We present a study of information flow that takes into account the observation that an item relevant to one person is more likely to be of interest to individuals in the same social circle than those outside of it. This is due to the fact that the similarity of node attributes in social networks decreases as a function of the graph distance. An epidemic model on a scale-free network with this property has a finite threshold, implying that the spread of information is limited. We tested our predictions by measuring the spread of messages in an organization and also by numerical experiments that take into consideration the organizational distance among individuals. The problem of information flows in social organizations is relevant to issues of productivity, innovation and the sorting out of useful ideas out of the general chatter of a community. How information spreads determines the speed with which individuals can act and plan their future activities. In particular, email has become the predominant means of communication in the information society. It pervades business, social and scientific exchanges and as such it is a highly relevant area for research on communities and social networks. Not surprisingly, email has been established as an indicator of collaboration and knowledge exchange [1, 2, 3, 4, 5] . Email is also a good medium for research because it provides plentiful data on personal communication in an electronic form. Since individuals tend to organize both formally and informally into groups based on their common activities and interests, the way information spreads is affected by the topology of the interaction network, not unlike the spread of a disease among individuals. Thus one would expect that epidemic models on graphs are relevant to the study of information flow in organizations. In particular, recent work on epidemic propagation on scale free networks found that the threshold for an epidemic is zero, implying that a finite fraction of the graph becomes infected for arbitrarily low transmission probabilities [6, 7, 8] . The presence of additional network structure was found to further influence the spread of disease on scale-free graphs [9, 10, 11]. There are, however, differences between information flows and the spread of viruses. While viruses tend to be indiscriminate, infecting any susceptible individual, information is selective and passed by its host only to individuals the host thinks would be interested in it. The information any individual is interested in depends strongly on their characteristics. Furthermore, individuals with similar characteristics tend to associate with one another, a phenomenon known as homophily [12, 13, 14] . Conversely, individuals many steps removed in a social network on average tend not to have as much in common, as shown in a study [15] of a network of Stanford student homepages and illustrated in Figure 1 . We therefore introduce an epidemic model with decay in the transmission probability of a particular piece of in- formation as a function of the distance between the originating source and the current potential target. In the following analysis, we show that this epidemic model on a scale-free network has a finite threshold, implying that the spread of information is limited. We further tested our predictions by observing the prevalence of messages in an organization and also by numerical experiments that take into consideration the organizational distance among individuals. Consider the problem of information transmission in a power-law network whose degree distribution is given by [16]  where α > 1, and C is determined by the normalization condition. The generating function of the distribution is Following the analysis in [17] for the SIR (susceptible, infected, removed) model, we now estimate the probability p (1) m that the first person in the community who has received a piece of information will transmit it to m of
Wu et al. REF study the propagation of information in email networks, showing that the transmissibility of a piece of information decays with network distance.
Information Flow in Social Groups
Mathematics Physics Biology