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1 Introduction to Networks 1.1 What are networks? Intuitively the networks that we encounter share the features that individual elements of the whole system are coupled by links that form a complicated system of interactions that determine the entire structure or the dynamics that the structure produces. However two basic questions are not addressed: 1. What systems can be modelled by networks? 2. What are interesting networks? If we think about it, almost all systems in nature can be considered as parts that interact by reciprocal forces, so why not model everything as a network? An example from astrophysics can illustrate why the network approach may not necessarily be the right approach for certain systems and should be reserved for those for which it is suitable. Consider a system as illustrated in Fig. 1.1. The figure shows a globular cluster of a couple of thousand stars. All these stars can be considered elements that interact by gravitational force. If we label the stars i then the interaction force between pairs at a given point in time is given the force of gravity Fi j = G Mi M j |xi − x j |2 (1.1) where G = 6.67 × 10−11 m3 /kg s2 is the graviational constant, M i and M j are the masses of stars i and j , and xi and x j are their positions. So why not consider this a network of nodes i with connections of strength F i j ? Why not investigate a system such as this one with network theory? In physics this system is called a many body problem and it seems strange that these systems aren’t the prime example for systems suitable for network science. The reason for this can be understood by looking at the network systems from another perspective: Networks are systems in which certain connections are missing. In fact, most of the most interesting things about networks emerge because from a set of potential connections some are missing or some are much weaker in effect than others and can be neglected. The important ingredient in networks is that only a subset of potential links are realized. Let’s assume we have a system of N interacting units and let’s assume we can measure their interaction strength A i j . Now we make a histogram of these interaction strengths. Let’s assume we find a scenario as depicted in Fig.1.2a : The A i j are distributed over some interval. It’s difficult to distinguish between interaction types and draw a line that could help us distinguish 1

1 Introduction to Networks Figure 1.1: A globular cluster of stars is a typical example of a many body system, not a network system. between them and potentially neglect a subset of interactions. This scenario is typically seen in the globular cluster system or other many body systems. There are other systems though that might exhibit distributions of interactions that look more like those depicted in Fig. 1.2b and c. In b, the distributions of interactions has two peaks, one centered around zero and the other at some typical value. In a system of this type the structure of interactions would suggest that we could model the system by either setting interactions to “existing” or “not existing”. If we were to draw a picture of the system we would draw a network. A similar situation is encountered in Fig. 1.2c. The difference to b is that the interactions not centered at zero have a broad distribution so that we may ignore the interactions around zero. However, the range of values for those that are in the right half of the distribution still must be considered. In this case we would model this by a weighted network. Many systems encountered in network science never really need to be addressed this way because in many systems there is either a link between nodes or there is not, computers are either connected or they are not, facebook users are either friends or they are not, phonecalls are exchanged or they are not. Yet, it’s important to keep in mind what certain networks mean, in particular in relation to social networks and operational definitions of networks. In quantitative science, quantities must be defined operationally: they are defined by the experiment that quantifies them. Let’s assume we would like to measure a social tie between dolphins (this is actually being done in research) by measuring how close they approach each 2

1 Introduction to Networks Figure 1.2: Distributions of interactions in many body systems. a) interactions range over a wide range b) basically two different interaction strengths exist, one fraction centeredaround A ≈ 0 the other fraction centered around a typical value (dashed line) and c) a peak around A ≈ 0 remains, followed by a gap and the nonzero fraction is distributed broadly. other. We define a distance d and if a pair of dophins approach each other below that distance for a minimum of number N of times each day we say they have a tie. This generates a network of social ties in a swarm of dolphins. This network definition depends on the parameters d and N . The network approach is only viable in this system if whatever conclusions we draw are robust agains small changes in these parameters. This is to be expected if for the interactions we measure something as illustrated in Fig. 1.2b and c, but it is doubtfull that this would be the case if interaction strengths were distributed according to Fig. 1.2a. Bottom line: • Networks have nodes and links • Interesting networks are those in which potential links are missing • Always think about how a network is defined and whether the network approach is the most suitable one for your system. 1.2 Notation and graph theoretic origin Our view of networks is that we’ve got nodes that are connected by links. Mathematicians call networks graphs because one can draw them. In graphing theory a graph G is a collection of nodes V and links E , i.e. G = G (V, E ). The reason why the set of nodes is called V is because nodes are called vertices by mathematicians and links are called edges. We will not use this notation, since our focus is on real networks. 3

1 Introduction to Networks Also the term edge is confusing. Here’s a table of typical expressions often used in place of nodes and links: label label field node vertex actor site link edge tie bond network science mathematics social science physics We will use the terms nodes and links throughout the class. 1.2.1 How to draw networks A key aspect of networks is that you can gain insight about their structure by drawing them onto a piece of paper but how to draw them doesn’t matter. All that counts is their connectivity structure. There’s one exception: spatial networks which we will discuss in detail later. In spatial networks each node as a location in a space, most frequently in a two-dimensional space. 1.2.2 The orgin of graph theory - The Königsberg problem The best way to see the value in abstracting in this way is by looking at the Königsbergproblem. Königsberg is a city in Prussia, formerly in Germany and now part of Russia. A river runs through it and 7 bridges cross to an island in the middle as illustrated in Fig. 1.3. A problem posed a number of centuries ago was: • Is it possible to cross all the bridges in any arbitrary sequence with crossing each bridge once. Euler solved this problem by using network (or rather graph theory), realizing that only the topological situation counts. We drew a picture as shown in Fig. 1.3b. Each node represents a piece of land and the links represent bridges. Euler realized that if you cross each bridge exactly once in a given path there are basically two possibilities: 1. The path starts on one node i and ends at the same node i in which case the number of bridges connecting to that node must be even. Likewise for all other nodes because for every path that goes to a node one must be leaving. 2. If we have a path that starts at node i and ends at node j those two nodes can have an odd number of connections, but all the other nodes must have an even number of connections. If we look at the graph closely, we see that the number of connections to each node is odd and therefore not path exists in which each bridge is crossed exactly once. 4

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