Supporting Information to Wylie, D.C. and W.M. Getz. Sick and Edgy: Walk-Counting
as a Metric of Epidemic Spreading on Networks.
Section 1. Structure Metrics I
Metric
Degree Distribution
[2]
Assortativity
[38-40]
Transitivity / Clustering
Coefficient
[1, 4]
Characteristic Path
Length / Radius /
Diameter
[4, 41]
Scope
Local
TwoNode
Threenode
Global
Random Graph
Relatively
Homogenous
Not assortative
Small-World
Very
Homogenous
Not assortative
Scale-Free
Power-law
(very skewed)
Not assortative
Not clustered
Highly
clustered
Not clustered
O(log(n))
Dependent on
Rewiring
probability p:
O(n) at p=0;
becomes
O(log(n)) at
small p>0
None
O(log(n))
(very short)
Community Structure
Global
Not much*
None
[42-47]
Table SI.1. Characteristics of canonical network structures as measured by the most
commonly used structural metrics.
Many different metrics have been suggested for consideration of the “structure”
of a graph/network – e.g., degree distribution, transitivity/clustering coefficients,
presence of certain small subgraphs (“motifs”), diameter, etc. [1-3]. Some of the most
commonly considered such metrics merit further discussion here, and will be described in
order of increasing “globality.”
Perhaps the most familiar, and most local, measure of network structure is the
degree distribution [2]. This measure may be described as the probability distribution for
the quantity k representing the number of links to other nodes of a node picked from the
network in question at random, with uniform probability. Of particular interest in this
regard are scale-free networks [5], which are defined by a power-law degree distribution,
P(k)~k-γ, with, in many relevant cases, the exponent γ<3 [2, 3]. This sort of distribution
may thus be so skewed that all moments above the first diverge, with the consequence
that the minimum transmissibility for possibility of epidemic outbreak declines to zero
[21, 22, 27].
There is a greater variety of slightly less local network metrics which might
collectively be described as “few-node correlation” measures. For example, the
assortativity (or dissortativity) of a network considers the properties of nodes taken in
pairs, measuring the correlation between the degree of a node and that of its neighbors
[38-40]. The transitivity [3] (or, alternately, the clustering coefficient [3, 4]) considers
triples of nodes by monitoring the prevalence of triangles (i.e., collections of three nodes
which are completely connected) in the network. The small-world networks of Watts and
Strogatz [4] have finite (rewiring probability p-dependent) transitivities/clustering
coefficients at all sizes [2].
Milo et. al. [48] proposed the examination of networks for the repeated presence
of certain few-node substructures (motifs) in significant excess of what might be
expected from a suitable randomized null-model. Any of the three network types taken
here as canonical might serve as such an appropriate null-model, depending on the
specific network to be analyzed for motifs.
In contrast to the above, the characteristic path length [4] (and similar measures
such as the diameter and radius [41]) of a network, defined as the longest distance
between any two nodes, is a truly global metric and cannot generally be computed by
considering only small components of the network one-by-one.
Finally, some networks may be divided into a set of subnetworks, or communities,
such that within each community, the density of interconnections between nodes is
appreciably higher than the connection density among nodes taken from distinct
communities [42]. The problem of subdividing networks into communities is clearly
global in nature; a number of methods for approaching this task have been suggested
[42-47].
Section 2. Structure Metrics II – Matrix Methods
Various matrices may be associated with networks, and there have been many
results indicating that the linear algebraic properties of these matrices can contain useful
structural information [37, 49-54]. For the purposes of this paper, one of the most
important such results is also one of the simplest, namely that the number of k-walks (i.e.,
walks of k steps) from node i to node j in a given network G is given by (Ak)ij [49], where
the adjacency matrix A = A(G) is defined by Aij = 1 if there is an edge connecting node i
to node j in G, and Aij = 0 otherwise. This may be proven so by a simple induction; after
noting that the base case for k = 1 is trivially true, consider:
1)
A k 1 ij   A k iq Aqj
q

 A 

k
q| q  j
iq
where, in the second line, a indicates the set of integers q such that node q neighbors
node j (symbolically represented by qj). Thus Equation (16) indicates that the number
of (k+1)-walks from i to j is equal to the sum of the numbers of k-walks from i to q over
all neighbors q of j, completing the inductive argument, since each (k+1)-walk from i to j
consists of a k-walk from i to a neighbor q of j, followed by the step from q to j.
Much more attention has been given to spectral properties of both the adjacency
matrix as defined above and the Laplacian matrix, for which alternative definitions exist
[51]. The simplest version, the combinatorial Laplacian, may be defined by the
component equation Lij = Diij - Aij, where Di is the degree of node i. The spectra of both
the adjacency matrix and the Laplacian matrix have particularly nice properties for
regular networks (that is, networks in which all nodes have the same degree) [50]. An
oft-mentioned property of the Laplacian matrix is that the multiplicity of the smallest
eigenvalue (0) is equal to the number of connected components of the network. For a
connected network, it is known that the magnitude of the second-smallest eigenvalue of L
is related to the number of edges that must be removed to split the network into pieces
[50].
With regard to the “canonical” network structures, a great deal has been learned
regarding their adjacency matrix spectrum [37, 52, 54, 55]. The most relevant features
from the point of view of this paper pertain to the largest (in magnitude) eigenvalues.
Scale-free networks generally have larger principal eigenvalues (scaling with the fourth
root of the number of nodes) than do the other two canonical network species, which, for
fixed average node degree, both feature principal eigenvalues asymptotically independent
of network size [37]. However, Watts-Strogatz small-world networks generally have the
largest number of eigenvalues clustered at approximately the same value as the principal
eigenvalue, while Barabasi-Albert scale-free networks generally have the largest
separation between the magnitude of the principal eigenvalue and the remainder of the
spectrum [37].
This last point is related to the high degree of clustering present in small-world
networks. Networks with community structure also tend to have multiple eigenvalues
clustered just below the principal eigenvalue, and in fact this property is exploited by
many of the algorithms designed to detect communities in networks [3, 44, 46].
Section 3. Construction of Adjustable Network Topologies
Network construction is divided into five phases:
1.) During the first phase, C distinct BA scale-free networks of (n/C) nodes with average
degree dsf (<d) are assembled.
2.) During the second phase, each remaining pair of edges within each network is
connected with probability ((d-dsf)/((n/C)-dsf-1)), so that the average degree of the nodes
is d in each of the C networks.
3.) The third phase of construction involves rearranging edges within each of the C
networks so as to adjust assortativity without altering the degree distribution; this is done
by an algorithm similar to that described in Xulvi-Brunet and Sokolov [40]. Described
briefly, this method involves repeatedly selecting, a particular number of times denoted
by Nsort-swap, pairs of node-disjoint edges (both within the same network) at random and
breaking them. The four nodes involved will then be rewired with two new edges,
chosen either at random (with probability 1-|psort|) or (with probability |psort|) according to
a scheme to adjust assortativity. If the latter option is chosen, the two new edges will be
either (if psort>0) a new edge connecting the two highest degree nodes of the four
involved in the original edges, as well as a connection between the two remaining lower
degree nodes, or (if psort<0) a new edge connecting the two extremal degree nodes
together, along with a new connection between the two nodes of middling degree. If
either of the putative new edges would be identical to a preexisting edge, all changes are
aborted and the iteration begun anew (unless a certain threshold (25*Nsort-swap) number of
total attempts have been made, successfully or unsucessfully, in which case no further
iterations are pursued).
4.) The C networks (which will now be referred to as communities) are finally connected
to each other during the fourth phase. This phase consists of an iterative procedure in
which, at each iteration, one edge is chosen from each of two distinct communities.
These two connections are broken and replaced by two new connections bridging the two
communities in question. One of the new connections connects the first node of the
broken edge in one community to either the first or second node (50% chance of either)
of the broken edge in the other community, while the other new connection connects the
remaining two nodes that were involved in the broken edges to each other. (If either of
the resulting new connections would be the same as a preexisting connection, the
procedure is begun anew.) Note that this procedure leaves the degree of all nodes
invariant, and hence does not affect the degree distribution (though it will generally
change the assortativity of the network). The selected number of iterations executed,
denoted by Ncluster-swap, then controls both how strongly clustered the resulting network is
and how well-defined the various communities will be.
5.) The final phase of “adjustable network” construction consists of a series of edge
swaps between edges which connect distinct communities in order to further adjust the
assortativity of the network, again in a manner very similar to that of Xulvi-Brunet [40].
If the network is composed of C>1 communities then the following procedure is then
repeated iteratively a number of times, denoted by Ncluster-sort-swap: randomly choose two
edges which bridge distinct communities. With probability |psort|, break these two edges
and replace them with one edge connecting the two highest degree nodes together and
one edge connecting the remaining (lower degree) nodes (if psort>0), or with one edge
connecting the two nodes of middling degree together and one connecting the nodes of
extremal degree together (if psort<0). Reject the rearrangement and begin the iteration
anew (unless 25*Ncluster-sort-swap total iteration attempts have been made, in which case no
further iterations are pursued) if either of the new edges would not bridge distinct
communities or if either of the new connections would be the same as a preexisting
connection.