1. Real world complex networks:
(a)
Supply chain networks
With globalization and the development of
technology for cost efficient operation of supply chains, supply chains are
evolving and becoming complex networks.
These networks connect suppliers to consumers, whose daily life depends
on these supply chains which distribute goods and services, such as fuel,
electricity, water, and food items. Any disruption in network may have impact
on only one or few nodes, but its impact may propagate further other connected
nodes. It is important to assess the impact of the events as the disruptions
could lead to catastrophic disasters. Prediction models, which assess the impact,
have to consider with enormous complexity of the networks.
(b)
Financial networks
Financial networks represent lending decisions
of the banks and the resulting relations between different banks and agencies.
In today’s world, financial network is highly fragile because the adverse
effects of on one of the node would impact on all the nodes in the networks as
each one of them inter connected each other, and would lead financial crisis.
How do we predict impact of actions/decisions of the bank leading to
catastrophic events in such complex network system?
(c)
Ecological networks
Eco systems are dynamic and changing. External
forces like climate change and pollution have huge impact on how ecosystems
adapt, in some cases many species would extinct. Darwin used the metaphor of a
‘tangled bank’ to describe the complex interactions between species, hinting
that complexity of these interactions is impossible to understand. With
advances in complex network theory, attempts are made to represent ecosystems as
complex networks where each node can be described as plant, species, or whole
population of one kind, and each link is weighted depending on their
interaction in that system.
2. Resiliency of the complex network system
2.1 What is it?
Consider a tiny ecosystem in Australia,
consisting of ants and plants. Plant species is guarded by the ant from the
insects, and the plant in return offers food and shelter to the ants. As shown
in the following picture, each ant species can visit multiple plant species,
and each plant species offers shelters for multiple ant species. By considering
interactions between just two entities, network already became complex.
Fig: (a). Interaction between each ant species
(left column) with different tree species (Right column). (b) Connection
between plant species if both plant species offer shelter to the same kind of
ant species.
In this ecological network, each plant species
is treated as node where size of the node represents abundance of the species
in that network, and each link represents common kind of ant species visitors.
The high abundance of one tree species (ability to shelter large number of ant
species) would help to more ants, and so these ants would help more number of
tree species. In this way, each tree is helping one another.
Fig (2): Network representation: Plants as
nodes (Size refers to abundance), and Link refers to both plants are visited by
same kind of Ant species (Link strength depending on number of common ants
species)
This network would adapt to the changes in the
networks depending on number of links between species, strength of the links,
and size of the nodes. Let us see how network dynamics change as we remove ant
species one by one randomly. As more kind of ant species disappear, some plant
species would disappear and abundance levels of those remaining plant species
would decrease as shown in the following picture.
(a)
(b)
(c)
(d)
Fig 3. From (a) to (d): Dynamics of networks
as ant kinds disappear randomly from the network. After certain stage,
population of plant species suddenly disappears from the ecosystem. Scale at the bottom refers to randomly removed ant species from the network at each step
There is key point where remaining population
suddenly crashes. Network can no longer compensate for the further changes, and
would lead to irreversible change on the ecosystem. How to predict those
triggering points, and ability of existing complex systems to compensate for
the changes before these networks collapse?
Resilience
is network property, and it can be defined as ability of the network to adapt
its activity and interactions so that it retains functionality without any
significant impact on its nodes or agents of the network.
Following
sections describes mathematical formulation for prediction of resilience of the
network. Resilience could be measure population of the species in the
ecological network, continuation of supply chain network in case of unwanted
event, power supply connectivity in power grid.
2.2 Single node one dimensional system
Although analysis of network resilience helps
in understanding consequences of the events on human health, world economy,
operations and services, and ecological systems, events leading loss of
resilience are rarely predicted in advance.
Let us consider simple one node system. The
traditional mathematical treatment of resilience approximates the behavior of a
complex system with non-linear equation:
β – Parameters that captures changing environmental conditions
(example: Temperature, rain fall)
x - refers to resilience function possible
states the possible states of the
system depending on β (example: plant
population)
f - refers function defining the system dynamics (like governing laws
that describe market conditions, plant species growth)
Solution for resilience function
x(β), which represents can be solved
by using following equations:
Equation 2 provides the systems steady state
condition at x0 . Equation 3 ensures its
linear stability. X vs β are plotted in
following graph: In the following plots, blue line refers preferred state where
as red line refers to undesired state. (For example, undesired state in
ecological system is extinction of the node)
Fig (4): All
three plots represent three different possible cases how resilience function
varies with β . (a) Tunable parameter, which
captures changes in the environment, reaches critical point the possible states
for x are two or more. Blue (desirable state) exists for β > βc. (blue) and two (or more) stable fixed points, a desired
(blue) and an undesired (red) forβ < βc .
(b) Possible state for X is only red below for β < βc (c)
Resilience function with a stable solution for β < βc and no solution
above βc, resulting in an uncontrolled divergent or chaotic behavior
Limitation
with 1D non-linear network:
Although it is powerful conceptually, complex network system is controlled by
large number of variables. Resiliency function which refers to possible states
should be defined by non linear equations that capture the interaction between
various nodes of the complex network system. The resulting resilience function
is therefore required to solve equations in multidimensional manifold over the
complex parameter space.
2.3 Multi-node multi dimensional system
Consider system
consisting of N components, and each component activity is explained as the
vector:
First term: Self dynamics of each component at the state, Xi
Second term: Interaction between component I and its interacting partners
G(xi, xj): Dynamical laws that govern systems components/nodes
(Factors that influence changes in the network like global warming, financial
market conditions, metabolism)
Matrix Aij: Interaction
between nodes/Link weights
In multi dimensional system, resilience
function is dependent on N
x N parameters of the weighted network Aij , each
referring to changes in the network.
Example for multi dimensional formulation: (plants species and pollinators interaction)
To understand above formulation, consider a
example where xi(t) refers to
the abundance
of species i, and equation(4) is written as follows:
The first term: The incoming migration of I
at a rate Bi from neighboring ecosystems.(Positive growth)
2nd
term: Logistic growth with the system carrying capacity Ki, and the Allee effect, according to which for low abundance
(xi < Ci) the system features negative growth
3rd
term: Mutual interactions. (Plant-pollinator
where such interaction helps in growth of the plant species to certain extent
depending on the state of xi and xj)
In this example, we can study randomly removing fraction of
nodes (deleting plant species from the network), or we can remove fraction of
pollinators (removing some of the pollinators from the network) or change the
weights of Aij to mimic the global environmental changes.
Limitations in multi-dimensional model:
When the changes in the network exceeds a certain threshold, a
bifurcation occurs and results in two stable fixed points: the desired state and an undesired low-abundance state (catastrophic event). Under these
conditions the system loses its resilience, potentially transitioning to the
undesired state. The precise
bifurcation point marking this loss of resilience is, however, highly
unpredictable. Limitation of our ability to predict the network resilience
could be due to transition depends on the network topology, the form, and the
nature of perturbation applied.
2.4 Single universal resilience function
Authors in the paper suggest single universal resilience
function to overcome ability predict precise bifurcation point where complex
network system loses its resilience.
The hypothesis behind this formulation is,
In a network
environment, the state of each node is affected by the state of its immediate neighbors. Therefore, the effective state of the system can be characterized
using the average nearest-neighbor activity.
where 
is
the vector of outgoing weighted degrees
is
the vector of outgoing weighted degrees
is
the vector of incoming weighted degrees,
, 
is
the average weighted degree, and 1 is the unit vector 1 = (1,…,1)T. 
where
averages over the
product of the outgoing and incoming degrees of all nodes. This reduction maps
the multi-dimensional complex system (4) into an effective 1D equation of the
form of equation (1), where
Mapping
of equation (4) to the 1D equation (7) allows taking advantage of the
theoretical tools developed for low-dimensional systems and applying them to a
broad range of complex systems.
In
summary of the resilience pattern of complex network systems is very difficult
to predict in manifold multi dimensional space (x, Aij). If the system is mapped
into β-space as explained above, it is possible to predict the system’s
response to diverse changes in the network and correctly identify the point
where system loses its resilience.
3. References:
1. Universal resilience patterns in complex
networks Jianxi Gao, Baruch
Barzel , Albert-László
Barabási http://www.nature.com/nature/journal/v530/n7590/full/nature16948.html
2. Video on
Network Earth http://www.mamartino.com/index.html
3. Notes on resilience: http://www.cse.unr.edu/~mgunes/cs765/cs790f09/Lecture16.ppt
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