Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain functional. In addition, we analyze different conditions of internal node failure due to disconnection from nodes within its own network. We show that several local internal failure conditions lead to similar nontrivial results. When there are no internal failures the model is equivalent to a bipartite system, which can be useful to model a financial market. We explore the rich behaviors of these models that include discontinuous and continuous phase transitions. Using the generating functions formalism, we analytically solve all the models in the limit of infinitely large networks and find an excellent agreement with the stochastic simulations.

Studying complex systems includes analyzing how the different components of a given system interact with each other and how this interaction affects the system’s global collective behavior. In recent years complex network research has been a powerful tool for examining these systems, and the initial research on isolated networks has yielded interesting results^{1–3}.

A network is a graph composed of nodes that represent interacting individuals, companies, or elements of an infrastructure. Node interactions are represented by links or edges. Real-world systems rarely work in isolation and often crucially depend on one another^{4–10}. Thus single-network models have been extended to more general models of interacting coupled networks, the study of which has greatly expanded our understanding of real-world complex systems. One intensive study of these “networks of networks” has focused on the propagation of failure among closely-related systems^{11–26}. The great blackout of Italy in 2003 and the earthquake of Japan in 2011 were catastrophic events that demonstrated that breakdowns in power grids strongly impact other systems such as communication and transport networks, and that the failure of these networks in turn accelerates the failure of the power grid. The propagation of these “failure cascades” has received wide study in recent years^{11–21,27,28}.

The simplest model of these systems consists of two interdependent networks in which nodes in one network are connected by a single bidirectional edge to nodes in a second network^{11}. In this model a node is functional (i) if it belongs to the largest connected component (the “giant component”) in its own network (the internal rule of functionality) and (ii) if its counterpart in the other network is also functional (the external rule of functionality). This original model has been extended to include localized and targeted attacks^{15,29–32} and mitigation^{13,25,26,33,34} and recovery strategies^{27,35}. Recently it was found that the giant component membership requirement can be replaced by a weaker requirement of belonging to a cluster of a size larger than or equal to a threshold ^{*}
^{28}. Alternatively, a heterogeneous k-core condition can be applied as an internal functionality condition in which node _{i} immediate neighbors remain functional^{36–40}. In this model the random failure of a critical fraction of nodes in an isolated network leads to an abrupt collapse of this network.

Although the original interdependent network model expanded our understanding of different coupled systems, the single-dependency relationship between nodes in different networks does not accurately represent what happens in real-world structures. A cascading failure model of a network of networks with multiple dependency edges has been applied to a scenario in which nodes fail only when they lose all their support nodes in the other network^{14,17}, but nodes in complex real-world systems can be so fragile that the loss of a single support link can cause them to shut down. More generally, each node may require a certain minimal number of supply links connected to the nodes in the other network to remain functional. In the world-wide economic system, for example, banks and financial firms lend money to non-financial companies who must pay the amount back with interest after a stated period of time. If a single non-financial company becomes insolvent, the bank that lent money to this company will likely not fail, but if the number of companies that cannot pay back their loans is sufficiently large, the possibility of bank failure becomes real. This resembles the k-core process in a single network described above.

Here we model the process of cascading failure in a system of two interdependent networks _{sX,i} supply nodes in the other network that are connected to node _{sX,i}
^{*} ≤ _{sX,i}. We call this the external functionality condition. We assume that a supply threshold is predefined for each node.

In principle, this model is non-trivial even if the survival of a node in network _{i} has a number of functional neighbors greater than or equal to _{i}
^{*} (“

We develop a theoretical model that is solved using the formalism of generating functions. We present numerical solutions and compare them with stochastic simulations. We find that for all internal rules of functionality, increasing the _{sX}.

We assume that the system consists of two networks _{A}(_{B}(_{sA,i} supply links from nodes in network _{sB,j} demand links that act as supply links for nodes in network _{sA}(_{sB}(^{13}. If this is the case, _{sA}(0) > 0.

The functionality of the nodes in both networks is related to their connections within their own network, which we call the internal rule of functionality. In addition, the state of the nodes also depends on the supply demand links that connect both networks, which we call the external rule of functionality.

We study three different internal rules of functionality:

Model I (The “giant component” rule): nodes that belong to the giant component in their own network are functional.

Model II (The “finite component” or “mass” rule): a finite component of size

Model III (The “k-core” rule): a node _{i} remains active if the number of its functional neighbors is greater than or equal to

The external rule of functionality states that nodes in network

We call _{sX}(_{sX} = _{sX}(

For example, in the case of a uniform supply threshold _{sX} is a step function, i.e., _{sX}(_{sX}(_{sX}(_{sX}(0, _{sX}) > 0.

Figure _{sA}(_{sB}(_{k,3}, but for simplicity in Fig. _{s,i}
^{*} = 1 for all _{s,i}
^{*} = 2 all nodes must have two functional supply nodes from the other network to remain functional. Nodes

Schematic of the rules of functionality of the model. Black links represent internal connections and orange links the supplies between networks. The state of the nodes varies according to their color: functional nodes (^{*} ≡ _{i}
^{*}), or (III) it must have a number of neighbors equal to or greater than ^{*} ≡ _{i}
^{*} (we show ^{*} = 2). In panel (b) we show the external rule of functionality for _{s}
^{*} = 1, and _{s}
^{*} = 2 in panel (c). In these cases _{sA}(_{sB}(_{k,3}, however, not all supplies are shown, nor are the internal connectivity links.

We construct a system of two randomly connected networks in which connectivity links within each network follow degree distributions _{A}(_{B}(_{sA}(_{sB}(_{A} and _{B}, where _{A} and _{B} are the number of nodes in networks _{A}〈_{sA} = _{B}〈_{sB} is satisfied, where 〈_{sA} and 〈_{sB} are the average degrees of the supply links in networks A and B respectively.

When we randomly remove a fraction 1 − _{X} of nodes from network _{X} for an isolated network _{X} = _{X}
_{X}(_{X}), where _{X}(_{X}) ≤ 1 is an exacerbation factor that takes into account additional node failures triggered by the random removal of a fraction of 1 − _{X} nodes. The explicit form of this factor is determined by the internal functionality rules of the model. The Supplementary Information presents equations for _{X} for Rules I, II, and III (see Supplementary Information: section _{X}(_{X}) = 1.

The cascading process begins with a random failure in network _{s,i} supply-demand links must have

External functionality failure is similar to heterogeneous k-core percolation^{37}. To describe this failure due to a lack of supply between networks _{sA}(_{sB}(_{sA}(_{sB}(_{sA} and _{sB} of supply-demand links and the distribution of the thresholds _{sA}(_{sB}(_{s}〉_{X} is the average number of supply links per node in network ^{41} for a variant of the Watts opinion model^{42}.

We next examine a theoretical approach to the temporal evolution of the cascading process. As explained above, initially a randomly selected fraction 1 − _{A,1} = _{A}(_{B}, which is the probability of randomly choosing a supply link that is connected to a functional node in the other network. When a node fails, all its demand links also fail. Thus _{B,1} = _{A,1}.

After applying the external functionality rule to network _{B,1} = _{sB}(_{B,1}). Because there are additional disconnected nodes in network _{B}, the number of functional nodes in network _{B,1} = _{B,1}
_{B}(_{B,1}). In the second stage of the cascade we cannot apply the same rules to obtain _{A,2}, because _{A,2} ≠ _{B,1}. If, for example, a supply-demand link connects nodes _{A,2} = _{sB}(_{B,1})_{B}(_{B,1}).

Thus the recursion relations for the stages _{X,n} nodes leaves the same number of functional nodes as in stage

The process begins with _{A,1} = 1 and _{A,1} =

We next present these theoretical results using several simple examples and verifying them with stochastic simulations.

To test the validity of the equations, Fig. _{c}, computed using the equations and stochastic simulations when the giant component functionality rule is applied (see Supplementary Information: subsections _{c}). Note that the plots show the simulation results are in total agreement with the theoretical results.

Temporal evolution, close to the critical threshold, of the giant component _{A}(_{B}(_{X}(_{k,5}, with _{sA}(_{sB}(_{k,5} and _{s}
^{*} = 2. The critical threshold for this system is _{c} = 0.381. (

Figure _{A} and _{B} in the steady state as a function of the initial fraction of surviving nodes _{X}(_{k,5}, with _{s,A}(_{s,B}(_{k,5}. For the external rule of functionality we use _{sX}(_{sX}(_{X}(_{k,5}, but _{s,A}(_{s,B}(_{k,1} and

Two random regular (RR) networks with _{A}(_{B}(_{k,5} and _{sA}(_{sB}(_{k,5} and system size ^{5} for different values of required supplies, _{A}(_{B}(_{k,5}, but _{sA}(_{sB}(_{k,1}, ^{11}. In panels (a) and (b) we show the order parameter of network _{A} and _{B}, respectively for the giant component rule.

Note that in network _{s}
^{*} is proportional to _{c}. This means that the depletion of the supply from network _{c}). Note also that, as expected, the behavior of network ^{11} in which the behaviors of network A and B are identical. In addition, note that the system is more resilient when _{s}
^{*} is smaller, i.e., when the supply level decreases. We also observe that the interdependent system with only one supply-demand link (the dashed-dotted line) is more resilient than a system with more connections between the two networks, but with large functionality thresholds

If instead of the giant component we apply the k-core as an internal functionality rule we get the same qualitative results. For different values of ^{*} and _{s}
^{*} the order parameters also undergo a discontinuous transition, and the system becomes more vulnerable when the threshold of internal links and the threshold of supply links increases (see Supplementary Information: section

When applying the “mass” rule, finite components of size _{X}(_{s} = 1 and _{s}
^{*} = 1, and all finite components of size greater than or equal to ^{28}. Here _{X}(1) = 1 and _{X}(_{s}
^{*} = 1 is fixed, the system becomes more resilient and the transition remains continuous. In contrast, if all the components of size _{X}(2) = 1] the transition becomes discontinuous irrespective of the number of supply-demand links connecting the networks. Nevertheless, not all the components of size _{A}(_{B}(

Order parameters for the “mass rule”, for a system of networks with internal distribution _{A}(_{B}(_{k,5}, supply distributions _{sA}(_{sB}(_{k,2} and thresholds _{A}
^{*} = _{B}
^{*} = 1. All the components of size _{max}) = 0 where _{max} is the maximum value of

Thus when _{s}
^{*} = 1 there is a critical value of _{c}(2) that separates the zone of continuous transition from the zone of discontinuous transition. Figure _{A}(_{B}(_{k,5} and supply distribution _{sA}(_{sB}(_{k,ks}. Note that the behavior of the critical probability as a function of the number of supply-links _{s} between the networks delimits these two zones. As _{s} increases the system becomes more robust, and more components must fail to cause an abrupt transition. In the limiting case _{s} → ∞ the curve reaches the value _{c}(2) = 1, but also _{c} → 0. On the other hand, when _{s}
^{*} > 1 the transition is always discontinuous for any value of _{s}.

Phase diagram that shows the continuous and discontinuous transitions zones when the “mass rule” is applied. The curve represents the critical probability of failure of the components of size _{A}(_{B}(_{k,5}, _{sA}(_{sB}(_{k,ks} and _{sA}
^{*} = _{sB}
^{*} = 1. For clarity, the _{s} axis is shown on a log scale.

What happens if no internal functionality rule is applied? This could be the case in a bipartite system in which nodes within each network do not interact but use nodes in the other network as bridges to establish connections. Here the exacerbation factor is simply _{X}(_{sX}(_{sX}(_{sX}(_{sX}(_{sX}) = _{sX}, there is again no transition because here functions _{s}(_{s}(_{sX} is by nonlinear, the behavior changes. Figure _{sX}(_{sX}) = 3(_{sX})^{2} − 2(_{sX})^{3} and for a supply-demand distribution _{s,X}(_{k,ks}. Note that for small values of _{s} the order parameter moves smoothly to zero but for _{s} = 8 the system undergoes a discontinuous transition. The existence of these transitions can be explained studying Eqs (_{c}).

Order parameter of network _{sX}(_{sX}) = 3(_{sX})^{2} − 2(_{sX})^{3}. The supply-demand distribution is single valued with _{sX} = 3 (▲), _{sX} = 5 (_{sX} = 7 (_{sX} = 8 (_{sX} = 10 (_{s} ≥ 8 there is a discontinuous transition. The curves were obtained from the equations.

Unlike the previous results, the transition here does not produce a total collapse of the system, and after the jump a small fraction of nodes remains functional for any _{s}〉_{X} = _{c} = 7.58465, _{c} = 0.728102 at which the first order phase transition emerges. For _{c} the transition is first order and for _{c} there is no phase transition for

We next analyze the limiting case of large _{s} values when all nodes in network _{sB}
^{*}, and we find that the collapse threshold _{c} converges to a value determined by the ratio _{sB}
^{*}/_{sB} given by

The _{c} value depends on _{sX}〉 → ∞ the functions _{sB}(_{sB}(_{c} depends solely on the topology of network _{c}, but when _{s}
_{s} for the derivation of Eq. (

Figure _{A} in network _{A}(_{k,zA}. Note that all curves go to _{c} = 1 when _{A} values have lower _{c} values because increased connectivity means increased resilience. In addition, when _{c} → 1/(_{A} − 1) as ^{43,44} in isolated RR networks. Similarly, for the “mass” rule we find that _{c} → 0 when

Critical threshold _{c} as a function of _{sB}
^{*}/_{sB} for different values of _{A}, the internal connectivity of network _{A}: _{A} = 3 (_{A} = 5 (_{A} = 10(_{X}
^{*} = 2 Note that in panel (b) _{c} ~^{1/4} when

If there is a Poisson internal degree distribution in network _{A}(_{A}]〈_{A}
^{k}/_{A} is the mean connectivity, we can write a closed-form expression for _{c} for the giant component rule,_{c} does not depend on the internal degree distribution of network _{s}
_{s}. On the other hand, if the system is bipartite then from Eq. (_{c} =

We have analyzed the cascading failure process in a system of two interdependent networks in which nodes within each network have multiple connections, or supply-demand links, with nodes from their counterpart network. In this model each node must have at least a given number of supply-links leading to functional nodes in the other network to remain active. We call this number the supply threshold and we call this condition the external functionality rule. We have studied the process under three internal functionality rules, (I) nodes must belong to the giant component in their own network, (II) nodes that belong to a finite component survive with a probability determined by the mass of the component, and (III) an internal version of the external functionality rule, known as heterogeneous k-core percolation. In addition, we have studied a system in the absence of any internal functionality rule, which is equivalent to a bipartite network. Our system is a generalization of the models of interdependent networks^{11,13} that represent a particular case of our model with _{sX}(^{11,13}.

We have found that for all the internal functionality rules the system is more robust when the supply threshold is lower. Under internal rules I and III there is a discontinuous transition at a collapse threshold _{c}. The main difference between our model and the previously studied models^{11,13} is that in the case of multiple supply links the initial attack on network _{c}. This makes the transition, when it occurs in network _{s}
^{*} = 1 the transition can be continuous depending on the probability that components of size _{s} there is a critical probability

When the model is applied to a bipartite system, the behavior is determined by function _{sX}. In particular, when this function is polynomial there is no transition in _{sX} ≤ 7, but when _{s} increases this curve breaks and becomes discontinuous.

Finally we have studied the asymptotic limit value of the number of supply-demand links, and find that when _{sB} is a step function there is an exact relationship between the ratio _{sB}
^{*}/_{sB} and the collapse threshold _{c}. We also find that in this limit the resilience of the interacting system is enhanced up to the point at which the critical threshold _{c} is solely dependent on the topology of network

For the stochastic simulations we use for both networks a system size of ^{6} to compute the steady state and ^{8} for the temporal evolution close to the critical threshold (See Fig. ^{45} for the construction of the networks. The simulation results are averaged over 1000 network realizations.

For model II, the “mass” rule, a finite component of size

In our theoretical analysis, to calculate the values of the order parameters at the steady state we iterate the temporal evolution Eqs (_{A} ≡ _{A,n} = _{A,n}
_{− 1} is satisfied. At this stage the magnitudes of all order parameters reach a steady state and no longer change.

Supplementary Information

The Boston University work was supported by DTRA Grant HDTRA1-14-1-0017, by DOE Contract DE-AC07-05Id14517, and by NSF Grants CMMI 1125290, PHY 1505000, and CHE-1213217. Yeshiva work was also supported by HDTRA1-14-1-0017. SVB acknowledge the partial support of this research through the Dr. Bernard W. Gamson Computational Science Center at Yeshiva College. MAD and LAB wish to thank to UNMdP, FONCyT and CONICET (Pict 0429/2013, Pict 1407/2014 and PIP 00443/2014) for financial support. HHAR wish to thanks to FAPEMA (UNIVERSAL 1429/16) for financial support.

All authors designed the research, analyzed data, discussed results, and contributed to writing the manuscript. M.A.D., L.D.V., S.V.B. and L.A.B. implemented and performed numerical experiments and simulations.

The authors declare that they have no competing interests.