Network Analysis For Sustainability Management 1 Cátia Vaz 1º Summer Course in E4SD
Outline Motivation Networks representation Structural network analysis Behavior network analysis 2
Networks Over the past decade has been a growing public fascination with the complex connectedness of modern society. The connectedness: Aggregate behavior of groups of people; Reflects the ways in which our decisions can have subtle consequences for others. 3
A network is Networks A pattern of interconnections among a set of things More precisely. A collection of objects in which some pair of objects are connected by links 4
Network Example 1: 5
Network Example 2: 6
Network Example 3: 7
Network Connectedness Network connectedness at two levels Structure level Who is linked to whom Behavior level Each individual s actions have implicit consequences for the outcomes of everyone in the system For analysing the network connectedness and inferring properties, we rely on Graph theory Game theory 8
Structural level analysis Detection of strong/weak ties Individuals centrality determine the relative importance of a individual within the graph Betweenness centrality, Closeness centrality, degree centrality Captures different aspects of the relevance of an individual Communities detection 9
10 Structural Analysis Example: unclecj.blogspot.com
Behavior level analysis How a group of people must simultaneously choose to act, knowing that the outcome will depend on the decisions made by all of them? 11 Traffic Network Example:
Strategic Interaction in Networks We can combine graph theory and game theory to produce richer models of behavior in networks 12 Usually, the network structure encodes a lot about the pattern of trade The success levels of different participants are affected by their positions in the network What factors determine a powerful position on networks?
Strategic Interaction in Networks 13 Seaching in Twitter within the topic new Finance Minister in Portugal
Network Dynamics: Population Effects The way in which new practices spread through a population depends in large part on the fact that people influence each other s behavior. This is a central issue for understanding networks and aggregate behavior 14 Taking network structure into account provides further insights into how such kinds of influence take place!
Network Dynamics: Population/Structural Effects When individuals have incentives to adopt the behavior of their neighbors there can be: Cascading Effects 15 Example: recycling in Portugal Sociedade Ponto Verde Video
Outline Motivation Networks representation Structural network analysis Behavior network analysis 16
Network Characterization 17 All networks consists of two primary building blocks: Vertices (or nodes, agents,..) Edges (or ties, arcs, connections, ) Networks Vertices Edges Twitter Users Follower/Following; Mentions; Replies Facebook Friends Friendship Relations Skype Contacts Messages/Conversations Traffic Cross Roads
Edges can be: Undirected Are reciprocated Directed Are asymmetric Network Characterization Have a origin and a destination Networks Vertices Edges Twitter Users Follower (directed) Facebook Friends Friendship Relations (undirected) Skype Contacts Messages/Conversations (undirected) Traffic Cross Roads (directed) 18
Network Characterization Edges can be classified into: Weighted 19 A edge has an associated value that indicate the strength or frequency of a tie Unweighted Only indicates if an edge exists or not Can be seen as a binary edge (with zero or one value) Networks Vertices Edges Twitter Users Follower (unweighted) Facebook Friends Friendship Relations (unweighted) Skype Contacts Messages/Conversations (weighted/unweighed) Traffic Cross Roads (usually weighted)
Graphs A graph is a way of specifying relationships among a set of nodes The relationships are the edges A graph serve as a mathematical model of network structures 20
Graph Example 1 Consider that Ann is a friend of Carol and Bob and Bob is a friend of Alice. How to visualize this graph? 21 Undirected and Unweighted Network
Graph Example 2 22 Consider the bideractional avenue Avenida da Liberdade between Marquês de Pombal and Rossio; the undirectional road Rua Braamcamp between Marquês de Pombal and the Praça Castilho and the undirectional road Rua Lisboa between Praça Castilho and Rossio. Directed and Weighted Network Q: What do values represents?
Network Representation 23 As a matrix Ann Bob Carol Alice Ann 0 1 1 0 Bob 1 0 0 1 Carol 1 0 0 0 Alice 0 1 0 0 As an Edge List Vertex 1 Vertex 2 Ann Bob Ann Carol Bob Ann Bob Alice Carol Ann Alice Bob
Network Representation 24 As a matrix Praça Castilho Marques De Pombal Praça Castilho Marques de Pombal Rossio 0 0 600 280 0 1600 Rossio 0 1600 0 As an Edge List Vertex 1 Vertex 2 Label Praça Castilho Marques de Pombal Marques de Pombal Rossio Rossio 600 Rossio 1600 Praça Castilho Marques de Pombal 280 1600 Q:What are the differences between this and the previous one?
Paths A path is a sequence of nodes with the property that each consecutive pair in the sequence is connected by an edge. 25 Q: Are there paths in our networks examples?
Connectivity and Connected Components A graph is connected if for every pair of nodes there is a path between them Not all graphs are connected! A connected component of a graph is a subset of nodes that: 26 Property 1: every node in the subset has a path to every other Property 2: is not contained in another subset with property 1. Q: What are our in network examples the connected components?
Small-World Phenomenon: are we really so near from each other? When analysing the connected component of large networks: Presumably the global network is not connected! Think about a social network Still, they have a giant component 27 A connected component that contains a significant fraction of all nodes The small work phenomenon occurs in the giant component of the network!
Small-World Phenomenon: are we really so near from each other? 28 Milgram Exp. MSN Facebook Twitter Twitter Vertex person user user user user Link selected person message exchange (conversation) friendship follower mention Symmetric no yes yes no no Avg. dist 6.2 6.6 4.74 4.12 6.63
Outline 29 Motivation Networks representation Structural network analysis Behavior network analysis
Triadic Closure It is also useful to think about how a network evolves over time An important principle to take into account is If two people in a social network have a friend in common, then there is an increased likelihood that they will become friends themselves at some point in the fuure (Anatole Rapaport 1953) Designated as Triadic Closure Reasons for triadic closure B Opportunity Trusting Incentive To capture is prevalence we can use the clustering coefficient A C 30
Bridges 31 C A B L D E K O An edge that joins nodes A and B in a graph is called a bridge if deleting the edge would cause A an B to lie in two different components. Q: Does bridges occurs in real social networks?
Local Bridges J G 32 M F N C A B L D E K O An edge that joins nodes A and B in a graph is called a local bridge if A and B does not have friends in common. Deleting such an edge would increase distance between A and B. Q: Does local bridges occurs in real social networks?
The strenght of weak ties Stronger links represents closer friendships and greater frequency of interaction We can classify links in: Stronger ties (corresponding to friends) Weaker ties (corresponding to acquantances) Thus, under the assumption of Triadic Closure property and a sufficient number of strong ties Local Bridges are necessary weak ties Therefore, weak ties: Connect to new sources of information, new opportunities Allow to reach other parts of the network 33
Betweeness Centrality Betweeness Centrality of a node is: 34 The number of shortest paths from all vertices to all others that pass through that node This measurement can be seen as a kind of a bridge score, since it measures: how much removing an individual of the network would disrupt the connections between other individuals in the network. The bigger is the betweeness centrality value for a node => The bigger is its importance as a broker between two otherwise separated groups
Example 1: Betweeness 35 Seaching in Twitter within the topic new Finance Minister in Portugal
Example 2:Betweeness 36 Seaching in Twitter within the topic: comprar produtos nacionais
Closeness Centrality The closeness centrality of a node is 37 the average distance between the node and the other nodes of the network Can be seen as a measurement of how close each individual is to the other individuals in the network The lower the closeness centrality value is => the more near this node is from the others
Example 1: Closeness 38 Seaching in Twitter Within the topic new Finance Minister in Portugal
Example 2: Closeness 39 Seaching in Twitter within the topic: comprar produtos nacionais
Communities detection A community in a network is a subset of the network nodes such that is densely connected internally 40 Q: How many communities in this network?
Communities detection And in this network, how many communities are? 41 A divisive method proposed by Newman and Girvan has been widely applied to networks, in particular to social networks.
Outline 42 Motivation Networks representation Structural network analysis Behavior network analysis
Diffusion in networks 43 When we want to analyze the processes by which new ideas and innovations are adopted by a population, the underlying social network can be considered at two different levels: (Level 1): The level of seeing the network as a relatively amorphous population of individual and look at effects in aggregate (Level 2): The level of seeing closer to the fine structure of the network, looking at how individuals are influenced by the particular network neighbors Level 1 Level 2 Network models Population models Structural models
Diffusions of ideas as behaviors There are clear connections between epidemic disease and the diffusion of ideas through social networks: 44 Both diseases and ideas can spread from one person to another It is known as social contagion With social contagion People make decisions on adopting a new idea or innovation SI, SIS and SIR are epidemical models Belong to the structural model group (level 2)
SI, SIS and SIR model 45 SI model SIS model S S λ λ I I S->Susceptible or Ignorant I -> Spreader R ->Stiflers (informed agents who don t spread information) δ SIR model S λ I δ R λ: contact infection rate δ: recovery rate
SI model 46 S λ I <k> : average number of contacts of a given individual x : fraction of infected in the population 1-x : fraction of susceptible λ<k> : Spread rate / force of infection i.e., an infected individual is able to transmit the disease with λ<k> others per unit time
SIS model 47 λ S I δ <k> : average number of contacts of a given individual x : fraction of infected in the population 1-x : fraction of susceptible λ<k>: Spread rate / force of infection δ: recovery rate
But the complexity of social behavior Spread of information, adoption of new trends, habits, opinions, etc., are all intentional acts, unlike disease spreading. Some behaviors, trends and ideas may bring more benefits than others... For instance, we are free to choose among different opinions and behaviors, or even create new ones... Contrary to disease spreading, there s much more around than contact processes. 48
How rational are we? 49
The role of social networks 50
Happiness is contagious 51 On average, the likelihood of I feel happy increases by 15% if I have a happy friend (distance 1)! Increases 10% if a friend of a friend happy (distance 2)! Increases 5% if a friend of a friend of a friend happy (distance 3)! Each unhappy friend decreases 7% this probability! Christakis & Fowler, NEJM, 2007, 2008
Game theory & rational behavior The cost-benefit dilemma: 52 Donor Pays a cost c Receiver Receives a benefit b If both play as a donor and as a receiver Rational Goal : Maximize your own payoff: If your opponent plays C: you better play D If your opponent plays D: you better play D But: CC is better than DD you your opponent C D C b-c -c D b 0 General Dilemma: Despite mutual cooperation (CC) being better than mutual defection (DD), individual rational choice evolves to (DD)
All symmetric 2-person dilemmas of cooperation symmetric 2-player games : 2 individuals meet 53 each player uses 1 of 2 strategies ( Cooperate or Defect ) each possible outcome has an associated payoff (tabulated in the payoff-matrix) R: mutual cooperation P : mutual defection S : sucker s payoff T : temptation to defect you your opponent C D C R S D T P
Case study: Recycling in Portugal 54 A buddy on the school A children go green Not go green go green b 0 not go green 0 0
For multiple players 55
Case study: Recycling in Portugal 56
For scale-free networks 57
Bibliography 58 David Easley and Jon Kleinberg, Networks and Markets: Reasoning about a Highly Connected World, Cambridge University press, 2010 Derek L. Hansen, Ben Shneiderman and Marc A. Smith: Analysing Social Media Networks with NodeXL: Insights from a Connected World, Elsevier,2011 Christakis & Fowler, The Collective Dynamics of Smoking in a Large Social Network, NEJM, 2007, 2008 Christakis & Fowler, Dynamic spread of happiness in a large social network: longitudinal analysis over 20 years in the Framingham Heart Study, NEJM, 2007, 2008
Thanks Professor Adjunto PhD cvaz@cc.isel.pt 59