A packed exponential connections (PEC) network is a grid-based network with connectivity and routing results that are competitive with hypercubic networks. The prior results are all empirical, since the structure of the network has been understood only through an indirect existence proof. In this paper we provide the first direct characterization of a PEC network.

The power law arises commonly in modeling the number of vertices of a given degree in large graphs. In estimating the degree of the power law, the typical approach is to truncate by eye the log-log plot, then fit a linear equation to the remaining log-transformed data. Here we formulate a hard-coded truncation rule to replace the visual truncation, justify it by showing that the truncation point goes to infinity and misses a vanishing fraction of the data with probability tending to one, and refine the subsequent regression with a weighting and a way to use the covariation between slope and intercept to optimize the slope estimate.

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions.

Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as “real” communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best” possible community—according to the conductance measure—over a wide range of size scales. We study over one hundred large real-world networks, ranging from traditional and online social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes.

Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually “blend into” the expander-like core of the network and thus become less “community-like,” with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number, which gives a limit to the size of a well-functioning community.

However, this behavior is not explained, even at a qualitative level, by any of the commonly used network-generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as test beds of community-detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network data sets.