Interconnections between networks act like an external field in first-order percolation transitions

TL;DR

This study explores how interconnections between networks influence first-order percolation transitions, revealing distinct critical exponents and their dependence on coupling strength, thus providing insights into the phase transition nature in interdependent networks.

Contribution

It introduces the concept of external field effects in first-order percolation transitions and characterizes the associated critical exponents, which vary with dependency coupling strength.

Findings

Critical exponents $$ and $ ext{delta}$ are defined for first-order transitions.

Exponents differ from those in second-order transitions and depend on coupling strength.

Both sets of exponents satisfy Widom's identity, supporting their validity.

Abstract

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we study if and how the effects of such interconnections can be described as an external field for interdependent networks experiencing first-order percolation transitions. We find that the critical exponents $\gamma$ and $\delta$, related to the external field can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents can be found even within a single model of interdependent networks, depending on the dependency coupling strength. Specifically, the exponent $\gamma$ in the first-order regime (high coupling) does not obey the fluctuation dissipation theorem, whereas in the continuous regime (for low coupling) it does. Nevertheless, in both cases they satisfy Widom's identity, $\delta - 1 = \gamma / \beta$ which further supports the validity of their definitions. Our results provide physical intuition into the nature of the phase transition in interdependent networks and explain the underlying reasons for two distinct sets of exponents.

Equations

$$S(0,p)-S(0,p_c)\sim (p-p_c{)}^{\beta }.$$

$$S(r,p_c)-S(0,p_c)\sim r^{1/\delta }.$$

$$\chi \equiv {\left(\frac{\partial S(r,p)}{\partial r}\right)}_{r\to 0}\sim |p-p_c{|}^{-\gamma }.$$

$$u=1-p\left[1-G_1^{intra}(u)(1-r+r{G}_0^{inter}(v))\right]\times$$

$$\times \left[1-q+qp(1-G_0^{intra}(u)(1-r+r{G}_0^{inter}(v))\right],$$

$$v=1-p\left[1-G_0^{intra}(u)G_1^{inter}(v)\right]\times$$

$$\times \left[1-q+qp(1-G_0^{intra}(u)G_0^{inter}(v))\right].$$

$$\frac{1-q+2qp-\sqrt{(1-q)^{2}+4qS}}{2qp}e^{zS}+(r-1)=r\exp\Bigg{(}\frac{\kappa p}{r}\bigg{[}(r-1)(e^{-zS}-1)\frac{1-q+\sqrt{(1-q)^{2}+4qS}}{2}-\frac{S}{p}\bigg{]}\Bigg{)}.$$

$$u=1-p+p{G}_1^{intra}(u)[1-r+r{G}_0^{inter}(v)],$$

$$v=1-p+p{G}_0^{intra}(u)G_1^{inter}(v).$$

$$\begin{gathered} e^{-zS}(r-1)+1-\sqrt{\frac{S}{p}}= \\ =r\exp \left[\frac{\kappa \sqrt{Sp}(e^{-zS}(r-1)+1-\sqrt{\frac{S}{p}}-r)}{r}-zS\right]. \end{gathered}$$

$$f(S,p,r,q)=\bigg{[}(1-r)(1-e^{-zS})\cdot\\ \cdot\frac{1-q+\sqrt{(1-q)^{2}+4qS}}{2}-\frac{S}{p}\bigg{]},$$

$$r+\frac{2f(S,p,r,q)}{1-q+\sqrt{(1-q)^{2}+4qS}}=\\ =r\exp\bigg{[}\frac{\kappa pf(S,p,r,q)}{r}-zS\bigg{]}.$$

$$\begin{array}{rl}& f(S,p,0,q)=f(S_c,p_c,0,q)+f_S(S_c,p_c,0,q)(S-S_c)+\\ & +f_p(S_c,p_c,0,q)(p-p_c)+...=0.\end{array}$$

Full text

In the past two decades, network theory has successfully described collective phenomena of many complex systems such as the brain, climate and infrastructures (7)-meunier2010modular ; (7)-morone2017model ; (7)-stam2010emergence ; (7)-yamasaki2008climate . The resilience of such networks is often studied under a percolation process where a fraction $1-p$ of nodes are removed randomly from the network and the size of the largest connected component, $S$ (the order parameter of the system) is measured bunde2012fractals ; albert2000error ; cohen-prl2000 ; cohenhavlin2010complex_book ; newman2011structure_book ; newmancallaway2000network . Several generalizations of percolation-like processes have also been developed and these processes also affect the values of the critical exponents, and the nature of the transition e.g., whether it is continuous or abrupt. Specifically, interdependent networks buldyrev-nature2010 ; gao2012interdependentnetworks ; rinaldi-ieee2001 ; gao2011robustnessNoN where one network depends on another, have drawn much interest. In these systems there exist several networks with the ordinary connectivity links within the networks, yet dependency links between the networks imply that if a node at one end of a dependency link fails than the node at the other end will also fail, even if it is still connected in its own network. This process leads percolation on interdependent networks to result in cascading failures leading to abrupt, first-order percolation transitions.

Several researchers have studied ‘interconnected networks’ where two networks each with many connections inside their own network, also have a smaller number of links between them saumell2012epidemic ; radicchi-naturephysics2013 ; de2014navigability ; brummitt2012suppressing_raissa_dsouza_pnas . Such networks have also been described in the literature as networks with community structure, since each network can be regarded as a separate community girvan2002community ; palla2005uncovering ; lancichinetti2009detecting ; mucha2010community .

Recently, a new model of community structure has been proposed where only some small fraction, $r$, of nodes are assumed a priori to be capable of having interlinks to other communities Gaogaodong2018resilience . It was found that the fraction $r$ of such nodes affects the percolation transition in a manner analogous to an external field in spin systems or a ghost field in percolation.

The effects of an external field are best characterized through the key exponents $\beta$, $\delta$, and $\gamma$ describing the behavior of the system near (and at) criticality bunde2012fractals ; (1-6)-huang2009introduction ; (1-4-6)-stanley1971phase ; (4-5)-stauffer2014introduction , which fulfill Widom’s identity $\delta-1=\gamma/\beta$ implying that there are only 2 degrees of freedom in determining these exponents. For percolation processes, these critical exponents are defined when the control parameter $p$ is near (and at) the percolation threshold $p_{c}$.

$\mathit{1)}$ The critical exponent $\beta$ describes the behavior of the order parameter ($S$) near the critical point with zero-field ($r=0$) and is given by

[TABLE]

$\mathit{2)}$ At the critical point, ($p=p_{c}$), the increase of the order parameter with the magnitude of the field, $r$, is given by the critical exponent $\delta$ as

[TABLE]

$\mathit{3)}$ The susceptibility of the system, $\chi$, is given by the partial derivative of the order parameter with respect to the field, $r$, and scales near the critical point with the exponent $\gamma$ as

[TABLE]

Here we study analytically and via simulations the percolation of this novel community structure in interdependent networks (see Fig. 1) with $q$ fraction of interdependent nodes i.e., $1-q$ fraction of nodes in each network are autonomous. We observe two distinct regimes in this single model, characterizing different values of the critical exponents and universality class. For small values of $q$, the network undergoes a continuous second-order phase transition as for isolated ER networks and has the corresponding critical exponent values. However, for large values of $q$ these systems undergo first-order phase transitions with a different set of exponent values. Moreover the fraction of interconnected nodes, $r$, can even for the case of an abrupt transition, be analogized to an external field. Indeed, prior work gave puzzling results in the first-order transition regime, where two alternative definitions of the exponent $\gamma$ were shown to lead to two distinct exponent values lee2016hybrid . Here, by using our new definition of relating interconnections to an external field, we demonstrate that this puzzle can be rectified by recognizing that these distinct values imply a violation of the fluctuation dissipation theorem.

Model.– Our network model assumes two communities where only a small fraction $r$ of nodes in each community are capable of having interlinks Gaogaodong2018resilience . A total of $M_{inter}$ links are then assigned among this small subset of nodes. Motivation for the interconnected structure of the model can be found in systems where additional resources are needed at a site in order to accommodate long-range links. At the same time, once such infrastructure exists, adding additional interlinks is of low cost. One example is the airport network where longer runways are needed for planes that fly transoceanic flights and thus some airports have such flights while others do not. Nonetheless, once this infrastructure exists, adding more transoceanic flights is easy. Similarly, power stations that transfer large load to long distances may require additional infrastructure in order to handle such load. Next, two networks constructed from this model are set to be partially interdependent with $q$ fraction of nodes in each network depending on nodes in the other network, as seen in Fig. 1.

Analytic solution.– We begin by developing an analytic solution for the effect of interlinks on percolation of interdependent networks of the type described above. We start by defining the generating functions for the degree distribution of intra- and inter- connected nodes. For intra nodes we obtain $G^{intra}_{0}(x)=\sum_{k}p^{intra}_{k}x^{k}$ and $G^{intra}_{1}(x)=\sum_{k}q^{intra}_{k}x^{k}$ where $p^{intra}_{k}$ is the probability for a node to have $k$ intra links and $q^{intra}_{k}=\frac{(k+1)p^{intra}_{k+1}}{z}$ is the intra excess degree distribution with $z$ being the average intra degree newmancallaway2000network ; newman2001random . We assume that interlinks are always assigned randomly and thus their generating functions are given by $G^{inter}_{0}(x)=G^{inter}_{1}(x)=e^{-\kappa(1-x)}$ where $\kappa=\frac{M_{inter}}{rN}$ is the average inter-degree of the $r$ fraction of nodes in the core and $N$ is the total number of nodes.

We next define $u$ and $v$, the probability that after removal of $1-p$ fraction of nodes from each network, an intra and inter edge respectively do not lead to a node connected to the giant component. They satisfy the equations:

[TABLE]

For ER networks (i.e., $p_{k}^{intra}=\frac{z^{k}e^{-z}}{k!}$), $G^{intra}_{0}(u)=G^{intra}_{1}(u)=e^{-z(1-u)}$ and $S=1-u$, leading to a single transcendental equation relating $S$, $q$, and $r$

[TABLE]

As seen in Fig. 2, for large values of $q$ the system undergoes an abrupt first order transition while for small values of $q$ (Fig. 2 inset and Gaogaodong2018resilience ; lee2016hybrid ) it experiences a continuous second-order transition. As $r$ increases the size of the mutual giant component at $p_{c}(r=0)$ increases with clear scaling relationships (see below and Fig. 2b,c,d) suggesting that $r$ can be analogized to an external field. Two sets of different critical exponents arise from Eq. (4). For strong dependency (i.e. large values of $q$) we obtain $\delta=2$ and $\beta=\gamma=1/2$ while for weak coupling we find $\delta=2$ and $\beta=\gamma=1$ Gaogaodong2018resilience . Both sets of critical exponents satisfy Widom’s identity $\delta-1=\gamma/\beta$.

For strong dependency, our results also support prior results by Lee. et al for the value of $\beta$ lee2016hybrid . However, our results differ from the results of lee2016hybrid who obtained $\gamma=1$ when using a definition of $\gamma$ based on the fluctuations in the size of the order parameter ($S$) near criticality indicating a violation of the fluctuation dissipation theorem (FDT). We resolve this contradiction by suggesting that the reason for this phenomena is fundamental since the FDT holds only in equilibrium while for systems out of equilibrium (e.g., undergoing an abrupt transition due to the spreading of damage) as here, there are no guarantees of its validity. Indeed, violations of the FDT have been also observed and studied widely in glassy systems crisanti2003violation_FDT ; grigera1999observation_VFDT ; marinari1998violation_FDT due to the aging phenomena (remaining out of equilibrium for all time). Indeed, our system exhibits a first order transition with a metastable state which while often treated as if it is an equilibrium state, it is not. Thus our result support the suggestion in baez2003fluctuation_FDT_metastable that first order transitions are off equilibrium. This therefore leads to a violation of the FDT.

In Fig. 3, the two sets of different critical exponents for strong and weak dependency are obtained from numerical analysis of Eq. (4). Later we present also the analytic derivation of the two sets of exponents based on Eq. (4). It can be seen that for large values of $q$ we obtain $\gamma=1/2$, the FDT is violated and the system undergoes an abrupt transition (Fig. 2). In contrast, for small values of $q$ we obtain $\gamma=1$ and the FDT holds with the system undergoing a continuous second-order transition (Fig. 2 inset and Gaogaodong2018resilience ; lee2016hybrid ). This shows how even in a single model (represented by a single equation (Eq. (4))) we can observe both sets of exponents and further supports our claim that the violation of the FDT is inherent to this type of transition and not related to differences between models.

Limit of $q=1$ (Fully interdependent Networks).– For the specific case of fully interdependent networks ($q=1$), we consider the case of removal of $1-p$ fraction of nodes from network A only in order to be consistent with buldyrev2010catastrophic ; gao2012interdependentnetworks . In this case $u$ and $v$ satisfy the equations Gaogaodong2018resilience :

[TABLE]

For the case of fully interdependent ER networks we use the framework from buldyrev2010catastrophic ; gao2012interdependentnetworks to arrive at the following equation for the mutual giant component (see SI),

[TABLE]

Note that for $r=0$, Eq. (5) recovers the well-known result for two interdependent networks $S=p(1-e^{-zS})^{2}$ and Eq. (4) is recovered with $p\rightarrow\sqrt{p}$ since there we removed $p$ fraction of nodes from both networks. Fig. 2 shows excellent agreement between the theory of Eq. (5) and simulations and the set of critical exponents for strong dependency is presented in Fig. 2b,c,d. Fig. 4 shows $S(r,p)$ for interdependent SF networks (i.e., $p_{k}^{intra}\sim k^{-\lambda}$) for different values of $\lambda$ with large $q$ and shows perfect agreement between the theory and the simulations. We also observe here a clear scaling relationship between $r$ and $S$ (Fig. 4d,e,f), further justifying the analogy to an external field.

Analytic Derivation of Critical Exponents.– Having solved the model and demonstrated that scaling relationships analogous to those of an external field can be found, here we seek to extract these scaling exponents analytically. This can be done by assuming $\kappa$ to be a constant. To derive the exponents for ER networks, we let $f(S,p,r,q)$ be

[TABLE]

and thus Eq. (4) takes the form

[TABLE]

In the limit of $r\to 0$, we recover $f(S,p,0,q)=0$ defining the giant component of two interdependent ER networks. Likewise, at criticality, $f_{S}(S_{c},p_{c},0,q)=0$, where $f_{S}$ refers to the derivative of $f$ with respect to $S$. Expanding $f(S,p,0,q)$ around $S=S_{c}$ and $p=p_{c}$ gives

[TABLE]

Further developing this expansion leads to $\beta=1/2$ for large $q$ and $\beta=1$ for small $q$ analytically. The exponents $\delta$ and $\gamma$ require different approximations in the expansion of $f(S,p,r,q)$ in order to arrive at their values analytically (we find $\delta=2$ and $\gamma=1/2$ for large $q$ and $\delta=2$ and $\gamma=1$ for small $q$), see SI for the detailed derivation.

Simulations and theory for the critical exponents for ER networks with large $q$ are in excellent agreement and are shown in Fig. 2b-d. The simulations support our derivations that $\delta=2$ and $\beta=\gamma=1/2$. These values also satisfy Widom’s identity. We also confirmed these values for $k$-core percolation dorogovtsev2006k ; goltsev2006k ; carmi2007model ; schwarz2006onset_kcore_field on networks with the community structure defined here (see SI) and found the same values– further supporting that interdependent networks and $k$-core percolation are in the same universality class. Simulations for the critical exponents for small $q$ are shown else where Gaogaodong2018resilience and also show excellent agreement with our derivation that $\delta=2$ and $\beta=\gamma=1$. Thses values also satesfy Widom’s identity.

We also measure the critical exponents of the external field for SF networks having different values of $\lambda$ for large $q$. The results for $S(r,p)$ and the critical exponents for different values of $\lambda$ are shown in Fig. 4. For $\lambda>4$ we have low heterogeneity and indeed as expected we find similar results as for ER networks (i.e, $\delta=2$ and $\beta=\gamma=1/2$). For high heterogeneity (i.e. $3<\lambda<4$ and $2<\lambda<3$) the system has different exponents than ER for the case of single layers Gaogaodong2018resilience , yet for interdependent networks we find that the critical exponents are the same as for interdependent ER networks, $\delta=2$ and $\beta=\gamma=1/2$. Thus, our results suggest that interdependent ER networks and interdependent SF networks with large $q$ are in the same universality class in contrast to small $q$ at which the exponents are different Gaogaodong2018resilience . The reasoning is most probably due to the fact that the random spread of damage due to interdependence does not distinguish between high and low degree nodes.

In summary, we have studied the effects of an external field on first-order percolation phase transitions by analyzing analytically and numerically, interdependent networks with interconnections. We find that a model of interdependent networks is able of expressing two characteristic behaviors depending on the level of interdependence coupling, $q$. For high-values of $q$ the critical exponents are the same for both ER and SF networks ($\delta=2$ and $\beta=\gamma=1/2$). These exponents satisfy Widom’s identity $\delta-1=\gamma/\beta$ and their common value suggests the existence of a single universality class describing these cascading phenomenon. Interestingly, we observe a violation of the fluctuation dissipation theorem (FDT) kubo1966fluctuation_FDT ; weber1956fluctuation_FDT , expressed through the contradiction between the values of the critical exponent $\gamma$ for different descriptions of the susceptibility $\chi$. This violation helps solving the puzzle of why a field response description yields $\gamma=1/2$ while a fluctuation description gives $\gamma=1$ lee2016hybrid . Our model displays both behaviors, as for strong coupling the FDT is violated, while for weaker coupling it holds, further demonstrating that this is the previously unknown reason for the different exponent values.

Acknowledgments.– B. G. and H. S. contributed equally to this work. We thank Ivan Bonamassa for very useful discussions related to this project. We also thank the Italian Ministry of Foreign Affairs and International Cooperation jointly with the Israeli Ministry of Science, Technology, and Space (MOST); the Israel Science Foundation, ONR, the Japan Science Foundation with MOST, BSF-NSF, ARO, the BIU Center for Research in Applied Cryptography and Cyber Security, and DTRA (Grant no. HDTRA-1-10-1-0014) for financial support.