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Diffuse Musings James Moody Duke University, Sociology Duke Network Analysis Center http://dnac.ssri.duke.edu/ SAMSI Complex Networks Workshop Aug/Sep 2010

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Consider two degree distributions: A long-tail distribution compared to one with no high-degree nodes. The scale-free networks signature is the long-tail So what effect does changes in the shape have on connectivity 1.Shape Matters Consequences of Degree Distribution Shape

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1.Shape Matters Consequences of Degree Distribution Shape

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Volume Dispersion x Skewness 1.Shape Matters Consequences of Degree Distribution Shape

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Search Procedure: 1)Identify all valid degree distributions with the given mean degree and a maximum of 6 w. brute force search. 2)Map them to this space 3)Simulate networks each degree distribution 4)Measure size of components & Bicomponents 1.Shape Matters Consequences of Degree Distribution Shape

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1.Shape Matters Consequences of Degree Distribution Shape

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1.Shape Matters Consequences of Degree Distribution Shape

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Consider targeting high-degree nodes by cracking down on commercial sex workers: Interventions can have very different effects depending on where you sit within this field Does this matter? 1.Shape Matters Consequences of Degree Distribution Shape

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Network Sub-Structure: Triads 003 (0) 012 (1) 102 021D 021U 021C (2) 111D 111U 030T 030C (3) 201 120D 120U 120C (4) 210 (5) 300 (6) Intransitive Transitive Mixed 1.Linking Shapes From motifs to structure

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P RC {300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster: M M N* M M M A* 1 1 1 1 1 1 1 1 1 0 1 1 1 10 0 0 0000 00 00 And many more... 1.Linking Shapes From motifs to structure

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Structural Indices based on the distribution of triads The observed distribution of triads can be fit to the hypothesized structures using weighting vectors for each type of triad. Where: l = 16 element weighting vector for the triad types T = the observed triad census T = the expected value of T T = the variance-covariance matrix for T 1.Linking Shapes From motifs to structure

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For the Add Health data, the observed distribution of the tau statistic for various models was: Indicating that a ranked-cluster model fits the best. 1.Linking Shapes From motifs to structure

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-Imagine a typical mixer party, where one of the guests knows a bit of gossip that everyone would like to know. -Assuming that people tell this gossip to the people they meet at the party: a)How many people would eventually hear the gossip? b)How long would it take to spread through the group? The Cocktail Party Problem 3. Time Matters Dynamics of affect dynamics on

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-Some specifics to narrow down the problem. A (seemingly) simple network problem: record who talks to who, and map the network. Mean distance: 1.99 Diameter: 4 steps The Cocktail Party Problem 3. Time Matters Dynamics of affect dynamics on

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-But such an image conflates many temporally distinct events. A more accurate image is something like this: In general, the graphs over which diffusion happens often: Have timed edges Nodes enter and leave Edges can re-occur multiple times Edges can be concurrent These features break transmission paths, generally lowering diffusion potential – and opening a host of interesting questions about the intersection of structure and time in networks. The Cocktail Party Problem 3. Time Matters Dynamics of affect dynamics on

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Source: Bender-deMoll & McFarland The Art and Science of Dynamic Network Visualization JoSS 2006 3. Time Matters Dynamics of affect dynamics on

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Relationship timing constrains diffusion paths because goods can only move forward in time. ab c d Standard graph: - Connected component - Everyone could diffuse to everyone else 3. Time Matters Dynamics of affect dynamics on

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Relationship timing constrains diffusion paths because goods can only move forward in time. ab Dynamic graph: - Edges start and end - Cant pass along an edge that has ended Time b c c d 3. Time Matters Dynamics of affect dynamics on

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Relationship timing constrains diffusion paths because goods can only move forward in time. ab Dynamic graph: - Edges start and end - Cant pass along an edge that has ended Diffusion is asymmetric: a can reach c (through b) and b and reach d (through c), but not the other way around. Time b c c d 3. Time Matters Dynamics of affect dynamics on

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Relationship timing constrains diffusion paths because goods can only move forward in time. Time abc c d Concurrency, when edges share a node at the same time, allows diffusion to move symmetrically through the network. This can have a dramatic effect on increasing the down-stream potential for any give tie. 3. Time Matters Dynamics of affect dynamics on

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Implied Contact Network of 8 people in a ring All relations Concurrent Edge timing constraints on diffusion Reachability = 1.0 3. Time Matters Dynamics of affect dynamics on

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Implied Contact Network of 8 people in a ring Serial Monogamy (3) 1 2 1 1 2 1 2 2 Reachability = 0.43 Edge timing constraints on diffusion 3. Time Matters Dynamics of affect dynamics on

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1 2 1 1 2 1 2 2 Timing alone can change mean reachability from 1.0 when all ties are concurrent to 0.42. In general, ignoring time order is equivalent to assuming all relations occur simultaneously – assumes perfect concurrency across all relations. Edge timing constraints on diffusion 3. Time Matters Dynamics of affect dynamics on

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Timing constrains potential diffusion paths in networks, since bits can flow through edges that have ended. This means that: Structural paths are not equivalent to the diffusion-relevant path set. Network distances dont build on each other. Weakly connected components overlap without diffusion reaching across sets. Small changes in edge timing can have dramatic effects on overall diffusion Diffusion potential is maximized when edges are concurrent and minimized when they are inter-woven to limit reachability. Combined, this means that many of our standard path-based network measures will be incorrect on dynamic graphs. 3. Time Matters Dynamics of affect dynamics on

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Solution? Turn time into a network! Time-Space graph representations Stack a dynamic network in time, compiling all node-time and edge- time events (similar to an event-history compilation of individual level data). Consider an example: a)Repeat contemporary ties at each time observation, linked by relational edges as they happen. b)Between time slices, link nodes to later selvesidentity edges 3. Time Matters Dynamics of affect dynamics on

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So now we: 1)Convert every edge to a node 2)Draw a directed arc between edges that (a) share a node and (b) precede each other in time. Solution? Turn time into a network! 3. Time Matters Dynamics of affect dynamics on

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So now we: 1)Convert every edge to a node 2)Draw a directed arc between edges that (a) share a node and (b) precede each other in time. 3)After the transformation, concurrent relations are easily seen as reciprocal edges in the line-graph. Becomes this: Solution? Turn time into a network! 3. Time Matters Dynamics of affect dynamics on

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4. Universal or Particular? When do we need to bring in domain-specific information? diffusion as transmission between nodes seems universal; but the content of the graph likely interacts with the structure. H W C C C Provides food for Romantic Love Bickers with How does information move here? Generality depends on: a)Transmission directionality: does passing the bit affect the sender? b)Relational Permeability: Does transmission move differently across different relations? c)Structural Reflexivity: does transmission affect the structure?

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http://dnac.ssri.duke.edu/

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