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Connectivity and the Small World Overview Background: •de Pool and Kochen: •Random & Biased networks •Rapoport’s work on diffusion Travers and Milgram •Argument •Method •Watts •Argument •Findings •Methods: •Biased Networks •Reachability Curves •Calculating L and C Connectivity and the Small World Started by asking the probability than any two people would know each other. Extended to the probability that people could be connected through paths of 2, 3,…,k steps Linked to diffusion processes: If people can reach others, then their diseases can reach them as well, and we can use the structure of the network to model the disease. The reachability structure was captured by comparing curves with a random network, which we will do later today. Connectivity and the Small World Travers and Milgram’s work on the small world is responsible for the standard belief that “everyone is connected by a chain of about 6 steps.” Two questions: Given what we know about networks, what is the longest path (defined by handshakes) that separates any two people? Is 6 steps a long distance or a short distance? Longest Possible Path: OH Hermit Store Owner Truck Driver Two Hermits on the opposite side of the country Manager About 12-13 steps. Corporate Manager Corporate President Congress Rep. Congress Rep. Corporate President Corporate Manager Manager Truck Driver Store Owner Mt. Hermit What if everyone maximized structural holes? Associates do not know each other: Results in an exponential growth curve. Reach entire planet quickly. What if people know each other randomly? Random graph theory shows us that we could reach people quite quickly if ties were random. Random Reachability: By number of close friends 100% Degree = 4 Degree = 3 Percent Contacted 80% Degree = 2 60% 40% 20% 0 0 1 2 3 4 5 6 7 8 Remove 9 10 11 12 13 14 15 Milgram’s test: Send a packet from sets of randomly selected people to a stockbroker in Boston. Experimental Setup: Arbitrarily select people from 3 pools: a) People in Boston b) Random in Nebraska c) Stockholders in Nebraska Milgram’s Findings: Distance to target person, by sending group. Most chains found their way through a small number of intermediaries. What do these two findings tell us of the global structure of social relations? Milgram’s Findings: Length of completed chains Duncan Watts: Networks, Dynamics and the Small-World Phenomenon Asks why we see the small world pattern and what implications it has for the dynamical properties of social systems. His contribution is to show that globally significant changes can result from locally insignificant network change. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon Watts says there are 4 conditions that make the small world phenomenon interesting: 1) The network is large - O(Billions) 2) The network is sparse - people are connected to a small fraction of the total network 3) The network is decentralized -- no single (or small #) of stars 4) The network is highly clustered -- most friendship circles are overlapping Duncan Watts: Networks, Dynamics and the Small-World Phenomenon Formally, we can characterize a graph through 2 statistics. 1) The characteristic path length, L The average length of the shortest paths connecting any two actors. (note this only works for connected graphs) 2) The clustering coefficient, C •Version 1: the average local density. That is, Cv = ego-network density, and C = Cv/n •Version 2: transitivity ratio. Number of closed triads divided by the number of closed and open triads. A small world graph is any graph with a relatively small L and a relatively large C. The most clustered graph is Watt’s “Caveman” graph: Ccaveman 6 1 2 k 1 Lcaveman n 2(k 1) Duncan Watts: Networks, Dynamics and the Small-World Phenomenon C and L as functions of k for a Caveman graph of n=1000 1.2 140 Clustering Coefficient 100 0.8 80 0.6 60 0.4 40 0.2 20 0 0 20 40 60 Degree (k) 80 100 0 120 Characteristic Path Length 120 1 Duncan Watts: Networks, Dynamics and the Small-World Phenomenon Compared to random graphs, C is large and L is long. The intuition, then, is that clustered graphs tend to have (relatively) long characteristic path lengths. But the small world phenomenon rests on just the opposite: high clustering and short path distances. How is this so? Duncan Watts: Networks, Dynamics and the Small-World Phenomenon A model for pair formation, as a function of mutual contacts formations. 1 mi , j k m Ri , j ki , j (1 p) p k mi , j 0, p mi , j 0 Using this equation, produces networks that range from completely ordered to completely random. (Mij is the number of friends in common, p is a baseline probability of a tie, and k is the average degree of the graph) Duncan Watts: Networks, Dynamics and the Small-World Phenomenon A model for pair formation, as a function of mutual contacts formations. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon C=Large, L is Small = SW Graphs Why does this work? Key is fraction of shortcuts in the network In a highly clustered, ordered network, a single random connection will create a shortcut that lowers L dramatically Watts demonstrates that Small world graphs occur in graphs with a small number of shortcuts Empirical Examples 1) Movie network: Actors through Movies Lo/Lr= 1.22 Co/Cr = 2925 2) Western Power Grid: Lo/Lr= 1.50 Co/Cr = 16 3) C. elegans Lo/Lr= 1.17 Co/Cr = 5.6 What are the substantive implications? Return to the initial interest in connectivity: disease diffusion 1) Diseases move more slowly in highly clustered graphs (fig. 11) - not a new finding. 2) The dynamics are very non-linear -- with no clear pattern based on local connectivity. Implication: small local changes (shortcuts) can have dramatic global outcomes (disease diffusion) Duncan Watts: Networks, Dynamics and the Small-World Phenomenon How do we know if an observed graph fits the SW model? Random expectations: For basic one-mode networks (such as acquaintance nets), we can get approximate random values for L and C as: Lrandom ~ ln(n) / ln(k) Crandom ~ k / n As k and n get large. Note that C essentially approaches zero as N increases, and K is assumed fixed. This formula uses the density-based measure of C. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon How do we know if an observed graph fits the SW model? One problem with using the simple formulas for most extant data on large graphs is that, because the data result from people overlapping in groups/movies/publications, necessary clustering results from the assignment to groups. G1 G2 G3 G4 G5 1 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1 . . . . LINES CUT . . . . . William 0 1 0 0 Xavier 0 1 0 1 Yolanda 1 0 1 0 Zanfir 0 1 1 1 12 14 9 14 Amy Billy Charlie Debbie Elaine Frank George 0 0 0 0 1 0 0 0 0 0 1 5 Duncan Watts: Networks, Dynamics and the Small-World Phenomenon How do we know if an observed graph fits the SW model? Newman, M. E. J.; Strogatz, S. J., and Watts, D. J. “Random Graphs with arbitrary degree distributions and their applications” Phys. Rev. E. 2001 This paper extends the formulas for expected clustering and path length using a generating functions approach, making it possible to calculate E(C,L) for graphs with any degree distribution. Importantly, this procedure also makes it possible to account for clustering in a two-mode graph caused by the distribution of assignment to groups. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon How do we know if an observed graph fits the SW model? Newman, M. E. J.; Strogatz, S. J., and Watts, D. J. “Random Graphs with arbitrary degree distributions and their applications” Phys. Rev. E. 2001 log( N / z1 ) l log( z 2 / z1 ) Where N is the size of the graph, Z1 is the average number of people 1 step away (degree) and z2 is the average number of people 2 steps away. Theoretically, these formulas can be used to calculate many properties of the network – including largest component size, based on degree distributions. A word of warning: The math in these papers is not simple, sharpen your calculus pencil before reading the paper… Duncan Watts: Networks, Dynamics and the Small-World Phenomenon How do we know if an observed graph fits the SW model? Since C is just the transitivity ratio, there are a number of good formulas for calculating the expected value. Using the ratio of complete to (incomplete + complete) triads, we can use the expected values from the triad distribution in PAJEK for a simple graph or we can use the expected value conditional on the dyad types (if we have directed data) using the formulas in SPAN and W&F. The line of work most closely related to the small world is that on biased and random networks. Recall the reachability curves in a random graph: Random Reachability: By number of close friends Percent Contacted 100% Degree = 4 Degree = 3 80% Degree = 2 60% 40% 20% 0 0 1 2 3 4 5 6 7 8 Remove 9 10 11 12 13 14 15 For a random network, we can estimate the trace curves with the following equation: Pi 1 (1 X i )(1 e api ) Where Pi is the proportion of the population newly contacted at step i, Xi is the cumulative number contacted by step i, and a is the mean number of contacts people have. This model describes the reach curves for a random network. The model is based on a, which (essentially) tells us how many new people we will reach from the new people we just contacted. This is based on the assumption that people’s friends know each other at a simple random rate. For a real network, people’s friends are not random, but clustered. We can modify the random equation by adjusting a, such that some portion of the contacts are random, the rest not. This adjustment is a ‘bias’ - I.e. a non-random element in the model -- that gives rise to the notion of ‘biased networks’. People have studied (mathematically) biases associated with: •Race (and categorical homophily more generally) •Transitivity (Friends of friends are friends) •Reciprocity (i--> j, j--> i) There is still a great deal of work to be done in this area empirically, and it promises to be a good way of studying the structure of very large networks. Figure 1. Connectivity Distribution for a large Jr. High School (Add Health data) P r o p o r tio n R e a c h e d "Pine Brook Jr. High" 1 Random graph Observed 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 Remove 9 10 11 12 13 14 How useful are C & L for characterizing a network? These two graphs both have high C To calculate Average Path Length and Clustering in UCINET Clustering Coefficient 1) 2) 3) Load the network To keep w. Watts, make the network symmetric • Transform > Symmetrasize > Maximum • Note what you saved the graph as Calculate clustering coefficient • Network > Network Properties > Clustering Coefficient • The local density version is the “overall clustering coefficient” • The transitivity version is the “weighted clustering coefficient” To calculate Average Path Length and Clustering in UCINET Average Length 1) 2) 3) Load the network To keep w. Watts, make the network symmetric • Transform > Symmetrasize > Maximum • Note what you saved the graph as Calculate Distance • Network cohesion Distance • Tools Statistics Univariate Matrix