MODELS FOR EVOLUTION AND JOINING OF SMALL WORLD NETWORKS

Transcription

1 MODELS FOR EVOLUTION AND JOINING OF SMALL WORLD NETWORKS By SURESH BABU THIPIREDDY Bachelor of Technology in Computer Science Jawaharlal Nehru Technological University Hyderabad, Andhra Pradesh, India 2007 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2009

2 MODELS FOR EVOLUTION AND JOINING OF SMALL WORLD NETWORKS Thesis Approved: Dr. Subhash Kak Thesis Adviser Dr. Venkatesh Sarangan Dr. Michel Toulouse Dr. A. Gordon Emslie Dean of the Graduate College. ii

3 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my graduate advisor Dr. Subhash Kak for his guidance and help in completing my work. He has been very inspiring and I am very grateful for his encouragement and support throughout my graduate studies. I would also like to thank my graduate committee members Dr. Venkatesh Sarangan and Dr. Michel Toulouse for their support. I would like to thank Abhishek, Sandeep and Krishna for their support in finishing my work and I would also like to thank all my friends who have given me moral support and friendship. Finally, I am very much grateful to my parents for their affection, support and encouragement throughout my life. iii

4 TABLE OF CONTENTS Chapter Page 1. INTRODUCTION Overview Problem description Proposed method Rest of the document 7 2. LITERATURE REVIEW Random networks Scale-free networks Small world phenomenon Existing model for growing a SWN METHODLOGY Model for joining networks Model for growing an existing SWN Other lattice structures RESULTS CONCLUSION.55 BIBLIOGRAPHY.56 iv

5 LIST OF TABLES Table Page 4.1 Joining of two small world networks with same number of nodes Joining of two small world networks with different number of nodes Joining of two random networks with same number of nodes Joining of two random networks with different number of nodes Joining of a SWN and a random network with same number of nodes Joining of a SWN and a random network with different number of nodes..46 v

6 LIST OF FIGURES Figure Page 1.1 A simple social network Watts and Strogatz model Ring lattice structure WS-Model with different probabilities OHO Model ZRG Model ZRC Model ZZSG Model SAV Model Adjacency matrix representation of a SWN with 10 nodes Adjacency matrix representation of a SWN with 20 nodes Adjacency matrix representation of a SWN with 30 nodes Joining of two networks Growth of an existing SWN Square lattice structure Rewired square lattice structure Hexagonal lattice structures Rewired hexagonal lattice structures 33 vi

7 4.1 Probability Vs Clustering coefficient for joint network formed by joining two SWNs with same number of nodes Probability Vs Average path length for joint network formed by joining two SWNs with same number of nodes Probability Vs Clustering coefficient for joint network formed by joining two SWNs with different number of nodes Probability Vs Average path length for joint network formed by joining two SWNs with different number of nodes Probability Vs Clustering coefficient for joint network formed by joining two random networks with same number of nodes Probability Vs Average path length for joint network formed by joining two random networks with same number of nodes Probability Vs Clustering coefficient for joint network formed by joining two random networks with different number of nodes Probability Vs Average path length for joint network formed by joining two random networks with different number of nodes Probability Vs Clustering coefficient for joint network formed by joining a SWN and a random network with same number of nodes Probability Vs Average path length for joint network formed by joining a SWN and a random network with same number of nodes Probability Vs Clustering coefficient for joint network formed by joining a SWN and a random network with different number of nodes 47 vii

8 4.12 Probability Vs Average path length for joint network formed by joining a SWN and a random network with same number of nodes Nodes being added Vs Clustering coefficient for a growing SW network Nodes being added Vs Average path length for a growing SW network Probability Vs Clustering coefficient for a rewired square lattice structure with 25 nodes and 100 nodes Probability Vs Average path length for a rewired square lattice structure with 25 nodes and 100 nodes Probability Vs Clustering coefficient for a rewired two layered and three layered hexagonal lattice structure Probability Vs Average path length for a rewired two layered and three layered hexagonal lattice structure 54 viii

9 CHAPTER 1 INTRODUCTION 1.1 OVERVIEW OF SOCIAL NETWROKS Complex networks serve as model of problems in fields of science ranging from biology to computer science. Such networks are structures consisting of nodes connected by edges. Some of the examples include the Internet which is a network of domains, the World Wide Web which is a network of websites, the brain which is a network of neurons, and an organization which is a network of people. A social network is a social structure represented by a set of nodes which are connected based on interdependencies such as friendship, business, like or dislike. Such a network is characterized by properties like clustering coefficient, degree, shortest path and diameter. A sample social network is shown below. Figure-1.1 A simple social network (showing nodes connected to each other) 1

10 Degree of a social network is the number of edges connected to a node. Local clustering is the extent to which an individual s friends are each other s friends. Diameter of a network is the maximum of the shortest paths between every node in a network. The randomness of a complex structure is an important area of research as it affects the properties of a network. For many years there was an assumption that networks can be represented as regular graphs [16]. But in the late 1950s, Erdos and Renyi (ER), made an important contribution to classical mathematical graph theory [17]. They described a network with complex topology by a random graph, and laid foundation for random network theory. Although most real world networks are neither completely regular nor completely random, the ER model proposed by Erdos and Renyi was a sensible one and it dominated research in several application areas for about half a century [16]. In the past few years, a large collection of data on various real world networks has been stored in huge databases because of the computerization of data entry and growth in computational power. These large databases have helped researchers to analyze the properties of complex networks. Two significant insights that have emerged from the study of these databases are the small-world effect and the scale-free feature of most complex networks [16]. In 1998, in order to describe the transition from a regular lattice to a random graph, Watts and Strogatz (WS) introduced the concept of small-world network [10], 2

11 which is characterized by small diameter. It is notable that the small-world phenomenon is very common. Regular lattices, which are the starting point for the Watts and Strogatz model, are clustered but do not exhibit the small-world effect i.e., the shortest path between every pair of nodes is high. Conversely, random graphs show the small-world effect, but do not show clustering i.e., the probability that the friends of a node are friends of each other is low. Thus, both the regular lattice model and the ER random model do not possess the important property of real networks. Most real-world networks are neither completely regular nor completely random. The reality is that people usually know their neighbors, but their circle of acquaintances may not be confined to those who live right next door, as the lattice model would imply [16]. And also networks formed on the internet are not created at random. So, the study of the small world networks has received a lot of attention within a short period of time. A sample small-world network is shown in the figure below: Figure-1.2 Watts and Strogatz model with probability p=0.32 3

12 Another significant discovery in the field of complex networks is the observation that many large-scale complex networks are scale-free and exhibit power law distribution [3]. It was argued by Barabasi and Albert that many existing models fail to take into account important properties of most real network. Real world networks are open, i.e., the numbers of nodes keep growing over a period of time, thereby increasing the size of the network. Furthermore, uniform probabilities are assumed for both the random graph and small-world models while creating new edges [16], which is not realistic. Scale free networks are formed based on two mechanisms: 1) Population growth, in this real networks grow in time as new members join the network; and 2) Preferential attachment, in this newly arriving node will tend to connect to already well-connected nodes. A significant recent discovery is the observation that a number of large-scale complex networks, including the Internet, WWW, and metabolic networks, are scale-free and their connectivity distributions have a power-law form. 1.2 PROBLEM DESCRIPTION Small-world networks describe many real-life networks, such as the World Wide Web, communication networks, the electric power grid, or social networks that achieve both a strong local clustering and a small average path length [12]. Small-world networks are characterized by two main properties. First, is the characteristic path length, L, which measures the separation between any two nodes in the graph, averaged over all pair of nodes; it has been found that L grows slowly as the size of the network increases. Second, is the clustering coefficient, C, which measures the cliquishness of a neighborhood. The clustering C i of node i is defined by C i = q i / [ k i (k i -1)/2] where q i is the total number of 4

13 links between the k i neighbors of node i and k i (k i -1)/2 is the maximum number of links that could exist between k i nodes [12]. The small world networks generated by the Watts and Strogatz (WS) model have been influential in the field of social networks. Since, the transition from a regular lattice to a small world network shown in WS was the first of its kind; researchers have modified this model in different ways. In the WS model rewiring was done on a ring lattice using a probability p. The main drawback of the Watts and Strogatz simulation was that the number of nodes in the network was fixed. Since real world networks grow over a period of time, the generation of growth algorithm is an important research topic. A few researchers have proposed different models to generate a growing network which exhibit the properties of small world. For example, the OHO model [11] is initially in the form of a circle consisting of three nodes which are connected to each other. Then, a new node is randomly placed on the circle and is connected to m geographically closest neighbors. Li Yong et al proposed a new algorithm which is known as SAV algorithm [15]. The purpose of this algorithm was to generate a small world network from a ring lattice structure. This algorithm offered a different method of generating a small world network, but growth of the network was not considered. The algorithm initially has a fixed number of nodes in the form of a ring lattice. The above models either generate a network using a ring lattice or grow a network with a basic assumption that the network starts with three nodes arranged in the form of a triangle. Not much research has been done to generate an evolving small world network in which a small world network is generated initially and then this small world network is 5

14 grown gradually. It is possible in the real world networks that instead of growing a small world network node by node, a group of nodes join to form a network and then get attached to a small world network. The network that is being attached could be of any kind. 1.3 PROPOSED METHOD In this study, we investigate different methods to join two existing networks. We consider different combinations of networks like small world-small world, small worldrandom and random-random. These combinations are considered with structural aspects like connecting to a well established member of a network, active communities in a network and finding new or old friends in a network. We also propose a model for growing an existing small world network. Here we initially generate a small world network and then grow it using a growth model as new nodes join the network. We also consider communities of nodes in which we establish connection between the new node and the existing nodes of a network. Finally, we look at new lattice models different form the ring lattice structure considered in the Watts and Strogatz small world simulation. 6

15 1.4 REST OF DOCUMENT: In chapter 2, we provide brief literature review and work done in this area. Chapter 3 gives a description of the proposed models and chapter 4 presents the results of our simulations. Chapter 5 presents our conclusions. 7

16 CHAPTER 2 LITERATURE REVIEW 2.1 RANDOM NETWORKS Erdos and Renyi introduced random graphs in 1950s [1],[2]. The Erdos-Renyi random graph consists of n isolated nodes and an edge is constructed between every pair of nodes with a probability p. A random graph can be represented by G(n,p), where n is the number of node in the network and p is the probability for an edge to be constructed between two nodes. In the Erdos-Renyi graph, the probability that the degree of a node t in a network is given by the binomial distribution: P (t) = C t n-1 p t (1-p) n-1-t The average degree of a node is expressed as z = (n-1)p. By expressing probability p in terms of z, after re-writing the above equation, the degree distribution is expressed as: P (t) = C t n-1 (z/n-1) t (1-(z/n-1)) n-1-t The above equation may be approximated to P (t) = (z t /t!) e -z. Thus the degree distribution for the nodes follows Poisson distribution. 8

17 2.2 SCALE-FREE NETWORKS Scale free networks were introduced by Barabasi and Albert (BA) in 1999 [3], [4] and the degree distribution of these graphs follows power law distribution. They proposed a simple mathematical model with two mechanisms: population growth and preferential attachment. The mechanism of the population growth is that real networks grow in time as new members join the network and the mechanism of the preferential attachment is that newly arriving node tends to connect to already well connected nodes rather than poorly connected nodes. Barabasi and Albert defined the probability p(k i ) of an existing node i with k i links receiving a new link as below: p(k i ) = ck i ; where c is normalized constant. In a BA graph, the probability that the degree of a node d is given by P(d)=c/d α for some positive constants c and α. Typically, 2 α 3. The probability that the new node is connected to a given existing node i is given by P(d i ) = d i / i-1 j d j where d j is the degree at the node j. 2.3 SMALL WORLD PHENOMENON The small world phenomenon was discussed first in the late 1950s by political scientist Ithiel de Sola Pool and mathematician Manfred Kochen [5]. Pool and Kochen formulated many questions that have come to define the field of social networks: 1) How many other people does each individual in a network know? In other words what is the person s degree in the network? 9

18 2) What is the probability that two people chosen at random from the population will know each other? 3) What is the chance that they have a friend in common? 4) What is the chance that shortest chain between them requires two intermediates? Or more than two? Pool and Kochen s work provided the inspiration for, among other things, the famous Small world experiment conducted in 1960s by Stanley Milgram [8]. The small world phenomenon, demonstrated experimentally by Stanley Milgram says that most humans are connected by chains of friendships that have roughly six individuals in them. This experiment is famously known as Six degree of separation. This experiment was carried out by asking people in American cities like Wichita and Omaha to send letters to specifically named East-coast individuals who were not known to the senders. A condition placed on each sender was that he or she could only mail the letter to someone with whom the sender was on a first-name basis. The same condition was placed on each recipient of the letter he or she could only mail the letter to a friend with whom he or she was on a first name basis and who appeared to be most likely to route the letter to its final destination. When Milgram examined the mail chains that were completed successfully in this manner, he discovered that, on the average, each letter took six steps between the original sender and the target. Although Milgram s experiments created a lot of excitement, it is important to note that only a few chains started in these experiments were completed. Non-completion of the chains does not mean that the small world phenomenon does not exist. It is easy to 10

19 understand why most chains would not be completed. Many people receiving a letter must have ignored them thinking as a chain letters and those who took the chain letters as serious might still have ignored them as it is too much of a bother to mail the letter again. In 1998, Duncan Watts and Steven Strogatz published a Small world computer simulation [10], which goes a step further. The simulations do capture the fact that any two individuals in a network are connected, on the average, by a small chain. However, in addition to measuring the average length of the shortest chain connecting any pair of individuals, Watts and Strogatz also measured the local clustering as would be perceived by any individual locally. By local clustering it means the extent to which an individual s friends are each other s friends. Structured and random graphs are characterized by two properties: 1) diameter of the graph; and 2) clustering coefficient of the graph. The diameter of a graph is the maximum value of the shortest path between any two nodes or is also the often considered as the average value of the shortest path between every pair of nodes in the graph. The clustering coefficient of a graph measures the average extent to which the immediate neighbors of any node are also each other s immediate neighbors. Watts and Strogatz carried out their simulations on a ring lattice. The total number of nodes in a network is assumed to be N and each node has k local contacts. For example, if the value of k is set as 4, then it implies that each node is connected to only 4 other nodes in the network. 11

20 The basic ring lattice network with N=20 nodes and k=4 neighbors is shown below Figure-2.1 Ring lattice structure A small world network is created by rewiring the basic diagram shown above in which each node is connected directly to its immediate neighbors and to a few additional neighbors in its vicinity. The extent of rewiring is controlled by the probability p. The small-world graph of Watts and Strogatz model with different probabilities are shown below: (a) (b) 12

21 (c) (d) (e) (f) Figure-2.2 WS-Model with different probabilities (In the above figure (Figure-2.2) small world graphs with different probabilities are shown. For (a) value of p is 0, for (b) p is 0.08, for (c) p is 0.32, for (d) p is 0.5, for (e) p is 0.75 and for (f) p is 1), where p is the probability. 13

22 When the value of p is 0, there wouldn t be any change in the basic ring structure and as the p value increases the edges are randomly rewired. When the value of p reaches 1 in the simulation, we get a random graph like an Erdos-Renyi graph. When p=1 the graph obtained will have a smaller diameter and the local clustering of the nodes will be destroyed. When p=0, the diameter of the graph becomes large which would be directly proportional to the number of nodes in the graph and the clustering of nodes will also be large. A network is a small world network if it has a large clustering coefficient and a small diameter. 2.4 EXISTING MODELS FOR GROWING SMALL WORLD NETWORKS The small world network constructed by Watts and Strogatz illustrates the small world properties in a simple way. However, the small world property is considered to be much more general than the network shown in previous examples. Ozik et al [11] introduced a simple mechanism (OHO model) for the evolution of the small world networks. In this model connections are made purely local and the network growth leads to stretching of old connections and to high clustering. In OHO model, the formation of links between nodes is formed based on geographically local processes [11]. That is, when a new node appears, it forms links only to those pre-existing nodes that are geographically close to it. In spite of the link formation being exclusively local, long range links will be shown to arise as a result of network growth. This in addition to the clustering induced by local connections yields the small world property. 14

23 Initially, OHO model has m+1 all-to-all connected nodes on the circumference of the circle. At each subsequent discrete time step network is grown according to the following prescription: a) new node is placed in a randomly chosen inter node interval along the circle circumference, where all intervals have the same probability of being chosen; b) the new node makes m links to its m previously existing nearest neighbors. Nearest here refers to the distance measured in number of intervals along the circumference of the circle. These steps are repeated until the network has grown to a desired number of nodes. The graphical representation of the above model is shown below: Figure-2.3 OHO model 15

24 Most previous models of small-world networks are stochastic. Zhong Zhi Zhang, Li Li Rong and Chong Hui Guo (ZRG) presented a model that generates a small-world network in a simple deterministic way [12]. This model is a growing network, whose size (the number of nodes) increases exponentially with time. Let network after t step evolution be denoted by N (t). The network is created by an iterative method. Its construction algorithm is the following: For t = 0, N (0) is a triangle whose three nodes connect to one another. For t 1, N (t) is obtained from N (t 1) by adding for each edge created at step t 1 a new node and attaching it to both end nodes of the edge. In the evolution process of the model, for each new node added, two new edges are created. And for each of the newly-created edges, a node will be created and connected to both the ends of the edge in the next step. The diagrammatic representation of the above ZRG model is shown in Figure-2.4. Figure-2.4 ZRG model 16

25 Z. Z. Zhang, L. L. Rong and F. Comellas (ZRC) came up with a model [13], which is a slight variation of the OHO model. It is a minimal extended evolving model for small-world networks which is controlled by a parameter q. Here a model for growing network, constructed in an iterative manner, is described. The network after t time steps is denoted by N (t). The network starts from an initial state (t = 0) of m + 1 (m even) nodes distributed on a ring, all of which are connected to one another by m (m + 1)/2 edges. For t = 1, N (t) is obtained from N (t 1) as follows: for each inter node interval, along the ring of N (t 1), with probability q, a new node is created and connected to its m nearest neighbors (m/2 on either side), previously existing at step t 1. Distance, in this case, refers to the number of intervals along the ring. When q = 1 and m = 2, the network is reduced to the deterministic ZRG model. If q <1, the network grows randomly. In particular, as q approaches zero (without reaching this value) the model coincides with the OHO model, where at each time step, only one interval is chosen. With every interval having the same probability of being selected, then a node is placed in the chosen inter node interval and linked to its m nearest. 17

26 The diagrammatic representation of a network generated from the above process with q=1 and m=2 is as below: Figure-2.5 ZRC model A growing model which interpolates between one-dimensional regular lattice and small-world networks was proposed by Zhong Zhi Zhang, Shuigeng Zhou, Zhen Shen and Jihong Guanc (ZZSG) [14]. The model undergoes an interesting phase transition from large to small worlds. The model is constructed in the following way: (i) Initial condition: Start from an initial state (t =2) of three nodes distributed on a ring, all of which form a triangle. (ii) Growth: At each increment of time, a new node is added, which is placed in a randomly chosen inter-node interval along the ring. Then perform the following two operations. a) Addition of edges: The new node is connected to its two nearest nodes (one on either side) previously existing. Nearest, in this case, refers to the number of intervals along the ring. 18

27 b) Removal of an edge: With probability q, remove the edge linking the two nearest neighbors of the new node. (iii) The growing processes are repeated until the network reaches the desired size. There are two limiting cases of the present model. When q=1, the network is reduced to the one dimensional ring lattice. For q=0, no edge is deleted, the model coincides with a special case m=2 of the OHO model for small-world networks. The diagrammatic representation of above model is as below: Figure-2.6 ZZSG model 19

28 Li Yong, Fang Jin-Qing, Liu Qiang, and Liang Yong proposed an algorithm known as spread all over vertices, which is used for generating small world networks from ring lattices [15]. During the rewiring in the ring lattice network the degree of the nodes does not change. The SAV algorithm for constructing small-world network is as below: Start with a ring of N nodes, each connected to its k nearest neighbors by undirected edges. For clarity, N = 10 and k = 4 are considered. At each time step perform one of the following three operations: Randomly select a node i. From the nodes connected with the node i, randomly select a node j and remove the edge connected to the node i. From the nodes unconnected with the node i, randomly select a node u and connect a new edge to the node i. From the nodes connected with the node u, randomly select a node v and remove the edge connected to the node u. For the node v connect a new edge to the node j. The diagrammatic representation of the above model is as shown below: Figure-2.7 SAV model 20

29 CHAPTER 3 METHODOLOGY In this chapter we provide a description of our main work. We will examine a model for joining two arbitrary networks and study its properties. We also observe the properties of joint network that is being generated. We then look at a model for growing an existing small world network and see how the properties of the network vary. Lastly we study the properties of a networks generated by rewiring square and hexagonal lattices. 3.1 MODEL FOR JOINING NETWORKS Some of the structural reasons for joining two social networks are: to connect to an established node on other network, to join interested groups, to meet new people and to find old friends. A normal way to achieve the above requirements is to connect the nodes of one network to the nodes of other network based on uniform probability. Here we generate the joint network assuming degree, clustering coefficient, average path length of the nodes in each network are known. 21

30 Below are the different conditions: 1) In order to connect to an established node, the nodes of one network are connected to the nodes of other network based on the degree of each node in the other network (say if degree of a node is greater than the average degree of the network, deg>average deg). 2) In order to join an interested group with active communication between its members, the nodes of one network are connected to the nodes of other network based on the clustering coefficient of each node in the other network (say if the clustering coefficient of a node is greater than the average clustering coefficient of the network, cc>average cc). 3) In order to meet new people or to find an old friend, the nodes of one network are connected to the nodes of other network based on the average path length of each node in the other network (say if average path length of a node is greater than the average of average path lengths of a network, apl<average apl). When two networks are joined based on the above structural aspects, a new network is evolved. The initial steps for joining two networks are same for all the above described criteria s. 1) Consider two networks, say network1, network2. These networks can be a small world networks or random networks or one each. 2) Let N1 be number of nodes in network1 and N2 be the number of nodes in network2. Assume that the network1 is represented by an adjacency matrix with N1 rows and N1 columns and network2 is represented by an adjacency matrix with N2 rows and N2 columns. 22

31 3) For each network, find the degree, clustering coefficient and Average path length at each node. And also the Average of degrees of all nodes, Average of clustering coefficient of all nodes and Average of average path length. 4) Now, construct a joint network with N nodes, where N= N1+N2. Joint network is represented by an adjacency matrix with N rows and N columns. The representation of networks in the form of a matrix is shown below: A small world network generated from a ring lattice with N1=10 nodes (each node connected to k=4 neighbors, k/2 on each side) and a rewiring probability 0.55 in the form of a matrix is as below: Figure-3.1 Adjacency matrix representation of a SWN with 10 nodes Another small world network generated from a ring lattice with N2=20 nodes (each node connected to k=4 neighbors, k/2 on each side) and a rewiring probability 0.55 in the form of a matrix is as below: 23

32 Figure-3.2 Adjacency matrix representation of a SWN with 20 nodes An initial joint network formed by joining the above two networks with N=30 nodes: Figure-3.3 Adjacency matrix representation of a SWN with 30 nodes 24

33 Evolution of a joint network in a normal way: Consider the initial joint network formed by joining the two networks. For each node in network1 based on a probability P, we connect an edge to a node selected preferentially from the network2. The preference for each node in netowrk2 is given by the below equation: Preference[i] = deg[i]/maximum degree in network2 For each node in network2 based on a probability P, we connect an edge to a node selected preferentially from the network1. The preference for each node in network1 is given by the below equation: Preference[i] = deg[i]/maximum degree in network1 Evolution of the joint network based on structural aspects: Consider the initial joint network formed joining the two networks. Select one method from the below structural aspects 1) Evolution of joint network based on degree Select the nodes from network1 which has a degree > average degree of the network1 and say these nodes as set1. Select the nodes form network2 which has a degree > average degree of the netowrk2 and say these nodes as set2. 2) Evolution of joint network based on clustering coefficient Select the nodes from network1 which has a clustering coefficient > average clustering coefficient of the network1 and say these nodes as set1. 25

34 Select the nodes form network2 which has a clustering coefficient > average clustering coefficient of the netowrk2 and say these nodes as set2. 3) Evolution of joint network based on average path length Select the nodes from network1 which has an average path length < average of average path length of the network1 and say these nodes as set1. Select the nodes form network2 which has an average path length < average of average path length of the netowrk2 and say these nodes as set2. For each node in network1 based on a probability P, we connect an edge to a node selected preferentially from the set2. The preference for each node in set2 is given by the below equation: Preference[i] = deg[i]/maximum degree in set2 For each node in network2 based on a probability P, we connect an edge to a node selected preferentially from the set1. The preference for each node in set1 is given by the below equation: Preference[i] = deg[i]/maximum degree in set1 The probability P for a node k in a network is given by P (k) = deg[k] /maximum degree in the network Then, the properties of the network after establishing the connections between the two networks are observed. 26

35 A small world network and a random network with 20 nodes each are considered initially. Then in each network nodes with its degree greater than the average degree of the network are selected (these nodes are shown red in color). Then for each node in a network based on a probability an edge is connected to one of the node selected above preferentially in other network. After going through every node in both the networks, a joint network is evolved. Figure-3.4 Joining of two networks 27

36 3.2 MODEL FOR GROWING AN EXISTING SMALL WORLD NETWORK Method description: Initially consider a small world network, say network1. Let N1 be number of nodes in network1. The network1 is represented by an adjacency matrix with N1 rows and N1 columns. Find the degree, clustering coefficient and Average path length at each node. And also the Average of degrees of all nodes, Average of clustering coefficient of all nodes and Average of average path length of the network. Below are possible conditions to grow a network: 1) Let n be new node, connect the new node preferentially to (Average degree) number of nodes from the network1. 2) Find the nodes with degree > Average degree of the network. Let n be new node, connect the new node preferentially to (Average degree) number of selected nodes. 3) Find the nodes with clustering coefficient > Average clustering coefficient of the network. Let n be new node, connect the new node preferentially to (Average degree) number of selected nodes. 4) Find the nodes with average path length > Average of average path length s of the network. Let n be new node, connect the new node preferentially to (Average degree) number of selected nodes. After considering one of the above methods, we then establish a link between the pairs of selected nodes using the linear function probability. 28

37 Diagrammatic representation of the growth of an existing small world network is shown in Figure-3.5. Initially a small world network with N nodes is generated, then the nodes with clustering coefficient greater than the average clustering coefficient of the network are selected (shown red in color in the below diagram). When a new node n arrives, it then connects to the (average degree) nodes selected above. Later an edge is connected between pairs of selected nodes (yellow line) based on a linear function probability. Figure-3.5 Growth of an existing SWN 29

38 3.3 OTHER LATTCIE STRUCTURES The basic structure of Watts and Strogatz model is a ring lattice structure, which is then rewired to form a small world network. Here we study the square lattice structure and as well as hexagonal structure. We find the initial properties of the square lattice and hexagonal lattice structure and see how these properties vary as we rewire those structures. SQUARE LATTICE We consider a square lattice with N-rows and N-rows, on which we can assume N square number of nodes. In the initial state nodes on the corners of the square structure are connected to 2 nearest neighboring nodes, nodes on the first, last row and first, last column except the nodes on the corner are connected to 3 nearest neighboring nodes and every other node is connected to 4 nearest neighboring nodes. A sample square lattice structure with nodes arranged on it is shown below: Figure-3.6 Square lattice structure 30

39 We then rewire every edge based on a probability P and see how the properties of the resulting structure is being affected. A sample figure showing resulting structure after rewiring the square lattice with a probability of P = 0.2 is shown in Figure-3.7. Figure-3.7 Rewired square lattice structure This resulting figure is more randomized in structure with a reduced average path length and increased clustering coefficient. 31

40 HEXAGONAL LATTICE STRUCTURE Initially, we have only one hexagon, on which we can place six nodes. Then we expand the hexagons outwards, which looks like a second layer of hexagons and with this second layer we can accommodate another 18 nodes. In a similar fashion we can have a third layer of hexagons; with this we can accommodate another 30 nodes. The structures discussed above are shown below: Figure-3.8 Hexagonal lattice structures Having a look at these structures, we can assume that the nodes on the inner most or central hexagon to be on circle and the nodes on second layer of hexagons to be on another circle on top of the inner circle and similarly the nodes on the third layer of hexagons to be on a third circle on top of both circles. Thus the hexagonal structure resembles the structure with three circles one on top of other with some interconnections between the circles. 32

41 We consider the above hexagonal structure and rewire every edge with a probability P. The resulting structure is more random and the properties of the resulting structure vary a bit. Below are the structure formed after rewiring the above two hexagonal structures with a probability P=0.5. Figure-3.9 Rewired hexagonal lattcie structures 33

42 CHAPTER 4 RESULTS In this chapter we show the results of joining two networks and see the properties of the joint network. We have run the simulations based on the structural aspects of joining the networks and different combinations of networks. We also show the results of growing an existing small world network. Then, we give brief description of the properties of rewired square and hexagonal lattice structures. JOINING OF TWO NETWORKS Two individual networks are joined to form a joint network based on different structural aspects such as connecting to a person with lots of friends, joining a group where there is an active interaction between the members of the group and connecting to an old friend or finding new friend. In order to achieve the above structural requirements while joining the two networks, we use the average degree, average clustering coefficient and average path length of the network. We join different combinations of small world networks and random networks, like small world-small world, random random and small world random. The characteristics of individual networks and the joint network for different combinations of the networks are described in the form of tables and graphs. Tables in each section show 34

43 the properties of the individual networks and joint networks with probability P for three different structural aspects. In all the tables N1 represents the number of nodes in network1, N2 represent nodes in network2 and N represents nodes in the joint network. The properties of each set of nodes are simulated thrice, first based on the average degree of the networks, second based on the average clustering coefficient of the networks and third based on the average path length of the networks. The graphs in all the sections below show how the properties of the joint network vary for different values of probability. Joining two small world networks: The different scenarios that we consider here are: 1) Two small world networks with equal number of nodes, and 2) Two small world networks with unequal number of nodes. Table-4.1 shows the properties of the individual small world networks with equal number of nodes and joint network formed by joining two small world networks with equal number of nodes. Small world network1 Small world network2 Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl

44 Table-4.1: Joining of two small world networks with equal number of nodes Based on CC Based on APL 50-50, , Figure-4.1: Probability Vs Clustering coefficient for joint network formed by joining two SWNs with equal number of nodes 36

45 Based on degree Based on CC Based on APL 50-50, , Figure-4.2: Probability Vs Average path length for joint network formed by joining two SWNs with equal number of nodes. Table-4.2 shows the properties of the individual small world networks with unequal number of nodes and joint network formed by joining two small world networks with unequal number of nodes. Small world network1 Small world network2 Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl

46 Table-4.2: Joining of two small world networks with unequal number of nodes Based on degree Based on CC Based on APL , , Figure-4.3: Probability Vs Clustering coefficient for joint network formed by joining two SWNs with unequal number of nodes Based on degree Based on CC Based on APL , , Figure-4.4: Probability Vs Average path length for joint network formed by joining two SWNs with unequal number of nodes 38

47 By looking at the properties of the joint network formed by joining two small world networks with equal numbers of nodes, we can see that the clustering coefficient of the joint network decreases initially for low probability values and increases as the probability increases. The average path length of the joint network decreases gradually as the probability increases. The same results were observed with unequal number of nodes also. These properties of a joint network formed in both cases are similar to that of the small world model. Joining two random networks: The different scenarios that we consider here are: 1) Two random networks with equal number of nodes, and 2) Two small random with unequal number of nodes. Table-4.3 shows the properties of the individual random networks with equal number of nodes and joint network formed by joining two random networks with equal number of nodes. Random network1 Random network2 Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl

48 Table-4.3: Joining of two random networks with equal number of nodes Based on degree Based on CC Based on APL 50-50, , Figure-4.5: Probability Vs Clustering coefficient for joint network formed by joining two random networks with equal number of nodes 40

49 Based on degree Based on CC Based on APL 50-50, , Figure-4.6: Probability Vs Average path length for joint network formed by joining two random networks with equal number of nodes Table-4.4 shows the properties of the individual networks with unequal number of nodes and joint network formed by joining random networks with unequal number of nodes. Random network1 Random network2 Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl

50 Table-4.4: Joining of two random networks with unequal number of nodes Based on degree Based on CC Based on APL , , Figure-4.7: Probability Vs Clustering coefficient for joint network formed by joining two random networks with unequal number of nodes 42

51 Based on degree Based on CC Based on APL , , Figure-4.8: Probability Vs Average path length for joint network formed by joining two random networks with unequal number of nodes The clustering coefficient of the resultant joint network formed by joining two random networks with equal number of nodes increases as the probability increases and the average path length of the joint network decreases as the probability increases. These results are same for the joint network formed by joining two random networks with unequal number of nodes. In both cases the properties of the joint network are similar to the properties of small world model. Joining small world network and random network The different scenarios that we consider here are: 1) A small world network and a random network with equal number of nodes, and 2) A small world and a random network with unequal number of nodes. 43

52 Table-4.5 shows the properties of the individual networks with equal number of nodes and also joint network formed by joining a small world network and a random network with equal number of nodes. Small world network Random network Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl Table-4.5: Joining of a SWN and a random network with equal number of nodes 44

53 Based on degree Based on CC Based on APL 50-50, , Figure-4.9: Probability Vs Clustering coefficient for joint network formed by joining a small world network and a random network with equal number of nodes Based on degree Based on CC Based on APL 50-50, , Figure-4.10: Probability Vs Average path length for joint network formed by joining a small world network and a random network with equal number of nodes Table-4.6 shows the properties of the individual networks with unequal number of nodes and joint network formed by joining a small world network and a random network with unequal number of nodes. 45

54 Small world network Random network Joint Network (P) N1 Avg_cus Avg_apl N2 Avg_cus Avg_apl N Avg_cus Avg_apl Table-4.6: Joining of a SWN and a random network with unequal number of nodes 46

55 Based on degree Based on CC Based on APL , , , , , Figure-4.11: Probability Vs Clustering coefficient for joint network formed by joining a small world network and a random network with unequal number of nodes Based on degree Based on CC Based on APL , , , , , Figure-4.12: Probability Vs Average path length for joint network formed by joining a small world network and a random network with unequal number of nodes 47

56 As we can see from the above graphs the clustering coefficient of the resultant joint network formed by joining a small world network and a random network with equal number of nodes decreases initially and then increases as the probability increases. The average path length of the joint network decreases gradually as the probability increases. The results are same for the joint network formed by joining a small world network and a random network with unequal number of nodes. 48

57 GROWING AN EXISTING SMALL WORLD NETWORK Initially we generate a small world network and then we grow that network by adding one node at each step until the network is grown to a desired size. While growing a network we consider the structural aspects such as connecting to a well established node and communities. The properties of the network as it is grown are shown in the form of graphs. In these graphs, networks with N number of nodes are grown until networks reaches N+100 number of nodes. Based on degree Based on CC Based on APL 50, 100, 150, 200 Figure-4.13: Nodes being added Vs Clustering coefficient for a growing SW network 49

58 Based on degree Based on CC Based on APL 50, 100, 150, 200 Figure-4.14: Nodes being added Vs Average path length for a growing SW network The average clustering coefficient of the growing small world network is better than that of the small world network considered initially. As the number of nodes in the starting network increases, the growth in the clustering coefficient with respect to the number of nodes being added is reduced. The average path length of the growing network is almost similar to the initial small world networks. Based on the above observations we could say that growth of the small world network using the proposed model will preserve the characteristics of the network considered. 50