A cluster algorithm for graphs

A cluster algorithm for graphs called the emph{Markov Cluster algorithm (MCL~algorithm) is introduced. The algorithm provides basically an interface to an algebraic process defined on stochastic matrices, called the MCL~process. The graphs may be both weighted (with nonnegative weight) and directed. Let~$G$~be such a graph. The MCL~algorithm simulates flow in $G$ by first identifying $G$ in a canonical way with a Markov graph $G_1$. Flow is then alternatingly expanded and contracted, leading to a row of Markov Graphs G_{(i). Flow expansion corresponds with taking the~k^{th~power of a stochastic matrix, where~$kinN$. Flow contraction corresponds with a parametrized operator~$Gamma_r$, $rgeq 0$, which maps the set of (column) stochastic matrices onto itself. The image~$Gamma_r M$ is obtained by raising each entry in~$M$ to the~r^{th~power and rescaling each column to have sum~$1$ again. The heuristic underlying this approach is the expectation that flow between dense regions which are sparsely connected will evaporate. The invariant limits of the process are easily derived and in practice the process converges very fast to such a limit, the structure of which has a generic interpretation as an overlapping clustering of the graph~$G$. Overlap is limited to cases where the input graph has a symmetric structure inducing it. The contraction and expansion parameters of the MCL~process influence the granularity of the output. The algorithm is space and time efficient and lends itself to drastic scaling. This report describes the MCL~algorithm and process, convergence towards equilibrium states, interpretation of the states as clusterings, and implementation and scalability. The algorithm is introduced by first considering several related proposals towards graph clustering, of both combinatorial and probabilistic nature. Revised version of the report~[1]. A more mathematically oriented account on the MCL~process is given in~[2], establishing that under certain weak conditions the iterands of the MCL~process posses structure admitting a cluster interpretation. Various experiments conducted on a wide range of test-graphs are described in~[3]. The latter report also describes a generic graph clustering performance measure and a distance defined on the space of partitions. The work was carried out under project INS-3.2, Concept Building from Key-Phrases in Scientific Documents and Bottom Up Classification Methods in Mathematics. [1] A new cluster algorithm for graphs. Technical report INS-R9814, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, 1998. [2] A stochastic uncoupling process for graphs. Technical report INS-R0011, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, 2000. [3] Performance criteria for graph clustering and Markov cluster experiments. Technical report INS-R0012, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, 2000

Change Mining in Adaptive Process Management Systems

The wide-spread adoption of process-aware information systems has resulted in a bulk of computerized information about real-world processes. This data can be utilized for process performance analysis as well as for process improvement. In this context process mining offers promising perspectives. So far, existing mining techniques have been applied to operational processes, i.e., knowledge is extracted from execution logs (process discovery), or execution logs are compared with some a-priori process model (conformance checking). However, execution logs only constitute one kind of data gathered during process enactment. In particular, adaptive processes provide additional information about process changes (e.g., ad-hoc changes of single process instances) which can be used to enable organizational learning. In this paper we present an approach for mining change logs in adaptive process management systems. The change process discovered through process mining provides an aggregated overview of all changes that happened so far. This, in turn, can serve as basis for all kinds of process improvement actions, e.g., it may trigger process redesign or better control mechanisms

Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation

The persistence probability, $P_C(t)$, of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. In the mean-field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For $\gamma \ge 0$ the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For $\gamma < 0$ the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For $0 < \gamma < 2$ the distribution is flat and, surprisingly, independent of $\gamma$.Comment: 11 pages, 6 figures, RevTeX4, submitted to Phys. Rev.

Saccadic eye movements estimate prolonged time awake

Prolonged time awake increases the need to sleep. Sleep pressure increases sleepiness, impairs human alertness and performance and increases the probability of human errors and accidents. Human performance and alertness during waking hours are influenced by homeostatic sleep drive and the circadian rhythm. Cognitive functions, especially attentional ones, are vulnerable to circadian rhythm and increasing sleep drive. A reliable, objective and practical metrics for estimating sleepiness could therefore be valuable. Our aim is to study whether saccades measured with electro-oculography (EOG) outside the laboratory could be used to estimate the overall time awake without sleep of a person. The number of executed saccades was measured in 11 participants during an 8-min saccade task. The saccades were recorded outside the laboratory (Naval Academy, Bergen) using EOG every sixth hour until 54 hr of time awake. Measurements were carried out on two occasions separated by 10 weeks. Five participants participated in both measurement weeks. The number of saccades decreased during sustained wakefulness. The data correlated with the three-process model of alertness; performance differed between participants but was stable within individual participants. A mathematically monotonous relation between performance in the saccade task and time awake was seen after removing the circadian rhythm component from measured eye movement data. The results imply that saccades measured with EOG can be used as a time-awake metric outside the laboratory.Peer reviewe

Kinetic Anomalies in Addition-Aggregation Processes

We investigate irreversible aggregation in which monomer-monomer, monomer-cluster, and cluster-cluster reactions occur with constant but distinct rates K_{MM}, K_{MC}, and K_{CC}, respectively. The dynamics crucially depends on the ratio gamma=K_{CC}/K_{MC} and secondarily on epsilon=K_{MM}/K_{MC}. For epsilon=0 and gamma<2, there is conventional scaling in the long-time limit, with a single mass scale that grows linearly in time. For gamma >= 2, there is unusual behavior in which the concentration of clusters of mass k, c_k decays as a stretched exponential in time within a boundary layer k<k* propto t^{1-2/gamma} (k* propto ln t for gamma=2), while c_k propto t^{-2} in the bulk region k>k*. When epsilon>0, analogous behaviors emerge for gamma<2 and gamma >= 2.Comment: 6 pages, 2 column revtex4 format, for submission to J. Phys.

Emergence in holographic scenarios for gravity

'Holographic' relations between theories have become an important theme in quantum gravity research. These relations entail that a theory without gravity is equivalent to a gravitational theory with an extra spatial dimension. The idea of holography was first proposed in 1993 by Gerard 't Hooft on the basis of his studies of evaporating black holes. Soon afterwards the holographic 'AdS/CFT' duality was introduced, which since has been intensively studied in the string theory community and beyond. Recently, Erik Verlinde has proposed that even Newton's law of gravitation can be related holographically to the `thermodynamics of information' on screens. We discuss these scenarios, with special attention to the status of the holographic relation in them and to the question of whether they make gravity and spacetime emergent. We conclude that only Verlinde's scheme straightfowardly instantiates emergence. However, assuming a non-standard interpretation of AdS/CFT may create room for the emergence of spacetime and gravity there as well

Communities of Local Optima as Funnels in Fitness Landscapes

We conduct an analysis of local optima networks extracted from fitness landscapes of the Kauffman NK model under iterated local search. Applying the Markov Cluster Algorithm for community detection to the local optima networks, we find that the landscapes consist of multiple clusters. This result complements recent findings in the literature that landscapes often decompose into multiple funnels, which increases their difficulty for iterated local search. Our results suggest that the number of clusters as well as the size of the cluster in which the global optimum is located are correlated to the search difficulty of landscapes. We conclude that clusters found by community detection in local optima networks offer a new way to characterize the multi-funnel structure of fitness landscapes