Ilkka Kivimäki

Predicting the continuum between corridors and barriers to animal movements using Step Selection Functions and Randomized Shortest Paths.

The loss, fragmentation and degradation of habitat everywhere on Earth prompts increasing attention to identifying landscape features that support animal movement (corridors) or impedes it (barriers). Most algorithms used to predict corridors assume that animals move through preferred habitat either optimally (e.g. least cost path) or as random walkers (e.g. current models), but neither extreme is realistic. We propose that corridors and barriers are two sides of the same coin and that animals experience landscapes as spatiotemporally dynamic corridor-barrier continua connecting (separating) functional areas where individuals fulfil specific ecological processes. Based on this conceptual framework, we propose a novel methodological approach that uses high-resolution individual-based movement data to predict corridor-barrier continua with increased realism. Our approach consists of two innovations. First, we use step selection functions (SSF) to predict friction maps quantifying corridor-barrier continua for tactical steps between consecutive locations. Secondly, we introduce to movement ecology the randomized shortest path algorithm (RSP) which operates on friction maps to predict the corridor-barrier continuum for strategic movements between functional areas. By modulating the parameter Ѳ, which controls the trade-off between exploration and optimal exploitation of the environment, RSP bridges the gap between algorithms assuming optimal movements (when Ѳ approaches infinity, RSP is equivalent to LCP) or random walk (when Ѳ → 0, RSP → current models). Using this approach, we identify migration corridors for GPS-monitored wild reindeer (Rangifer t. tarandus) in Norway. We demonstrate that reindeer movement is best predicted by an intermediate value of Ѳ, indicative of a movement trade-off between optimization and exploration. Model calibration allows identification of a corridor-barrier continuum that closely fits empirical data and demonstrates that RSP outperforms models that assume either optimality or random walk. The proposed approach models the multiscale cognitive maps by which animals likely navigate real landscapes and generalizes the most common algorithms for identifying corridors. Because suboptimal, but non-random, movement strategies are likely widespread, our approach has the potential to predict more realistic corridor-barrier continua for a wide range of species.

Developments in the theory of randomized shortest paths with a comparison of graph node distances

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.

A bag-of-paths framework for network data analysis

This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness, or more precisely accessibility, between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest-path distance and the commute-cost distance. In addition, it extends the Bellman-Ford formula for computing the shortest-path distance in order to integrate sub-optimal paths (exploration) by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification tasks show that both of the new distance families are competitive with other state-of-the-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.

Two betweenness centrality measures based on Randomized Shortest Paths

This paper introduces two new closely related betweenness centrality measures based on the Randomized Shortest Paths (RSP) framework, which fill a gap between traditional network centrality measures based on shortest paths and more recent methods considering random walks or current flows. The framework defines Boltzmann probability distributions over paths of the network which focus on the shortest paths, but also take into account longer paths depending on an inverse temperature parameter. RSP’s have previously proven to be useful in defining distance measures on networks. In this work we study their utility in quantifying the importance of the nodes of a network. The proposed RSP betweenness centralities combine, in an optimal way, the ideas of using the shortest and purely random paths for analysing the roles of network nodes, avoiding issues involving these two paradigms. We present the derivations of these measures and how they can be computed in an efficient way. In addition, we show with real world examples the potential of the RSP betweenness centralities in identifying interesting nodes of a network that more traditional methods might fail to notice.

Random walks based modularity: application to semi-supervised learning

Although criticized for some of its limitations, modularity remains a standard measure for analyzing social networks. Quantifying the statistical surprise in the arrangement of the edges of the network has led to simple and powerful algorithms. However, relying solely on the distribution of edges instead of more complex structures such as paths limits the extent of modularity. Indeed, recent studies have shown restrictions of optimizing modularity, for instance its resolution limit. We introduce here a novel, formal and well-defined modularity measure based on random walks. We show how this modularity can be computed from paths induced by the graph instead of the traditionally used edges. We argue that by computing modularity on paths instead of edges, more informative features can be extracted from the network. We verify this hypothesis on a semi-supervised classification procedure of the nodes in the network, where we show that, under the same settings, the features of the random walk modularity help to classify better than the features of the usual modularity. Additionally, the proposed approach outperforms the classical label propagation procedure on two data sets of labeled social networks.

Effect of Dimensionality Reduction on Different Distance Measures in Document Clustering

In document clustering, semantically similar documents are grouped together. The dimensionality of document collections is often very large, thousands or tens of thousands of terms. Thus, it is common to reduce the original dimensionality before clustering for computational reasons. Cosine distance is widely seen as the best choice for measuring the distances between documents in k-means clustering. In this paper, we experiment three dimensionality reduction methods with a selection of distance measures and show that after dimensionality reduction into small target dimensionalities, such as 10 or below, the superiority of cosine measure does not hold anymore. Also, for small dimensionalities, PCA dimensionality reduction method performs better than SVD. We also show how l 2 normalization affects different distance measures. The experiments are run for three document sets in English and one in Hindi.

Semisupervised Classification Through the Bag-of-Paths Group Betweenness

This paper introduces a novel and well-founded betweenness measure, called the bag-of-paths (BoP) betweenness, as well as its extension, the BoP group betweenness, to tackle semisupervised classification problems on weighted directed graphs. The objective of semisupervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeled nodes at our disposal. The BoP betweenness relies on a BoP framework, assigning a Boltzmann distribution on the set of all possible paths through the network such that long (high-cost) paths have a low probability of being picked from the bag, while short (low-cost) paths have a high probability of being picked. Within that context, the BoP betweenness of node j is defined as the sum of the a posteriori probabilities that node j lies in between two arbitrary nodes (i, k) when picking a path starting in i and ending in k. Intuitively, a node typically receives a high betweenness if it has a large probability of appearing on paths connecting two arbitrary nodes of the network. This quantity can be computed in closed form by inverting an n×n matrix where n is the number of nodes. For the group betweenness, the paths are constrained to start and end in nodes within the same class, thereby defining a within-class group betweenness for each class. Unlabeled nodes are then classified according to the class showing the highest group betweenness. Experiments on various real-world datasets show that the BoP group betweenness performs competitively compared to all the tested state-of-the-art methods. The benefit of the BoP betweenness is particularly noticeable when only a few labeled nodes are available.

Using correlation dimension for analysing text data

In this article, we study the scale-dependent dimensionality properties and overall structure of text data with a method that measures correlation dimension in different scales. As experimental results, we present the analysis of text data sets with the Reuters and Europarl corpora, which are also compared to artificially generated point sets. A comparison is also made with speech data. The results reflect some of the typical properties of the data and the use of our method in improving various data analysis applications is discussed.