Introduction to the Modeling and Analysis of Complex Systems

15.2. TERMINOLOGIES OF GRAPH THEORY 297scientists use “network/node/link,” social scientists use “network/actor/tie,” etc. This is atypical problem when you work in an interdisciplinary research area like network science.We just need to get used to it. In this textbook, I mostly use “network/node/edge” or“network/node/link,” but I may sometimes use other terms interchangeably as well.By the way, some people ask what are the differences between a “network” and a“graph.” I would say that a “network” implies it is a model of something real, while a“graph” emphasizes more on the aspects as an abstract mathematical object (which canalso be used as a model of something real, of course). But this distinction isn’t so essentialeither.To represent a network, we need to specify its nodes and edges. One useful term isneighbor, defined as follows:Node j is called a neighbor of node i if (and only if) node i is connected to node j.The idea of neighbors is particularly helpful when we attempt to relate network modelswith more classical models such as CA (where neighborhood structures were assumedregular and homogeneous). Using this term, we can say that the goal of network representationis to represent neighborhood relationships among nodes. There are manydifferent ways of representing networks, but the following two are the most common:Adjacency matrix A matrix with rows and columns labeled by nodes, whose i-throw, j-th column component a ij is 1 if node i is a neighbor of node j, or 0otherwise.Adjacency list A list of lists of nodes whose i-th component is the list of node i’sneighbors.Figure 15.1 shows an example of the adjacency matrix/list. As you can see, the adjacencylist offers a more compact, memory-efficient representation, especially if the network issparse (i.e., if the network density is low—which is often the case for most real-worldnetworks). In the meantime, the adjacency matrix also has some benefits, such as itsfeasibility for mathematical analysis and easiness of having access to its specific components.Here are some more basic terminologies:Degree The number of edges connected to a node. Node i’s degree is often writtenas deg(i).