Weighted percolation on directed networks

TL;DR

This paper develops a theoretical criterion for network disintegration under node removal in directed networks, based on the largest eigenvalue of a weighted adjacency matrix, applicable to various removal strategies without needing a Markov model.

Contribution

It introduces a novel eigenvalue-based percolation criterion for directed networks that does not require a Markov network model, broadening applicability.

Findings

The percolation threshold is predicted by the largest eigenvalue condition.

Numerical tests confirm the accuracy of the theoretical criterion.

The method applies to various node removal strategies in directed networks.

Abstract

We present an analysis of the percolation transition for general node removal strategies valid for locally tree-like directed networks. On the basis of heuristic arguments we predict that, if the probability of removing node $i$ is $p_i$, the network disintegrates if $p_i$ is such that the largest eigenvalue of the matrix with entries $A_{ij}(1-p_i)$ is less than 1, where $A$ is the adjacency matrix of the network. The knowledge or applicability of a Markov network model is not required by our theory, thus making it applicable to situations not covered by previous works. We test our predicted percolation criterion against numerical results for different networks and node removal strategies.

Figures

Equations

$${\hat{C}}_{{\mathbf{zz}}^{\prime }}=(d^{out}{)}^{\prime }P({\mathbf{z}}^{\prime }|\mathbf{z}),$$

Full text

There has been much recent interest in the structure and function of complex networks newman1 . One aspect that has received considerable attention is the resilience of networks to the removal of some of their nodes callaway ; cohen ; cohen1 ; cohen2 ; newman2 ; moreno ; serrano ; ser . This problem is related, for example, to the robustness of transportation and information networks to disturbances like random failures or targeted attacks, or to the resistance of biological networks to the action of drugs. Another related problem is determining the threshold for epidemic spreading grassberger . An important objective is to determine what characteristics of the structure of the network determine the proportion of nodes that can be removed before the network disintegrates.

A model often considered is one in which nodes are removed from a network of size $N$ with a uniform probability $p$. In the large $N$ limit, for probabilities less than a critical value $p_{c}$, there is a connected component of the network of size of order $N$ (the giant component), and for values of $p$ larger than $p_{c}$, there is no connected component of size of order $N$ newman1 ; callaway ; cohen ; cohen1 ; cohen2 ; newman2 ; moreno ; serrano . The critical probability $p_{c}$ at which this percolation transition occurs has been the subject of several theoretical works, and various approximations have been put forward.

In what follows we define the in and out degrees of a node $i$ by $d^{out}_{i}=\sum_{j=1}^{N}A_{ij}$ and $d^{in}_{i}=\sum_{j=1}^{N}A_{ji}$. Here $A_{ij}$ is the network adjacency matrix; $A_{ij}=1$ if there is a directed link from $i$ to $j$ and [math] otherwise. If $A=A^{T}$ the network is said to be undirected and $d_{i}^{out}=d_{i}^{in}=d_{i}$. For undirected, degree uncorrelated networks (the number of connections per node for neighboring nodes is not correlated), Cohen et al. have shown cohen that the critical probability is approximately given by $(1-p_{c})\left[(\langle d^{2}\rangle/\langle d\rangle)-1\right]=1$, where $\langle\cdot\rangle$ denotes an average over nodes. Reference  moreno treats the case of undirected networks with correlations for degree Markovian networks, i.e., networks in which all nontrivial correlations are captured by the probability $P(d^{\prime}|d)$ that a randomly chosen link from a node with degree $d$ is connected to a node with degree $d^{\prime}$. Reference moreno generalizes the result in Ref. cohen and obtains a critical value of $p_{c}$ given by $(1-p_{c})\Lambda=1,$ where $\Lambda$ is the largest eigenvalue of the matrix with entries $C_{dd^{\prime}}=(d^{\prime}-1)P(d^{\prime}|d)$. Other works on undirected networks have extended the Markovian approach to include the effect of clustering (e.g., often present in social networks); in particular Ref. ser presents a general analysis for the case of weak clustering.

Reference serrano studies the percolation transition in directed degree Markovian networks. The types of components studied are a strongly connected component (SCC), defined as a set of nodes such that every node in the SCC is reachable from any other node in the SCC by a directed path, its associated in-component (IN), defined as the set of nodes from which the SCC is reachable by a directed path, and out-component (OUT), defined as the set of nodes reachable from the SCC by a directed path. Notice that there might be several such components in a particular network. Of interest is the largest strongly connected component which, if its size is of order $N$, is called the giant strongly connected component, GSCC. The out and in components of the GSCC are denoted GOUT and GIN.

It was found in Ref. serrano that as the probability of node removal $p$ is increased, GSCC, GOUT, and GIN disappear at the same critical value $p_{c}$. This value was found to be determined by the largest eigenvalue of a matrix expressed in terms of $P_{o}\left({\bf y}^{\prime}|{\bf y}\right)$ and $P_{b}\left({\bf y}^{\prime}|{\bf y}\right)$ where ${\bf y}=(d_{p}^{in},d_{p}^{out},d_{p}^{bi})$, and $d_{p}^{in}$, $d_{p}^{out}$ and $d_{p}^{bi}$ are the number of purely incoming, purely outgoing and bidirectional edges for a given node (i.e., an edge $A_{ij}=1$ is purely outgoing from node $i$ if $A_{ji}$ = 0). $P_{o}\left({\bf y}^{\prime}|{\bf y}\right)$ and $P_{b}\left({\bf y}^{\prime}|{\bf y}\right)$ are the probabilities of reaching a node of degree ${\bf y}^{\prime}$ from a node of degree ${\bf y}$ by following an outgoing and a bidirectional edge, respectively.

One of our aims in this paper is to remove the need for the applicability and knowledge of a Markov network model. In order to do so, in this paper we will focus on a class of directed networks that are locally tree-like in the sense that they have few short loops. More precisely, we assume that for each node $i$ and not too large $L$, the number of different nodes reachable by paths of length $L$ or less starting at node $i$ is close to the total number of paths of length $L$ or less starting from node $i$. In particular ($L=2$) we will assume that bidirectional edges are absent or negligible in number. Under this assumption, ${\bf y}=(d^{in},d^{out},0)$, and the matrix in Ref. serrano whose eigenvalue determines the percolation transition reduces to

[TABLE]

where ${\bf z}=(d^{in},d^{out})$. We note that our locally tree-like condition for directed networks is analogous to assuming negligible clustering.

In many situations the node removal probability is not a constant. For example, airports might have different security measures, or differ in their vulnerability to an attack or weather related shutdown due to their geographical location. Also, we have noted recently imp that a measure of the dynamical importance of nodes is proportional to $v_{i}u_{i}$, where $u$ and $v$ are the right and left eigenvectors corresponding to the largest eigenvalue $\lambda$ of the adjacency matrix of the network, $A$: $Au=\lambda u$, $v^{T}A=\lambda v^{T}$. Thus, a potential removal strategy (to be used in an example later in this paper) is that in which nodes are removed based on the value of $v_{i}u_{i}$. More generally, we would like to study the effect of node removal strategies that assign a probability $p_{i}$ to the removal of node $i$, and we refer to this problem as weighted percolation. (Other previously considered possibilities are that highly connected nodes are preferentially removed from the network, e.g., $p_{i}$ is proportional to the degree of node $i$ cohen1 , or to a power of the degree of node $i$ cohen2 .)

Our objective is to present a simple heuristic method for treating general weighted percolation removal strategies (i.e., general $p_{i}$) on directed networks without the need for a Markovian network model. While we do not use the Markovian assumption nor a specific node removal probability, we assume a locally tree-like network structure (we will discuss the validity of this assumption below), and we also require knowledge of the network adjacency matrix $A$. We find that the network disintegrates, as defined by the disappearance of the giant connected components, when the node removal strategy is such that the largest eigenvalue $\hat{\lambda}$ of the matrix $\hat{A}$ with entries $\hat{A}_{ij}=A_{ij}(1-p_{i})$ is less than $1$.

To obtain the above result we adapt the mean field arguments given for example in newman2 and moreno to our case. Consider first the disappearance of the giant in-component GIN. Let $\eta_{i}$ be the probability that node $i$ is not in the giant in-component GIN. Node $i$ is not in GIN either if it has been removed (with probability $p_{i}$), or if it has not been removed and none of its out-links point to nodes in GIN. Consider two such nodes $j_{1}$ and $j_{2}$ that $i$ points to. We argue that it is reasonable to make the approximation that whether $j_{1}$ belongs to GIN is independent of whether $j_{2}$ belongs to GIN. Whether $j_{1}$ is in GIN depends on whether the nodes it points to are in GIN, which depends on the nodes they point to, and so on. Our locally tree-like assumption implies that the nodes that can be reached from $j_{1}$ by a short path are essentially independent of the nodes that can be reached from $j_{2}$ by a short path. Based on this independence assumption, $\eta_{i}$ is given by (recall that $A_{ij}=0$ or $1$) $\eta_{i}=p_{i}+(1-p_{i})\prod_{j=1}^{N}(\eta_{j})^{A_{ij}}$. This equation always has the trivial solution $\eta_{i}=1$. The presence of a giant in-component requires a solution for which the expected size $s=N-\sum_{j=1}^{N}\eta_{i}$ is positive. Setting $\eta_{i}=e^{-z_{i}}$ and assuming $0\leq z_{i}$ and $\sum_{j=1}^{N}A_{ij}z_{j}\ll 1$, we obtain the approximation $z_{i}=\sum_{j=1}^{N}A_{ij}(1-p_{i})z_{j}$. When $p_{i}=1$ (i.e., all nodes are removed), the only solution is the trivial solution $z_{i}=0$. As we decrease the $p_{i}$’s, a nontrivial solution (corresponding to a giant in-component) first appears when the largest eigenvalue $\hat{\lambda}$ of the matrix $\hat{A}$ with entries $\hat{A}_{ij}=A_{ij}(1-p_{i})$ is $1$. Note that, as required, we can satisfy $\eta_{i}\leq 1$ since the components $z_{i}$ of the eigenvector corresponding to $\hat{\lambda}$ are, by the Frobenius theorem siamlambda , nonnegative. Applying the same reasoning to the out-component GOUT we find that it appears when the largest eigenvalue of the matrix with entries $B_{ij}=A_{ji}(1-p_{j})=(\hat{A}^{T})_{ij}$ is $1$. Since the transpose of $\hat{A}$ and $\hat{A}$ have the same spectrum, the giant in and out-components appear simultaneously.

The above can also be understood by the following heuristic argument. Our previous discussion applies not only to GOUT (GIN), but more generally to sets generated by repeatedly following outgoing (incoming) links starting from a given node. Therefore, one can estimate the size of such sets and locate the transition as the point at which one of them has macroscopic size. In doing so, it is essential not to overcount the number of nodes, and it is here where our assumption of locally tree-like network structure allows us to simplify the problem. The number of directed paths of length $m$ starting from node $i$ can be estimated using this assumption, for not too large $m$, as the sum of the components of the vector ${\hat{A}}^{m}e^{i}$, where $e^{i}$ is the unit vector for coordinate $i$. If the largest eigenvalue of $\hat{A}$ is larger than $1$, the number of paths of length $m$ grows exponentially with $m$ for some starting node $i$. Under our assumptions, these paths traverse different nodes, and thus the out-component of $i$ has large size, in agreement with our previous result.

In order to motivate the locally tree-like assumption, we consider the illustrative case of uncorrelated networks. (While our concern in this paper is not with uncorrelated networks, they, nevertheless, provide a useful example of why the locally tree-like assumption might be valid in some cases.) In particular, we estimate the fraction of bidirectional edges. The probability $p_{ij}$ that nodes $i$ and $j$ share a bidirectional edge is given by $p_{ij}=d_{i}^{out}d_{i}^{in}d_{j}^{in}d_{j}^{out}/(N^{2}\langle d\rangle^{2})$, where, since $\langle d^{in}\rangle=\langle d^{out}\rangle$, we use $\langle d\rangle$ to denote either of these averages. In the case that the degrees at nodes $i$ and $j$ are uncorrelated, on average, $\langle p_{ij}\rangle=\left(\langle d_{i}^{out}d_{i}^{in}\rangle/(N\langle d\rangle)\right)^{2}\approx\left(\lambda/N\right)^{2},$ where we have used the mean field approximation for the maximum eigenvalue $\lambda$ of $A$ (see below). Not too far above the percolation transition $\lambda$ is of order $1$, and we thus expect the locally tree-like assumption to be valid.

We next discuss how our results compare to those of Ref. serrano in the case of negligibly few bidirectional edges (i.e., Eq.(1)). If $p_{i}=p$, our result for the critical probability $p_{c}$ reduces to $(1-p_{c})\lambda=1$, where $\lambda$ is the largest eigenvalue of the adjacency matrix $A$. If the network is degree Markovian, and we let $\psi_{\bf z}^{(m)}$ be the average number of directed paths of length $m$ starting from nodes of degree ${\bf z}$, we have $\psi_{{\bf z}}^{(m+1)}=d^{out}\sum_{{\bf z}^{\prime}}P({\bf z}^{\prime}|{\bf z})\psi_{{\bf z}^{\prime}}^{(m)}$. Since, for large $m$, the number of paths of length $m$ grows like $\lambda^{m}$, we associate to the previous equation the eigenvalue problem $\lambda_{M}\psi_{{\bf z}}=d^{out}\sum_{{\bf z}^{\prime}}P({\bf z}^{\prime}|{\bf z})\psi_{{\bf z}^{\prime}}$, where $\lambda_{M}$ is the Markovian approximation to $\lambda$. The previous result agrees with Eq. (1) [the matrices $d^{out}P({\bf z}^{\prime}|{\bf z})$ and $(d^{out})^{\prime}P({\bf z}^{\prime}|{\bf z})$ have the same spectrum]. We note that, in the absence of degree-degree correlations, we have $P({\bf z}^{\prime}|{\bf z})=d^{in}P({\bf z}^{\prime})/\langle d\rangle$, which when inserted in the eigenvalue equation yields the mean field approximation for the eigenvalue, $\lambda_{mf}=\langle d^{in}d^{out}\rangle/\langle d\rangle$.

We now illustrate our theory with two numerical examples. The first example (Example 1) illustrates the flexibility of our approach to address various weighted percolation node removal strategies, while the second example (Example 2) illustrates the point that our approach does not require the knowledge or applicability of a Markov network model.

Example 1. For simplicity, we consider uncorrelated random networks with degree distributions $P(d^{in},d^{out})$ in which $d^{in}$ and $d^{out}$ are independent and have the same distribution $\tilde{P}(d)$, that is, $P(d^{in},d^{out})=\tilde{P}(d^{in})\tilde{P}(d^{out})$. We use a generalization of the method in Ref. chung in order to generate networks with a power law degree distribution, $\tilde{P}(d)\propto d^{-\gamma}$. We choose the sequence of expected degrees ${\tilde{d}^{in}_{i}}=c(i+i_{0}-1)^{-1/(\gamma-1)}$ for the in-degrees, and a random permutation of this sequence for the out-degrees, where $i=1,\dots,N$, and $c$ and $i_{0}$ are chosen to obtain a desired maximum and average degree. Then, the adjacency matrix is constructed by setting $A_{ij}=1$ for $i\neq j$ with probability $\tilde{d}_{i}^{out}\tilde{d}_{j}^{in}/(N\langle d\rangle)$ and zero otherwise ($A_{ii}=0$). The ensemble expected value of the resulting network degree distribution is given by $P(d^{in},d^{out})$. (Note that we assume $\tilde{d}_{i}^{out}\tilde{d}_{j}^{in}<N\langle d\rangle$.) In Fig. 1(a) we show, for a $N=2000$ scale free network with exponent $\gamma=2.5$ and $\langle d\rangle=3$, the size of GIN as a function of the number of removed nodes $R$, when nodes are removed in order of decreasing $d_{i}^{in}d_{i}^{out}$ (thick solid line), decreasing dynamical importance imp ($v_{i}u_{i}/v^{T}u$) (thin solid line), and randomly (dashed line) foot . The removal probabilities in the first two cases are given by $p_{i}=1$ if $i\in S$ and [math] otherwise for a subset $S$ of nodes. (For example, in the first case $S=\{i:d_{i}^{in}d_{i}^{out}>d_{*}^{2}\}$, and $\hat{A}$ reduces to the matrix obtained by removal of all nodes in $A$ for which $d_{i}^{in}d_{i}^{out}>d_{*}^{2}$.) We also show in Fig. 1(b) the largest eigenvalue $\hat{\lambda}$ of $\hat{A}$ which in this case is equivalent to the adjacency matrix of the network resulting from the removal of the nodes. We observe for all three cases that the network disintegrates, as predicted, when $\hat{\lambda}=1$ (indicated with the vertical arrows in Fig. 1(b)). The number of removed nodes necessary to disintegrate the network is less when removing nodes by dynamical importance. Removal of nodes by $d_{i}^{in}d_{i}^{out}$ requires somewhat more nodes to disintegrate the network, while random removal requires removal of the most.

The simplest Markovian network assumption is that correlations between connected nodes depend solely on the degree of these nodes and not on any other property the nodes might have. While this is a useful analytical framework, probably applicable to many cases, it is also likely to fail in other cases (e.g., kim ). In our next example we will consider a network in which the node statistics depends on variables additional to the degrees and for which, therefore, a degree based Markovian approximation is not a priori expected to apply.

Example 2. We start with a network generated as in Example 1 with $N=10^{5}$, $\gamma=2.5$, and $\langle d\rangle=3$. Then, we first specify a division of the nodes in the network into two groups of the same size, $A$ and $B$, ($A\bigcup B=\{1,2,\dots,N\}$, $A\bigcap B=\emptyset$). We define a measure of the degree-degree correlations $\rho=\langle d_{i}^{in}d_{j}^{out}\rangle_{e}/\langle d_{i}\rangle_{e}^{2}$, where $\langle\dots\rangle_{e}$ denotes an average over edges, $\langle Q_{ij}\rangle_{e}\equiv\sum_{i,j}A_{ij}Q_{ij}/\sum_{i,j}A_{ij}$. The following (an adaptation of the method in Ref. mixing ) is repeated until the network has the desired amount of degree-degree correlations as evidenced in the value of $\rho$: Two edges are chosen at random, say connecting node $i$ to node $j$ and node $n$ to node $m$. If $i,j,n,m$ are all in $A$, the edges are replaced with two edges connecting node $i$ to node $m$ and node $n$ to node $j$ if $(d_{n}^{in}d^{out}_{m}+d_{i}^{in}d^{out}_{j}-d_{n}^{in}d^{out}_{j}-d_{i}^{in}d^{out}_{m})<0$. If $i,j,n,$ and $m$ are all in $B$, the edges are replaced with two edges connecting node $i$ to node $m$ and node $n$ to node $j$ if $(d_{n}^{in}d^{out}_{m}+d_{i}^{in}d^{out}_{j}-d_{n}^{in}d^{out}_{j}-d_{i}^{in}d^{out}_{m})>0$. Otherwise the edges are unchanged. The effect of this division is to create two subnetworks, $A$ and $B$, with positive and negative degree-degree correlations. Starting from such a network, we successively remove a randomly chosen node and compute the size of the GIN relative to its initial size. In Fig. 2 we plot this normalized size of the GIN as a function of the fraction of remaining nodes for ten realizations of the node removal sequence. The vertical arrow represents the prediction from the eigenvalue. Although the transition points of individual realizations have some spread, the arrow gives a good approximation of their mean (see inset).

We now discuss the advantages and disadvantages of the eigenvalue approach when compared to the Markov approximation. As opposed to the Markov approximation, the eigenvalue approximation allows the easy treatment of general node removal strategies (‘weighted percolation’). Furthermore, it does not require the assumption that the node correlations depend only on their degree and are only to nearest neighbors. In addition, the construction of the matrix $d^{out}P({\bf z}^{\prime}|{\bf z})$ and the determination of its largest eigenvalue is in some cases harder than the direct determination of the largest eigenvalue of the adjacency matrix $A$. On the other hand, in many cases the adjacency matrix of the network is not known, and local sampling methods from which an approximation to the matrix $d^{out}P({\bf z}^{\prime}|{\bf z})$ can be constructed must be used. Additionally, the eigenvalue approach is valid only when the network has locally tree-like structure.

In conclusion, we have presented a simple eigenvalue-based criterion for percolation on directed networks. Our method should be viewed as complementary to previous studies in that it does not require knowledge or applicability of a Markov network model and can treat general node removal strategies, but requires knowledge of the network adjacency matrix $A$ and only applies when the network has locally tree-like structure.

This work was supported by ONR (Physics), by the NSF (PHY 0456240 and DMS 0104-087), and by AFOSR FA95500410319.