Processes of Diffusion in Networks

Posted on Tue 14 December 2010 in Rumination

Social diffusion processes, real and alleged, have been the focus of intense study for many years, reaching into the early days of anthropology. Because so many different things can be thought to diffuse or flow through a population-- culture, ideas, practices, behaviors, technologies and innovations, opinions, commodities, money, influence, diseases, and information, to name just a few-- the topic has been studied in many different disciplines-- anthropology, economics, epidemiology, marketing, mathematics, sociology, computer science, and communication studies, again to just name a few-- from a great number of perspectives, and with many different objectives. Reviews of some of these research programmes may be found in (Brown 1981; Rogers 2003; Valente 2005; Vega-Redondo 2007...). They are too many and their respective literatures too deep to review here. That said, much research has tended to cluster around two sometimes overlapping concerns: diffusion of innovations and the diffusion of news and other informational content (e.g. Dynamics of the News Cycle).

Everett Rogers (2003 p.94-100) describes eight or nine research programs[1] ranging over many of the possible sociological concerns relating to innovation. These include interest in the reasons that innovations are adopted or not, the causes of different rates of adoption, the consequences of innovation adoption when it occurs, and broadly speaking, diffusion of innovations in social networks.

Early theoretical work in the diffusion of innovations literature emphasized modeling the general rate of diffusion in unstructured populations or on spatial-temporal models using spatial autocorrelation (Valente 2005). Later more sophisticated network models of the diffusion of innovations became (and remain) popular.

Models of social contagion or diffusion typically follow three (not necessarily network-oriented) epidemiological formats (Vega-Redondo 2007 p.75-76):

  • SI: susceptible → infected
  • SIS: susceptible → infected → susceptible
  • SIR: susceptible → infected → recovered/removed

There are important differences between these model formats. In a SI model, individuals who become infected (or adopt an innovation) stay infected and may continue to act as catalysts for the infection of others. In a SIR model, individuals who are infected eventually recover and are immune or are removed (e.g. by dying). Recovered and removed individuals cannot infect others. Finally, a SIR model allows individuals to move back and forth between infected and susceptible states.

Network diffusion models may also be categorized by the way in which the probability of successful transmission depends on the number of infected neighbors. Vega-Redondo (2007) distinguishes between what he calls epidemiological models, in which the probability of infection is given by the net exposure to infected individuals, from models exhibiting so-called neighborhood effects, in which the probability of infection depends in some manner on the proportion of neighbors that are infected.

For example, suppose that individual A has four neighbors, three of whom have adopted a particular innovation, and another individual B who has three neighbors, all of whom have adopted the innovation. Assuming that neither A nor B have yet adopted the innovation, in a simple epidemiological model, individual A's and individual B's propensity to adopt the innovation is independent of how popular that innovation is in their respective neighborhoods, but in a model with neighborhood effects, A and B might have a different propensity to adopt the innovation.

If in the neighborhood effect model the probability of adoption is equal to the proportion of one's neighbors who have adopted, then since only 75% of A's neighbors have adopted, while all of B's neighbors have adopted, B is more likely than A to adopt the innovation. In some cases, depending on the model, the propensity to adopt might decrease with the proportion of one's neighbors adopting. Neighborhood-effect models of diffusion might be seen as games played between neighbors in a network and thus might be viewed as an sociological application of evolutionary game theory.

Research into diffusion processes crosses disciplines as diverse as computer science, anthropology, sociology, economics, epidemiology, chemistry, and physics. Much recent work, in the last fifty years or so, has been explicitly network oriented and has emphasized the interrelationships between network topology and transmission mechanisms in determining the nature of diffusion processes such as the rate of diffusion and the various thresholds at which diffusion processes become self-sustaining.

Diffusion of Semantic Content

Despite the many apparent similarities between different diffusion processes, it is important to be attentive to the particulars of each kind of diffusion process. Commodities diffuse among a population of consumers in ways different than do news topics among blogs, behaviors among a network of friends differently than diseases, and routing information in a sensor network differently than routing information in a mobile phone network, for example.

The diffusion of information or more generally semantic content has been a cross-disciplinary concern, and has been treated in a variety of ways, depending on the domain of application. It is generally recognized that such content exhibits properties that distinguishes it from the diffusion of other phenomena. For example, it is recognized that the sharing of semantic content, unlike commodities, does not necessarily incur a consequent loss of that content for the sharer, and that information is often shared preferentially with those for whom it may be of interest or desired[2].

Nonetheless, in more general settings the implications of the properties of semantic content for its diffusion has to my knowledge not yet been formally investigated. Content is typically treated as a non-relational item whose diffusion-mechanism is essentially content-neutral, except perhaps in its differential transmissibility or mutability. Furthermore, it frequently restricts its models to the diffusion of isolated pieces of content in an otherwise content-less context. Consequently, it confounds the diffusion of content vehicles with the diffusion of the semantic content itself, treating content vehicles as having an intrinsic meaning or significance.

This is, of course, quite forgivable in many instances. To be fair, this simple approach probably does a fair job of approximating the diffusion of semantic content at a unit of content or level of abstraction at which the applicability of a more rigorous approach may not be either readily apparent or especially necessary. Yet it has the unfortunate effect of potentially blinding us to the way in which the relation between contents and cognition (or computation) can generate a second means of content diffusion, for example by allowing multiple agents in a network to independently infer the same piece of information without that specific piece of information ever being explicitly communicated to them.

[1] These are: (1) "Earliness of knowing about innovations" (2)"Rate of adoption of different innovations in a social system" (3)"Innovativeness" (4)"Opinion leadership in diffusing innovations" (5)"Diffusion networks" (6)"Rate of adoption of innovations in different social systems" (7)"Communication channel usage" (8)"Consequences of an innovation" and (9)"Others" (Rogers 2003 p.94-100).

[2] In contrast to the communication of disease, which is not desirable for those involved!


Brown, Lawrence A. Innovation diffusion. Routledge, 1981.

Rogers, Dr Everett M. Diffusion of Innovations, Fifth Edition. Simon and Schuster, 2003.

Valente, Thomas W. Network Models and Methods for Studying the Diffusion of Innovations. In Models and methods in social network analysis, edited by Peter J. Carrington, John Scott, and Stanley Wasserman, 98-116. Cambridge University Press, 2005.

Vega-Redondo, Fernando. Complex social networks. Cambridge University Press, 2007.