is Associate Professor of Sociology at Columbia University and an external faculty member of the Santa Fe Institute. He holds a Ph.D. in theoretical and applied mechanics from Cornell University and is the author of Six Degrees: The Science of A Connected Age. He lives in New York City.

"embodies the random rewiring concept explicity and which unifies the properties of the alpha- and beta-models" (c. t. p. 6)

"motIvated by consideration of real social networks and how people actually make new acquaintances" (c. t. p. 6)

"motivated by simplicity of construction and the clean interpretation of its limiting cases" (c. t. p. 6)

"defining probability distributions have (effectively, for small n and k) a finite cutoff" p. 100

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

unnecessary of the strength of the human connections (also a weakness of the concept) in contrast to Granovetter's weak ties (inspiration p. 14)

alpha-, beta- and ostrich-model are included in relational graphs (c. t. p. 42)

"highly clustered graphs with small characteristic path lenghts" (p. 42)

"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)

"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)

"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)

1. "Assume that a network can be represented purely in terms of the connections between its elemets"

2. "Assume that the likelihood of a new connection being created is determined in some fashion by the already-existing pattern of connections"

a world that is "disconnected into many isolated 'caves' " with only some common connecting nodes/people between them (p. 44)

"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)

"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)

"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

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)

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

additional diagram for alpha-graphs constructed on a ring substrate

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

"broad features of a clustering coefficient (alpha) are similiar to those of L (alpha)" p. 58

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

"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

1. important to have graphs that are connected for all values of alpha (for comparison of L or cc)

3. allows for the greatest variation in L(alpha) over the full range of alpha in comparison to other substrates

5. for sufficiently high alpha, all differences between substrates are erased. indistinguishable results

corresponding probability: "a function of the Euclidean distance between the vertices"

"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

"Relational graphs admit a particular class of graphs that exhibit characteristic path lengths approximately the same as equivalent random graphs (L - ln(n)/ln(k) ), but with much greater clustering" -> small world graphs (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

"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

shortcut edges: "connect vertices that would otherwise be widely seperated" (p. 71)

supersede a path with a length that was only two in the first place, as the shortest path between i and j

"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

"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

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

"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

important for this process this the random number r. if r >= ß -> edge unaltered, if r<ß gets rewired (c. t. p. 67)

Watts, Duncan J. (2018). Small Worlds : The Dynamics of Networks between Order and Randomness /. Princeton Studies in Complexity ; 9. Princeton, NJ : : Princeton University Press, .

small world can not be enhanced: "However, simulation of three models prove thateven networkshaving similar clustering, density, size and average path length may lead to qual-itatively and quantitatively different results." (p. 23)

criteria defined by the small-world phenomenon don’tconstrainst the dynamics enough for simulation results to be conclusive" (p. 23)

none of these algorithms is representative of the behavior of themodel in the whole space of small-world models: "Moreover, the similarity of results observed in some cases (spacesX6andX0for epidemics dynamics,X3for opinion dynamics andX6for Axelrod’smodel) when using Watts-Strogatz and Barab ́asi-Albert networks could confusemodelers, letting them belief that these results are representative of the modelbehavior, while our experiments clearly reveal important differences when ex-ploring other networks. " (p. 22)

summary: In this paper, we underlined a potential flaw in common practices in compu-tational simulation: many modelers argue that their simulations make sensebecause they use plausible networks, but actually study the dynamics of theirmodels over specific samples of these plausible networks. We first identified threepossible problematics related to this fact: (i) the possiblelack of representativ-ity of a generator to the class of networksof interest, (ii) the possiblelack ofconclusiveness of simulations over a class of networksand (iii) thepossibility torefine the criteria of network choicefor reaching (more) conclusive results. Weformalized these problematics using the concept of criteria of network choice,spaces of networks and spaces of dynamics. We argued that the criteria fornetwork choice should not only be based on the plausibility of networks butshould also limit the space of plausible networks enough for simulation resultsto be conclusive.We proposed an experimental protocol to tackle these three problematics,and applied it to the space of small-world networks, and the two Watts-Strogatzand Barab ́asi-Albert networks that are often used in agent-based modelling (p. 22)