# UNIVERSITA DEGLI STUDI DI CATANIA FACOLTA DI INGEGNERIA

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1 UNIVERSITA DEGLI STUDI DI CATANIA FACOLTA DI INGEGNERIA PhD course in Electronics, Automation and Complex Systems Control-XXIV Cycle DIPARTIMENTO DI INGEGNERIA ELETTRICA ELETTRONICA E DEI SISTEMI ing. Angelo Sarra Fiore Tutor: prof. ing. Luigi Fortuna ing. Mattia Frasca

2 Outline Networks Theory Friends Network How a friends network grows? Courses

3 A network is a system consisting of many entities, called nodes, linked to each other and interacting through connections (edges). A general network may be represented as a graph G = (N, E), where N is the set of nodes and E the set of edges = = {(,2),(,5),(2,3),(2,5),(3,6),(4,5)} {,2,3,4,5,6} ) :, ( E edges N nodes E N G Networks Theory WHAT IS A NETWORK? = A Mathematically a graph G (N, E) can be represented by a matrix (called adjacency matrix A) that has nodes as elements of rows and columns and the elements are different from zero (if different from it is a weight" of the link: cost, velocity, energy etc.) if the two nodes are connected.

4 Networks Theory EXAMPLES Many complex systems can be represented as networks of interacting elements. -Social Networks Coauthors Networks Actors Networks /IM Networks -Technological Networks Telephone Networks Internet Transport Networks -Knowledge (Information) Networks References Network World Wide Web -Biological Networks Proteins Networks Neural Networks

5 Networks Theory CHARACTERISTIC PARAMETERS OF NETWORKS Node degree: number of the edge of a node k i = a ij j N Shortest path length: the average of the shortest paths connecting each pair of nodes L = N( N ) d ij i, j N, i j where d ij is the element of the D matrix which contains the shortest paths connecting each pair of nodes Clustering coefficient of a node: quantifies the importance of a node by evaluating the number of connections that remain if the node was removed c i = 2 arcs' number k ( k ) i i of G i

6 Networks Theory HISTORY The classical network theory goes back to 736 with Eulero's study to solve the Koningsberg Bridges Problem. In 959 P. Erdos and A. Rényi, two Hungarian mathematicians, introduced random network In 998 S. Strogatz e D. Watts, studying actors' networks, energy distribution network and the neural network of the worm C.elegans, saw a deviation from random networks and introduced the small world networks In 999 Barabási e R. Albert saw that many networks had a power law degree distribution N(k) ~ k - γ and defined the scale-free networks and very popular is the sentence "the rich get richer"

7 Friends Network We re analyzing the friends networks of some user of the popular social network Facebook (originally known as Thefacebook) is the most popular social network site today and was founded on 4 February 24 by Mark Zuckerberg, a nineteen student of the Harvard s University, with the help of Andrew McCollum and Eduardo Saverino. In April 29 the number of active users has reached 2 million and the average number of friends per user is 2.

8 Friends Network Start with a star network where the user is the central node

9 Friends Network Remove the central node to obtain the friend s network

10 Friends Network Remove isolated nodes and Extract the Main Component

11 Friends Network Analysis of this network

12 Friends Network Comparison with other network topologies

13 Degree Distribution of Four Different Networks L=3,488 C=,4528 K=8,398 nodes: 75 edges: L=2,5534 C=,5993 K=,2745 nodes: 3 edges: k k L=2,273 C=,436 K=24,292 nodes: 37 edges: L=2,639 C=,574 K=2,37 nodes: 262 edges: k k

14 Power Law Degree Distribution 6 Erdos L=2,28 C=,493 K=25,8824 Random L=2,944 C=,52 K=26,7324 Real L=3,2683 C=,422 K=26,7324 nodes: 528 edges: P(k)=a*k -γ a=48 γ= k k

15 Results of the Analysis L C networks erdos random real,8,7,6,5,4,3,2, networks erdos random real K networks In comparison with the random and the Erdos-Renyi networks friends networks present an higher clustering coefficient and a lower path length, typical characteristics of small-world networks, but the degree distributions aren t power law erdos random real

16 How a Friends Network Grows? Study of a social network that evolves in time A dynamic network is a particular network in which the topology changes in time for the variations in the sets of edges and nodes

17 How a Friends Network Grows? Edges number Nodes number sample sample

18 How a Friends Network Grows? L = C = sample sample 25 2 K 5 5 K = sample

19 How a Friends Network Grows? 9 DEGREE DISTRIBUTION k

20 How a Friends Network Grows? 62 nodes 4354 edges

21 Courses Fondamenti di Bioingegneria Elettronica Misure Elettroniche

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