FIVE YEARS AFTER audiences flocked to the movie Six Degrees of Separation, scientists rediscovered social networks. In June 1998, the leading scientific journal, Nature, published an article by two physicists, Duncan Watts and Steven Strogatz, that addressed the so-called small-world problem. This is the possibility, first suggested by Stanley Milgram’s experiments in the 1940s, that the United States population was connected by only six degrees of separation.

The “small world” network model Watts produced unravelled this sociological conundrum and extended it to explain some mysteries of the biological world. Watts discovered small-world architectures in the network of connections between actors co-starring in the same movies, the US western states electricity grid and the neural connections of the “celebrated and much studied worm Caenorhabditis elegans”. He applied the model to other areas – the spread of infections and the synchronisation of crickets’ chirping. Since then connectivity and small-world networks have been discovered in many types of self-organising systems from ecology to computer programming.

Watts’s small world showed that surprisingly few connections and connectors are needed to create a network. Sociologists had been studying networks since the 1930s, but Watts and Strogatz turned their thinking around.

Last year Lucy Butcher and I examined the boards of companies, public organisations and community organisations in Brisbane to see what a small-world social network looks like. We found a social network created by prominent citizens’ participation on the boards of these organisations. In social-network analysis this is known as an affiliation network since people are associated not by direct links but by their affiliation with a corporate entity. The network connections between members of these boards are made by joint memberships (or “interlocking directors”) of those holding positions on two or more boards. These are the network connectors.

In Brisbane there are 314 connectors who link a population of 1930 board members at just 4.68 degrees of separation. The network is grounded in public-sector advisory boards rather than private companies. This suggests how political patronage interacts with social prestige to weave the web of civil society into stable, but not permanent, structures of community power.

THE SMALL-WORLD model highlights unexpected consequences of small but familiar coincidences of social life. Most people have acquaintances or distant relatives who have contact with celebrity circles or public life. Such connections are a tiny, often inactive, part of our total social networks. Watts’s small-world theory demonstrated how, like yeast in bread, this very small proportion of “long-distance” links can, in theory, connect very large populations at six degrees of separation or less. Long-distance connections of this kind jump across social distance and can traverse entire populations. It is conceivable that, somehow, they link everyone in the world to everyone else. The average degree of separation among the world’s population is probably greater than the six degrees of separation suggested by Milgram, but not significantly greater. In practical terms, most people are unlikely to use such a chain beyond its second point, a friend of a friend.

Social networks model as mathematical graphs. In social networks people are the nodes and the connections between them the edges. Theorists found that random graphs connected quite suddenly, like water turning into ice, when the number of connections per person reaches a certain level. The number of edges needed to connect a large group does not increase directly with size: the proportion of edges necessary to connect 100 nodes is 4.6, yet 1000 nodes requires 6.9 and 1 billion nodes needs only 20.7. This explains why there may only be a few degrees of separation in very large populations.

Prior to Watts’s work, random graphs had not seemed relevant to social networks or the small-world problem because people do not make social connections randomly. On the contrary, social connections are locally clustered and interactions within a circle of friends are reinforcing. It is common, when we meet a new person, to establish rapport by talking about “mutual” acquaintances. We explore and reinforce these connections rather than ones that go beyond the group. These are our “strong” ties or “bonds”. Before Watts, social-network theorists assumed that these strong social ties precluded significant development of long-distance or “weak” social ties. Similarly, ties across distinct populations were considered rare. Milgram wondered how social paths would cross from black to white populations in the US of the 1940s. We might wonder how an average Tasmanian would link to someone in Namibia. Localism, parochialism, in-group conformity and similar social habits were thought to override the potential for network connectivity.

Watts’s model shattered this barrier. He simulated graphs where all the points began by being clustered together in local groups. He then relaxed the clustering constraint creating links “rewired” at random, as in a random graph. He discovered that astonishingly small numbers of links need to be rewired to achieve short paths and fewer degrees of separation. A random rewiring would mean that while the population of Tasmania is clustered in local groups that connect with each other, a chance connection that links a Tasmanian to a Namibian links these two populations.

Thus, small-world connectivity can occur even when most time and energy is engaged with local circles and friendships. There is no trade-off between building local bonds and long-distance ties. One does not displace the other.

IN AN ACTIVE community, networks are part of public life. Advisory boards, consultative committees and community organisations draw people into the public realm and provide meeting places for active citizens. Seen in isolation, these boards are small, localised groups whose members interact and come to know one another. However, there are a few people who serve on two or more boards. These are the networkers who, potentially, join otherwise separated groups of board members into a network.

Social-network analysts have worked with affiliation networks for many years. First they recognised the duality in these networks. Affiliation networks are, simultaneously, links between organisations and links among people. Because of the power and prominence of large corporations in business and public life, the most exhaustive studies of affiliation networks were by researchers dealing with networks of interlocking company directors. These studies focused almost exclusively on links between organisations. Recent interest in small-world networks has encouraged researchers to investigate corporate interlocking as a network of interpersonal connections. The results are striking, showing that connections between organisations are likely to follow from links between people, rather than the other way around.

Our investigation of the affiliation network of public boards and committees in Brisbane illustrates this duality. At one level, it is a network study. It is also the beginnings of a community-power study that seeks to identify decision-making elites.

TO SEARCH FOR a network, we gathered information about 365 organisations in Brisbane. This included all the Brisbane-based Australian Stock Exchange-listed companies and some interlocked non-listed companies, industry and business associations, government-owned corporations, Brisbane City Council business entities, public institutions whose board members are listed on the State Government’s Register of Queensland Government bodies and large Queensland-based community organisations.

The members of these boards constitute Brisbane’s “elite” of publicly active people. Potentially, they are all networked. We identified 2619 people holding 3118 board positions. Three hundred and thirty of them were networkers who held two or more positions creating 803 inter-organisational links. This number of links is slightly below the critical threshold for full connectivity of a random network but above the subcritical level necessary for a large component to emerge. We thus found a large connected component of 253 organisations, a few small components and many organisations that remained unconnected.

These 253 organisations are the base of the network of Brisbane’s public elites. They differ from the original list in two ways – the percentage of companies falls by about 10 per cent while the percentage of state government bodies increases by a similar percentage. This suggests that networking and interlocking are more general in the public sector than in the company sector.

The 1930 people on these 253 boards are connected in an affiliation network. The average size of these boards is 9.5 members, the 1930 board members hold 2410 seats, an average of 1.25 each. Three hundred and fourteen (16.3 per cent) of the board members are connectors who sit on two or more boards.

Since the organisations are all connected, there is a path connecting every person to everyone else. Counting just the shortest paths, there are about 1.85 million paths in this network. The longest of these entails 15 degrees of separation. The average path length is 4.68 and the standard deviation 1.48. Thus the “degree of separation” in this small world is about 4.5.

How “small” is this average degree of separation? The distance that a random graph would suggest for the size of this network (253 nodes) is 2.81, somewhat less than the actual length of 4.68 that we found in Brisbane.

The degree of separation we find is about 50 per cent longer than a random graph would produce but significantly shorter than a minimally connected decentralised graph, which would predict 65 degrees of separation. Connections in this network definitely have small-world characteristics but probably still have elements of localisation that stop them being truly random.

THE 12 BRISBANE boards with the highest number of connections to other boards are the core of the affiliation network. The key connectors (networkers) are people with multiple memberships across these boards. The 12 boards are Brisbane City Council Holding Entity Advisory Board, The Brisbane Institute, Electricity Supply Industry Superannuation, Energex, Port of Brisbane Corporation, Queensland Investment Corporation, Griffith University Council, Queensland Library Foundation, Queensland Council of Unions, Sunsuper, University of Queensland Senate and Workcover Queensland.

Advisory boards of major public institutions are particularly important to the creation of a community network. Without them, the network would be very sparse and possibly not even connected. Listed companies, by contrast, do not enter this core group of boards. (The most connected listed company board is Bank of Queensland.) This is not surprising for two reasons. Advisory boards are larger than company boards; their members are seldom paid and serve because of their prior community contribution and add to the boards from that base. To be attractive, boards have to provide them with ways to enhance their contacts and networks. Furthermore, as advisory boards, they need to draw in expertise and experience from a wide range of stakeholders. The evidence of their linkages to other organisations is a reflection of this role.

An indication of a board’s significance in a network of interpersonal connections is suggested by the “efficiency” of a board to an individual wishing to make the most connections through the least number of people. We find this indicator by taking the number of connections made by a board and dividing it by the number of networkers on that board. For someone wishing to connect as efficiently as possible this indicator shows where the most well-connected networkers gather.

In Brisbane, they are (in rank order) the boards of the Brisbane City Council Holding Entity, Workcover Queensland, Sunsuper, Queensland Library Foundation, Queensland Investment Corporation, The Brisbane Institute, Energex, Port of Brisbane Corporation, Griffith University Council, Queensland Council of Unions, Electricity Supply Industry Superannuation and the Senate of the University of Queensland. The “efficiency’ suggested by this indicator means that the Brisbane City Council Holding Entity advisory board has the most highly connected group of people. It has gathered people with multiple connections, increasing the connections available through each member. The Senate of the University of Queensland, by contrast, has many networkers on it but these networkers do not have a lot of further contacts. The Senate itself is the forum for the links to be made rather than its members being highly connected networkers.

THE CONNECTIONS AMONG Brisbane’s public elites exemplify small-world network architecture, a short path length comparable to that of a random graph but the persistence of discrete clusters. We also found that the Brisbane network was grounded on public advisory boards rather than private-sector companies and boards. The public-sector bodies attract people active on other boards, including company boards. They are the forums for publicly active persons to meet one another and network. There is no institutionalised centralisation in this network. However, if sustained over a period of time, locally determined patterns of appointment, and attraction, will produce a centrality of organisations and persons.

Our snapshot revealed a structured and centralised community network. It is likely, however, that when the state government and city council change, this particular structure will be disrupted. Like natural systems and ecologies, social networks are self-organising. They adapt to changing environments, simply reorganising their basic properties of connectivity and centralisation into a new configuration.

Appendix : The science of small world networks

Scientists have discovered social networks. In June 1988 the leading scientific journal, Nature, published an article by two physicists, Duncan Watts and Steven Strogatz (Watts and Strogatz 1998). They started with a social network problem, the so-called small world problem. This is the possibility, suggested by Milgram’s experiments in the 1940s, that the population of the US was connected at only six degrees of separation (Milgram 1967; Kochen 1989).

The ‘small world’ network model Watts had produced unravelled this sociological paradox but it also had applications in the natural sciences. Watts’ discovered ‘small world’ architectures in the network of connections between actors co-starring in the same movies, the US Western States electricity grid and the neural connections of the ‘celebrated and much studied worm Caenorhabditis elegans.’ (Watts 1999) In addition he applied the model to several other areas of scientific interest such as the spread of infections and the synchronisation of crickets’ chirping. Since 1998 connectivity and small world networks are being discovered in many areas from ecology to computer systems(Buchanan 2002).

What makes small world networks sexy? Sociologists have been studying networks since the 1930s. Watts and Strogatz must have touched on something new. In fact, the small world model posits nothing spectacular about the social world we know, it merely highlights some unexpected consequences of the small coincidences of social life with which we are all familiar.

Long Distance Social Connections and Random Graphs

Most people have some acquaintances or distant relatives who have contact with, or move in, celebrity circles or public life. Such connections are a tiny, often inactive, part of our total social networks. Watts’ small world theory demonstrated how, like yeast in bread, this very small proportion of ‘long-distance’ links can, in theory, connect very large populations at six degrees of separation or thereabouts. Long-distance connections of this kind jump across social distance or can traverse entire populations. It is conceivable that, somehow, they link everyone in the world to everyone else. The average degrees of separation among the world’s population are probably greater than the six degrees of separation suggested by Milgram, but not significantly greatly.

Social networks model as mathematical graphs, constructions of points (nodes) linked by connecting lines (edges). In social networks persons are the nodes and the connections between them are the edges. Graph theory is the basis of Watts’ model. Since the first model of a random graph was presented (Erdos and Renyi 1959) graph theorists and social network analysts had done much work on random graphs. Random graphs are graphs with a given number of nodes to which edges are added at random. Theorists knew that random graphs become connected quite suddenly when the ratio of edges to nodes (the number of connections per person) reaches a certain level. This is a phase transition similar that when water turns to ice. At this point also, the random graph has a short path length (degrees of separation) since it has many long-distance links.

The number of edges needed to connect a large graph does not increase directly with the size of the graph, rather it is related to the natural logarithm of graph size. This natural log increases only very slowly as the real size of graph increases. Thus the amount of connection necessary to connect very large graphs is only a small increase over that necessary to connect a small graph. Whereas the proportion of edges to lines necessary to connect 100 nodes is 4.6, the proportion for 1,000 nodes is 6.9 and the proportion for one billion (1,000,000,000) nodes is 20.7.

In essence, therefore, the properties of the random graph explain the small world problem of short average path lengths in very large populations.

Random Graphs and Real Social Networks

Random graphs did not seem relevant to social networks or the small world problem since people do not make social connections randomly. On the contrary our social connections are locally clustered. Social interactions within a circle of a circle of friends are reinforcing. It is common to when we meet a new person to establish rapport by talking about the ‘mutual’ acquaintances we have in common. We explore and reinforce these connections rather than ones that go outside the group. These are our ‘strong’ ties (Granovetter 1973) or ‘bonds. Before Watts, social network theorists assumed that the construction and reinforcement of these ‘strong’ social ties would preclude significant development of long distance or ‘weak’ social ties. Similarly, ties across distinct populations are rare. Milgram wondered how social paths would cross from black to white populations in the US of the 1940s. We might wonder how an average Tasmanian would link to someone in Namibia. Localism, parochialism, in-group conformity and similar social habits were thought to override the potential for network connectivity.

Watts’ model shattered this apparent barrier. Watts simulated graph formations where all the points began by being clustered together in localised groups. He then relaxed the clustering constraint in small steps creating links ‘rewired’ at random, as in a random graph (Figure 1).

What he discovered was that the amount of links needing to be rewired to achieve the short path length of the random graph was astonishingly small. Connectivity and short path length were achieved when just one percent (or even less) of links were rewired in this way. Local clustering, however, was hardly affected although path length dropped dramatically (Figure 2).

Thus, small world connectivity can still arise in circumstances where the vast majority of peoples’ time and energy is engaged with local circles and friendships. There is no trade off between building local ‘bonds’ and long-distance ‘ties’. One does not displace the other. Indeed, it would be hard to imagine local circumstances extreme enough to displace the low level of long-distance connections necessary for network wide connectivity in Watts’ model.

The small world of public connections in Brisbane: four degrees of separation

What does a small world network look like? In an active community, networks are part of public life. Advisory boards, consultative committees, community organisations and the like draw people into the public realm and provide meeting places for active citizens. Seen in isolation, these boards are small, localised groups whose members interact and come to know one another. However there are a minority of people who serve on two or more boards. These persons are the networkers who, potentially, join otherwise separated groups of board members into a network.

Social network analysts have worked with these affiliation networks for many years. First they recognised the duality in these networks. Affiliation networks are simultaneously networks of interorganizational links and, on the flip side, networks of links among persons (Breiger 1974). Because of the power and prominence of large corporations in business and public life the most exhaustive studies of affiliation networks were made by researchers dealing with networks of interlocking company directors (Alexander 1998). But these studies focused almost exclusively at the level of the interorganizational network. Only recently has the general interest in small world networks promoted researchers to investigate corporate interlocking as a network of interpersonal connections (Alexander 2003)

We have recently investigated the affiliation network of public boards and committees in Brisbane. At one level this is a network study. At another level, however, it is the beginnings of a community power study (Caulfield, Wanna et al. 1995). Community power studies seek to identify the decision-making elites of a community. The identification of the ‘elites’, top position holders in important institutions and organizations, such as we have done here, is a common starting point for these studies.

To see a network we gathered information about 365 organizations in Brisbane. These included all the ASX-listed companies which nominate Brisbane as their home exchange and some non-listed companies interlocked to them, industry and business associations (including the Queensland Council of Unions), all (State) government-owned corporations, the Brisbane City Council business entities and their holding company, all bodies listed in the Queensland registry of state bodies and the large Queensland based community organisations appearing the Australian Directory of Associations.

The members of all these boards constitute Brisbane’s ‘elite’ of publicly active persons. They are all networked, potentially, through the connections across these boards. In total we identified 2,619 persons holding 3,118 board positions. 330 of these people were networkers who held two or more positions. The array of joint positions they held created 803 interorganizational links. Random graph theory predicts that the critical threshold for full connectivity among 365 organisations requires round about 1,077 or more such interorganizational links. The interorganizational graph of our affiliation network is under this threshold but above that of the sub-critical phase. Our graph has the characteristics of the sub-critical phase. It has one large, connected component containing 253 organizations, a few small components and many organisations that remain unconnected.

The large component of 253 organizations constitutes the network of the Brisbane public elites. We will focus on this connected component for the rest of this paper. It has much the same composition of organization types as the original list with two exceptions. The percentage of companies falls by about 10 percent while the percentage of state government bodies increases by about the same percent. This indicates that networking and interlocking are more general in the latter group of organizations than in the company sector.

Table One: Brisbane Organizations by Organization Type – Full List and Large Component

The Interpersonal Network: Degrees of separation

It is only the persons on the boards of organizations in our large component who can be connected in an affiliation network. There are 1,930 persons in this affiliation network and they hold 2,410 board seats, an average of 1.25 positions each. Thus, on average, one in five of these board members holds a second position. The average size of these boards is 9.5 members. 314 (16.3%) of these people are networkers, sitting on two or more boards.

Since the organizations in the large component are all connected, there is a path connecting every person to every one else. Counting just the shortest paths (the geodesics) there are about 1.85 million paths in this network. The longest of these paths entails 15 degrees of separation. The average length of these paths in this affiliation network is 4.68 and the standard deviation 1.48.

How ‘small’ is this average degree of separation? How does it compare with the distance that a random graph would have. (Albert and Barabasi 2002: p. 58) suggest that the average path length of a random graph is approximated by division of the natural log of its size and the natural log of the average number of contacts that each person in the network has. This gives an approximation of average path length for random graphs of the size of our network (253 nodes) of 2.81, somewhat less that the actual length of 4.68 that we have found.

We thus find degrees of separation among Brisbane public elites that are about 50 percent longer than a random graph would produce. On the one hand this supports Watts’ claim that small world distance is infinitely closer to a random graph than a minimally connected, decentralized graph would be. The latter would have an average path length of about 65 degrees of separation for 253 organizations. On the other hand, this network is not as ‘efficiently’ connected as a random graph would be. It is probably the case therefore that the making of connections in this network still has elements of localisation that stop it being truly random.

Patterns of Connection: Mapping the network

We can map affiliation networks in a way that shows how connections between persons and connections between organizations combine and overlap. In Figure 3 we map the connections between the 12 Brisbane boards with the most interorganizational connections. This is the core of our affiliation network. In Figure 3 we show all persons on these 12 boards who are networkers. However, since we select on 12 boards, the networkers in the diagram are only the persons with multiple memberships across these 12 boards. In the diagram the groups are enclosed in circles and interpersonal connections made within groups are shown as grey edges. Interorganizational connections made by networkers are black lines. Each membership of the networkers is shown as a separate node and thus each networker is a bundle of two, three or four memberships. Persons holding just one position are the lightest coloured nodes while there are heavier colours for the memberships held by networkers with 2, 3 and 4 memberships.

The core network of Brisbane’s public elites

Figure Three maps the core network of our survey of public elites in Brisbane. By selecting the 12 boards with the highest number of connections to other boards, we highlight the boards central to the network. These are the nodes through which will be involved with the greatest number of paths within the network. We have simplified the diagram however to show only the connections among these 12 boards themselves and their positioning relative to each other.

Table Two gives the names and organizational type of the boards in this core group.

Table 2: Boards in network core (15 or more outside connections)

We see how important the advisory boards of major public institutions are on this list. They are crucial to the creation of a community network. Without them, the network would be very sparse and possibly not even connected. Listed companies, by contrast, do not enter this core group of boards. (The most connected listed company board is Bank of Queensland with 13 outside connections.) This is not surprising however. Advisory boards are larger than company boards and their members are seldom paid (certainly not at the scale of private company director’s fees). They serve because of their recognised prior contribution to the community and contribute to the boards from that base. To be attractive boards have to provide them with ways to enhance their contacts and networks. Furthermore, as advisory boards, they need to draw in expertise and experience from a wide range of stakeholders. The evidence of their linkages to other organizations is a reflection of this role.

The placement of each organization in Figure 3 indicates its centrality relative to other organizations in the core. This is a valid but limited view of an organization’s place in the overall network. An indication of a board’s significance in network of interpersonal connections is suggested by the ‘efficiency’ of a board to an individual wishing to make the maximum number of connections through the least number of people. We find this indicator by dividing the number of connections made by a whole board by the number of networkers on that board. For someone wishing to connect as efficiently as possible, this indicator would say where the most well connected networkers are gathered.

Table Three ranks the core organizations on this indicator.

Table Three: Core organizations ranked by ‘efficiency’ indicator

We can see the ‘efficiency’ suggested by this indicator by comparing the top and bottom boards. The Brisbane City Council advisory board (BCHO) is a highly connected group of persons. It has gathered people to it that already have multiple connections and hence has many connections available through each member. The Senate of the University of Queensland, by contrast, has many networkers on it but these networkers do not have a lot of further contacts. The Senate itself is the forum for the links to be made rather than its members themselves being highly connected networkers.

Summary and Conclusion

Scientific interest in small world networks suggests how they arise from the combination of localised activity (‘bonds’) with only a smattering of long-distance ‘ties’. Watts’ model showed how the phenomena of short average path lengths, the six degrees of separation of small world theory, emerges far sooner than was originally thought possible. The connections among Brisbane’s public elites exemplify small world network architecture. For members of boards connected to the large component of 253 organizations, the average distance between them is only 4.68 steps.

It is, primarily, the public advisory boards that make a network happen rather than private sector companies and boards. The State government and City Council advisory bodies attract people active on other boards, including company boards. They are the forums for publicly active persons to meet one another and network. There is no institutionalised centralization in this network. However, if sustained over a period of time, locally determined patterns of appointment, and attraction, will produce a centrality of organisations and persons such as we map here. Our snapshot, frozen in time, produces a structured picture of the community network. It is likely, however, that when the State government and City Council change, the particular structure we have found will be disrupted. Like natural systems and ecologies social networks change and adapt to changing environments re-organising their basic properties of connectivity and centralization into different configurations.

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