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Small World Networks Jean Vaucher Ift Avril 2005

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**Contents Pertinence of topic Characterization of networks**

Regular, Random or Natural Properties of networks Diameter, clustering coefficient Watt’s network models (alpha & beta) Power Law networks Clustered networks with short paths Can these short paths be found ? ift6802

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Duncan J. Watts Six degrees - the science of a connected age, 2003, W.W. Norton. I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everybody on this planet. Six degrees of separation by John Guare ift6802

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**Networks Networks are everywhere Networks have been studied long time**

Internet Neurons is brains Social networks Transportation Networks have been studied long time Euler (1736): Bridges of Königsberg theory of graphs, which is now a major (and difficult! – or almost obvious) branch in mathematics ift6802

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**So what is new? Global interconnections Internet Power grids**

Mass travel, mass culture FAILURES Computer Viruses Power Blackouts Epidemics Modeling & analysis ift6802

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Milgram’s Experiment Found short chains of acquaintances linking pairs of people in USA who didn’t know each other; Source person in Nebraska Target person in Massachusetts. Sends message by forwarding to people they knew personally (who should be closer to target) Average length of the chains that were completed was between 5 and 6 steps “Six degrees of separation” principle ift6802

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**Correct question Why is this surprising**

WHY are there short chains of acquaintances linking together arbitrary pairs of strangers??? Or Why is this surprising ift6802

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Random networks In a random network, if everybody has 100 friends distributed randomly in the world population, this isn’t strange In 6 hops, you can reach 1006 people - a million million > 6,000 million (world pop.) BUT: our social networks tend to be clustered. ift6802

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**Social networks Not random But Clustered**

Most of our friends come from our geographical or professional neighbourhood. Our friends tend to have the same friends BUT In spite of having clustered social networks, there seem to exist short paths between any random nodes. ift6802

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**Social network research**

Devise various classes of networks Study their properties ift6802

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**Network parameters Network type Size: # of nodes Number of connexions:**

Regular Random Natural Size: # of nodes Number of connexions: average & distribution Selection of neighbours ift6802

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**REGULAR Network Topologies**

STAR TREE GRID BUS RING ift6802

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**Connectivity in Random graphs**

Nodes connected by links in a purely random fashion How large is the largest connected component? (as a fraction of all nodes) Depends on the number of links per node (Erdös, Rényi 1959) ift6802

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Connecting Nodes ift6802

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Random Network (1) add random paths ift6802

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Random Network (2) paths trees ift6802

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Random Network (3) paths trees networks ift6802

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Random Network (3+) paths trees networks ….. ift6802

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**Network Connectivity (4)**

paths trees networks fully connected ift6802

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**Connectivity of a random graph**

1 Fraction of all nodes in largest component Disconnected phase Conected phase 1 Average number of links per node ift6802

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**Regular or Ordered Network**

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**Network measures Connectivity is not main measure.**

Characteristic Path Length (L) : the average length of the shortest path connecting each pair of agents (nodes). Clustering Coefficient (C) is a measure of local interconnection if agent i has ki immediate neighbors, Ci, is the fraction of the total possible ki*(ki-1) / 2 connections that are realized between i's neighbors. C, is just the average of the Ci's. Diameter: maximum value of path length ift6802

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**Regular vs Random Networks needed to fully connect**

Average number of connections/node few, clustered fewer, spread Number of connections needed to fully connect many fewer (<2/3) Diameter large moderate ift6802

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**Natural networks Between regular grids and totally random graphs**

Need for parametrized models: Regular -> natural -> random Watts Alpha model ( not intuitive) Beta rewiring model ift6802

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Clustering Clustering measures the fraction of neighbors of a node that are connected themselves Regular Graphs have a high clustering coefficient but also a high diameter Random Graphs have a low clustering coefficient but a low diameter Both models do match the properties expected from real networks! Regular Graph (k=4) Long paths L ~ n/(2k) Highly clustered C~3/4 Random Graph (k=4) Short path length L~logkN Almost no clustering C~k/n Base metwork is circle ift6802

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Small-World Networks Random rewiring of regular graph (by Watts and Strogatz) With probability p (or ) rewire each link in a regular graph to a randomly selected node Resulting graph has properties, both of regular and random graphs High clustering and short path length FreeNet has been shown to result in small world graphs ift6802

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**Example: 4096 node ring ift6802 K=4**

A possible explanation of what is happening can be found in the graph structure of FreeNet. Recently a class of graphs has been characterized, that exhibits so-called small world characteristics. This was motivated by the observation that the distance of acquaintances in social networks is rather small on average (For example, a famous experiment performed by Milgram suggests that this distance is for the US 6, therefore this phenomenon is also called the 6 degrees of separation). An explanation for this phenomenon can be given as follows: acquaintances are normally local in nature. Therefore an acquaintance graph would typically have a very regular structure such as on the left (Note that this is the structure of the graph used for the initial network topology on the previous slide). This graph has the following properties: it is highly clustered, which means that the probability that two neighbors of a specific node are also connected with high probability. E.g. among the 4 neighbors of a node on the left hand side, only one pair is not connected. The average distance among nodes is however high. On the other hand, from the theory of random graphs it is known, that the average path length is logarithmic in the number of nodes, but the clustering is very low. Such a graph is shown on the right. A low diameter (average path length) is however good for search, e.g. using flooding. The interesting discovery was, that there exists a class of graphs which has the high clustering effect of regular graphs, but already has the property of short diameter of random graphs. Such a graph is shown in the middle. Essentially, it can be obtained, by introducing a few "short-cuts" randomly. One can think about people making visits to far away places, and thus introducing shortcuts in the graph of acquaintances. In Freenet this effect is obtained by the rewiring of graph during insertion and search. Regular graph: n nodes, k nearest neighbors path length ~ n/2k 4096/16 = 256 Rewired graph (1% of nodes): path length ~ random graph clustering ~ regular graph Small World Graph Random graph: path length ~ log (n)/log(k) ~ 4 ift6802

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**Beta network 1 C L 1 Small- world networks Rewiring probability **

L 1 Rewiring probability ift6802

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More exactly …. (p = ) C L Small world behaviour ift6802

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**Effect of short-cuts Huge effect of just a few short-cuts.**

First 5 rewirings reduces the path length by half, regardless of size of network Further 50% gain requires 50 more short-cuts ift6802

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**The strength of weak ties**

Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties this is because valuable information comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network) ift6802

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**Two ways of constructing**

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**Alpha model Watts’ first Model (1999)**

Inspired by Asimov’s “I, Robot” novels R. Daneel Olivaw Elijah Baley Caves of Steel (Earth) Solaria ift6802

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**Two extreme types of social networks**

Caveman’s world people live in isolated communities probability meeting a random person is high if you have mutual friends and very low if you don’t Solaria people live isolated from each other but with supreme communication capabilities your social history is irrelevant to your future ift6802

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**Alpha network Alpha () distance parameter**

=0 : if A and B have a friend in common, they know each other (Caveman world) =∞ : A & B don’t know each other, no matter how many common friends they have (Solarian world) ift6802

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**Likelihood that A meets B**

Caveman world =0 = =1 Likelihood that A meets B Solaria world Number of mutual friends shared by A and B ift6802

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**Alpha network Clustering coefficient C Path length L Small- Fragmented**

networks Small- world net- works Path length L critical L drops because we only count nodes that are connected ift6802

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**How about real networks**

All nodes in alpha and beta networks are equal in the sense that the number of connections each nodes has is not very far from the average Watts and Strogatz had used normal distribution Real world is not like that Sizes of cities, Wealth of individuals in USA, Hubs in transportation systems Barabási and Albert (1999) Scale-free networks, whose connectivity is defined by a power-law distribution ift6802

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**Random Networks Each node is connected to a few other nodes.**

The number of connections per node forms a Poisson distribution, with a small average of number of connections per node. This & three following graphics from: Linked: The New Science of Networks by Albert-Laszlo Barabasi; 2002 ift6802

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**Scale-Free Networks Each node is connected to**

at least one other; most are connected to only one, while a few are connected to many. The number of connections per node forms a hyperbolic distribution, with no meaningful average number of connections per node. ift6802

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**Random Scale-Free Scale-free networks are associated with**

networks that grow by “natural” processes in which the number of nodes increases with time not just the number of connections. ift6802

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**Power law phenomena Average & median are far apart**

Whales and minnows Average from a few large nodes Median governed by majority of small nodes ift6802

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**Performance Real power law networks also have short distances**

Existence of central backbone of highly connected HUBS nodes Similar phenomena noted in linguistics and economics Zipf Pareto ift6802

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**Zipf's law - linguistics**

Zipf, a Harvard linguistics professor, sought to determine the frequency of use of the 3rd or 8th or 100th most common words in English text. Zipf's law states that the frequency y is inversely proportional to it's rank r: Y ~ r -b, with b close to unity. Zipf Presentations ift6802

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**The Pareto Income Distribution**

The Pareto distribution gives the probability that a person's income is greater than or equal to x and is expressed as ift6802

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**Vilfredo Pareto, 1848-1923 Italian economist Born in Paris**

Polytechnic Institute in Turin in 1869, Worked for the railroads. Pareto did not study economics seriously until he was 42. In 1893 he succeeded his mentor, Walras, as chair of economics at the University of Lausanne. ift6802

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**Pareto’s contributions**

Pareto optimality. A Pareto-optimal allocation of resources is achieved when it is not possible to make anyone better off without making someone else worse off. Pareto's law of income distribution. In 1906, Italian economist Vilfredo Pareto created a mathematical formula to describe the unequal distribution of wealth in his country, observing that 20% of the people owned 80% of the wealth. ift6802

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**Pareto distribution is said to be scale-free because **

log-log plot Pareto distribution, m=10000, k=1 Pareto distribution is said to be scale-free because it lacks a characteristic length scale ift6802

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**Building Power-law networks**

It is easy to create PL networks Build network node by node Connect new node to an existing node Probability of connection proportional to its number of links The rich get richer The poor get poorer ift6802

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**Structure and dynamics**

The case of centrality centers are in networks by design (central control, dictatorship) by non-design (unnoticed critical resources, informal groups) or they emerge as a consequence of certain events ”he was at the right place at a right time” clapping in unison ift6802

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**Further applications Search in networks Epidemics: medical & software**

Short paths are not enough Epidemics: medical & software Danger of short-cuts Paths + infectiousness Infection by ideas Fads & Economic Bubbles Individual rationality Peer pressure ift6802

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**Getting practical: search in networks**

A node may be linked to another node via a short path but what does it matter if you cannot find the path? In alpha and beta networks there is no notion of distance, therefore directed searches cannot recognize shortcuts Kleinberg’s (gamma) networks (2000) ift6802

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**Kleinberg’s Small-World Model**

Embed the graph into an r-dimensional grid (2D in examples) constant number p of short range links (neighborhood) q long range links: choose long-range links such that the probability to have a long range contact is proportional to 1/dr Importance of r ! Decentralized (greedy) routing performs best iff. r = dimension of space (here=2) r = 2 ift6802

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Influence of “r” (1) Each peer u has link to the peer v with probability proportional to where d(u,v) is the distance between u and v. Optimal value: r = dim = dimension of the space If r < dim we tend to choose more far away neighbors (decentralized algorithm can quickly approach the neighborhood of target, but then slows down till finally reaches target itself). If r > dim we tend to choose more close neighbors (algorithm finds quickly target in it’s neighborhood, but reaches it slowly if it is far away). When r = 0 – long range contacts are chosen uniformly. Random graph theory proves that there exist short paths between every pair of vertices, BUT there is no decentralized algorithm capable finding these paths ift6802

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**p(r) 2 r short paths cannot be found no short =0**

Typical length of directed search 2 short paths cannot be found no short =0 p(r) (log scale) increasing r (log scale) ift6802

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**Influence of “r” (or ) A4 A2 A3 A1**

Given node u if we can partition the remaining peers into sets A1, A2, A3, … , AlogN , where Ai, consists of all nodes whose distance from u is between 2i and 2i+1, i=0..logN-1. Then given r = dim each long range contact of u is nearly equally likely to belong to any of the sets Ai A4 A3 A2 A1 ift6802

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The New Yorker View When gamma is at its critical value two, the resulting network has the peculiar property that nodes possess the same number of ties at all length scales (in 2D world) ift6802

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**DHTs (distributed hash tables) and Kleinberg model**

P-Grid’s model Kleinberg’s model Balanced n-ary search ift6802

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More hierarchy Kleinberg’s model has only one distance measure, geographical (2D) In human society the social distance is multidimensional if A is close to B and C is close to B but in different dimension then A and C can be very far from each other ”violation of the triangle inequality” but multidimensionality may enable messages to be transmitted in networks very efficiently ift6802

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**Watts et al (2002) search in social networks**

= homophily, the tendency of like to associate with like H=number of dimensions along which individuals measure similarity 6 Searchable networks 1 10 Kleinberg condition H ift6802

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**Small Worlds & Epidemic diseases Nodes are living entities**

Link is contact 3 States Uninfected Infected Recovered (or dead) ift6802

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Epidemic diseases Level of infectiousness needed to start an epidemic varies with presence of shortcuts In regular grid, disease may die out due to lack of victims In small world, pandemics are facilitated SRAS Mad cow disease in England ift6802

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**Failures in networks Fault propagation or viruses**

Scale-free networks are far more resistant to random failures than ordinary random networks because of most nodes are leaves But failure of hubs can be catastrophic vulnerable or targets of deliberate attacks which may make scale-free networks more vulnerable to deliberate attacks Cascades of failures ift6802

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**Back to Social Networks**

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**Spread of ideas Messages in social networks Fads & fashions**

Body piercing, baseball caps Harry Potter, Amélie Poulin Innovation, scientific revolutions Solar-centric universe Plate tectonics Is it like the spread of disease ? ift6802

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**Effect of peers & pundits**

People’s decisions are affected by what others do and think Presure to conform ? Efficient strategy when insufficient knowledge or expertise Ex: picking a restaurant ift6802

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** Equilibrium ??? Economic models Selfish agents**

Individual rationality Markets Equilibrium ??? Many agents are trend followers Speculation crashes ift6802

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**Factors which affect decisions Milgram Asch**

Social Experiments Factors which affect decisions Milgram Asch ift6802

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**Stanley Milgram (1933-1984) Controversial social psychologist**

Yale & Harvard Small world experiment, 1967 6 degrees of separation Obedience to authority ift6802

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**Validity of Milgram’s experiment**

Global connectivity ? US: Omaha Boston stockbroker Only 96 valid subjects (out of 300) 100 from Boston 100 big investors 96 picked at random in Nebraska Success? 18 out of 96 Other experiments: 3 out of 60 Worse…. ift6802

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Conformity Other presentation ift6802

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**Threshold models of decisions**

Number of infected neighbors 1 Probability of infection Probability of choosing option A 1 Critical Threshold Fraction of neighbors choosing A over B Standard disease spreading model Social decision making ift6802

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Global Cascades Idea catches on…. ift6802

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Fin ift6802

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