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Self-Organization in Networks Ron Eglash, RPI

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Network theory origins in graph theory Network theory was first introduce as “graph theory” by Euler. Rather than “links” and “nodes” he called them “vertices” and “edges.” The city of Konigsberg (Russia) has seven bridges. A popular pasttime in the city was attempting to find a walk such that you cross each bridge once and only once, without re-tracing steps. Euler proved that is impossible. He was a wet blanket at bridge parties. First, he shows that you can treat the landmass as vertices (nodes) and the bridges as edges (links). Thus its the origin of topology as well.

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Network theory origins in graph theory a.The start and end are “terminal nodes” – all others are “non-terminal nodes” b.Since you never re-trace your steps, you have to enter each non-terminal node by one bridge, and then leave by different bridge. c.The non-terminal nodes must therefore have an even number of bridges d.Terminal nodes can either be both odd or none odd both odd works none odd works

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Network theory origins in graph theory The number of edges (links) is the “degree” of a node. Euler showed that a necessary condition for the walk is that the graph have exactly zero or two nodes of odd degree. Since the graph of Königsberg has four nodes of odd degree, it cannot be an Eulerian path.

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Network theory origins in Lusona? Recall that lusona, the African design practice of drawing graphs in the sand, had exactly the same constraints: a connected path that never re-traces. Could we also credit the Tchokwe with the origins of network theory? What could you ask a Lusona expert to help make that determination? And if we do it for Lusona, what about Celtic knots and Kolam in India? Sequence of Lusona, “trees of the ancestors”

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Milgram’s small world experiment a.Travers and Milgram’ s work on the small world is responsible for the folklore that “everyone is connected by a chain of about six steps” b.Their experiment “Send a packet from sets of randomly selected people to a stock broker in Boston” c.Experimental setup: Arbitrarily select people from 3 pools: d.People in Boston e.Random In Nebraska f.Stockholders in Nebraska g.Most letters never actually made it, but the ones that did accomplished the task in surprisingly few steps, on average only 5.

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Network vocabulary Translating from Euler: Graph = network, vertices = nodes, edges = links A network is composed of nodes and links between them. A “connected graph” means that no nodes are unreachable by another node. The “degree” of a node is the number of links connecting that node. The “path length” is the number of links from a given node to another given node. An “undirected” network ignores the difference between incoming and outgoing. This may be an important difference (eg website typically hope to gain large numbers of inbound links)

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Network degree distribution In Melanie Mitchel’s example, her node has the highest degree (10). In the degree distribution, she is at the upper extreme.

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Network degree distribution A node with much higher degree than most is a “hub.” Various measures of the “centrality” of a node exist. “Degree centrality”: simply the node with the most links. Google’s website is currently the highest degree hub online.

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Clustering of nodes A “clustering” is a group of nodes that are more likely to link to each other than to other nodes in the graph they are connected to. Clusters can have a fractal topology, with clusters of clusters and so on down.

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Clustering coefficient The clustering coefficient of a selected node is defined as the probability that two randomly selected neighbors are connected to each other. For example, how likely is it that two of your friends are also friends with each other? cc1(v) = number of pairs of neighbors actually connected by edges number of pairs of neighbors Vertex v has 5 neighbors, which means 10 possible pairs. 3 are actually connected to each other. CC = 5/10 = 0.5

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Small World Network: original hypothesis Regular network of 60 nodes: average path length is 15 Randomly add links to only 5% of nodes: average path length drops to 9 A network has the “small world” property if it has a small average path-length relative to the total number of nodes. Specifically typical path length L between two randomly chosen nodes grows as the logarithm of the number of nodes N in the network

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Better model than adding random links: scale-free version Regular network has high average path length (“number of hops”), high clustering. Small world has low average path length, high clustering Random network has low average path length, low clustering

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Scale-free networks similar to fractals Recall from complexity theory: the most complex is between ordered and random

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Small World Social Networks Actor network: example of a small world

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Small World Social Networks Other social network examples: Any two documents on the Web are around 19 clicks away (directed) on average Mathematicians and papers (Erdös number) Musicians and rock groups Baseball players and teams Friends Memes?

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Small World Biological Networks C. Elegans (nematode) neurons

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Small World Biological Networks Disease transmission (virus links organisms) Metabolite processing networks (catalyst chains) DNA transcription (nodes are genes)

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Small World Technological Networks Airline hubs

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Small World Technological Networks Power grid

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Small World Technological Networks

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Small World Network Empirical test: measure CC of small world network in real world Measure CC for same nodes after reconnecting them randomly Examples: Kevin Bacon Graph (KBG) Power Grid (Western US) C. elegans Worm

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Google page rank as measure Directed graph, so you only measure “in-links”: the “importance” of your webpage is who links to you, not who you link to. Measure the in-degree of each node.

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Google page rank as measure Number of Web pages with in- degree k is proportional to 1/K 2 No matter how far we “zoom” out, you see the same relation! Typical of scale-free networks

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Scale-free networks are power laws Number of Web pages with in-degree k is proportional to 1/K 2 i.e. written as Number = K −2 Power law networks are: Resilient to random deletion Vulnerable to hub deletion

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We need some skepticism about the ubiquity of power laws Cosma Shalizi: “Our tendency to hallucinate power laws is a disgrace.”

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Nonetheless, scale-free networks are at least common--why? Neural connectivity: resilience, local-global compromise (too bulky to connect all) Genetic connectivity: resistance to viruses Scientific papers: preferential attachment

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Why are scale-free networks common? Because self-organization is common! Toss a handful of particles in the air: “self-organized” but without order. Trival case Sand waves from wind action: a quasi-ordered emergent pattern. Significant case. Self-organization tends to produce scale-free structures in networks too! Salt crystal forms from evaporating water. Completely ordered. Trivial case.

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Branching structures are both geometrically and topologically scale-free Lungs fern algae

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Tree variation as variation in (geometric) fractal dimension Df = 1.05 Sparse conversation Df = 1.50 Conversation at the "edge of chaos" Df = 1.70 Dense conversation

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Fractal structures are common not just in nature but in artifical and social systems as well Tree representation of threaded online conversation

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Network Theory example: Measuring the fractal dimension of conversation trees Visually intuitive: Sparse trees = low fractal dimension, lush trees = high fractal dimension. The dimension number can take over when our intuition fails. There is a powerful connection to complexity theory…

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How to counter the reductive tendancy in network modeling? Reductive example: Linked: The New Science of Networks by Albert-László Barabási. The Internet’s “highly connected hubs” (due to fractal structure) greatly increase vulnerability to planned attack. Used data showing that networks of human sexual contact have a fractal structure. Concluded that HIV infection rates could be greatly reduced by targeting the same “highly connected hubs” – sexually promiscuous individuals.

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Targeting “highly connected hubs” in sexual networks can increase HIV rates In Africa, for example, people connected with AIDS risk have been subject to harassment, violence, and even murder. As a result, communication about HIV is very poor Lack of communication greatly increases transmission rates. Fixations on sexual promiscuity in Africa have been closely linked to right-wing religious opposition to condom use.

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Conclusion Thus the reflexive or recursive engagements of STS are not merely negative barriers to truth claims. They can be a positive force in generating representations that are better accounts of the worlds we inhabit. Complexity theory can open up representations to the “trielectic” between natural, social, and technological worlds

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