## Standard Model

Representational framework:

• vertex set: a finite set of identifiable entities
• node: a single entity
• edge: relation between two entities

This representational framework is quite restrictive, as its core assumptions are strong.

Core Assumptions:

1. [Finite Components] System can be reduced to a well-defined set of discrete components.
3. [Dichotomous Relations] For any given pair of such components, the relationship is dichotomous, either present or absent.
4. [Static representation] Components of the system and their relationships are essentially static on the time scale of the processes of interest.

## Inappropriate Uses and Relaxations

The section is largely based on {Carter, 2009}

### Node

Pooling: collapsing all potentially interacting elements into a single unit such as groups, households, or organizations may be a very poor approximation of reality. Because

1. the set of interacting units are fluid
2. subunits of a larger unit may themselves interact with others both within and without the parent.

To avoid misleading conclusions, the set of nodes should be defined so as to include all distinct entities that are capable of participating in the relationship under study.

### Edge

Dichotomization: the choice of a threshold level can substantially alter the properties of the resulting network, through

1. selective tie removal
2. changes in network density

Rather, one must determine whether the relationship under study is sufficiently stable to be well-approximated by a constant function over the period of interest and whether the values taken by this function across pairs are sufficiently constrained to be approximately dichotomous.

Relaxation:

1. directed edge
2. weighted edge
3. hyperedge

### Time Scales

Time-aggregation: nodes are aggregated over an extended period when real relationships are short and sequential.

Relaxation:

1. network time series

## Applications

• species interactions in ecosystem {Jordi Bascompte, Disentangling the Web of Life. Science, 2009.}
• banking network [Schweitzer, et al. Economic Networks: The New Challenges. Science, 2009.]

### Diffusion of Information Through Networks

• T. Schelling. Micromotives and Macrobehavior. Norton, 1978.
• M. Granovetter. Threshold models of collective behavior. American Journal of Sociology 83(6):1420-1443, 1978.
• S. Morris. Contagion. Review of Economic Studies 67 (2000), 57-78.
• E. Berger. Dynamic Monopolies of Constant Size. Journal of Combinatorial Theory Series B 83(2001), 191-200.
• D. Kempe, J. Kleinberg, E. Tardos. Maximizing the Spread of Influence in a Social Network. In Proceedings of KDD 2003.
• H. Peyton Young. The Diffusion of Innovations in Social Networks. Santa Fe Institute Working Paper 02-04-018.
• C. Asavathiratham. The Influence Model: A Tractable Representation for the Dynamics of Networked Markov Chains. Ph.D. Thesis, MIT 2000.
• M. Richardson, P. Domingos. Mining the Network Value of Customers. Proc. KDD 2001.
• P. Domingos, M. Richardson. Mining Knowledge-Sharing Sites for Viral Marketing. Eighth International Conference on Knowledge Discovery and Data Mining, KDD-2002.

Resources:

## References

{^1} Carter T. Butts, Revisiting the Foundations of Network Analysis. Science, 2009.