Percolation in Random Graphs: A Finite Approach
We propose an approach to calculate the critical percolation threshold for finite-sized Erdős-Rényi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available.
The exempla that eventually became the inspiration for percolation theory in random graphs were originally defined in terms of paths HammersleyMorton54 . Consider for example a porous medium with a liquid flowing downwards along paths probabilistically connecting the top to bottom. Although intuitive, a mathematically rigorous treatment of such a problem quickly runs into a combinatorical explosion Broadbent64 . Thus, despite its practical importance, to this date analytically exact results remain sparse.
To overcome these problems, several simplifications were proposed in the study of percolation in random graphs, most notably the consideration of infinite system size (for reviews, see Hofstad10 ; Grimmett99 ). In this limit, the search for spanning paths becomes meaningless, however, and is commonly abstracted by asking whether a significantly large connected cluster exists. The percolation transition is then identified with the appearance of a spanning giant connected component containing nodes, where is the number of nodes in the graph.
Various analytical methods have been introduced to assess the size distribution of connected components in random graphs in the asymptotic limit. An approach using generating functions to assess the emergence of connected components of specific size assumes infinite but locally finite quasi-transitive graphs in which no closed paths, or cycles, exist Newman01 . Here, the search for the giant connected component is equivalent to the search for a spanning tree. Statistical AlbertBarabasi02 or mean-field approaches HaraSlade90 generally provide only scaling results in the asymptotic limit. Independent of the methods utilized, however, the percolation threshold, i.e. the critical connection probability at which a giant connected component appears, was found to be in random graphs, with the size of the giant component scaling with below and at Hofstad10 ; Grimmett99 ; AlbertBarabasi02 .
Here, we follow an operator graph-theoretic method introduced in RudolphLilithMuller14 , which allows to calculate algebraically well-defined graph measures exactly in the non-asymptotic limit. We propose an approach to percolation in finite random graphs based on closed paths, or cycles. This notion is not only closer to the original conception of the percolation phenomenon, but also mathematically tractable in finite directed Erdős-Rényi random graphs. Specifically, we define the percolation threshold as the critical connection probability at which the first minimal Hamiltonian cycle of length , defined as a closed walk that visits each node exactly once, occurs. We note that the occurrence of this minimal Hamiltonian cycle is a sufficient condition for the emergence of a giant connected component spanning the full graph.
We start by constructing a non-self-looped Erdős-Rényi digraph algebraically. To that end, we introduce a binomial random annihilation operator, defined as
and understood statistically, i.e. the sum over applications of on returns . It can easily be demonstrated that the set of these operators form a linear algebra which is both commutative and associative under multiplication, as well as distributive. Furthermore, we introduce an circulant matrix
with denoting a matrix of independent random annihilation operators .
To demonstrate the application of Eq. (3), we calculate the number of closed walks of length , defined as the trace over the th power of an adjacency matrix,
where in the last step the statistical nature of was employed. As is a circulant matrix, application of the circulant diagonalization theorem yields
for its th power. Inserting the latter into (4), one obtains for the total number of closed walks of length in a non-self-looped Erdős-Rényi digraph with connectedness and size
Figure 1 compares the numerical result and corresponding analytical solution for a small graph of nodes.
Next we consider general walks of length visiting distinct nodes. To that end, let with being a set of indices. We consider labelled partitions of this set into two unordered subsets and with and , respectively, such that
Denoting by the set of all such partitions, we can define a generalized
where for given index sets and
Here, denotes the set of all unordered pairs with , and the Kronecker delta. Given two index sets and , Eqs. (7) and (Percolation in Random Graphs: A Finite Approach) algebraically formulate that all indices in are mutually distinct, all indices in are mutually equal, and each index from is distinct from each index in .
Generalizing Eq. (4), the number of walks and closed walks, and , respectively, of length visiting distinct nodes is given by the th power of the graph’s adjacency matrix, with restrictions imposed on the indices to ensure that only distinct nodes are visited. With (7), we have
We note that the latter is constructed from open walks of length visiting nodes by adding one more edge connecting the last node in the walk with its first node.
Restricting to the special case , an analytically closed form for can be obtained by observing the recursion
The term denotes the number of walks of length 1 visiting 2 distinct nodes, which, in a non-self-looped digraph, is equivalent to the graph’s total adjacency , thus yielding finally
for the total number of walks of length visiting distinct nodes in an Erdős-Rényi digraph with connectedness and size .
where denotes the number of closed walks of length 2 visiting 2 distinct nodes. The latter is equivalent to twice the number of bidirectional connected node pairs in non-self-looped random digraphs, , thus yielding
for the total number of closed walks of length visiting distinct nodes. Figure 2 compares the numerical result and corresponding analytical solution for a small graph of nodes. We note that the number of nodes in the numerical analysis was kept small as the search for specific walks constitutes an NP-hard problem, and requires significant computational resources for larger graphs.
With Eq. (15), we can now proceed to address the percolation threshold in terms of minimal Hamiltonian cycles, given by . Percolation is here defined to occur when there is at least one such cycle. Due to the symmetry of cycles of length , if one such cycle emerges, there are such cycles present in the graph. Thus, the critical connectedness can be defined as the connectedness for which . With (15), we obtain
which yields, with for random graphs,
for the critical percolation threshold for Erdős-Rényi digraph of size .
In order to compare the result (16) with the emergence of a dominant giant connected component, which is most commonly used to characterize the percolation transition, we numerically generated Erdős-Rényi digraphs of various size and connectedness, and investigated the average size of their respective giant connected components (Fig. 3, solid). The critical threshold [Eq. (16); Fig. 3, dashed] lies within the sharp percolation transition, marked by the emergence of a dominant giant component. Moreover, the numerical analysis indicates that consistently coincides with the emergence of a giant component covering about 80-85% of the graph (Fig. 3, gray bar), independent of the graph size within the investigated parameter regime. This finding suggests that, at percolation threshold , the size of the giant component will scale linearly with the graph size .
We note that the latter stands in stark contrast to the classical result, which finds a scaling with Hofstad10 ; Grimmett99 ; AlbertBarabasi02 . However, this classical result, which uses the emergence of a giant component as marking the percolation transition, must be viewed with care, as it yields a relative size of the giant connected component which scales as for . Thus, for infinite graphs, the giant component would occupy an infinitesimal fraction of the whole graph and not , as required.
Finally, we investigated the asymptotical behavior of . Using Stirling’s approximation, Eq. (16) yields
In this paper, we have investigated the percolation transition for finite-size simple random digraphs in a context close to its original conception HammersleyMorton54 , defined as the first occurrence of a path, or walk, spanning the whole system. To that end, we have calculated the expected total number of closed walks of length [; Eq. (5)] and total number of closed walks of length visiting distinct nodes [; Eq. (15)] in Erdős-Rényi digraphs of connectedness and size . The latter expression was then used to calculate the critical connectedness at which the first minimal Hamiltonian cycle emerges, thus quantifying non-asymptotically and analytically exact the percolation threshold .
In contrast to the classical definition of percolation in random graphs, which is meaningful only for infinite systems and uses the emergence of the giant component of size to mark the percolation transition, walks on graphs are an algebraically well-defined quantity and can be calculated exactly in cases where an explicit algebraic form of the adjacency matrix of the graph is available. Our approach is general and can be applied to characterize the percolation transition in other graph models for which an algebraic formulation of the adjacency matrix is available. A mathematically rigorous presentation of this framework is in active development.
Acknowledgements.The authors wish to thank OD Little for comments. This work was supported by CNRS, the European Community (BrainScales Project No. FP7-269921), and École des Neurosciences de Paris Ile-de-France.
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