Kemeny Constant Research Articles

Kemeny Constant-Based Optimization of Network Clustering Using Graph Neural Networks.

The recent trend in using network and graph structures to represent a variety of different data types has renewed interest in the graph partitioning (GP) problem. This interest stems from the need for general methods that can both efficiently identify network communities and reduce the dimensionality of large graphs while satisfying various application-specific criteria. Traditional clustering algorithms often struggle to capture the complex relationships within graphs and generalize to arbitrary clustering criteria. The emergence of graph neural networks (GNNs) as a powerful framework for learning representations of graph data provides new approaches to solving the problem. Previous work has shown GNNs to be capable of proposing partitionings using a variety of criteria. However, these approaches have not yet been extended to Markov chains or kinetic networks. These arise frequently in the study of molecular systems and are of particular interest to the biomolecular modeling community. In this work, we propose several GNN-based architectures to tackle the GP problem for Markov Chains described as kinetic networks. This approach aims to maximize the Kemeny constant, which is a variational quantity and it represents the sum of time scales of the system. We propose using an encoder-decoder architecture and show how simple GraphSAGE-based GNNs with linear layers can outperform much larger and more expressive attention-based models in this context. As a proof of concept, we first demonstrate the method's ability to cluster randomly connected graphs. We also use a linear chain architecture corresponding to a 1D free energy profile as our kinetic network. Subsequently, we demonstrate the effectiveness of our method through experiments on a data set derived from molecular dynamics. We compare the performance of our method to other partitioning techniques, such as PCCA+. We explore the importance of feature and hyperparameter selection and propose a general strategy for large-scale parallel training of GNNs for discovering optimal graph partitionings.

Modeling spatial networks by contact graphs of disk packings

Spatial networks are ubiquitous in the real world, with examples including Internet, power grids, neural networks, and so on. In this paper, based on a proposed disk packing, we present a contact graph as an exactly solvable model for spatial networks. We then derive analytically some critical structural properties of the model, including degree distribution, clustering coefficient, and diameter, and show that it displays simultaneously the striking scale-free small-world phenomena observed in most real networks. We also determine all the eigenvalues and their multiplicities of the transition probability matrix for the contact graph, and exploit the eigenvalues to derive closed-form expressions for Kemeny constant of random walks and the number of spanning trees on the contact graph. Finally, we derive closed-form expressions for some interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance. As an application of resistance distances, we study the coherence of the first-order consensus dynamics, which tends to a small constant as the graph grows, implying that the graph is resistant to noise.

Oriented spanning trees and stationary distribution of digraphs

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.

Kemeny's constant and Wiener index on trees

On trees of fixed order, we show a direct relation between Kemeny's constant and Wiener index, and provide a new formula of Kemeny's constant from the relation with a combinatorial interpretation. Moreover, the relation simplifies proofs of several known results for extremal trees in terms of Kemeny's constant for random walks on trees. Finally, we provide various families of co-Kemeny's mates, which are two non-isomorphic connected graphs with the same Kemeny's constant, and we also give a necessary condition for a tree to attain maximum Kemeny's constant for trees with fixed diameter.

Mean first passage time and Kemeny's constant using generalized inverses of the combinatorial Laplacian

In the field of random walks, the mean first passage time matrix and the Kemeny's constant allow us to deepen into the study of networks. For a transition matrix , we can observe in the literature how the authors characterize mean first passage time using generalized inverses of and its associated group inverse. In this paper, we focus on obtaining expressions for the mentioned parameters in terms of generalized inverses of the combinatorial Laplacian. For that, we first analyse the structure and the relations between any generalized inverse and the group inverse of the combinatorial Laplacian. Then, we get closed-formulas for mean first passage matrix and Kemeny's constant based on the group inverse of the combinatorial Laplacian. As an example, we consider wheel networks.

Spectral techniques and mathematical aspects of K 4 chain graph

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them. An electrical network can be constructed from a graph by replacing each edge with a unit resistor. Resistance distances are computed by methods of the theory of resistive electrical networks (based on Ohm’s and Kirchhoff’s laws). The standard method to compute resistance distance is via the Moore-Penrose generalized inverse of the Laplacian matrix of the underlying graph G. In this article, we used the electric network approach and the combinatorial approach, to derive the exact expression for resistance distances between any two vertices of the ring model. By employing a recursive connection, firstly we determined all the eigenvalues and their multiplicities in relation to the corresponding Laplacian and normalised matrices. Secondly, we obtained the Kirchhoff index and multiplicative degree Kirchhoff index for . Finally, we calculated the mean first passage time and Kemeny constant of . Our computed results provide a comprehensive approach for exploring random walks on complex networks, especially biased random walks, which may also help to better understand and tackle some practical problems such as search and routing on networks.

Variational kinetic clustering of complex networks.

Efficiently identifying the most important communities and key transition nodes in weighted and unweighted networks is a prevalent problem in a wide range of disciplines. Here, we focus on the optimal clustering using variational kinetic parameters, linked to Markov processes defined on the underlying networks, namely, the slowest relaxation time and the Kemeny constant. We derive novel relations in terms of mean first passage times for optimizing clustering via the Kemeny constant and show that the optimal clustering boundaries have equal round-trip times to the clusters they separate. We also propose an efficient method that first projects the network nodes onto a 1D reaction coordinate and subsequently performs a variational boundary search using a parallel tempering algorithm, where the variational kinetic parameters act as an energy function to be extremized. We find that maximization of the Kemeny constant is effective in detecting communities, while the slowest relaxation time allows for detection of transition nodes. We demonstrate the validity of our method on several test systems, including synthetic networks generated from the stochastic block model and real world networks (Santa Fe Institute collaboration network, a network of co-purchased political books, and a street network of multiple cities in Luxembourg). Our approach is compared with existing clustering algorithms based on modularity and the robust Perron cluster analysis, and the identified transition nodes are compared with different notions of node centrality.

Kemeny's constant for nonbacktracking random walks

AbstractKemeny's constant for a connected graph is the expected time for a random walk to reach a randomly chosen vertex , regardless of the choice of the initial vertex. We extend the definition of Kemeny's constant to nonbacktracking random walks and compare it to Kemeny's constant for simple random walks. We explore the relationship between these two parameters for several families of graphs and provide closed‐form expressions for regular and biregular graphs. In nearly all cases, the nonbacktracking variant yields the smaller Kemeny's constant.

The normalized Laplacian spectrum of n-polygon graphs and applications

Given an arbitrary connected graph G, the n-polygon graph τ n ( G ) is obtained by adding a path with length n ( n ≥ 2 ) to each edge of graph G, and the iterated n-polygon graphs τ n g ( G ) ( g ≥ 0 ) are obtained from the iteration τ n g ( G ) = τ n ( τ n g − 1 ( G ) ) , with the initial condition τ n 0 ( G ) = G . In this paper, a method for calculating the eigenvalues of the normalized Laplacian matrix for graph τ n ( G ) is presented if the eigenvalues of a normalized Laplacian matrix for graph G is first given. The normalized Laplacian spectra for the graph τ n ( G ) and graphs τ n g ( G ) ( g ≥ 0 ) can also then be derived. Finally, as applications, we calculate the multiplicative degree-Kirchhoff index, Kemeny's constant, and the number of spanning trees for the graph τ n ( G ) and graphs τ n g ( G ) by exploring their connections with the normalized Laplacian spectrum, and obtain exact results for these quantities.

Families of graphs with twin pendent paths and the Braess edge

In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to Kemeny's constant: in particular, we examine graphs with a cut-vertex at which at least two branches are paths, regarding whether the insertion of a particular edge into a graph results in an increase of Kemeny's constant. We provide several tools for identifying such an edge in a family of graphs and for analysing asymptotic behaviour of the family regarding the tendency to have that edge; and classes of particular graphs are given as examples. Furthermore, asymptotic behaviours of families of trees are described.

Perron value and moment of rooted trees

The Perron value ρ(T) of a rooted tree T has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for T. A different, combinatorial weight notion for T – the moment μ(T) – emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that μ(T) is “almost” an upper bound for ρ(T) and the ratio μ(T)/ρ(T) is unbounded but at most linear in the order of T. Furthermore, we introduce two new objects associated with T – the Perron entropy and the neckbottle matrix – and we investigate how different operations on the set of rooted trees affect the Perron value and the moment.

A 1-separation formula for the graph Kemeny constant and Braess edges

Kemeny’s constant of a simple connected graph G is the expected length of a random walk from i to any given vertex \(j \ne i\). We provide a simple method for computing Kemeny’s constant for 1-separable graphs via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on n vertices maximizes Kemeny’s constant for the class of undirected trees on n vertices. Applying this method again, we simplify existing expressions for the Kemeny’s constant of barbell graphs and demonstrate which barbell maximizes Kemeny’s constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland and Zeng (Electron J Linear Algebra 31(1):444–464, 2016) and Ciardo (Linear Algebra Appl, 2020).

Measuring the Network Vulnerability Based on Markov Criticality

Vulnerability assessment—a critical issue for networks—attempts to foresee unexpected destructive events or hostile attacks in the whole system. In this article, we consider a new Markov global connectivity metric—Kemeny constant, and take its derivative called Markov criticality to identify critical links. Markov criticality allows us to find links that are most influential on the derivative of Kemeny constant. Thus, we can utilize it to identity a critical link ( i , j ) from node i to node j , such that removing it leads to a minimization of networks’ global connectivity, i.e., the Kemeny constant. Furthermore, we also define a novel vulnerability index to measure the average speed by which we can disconnect a specified ratio of links with network decomposition. Our method is of high efficiency, which can be easily employed to calculate the Markov criticality in real-life networks. Comprehensive experiments on several synthetic and real-life networks have demonstrated our method’s better performance by comparing it with state-of-the-art baseline approaches.

Kemeny-based testing for COVID-19.

Testing, tracking and tracing abilities have been identified as pivotal in helping countries to safely reopen activities after the first wave of the COVID-19 virus. Contact tracing apps give the unprecedented possibility to reconstruct graphs of daily contacts, so the question is: who should be tested? As human contact networks are known to exhibit community structure, in this paper we show that the Kemeny constant of a graph can be used to identify and analyze bridges between communities in a graph. Our 'Kemeny indicator' is the value of the Kemeny constant in the new graph that is obtained when a node is removed from the original graph. We show that testing individuals who are associated with large values of the Kemeny indicator can help in efficiently intercepting new virus outbreaks, when they are still in their early stage. Extensive simulations provide promising results in early identification and in blocking the possible 'super-spreaders' links that transmit disease between different communities.

On Kemeny's constant for trees with fixed order and diameter

ABSTRACT Kemeny's constant of a connected graph G is a measure of the expected transit time for the random walk associated with G. In the current work, we consider the case when G is a tree and, in this setting, we provide lower and upper bounds for in terms of the order n and diameter δ of G by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from G. By considering a specific family of trees – the broom-stars – we show that the upper bound is asymptotically sharp.

Kemeny's constant for countable Markov chains

In this paper, we aim to investigate Kemeny's constant for discrete-time or continuous-time, irreducible and positive recurrent Markov chains on a countable state space. We show that Kemeny's constant is infinite for a discrete-time Markov chain on an infinitely countable state space, which confirms a conjecture in page 1031 of [3]. Moreover, it is shown that the study on Kemeny's constant for a continuous-time Markov chain with a bounded generator can be converted to that for a discrete-time Markov chain through the uniformization technique. It is different and challenging to analyze Kemeny's constant for a chain with unbounded generator. Finally, a upward skip-free process is considered and some discussions for Kemeny's constant are given.

Correlation functions, mean first passage times, and the Kemeny constant.

Markov processes are widely used models for investigating kinetic networks. Here, we collate and present a variety of results pertaining to kinetic network models in a unified framework. The aim is to lay out explicit links between several important quantities commonly studied in the field, including mean first passage times (MFPTs), correlation functions, and the Kemeny constant. We provide new insights into (i) a simple physical interpretation of the Kemeny constant, (ii) a relationship to infer equilibrium distributions and rate matrices from measurements of MFPTs, and (iii) a protocol to reduce the dimensionality of kinetic networks based on specific requirements that the MFPTs in the coarse-grained system should satisfy. We prove that this protocol coincides with the one proposed by Hummer and Szabo [J. Phys. Chem. B 119, 9029 (2014)], and it leads to a variational principle for the Kemeny constant. Finally, we introduce a modification of this protocol, which preserves the Kemeny constant. Our work underpinning the theoretical aspects of kinetic networks will be useful in applications including milestoning and path sampling algorithms in molecular simulations.

Random walk hitting times and effective resistance in sparsely connected Erdős‐Rényi random graphs

AbstractWe prove a bound on the effective resistance between two vertices of a connected graph which contains a suitably well‐connected subgraph. We apply this bound to the Erdős‐Rényi random graph with , proving that concentrates around , that is, the sum of reciprocal degrees. We also prove expectation and concentration results for the random walk hitting times, Kirchoff index, cover cost, and the random target time (Kemeny's constant) on in the sparsely connected regime .

Study on the normalized Laplacian of a penta‐graphene with applications

AbstractLet L n denote a linear pentagonal chain with 2n pentagons. The penta‐graphene (penta‐C), denoted by R n is the graph obtained from L n by identifying the opposite lateral edges in an ordered way, whereas the pentagonal Möbius ring is the graph obtained from the L n by identifying the opposite lateral edges in a reversed way. In this paper, through the decomposition theorem of the normalized Laplacian characteristic polynomial and the relationship between its roots and the coefficients, an explicit closed‐form formula of the multiplicative degree‐Kirchhoff index (resp. Kemeny's constant, the number of spanning trees) of R n is obtained. Furthermore, it is interesting to see that the multiplicative degree‐Kirchhoff index of R n is approximately of its Gutman index. Based on our obtained results, all the corresponding results are obtained for .

Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

AbstractSubdivision, triangulation, Kronecker product, corona product and many other graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its $q$-subdivision graph $S_q(G)$ is obtained from $G$ through replacing every edge $uv$ in $G$ by $q$ disjoint paths of length 2, with each path having $u$ and $v$ as its ends. We derive explicit formulas for many quantities of $S_q(G)$ in terms of those corresponding to $G$, including the eigenvalues and eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny constant, two-node resistance distance, Kirchhoff index, additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index. We also study the properties of the iterated $q$-subdivision graphs, based on which we obtain the closed-form expressions for a family of hierarchical lattices, which has been used to describe scale-free fractal networks.