Please enable JavaScript.

Coggle requires JavaScript to display documents.

Ausarbeitung: Small Worlds - Watts - Coggle Diagram

- - - - low average distance between nodes
      - low average clustering
      - random way of clustering
    - - high average distance between nodes
      - high average clustering
      - uniform way of clustering
  - - - three main models
        
        third like omega
        
        explanation
        
        "embodies the random rewiring concept explicity and which unifies the properties of the alpha- and beta-models" (c. t. p. 6)
        
        "is motivated by a desire to unify the observed properties of the alpha- and beta-models, in terms of a model-independent parameter ostrich" (p. 42)
        
        ostrich: model-independent parameter
        
        "different models of relational graphs can be unified by considering their dependence upon a model-independent parameter (ostrich, little omega), which quantifies the fraction of shortcut edges" p. 100
        
        shortcut edges: "edges that connect verteices that would otherwise be widely seperated" // a slightly more general mechanism is that of vertex contraction, quantified by the parameter ustrich
        
        alpha graph
        
        explanation
        
        "motIvated by consideration of real social networks and how people actually make new acquaintances" (c. t. p. 6)
        
        "is designed to represent the construction of a network in a fashion reminiscent of how a real social network is formed, that is, as a function of the currently existing network" (p. 42)
        
        distinction between structures of Real World Networks
        
        connected-caveman world
        
        tendence: alpha -> 0
        
        a world that is "disconnected into many isolated 'caves' " with only some common connecting nodes/people between them (p. 44)
        
        scenario: alpha -> 0
        
        "propensity of two unrelated people (which means that they are no mutual friends) to be connected is very small" (p. 44)
        
        "they share just one friend in common" (p. 44)
        
        "however, their propensity to be acquainted immediately becomes very high and stays that way regardless of how many additional mutual friends they may have" (p. 44)
        
        "propensity to become friends against fraction of current mutual freinds, it would start near zero, rise very rapidly to some relatively large number (which is normalised to one), and then plateau" (p.45)
        
        additional explanation on p. 47 and 48
        
        world Salaria
        
        tendence: a -> infinity
        
        "influence of current friendships over new friendships to be so slight as to be indistinguishable from random chance" (p. 44)
        
        inspired by a novel from Isaac Asimov "where future humans live in isolation and interact, via robots and computers, as readily across the planet as they do with their spouses" (p. 44)
        
        scenario: alpha -> infinity
        
        "no one has much propensity to connect to anyone in particular" p. 45
        
        "propensity versus mutual freinds curve would start near zero and stay near zero right up until the point at which all friends and mutual friends and then suddenly jump to one" p. 45
        
        parameter of the mathematical formula for the vertex i's propensity to connect to vertex j (zero if they are already connected) (c. t. p. 46)
        
        number of vertices which are adjacent both to i and j ( m i,j )
        
        the averge degree of the graph; = average number of connections/edges of any vertex in the graph (k)
        
        random probability of an edge (i,j) existing (p << (n 2) -1)2) (p)
        
        tunable parameter to regulate between propensities for new connections (alpha)
        
        observations
        
        Characteristic Path Length
        
        smallest L with a small alpha (c. t. p. 49) at this point the whole graph is not completely connected
        
        "L rises to a maximum, at which point the entire graph becomes connected, and then decreases with increasing alpha" p. 49
        
        "decreases with increasing alpha" p. 49
        
        additional diagram for alpha-graphs constructed on a ring substrate
        
        Phase Transition (p. 53, p. 54)
        
        low values of alpha (alpha<1) "L starts at roughly two-thirds the value of an equivalent 1-lattice (with the same n and k) and then irses in a distinct hump
        
        high values of alpha (alpha>11), L is much smaller, and for alpha increasing much past 10
        
        intermediate alpha, smooth transition from "big" to "small" graphs
        
        Clustering Coefficient
        
        " 'clustering cliff' occurs distinctly after the 'length cliff' " p. 58
        
        "broad features of a clustering coefficient (alpha) are similiar to those of L (alpha)" p. 58
        
        diagram p. 59
        
        distinguishing between different structures of an alpha-graph
        
        tree substrate can display organisational Real World hierarchies
        
        "tree-based graphs are frequently encountered in problems to do with the lenghts of graphs and have substantial applications to social systems" p. 62
        
        Cayley tree substrate
        
        "once alpha becomes sufficiently large, it is no longer possible to distinguish the length of an alpha-graph built on a tree from that built on a ring" p. 62
        
        concentration on Ring substrates argumentation p. 65
        
        important to have graphs that are connected for all values of alpha (for comparison of L or cc)
        
        all substrates qualitatively similiar L(alpha) curves
        
        allows for the greatest variation in L(alpha) over the full range of alpha in comparison to other substrates
        
        can compared well to non-substrate statistics
        
        for sufficiently high alpha, all differences between substrates are erased. indistinguishable results
        
        beta ß: high beta, high randomness
        
        explaination
        
        "motivated by simplicity of construction and the clean interpretation of its limiting cases" (c. t. p. 6)
        
        "is an attempt to investigate the generality of the phenomena observed for a the alpha-model, by removing any pretence of a social network, but spanning a similiar range of graph structures" (p. 42)
        
        "no talk of mutual friends or clusters of acquaintance circles" p. 66
        
        "simply a perfect ring structure that, by virtue of a single parameter, metamorphoses into a random graph" p. 66
        
        motivations p. 66
        
        "properties of alpha-graphs near the alpha=0 extreme are dominated by the properties of the substrate upon which they are built" p. 66
        
        "length of alpha-graphs are largely independent of the substrate chosen"
        
        "no sociological apparatus required to this model, which is precisely the point" p. 66
        
        "intermediate cases (0< ß < 1) are less easily understood, but their interpretation seems much clearer than the equivalent graphs constructed using the alpha-model" p. 67
        
        The Model
        
        starting position (ß=0): perfect 1-lattice
        
        with increasing ß: nearest neighbours get randomly "rewired to another vertex j" p. 67 (excluding self-connections and repeated connections); same with next-nearest neighbours
        
        important for this process this the random number r. if r >= ß -> edge unaltered, if r<ß gets rewired (c. t. p. 67)
        
        ß = 1: all edges are rewired randomly
        
        both: alpha and beta
        
        new edges (p. 70)
        
        "tend to form triads"
        
        supersede a path with a length that was only two in the first place, as the shortest path between i and j
      - classes of graphs
        
        spatial graphs
        
        "defining probability distributions have (effectively, for small n and k) a finite cutoff" p. 100
        
        corresponding probability: "a function of the Euclidean distance between the vertices"
        
        "different models of spatial graphs can be unified with respect to a different, model-independent parameter (big Eue), which characterises an externally imposed, spatial length scale, beyond which edges cannot connect" p. 100
        
        relational graphs
        
        alpha-, beta- and ostrich-model are included in relational graphs (c. t. p. 42)
        
        "depends on preexisting conditions" p. 100
        
        "exhibit a peak in their structural complexity (as defined by the average variation in the range of their edges) for omegastrich < 1, which lies in the small-world regime" p. 100
        
        topologically similiar
    - - m: neighbours
      - M: number of links
      - N: number of nodes
      - see sheet
      - minimally structured p. 50
      - minimally connected p. 50
      - shortcut edges: "connect vertices that would otherwise be widely seperated" (p. 71)
  - - - low average distance between nodes
      - high average clustering
      - two generalisation (p. 43)
        
        "Assume that a network can be represented purely in terms of the connections between its elemets"
        
        "Assume that the likelihood of a new connection being created is determined in some fashion by the already-existing pattern of connections"
  - - - emergence and evolution of cooperative behaviours in large organisations, whose structural nature is allowed to vary
      - The formation and spread of fame, fads, and social movements
      - Told in a funny way: 'A rigorous proof that Kevin Bacon really is the centre of the civilised world, and, if not, who is?'
      - The spread of everything from computer viruses to infectious or sexually transmitted diseases
- - - - "Milgram sent a number of packets to agreeable "sources" in Nebraska and Kansas,
      - with instructions to deliver these packets to one of two specific "target" persons in Massachusetts"
      - "targets were named and described in terms of approximate location, profession, and demography,
      - but the sources were only allowed to send the packets directly to someone they knew by first name"
      - "each link was supposed to record, in the packet, details about themselves corresponding to those provided about the target, enabling the experimenters to track the progress of the packet and the demographic nature of chain along which it passed"
      - "Milgram's methodoloy (sometimes referred to as 'the smalle-world method')" ," 'random biased walk' in a network" p. 23
  - - - "statistical analysis of pathways through networks with varying degrees of local structure"
      - "qualitative description of the structure of networks in terms of local (e. g. clustering) and nonlocal (e. g., weak ties) features.!
      - "renormalisation of networks, viewed as meta-networks of highly clustered or equivalent subnetworks"
      - embedding of networks into spaces where the coordinates are readily interpretable and relationships where the coordinates are readily interpretable and relationships between members can be more easily visualised
      - challenges in Real World Networks (c. t. p. 21)
        
        "social networks exhibit structural characteristics that are inherently nonlocal (Granovetter's 'bridges')" --> "no purely local analysis can predict their global statistical features"
        
        Analytical difficulties: "none of the work has been tested for large population size (n) with sparce connectivity under any but the most restrictive conditions"
        
        "no treatment has been given to the properties of continuous families of networks, whose structural properties vary all the way from one extreme to the other"
        
        additional empirical challenges on page 23
        
        "Most people seems to be quite poor at estimating their number of friends reliably"
        
        definition of a 'meaningful' or 'substantive' contact or relationship
      - dimensions/ two relations of theory of social networks in Sociology (c. t. p. 21)
        
        "how 'far' each pair of vertices is from each other in the (unknown) metric of the (unknown) 'social space' "
        
        "whether or not they are connected and (perhaps) how strongly"
      - additional challenges. from Real World to a graph
        
        "apparently no consistent and comprehensive method of characterising all human values, idiosyncrasies, and foibles that then aggregate to produce equally enigmatic human relationships" p. 43
        
        challenge of creating a model: "to abstract the essence of the phenomenon to be described by stripping away detail without stripping away essential detail" p. 43
        
        "apparently no consistent and comprehensive method of characterising all human values, idiosyncrasies, and foibles that then aggregate to produce equally enigmatic human relationships" p. 43
- - - - criteria defined by the small-world phenomenon don’tconstrainst the dynamics enough for simulation results to be conclusive" (p. 23)