The total variation distance measures the distance between two probability distributions μ and η: | | μ - η | | : = 1 2 ∑ x ∈ Ω | μ ( x ) - η ( x ) | . (1) We define τ a ( ϵ ) : = min { t ∈ N : | | p a ( t ) - π | | ≤ ϵ } as the minimal number of steps a random walk has to take to reach a distribution which is close to ϵ to its stationary distribution when starting at state a ∈ Ω. The total mixing time τ(ϵ) of a Markov chain is defined as τ ( ϵ ) : = max a ∈ Ω { τ a ( ϵ ) } . (2) A Markov chain is known as rapidly mixing, if τ(ϵ) can be bounded from above by a polynomial which depends on the input size n and the parameter ϵ⁻¹. The total mixing time can be bounded by several techniques. We briefly present two widely used bounding methods which we investigate in the remainder of this article.

Let 1 = λ₁ > λ₂ ≥ … ≥ λ_|Ω| > −1 be the real eigenvalues of the transition matrix P and let λ_max be defined as λ_max = max{|λ₂|, |λ_Ω|}. The total mixing time can be bounded by the lower and upper spectral bound[5], i.e., τ ( ϵ ) ≤ 1 - λ max - 1 · ln ( ϵ - 1 ) + ln ( π min - 1 ) (3) τ ( ϵ ) ≥ 1 2 λ max ( 1 - λ max ) - 1 ln ( 2 ϵ ) - 1 , (4) where π_min denotes the smallest component of π. The transition matrix P is not necessarily symmetric. However, it can be transformed into a symmetrical matrix: P s y m = D - 1 / 2 · P · D 1 / 2 , (5) where D is the |Ω| × |Ω| diagonal matrix with the components of π on its main diagonal. This transformation relies on the reversibility of a Markov chain. It is a classical result that P_sym has the same eigenvalues as P.

Sinclair’s multi commodity flow method[5] is often used for bounding the total mixing time. Let P = ⋃ x ≠ y P x y be a family of simple paths in Γ, each P x y consisting of simple paths between x and y ∈ Ω. The maximum load congestion ρ, with respect to P, is then defined as ρ ( P ) = max ( u , v ) ∈ Ψ ∑ p ∈ P : ( u , v ) ∈ p π ( x ) π ( y ) | p | π ( u ) P ( u , v ) . (6) For any system of paths P the total mixing time of a reversible Markov chain can be bounded by the congestion bound, i.e., τ ( ϵ ) ≤ ρ ( P ) · ln ( ϵ - 1 ) + ln ( π min - 1 ) . (7) The quality of the congestion bounds depend on the path construction scheme P. The congestion bound is often used to gain theoretical bounds on the total mixing time.

Given an instance representation (e.g. a degree sequence), the first step of the analysis is the construction of the state graph for the given instance. The purpose of this step is to gain a sparse representation of the state graph Γ = (Ω, Ψ). Initializing Ω with an arbitrary state generated by the MarkovChain::arbitraryState method, we perform a full graph scan to iteratively enumerate the set of states and transition arcs. In each step, we select a state u ∈ Ω and enumerate all its adjacent states N_u via the MarkovChain::neighbours method. We add a transition arc (u, v) with proposal probability κ(u, v) to Ψ for each adjacent state v ∈ N_u and add v to Ω, if not already included. At the end of this step, we have a complete list Ω of states and an adjacency list representation of the transition arcs Ψ. The probability of each transition (u, v) ∈ Ψ is subsequently computed as κ(u, v) ⋅ min(1, w(v)/w(u)), where the function w:Ω→ℝ+ can optionally be overridden to enable the Metropolis rule (see Algorithm 1). The construction of the state graph relies on connectedness of the state graph, which is given whenever the Markov chain is irreducible. The construction of the state graph is a sequential process, consuming O(|Ω| + |Ψ|) time and memory.

We compute the total mixing time τ(ϵ) of a state graph based on the following idea: Let P be the transition matrix of an ergodic Markov chain and let a ∈ Ω be an initial state. One step of the random walk changes the probability distribution p a ( t ) to p a ( t + 1 ) = p a ( t ) · P. Iterating this argument, the distribution after t steps is given by p a ( t ) = p a ( 0 ) · P t , (8) where p a ( 0 ) is a row vector whose components are p a ( 0 ) ( a ) = 1 for the initial state a and p a ( 0 ) ( i ) = 0, for i ≠ a. So, each row of the matrix P^t gives the distribution to a certain p a ( t ) for a ∈ Ω. The total mixing time τ(ϵ) can therefore be computed from P by iterated matrix multiplication. The principle can be described as follows: We transform the adjacency list representation of Γ into its adjacency matrix representation. The entries P_{i, j} of this matrix represent the transition probabilities for going from state i to state j, so this matrix is the transition matrix described above. Since the total variation distance decreases monotonically with t, we can find τ(ϵ) with a two-step procedure.

Computing the total mixing time requires O(|Ω|²) memory and O(log(τ(ϵ)) ⋅ |Ω|^ω) time, where ω is the exponent of the matrix multiplication algorithm. (In most libraries, ω is equal to 3. However, the matrix multiplication implementation can be exchanged by using another implementation when building the library.)

Computing the total mixing time is currently by far the most time and memory consuming operation in our library. Due to quadratic memory requirement, it is applicable only for small state graphs with |Ω| ≤ 30000 (depending on main memory). Since this method is also very time-consuming, we provide three implementations, which can be used if the appropriate hardware is available:

The three implementations share the same asymptotical running time, but differ greatly in practice (see Fig 1 for a comparison of performance). Since most of the work is matrix multiplication, this comparison almost directly maps to a performance comparison between the corresponding libraries for matrix multiplication. The O(log(τ(ϵ))) invocations of the total variation distance computation, each with time complexity of O(|Ω|²), contribute only a little to the running time.

The difference between the largest and the second largest eigenvalue (1 − λ_max) of a Markov chain’s transition matrix is known as the spectral gap and is of central interest in mixing time analysis. To compute this quantity, we use the well-known ARPACK++[14] library to compute the two real eigenvalues with the largest magnitude of the transition matrix P. In case of non-symmetric transition rules, we first transform the transition matrix P into the symmetrical matrix P_sym via Eq (5). After computing both eigenvalues with the largest magnitude, we use the second eigenvalue in order to compute the upper and lower spectral bound via Inequalities Eqs (3) and (4). Since this method requires only a sparse representation of the transition matrix, it is also applicable for large state graphs.

As stated in the introduction, we are also interested in the quality of other bounding techniques. In particular, we are interested in the canonical path method which is often used to gain theoretical bounds. We implemented a method that can be used to gain an upper bound on the mixing time by computing the maximal congestion of a user-definable path construction scheme. Four each pair of states (u, v) ∈ Ω × Ω we apply the path construction scheme to construct a path p from state u to state v. The congestion of each transition arc lying on this path is increased by |p|π(u)π(v) (see Eq (6)). Since the construction of |Ω|² paths can be done completely independently of each other, we use OpenMP to construct all paths in parallel. The maximal congestion of any transition arc is used for computing the upper bound via Inequality Eq (7). This method has a time complexity of O(|Ω|²) and a memory complexity of O(|Ψ|).

As state graphs can be seen as weighted directed graphs, all kinds of graph algorithms can be integrated into marathon. As an example, we added functions for computing the diameter of a state graph, as well as functions for computing the average path length.

Broder [15] introduced a Markov chain which can be described as follows. In state M ∈ Ω_M we pick a candidate state M′ (line three in Algorithm 1) by choosing an edge e = {u, v} from E uniformly at random. One of five cases occurs:

The state M′ is proposed as a candidate state with proposal probability κ(M, M′) = 1/|E| in line three of Algorithm 1. The proposed state M′ is accepted or refused by line four of Algorithm 1, using unit weights. So, this chain converges to a uniform stationary distribution. In the remainder of this article, we refer to this Markov chain as Matching Chain One. Jerrum and Sinclair [16] proved that Matching Chain One is rapidly mixing for a graph class where the ratio of near-perfect matchings N ( G ) and perfect matchings M ( G ) in G is polynomially bounded. Being specific, its total mixing time can be bounded from above: τ ( ϵ ) ≤ 16 2 | E | 2 | N ( G ) | | M ( G ) | 4 ln ( | Ω M | · ϵ - 1 ) . (9) The class of dense bipartite graphs, i.e. bipartite graphs G = (U, W, E) with |U| = |W| = n and with minimal vertex degree of at least n/2, was shown as a class for which the ratio of near-perfect and perfect matchings is polynomially bounded [16]. For this class of graphs, the following upper bound on the total mixing time can be gained [17]: τ ( ϵ ) ∈ O ( n 7 ln ( | Ω M | · ϵ - 1 ) ) . (10) A main problem of Matching Chain One is the fact that it does not only generate perfect matchings but also near-perfect matchings. In many graphs with 2n vertices the ratio of the number of near-perfect and perfect matchings cannot be bounded by a polynomial which is dependent on n. In such cases, the sampling of a perfect matching needs an exponential number of trials to see a perfect matching at all.

One way to overcome the problem of an exponential total mixing time has been given by Jerrum, Sinclair and Vigoda [18]. Their main idea is to use a carefully chosen weight function w to sample from a non-uniform distribution. This way, they gained a rapidly mixing Markov chain which can be used to sample perfect matchings from any given bipartite graph. In addition, the transition rules were changed to the following: In a state M ∈ Ω_M:

The proposal probability κ(M,M′) for two states M,M′ ∈ Ω_M is therefore 1/n when moving between perfect and near-perfect matchings and 1/(2n) when moving between near-perfect matchings. In connection with Algorithm 1, this Markov chain converges to a stationary distribution π which is proportional to the weight function w. Defining M as the set of perfect matchings of G, and N u , v as the set of near perfect matchings in G where u and v are unmatched, the weight function suggested in [18] is defined as: w ( M ) = 1 , M ∈ M | M | / | N u , v | , M ∈ N u , v . (11) Knowing the values of | M | and | N u , v |, the total mixing time of this chain can be bounded from above by the following polynomial [19]: τ ( ϵ ) ∈ O ( n 4 ln ( ( π m i n · ϵ ) - 1 ) . (12) Unfortunately, | M | and | N u , v | are usually not known in practice. Jerrum, Sinclair and Vigoda gave a description of a procedure for approximating these quantities in polynomial time [18]. In our experiments though, we knew the exact values since we had a list of all states. In combination with Algorithm 1, this Markov chain is rapidly mixing for all bipartite graphs and can be used for sampling of perfect matchings. We refer to this chain as Matching Chain Two.

At a state G = (U, V, E) ∈ Ω_R, choose four integers i, j, k, l with 1 ≤ i ≤ k ≤ n and 1 ≤ j ≤ l ≤ n′ uniformly at random. One of three cases occur:

Such a switch, i.e. changing the endpoints of two edges, does not change the degree sequence and thus constructs a new bipartite graph realization of S. It turns out, all graph realizations can be generated by starting with a given graph realization and applying a number of switches [22]. The proposal probability κ(G,G′) of a non-loop transition is 2 n ( n + 1 ) · 2 n ′ ( n ′ + 1 ). Using unit weights w(x) = 1 for all x ∈ Ω_R Algorithm 1 converges to a uniform stationary distribution. Kannan et al. [21] proved polynomially bounded mixing times of this chain for regular sequence pairs. Miklós et al. [23] extended the proof to half-regular sequence pairs. For all other sequence classes, the problem of the rapid mixing property is still open. We will refer to this Markov chain as Switch Chain One. Recently, Greenhill gave a proof that the total mixing time of the directed, non-bipartite version of Switch Chain One is bounded by the following polynomial when the largest degree d_max of the input sequence lies in the range 3 ≤ d max ≤ 1 4 | E |, where |E| is the number of edges in the realization [24]: τ ( ϵ ) ≤ 1 10 d max 14 | E | 9 ( | E | ln | E | + ln ( ϵ - 1 ) ) . (13)

A modification of Switch Chain One, given by Berger et al. [25] for directed graph sequences, can also be used for the bipartite version. The main idea is to reduce the number of transition loops in the state graph. To avoid non-convergence, an additional additional artificial edge {u₀, v₀}, which connects two additional vertices u₀ and v₀, is added to the set of edges. In a state G = (U, V, E) ∈ Ω_R select a pair of non-adjacent edges e₁ = {u_i, v_j} and e₂ = {u_k, v_l} from the set E∪{{u₀, v₀}}.

Without the additional edge, a problem would occur if no adjacent edge pairs existed for each graph realization of a sequence; then, the state graph would not possess loops. Similar arguments like those of Ref. [25] show that this chain converges to the uniform distribution. This chain reduces the average probability of transition loops and, as it will turn out, the average total mixing time. However, the asymptotical total mixing time is the same as for Switch Chain One. We will refer to this chain as Switch Chain Two.

Firstly, we enumerated all small input instances up to a given limit for each chain. In particular, we enumerated 19,378 bipartite degree sequence pairs with up to 6+6 vertices as input for the switch chains and 89,242 connected non-isomorphic bipartite graphs with 6+6 vertices as input for the matching chains. While the first task was done with a simple backtracking procedure, the latter task was done by the command line tool nauty[26]. These combinatorial objects served as input instances for the Markov chains. Even though the size of input is small, the resulting state graphs can be large. For example, the largest state graph processed in this experiment had 297,200 states and corresponds to the regular degree sequence pair (3, 3, 3, 3, 3, 3),(3, 3, 3, 3, 3, 3). Since both the number of instances, as well as the size of the state graphs grow exponentially, we were not able to perform such an experiment for a much larger instance size. For each instance, we constructed the corresponding state graph. For all state graphs with a maximum of 20,000 states, we computed the following properties: a) total mixing time, b) spectral bound, and c) canonical path congestion bound. Although we could easily compute the latter bounds for much larger state graphs, we did not do so, since we need the total mixing time for comparison.

In Fig 3, we show the results of the enumeration experiment. Each input instance corresponds to two data points, showing the ratio of the upper spectral bound (green), and congestion bound (blue) with the total mixing time. We immediately observe that the spectral bound is very close to the total mixing time for all instances, in contrast to the congestion bound. More precisely, in case of Matching Chain Two, the congestion bound is up to 800 times larger than the total mixing time. Similar observations can be made for the other chains.

In Fig 4, we highlight the instances which are known to have a polynomial congestion bound. In the case of the switch chains, these are regular and half-regular degree sequences. In the case of Matching Chain One, these are dense bipartite graphs. Matching Chain Two is known to be rapidly mixing for all instances. We find that the observations can also be confirmed within this set.

We picked a half-regular sequence pattern (n − 1, n − 2, 2, 1), (2, 2, …, 2), where the number of two’s in the second sequence is n. The parameter n can be used to scale the instance to different sizes. In doing so, a bipartite graph which realizes this degree sequences has n + 4 vertices and 2n edges. These sequence pairs are half-regular and so, it is known that the switch chains are rapidly mixing. For each n between 4 and 50, we constructed the corresponding state graph and computed the total mixing time and its bounds. Again, due to the large memory consumption, we stopped computing the total mixing time when |Ω| exceeded 20,000. To predict missing values of total mixing time for larger state graphs up to about 600,000 states, we used a linear model based on the following observation.

We observed that the lower spectral bound seems to have a linear relationship with the total mixing time. This allowed us to predict the missing total mixing time values using a simple linear model (see Fig 5 for details).

To quantify the gap between the bounds and total mixing time, we again divided each bound through the corresponding total mixing time or its predicted value. Fig 6 shows the result of this experiment. We observed that the gap between the congestion bound and the total mixing time grows with input size n. Furthermore, we observed that the same is true for the upper spectral bound, but in a much slower way.

We repeated this experiment with other half-regular sequence patterns and observed similar effects. In particular, we changed our sequence pattern (n − 1, n − 2, 2, 1), (2, 2, …, 2), which we call type A, slightly to the sequences patterns (n − 1, n − 2, 3), (2, 2, …, 2) (type B) and (n − 1, n − 2, 1, 1, 1), (2, 2, …, 2) (type C). Fig 7 shows the, partially estimated, total mixing time for each sequence type. We observed that each type has its own growing asymptotic. The data suggests, in the case of Switch Chain One, the total mixing time of sequence type A grows with O(n^2.27), of type B with O(n^1.16), and of type C with O(n^2.43). In the case of Switch Chain Two, we observe similar effects.

These results are interesting because these bounds are lower bounds on the mixing time of the switch chains for all half-regular sequence pairs. Since two of them are clearly super-linear, a random walk with a linear number of steps must ultimately fail and will result in a non-random sample.

We repeated the enumeration experiment but made some artificial modifications on each state graph before computing its properties. For each state graph we determined the minimal loop probability over all states P_min: = min_i∈Ω P_ii. We wanted to reduce the transition probability of each loop in the state graph by this quantity. However, to avoid the removal of all loops and possible problems with convergence, we removed only 99% of this probability from the loops. This way, we gained new transition probabilities of P i i ′ ← P i i - . 99 P min  for all  i ∈ Ω . (14) The amount of.99P_min which was reduced from the probability mass of each state, was restored to the network through rescaling of the remaining transition arcs: P i j ′ ← P i j 1 - . 99 P min  for all  i , j ∈ Ω . (15) By subtracting the same amount of probability from each loop, we kept the symmetry of the transition matrix so that the chain’s stationary distribution was still uniform. The result was a state graph with the same structure but drastically reduced loop probability. In such state graphs, the mixing times are much more dependent on the structure of the state graph than on the number of the loops. We repeated the computation of the total mixing time and its upper bounds with those modified state graphs for Switch Chain One.

We first observed that the modification indeed had a strong effect on the total mixing time (see Fig 9).

While the largest total mixing time in our instance set is about 1400, in the set of reduced state graphs, it is 63. In addition, we observed that the total mixing time after the reduction steps no longer correlates with the average loop probability. So, the effects we observe now are more or less detached from the influence of the loops. We observed, that the effect we observed in the enumeration experiment still remain (see Fig 10). The quality of the congestion bound is still bad while the largest total mixing time occurs at relatively small state graphs.

We finished our experiments with the observation that the upper spectral bound as well as the congestion bound heavily depend on the average vertex degree of the state graph (see Fig 11). We concluded that the average degree is a structural property of a state graph which has a stronger influence on the total mixing time than its size.