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Master of Science in Engineering in Computer Science (MSE-CS)

DIPARTIMENTO DI INGEGNERIA INFORMATICA AUTOMATICA E GESTIONALE ANTONIO RUBERTI

(MSE-CS)

Seminars in Software and Services for the Information Society

Umberto Nanni

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Social Networks – Basic concepts

An old paper

1955

No one supposes that there is any connexion between horse-kicks suffered by soldiers in the German army and blood cells on a microscope slide other than that the same urn scheme provides a satisfactory abstract model of both phenomena.

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A quote

“If we make a chart of social interactions, of who talks to whom, the clusters of dense interaction in the chart will identify a rather well-defined hierarchic structure. will identify a rather well-defined hierarchic structure. The groupings in this structure may be defined

• perationally by some measure of frequency of

interaction in this sociometric matrix.” The Sciences of the Artificial, Cambridge, MA, MIT Press, 1969.

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Herbert Alexander Simon [1916 – 2001]

• 1975: Turing award
• 1978: Nobel award (Economy)

Power Law (Pareto’s law, Zipf’s law)

• population of cities
• magnitude of earthquakes
• craters of the moon
• sunspots

y = c x−α + ε(...)

• sunspots
• file size
• “size" of the wars
• frequency of use of the words
• frequency of proper names
• number of articles written by scientists
• number of citations of articles
• per capita income

y = c x + ε(...)

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• per capita income
• number of species in taxonomies
• accesses to pages on the web
• sales of: books, music pieces, many products sold on the web
• ...

Some examples

number of web sites number of web sites 5 Seminars of Software and Services for the Information Society Umberto Nanni number of users number of users

A day of web accesses by AOL users

Some examples

percentage

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population of cities

Some examples

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Some examples

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Parameters in the power law

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Social Network

NODES: set of subjects ARCS: binary relation between instances that are part of a given set The relation giving rise to arcs:

• may NOT be symmetrical

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• may NOT be symmetrical
• can be derived from relationships with other

entities

Social networks

• membership of a social network can be:

– conscious, with an explicit declaration of links – unconscious, placement into groups depends on a relatively homogeneous behavior

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Social Networks – more examples

• World Wide Web
• Internet

Internet

• citations between scientific papers
• citations between authors
• direct knowledge between individuals
• business relationships between companies
• sending e-mail

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• sending e-mail
• (biology) involvement of proteins in a process
• telephone communications
• ...

Personal acquaintedness

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Sociogram

Graphical representation of a social graph with a metaphor that provides quantitative evidence of metaphor that provides quantitative evidence of the weight of the arcs, together with the verse.

Jacob L. Moreno, 1934 (!)

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Historical Experiments on a Social Network

1967: Milgram Experiment

• letters sent to specific recipient
• it can be sent only to a recipient personally
• it can be sent only to a recipient personally

known

• 5% success; distance-average = 6

1969 Travers-Milgram Experiment

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• compared to the previous more information

about the recipient

• 29% success; distance-average = 5.2

Small World Phenomenon

Communities in social graphs

Bipartite components:

• HUB: nodes with many leaving arcs
• AUTHORITIES: nodes with many entering arcs
• AUTHORITIES: nodes with many entering arcs

hub authority componente bipartita completa

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Social Networking Potential (SNP)

• alpha subject

Social Networks – some metrics and properties

• in/out degree gin, gout: number of entering/leaving arcs
• diameter d: maximum distance between a pair of nodes

→ Small World Phenomenon

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• neighborhood N(h): number of nodes within distance h

More metrics on Social Networks

• Betweenness: Degree an individual lies between other individuals in the network; the

extent to which a node is directly connected only to those other nodes that are not directly connected to each other; an intermediary; liaisons; bridges.

• Closeness: The degree an individual is near all other individuals in a network (directly or

indirectly). It reflects the ability to access information through the "grapevine" of network indirectly). It reflects the ability to access information through the "grapevine" of network

• members. Thus, closeness is the inverse of the sum of the shortest distances between

each individual and every other person in the network.

• (Degree) centrality: The count of the number of ties to other actors in the network. See

also degree (graph theory).

• Flow betweenness centrality: The degree that a node contributes to sum of maximum

flow between all pairs of nodes (not that node).

• Eigenvector centrality: a measure of the importance of a node in a network. It assigns

relative scores to all nodes in the network based on the principle that connections to nodes having a high score contribute more to the score of the node in question.

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nodes having a high score contribute more to the score of the node in question.

• Centralization: The difference between the n of links for each node divided by maximum

possible sum of differences. A centralized network will have many of its links dispersed around one or a few nodes, while a decentralized network is one in which there is little variation between the no. of links each node possesses

• Clustering coefficient: A measure of the likelihood that two associates of a node are

associates themselves. A higher clustering coefficient indicates a greater 'cliquishness'.

More metrics on Social Networks

• Cohesion: The degree to which actors are connected directly to each other by cohesive
• bonds. Groups are identified as ‘cliques’ if every actor is directly tied to every other

actor, ‘social circles’ if there is less stringency of direct contact, which is imprecise, or as structurally cohesive blocks if precision is wanted

• (Individual-level) density: the degree a respondent's ties know one another/ proportion
• (Individual-level) density: the degree a respondent's ties know one another/ proportion
• f ties among an individual's nominees. Network or global-level density is the

proportion of ties in a network relative to the total number possible (sparse versus dense networks)

• Path Length: The distances between pairs of nodes in the network. Average path-length

is the average of these distances between all pairs of nodes

• Radiality: Degree an individual’s network reaches out into the network and provides

novel information and influence

• Reach: The degree any member of a network can reach other members of the nework
• Structural cohesion: The minimum number of members who, if removed from a group,

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• Structural cohesion: The minimum number of members who, if removed from a group,

would disconnect the group

• Structural equivalence: Refers to the extent to which actors have a common set of

linkages to other actors in the system. The actors don’t need to have any ties to each

• ther to be structurally equivalent
• Structural hole: Static holes that can be strategically filled by connecting one or more

links to link together other points. Linked to ideas of social capital: if you link to two people who are not linked you can control their communication

Some Social Networks

• Gint Internet Graph

– degrees according Power Law

• Gweb Web Graph

– degrees according Power Law – diameter aboput 20 (Small World Phenomenon)

• Gmail e-mail Graph

– degrees according Power Law

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– degrees according Power Law

• Facebook Graph
• ...

Web graph

• "Importance" of pages → concept exploited by

Google (Page Rank)

• bipartite cores (hub / authority)
• general structure: butterfly (bow-tie)

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in

• ut

nucleus

products

Heterogeneous networks

Bipartite graph customers

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Cooperative Filtering