“Tragedy of the Commons” was published for the first time by Garrett Hardin in 1968: theory concerns a dilemma in which multiple individuals, acting independently according to their self-interest, ultimately destroy shared limited resources (commons) even if it is clear that it is not in anyone’s interest to happen. However, when the economists start looking at biological ecosystems (humans a part), he then discovered that they work very well.

So let’s focus for a while on biological ecosystem.

Ecosystems: a delicate interplay between Complexity and Stability

When talking about ecosystems we know that an assembly of a certain number of species, each of which with feedback mechanisms that would ensure the population’s stability were it alone, can show sharp transition from overall stability to instability as the number and strength of interactions among species increase. In fact, this is a systemic behavior happening in complex systems, of which biological ecosystems are a category.

Well, this is the same kind of systemic risk of the recent financial crisis. In this paper “Systemic Risk in Banking Ecosystems” they draw analogies with the dynamics of ecological food webs and viral networks to explore the interplay between complexity and stability in simplified models of financial networks. They conclude the paper with some lessons that can be drawn for minimizing financial networks meltdowns. First more efforts are needed for assessing the system-wide characteristics of the financial network (e.g. risk management models). Another important aspect is need of having modular configurations to prevent instability contagion infecting a whole network: in fact, this is limiting the potential for cascades.

Now the third piece of the puzzle. Prof. Elinor Ostrom, (Indiana University) was awarded with the 2009 Nobel Prize in Economic Sciences (shared with Oliver E. Williamson) for the results she achieved in analysing how ecosystems communities are managing resources to their advantage. Prof. Elinor Ostrom and her collaborators have argued that consensual, self-generated governance can limit use of resources to sustainable levels, maintaining ecosystems in equilibrium. To learn more have a look at this paper about the Polycentric Governance of Complex Economic Systems. Self-governance, in ecosystems with humans, is then possible.

Well, we’ve elaborated several time about the vision of future networks being the communication fabric of ecosystems, where different Players are competing and cooperating. Let’s apply above reasoning to this. The (Nobel) conclusion is that future networks’ self-governance is possible provided we find and apply those autonomic rules governing the delicate interplay between complexity and stability. And these rules may have far reaching implications and impacts from a socio-economic viewpoint.

In this, Nature is ahead of us.

Network are becoming more and more complex and dynamic, capable of interconnecting large numbers of resources (e.g., routers, switches, transport nodes, servers…), Users’ devices (e.g., smart phones, etc) and, in the future, any machines (e.g. sensors, smart things, etc) embedding communication capabilities.

Future networks will be similar to complex systems where global properties and effects can emerge abruptly at a critical level of interactions between their components. In these dynamics, there is the hidden risk of instabilities. Overall, instability may have primary effects both jeopardizing the network performance  and compromising an optimized use of resources. In the worst case, an instability may create even a meltdown of a portion of network.

This is the main problem which I’ve proposed for study (more or less on year ago) in the EU project Univerself as part of the Telecom Italia participation in the project. Basically, we’re looking for methods and systems able to ensure network stability through local self-adaptation of nodes and, if-when not sufficient, via centralized policy based control.

This morning I’ve been very pleased to read this interesting paper Icebergs in the Clouds: the Other Risks of Cloud Computing addressing the risk of instabilities on the Cloud, which is essentially a metaphor for a network of computing and storage entities in which tasks and resources can be shared.

Example instability risk from unintended coupling of independently developed reactive controllers

Paper points out that complex systems can fail in many unexpected ways and outlines various simple scenarios. In the worst case, a cloud could experience a full meltdown that could seriously threaten any business that relies on it. Well, this is very much the same for future networks!

A growing number of researchers are beginning to see this problem: unpredictable behaviors often emerges in systems made up of “networks of networks”.

Paper concludes with the following: “We should study [these unrecognised risks] before our socioeconomic fabric becomes inextricably dependent on a convenient but potentially unstable computing model.”

Universality is one of the most studied and fascinating challenges in physics.

Universality originated in the study of phase transitions in statistical mechanics. Loosly speaking, a phase transition is a change of state of a nonlinear system occuring when a control parameter  is varied across a critical point. For systems exhibiting universality, the closer the parameter is to its critical value, the less sensitively the state depends on “the details” of the system.

This concept is applicable to networks as well. Considering a network (whatever, from a neural network to protein-protein interaction network, from the Internet to a social network) the question is whether there exist certain characteristics making the networks to exhibit universal dynamics, regardless of their difference in topology, structural changes or perturbations.

Human Protein Interaction Network

Let’s suppose that we wish to design a network that is stable and robust to external perturbations. Looking for the universality characteristics of the network means finding the way to make network dynamics stable against any structural changes that may be caused by attacks or failures.

Amazingly, this seems to be possible. Have a look at this paper.

In particular, in  the paper they claim they have found “weighting schemes for which the details of various real-world networks, whether biological, technological, or social, have little influence on typical dynamical processes such as synchronization, epidemic spreading, and percolation”.

Imagine the impact of discovering and applying universality principles to future networks, and not only… !

We’ve mentioned several times that ubiquity and “complexity” of networks will continuously grow in the future. Network of Netwoks will disappear into the fabric of reality creating a pervasive cyberspace. In this dreaming scenario however, it should be noted that current TPC-IP protocols may be ill-suited for several reasons, included availability of high capacity (even if this sounds apparently incredible; see this paper for details).

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=980245

 This can result in potential instabilities in network of networks, which can have primary effects both in jeopardizing the performance and compromising an optimized use of resources. Even if resource ubiquity will naturally make the network robust and resilient against attack, on the other hand the expected level of dynamicity and complexity will imply the dependence of network’s global characteristics (e.g., connectivity and average delay rate) on some local parameters (e.g., communication path and transmission probability). In other words, network behaviour may be influenced by state-phase transitions (i.e. an abrupt change in some operating characteristics may take place with a relatively small variation of certain parameters). As such, I believe that assuring network stability will be a hot and competitive issue for future networks (it could become even a new service: beyond “best stability”). What does network stabilization mean ?

Network stabilization can be defined as an adaptive feature of a network which, given certain objectives and constraints, should be able to self-converge, within given time requirements, to stable states, characterized by specific performance levels.

Suggestive example of a dynamic system trajectories in its phase space

Let’s make an example. Overall we can model a network as the interconnection of resources/links carrying the data generated by users/sources. Associated with each source is a route, which is the collection of links through which information from that source is flowing. In tradition congestion control, each link sets a price per unit of flow, based on the aggregate flow crossing that link, and the sources set their transmission rates based on the aggregate price they detect. In the absence of delays, this scheme is globally stable. In the presence of delays (or large bandwidth) the scheme can become unstable. Sources try maximizing individual profit based on their own utility functions, on the other hand, links uses prices to align, exactly or approximately, sources “selfish” behaviour.

One way to approach stability (for this example) is to define the Lagrangian function, so that the global optimization problem (flow control) can be turned into the search of saddle point problem. State stability, in this case, can be defined by a Lyapunov function. Actually Lyapunov’s direct method (or Lyapunov’s second method) can be easily and systematically applied to validate the existence (or non-existence) of stable states in an adaptive system. Loosely speaking a network is said to be stable near a given state if one can construct a Lyapunov function (scalar function) that identifies the regions of the state space over which such functions decrease along some smooth trajectories near the solution. In mechanical systems a Lyapunov function is considered as an energy minimization term, or overall it can be considered as a cost-minimization term or an error-minimization term. On the other hand, it should be noted that an inadequate Lyapunov function may cause the excess of false-positive warnings, a risk that cannot be avoided. Moreover even if we obtain an asymptotic stability, it is not clear what may happen during transients (typical of control feedbacks)

Indeed, another interesting perspective – as Roberto mentioned in a past comment – is understanding the relationship between global network state vs local networks states and how they influence each another. Actually each (global or local) network state is characterized by its associated data, so mining said distribution of data will be a valuable instrument for communication control and assuring stability. A phase space can represent the network behavior in terms of trajectories changing over time. So basically studying the structural stability of a complex dynamic network involves analysing how domains of attractions, particularly at the edges, are modified by alteration in the value of the network parameters.

Control and management of future Network of Networks (or Systems of Systems) will be a very complicated task. Actually, their structure is expected to change dynamically and its extension to grow rapidly. If you think that the potential number of non-linearities (due to configuration errors or nesting of loops) that may occur is proportional to the size of the network, it becomes clear that the larger and the more dynamic the network is, the lower probability that it will be in stable conditions. In this sense, future networks can be treated as dynamical system, where the notion of “stability” becomes highly important. One of the main challenges of control and management of future Network of Networks will be assuring that stability, avoiding phase transition in scenarios where there will be several concurrent (competing and cooperating) systems and processes.

In the Univerself project (http://www.univerself-project.eu/) I’ve proposed to approach this problem by modelling the logic of a Network of Networks in terms of trajectory-dynamics properties. It is an alternative approach, where the traditional optimization cost function is replaced by a cost-to-target function, i.e. a function tracking the net, whose optimization can be done using Lypunov-like scalar function. In this sense, the Lyapunov method induces a new more general equilibrium concept, that just in some circumstances is equivalent to the Nash one (Nash equilibrium point appears to be a steady state of the strategic interaction: when every individual is acting in agreement with the Nash equilibrium, no one has the need to take another strategy).

Petri Nets (PNs) is the powerful instrument that will be used for making these investigations. Presented for the first time in the 1962 in a PhD thesis “Communication with Automata” by Carl Petri (1926 – 2010), PNs are most suitable for modelling and analyzing the dynamics of concurrent systems whose behaviour could be described by finite sets of atomic processes and states. In other words, PNs allow representing any system behavior even when the mechanism is not fully understood, by combining different levels of abstraction in a single model.

PNs have been used for numerous applications to include also ecosystems and biological systems dynamics modelling and simulations. There are also studies proposing the use of PNs to model brain networks, which actually are very much similar to network of networks.

I’ll keep you posted on the outcomes.

We know that the behaviour of electrons in an electrical network can be described in terms of random walks; amazingly this local behaviour allows the network, as a whole system, to solve a complex optimization problem, which is minimizing heat dissipation for a given level of current flow. One might be tempted to extend this metaphor to communication networks.

Well, apparently, it’s even better: while the rules governing physical systems are fixed, for future communication networks we may be able, in principle, to engineer microscopic “autonomic rules” so as to achieve the desired macroscopic consequences at the network level. Specifically, as future communication networks will dramatically grow in size, I think that phase-transitions and meta-stability will be a highly interesting issues for the implications that these phenomena might have in network management and control.

As an example, in the project Univerself (http://www.univerself-project.eu/), I’m working on a microscopic description of a communication network as a Markov process with a large number of locally interacting components. Even if the number of possible states of each component is finite, there is the possibility of having meta-stable, i.e., persistent, states on the time scale of practical interest. This possibility is not just theoretical mathematics, but it may have a number of nice practical implications for management and control of network flows. For example, have a look at this paper:

http://www.statslab.cam.ac.uk/~frank/PAPERS/kmbk1.pdf

it shows how the integration of streaming traffic and file transfers can have a stabilizing effect on the variability of the number of flows in a network. Other examples show the impact of flow admission control strategies, or dynamic routing on the instability regions of a network.  A solid mathematical modelling of future networks will allow designing the said “autonomic rules” without compromising network stability.