Failure behavior in a connected configuration model
under a critical loading mechanism

Let $G=(V,E)$ denote a graph, where $V$ denotes the vertex set with $|V|=n$, and $E$ denotes the edge set with $|E|=m$. Typically, we consider graphs that can be scaled in the number of vertices/edges. Suppose that each edge in the network is subjected to a load demand that is initially exceeded by its edge capacity. The difference between the initial load and the edge capacity is called the surplus capacity of the edge, and we assume the surplus capacities to be independently and uniformly distributed on $[0,1]$ at the various edges. The failure process is triggered by an initial disturbance: all edges are additionally loaded with $\theta/m$ for some constant $\theta>0$. If the total load increase surpasses the surplus capacity of one or more edges, these edges fail and in turn, cause an additional load increase on all surviving edges that are in the same component upon failure. We call these load increments the load surges. We make two assumptions: edge failures occur subsequently, and all edges in the same component upon failure experience the same load surge. We continue with the failure of edges that have insufficient surplus capacity to handle the load surges till there are no more. We are interested in the number of edge failures, also referred to as the failure size.

We are interested in the setting where the failure size exhibits scale-free behavior. To this purpose, we define the critical load surge function $l_{j}^{m}(i)$ at edge $j$ after experiencing $i$ load surges to be

where $|E_{j}^{m}(i)|$ is the number of edges in the component that contains edge $j$ after perceiving $i$ edge failures in that component.

We observe that as long as two edges remain in the same component during the cascade, this recursive relation implies that the load surges are the same at both edges. Moreover, as long as all edges remain to be in a single component, it holds that $|E_{j}^{m}(i)|=m-i$ at every surviving edge, and hence (3) is solved by

In particular, applying (3) to a star topology with $n+1$ nodes and $m=n$ edges yields load surge function (4) for all surviving edges.

We point out that the load surges defined in (3) are typically non-deterministic, and are only well-defined as long as edge $j$ has not yet failed throughout the cascade. Edge failures may cause the network to disintegrate in multiple components, which affects $|E_{j}^{m}(i)|$ in a probabilistic way that depends on the structure of the graph. In contrast to processes in epidemiology or percolation models, the propagation of failures thus occurs non-locally and depends on the global structure of the graph throughout the cascade.

To provide an intuitive understanding why (3) gives rise to scale-free behavior for the failure size, consider the following. For a scale-free failure size to appear, the cascade propagation should occur in some form of criticality. This happens when the load surges are close to the expected values of the ordered surplus capacities. More specifically, since the surplus capacities are independently and uniformly distributed on $[0,1]$, the mean of the $i$-th smallest surplus capacity is $i/(m+1)$. If no disconnections occurred during the cascade, this would imply that the additional load surges at every edge should be close to $1/(m+1)$ after every edge failure. This explains intuitively why (4) leads to scale-free behavior for the star-topology, as is shown rigorously in [33]. Yet, whenever the network disintegrates in different components, the consecutive load surges need to be of a size such that the load surge is close the smallest expected surplus capacity of the remaining edges in order for the cascade to remain in the window of criticality. More specifically, suppose that the first disconnection occurs after $k$ edge failures and splits the graph in two components of edge size $l$ and $m-k-l$, respectively. Due to the properties of uniformly distributed random variables, we note that the expectation of the smallest surplus capacity in the first component equals $k/(m+1)+(1-k/(m+1))/l$. In other words, in order to stay in a critical failure regime, the additional load surges should be close to $(1-k/(m+1))/l$ after every edge failure until the cascade stops in that component, or another disconnection occurs. This process can be iterated, and gives rise to load surge function (3).

In this paper, our main focus is to apply failure mechanism (3) to the (connected) configuration model $CM_{n}(\mathbf{d})$ on $n$ vertices with a prescribed degree sequence $\mathbf{d}=(d_{1},...,d_{n})$. The configuration model is constructed by assigning $d_{v}$ half-edges to each vertex $v\in[n]:=\{1,...,n\}$, after which the half-edges are paired randomly: first we pick two half-edges at random and create an edge out of them, then we pick two half-edges at random from the set of remaining half-edges and pair them into an edge, and we continue this process until all half-edges have been paired. For consistency, we assume that the total degree $\sum_{i=1}^{n}d_{i}$ is even. The construction can give rise to self-loops and multiple edges between vertices, but these events are relatively rare when $n$ is large [16, 19, 20]. We point out that the number of edges is thus a function of $n$ and $\mathbf{d}$, i.e. $m:=m_{n}(\mathbf{d}):=\sum_{i=1}^{n}d_{i}/2$, where we suppress the dependency on $n$ and $\mathbf{d}$ for the sake of exposition.

Define $n_{i}$ as the number of vertices of degree $i$, and let $D_{n}$ denote the degree of a vertex chosen uniformly at random from $[n]$. We assume the following condition on the degree sequence.

In Appendix A, we provide an overview of the quantities that are commonly used throughout the paper. Unless stated otherwise, all limits are taken as $n\rightarrow\infty$, or equivalently by Condition 1.1, as $m\rightarrow\infty$. A sequence of events $(A_{n})_{n\in\mathbb{N}}$ happens with high probability (w.h.p.) if $\mathbb{P}(A_{n})\rightarrow 1$. For random variables $(A_{n})_{n\in\mathbb{N}}$, we write $X_{n}\overset{d}{\rightarrow}X$ and $X_{n}\overset{\mathbb{P}}{\rightarrow}X$ to denote convergence in distribution and in probability, respectively. For real-valued sequences $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$, we write $a_{n}=o(b_{n})$ and $a_{n}=O(b_{n})$ if $\lim_{n\rightarrow\infty}a_{n}/b_{n}$ tends to zero or is bounded, respectively. Similarly, we write $a_{n}=\omega(b_{n})$ and $a_{n}=\Omega(b_{n})$ if $\lim_{n\rightarrow\infty}b_{n}/a_{n}$ tends to zero or is bounded, respectively. We write $a_{n}=\Theta(b_{n})$ if both $a_{n}=O(b_{n})$ and $a_{n}=\Omega(b_{n})$ hold. We adopt an analogue notation for random variables, e.g. for sequences of random variables $(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$, we denote $X_{n}=o_{\mathbb{P}}(Y_{n})$ if $X_{n}/Y_{n}\overset{\mathbb{P}}{\rightarrow}0$. For convenience of notation, we denote $a_{n}\ll b_{n}$ if $a_{n}=o(b_{n})$. Finally, $\textrm{Poi}(\lambda)$ always denotes a Poisson distributed random variable with mean $\lambda$, $\textrm{Exp}(\lambda)$ denotes an exponentially distributed random variable with parameter $\lambda$, and $\textrm{Bin}(n,p)$ denotes a binomial distributed random variable with parameters $n$ and $p$.

We are interested in the probability that the failure size exceeds a threshold $k$. In this paper, we mainly focus on thresholds satisfying $1\ll k\ll m^{1-\delta}$ for some $\delta\in(0,1)$, which we refer to as the sublinear case. Our main result shows that the failure size has a power-law distribution.

To see why (5) holds, we need to understand the typical behavior of the failure process as the cascade continues. A first result we show is that it is likely that the number of edge failures that need to occur for the network to become disconnected is of order $\Theta(\sqrt{m})$. This suggests that as long as the threshold satisfies $k=o(\sqrt{m})$, the tail is the same as in the case of a star topology. The latter case is known to satisfy (5), as is shown in [33].

As long as the cascade continues, we show that it typically disconnects small components. This suggests that up to a certain point, the cascading failure mechanism creates a network with a single large component that contains almost all vertices and edges, referred to as the giant component, and some small disconnected components with few edges. It turns out that the total number of edges that are outside the giant component is sufficiently small, and hence the dominant contribution to the failure size comes from the number of edges that are contained in the giant component upon failure. Moreover, due to small sizes of the components outside the giant component, the load surge function for the edges in the giant as prescribed by (3) is relatively close to (4). We show that as long as threshold $k\ll m^{1-\delta}$ for some $\delta\in(0,1)$, the perturbations are sufficiently small such that the failure size behavior in the giant component satisfies (5).

We heuristically explain in Section 5 that the power-law tail property prevails beyond the sublinear case up to a certain critical point. However, the prefactor is affected in this case, i.e. the failure size tail is different from the star topology. Moreover, this notion holds for a broader set of graphs, and we validate this claim by extensive simulation experiments in Section 5.

We would like to remark that the disintegration of the network by the cascading failure mechanism is closely related to percolation, a process where each edge is failed/removed with a corresponding removal probability. In fact, percolation results will be crucial in our analysis to prove Theorem 1.3, combined with first-passage theory of random walks over moving boundaries. Before we lay out a more detailed road map to prove Theorem 1.3, we provide an outline of the remainder of this paper.

The proof of Theorem 1.3 requires many different steps, and therefore we provide a road map of the proof in Section 2. We explain that we need to derive novel percolation results for the sublinear case, which we show in Section 3. We rigorously identify the impact of the disintegration of the network on the cascading failure process in Section 4, where we use first-passage theory for random walks over moving boundaries to conclude our main result. Finally, we study the failure size behavior beyond the sublinear case in Section 5, as well as the failure size behavior for other graphs.

There are two elements of randomness involved in the cascading failure process: the sampling of the surplus capacities of the edges, and the way the network disintegrates as edge failures occur. The second aspect determines the values of the load surges, and only when the surplus capacity of an edge is insufficient to handle the load surge, the cascading failure process continues. Recall that as long as edges remain in the same component, they experience the same load surge. Since the surplus capacities are i.i.d., it follows that every edge in the same component is equally likely to be the next edge to fail as long as the failure process continues in that component. This is a crucial observation, as this provides a relation with the sequential edge-removal process. That is, suppose that we sequentially remove edges uniformly at random from a graph. Given that a new edge removal occurs in a certain component, each edge in that component is equally likely to be removed next, just as in the cascading failure process. Consequently, this observation gives rise to a coupling of the disintegration of the network through the cascading failure process to one that is caused by sequentially removing edges uniformly at random.

More specifically, suppose that sequentially removing edges uniformly at random yields the permutation $\{e_{(1)},...,e_{(m)}\}$. For the cascading failure process, sample $m$ uniformly distributed random variables and order them so that $U_{(1)}^{m}\leq...\leq U_{(m)}^{m}$ and assign to edge $e_{(j)}$ surplus capacity $U_{(j)}^{m}$. In particular, this implies that if the cascading failure process would always continue until all edges have failed, then the edges would fail in the same order as the sequential edge-removal process. Moreover, it is possible to exactly compute the load surge function from the order $\{e_{(1)},...,e_{(m)}\}$ over the edge set. We illustrate this claim by the example in Figure 1. In this example, we observe that $m=11$, and we consider the sequential removal process up to five edge removals. We see that the first three edge failures do not cause the graph to disconnect. It holds for every dotted edge $e_{j}\in[m]$,

and for every dashed edge $e_{j}\in[m]$,

Recursion (3) yields the corresponding load surge values.

This example illustrates that using our coupling, the sequential removal process gives rise to the load surge values as prescribed in (3). We point out that due to our coupling, if after step $j-1$ an edge $e_{(j)}$ for some $j\in[m]$ has sufficient surplus capacity to deal with the load surge, then so do all the other edges in that component. In other words, the cascade stops in that component. To determine the failure size of the cascade, one needs to subsequently compare the load surge values to the surplus capacities in all components until the surplus capacities at all the surviving edges are sufficient to deal with the load surges.

In summary, sequentially removing edges uniformly at random gives rise to the load surge values in every component. This idea decouples the two sources of randomness: first one needs to understand the disintegration of the network through the sequential removal process, leading to the load surge values through (3). Then, we can determine the failure size by comparing the surplus capacities to the load surges up to the point where the cascade stop in every component.

The next step is to determine the typical behavior of the sequential removal process on the connected configuration model $\overline{CM}_{n}(\mathbf{d})$. A first question that needs to be answered is how many edges need to fail, or equivalently, need to be removed uniformly at random from the network $\overline{CM}_{n}(\mathbf{d})$ to become disconnected. It turns out that this is likely to occur when $\Theta(\sqrt{m})$ edges are removed.

After this point, more disconnections start to occur as more edges are removed uniformly at random from the graph. In Section 3.3, we focus on the typical network structure if $\sqrt{m}\ll i\ll m^{1-\delta}$ edges have been removed uniformly at random for some $\delta\in(0,1/2)$. Typically, there is a giant component that contains almost all edges and vertices. The components that detach from the giant component are isolated nodes, line components, and possibly isolated cycles. We show that the total number of edges contained in these components are likely to be of order $\Theta(i^{2}/m)$, while the number of edges in more complex component structures are negligible in comparison. This leads to the following result, which is proved in Section 3.6.

We stress that determining the typical behavior of the network disintegration is not enough to prove our main result. In addition, for each of our results, we need to show that it is extremely unlikely to be far from its typical behavior, which we consider in Section 3.4. Moreover, we combine these large deviations results to show the following result in Section 3.6.

The sequential removal process gives rise to the load surge function at every edge, and we need to compare these values to the surplus capacities in every component until the cascade stops. To keep track of the cascading failure process in every component may seem cumbersome at first glance. However, due to the way the connected configuration model is likely to disintegrate, it turns out that it only matters what happens in the giant component.

Intuitively, this can be understood as follows. By Theorem 2.1 and 2.2, removing any sublinear number $i_{m}$ of edges always leaves $o(i_{m})$ edges outside the giant. Therefore, if $k:=k_{m}\ll m$ edges are sequentially removed, then only $o(k)$ of these edges were contained outside the giant upon removal. Moreover, even if the cascading failure process struck every edge of the components outside the giant, the contribution to the failure size would be at most $o(k)$. This is negligible with respect to the $k-o(k)$ failures that occur in the giant. The failure size $A_{n,\mathbf{d}}$ should therefore be well-approximated by the failure size in the giant. We formalize these ideas in Section 4.

More specifically, write

and hence $A_{n,\mathbf{d}}=\hat{A}_{n,\mathbf{d}}+\tilde{A}_{n,\mathbf{d}}$. Moreover, define

The main idea is that since the number of edges outside the giant is likely to be $o(i)$ when a sublinear number of $i$ edges are removed uniformly at random, the contribution of edge failures that occur outside the giant is asymptotically negligible. That is, we bound the probability that $\{A_{n,\mathbf{d}}\geq k\}$ to occur due to the cascading failure process by the same event in two related processes. For the upper bound, we consider the scenario where all edges in the small components that disconnect from the giant immediately fail upon disconnection. For the lower bound, we consider the scenario where none of the edges in the small components that disconnect from the giant fail.

To make this rigorous, we first consider the probability that the number of edge failures in the giant exceeds $\kappa(k)$. That is, if we sequentially remove $k$ edges uniformly at random, a set of $\kappa(k)$ edges were contained in the giant upon failure. We are interested in the probability that the coupled surplus capacities of all these edges are insufficient to deal with the corresponding load surges. By translating this setting to a first-passage problem of a random walk bridge over a moving boundary, we show the following result in Section 4.3.

Second, we use this result to derive an upper bound for the failure size tail. Trivially, the failure size is bounded by the number of edges that are contained in the giant component upon failure plus all the edges that exist outside the giant after the cascade has stopped. We introduce the stopping time

the minimum number of edges that need to be removed uniformly at random for the number of edges outside the giant to exceed $k-\upsilon(k)$. In other words, after we have removed $\upsilon(k)$ edges uniformly at random, the sum of the number of edges outside the giant and the number of removed edges exceeds $k$. The number of edge removals in the giant is given by $\kappa(\upsilon(k))$, and hence

We prove that $\upsilon(k)=k-o(k)$ with sufficiently high probability, and hence

Third, we derive a lower bound. We note that $\hat{A}_{n,\mathbf{d}}\leq A_{n,\mathbf{d}}$, and hence

where

That is, $\varrho(k)$ is the number of edges that need to be removed uniformly at random for $k$ failures to have occurred in the giant. We show that that $\varrho(k)=k+o(k)$ with sufficiently high probability, and hence

Since the upper and lower bounds coincide, this yields Theorem 1.3. We formalize this in Section 4.4.

To prove our results regarding the sequential removal process, we relate this process to another one where each edge is removed independently with a certain removal probability, also known as percolation. This is illustrated by the following lemma.

Lemma 3.1 is a direct consequence of the concentration of the binomial random variable, and the proof is given in Appendix B. This lemma establishes that sequentially removing $i:=i_{m}=\omega(1)$ edges uniformly at random is well-approximated by a percolation process with removal probability $q=i/m$. In particular, this holds for the (connected) configuration model. This allows us to study many questions involving the connected configuration model subject to uniformly removing edges in the setting of percolation. The study of percolation processes on finite (deterministic or random) graphs is a well-established research field, which dates back to the work of Gilbert [14] and is still very active these days. In particular, percolation on the configuration model is a fairly well-understood process, as established by Janson in [18].

It is known that there exists a critical parameter $q_{c}:=1-\mathbb{E}[D]/\mathbb{E}[D(D-1)]$ such that if $q<q_{c}$, then the largest component of the percolated (connected) configuration model contains a non-vanishing proportion of the vertices and edges [25, 18]. In particular, it is implied by the formula for $q_{c}$ that in order for a phase transition to appear, it is necessary that $\mathbb{E}[D(D-1)]/\mathbb{E}[D]\in(1,\infty)$. If $\mathbb{E}[D(D-1)]/\mathbb{E}[D]\leq 1$, then the largest component already has a sublinear size in $n$ even before percolation, while if $\mathbb{E}[D(D-1)]/\mathbb{E}[D]=\infty$, then there exists a connected component of linear size in the percolated graph for every $q\in(0,1)$. Typically, the (w.h.p. unique) component of linear size is referred to as the giant component. All other components are likely to be much smaller, i.e. of size $O_{\mathbb{P}}(\log n)$ under some regularity conditions on the limiting degree sequence [25]. If $q\geq q_{c}$, there is no giant component in the percolated graph and all the components have sublinear size [18].

In order to derive Theorem 1.3, these results in the literature are not sufficient. In particular, in view of Theorems 2.1 and 2.2, a more detailed description of the network structure is needed in case that $q=o(m)$. A critical tool to derive the results in this setting is the explosion algorithm, as we describe next.

Key to prove Theorems 2.1 and 2.2 is the explosion algorithm, designed by Janson in [18]. This algorithm prescribes that instead of applying percolation on $CM_{n}(\mathbf{d})$ with removal probability $q$, we can run the procedure as described in Algorithm 1.

Janson proved in [18] that the algorithm produces a random graph that is statistically indistinguishable from $CM_{n}(\mathbf{d},q)$, the graph obtained by removing every edge in (not necessarily connected) $CM_{n}(\mathbf{d})$ with probability $q\in[0,1]$. Yet, the graph obtained by the explosion method is significantly easier to study as it is simply a configuration model with a new degree sequence and a couple of vertices of degree $1$ that have been removed, where the latter operation does not significantly change the structure of the graph. Since the graphs obtained from the percolation process and the explosion method are identically distributed, we use the denomination $CM_{n}(\mathbf{d},q)$ both for the configuration model after percolation and for the graph we obtain via the explosion algorithm.

Next, we point out some properties we use extensively in this section. First, we observe that if $q=o(1)$, then by Taylor expansion, the removal probability in the explosion algorithm satisfies $1-(1-q)^{1/2}=(q/2)(1+o(1))$. Therefore, the probability of a vertex of degree $l$ to retain $j$ half-edges satisfies

if $q=o(1)$ and $j\leq l$. Moreover, let $n_{l,j}$ represent the number of vertices of degree $l$ that retain $j$ half-edges. That is, $n_{l,j}$ are random variables with distribution $n_{l,j}\overset{d}{=}Bin(n_{l},p_{l,j})$. Due to Markov’s inequality, it holds that

Moreover,

due to the Poisson limit theorem and the law of large numbers, respectively. Note that

By Markov’s inequality, it holds for every $\epsilon>0$

Similarly, it follows that for all $q=o(1)$,

and hence for every $\epsilon>0$,

Finally, we observe that for every $1/m\ll q\ll 1$, by the law of large numbers,

We recall that our focus is on the case where the number of edges that are removed from the giant is of order $o(m)$. In view of Lemma 3.1, this number is well-approximated by the number of edge removals in a percolation process with removal probability $q=o(1)$. In particular, in this regime there is a unique giant component w.h.p. and other components are likely to be much smaller, i.e. w.h.p. the number of vertices and edges ourside the giant is of order $o(m)$. Even more can be said about the structure of these components. In this section, we show in that these components are typically isolated nodes, line components or potentially isolated circles. More complex structures are relatively rare.

First, we explore the degree sequence $\mathbf{d}^{\prime}$ of $CM_{N}(\mathbf{d}^{\prime})$ in the explosion method from Algorithm 1. Analogously to the notation for the original degree sequence $\mathbf{d}$, we write $n_{i}^{\prime}$ for the number of vertices of degree $i$ in $\mathbf{d}^{\prime}$ and $p_{i}^{\prime}:=\lim_{n\rightarrow\infty}n_{i}^{\prime}/n$ as the limiting fraction.

We use the degree sequence $\mathbf{d}^{\prime}$ to study the structure of the components outside the giant in $\overline{CM}_{n}(\mathbf{d},q)$. Again, we will show that these components are either isolated nodes, line components or possibly isolated cycles. More complex structures are rather unlikely to appear as the network disintegrates.

We begin by proving a bound on the number of edges belonging to components that have a more complex structure, i.e. contain a vertex of degree at least three when $q\ll m^{-\delta}$ for some $\delta>0$. We show that this is not the leading-order term for the number of edges outside the giant, since the number of lines and isolated vertices is much larger. For a graph $G$, define $d_{\max}(G)$ as the largest degree of any vertex in $G$ and $E(G)$ as the edge set of $G$. For every edge $e$ and vertex $v$, write $\mathcal{C}(e)$ and $\mathcal{C}(v)$ for the connected component that contains $e$ or $v$, respectively. Finally, let $\mathcal{C}_{\rm max}$ denote the largest component in a graph $G$.