1. Keynote address Suraj Bandyopadhyay I. Genesis of social network perspective: Brief sketch Metaphorically, in history, we can trace the notion of social network a long way back formed thru alliances by marriage, circulation of gifts, exchange of goods and merchandise, and so on. During 18th and 19th centuries founding fathers of sociological thinking in Europe in their endeavor to comprehend how society was formed and remained self-perpetuated without any “omnipotent” religious or royal figure discovered its roots embedded in various kinds of “ties” arising out of kinship, division of labour, sharing common culture and values and so on binding individuals together. In this venture to locate the roots of society we observe three broad streams of conceptualization ― society viewed as a summation of individuals as independent and self-sustaining actors (JS Mill); society as an organic whole (Durkheim); and Tonies’ model to retain these two under one umbrella of gemeinschaft and geselschaft.

2. Distinctly different from the above streams of thought we find the above streams of thought we find Simmels explication of society as a manifestation of interactions among individuals (Martindale, Don: The nature and types of sociological theory, Houghton Mifflin Company Boston, 1960, 236-247). Very emphatically he has argued about the significance of “numbers” for social life while studying “combinations and interactions”. I shall just refer to his distinction between “The isolated individuals”, “Dyad” and “Triad”. A “dyad” provides “the simplest sociological formation “which” contains the scheme, germ and material of innumerable more complex forms. …. The dyad has different relation to each of its two elements than have larger groups to their members “other than a triad” where “the third element” serves “ the function of holding the whole together” (Wolff, KH ed.: The sociology of Georg Simmel, The Free Press of Glencoe, New York, 1950, 105-189). W have quoted these to show how Simmel visualized sociological significance of “dyad census” or “triad census” we do now-a-days. As Aron has synthesized according to Simmel society was bound by “reciprocal action of the parts” (German Society, The Free Press, 1957).

3. From around the beginning of 20th century Moreno’s sociometry came up as the link between Simmel’s sociological postulates and to-day’s methodology of SNA. Subsequently we find a sharp growth in application of social network perspective in diverse fields of social science research as described Barry Wellman, Wasserman and Faust, Bhaskar Dutta and M.O. Jackson, J. Mitchell and so on. Globally, however, the mainstream of analytical social network studies has remained ego-centric. Classic studies of J A Barnes of a Norwegian island parish and Elizabeth Bott’s family study were mainly descriptive. Most of the studies have examined actor’s behavior against pattern of interaction with others in a web of connections and combinations of “ties”, the pattern treated as one of its attributes.

4. II. Indian scenario In India, after Independence, the processes of urbanization and industrialization supplemented by measures of community development and agricultural growth generated forces of social mobility which accelerated dissolution of traditionally rigid, segmental and hierarchically organized social structure. Srinivas and Beteille put social network study in India in this perspective (M N Srinivas and Andre Beteille: “Networks in Indian social structure”, MAN, No. 212, November-December 1964, 165-168). Their perspective was, no doubt, ego-centric. But in Indian context it brought out the need for a social network analysis of multilayered linkages of Ego with others. Obviously, it would be a complex exercise in methodology. The issues, however, remained unattended for decades.

5. Paradoxically, at ISI Kolkata, we have not initiated our present research as a matter of social network study. It originated as a corollary to a project undertaken during 1971-74 by myself at ISI, Kolkata and Professor Donald von Eschen (McGill University, Montreal, Canada) on The Conditions of Rural Progress in India funded by Canadian International Development Agency, Ottawa, Canada. The study was based on data collected by a survey of 2697 households in 21 villages of Md. Bazaar C.D. Block in Birbhum of West Bengal. An extended summary of the findings has been published by us in Dipankar Gupta ed. Social Stratification, Oxford University Press, 1991, 353-368. We found the villages were steeply stratified by caste, class and power confounded with one another. The situation was further aggravated by the fact that a major source of livelihood of the local elites was by way of appropriation, not production. The ethos of distrust and hopelessness about change in the conditions for better in the near future prevailed the then world-view of the villagers. Could a village community sustain itself and move ahead under these conditions ? ─ How ? We sought to find an answer to these queries.

6. In this endeavor a part of our data collected to supplement the basic survey data consisted of requests made by household to another at the time of an emergency or crisis for help and cooperation of any type ─ material, financial, physical, advice and so on and the response received. Articulation of social relation of “help” brought out social network of help among the villagers. Sociologically, our primary concern was to what extent the villagers helped each other ? In formal academic parlance, “reciprocity”. How to obtain its measure from the network of help ? We translated such networks as di-graphs and decided to approach competent scientists from pertinent disciplines W have advanced step-by-step. Firstly we formulated our queries precisely as social researchers; Secondly we had communication and dialogues with potential collaborating mathematicians / statisticians (in our case the first one was graph-theoretician, late Professor A.R. Rao); Thirdly we received feed backs and collaboration was enriched by participation of Professor Bikas K. Sinha (Statistician); and Lastly our group has extended to include late Dr. Arun Chatterjee, Dr. Rabindranath Jana, Dr. Anil K. Choudhuri and very recently Suchismita Roy. Our latest “guest” is Dr. Prabir Ghosh Dostidar.

7. To begin with we have derived various measures of reciprocity, both graph-theoretic and statistical. We compared their properties and chose: (s – smin)/(smax – smin) as valid for our purpose where s is number of reciprocally tied pairs in a network and smin, smax are its minimum, maximum values given total number of vertices and arcs in the network. We have computed the values of all these measures of reciprocity for those 21 villages we have surveyed earlier. We have published the details in Sankhya, The Indian Journal of Statistics, Vol. 49, Series A, Pt. 2, 1987, 141-188 (A. Ramachandra Rao and Suraj Bandyopadhyay: Measures of reciprocity in a social network). Subsequently, two other research articles were published: Arun Chatterjee, Suraj Bandyopadhyay and A.R. Rao: “Relative importance of different factors for boundary of reciprocity: An illustration”, Connections, Vol. 16, Nos. 1-2, 1993, 15-22; and A.R. Rao, Rabindranath Jana and Suraj Bandyopadhyay: “A Markov chain Monte Carlo method for generating random (0,1)-matrices with given marginals”, Sankhya, Series A, Vol. 58, Pt. 2, 225-242. Later we took up studies to drive measures of different characteristics of a social network, such as connectedness, fragmentation, hierarchy, etc. The primary objective of our social network study is to examine the patterns of social structure of different communities ─ to compare and contrast them. Hence, the unit of our study is the whole network and our study is a global network study.

8. III. Constraints of analysis of global characteristics of social network About two decades earlier Barry Wellman has cautioned about methodological computational constraints of whole network analysis. On the other hand, Wasserman and Faust have pointed out SNA as a base for multidisciplinary collaboration among particularly social scientists with Statisticians, Mathematicians and Computational scientists. I think this to be the most important pre-requisite for a whole network study due to few major constraints specific to global characteristics obtained from a whole network. I submit two observations to explain. (1) Untenability of the assumption of Normality is one such constraint. Sociologically we may surmise that the observed deviation from Normality indicates internal articulation of ties within a social network not equalitarian, but is hierarchical. It came out in course of a survey of some village communities that Power curve appeared to “fit” best the in-degree sequence ( ) of most of their social networks. Power curve also captured the distribution of out- degree sequence ( ) of a large number of them.

9. Hence, it follows that what is required is to first examine the pattern of the variation of the global characteristics and subsequently to decide about exact method that should be applied for its statistical analysis and inference. We also ask that since the pattern of distributions of such intra- social network characteristics which deviate from Normality, is efficiently captured by Power curve, does it suggest the observed social network to be virtual Scale-Free network, if we may use the concept of Barabasi, A.L. and Bonabeau, E. (“Scale-Free networks”, Scientific American, Vol. 288, No.5, 60-69) ?

10. (2) The other constraint concerns estimation of measures of global parameters of social network for statistical analysis. Since the social network of a community or group is a “single case” by itself, it provides only a single value of a global measure such as reciprocity, fragmentation, hierarchy, and so on. How can these measures be compared and studied across communities ? Applying methodology of simulation of computer science a super-population of social networks can be generated under model-based assumptions, e.g., holding (N, m) or (N, and/or ) as given, where N = number of vertices in a social network and m = its total number of arcs. From such a super-population virtual parameters of a social network can be obtained and used for statistical analysis of global characteristics as mentioned above.

19. To illustrate we give below simulated values of few parameters of social network of the Muslim community of two urbanized villages Baghra and Harsingraidih adjacent to the town of Giridih in Jharkhand state and two commercial agricultural villages, Chitmadih (in Jharkhand) and Maladanga (in Birbhum district, W.B., (in the same ecological zone). The required table is in the next slide.

20. Under the model of given (N,m), simulated values of parameters Observed values VillageN,m Simulated/ Theoretical s*pq E(s)σsσs E(p)σpσp E(q)σqσq spq Bgr14,27 Sim.1.931.205.732.461.160.38 421 Th.1.941.20 Hrs34,16 Sim.0.940.9226.354.873.191.25 8268 Th.0.92 Chm53,67 Sim.0.790.8644.516.075.381.70 143812 Th.0.800.87 Mld22,9 Sim.0.080.2721.880.4213.170.39 22015 Th.0.080.27 * E(s) = m(m-1) / 2[N(n-1)-1] and σs =√[E(s) (1-E(s) + ((m-2)(m-3)) / 2(N(N-1)-3))], where N = n0o. of vertices and m = no. of arcs of a social network. Using the simulated values we have standarised the observed values of s, p and q. The standarised value show different ordering of the villages. Foe example: observed value of s of Maladanga is lowest, but on standarisation it becomes almost same as Harsingridih, Bagrah’s is lowest while Chitmadih’s remains at the top.

21. However, the results are model specific. Again, the simulated values are also affected by the distribution of out-degrees (di’s) inspite of total no. of arcs (m) remaining the same. The table below is an example. We refer to “Introduction to simulation” by R.E. Dawson in H. Guetzkow ed.: Simulation in Social Sciences Readings, Prentice-Hall Inc., 1962, 1-15. We shall not label these as limitations of the methodology of simulation, but as a caution to specify and enter the necessary attributes while we simulate. d’sS 2 = ∑d 2 E(s)σsσs 3 6, 2 1 584.75701.1444 5 1,4 1,3 1,2 4 664.63281.1681 5 2,2 5 704.56791.1552 6 1,3 2,2 4 704.57321.1706 4 3,3 2,2 1,0 1 704.58841.1576 4 4,2 1,1 2 704.57531.1521 5 1,3 5,0 1 704.58301.1691 5 1,4 2,3 1,2 1,1 2 724.57091.1563 5 3,4 1,1 1,0 2 924.27721.0238 x y = out-degree x occurring for y vertices Before I conclude I must express my thanks to Professor Bikas K. Sinha, for his suggestions and advice at different stages. I am also thankful to Dr. Rabindranath Jana and Dr. Anil K. Choudhuri for all technical help to prepare the manuscript.