## ⓘ Interdependent networks

The study of interdependent networks is a subfield of network science dealing with phenomena caused by the interactions between complex networks. Though there may be a wide variety of interactions between networks, dependency focuses on the scenario in which the nodes in one network require support from nodes in another network. For example of infrastructure dependency see Fig. 1.

## 1. Motivation for the model

In nature, networks rarely appear in isolation. They are typically elements in larger systems and can have non-trivial effects on one another. For example, infrastructure networks exhibit interdependency to a large degree. The power stations which form the nodes of the power grid require fuel delivered via a network of roads or pipes and are also controlled via the nodes of communications network. Though the transportation network does not depend on the power network to function, the communications network does. Thus the deactivation of a critical number of nodes in either the power network or the communication network can lead to a series of cascading failures across the system with potentially catastrophic repercussions. If the two networks were treated in isolation, this important feedback effect would not be seen and predictions of network robustness would be greatly overestimated.

## 2. Dependency links

Links in a standard network represent connectivity, providing information about how one node can be reached from another. Dependency links represent a need for support from one node to another. This relationship is often, though not necessarily, mutual and thus the links can be directed or undirected. Crucially, a node loses its ability to function as soon as the node it is dependent on ceases to function while it may not be so severely effected by losing a node it is connected to.

In percolation theory, a node is considered active as long as it is connected to the giant component. The introduction of dependency links adds another condition: that the node that it depends on is also active.

Dependency can be defined between different networks and also within the same network.

## 3. Percolation properties and phase transitions

Interdependent networks have markedly different percolation properties than single-networks.

If a single network is subjected to random attack 1 − p {\displaystyle 1-p}, the largest connected component decreases continuously with a divergence of its derivative d P ∞ d p {\displaystyle {\frac {dP^{\infty }}{dp}}} at the percolation threshold p c {\displaystyle p_{c}}, a second-order phase transition. This result is established for ER networks, lattices and other standard topologies.

However, when multiple networks are interdependent, cascading failures emerge due to the positive feedback caused by dependency links. This family of processes causes a discontinuous or first order phase transition. This has been observed for random networks as well as lattices. Furthermore, for embedded interdependent networks the transition is particularly precipitous without even a critical exponent for p > p c {\displaystyle p> p_{c}}.

Surprisingly, it has been shown that - contrary to the results for single networks - interdependent random networks with broader degree distributions are more vulnerable than those with narrow degree distributions. The high degree which is an asset in single networks can be a liability in interdependent networks. This is because the hubs which increase the robustness in single networks can be dependent on vulnerable low-degree nodes. The removal of the low-degree node then removes the hub and all of its links.

## 4. Dynamics of cascading failure

A typical cascading failure in a system of interdependent networks can be described as follows: We take two networks A {\displaystyle A} and B {\displaystyle B} with N {\displaystyle N} nodes and a given topology. Each node A i {\displaystyle A_{i}} in A {\displaystyle A} relies on a critical resource provided by a node B i {\displaystyle B_{i}} in B {\displaystyle B} and vice versa. If A i {\displaystyle A_{i}} stops functioning, B i {\displaystyle B_{i}} will also stop functioning and vice versa. The failure is triggered by the removal of a fraction 1 − p {\displaystyle 1-p} of nodes from A {\displaystyle A} along with the links in A {\displaystyle A} which were attached to each of those nodes. Since every node in B {\displaystyle B} depends on a node in A {\displaystyle A}, this causes the removal of the same fraction 1 − p {\displaystyle 1-p} of nodes in B {\displaystyle B}. In network theory, we assume that only nodes which are a part of the largest connected component can continue to function. Since the arrangement of links in A {\displaystyle A} and B {\displaystyle B} are different, they fragment into different sets of connected components. The smaller components in A {\displaystyle A} cease to function and when they do, they cause the same number of nodes but in different locations in B {\displaystyle B} to cease to function as well. This process continues iteratively between the two networks until no more nodes are removed. This process leads to a percolation phase transition at a value p c {\displaystyle p_{c}} which is substantially larger than the value obtained for a single network.

### * 4.1. Dynamics of cascading failure * Effect of network topology

In interdependent random networks in which a fraction q {\displaystyle q} of the nodes in one network are dependent on another, it is found that there is a critical value q c {\displaystyle q_{c}} above which first-order phase transitions are possible.

In spatially embedded interdependent networks, a new kind of failure has been observed in which a relatively small failure can propagate through space and destroy an entire system of networks.

## 5. Localized attacks

A new percolation process, localized attack was introduced by Berezin. Localized attack is defined by removing a node, its neighbors and next nearest neighbors until a fraction of 1-p is removed. The critical p_c where the system collapse for random networks was studied by Shao. Surprisingly, for spatial interdependent networks there are cases in which a finite number independent on the size of the system of nodes causes cascading failures through the whole system and the system collapses. For this case p_c=1.

## 6. Recovery of nodes and links

The concept of recovery of elements in a network and its relation to percolation theory was introduced by Majdandzic. In percolation it is usually assumed that nodes or links fail but in real life e.g, infrastructure nodes can also recover. Majdandzic et al introduced a percolation model with both failures and recovery and found new phenomena such as hysteresis and spontaneous recovery of systems. Later the concept of recovery was introduced to interdependent networks. This study, besides finding rich and novel critical features, also developed a strategy to optimally repair a system of systems.

## 7. Comparison to many-particle systems in physics

In statistical physics, phase transitions can only appear in many particle systems. Though phase transitions are well known in network science, in single networks they are second order only. With the introduction of internetwork dependency, first order transitions emerge. This is a new phenomenon and one with profound implications for systems engineering. Where system dissolution takes place after steady if steep degradation for second order transitions, the existence of a first order transition implies that the system can go from a relatively healthy state to complete collapse with no advanced warning.

## 8. Reinforced nodes

In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning after being disconnected from the giant component. Yuan et al generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. The critical fraction of reinforced nodes needed to avoid catastrophic failures have been found.

The original model of interdependent networks regarded only structural dependencies, i.e., if a node in network A depends on a node in network B and this node in network B fails also the node in A fails. This lead to cascading failures and abrupt transitions. Danziger et al studied the case where a node in one depends on the dynamic on the other network. They four rich dynamical phenomena in such systems.

## 9. Examples

- Climate networks. Spatial measurements of different climatological variables define a network. The networks defined by different sets of variables are interdependent.
- Infrastructure networks. The network of power stations depends on instructions from the communications network which require power themselves.
- Protein networks. A biological process regulated by a number of proteins is often represented as a network. Since the same proteins participate in different processes, the networks are interdependent.
- Economic/financial networks. Availability of credit from the banking network and economic production by the network of commercial firms are interdependent. In October 2012, a bipartite network model of banks and bank assets was used to examine failure propagation in the economy at large.
- Ecological networks. Food webs constructed from species which depend on one another are interdependent when the same species participates in different webs.
- Physiological networks. The nervous and cardiovascular system are each composed of many connected parts which can be represented as a network. In order to function, they require connectivity within their own network as well as resources available only from the other network.
- Transportation networks. The networks of airports and seaports are interdependent in that in a given city, the ability of that citys airport to function is dependent upon resources obtained from the seaport or vice versa.