes, 28959 undirected links component at node 1: 9200 nodes, 28957 undirected links component at node 21372: 2 nodes, 1 undirected links component at node 21437: 2 nodes, 1 undirected links 3 components; largest at node ID 1