The maximum steps required to cross the network is three, which would seem to indicate a network without a lot of clustering. This seems to be a fairly low value relative to the total number of nodes (236).

About 20 percent of the nodes have an eccentricity value of 2, with the remaining 80 percent requiring three steps to reach the furthest member of the graph. This provides further confirmation that this network is relatively evenly distributed and easily traversed by all nodes. We do not have any outliers at the fringes of the network who require four or five steps to cross the graph (this would require a higher network diameter, of course).

About 21 percent of nodes are connected with one another, based on the output value of 0.213. This is neither an exceptionally high number, nor could it be considered very low. If we reflect on the parameters of this network (grade levels, individual classes), the number appears to make sense in its context. The next step would be to compare this with the same measure from a similar network to judge whether this value is within an expected range (as it seems), or if this network is typically well connected or poorly connected.

It takes nodes close to two steps (1.86) on average to reach any other node in the network. This feels like a reasonable number given the prior measures we have discussed. If our graph density figure was considerably higher, we would then anticipate a lower average path length, as a higher proportion of members would have first degree connections.

As expected in a well-connected network, there are often multiple paths available to reach specific nodes, resulting in only a few edges with very high values; for instance, 97 percent of all edges have values below 0.5, with the range running between 0.34 and 1.0 The literal translation is that there are just 3 percent of cases where more than 50 percent of all nodes must travel along a single path to reach another node most efficiently (that is, shortest path). This informs us that we have a very robust network for information flow, and that even if a number of links were broken, the network would still be well-connected and efficient.

So what have we learned about the structure of this particular network?

Next, we'll look at a handful of centrality statistics from the same network, and determine what sort of story they are telling us about the behavior patterns of individual nodes inside the graph.

There is a considerable variation in the degree ranges within this network (from 18 through 98), as is the case in the majority of network graphs. This is not an extreme variation in the sense of measuring nodes on the Web, but given the total potential number of degrees (n-1 = 235), it does represent a significant variance. To help illustrate these differences, let's have a look at a graph where we have colored the nodes based on degree level—darker colors being indicative of higher degree counts:

Degree centrality expressed using node coloring

We can now see how the high degree nodes tend toward the center of the graph, while those with low degree values are forced to the perimeter; a typical pattern for many layout algorithms. It is also evident that many of the high degree members are in close proximity to one another, an observation which will prove consistent with the lack of significant clustering in this network. This fact will also influence some of the other centrality measures.

This is a much tighter distribution (1.583 through 2.209) than we saw for degrees, largely attributable to the generally close-knit structure of this network. As you can recall, the eccentricity level for all nodes was either 2 or (mostly) 3, an indication that there are no far-flung outliers residing at a great distance from the center of the graph. In cases where there was a greater distribution of eccentricity, we would anticipate a more dispersed distribution of values for this statistic. Visually, this is what we see, again coloring the values based on the statistic, with higher closeness scores (that is, greater distance) indicated by darker nodes:

Closeness centrality expressed using node coloring

Remember that this is a sort of inverse measure—higher values indicate more distant levels of proximity to other members of the network. So the same nodes that had fewer direct connections are also associated with higher closeness measurements, which should come as no surprise. Those in the center of the graph can more easily travel to all other nodes in the network (on average), while members on the perimeter cannot do so quite so easily.

In contrast to the narrowly distributed values within the closeness centrality statistic, we see a far greater range for the eigenvector measurement (0.108 through 1). Now we are attempting to understand the relative importance of connections to gauge whether influential members are likely to be connected with other members of similar standing. Likewise, this measure will tell us whether nodes with lower influence are more likely to be connected with their peers. In this case, both of these patterns appear to be true, giving us a wide range of values. Our example looks like this:

Eigenvector centrality expressed using node coloring

Here, we have a diagram that resembles, but differs slightly from the degree centrality image shown a moment ago. As expected, the higher levels of centrality are concentrated in the center of the graph, with a general progression toward lower levels as we approach the perimeter. Influential nodes are surrounded by other influential nodes, with the inverse also true, that is, low degree members are largely enmeshed with similarly behaving nodes.

There is a rather large spread in betweenness centrality from top to bottom (2.7 to 396.7), with a few nodes having values near zero located around the perimeter of the network. At the other extreme is a total of 8 nodes with betweenness measures of 300 or greater. These members are important players that help to connect diverse portions of the network, although we don't have any that fill the classic role of a bridge between clusters. A view of this statistic seems to confirm what we have already learned about the network:

Betweenness centrality expressed using node coloring

As expected, nodes at the perimeter have very low betweenness scores—there is nowhere to go that requires traveling through these members. However, we can detect a handful of critical members that facilitate network traversal, and notice how they are spread out to a much greater degree than we saw for some of the other measures. This is actually quite logical, as it would make little functional sense to have two nodes with very high betweenness properties sitting adjacent to one another; this would be highly redundant. Instead, we can see a more scattered pattern with one or two high scoring nodes strategically located within the graph, which help to create a more efficient network.

Based on these statistics, we can safely conclude the following:

Our third and final section will examine a few of the clustering measures available in the Statistics tab, and what they can tell us about the behavior patterns within this network.

As we discussed earlier, this is a measure that determines the percentage of available triplets that are fully closed. The statistical measures are 0.502 for the network, and 0.338 to 0.877 for individual nodes. In the case of our school network, the total network number is almost exactly half closed, with the remaining 50 percent still open—two of three edges are connected, but the third edge is missing.

Is this a high or low number? Again, this is somewhat subjective as well as dependent on the individual network, but it does seem to be a rather high figure. If we reflect back to what this network is measuring, this should not come as a complete surprise; we are not looking at long-term relationships or collaborations, but simply interactions between individuals within and across classrooms. The number will almost certainly grow if we restrict it to a single grade level or classroom, as it will reflect the proximity of members throughout the course of a day.

When we examine patterns at the individual member level, we see a wide range from less than 35 percent up to nearly 90 percent of all triplets being closed. Remember that this statistic is inversely correlated to the number of degrees and it begins to make sense. Nodes with relatively few connections are considerably more likely to have a high percentage of closed triplets, while the lowest coefficients come from some of the most aggressively connected members of the network.

There is a very wide range of triangles from the minimum (96) to maximum (1,616) number of triangles at the node level. At first, this variance seems extreme, but if we step back and look at a couple of other statistics, these numbers make complete sense. First, consider that our degree ranges ran from 18 to 98, and second, remember that many of the most popular nodes are connected with one another (the same is true for the least popular ones). This creates a geometric effect that favors the most highly connected nodes, at least in terms of the number of available triplets. However, as we learned from the clustering coefficient figures, a majority of these triangles will remain incomplete.

The modularity statistic splits this graph into five distinct clusters, numbered from 0 through 4. Doing some simple math tells us that each cluster will average close to 50 nodes. This might be satisfactory for our purpose, or if we need further splits, we can employ one of the dedicated clustering algorithms. A quick way to determine if this number is adequate is to color the graph using the modularity class, as we have previously done with the centrality measures.

This network has many Link Communities ranging in size from a single edge to well over 100 edges. What this tells us is not clear from a purely statistical viewpoint, but it can help us decipher interactions between multiple nodes, as defined by their connecting edges. In fact, the two most prominent communities in this graph account for greater than 20 percent of all connections, providing a new insight about the common behaviors that might be less clear if we rely solely on node-based statistics.

Our final clustering-oriented statistic to measure is embeddedness, which will aid us in understanding the depth at which nodes are affiliated with specific groupings. This measure can help explain how specific nodes or groups of nodes cooperate with one another beyond simply being superficially connected. In our school network, we have an average embeddedness level of 24.6 with individual edge values ranging from 0 to 64. If we examine the data further, it tells us that those edges with very low counts (infrequent interactions) are found at the lowest embeddedness levels; conversely, the highest scores typically correspond to frequent interactions between two nodes.

In the first case, this would represent the sort of superficial connection just referenced; perhaps this contact was random or incidental to a certain time and space. As the frequency of contacts between two parties grows (and with it, the embeddedness score) we can be fairly certain that this is not a random pattern, but something more purposeful and structured.

Now that you have learned how to work with a variety of useful statistics, it's time to take them to the next level by combining them with the powerful filtering capabilities in Gephi. We'll look at a number of graphs in this section so that you get a feel of both, the visual output of the various statistics as well as the multiplicative power delivered by the filtering capabilities of Gephi.