Evolving Complex Networks in Constitutional Republics

Jon Roland

2003 June 14

[Preliminary draft. Check back for further developments.]

The Internet, with its ability to route data packets around bad nodes, and the World Wide Web, with its evolving complex network of hyperlinks between web pages, have emerged as a dominant presence in today's world, on which many people now depend for their livelihoods. They present a model for many other kinds of evolving complex networks that grow and make connections between their nodes based on such factors as the number of connections that have already been made between nodes.

The classic approach to modeling complex systems has been to analyze them into a set of variables connected by causal links, which can be represented by dots, or "nodes", connected by line segments, or "edges", having direction, represented by an arrowhead. The causal link would be associated with a function that increases or decreases the value of the target variable based on changes in the value in the source variable. such causal links could loop back to previous variables in a chain of causality, resulting in "feedback" loops. There could also be causal links coming in from outside the system, called "inputs", and leaving the system, called "outputs". The model could be "run" by setting initial values for the variables and for the inputs and then observing how the values of the variables and outputs vary with time. This is the approach used by such research efforts at the MIT Systems Dynamics Laboratory.[1]

The difficulty with this approach is that as the number of variables and causal links becomes large, the model becomes difficult to run on even the most powerful computers. Many real-world systems contain millions or billions of nodes and causal links, far beyond the ability of existing computer systems to simulate by computing the varying values of the nodes.

Evolving complex networks

But some researchers[2] have developed a simpler modeling approach which considers only networks of elementary nodes having no value defined on them, with simple directed links between them, which grows over time by adding additional links to other nodes, randomly, at rates that depend only on the number of links already established to or from those nodes. An example of such a network is the Web, with its hyperlinks connecting billions of pages.

What this line of research has revealed is how the networks evolve over time, depending on the rules for adding new links. If links are added randomly, without regard for how many links already exist, a plot of the number of nodes as a function of the number of links tends to converge on a bell-shaped curve, called a Poisson distribution. This is called a "random network". However, if the probability of a link being added connecting to a target node is proportional to the number of links to or from that target node, then the plot tends to converge on what is called a power law distribution, starting with a high number of nodes for a small number of links, falling off exponentially as the number of links is increased, for what is called a "scale-free network". The evolution of such a network is toward the emergence of a few "hubs" with large numbers of links, leaving most nodes with few links.

If the preference of a node for adding new links to other nodes is to do so at an even higher rate than in proportion to the number of links to or from the target node, then the network may evolve toward all hubs merging into a single hub, and a network with a star topology, with all links emanating from a single node.

An example of a random network is the network of roads connecting destinations that are initially scattered randomly over the landscape. Initially, each destination will tend to acquire about four roads connecting it to other destinations, and if the destinations don't increase, this will tend to become a stable configuration. However, if we consider the numbers of trips between destinations, and allow the destinations to increase, depending on the number of trips to nearby destinations, then the destinations will tend to cluster, first into towns, then into cities, and then the cities will tend to merge into megalopolises, covering entire regions.

This modeling approach can be applied to an initial network consisting of a large number of nodes that are small farmers, craftsmen, and merchants who buy from and sell to one another based on price, quality, and physical proximity. Each will tend to develop habitual business relationships with a small number of neighboring vendors and customers, and the volume of business and prices of products and services will tend to a fairly stable equilibrium, following the classic free-market model. But if the cost of travel and transport is reduced, so that it becomes profitable to do business with distant vendors and customers, then the system will tend to evolve as a few nodes grow at the expense of the others. If the preference of each business is to buy from or sell to others at a volume approximately proportional to how much business the other does, then the economy will tend to stabilize into a pattern of a few large businesses, falling off in size while increasing in number. But is the preference is greater than proportional, one of the businesses will tend to emerge as a dominant monopoly.

This simple modeling approach reveals why actual economic markets tend to instability and monopoly. "The rich get richer" because economies of scale permit larger vendors to reduce their costs and drive smaller competitors out of business, and the process is accelerated by the tendency of customers to prefer larger vendors even when prices are equal. A similar evolution occurs in the political marketplace, where more and more power tends to gravitate to fewer and fewer individuals, until one individual is dominant.

Constitutional republics

The Founders of the United States of America understood this problem, and tried to address it in the political realm by establishing constitutional barriers to excessive or unbalanced concentrations of power. Unfortunately, they did not address the economic realm, or the tendency for economic power to be translated into political power. Some attempt to do this was made, beginning in the late 19th century, with anti-trust legislation and litigation, but the demands for being able to compete in the international marketplace, along with constitutional protections of contract and association, and the dependence of the political process on donations of money, have allowed the emergence of a network of corporations that challenge the power of nation-states, and are often only minimally accountable to even their own shareholders. The ability to cast a vote becomes less important than the ability to control what people get to vote on or how the votes are counted.

This problem may emerge on the Web if AOL, Google, Ebay, Yahoo, Amazon, Microsoft, and the phone and cable companies all merge into a single hub and access provider that is the center of almost all connections by Internet users. We have already seen how Google has come to displace and dominate the search engines, and Ebay the online auction brokers. Until now, the Internet has been a force for the distribution of power and increased democratic accountability, but that could change if the primary providers become excessively concentrated and unbalanced.

This modeling approach indicates that the way to prevent the collapse of a network into a star configuration with a single center is to prevent link-building preferences from exceeding proportionality to existing links to and from the target nodes. Some actual networks are constrained by physical limitations from such collapse, but finding ways to constrain the independent behavior of human beings in the economic or political marketplace will be a challenge, because while people may accept impersonal physical constraints, they will tend to resist or work around social or legal constraints.

The value of this approach to analysis and modeling can be seen by applying it to some of the subsystems in a constitutional republic.

Custom and precedent

The defining characteristic of a constitutional republic is a constitution, which should be written, that is supposed to be the source for all legitimate authority, and which is superior over all official acts in conflict with it. However, before constitutions and statutes based on them became a body of "black letter" law, most court decisions were mainly guided by the precedents of other court decisions made in similar cases, and the doctrine according to which court decisions should not be made which conflict with precedents is called stare decisis, from the Latin, "to let stand the decision". With the advent of black letter law, courts were confronted with ambiguities in the text and legislative history of that black letter law, and uncertainties in the factual situations to which it was to be applied, so their decisions came to be seen as supplementing the black letter law, and clarifying it. However, with a flood of cases demanding scarce judicial resources, and the high cost of re-examining black letter text and legislative history in every case, the doctrine of stare decisis came to be invoked to justify making decisions only on the basis of a few recent precedents, rather than on the original law as originally understood. This has resulted in judicial precedents drifting beyond the limits of what that original law and understanding allowed.[3]

Researchers in evolving networks examined how the citations in research reports to other research reports tended to result in the cited reports attracting other cites based in part on how often they had already been cited, resulting in a tendency for a few reports to become hubs in a network that attract unusually large number of links. This behavior can more easily be measured now that so many research reports are appearing on the Web, citing other reports on the Web.

A similar process seems to be occurring in judicial citations to precedents, with precedents that favor certain political interests, including interests in departing from the bounds of original understanding, tending to attract cites that reinforce their influence and draw further precedents in a trajectory that leads outside of constitutional and statutory bounds. Relying on precedent may contribute to a kind of short-term stability and reduce the cost of judicial decisionmaking, but in the longer term it can have serious negative social impacts. Examining how the dynamics of evolving networks apply to judicial decisionmaking may be able to reveal corrective measures that might be taken to avoid clustering that is pathological.

The use and misuse of custom and precedent is not limited to the judicial branch, but can be found in the practices of legislative and executive branches as well, and in a host of other political, social, and economic institutions. The driving factor is usually short-term reduction in decisionmaking costs, at the expense of others, both contemporary and future, onto whom the costs are shifted. This kind of cost-shifting and "rent-seeking" behavior is the subject of public choice theory.[4]

Future directions

This line of inquiry leads to investigation of competitive linking by hubs, their tendency to divide the network into subnets of dominance that compete with one another, and the tendency under some conditions for the competition to result in victors that extinguish their rivals and prevent the emergence of other rivals. We can see this in ecology, where a niche comes to be occupied by a single species, which drives its competitors to extinction, and the emergence of a single species that can occupy many niches can have an impact comparable to an asteroid impact. It appears we may be able to learn a great deal about the behavior of complex systems by simulating their evolving structures without having to examine the natures of the nodes or the causal functions of the links. This has an obvious connection to previous work done in the theory of diffusion of innovations, especially competitive diffusion processes.[5]

[1] See "The Counterintuitive Behavior of Social Systems", Jay Forrester (1970), at http://constitution.famguardian.org/ps/cbss.htm

[2] See the work of Albert-László Barabási and others, Study of Self-Organized Networks at Notre Dame, http://www.nd.edu/~networks/ .

[3] See "How stare decisis Subverts the Law", Jon Roland (2000), at http://constitution.famguardian.org/col/0610staredrift.htm .

[4] See Understanding Democracy: An Introduction to Public Choice, J. Patrick Gunning (2003), at http://constitution.famguardian.org/pd/gunning/votehtm/cont.htm .

[5] See Diffusion of innovations, Jon Roland (2003), at http://constitution.famguardian.org/col/03317_diffusion.htm