Mathematically, a one-mode projection can be written in terms of the inci- dence matrixBof the original bipartite network as follows. The productBkiBk j will be 1 if and only ifiand jboth belong to the same groupkin the bipartite network. Thus, the total numberPi jof groups to which bothiandjbelong is
Pi j
1
Õ
k1
BkiBk j
1
Õ
k1
BTikBk j, (6.10) where BTik is an element of the transpose BT of the incidence matrix. Equa- tion (6.10) can be written in matrix notation asPBTB, and then×nmatrixP plays a role similar to an adjacency matrix for the weighted one-mode projec- tion onto thennodes. Its off-diagonal elements are equal to the weights in that network, the number of common groups shared by each node pair. Pis not quite an adjacency matrix, however, since its diagonal elements are non-zero, even though the one-mode network itself, by definition, has no self-edges. The diagonal elements have values
Pii
1
Õ
k1
Bki2
1
Õ
k1
Bki, (6.11)
where we have made use of the fact that Bki only takes the values 0 and 1, so thatB2ki Bki. ThusPii is equal to the number of groups to which nodei belongs.
To derive the adjacency matrix of the weighted one-mode projection, there- fore, we would calculate the matrixP BTBand set the diagonal elements equal to zero. To derive the adjacency matrix of the unweighted projection, we would take the weighted matrix and replace every non-zero matrix element with a 1.
By a similar derivation, it is straightforward to show that the other one-mode projection, onto the groups, is represented by a1×1matrixP0BBT, whose off- diagonal elementP0i jgives the number of common members of groupsiand j, and whose diagonal elementPii0 gives the number of members of groupi.
6.7 | M
ultilayer and dynamic networks
(a) (b)
Figure 6.7: Multilayer and multiplex networks.(a) A multilayer network consists of a set of layers, each containing its own network, plus interlayer edges connecting nodes in different layers (dashed lines). An example is a transportation network with layers corresponding to airlines, trains, buses, and so forth. (b) A multiplex network is a special case of a multilayer network in which the nodes represent the same set of objects or people in each layer. For instance, a social network with several different types of connections could be represented as a multiplex network with one layer for each type. Dynamic or temporal networks are another example, where the layers represent snapshots over time of the structure of a single, time-varying network. In principle one can include interlayer edges in a multiplex network, as here, to represent the equivalence of nodes in different layers, although in practice these are often omitted.
A multilayer network is a set of individual networks, each representing nodes of one particular type and their connections, plus interlinking edges between networks—see Fig. 6.7a. The individual networks are referred to as layers. Thus, in our transportation example there might be a layer representing airports and flights, a layer representing train stations and train routes, and so on. Connections between layers could then be used to join nodes that are in the same geographical location, or at least close enough for easy walking.
Many airports have train stations, for instance, and many train stations have bus stops. Paths through the resulting multilayer transportation network then represent possible passenger journeys: a passenger catches the bus to the train station for instance (represented by an edge in the bus route layer), walks from the bus stop into the station (an interlayer edge), then takes a train to their destination (an edge in the train route layer).
An important special case of a multilayer network occurs when the nodes in each layer represent the same set of points, objects, or individuals. Such networks, which are calledmultiplex networks, represent systems in which there
is only one type of node but more than one type of edge. An archetypal example is a social network, in which the nodes represent people, but there are many different kinds of connections between them—friendship connections, family connections, business connections, and so forth—each represented by a separate layer. The fact that the nodes represent the same people in every layer can be captured by interlayer edges connecting each node to its copies in other layers, although in practice such interlayer edges are often omitted for the sake of simplicity.
Another special case of a multilayer network is adynamicortemporal network, a network whose structure changes over time. Most networks in fact do change over time—the Internet, the Web, social networks, neural networks, ecological networks, and many others change on a range of different time-scales. Most studies of networks ignore this fact and treat networks as static objects, which may be a reasonable approximation in some cases, but in others there is much to be learned by observing, analyzing, and modeling the time variation. Many empirical studies have been done on the way networks change over time. The usual approach is to measure the structure of the network repeatedly at distinct time intervals, resulting in a sequence of snapshots of the system, individual networks that can be thought of collectively as a multilayer network in which the layers have a specific ordering in time. In some examples only the edges change over time and not the nodes, in which case we have a multiplex network.
In others the nodes can also appear or disappear, in which case a full multilayer network is needed to capture the structure, with different sets of nodes in dif- ferent layers and interlayer edges between consecutive layers to indicate which nodes are equivalent. In some cases it may be useful to make the interlayer edges directed, pointing forward in time. For instance, when considering the spread of a disease over a time-varying contact network between individuals, possible routes the disease can take are represented by paths through the cor- responding multilayer network, but the interlayer edges can be traversed only in the forward direction in time—catching the flu today can make you sick tomorrow but it cannot make you sick yesterday.
Mathematically, a multiplex network can be represented by a set ofn×n adjacency matricesAα, one for each layerα(or each time point in the case of a dynamic network). Equivalently, one can think of the elementsAαi j of these matrices as forming a three-dimensional tensor, and tensor analysis methods can usefully be applied to multiplex networks [264].
A multilayer network is more complicated. For a multilayer network, one must represent both the intralayer and interlayer edges, with potentially vary- ing numbers of nodes in each layer. The intralayer edges can again be repre- sented with a set of adjacency matricesAα, although the matrices need not all
6.8 | T
rees
be the same size now. If there arenαnodes in layerαthen the corresponding adjacency matrix has sizenα×nα. The interlayer edges can be represented by a set of additionalinterlayer adjacency matrices. The interlayer adjacency ma- trixBαβis an nα×nβrectangular matrix with elementsBi jαβ 1 if there is an edge between nodeiin layerαand nodejin layerβ.
There are many examples of multilayer networks in empirical network stud- ies. As we have said, most real-world networks are time-varying, and hence can be thought of as multiplex or dynamic networks [239]. Many social networks incorporate more than one type of interpersonal interaction and hence can be represented as multiplex networks. There have been a number of studies of transportation networks of the type discussed above, which are true multilayer networks [135, 198], and a range of other examples can be found in the liter- ature. We refer the interested reader to the reviews by Boccaletti et al.[67], De Domenicoet al.[134], and Kiveläet al.[264], and the book by Bianconi [60].
6.8 T
rees
Atreeis a connected, undirected network that contains no loops—see Fig. 6.8a.9
By “connected” we mean that every node in the network is reachable from every The disconnected parts of a network are called
“components”—see Sec- tion 6.12.
other via some path through the network. A network can also consist of two or more parts, disconnected from one another, and if an individual part has no loops it is also called a tree. If all the parts of the network are trees, the complete network is called aforest.
All trees are necessarily simple networks, with no multiedges or self-edges, since if they contained ei- ther then there would be loops in the network, which is not allowed.
Trees are often drawn in arootedmanner, as shown in Fig. 6.8b, with aroot nodeat the top and a branching structure going down. The nodes at the bottom that are connected to only one other node are called leaves.10 Topologically, a tree has no particular root—the same tree can be drawn with any node, including a leaf, as the root node, but in some applications there are reasons for designating a specific root. A dendrogram is one example (see below).
Not many of the real-world networks that we encounter in this book are trees, although a few are. A river network is an example of a naturally occurring
9In principle, one could have directed trees as well, but the definition of a tree as a loopless network ignores edge directions if there are any. This means that a tree is not the same thing as a directed acyclic graph (Section 6.4.1), since the definition of a loop in a directed acyclic graph takes the directions of the edges into account. A directed acyclic graph may well have loops in it if we ignore directions (see, for example, Fig. 6.3).
10It may seem a little odd to draw a tree with the root at the top and the leaves at the bottom.
Traditional trees of the wooden kind are, of course, the other way up. The upside-down orientation has, however, become conventional in mathematics and computer science, and we bow to that convention here.
(a) (b)
Figure 6.8: Two sketches of the same tree. The two panels here show two different depictions of a tree, a network with no closed loops. In (a) the nodes are positioned on the page in any convenient position. In (b) the tree is a laid out in a “rooted” fashion, with a root node at the top and branches leading down to “leaves” at the bottom.
tree (see Fig. 2.6 on page 31). Trees do nonetheless play several important roles in the study of networks. In Chapter 11, for instance, we will study the network model known as the “random graph.” In this model local groups of nodes form trees and we can exploit this property to derive a variety of mathematical results about random graphs. In Section 14.5.1 we introduce the “dendrogram,”
a useful tool that portrays a hierarchical decomposition of a network as a tree.
Trees also occur commonly in computer science, where they are used as a basic building block for data structures such as AVL trees and heaps [9, 122] and in other theoretical contexts like minimum spanning trees [122], Cayley trees or Bethe lattices [388], and hierarchical models of networks (see Sections 14.7.2 and 18.3.2 and Refs. [109, 268, 465]).
Perhaps the most important property of trees for our purposes is that, since they have no closed loops, there is exactly one path between any pair of nodes.
(In a forest there is at most one path, but there may be none.) This is clear since if there were two paths between a pair of nodes A and B then we could go from A to B along one path and back along the other, making a loop, which is forbidden.
This property of trees makes certain kinds of calculations particularly sim- ple, and trees are sometimes used as a basic model of a network for this reason.
For instance, the calculation of a network’s diameter (Section 6.11.1), the be- tweenness centrality of a node (Section 7.1.7), and certain other properties based on shortest paths are all relatively easy with a tree.